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R E F E R E N C E PA G E 1
Cut here and keep for reference
ALGEBRA
GEOMETRY
Arithmetic Operations
Geometric Formulas a c ad bc 苷 b d bd a d ad b a 苷 苷 c b c bc d
a共b c兲 苷 ab ac a c ac 苷 b b b
Formulas for area A, circumference C, and volume V: Triangle
Circle
Sector of Circle
A 苷 12 bh
A 苷 r 2
A 苷 12 r 2
C 苷 2 r
s 苷 r 共 in radians兲
苷 12 ab sin
a
Exponents and Radicals xm 苷 x mn xn 1 xn 苷 n x
x m x n 苷 x mn 共x m兲n 苷 x m n
冉冊 x y
共xy兲n 苷 x n y n
n
苷
xn yn
n n x m兾n 苷 s x m 苷 (s x )m
n x 1兾n 苷 s x
冑
n n n xy 苷 s xs y s
n
r
h
¨
r
s
¨
b
r
Sphere V 苷 43 r 3
Cylinder V 苷 r 2h
Cone V 苷 13 r 2h
A 苷 4 r 2
A 苷 rsr 2 h 2
n x x s 苷 n y sy
r r
h
h
Factoring Special Polynomials
r
x 2 y 2 苷 共x y兲共x y兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲
Distance and Midpoint Formulas
Binomial Theorem 共x y兲2 苷 x 2 2xy y 2
共x y兲2 苷 x 2 2xy y 2
Distance between P1共x1, y1兲 and P2共x 2, y2兲: d 苷 s共x 2 x1兲2 共 y2 y1兲2
共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲n 苷 x n nx n1y
where
冉冊
n共n 1兲 n2 2 x y 2
冉冊
n nk k x y nxy n1 y n k
n共n 1兲 共n k 1兲 n 苷 k 1 ⴢ 2 ⴢ 3 ⴢ
ⴢ k
Midpoint of P1 P2 :
冉
x1 x 2 y1 y2 , 2 2
Lines Slope of line through P1共x1, y1兲 and P2共x 2, y2兲: m苷
Quadratic Formula If ax 2 bx c 苷 0, then x 苷
冊
b sb 2 4ac . 2a
y2 y1 x 2 x1
Pointslope equation of line through P1共x1, y1兲 with slope m:
Inequalities and Absolute Value
y y1 苷 m共x x1兲
If a b and b c, then a c.
Slopeintercept equation of line with slope m and yintercept b:
If a b, then a c b c. If a b and c 0, then ca cb.
y 苷 mx b
If a b and c 0, then ca cb. If a 0, then
ⱍxⱍ 苷 a ⱍxⱍ a ⱍxⱍ a
means
x 苷 a or
x 苷 a
means a x a means
xa
or
x a
Circles Equation of the circle with center 共h, k兲 and radius r: 共x h兲2 共 y k兲2 苷 r 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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R E F E R E N C E PA G E 2
TRIGONOMETRY Angle Measurement
Fundamental Identities
radians 苷 180⬚ 1⬚ 苷
rad 180
1 rad 苷
s
r
180⬚
r
共 in radians兲
Right Angle Trigonometry hyp csc 苷 opp
cos 苷
adj hyp
sec 苷
hyp adj
tan 苷
opp adj
cot 苷
adj opp
hyp
y r
csc 苷
¨ adj
x r
sec 苷
r x
tan 苷
y x
cot 苷
x y
cot 苷
cos sin
cot 苷
1 tan
sin 2 ⫹ cos 2 苷 1
1 ⫹ tan 2 苷 sec 2
1 ⫹ cot 2 苷 csc 2
sin共⫺兲 苷 ⫺sin
cos共⫺兲 苷 cos
tan共⫺兲 苷 ⫺tan
sin
⫺ 苷 cos 2
tan
⫺ 苷 cot 2
冉 冊 冉 冊
⫺ 苷 sin 2
B
sin A sin B sin C 苷 苷 a b c
(x, y)
a
r
C c
¨
The Law of Cosines
x
b
a 2 苷 b 2 ⫹ c 2 ⫺ 2bc cos A b 2 苷 a 2 ⫹ c 2 ⫺ 2ac cos B y
A
c 2 苷 a 2 ⫹ b 2 ⫺ 2ab cos C
y=tan x
y=cos x
1
1 π
sin cos
The Law of Sines
y
y y=sin x
tan 苷
冉 冊
Graphs of Trigonometric Functions y
1 cos
cos
r y
cos 苷
sec 苷
opp
Trigonometric Functions sin 苷
1 sin
¨
s 苷 r
opp sin 苷 hyp
csc 苷
2π
Addition and Subtraction Formulas
2π x
_1
π
2π x
π
x
sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y
_1
cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y y
y
y=csc x
y
y=sec x
cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y
y=cot x
1
1 π
2π x
π
2π x
π
2π x
tan共x ⫹ y兲 苷
tan x ⫹ tan y 1 ⫺ tan x tan y
tan共x ⫺ y兲 苷
tan x ⫺ tan y 1 ⫹ tan x tan y
_1
_1
DoubleAngle Formulas sin 2x 苷 2 sin x cos x
Trigonometric Functions of Important Angles
cos 2x 苷 cos 2x ⫺ sin 2x 苷 2 cos 2x ⫺ 1 苷 1 ⫺ 2 sin 2x
radians
sin
cos
tan
0⬚ 30⬚ 45⬚ 60⬚ 90⬚
0 兾6 兾4 兾3 兾2
0 1兾2 s2兾2 s3兾2 1
1 s3兾2 s2兾2 1兾2 0
0 s3兾3 1 s3 —
tan 2x 苷
2 tan x 1 ⫺ tan2x
HalfAngle Formulas sin 2x 苷
1 ⫺ cos 2x 2
cos 2x 苷
1 ⫹ cos 2x 2
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CA L C U L U S EARLY TRANSCENDENTALS SEVENTH EDITION
JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO
Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States
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Calculus: Early Transcendentals, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial Assistant: Jennifer Staller Media Editor : Maureen Ross Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Cheryll Linthicum Art Director: Vernon T. Boes Print Buyer: Becky Cross Rights Acquisitions Specialist: Don Schlotman Production Service: TECH· arts Text Designer: TECH· arts Photo Researcher: Terri Wright, www.terriwright.com Copy Editor: Kathi Townes Cover Designer: Irene Morris Cover Illustration: Irene Morris Compositor: Stephanie Kuhns, TECH· arts
© 2012, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 18003549706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be emailed to
[email protected] Library of Congress Control Number: 2010936599 Student Edition: ISBN13: 9780538497909 ISBN10: 0538497904 Looseleaf Edition: ISBN13: 9780840058850 ISBN10: 0840058853 Brooks/Cole 20 Davis Drive Belmont, CA 940023098 USA Cengage Learning is a leading provider of customized learning solutions with oﬃce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local oﬃce at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole.
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Contents Preface
xi
To the Student
xxiii
Diagnostic Tests
xxiv
A PREVIEW OF CALCULUS
1
Functions and Models 9 1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms Review
10 23
36 44
51 58
72
Principles of Problem Solving
2
1
75
Limits and Derivatives 81 2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Derivatives and Rates of Change
87
N
Problems Plus
108 130
143
Early Methods for Finding Tangents
The Derivative as a Function Review
99
118
Writing Project
2.8
82
153
154
165 170 iii
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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iv
CONTENTS
3
Differentiation Rules 173 3.1
Derivatives of Polynomials and Exponential Functions Applied Project
N
Building a Better Roller Coaster
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule Applied Project
3.5
184 191
Where Should a Pilot Start Descent?
Implicit Differentiation N
Families of Implicit Curves
217
Derivatives of Logarithmic Functions
3.7
Rates of Change in the Natural and Social Sciences
3.8
Exponential Growth and Decay
3.9
Related Rates
3.10
Linear Approximations and Differentials
Problems Plus
218 224
237
244
N
Taylor Polynomials
Hyperbolic Functions Review
208
209
3.6
Laboratory Project
4
184
198 N
Laboratory Project
3.11
174
250
256
257
264 268
Applications of Differentiation 273 4.1
Maximum and Minimum Values Applied Project
N
274
The Calculus of Rainbows
282
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and l’Hospital’s Rule Writing Project
N
284
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems Applied Project
N
4.8
Newton’s Method
4.9
Antiderivatives Review
Problems Plus
301
The Origins of l’Hospital’s Rule
4.5
290 310
310 318
325
The Shape of a Can
337
338 344
351 355
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CONTENTS
5
Integrals 359 5.1
Areas and Distances
360
5.2
The Definite Integral
371
Discovery Project
385
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
5.5
N
Problems Plus
386 397
Newton, Leibniz, and the Invention of Calculus
The Substitution Rule Review
406
407
415 419
Applications of Integration 421 6.1
Areas Between Curves Applied Project
N
422
The Gini Index
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
429
430 441
446 451
Applied Project
N
Calculus and Baseball
Applied Project
N
Where to Sit at the Movies
Review Problems Plus
7
Area Functions
5.3
Writing Project
6
N
455 456
457 459
Techniques of Integration 463 7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Computer Algebra Systems Discovery Project
N
464 471 478 484
494
Patterns in Integrals
500
505
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
v
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vi
CONTENTS
7.7
Approximate Integration
7.8
Improper Integrals Review
Problems Plus
8
519
529 533
Further Applications of Integration 537 8.1
Arc Length
538
Discovery Project
8.2
8.3
N
Arc Length Contest
Area of a Surface of Revolution Discovery Project
N
545
545
Rotating on a Slant
551
Applications to Physics and Engineering Discovery Project
N
Applications to Economics and Biology
8.5
Probability
Problems Plus
552
Complementary Coffee Cups
8.4
Review
9
506
562
563
568 575
577
Differential Equations 579 9.1
Modeling with Differential Equations
9.2
Direction Fields and Euler’s Method
9.3
Separable Equations
580 585
594
Applied Project
N
How Fast Does a Tank Drain?
Applied Project
N
Which Is Faster, Going Up or Coming Down?
9.4
Models for Population Growth
9.5
Linear Equations
9.6
PredatorPrey Systems Review
Problems Plus
603 604
605
616 622
629 633
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CONTENTS
10
Parametric Equations and Polar Coordinates 635 10.1
Curves Defined by Parametric Equations Laboratory Project
10.2
N
Polar Coordinates
Bézier Curves
645 653
N
Families of Polar Curves
10.4
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates Review
Problems Plus
644
654
Laboratory Project
11
Running Circles around Circles
Calculus with Parametric Curves Laboratory Project
10.3
N
636
664
665
670 678
685 688
Infinite Sequences and Series 689 11.1
Sequences
690
Laboratory Project
N
Logistic Sequences
703
11.2
Series
703
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Absolute Convergence and the Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
11.10
Taylor and Maclaurin Series
11.11
722
727 739
N
N
Review Problems Plus
N
746
753
An Elusive Limit
767
How Newton Discovered the Binomial Series
Applications of Taylor Polynomials Applied Project
732
741
Laboratory Project Writing Project
714
Radiation from the Stars
767
768 777
778 781
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
vii
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viii
CONTENTS
12
Vectors and the Geometry of Space 785 12.1
ThreeDimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
791 800
Discovery Project
12.5
808
Equations of Lines and Planes
Problems Plus
816
816
Putting 3D in Perspective
826
827
834 837
Vector Functions 839 13.1
Vector Functions and Space Curves
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration Applied Project
Review Problems Plus
14
N
Cylinders and Quadric Surfaces Review
13
The Geometry of a Tetrahedron
N
Laboratory Project
12.6
786
N
840 847
853
Kepler’s Laws
862
872
873 876
Partial Derivatives 877 14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values Applied Project
878
892 900 915
924
N
Discovery Project
946
Designing a Dumpster N
933
956
Quadratic Approximations and Critical Points
956
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CONTENTS
14.8
Lagrange Multipliers Applied Project
N
Rocket Science
Applied Project
N
HydroTurbine Optimization
Review Problems Plus
15
964 966
967 971
Multiple Integrals 973 15.1
Double Integrals over Rectangles
15.2
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
997
1003
1017 N
Volumes of Hyperspheres
1027
Triple Integrals in Cylindrical Coordinates 1027 N
The Intersection of Three Cylinders
Triple Integrals in Spherical Coordinates Applied Project
15.10
988
1013
Discovery Project
15.9
974
982
Discovery Project
N
Roller Derby
Problems Plus
1032
1033
1039
Change of Variables in Multiple Integrals Review
16
957
1040
1049 1053
Vector Calculus 1055 16.1
Vector Fields
1056
16.2
Line Integrals
1063
16.3
The Fundamental Theorem for Line Integrals
16.4
Green’s Theorem
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
1110
16.8
Stokes’ Theorem
1122
Writing Project
N
1075
1084 1091 1099
Three Men and Two Theorems
1128
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
ix
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x
CONTENTS
16.9
The Divergence Theorem
16.10
Summary
1135
Review Problems Plus
17
1128
1136 1139
SecondOrder Differential Equations 1141 17.1
SecondOrder Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of SecondOrder Differential Equations
17.4
Series Solutions Review
1142 1148 1156
1164
1169
Appendixes A1 A
Numbers, Inequalities, and Absolute Values
B
Coordinate Geometry and Lines
C
Graphs of SecondDegree Equations
D
Trigonometry
E
Sigma Notation
F
Proofs of Theorems
G
The Logarithm Defined as an Integral
H
Complex Numbers
I
Answers to OddNumbered Exercises
A2
A10 A16
A24 A34 A39 A50
A57 A65
Index A135
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Preface A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. GEORGE POLYA
The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first six editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement. The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation: Focus on conceptual understanding. I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. The Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well. In writing the seventh edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum.
Alternative Versions I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions. ■
Calculus: Early Transcendentals, Seventh Edition, Hybrid Version, is similar to the present textbook in content and coverage except that all endofsection exercises are available only in Enhanced WebAssign. The printed text includes all endofchapter review material.
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Calculus, Seventh Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the second semester. xi
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Calculus, Seventh Edition, Hybrid Version, is similar to Calculus, Seventh Edition, in content and coverage except that all endofsection exercises are available only in Enhanced WebAssign. The printed text includes all endofchapter review material.
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Essential Calculus is a much briefer book (800 pages), though it contains almost all of the topics in Calculus, Seventh Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website.
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Essential Calculus: Early Transcendentals resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3.
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Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters.
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Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking Engineering and Physics courses concurrently with calculus.
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Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences.
What’s New in the Seventh Edition? The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition: ■
Some material has been rewritten for greater clarity or for better motivation. See, for instance, the introduction to maximum and minimum values on page 274, the introduction to series on page 703, and the motivation for the cross product on page 808.
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New examples have been added (see Example 4 on page 1021 for instance). And the solutions to some of the existing examples have been amplified. A case in point: I added details to the solution of Example 2.3.11 because when I taught Section 2.3 from the sixth edition I realized that students need more guidance when setting up inequalities for the Squeeze Theorem.
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The art program has been revamped: New figures have been incorporated and a substantial percentage of the existing figures have been redrawn.
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The data in examples and exercises have been updated to be more timely.
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Three new projects have been added: The Gini Index (page 429) explores how to measure income distribution among inhabitants of a given country and is a nice application of areas between curves. (I thank Klaus Volpert for suggesting this project.) Families of Implicit Curves (page 217) investigates the changing shapes of implicitly defined curves as parameters in a family are varied. Families of Polar Curves (page 664) exhibits the fascinating shapes of polar curves and how they evolve within a family.
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The section on the surface area of the graph of a function of two variables has been restored as Section 15.6 for the convenience of instructors who like to teach it after double integrals, though the full treatment of surface area remains in Chapter 16.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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I continue to seek out examples of how calculus applies to so many aspects of the real world. On page 909 you will see beautiful images of the earth’s magnetic field strength and its second vertical derivative as calculated from Laplace’s equation. I thank Roger Watson for bringing to my attention how this is used in geophysics and mineral exploration.
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More than 25% of the exercises in each chapter are new. Here are some of my favorites: 1.6.58, 2.6.51, 2.8.13–14, 3.3.56, 3.4.67, 3.5.69–72, 3.7.22, 4.3.86, 5.2.51–53, 6.4.30, 11.2.49–50, 11.10.71–72, 12.1.44, 12.4.43–44, and Problems 4, 5, and 8 on pages 837–38.
Technology Enhancements ■
The media and technology to support the text have been enhanced to give professors greater control over their course, to provide extra help to deal with the varying levels of student preparedness for the calculus course, and to improve support for conceptual understanding. New Enhanced WebAssign features including a customizable Cengage YouBook, Just in Time review, Show Your Work, Answer Evaluator, Personalized Study Plan, Master Its, solution videos, lecture video clips (with associated questions), and Visualizing Calculus (TEC animations with associated questions) have been developed to facilitate improved student learning and flexible classroom teaching.
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Tools for Enriching Calculus (TEC) has been completely redesigned and is accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.
Features CONCEPTUAL EXERCISES
The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a TrueFalse Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35–40, 2.8.43–46, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–42, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.5–10, 16.1.11–18, 16.2.17–18, and 16.3.1–2). Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.10, 2.8.58, 4.3.63–64, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.39– 40, 3.7.27, and 9.4.2).
GRADED EXERCISE SETS
Each exercise set is carefully graded, progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems involving applications and proofs.
REALWORLD DATA
My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting realworld data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.36 (percentage of the population under age 18), Exercise 5.1.16 (velocity of the space
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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shuttle Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the windchill index as a function of air temperature and wind speed (Example 2 in Section 14.1). Partial derivatives are introduced in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 3 in Section 14.4). Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 4 in Section 15.1). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns. PROJECTS
One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity. Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare presentday methods with those of the founders of calculus—Fermat’s method for finding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.7), and intersections of three cylinders (after Section 15.8). Additional projects can be found in the Instructor’s Guide (see, for instance, Group Exercise 5.1: Position from Samples).
PROBLEM SOLVING
Students usually have difficulties with problems for which there is no single welldefined procedure for obtaining the answer. I think nobody has improved very much on George Polya’s fourstage problemsolving strategy and, accordingly, I have included a version of his problemsolving principles following Chapter 1. They are applied, both explicitly and implicitly, throughout the book. After the other chapters I have placed sections called Problems Plus, which feature examples of how to tackle challenging calculus problems. In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: “A mathematical problem should be difficult in order to entice us, yet not inaccessible lest it mock our efforts.” When I put these challenging problems on assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problemsolving principles are relevant.
TECHNOLOGY
The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required. The icon ; indicates an exercise that definitely requires the use of such technology, but that is not to say that it can’t be used on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Derive, Maple, Mathematica, or the TI89/92) are required. But technology doesn’t make pencil and paper
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate. TOOLS FOR ENRICHING™ CALCULUS
TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules.
HOMEWORK HINTS
Homework Hints presented in the form of questions try to imitate an effective teaching assistant by functioning as a silent tutor. Hints for representative exercises (usually oddnumbered) are included in every section of the text, indicated by printing the exercise number in red. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress, and are available to students at stewartcalculus.com and in CourseMate and Enhanced WebAssign.
ENHANCED W E B A S S I G N
Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the seventh edition we have been working with the calculus community and WebAssign to develop a more robust online homework system. Up to 70% of the exercises in each section are assignable as online homework, including free response, multiple choice, and multipart formats. The system also includes Active Examples, in which students are guided in stepbystep tutorials through text examples, with links to the textbook and to video solutions. New enhancements to the system include a customizable eBook, a Show Your Work feature, Just in Time review of precalculus prerequisites, an improved Assignment Editor, and an Answer Evaluator that accepts more mathematically equivalent answers and allows for homework grading in much the same way that an instructor grades.
www.stewartcalculus.com
This site includes the following. ■ Homework Hints ■ Algebra Review ■ Lies My Calculator and Computer Told Me ■ History of Mathematics, with links to the better historical websites ■ Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes ■ Archived Problems (Drill exercises that appeared in previous editions, together with their solutions) ■ Challenge Problems (some from the Problems Plus sections from prior editions) ■ Links, for particular topics, to outside web resources ■ Selected Tools for Enriching Calculus (TEC) Modules and Visuals
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Content Diagnostic Tests
The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.
A Preview of Calculus
This is an overview of the subject and includes a list of questions to motivate the study of calculus.
1 Functions and Models
From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view.
2
Limits and Derivatives
The material on limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 2.4, on the precise  definition of a limit, is an optional section. Sections 2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the examples and exercises explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.8.
3 Differentiation Rules
All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponential growth and decay are covered in this chapter.
4 Applications of Differentiation
The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow.
5 Integrals
The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed. (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.
6 Applications of Integration
Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration. General methods are emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral.
7 Techniques of Integration
All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I present a strategy for integration. The use of computer algebra systems is discussed in Section 7.6.
8 Further Applications of Integration
Here are the applications of integration—arc length and surface area—for which it is useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also included a section on probability. There are more applications here than can realistically be covered in a given course. Instructors should select applications suitable for their students and for which they themselves have enthusiasm.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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9 Differential Equations
Modeling is the theme that unifies this introductory treatment of differential equations. Direction fields and Euler’s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal consideration. These methods are applied to the exponential, logistic, and other models for population growth. The first four or five sections of this chapter serve as a good introduction to firstorder differential equations. An optional final section uses predatorprey models to illustrate systems of differential equations.
10 Parametric Equations and Polar Coordinates
This chapter introduces parametric and polar curves and applies the methods of calculus to them. Parametric curves are well suited to laboratory projects; the three presented here involve families of curves and Bézier curves. A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13.
11 Inﬁnite Sequences and Series
The convergence tests have intuitive justifications (see page 714) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove convergence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices.
12 Vectors and The Geometry of Space
The material on threedimensional analytic geometry and vectors is divided into two chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces.
13 Vector Functions
This chapter covers vectorvalued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws.
14 Partial Derivatives
Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity.
15 Multiple Integrals
Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals.
16 Vector Calculus
Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.
17 SecondOrder Differential Equations
Since firstorder differential equations are covered in Chapter 9, this final chapter deals with secondorder linear differential equations, their application to vibrating springs and electric circuits, and series solutions.
Ancillaries Calculus, Early Transcendentals, Seventh Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student understanding and to facilitate creative instruction. With this edition, new media and technologies have been developed that help students to visualize calculus and instructors to customize content to better align with the way they teach their course. The tables on pages xxi–xxii describe each of these ancillaries. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Acknowledgments The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken. I have learned something from each of them. SEVENTH EDITION REVIEWERS
Amy Austin, Texas A&M University Anthony J. Bevelacqua, University of North Dakota ZhenQing Chen, University of Washington—Seattle Jenna Carpenter, Louisiana Tech University Le Baron O. Ferguson, University of California—Riverside Shari Harris, John Wood Community College Amer Iqbal, University of Washington—Seattle Akhtar Khan, Rochester Institute of Technology Marianne Korten, Kansas State University Joyce Longman, Villanova University Richard Millspaugh, University of North Dakota Lon H. Mitchell, Virginia Commonwealth University Ho Kuen Ng, San Jose State University Norma OrtizRobinson, Virginia Commonwealth University Qin Sheng, Baylor University Magdalena Toda, Texas Tech University Ruth Trygstad, Salt Lake Community College Klaus Volpert, Villanova University Peiyong Wang, Wayne State University
TECHNOLOGY REVIEWERS
Maria Andersen, Muskegon Community College Eric Aurand, Eastfield College Joy Becker, University of Wisconsin–Stout Przemyslaw Bogacki, Old Dominion University Amy Elizabeth Bowman, University of Alabama in Huntsville Monica Brown, University of Missouri–St. Louis Roxanne Byrne, University of Colorado at Denver and Health Sciences Center Teri Christiansen, University of Missouri–Columbia Bobby Dale Daniel, Lamar University Jennifer Daniel, Lamar University Andras Domokos, California State University, Sacramento Timothy Flaherty, Carnegie Mellon University Lee Gibson, University of Louisville Jane Golden, Hillsborough Community College Semion Gutman, University of Oklahoma Diane Hoffoss, University of San Diego Lorraine Hughes, Mississippi State University Jay Jahangiri, Kent State University John Jernigan, Community College of Philadelphia
Brian Karasek, South Mountain Community College Jason Kozinski, University of Florida Carole Krueger, The University of Texas at Arlington Ken Kubota, University of Kentucky John Mitchell, Clark College Donald Paul, Tulsa Community College Chad Pierson, University of Minnesota, Duluth Lanita Presson, University of Alabama in Huntsville Karin Reinhold, State University of New York at Albany Thomas Riedel, University of Louisville Christopher Schroeder, Morehead State University Angela Sharp, University of Minnesota, Duluth Patricia Shaw, Mississippi State University Carl Spitznagel, John Carroll University Mohammad Tabanjeh, Virginia State University Capt. Koichi Takagi, United States Naval Academy Lorna TenEyck, Chemeketa Community College Roger Werbylo, Pima Community College David Williams, Clayton State University Zhuan Ye, Northern Illinois University
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREVIOUS EDITION REVIEWERS
B. D. Aggarwala, University of Calgary John Alberghini, Manchester Community College Michael Albert, CarnegieMellon University Daniel Anderson, University of Iowa Donna J. Bailey, Northeast Missouri State University Wayne Barber, Chemeketa Community College Marilyn Belkin, Villanova University Neil Berger, University of Illinois, Chicago David Berman, University of New Orleans Richard Biggs, University of Western Ontario Robert Blumenthal, Oglethorpe University Martina Bode, Northwestern University Barbara Bohannon, Hofstra University Philip L. Bowers, Florida State University Amy Elizabeth Bowman, University of Alabama in Huntsville Jay Bourland, Colorado State University Stephen W. Brady, Wichita State University Michael Breen, Tennessee Technological University Robert N. Bryan, University of Western Ontario David Buchthal, University of Akron Jorge Cassio, MiamiDade Community College Jack Ceder, University of California, Santa Barbara Scott Chapman, Trinity University James Choike, Oklahoma State University Barbara Cortzen, DePaul University Carl Cowen, Purdue University Philip S. Crooke, Vanderbilt University Charles N. Curtis, Missouri Southern State College Daniel Cyphert, Armstrong State College Robert Dahlin M. Hilary Davies, University of Alaska Anchorage Gregory J. Davis, University of Wisconsin–Green Bay Elias Deeba, University of Houston–Downtown Daniel DiMaria, Suffolk Community College Seymour Ditor, University of Western Ontario Greg Dresden, Washington and Lee University Daniel Drucker, Wayne State University Kenn Dunn, Dalhousie University Dennis Dunninger, Michigan State University Bruce Edwards, University of Florida David Ellis, San Francisco State University John Ellison, Grove City College Martin Erickson, Truman State University Garret Etgen, University of Houston Theodore G. Faticoni, Fordham University Laurene V. Fausett, Georgia Southern University Norman Feldman, Sonoma State University Newman Fisher, San Francisco State University José D. Flores, The University of South Dakota William Francis, Michigan Technological University James T. Franklin, Valencia Community College, East Stanley Friedlander, Bronx Community College Patrick Gallagher, Columbia University–New York Paul Garrett, University of Minnesota–Minneapolis Frederick Gass, Miami University of Ohio
Bruce Gilligan, University of Regina Matthias K. Gobbert, University of Maryland, Baltimore County Gerald Goff, Oklahoma State University Stuart Goldenberg, California Polytechnic State University John A. Graham, Buckingham Browne & Nichols School Richard Grassl, University of New Mexico Michael Gregory, University of North Dakota Charles Groetsch, University of Cincinnati Paul Triantafilos Hadavas, Armstrong Atlantic State University Salim M. Haïdar, Grand Valley State University D. W. Hall, Michigan State University Robert L. Hall, University of Wisconsin–Milwaukee Howard B. Hamilton, California State University, Sacramento Darel Hardy, Colorado State University Gary W. Harrison, College of Charleston Melvin Hausner, New York University/Courant Institute Curtis Herink, Mercer University Russell Herman, University of North Carolina at Wilmington Allen Hesse, Rochester Community College Randall R. Holmes, Auburn University James F. Hurley, University of Connecticut Matthew A. Isom, Arizona State University Gerald Janusz, University of Illinois at UrbanaChampaign John H. Jenkins, EmbryRiddle Aeronautical University, Prescott Campus Clement Jeske, University of Wisconsin, Platteville Carl Jockusch, University of Illinois at UrbanaChampaign Jan E. H. Johansson, University of Vermont Jerry Johnson, Oklahoma State University Zsuzsanna M. Kadas, St. Michael’s College Nets Katz, Indiana University Bloomington Matt Kaufman Matthias Kawski, Arizona State University Frederick W. Keene, Pasadena City College Robert L. Kelley, University of Miami Virgil Kowalik, Texas A&I University Kevin Kreider, University of Akron Leonard Krop, DePaul University Mark Krusemeyer, Carleton College John C. Lawlor, University of Vermont Christopher C. Leary, State University of New York at Geneseo David Leeming, University of Victoria Sam Lesseig, Northeast Missouri State University Phil Locke, University of Maine Joan McCarter, Arizona State University Phil McCartney, Northern Kentucky University James McKinney, California State Polytechnic University, Pomona Igor Malyshev, San Jose State University Larry Mansfield, Queens College Mary Martin, Colgate University Nathaniel F. G. Martin, University of Virginia Gerald Y. Matsumoto, American River College Tom Metzger, University of Pittsburgh
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
Michael Montaño, Riverside Community College Teri Jo Murphy, University of Oklahoma Martin Nakashima, California State Polytechnic University, Pomona Richard Nowakowski, Dalhousie University Hussain S. Nur, California State University, Fresno Wayne N. Palmer, Utica College Vincent Panico, University of the Pacific F. J. Papp, University of Michigan–Dearborn Mike Penna, Indiana University–Purdue University Indianapolis Mark Pinsky, Northwestern University Lothar Redlin, The Pennsylvania State University Joel W. Robbin, University of Wisconsin–Madison Lila Roberts, Georgia College and State University E. Arthur Robinson, Jr., The George Washington University Richard Rockwell, Pacific Union College Rob Root, Lafayette College Richard Ruedemann, Arizona State University David Ryeburn, Simon Fraser University Richard St. Andre, Central Michigan University Ricardo Salinas, San Antonio College Robert Schmidt, South Dakota State University Eric Schreiner, Western Michigan University Mihr J. Shah, Kent State University–Trumbull Theodore Shifrin, University of Georgia
Wayne Skrapek, University of Saskatchewan Larry Small, Los Angeles Pierce College Teresa Morgan Smith, Blinn College William Smith, University of North Carolina Donald W. Solomon, University of Wisconsin–Milwaukee Edward Spitznagel, Washington University Joseph Stampfli, Indiana University Kristin Stoley, Blinn College M. B. Tavakoli, Chaffey College Paul Xavier Uhlig, St. Mary’s University, San Antonio Stan Ver Nooy, University of Oregon Andrei Verona, California State University–Los Angeles Russell C. Walker, Carnegie Mellon University William L. Walton, McCallie School Jack Weiner, University of Guelph Alan Weinstein, University of California, Berkeley Theodore W. Wilcox, Rochester Institute of Technology Steven Willard, University of Alberta Robert Wilson, University of Wisconsin–Madison Jerome Wolbert, University of Michigan–Ann Arbor Dennis H. Wortman, University of Massachusetts, Boston Mary Wright, Southern Illinois University–Carbondale Paul M. Wright, Austin Community College Xian Wu, University of South Carolina
In addition, I would like to thank Jordan Bell, George Bergman, Leon Gerber, Mary Pugh, and Simon Smith for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Straffin, and Klaus Volpert for ideas for projects; Dan Anderson, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript. In addition, I thank those who have contributed to past editions: Ed Barbeau, Fred Brauer, Andy BulmanFleming, Bob Burton, David Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey Keynes, E.L. Koh, Zdislav Kovarik, Kevin Kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Doug Shaw, Dan Silver, Norton Starr, Saleem Watson, Alan Weinstein, and Gail Wolkowicz. I also thank Kathi Townes, Stephanie Kuhns, and Rebekah Million of TECHarts for their production services and the following Brooks/Cole staff: Cheryll Linthicum, content project manager; Liza Neustaetter, assistant editor; Maureen Ross, media editor; Sam Subity, managing media editor; Jennifer Jones, marketing manager; and Vernon Boes, art director. They have all done an outstanding job. I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, and now Liz Covello. All of them have contributed greatly to the success of this book. JAMES STEWART
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Ancillaries for Instructors PowerLecture ISBN 0840054211
This comprehensive DVD contains all art from the text in both jpeg and PowerPoint formats, key equations and tables from the text, complete prebuilt PowerPoint lectures, an electronic version of the Instructor’s Guide, Solution Builder, ExamView testing software, Tools for Enriching Calculus, video instruction, and JoinIn on TurningPoint clicker content. Instructor’s Guide by Douglas Shaw ISBN 0840054181
Each section of the text is discussed from several viewpoints. The Instructor’s Guide contains suggested time to allot, points to stress, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework assignments. An electronic version of the Instructor’s Guide is available on the PowerLecture DVD. Complete Solutions Manual Single Variable Early Transcendentals By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 0840049366
Multivariable By Dan Clegg and Barbara Frank ISBN 0840049471
Includes workedout solutions to all exercises in the text. Solution Builder www.cengage.com /solutionbuilder This online instructor database offers complete worked out solutions to all exercises in the text. Solution Builder allows you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. Printed Test Bank By William Steven Harmon ISBN 084005419X
Contains textspecific multiplechoice and free response test items. ExamView Testing Create, deliver, and customize tests in print and online formats with ExamView, an easytouse assessment and tutorial software. ExamView contains hundreds of multiplechoice and free response test items. ExamView testing is available on the PowerLecture DVD.
■ Electronic items
■ Printed items
Ancillaries for Instructors and Students Stewart Website www.stewartcalculus.com Contents: Homework Hints ■ Algebra Review ■ Additional Topics ■ Drill exercises ■ Challenge Problems ■ Web Links ■ History of Mathematics ■ Tools for Enriching Calculus (TEC)
TEC Tools for Enriching™ Calculus By James Stewart, Harvey Keynes, Dan Clegg, and developer Hu Hohn Tools for Enriching Calculus (TEC) functions as both a powerful tool for instructors, as well as a tutorial environment in which students can explore and review selected topics. The Flash simulation modules in TEC include instructions, written and audio explanations of the concepts, and exercises. TEC is accessible in CourseMate, WebAssign, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.
Enhanced WebAssign www.webassign.net WebAssign’s homework delivery system lets instructors deliver, collect, grade, and record assignments via the web. Enhanced WebAssign for Stewart’s Calculus now includes opportunities for students to review prerequisite skills and content both at the start of the course and at the beginning of each section. In addition, for selected problems, students can get extra help in the form of “enhanced feedback” (rejoinders) and video solutions. Other key features include: thousands of problems from Stewart’s Calculus, a customizable Cengage YouBook, Personal Study Plans, Show Your Work, Just in Time Review, Answer Evaluator, Visualizing Calculus animations and modules, quizzes, lecture videos (with associated questions), and more!
Cengage Customizable YouBook YouBook is a Flashbased eBook that is interactive and customizable! Containing all the content from Stewart’s Calculus, YouBook features a text edit tool that allows instructors to modify the textbook narrative as needed. With YouBook, instructors can quickly reorder entire sections and chapters or hide any content they don’t teach to create an eBook that perfectly matches their syllabus. Instructors can further customize the text by adding instructorcreated or YouTube video links. Additional media assets include: animated figures, video clips, highlighting, notes, and more! YouBook is available in Enhanced WebAssign.
(Table continues on page xxii.)
xxi Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CourseMate www.cengagebrain.com CourseMate is a perfect selfstudy tool for students, and requires no set up from instructors. CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes: an interactive eBook, Tools for Enriching Calculus, videos, quizzes, flashcards, and more! For instructors, CourseMate includes Engagement Tracker, a firstofitskind tool that monitors student engagement. Maple CDROM Maple provides an advanced, high performance mathematical computation engine with fully integrated numerics & symbolics, all accessible from a WYSIWYG technical document environment. CengageBrain.com To access additional course materials and companion resources, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where free companion resources can be found.
Ancillaries for Students Student Solutions Manual Single Variable Early Transcendentals By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 084004934X
Multivariable By Dan Clegg and Barbara Frank ISBN 0840049455
Provides completely workedout solutions to all oddnumbered exercises in the text, giving students a chance to check their answers and ensure they took the correct steps to arrive at an answer. Study Guide Single Variable Early Transcendentals By Richard St. Andre
well as summary and focus questions with explained answers. The Study Guide also contains “Technology Plus” questions, and multiplechoice “On Your Own” examstyle questions. CalcLabs with Maple Single Variable By Philip B. Yasskin and Robert Lopez ISBN 084005811X
Multivariable By Philip B. Yasskin and Robert Lopez ISBN 0840058128
CalcLabs with Mathematica Single Variable By Selwyn Hollis ISBN 0840058144
Multivariable By Selwyn Hollis ISBN 0840058136
Each of these comprehensive lab manuals will help students learn to use the technology tools available to them. CalcLabs contain clearly explained exercises and a variety of labs and projects to accompany the text. A Companion to Calculus By Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers ISBN 049501124X
Written to improve algebra and problemsolving skills of students taking a Calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use. Linear Algebra for Calculus by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 0534252486
This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra.
ISBN 0840054203
Multivariable By Richard St. Andre ISBN 0840054106
For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, as
■ Electronic items
■ Printed items
xxii Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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To the Student
Reading a calculus textbook is different from reading a newspaper or a novel, or even a physics book. Don’t be discouraged if you have to read a passage more than once in order to understand it. You should have pencil and paper and calculator at hand to sketch a diagram or make a calculation. Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises. In particular, you should look at the definitions to see the exact meanings of the terms. And before you read each example, I suggest that you cover up the solution and try solving the problem yourself. You’ll get a lot more from looking at the solution if you do so. Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, stepbystep fashion with explanatory sentences—not just a string of disconnected equations or formulas. The answers to the oddnumbered exercises appear at the back of the book, in Appendix I. Some exercises ask for a verbal explanation or interpretation or description. In such cases there is no single correct way of expressing the answer, so don’t worry that you haven’t found the definitive answer. In addition, there are often several different forms in which to express a numerical or algebraic answer, so if your answer differs from mine, don’t immediately assume you’re wrong. For example, if the answer given in the back of the book is s2 ⫺ 1 and you obtain 1兾(1 ⫹ s2 ), then you’re right and rationalizing the denominator will show that the answers are equivalent. The icon ; indicates an exercise that definitely requires the use of either a graphing calculator or a computer with graphing software. (Section 1.4 discusses the use of these graphing devices and some of the pitfalls that you may encounter.) But that doesn’t mean that graphing devices can’t be used to check your work on the other exercises as well. The symbol CAS is
reserved for problems in which the full resources of a computer algebra system (like Derive, Maple, Mathematica, or the TI89/92) are required. You will also encounter the symbol , which warns you against committing an error. I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake. Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can be accessed in Enhanced WebAssign and CourseMate (selected Visuals and Modules are available at www.stewartcalculus.com). It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful. Homework Hints for representative exercises are indicated by printing the exercise number in red: 5. These hints can be found on stewartcalculus.com as well as Enhanced WebAssign and CourseMate. The homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer. You need to pursue each hint in an active manner with pencil and paper to work out the details. If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint. I recommend that you keep this book for reference purposes after you finish the course. Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses. And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful. JAMES STEWART
xxiii Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Diagnostic Tests Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.
A
Diagnostic Test: Algebra 1. Evaluate each expression without using a calculator.
(a) 共3兲4 (d)
(b) 34
5 23 5 21
(e)
冉冊 2 3
(c) 34
2
(f ) 16 3兾4
2. Simplify each expression. Write your answer without negative exponents.
(a) s200 s32 (b) 共3a 3b 3 兲共4ab 2 兲 2 (c)
冉
3x 3兾2 y 3 x 2 y1兾2
冊
2
3. Expand and simplify.
(a) 3共x 6兲 4共2x 5兲
(b) 共x 3兲共4x 5兲
(c) (sa sb )(sa sb )
(d) 共2x 3兲2
(e) 共x 2兲3 4. Factor each expression.
(a) 4x 2 25 (c) x 3 3x 2 4x 12 (e) 3x 3兾2 9x 1兾2 6x 1兾2
(b) 2x 2 5x 12 (d) x 4 27x (f ) x 3 y 4xy
5. Simplify the rational expression.
(a)
x 2 3x 2 x2 x 2
(c)
x2 x1 2 x 4 x2
2x 2 x 1 x3 ⴢ x2 9 2x 1 y x x y (d) 1 1 y x (b)
xxiv
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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DIAGNOSTIC TESTS
6. Rationalize the expression and simplify.
(a)
s10 s5 2
(b)
s4 h 2 h
7. Rewrite by completing the square.
(a) x 2 x 1
(b) 2x 2 12x 11
8. Solve the equation. (Find only the real solutions.)
2x 2x 1 苷 x1 x (d) 2x 2 4x 1 苷 0
(a) x 5 苷 14 2 x 1
(b)
(c) x2 x 12 苷 0
ⱍ
(e) x 4 3x 2 2 苷 0 (g) 2x共4 x兲1兾2 3 s4 x 苷 0
ⱍ
(f ) 3 x 4 苷 10
9. Solve each inequality. Write your answer using interval notation.
(a) 4 5 3x 17 (c) x共x 1兲共x 2兲 0 2x 3 (e) 1 x1
(b) x 2 2x 8 (d) x 4 3
ⱍ
ⱍ
10. State whether each equation is true or false.
(a) 共 p q兲2 苷 p 2 q 2
(b) sab 苷 sa sb
(c) sa 2 b 2 苷 a b
(d)
1 TC 苷1T C
(f )
1兾x 1 苷 a兾x b兾x ab
(e)
1 1 1 苷 xy x y
Answers to Diagnostic Test A: Algebra 1. (a) 81
(d) 25 2. (a) 6s2
(b) 81
(c)
9 4
(f )
(e)
(b) 48a 5b7
(c)
1 81 1 8
x 9y7
3. (a) 11x 2
(b) 4x 2 7x 15 (c) a b (d) 4x 2 12x 9 (e) x 3 6x 2 12x 8
4. (a) 共2x 5兲共2x 5兲
(c) 共x 3兲共x 2兲共x 2兲 (e) 3x1兾2共x 1兲共x 2兲 x2 x2 1 (c) x2
5. (a)
(b) 共2x 3兲共x 4兲 (d) x共x 3兲共x 2 3x 9兲 (f ) xy共x 2兲共x 2兲 (b)
x1 x3
(d) 共x y兲
6. (a) 5s2 2s10 7. (a) ( x
1 2 2
)
34
8. (a) 6
(d) 1 s2 1 2
(g)
(b)
1 s4 h 2
(b) 2共x 3兲2 7 (b) 1
(c) 3, 4
(e) 1, s2
2 22 (f ) 3 , 3
12 5
9. (a) 关4, 3兲
(c) 共2, 0兲 傼 共1, 兲 (e) 共1, 4兴
10. (a) False
(d) False
(b) True (e) False
(b) 共2, 4兲 (d) 共1, 7兲
(c) False (f ) True
If you have had difficulty with these problems, you may wish to consult the Review of Algebra on the website www.stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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B
DIAGNOSTIC TESTS
Diagnostic Test: Analytic Geometry 1. Find an equation for the line that passes through the point 共2, 5兲 and
(a) (b) (c) (d)
has slope 3 is parallel to the xaxis is parallel to the yaxis is parallel to the line 2x 4y 苷 3
2. Find an equation for the circle that has center 共1, 4兲 and passes through the point 共3, 2兲. 3. Find the center and radius of the circle with equation x 2 y2 6x 10y 9 苷 0. 4. Let A共7, 4兲 and B共5, 12兲 be points in the plane.
(a) (b) (c) (d) (e) (f )
Find the slope of the line that contains A and B. Find an equation of the line that passes through A and B. What are the intercepts? Find the midpoint of the segment AB. Find the length of the segment AB. Find an equation of the perpendicular bisector of AB. Find an equation of the circle for which AB is a diameter.
5. Sketch the region in the xyplane defined by the equation or inequalities.
ⱍ ⱍ
ⱍ ⱍ
(a) 1 y 3
(b) x 4 and y 2
(c) y 1 x
(d) y x 1
(e) x y 4
(f ) 9x 2 16y 2 苷 144
1 2
2
2
2
Answers to Diagnostic Test B: Analytic Geometry 1. (a) y 苷 3x 1
(c) x 苷 2
(b) y 苷 5
5. (a)
(d) y 苷 x 6
(b)
y
(c)
y
y
3
1 2
1
2
2. 共x 1兲2 共 y 4兲2 苷 52
1
y=1 2 x
0
3. Center 共3, 5兲, radius 5
x
_1
0
_4
4x
0
2
x
_2
4. (a) 3
4
(b) (c) (d) (e) (f )
4x 3y 16 苷 0; xintercept 4, yintercept 163 共1, 4兲 20 3x 4y 苷 13 共x 1兲2 共 y 4兲2 苷 100
(d)
(e)
y
(f)
y 2
≈+¥=4
y 3
0 _1
1
x
0
2
x
0
4 x
y=≈1
If you have had difficulty with these problems, you may wish to consult the review of analytic geometry in Appendixes B and C.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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DIAGNOSTIC TESTS
C
Diagnostic Test: Functions 1. The graph of a function f is given at the left.
y
1 0
x
1
State the value of f 共1兲. Estimate the value of f 共2兲. For what values of x is f 共x兲 苷 2? Estimate the values of x such that f 共x兲 苷 0. State the domain and range of f .
(a) (b) (c) (d) (e)
2. If f 共x兲 苷 x 3 , evaluate the difference quotient
f 共2 h兲 f 共2兲 and simplify your answer. h
3. Find the domain of the function.
FIGURE FOR PROBLEM 1
2x 1 x2 x 2
(a) f 共x兲 苷
(b) t共x兲 苷
3 x s x2 1
(c) h共x兲 苷 s4 x sx 2 1
4. How are graphs of the functions obtained from the graph of f ?
(a) y 苷 f 共x兲
(b) y 苷 2 f 共x兲 1
(c) y 苷 f 共x 3兲 2
5. Without using a calculator, make a rough sketch of the graph.
(a) y 苷 x 3 (d) y 苷 4 x 2 (g) y 苷 2 x 6. Let f 共x兲 苷
再
1 x2 2x 1
(b) y 苷 共x 1兲3 (e) y 苷 sx (h) y 苷 1 x 1
(c) y 苷 共x 2兲3 3 (f ) y 苷 2 sx
if x 0 if x 0
(a) Evaluate f 共2兲 and f 共1兲.
(b) Sketch the graph of f .
7. If f 共x兲 苷 x 2x 1 and t共x兲 苷 2x 3, find each of the following functions. 2
(a) f ⴰ t
(b) t ⴰ f
(c) t ⴰ t ⴰ t
Answers to Diagnostic Test C: Functions 1. (a) 2
(b) 2.8 (d) 2.5, 0.3
(c) 3, 1 (e) 关3, 3兴, 关2, 3兴
(d)
(e)
y 4
0
2. 12 6h h 2 3. (a) 共 , 2兲 傼 共2, 1兲 傼 共1, 兲
(g)
(b) 共 , 兲 (c) 共 , 1兴 傼 关1, 4兴
0
x
2
(h)
y
(f)
y
1
x
1
x
y
0
1
x
y 1
0
4. (a) Reflect about the xaxis
0
x
1
_1
(b) Stretch vertically by a factor of 2, then shift 1 unit downward (c) Shift 3 units to the right and 2 units upward 5. (a)
(b)
y
1 0
(c)
y
x
_1
(b)
7. (a) 共 f ⴰ t兲共x兲 苷 4x 2 8x 2
(b) 共 t ⴰ f 兲共x兲 苷 2x 2 4x 5 (c) 共 t ⴰ t ⴰ t兲共x兲 苷 8x 21
y
(2, 3)
1 1
6. (a) 3, 3
y
1 0
x 0
x
_1
0
x
If you have had difficulty with these problems, you should look at Sections 1.1–1.3 of this book.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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D
DIAGNOSTIC TESTS
Diagnostic Test: Trigonometry 1. Convert from degrees to radians.
(b) 18
(a) 300
2. Convert from radians to degrees.
(a) 5 兾6
(b) 2
3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of
30 . 4. Find the exact values.
(a) tan共 兾3兲
(b) sin共7 兾6兲
(c) sec共5 兾3兲
5. Express the lengths a and b in the figure in terms of . 24
6. If sin x 苷 3 and sec y 苷 4 , where x and y lie between 0 and 2, evaluate sin共x y兲. 1
a
5
7. Prove the identities.
¨
(a) tan sin cos 苷 sec
b FIGURE FOR PROBLEM 5
(b)
2 tan x 苷 sin 2x 1 tan 2x
8. Find all values of x such that sin 2x 苷 sin x and 0 x 2 . 9. Sketch the graph of the function y 苷 1 sin 2x without using a calculator.
Answers to Diagnostic Test D: Trigonometry 1. (a) 5 兾3
(b) 兾10
6.
2. (a) 150
(b) 360 兾 ⬇ 114.6
8. 0, 兾3, , 5 兾3, 2
1 15
(4 6 s2 )
9.
3. 2 cm 4. (a) s3
(b) 12
5. (a) 24 sin
(b) 24 cos
y 2
(c) 2 _π
0
π
x
If you have had difficulty with these problems, you should look at Appendix D of this book.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A Preview of Calculus
© Pichugin Dmitry / Shutterstock
© Ziga Camernik / Shutterstock
By the time you finish this course, you will be able to estimate the number of laborers needed to build a pyramid, explain the formation and location of rainbows, design a roller coaster for a smooth ride, and calculate the force on a dam.
© Brett Mulcahy / Shutterstock
© iofoto / Shutterstock
Calculus is fundamentally different from the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems.
1 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A PREVIEW OF CALCULUS
A¡
The Area Problem A∞
A™ A£
The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.
A¢
A=A¡+A™+A£+A¢+A∞ FIGURE 1
A£
A¢
A∞
Aß
⭈⭈⭈
A¶
⭈⭈⭈
A¡™
FIGURE 2
Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write TEC In the Preview Visual, you can see how areas of inscribed and circumscribed polygons approximate the area of a circle.
A lim An nl⬁
The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century BC) used exhaustion to prove the familiar formula for the area of a circle: A r 2. We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y
y
y
(1, 1)
y
(1, 1)
(1, 1)
(1, 1)
y=≈ A 0
FIGURE 3
1
x
0
1 4
1 2
3 4
1
x
0
1
x
0
1 n
1
x
FIGURE 4
The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.
The Tangent Problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y f 共x兲 at a given point P. (We will give a precise definition of a tangent line in
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A PREVIEW OF CALCULUS y
Chapter 2. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on t. To get around the problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant line PQ. From Figure 6 we see that
t y=ƒ P
0
x
FIGURE 5
mPQ
1
f 共x兲 ⫺ f 共a兲 x⫺a
Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write
The tangent line at P y
t
m lim mPQ Q lP
Q { x, ƒ} ƒf(a)
P { a, f(a)}
and we say that m is the limit of mPQ as Q approaches P along the curve. Since x approaches a as Q approaches P, we could also use Equation 1 to write
xa
a
0
3
x
x
m lim
2
xla
f 共x兲 ⫺ f 共a兲 x⫺a
FIGURE 6
The secant line PQ y
t Q P
0
FIGURE 7
Secant lines approaching the tangent line
x
Specific examples of this procedure will be given in Chapter 2. The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716). The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in Chapter 5.
Velocity When we look at the speedometer of a car and read that the car is traveling at 48 mi兾h, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 mi. But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 mi兾h? In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in feet) at lsecond intervals as in the following chart: t Time elapsed (s)
0
1
2
3
4
5
d Distance (ft)
0
2
9
24
42
71
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A PREVIEW OF CALCULUS
As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 艋 t 艋 4: average velocity
change in position time elapsed 42 ⫺ 9 4⫺2
16.5 ft兾s Similarly, the average velocity in the time interval 2 艋 t 艋 3 is average velocity
24 ⫺ 9 15 ft兾s 3⫺2
We have the feeling that the velocity at the instant t 2 can’t be much different from the average velocity during a short time interval starting at t 2. So let’s imagine that the distance traveled has been measured at 0.lsecond time intervals as in the following chart: t
2.0
2.1
2.2
2.3
2.4
2.5
d
9.00
10.02
11.16
12.45
13.96
15.80
Then we can compute, for instance, the average velocity over the time interval 关2, 2.5兴: average velocity
15.80 ⫺ 9.00 13.6 ft兾s 2.5 ⫺ 2
The results of such calculations are shown in the following chart: Time interval
关2, 3兴
关2, 2.5兴
关2, 2.4兴
关2, 2.3兴
关2, 2.2兴
关2, 2.1兴
Average velocity (ft兾s)
15.0
13.6
12.4
11.5
10.8
10.2
The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t 2 is about 10 ft兾s. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals. In Figure 8 we show a graphical representation of the motion of the car by plotting the distance traveled as a function of time. If we write d f 共t兲, then f 共t兲 is the number of feet traveled after t seconds. The average velocity in the time interval 关2, t兴 is
d
Q { t, f(t)}
average velocity
which is the same as the slope of the secant line PQ in Figure 8. The velocity v when t 2 is the limiting value of this average velocity as t approaches 2; that is,
20 10 0
change in position f 共t兲 ⫺ f 共2兲 time elapsed t⫺2
P { 2, f(2)} 1
FIGURE 8
2
3
4
v lim 5
t
tl2
f 共t兲 ⫺ f 共2兲 t⫺2
and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P.
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A PREVIEW OF CALCULUS
5
Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences.
The Limit of a Sequence In the fifth century BC the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning space and time that were held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position a 1 and the tortoise starts at position t1 . (See Figure 9.) When Achilles reaches the point a 2 t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 t2 , the tortoise is at t3 . This process continues indefinitely and so it appears that the tortoise will always be ahead! But this defies common sense. a¡
a™
a£
a¢
a∞
...
t¡
t™
t£
t¢
...
Achilles FIGURE 9
tortoise
One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles 共a 1, a 2 , a 3 , . . .兲 or the successive positions of the tortoise 共t1, t2 , t3 , . . .兲 form what is known as a sequence. In general, a sequence 兵a n其 is a set of numbers written in a definite order. For instance, the sequence
{1, 12 , 13 , 14 , 15 , . . .} can be described by giving the following formula for the nth term: an a¢ a£
a™
0
1 n
We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n 1兾n are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing
a¡ 1
(a) 1
lim
nl⬁
1 2 3 4 5 6 7 8
1 0 n
n
In general, the notation
(b) FIGURE 10
lim a n L
nl⬁
is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large.
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A PREVIEW OF CALCULUS
The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if a 1 3.1 a 2 3.14 a 3 3.141 a 4 3.1415 a 5 3.14159 a 6 3.141592 a 7 3.1415926 ⭈ ⭈ ⭈ lim a n
then
nl⬁
The terms in this sequence are rational approximations to . Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise form sequences 兵a n其 and 兵tn 其, where a n ⬍ tn for all n. It can be shown that both sequences have the same limit: lim a n p lim tn
nl⬁
nl⬁
It is precisely at this point p that Achilles overtakes the tortoise.
The Sum of a Series Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall. In order to do so, he would first have to go half the distance, then half the remaining distance, and then again half of what still remains. This process can always be continued and can never be ended.” (See Figure 11.)
1 2
FIGURE 11
1 4
1 8
1 16
Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3
1
1 1 1 1 1 ⫹ ⫹ ⫹ ⫹ ⭈⭈⭈ ⫹ n ⫹ ⭈⭈⭈ 2 4 8 16 2
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A PREVIEW OF CALCULUS
7
Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. For instance, in decimal notation, the symbol 0.3 0.3333 . . . means 3 3 3 3 ⫹ ⫹ ⫹ ⫹ ⭈⭈⭈ 10 100 1000 10,000 and so, in some sense, it must be true that 3 3 3 3 1 ⫹ ⫹ ⫹ ⫹ ⭈⭈⭈ 10 100 1000 10,000 3 More generally, if dn denotes the nth digit in the decimal representation of a number, then 0.d1 d2 d3 d4 . . .
d1 d2 d3 dn ⫹ 2 ⫹ 3 ⫹ ⭈⭈⭈ ⫹ n ⫹ ⭈⭈⭈ 10 10 10 10
Therefore some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. Thus s1 12 0.5 s2 12 ⫹ 14 0.75 s3 12 ⫹ 14 ⫹ 18 0.875 s4 12 ⫹ 14 ⫹ 18 ⫹ 161 0.9375 s5 12 ⫹ 14 ⫹ 18 ⫹ 161 ⫹ 321 0.96875 s6 12 ⫹ 14 ⫹ 18 ⫹ 161 ⫹ 321 ⫹ 641 0.984375 s7 12 ⫹ 14 ⭈ ⭈ ⭈ s10 12 ⫹ 14 ⭈ ⭈ ⭈ 1 s16 ⫹ 2
1 ⫹ 18 ⫹ 161 ⫹ 321 ⫹ 641 ⫹ 128 0.9921875
1 ⫹ ⭈ ⭈ ⭈ ⫹ 1024 ⬇ 0.99902344
1 1 ⫹ ⭈ ⭈ ⭈ ⫹ 16 ⬇ 0.99998474 4 2
Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to write 1 1 1 1 ⫹ ⫹ ⫹ ⭈⭈⭈ ⫹ n ⫹ ⭈⭈⭈ 1 2 4 8 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A PREVIEW OF CALCULUS
In other words, the reason the sum of the series is 1 is that lim sn 1
nl⬁
In Chapter 11 we will discuss these ideas further. We will then use Newton’s idea of combining infinite series with differential and integral calculus.
Summary We have seen that the concept of a limit arises in trying to find the area of a region, the slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In each case the common theme is the calculation of a quantity as the limit of other, easily calculated quantities. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. After Sir Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore some of these uses of calculus in this book. In order to convey a sense of the power of the subject, we end this preview with a list of some of the questions that you will be able to answer using calculus: 1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation
rays from sun
2. 138° rays from sun
42°
3. 4. 5.
observer FIGURE 12
6. 7. 8. 9. 10. 11. 12.
from an observer up to the highest point in a rainbow is 42°? (See page 282.) How can we explain the shapes of cans on supermarket shelves? (See page 337.) Where is the best place to sit in a movie theater? (See page 456.) How can we design a roller coaster for a smooth ride? (See page 184.) How far away from an airport should a pilot start descent? (See page 208.) How can we fit curves together to design shapes to represent letters on a laser printer? (See page 653.) How can we estimate the number of workers that were needed to build the Great Pyramid of Khufu in ancient Egypt? (See page 451.) Where should an infielder position himself to catch a baseball thrown by an outfielder and relay it to home plate? (See page 456.) Does a ball thrown upward take longer to reach its maximum height or to fall back to its original height? (See page 604.) How can we explain the fact that planets and satellites move in elliptical orbits? (See page 868.) How can we distribute water flow among turbines at a hydroelectric station so as to maximize the total energy production? (See page 966.) If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which of them reaches the bottom first? (See page 1039.)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Functions and Models
Often a graph is the best way to represent a function because it conveys so much information at a glance. Shown is a graph of the ground acceleration created by the 2008 earthquake in Sichuan province in China. The hardest hit town was Beichuan, as pictured.
Courtesy of the IRIS Consortium. www.iris.edu
© Mark Ralston / AFP / Getty Images
The fundamental objects that we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of realworld phenomena. We also discuss the use of graphing calculators and graphing software for computers.
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CHAPTER 1
1.1
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FUNCTIONS AND MODELS
Four Ways to Represent a Function
Year
Population (millions)
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870
Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A 苷 r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t . The table gives estimates of the world population P共t兲 at time t, for certain years. For instance, P共1950兲 ⬇ 2,560,000,000 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C , the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a. a {cm/
[email protected]} 100
50
5
FIGURE 1
Vertical ground acceleration during the Northridge earthquake
10
15
20
25
30
t (seconds)
_50 Calif. Dept. of Mines and Geology
Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number ( A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number. A function f is a rule that assigns to each element x in a set D exactly one element, called f 共x兲, in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f 共x兲 is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f 共x兲 as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.1
x (input)
f
ƒ (output)
FIGURE 2
Machine diagram for a function ƒ
x
ƒ a
f(a)
f
D
FOUR WAYS TO REPRESENT A FUNCTION
11
It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f 共x兲 according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or s x ) and enter the input x. If x ⬍ 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x 艌 0, then an approximation to s x will appear in the display. Thus the s x key on your calculator is not quite the same as the exact mathematical function f defined by f 共x兲 苷 s x . Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of D to an element of E. The arrow indicates that f 共x兲 is associated with x, f 共a兲 is associated with a, and so on. The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs
ⱍ
兵共x, f 共x兲兲 x 僆 D其
E
(Notice that these are inputoutput pairs.) In other words, the graph of f consists of all points 共x, y兲 in the coordinate plane such that y 苷 f 共x兲 and x is in the domain of f. The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the ycoordinate of any point 共x, y兲 on the graph is y 苷 f 共x兲, we can read the value of f 共x兲 from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the xaxis and its range on the yaxis as in Figure 5.
FIGURE 3
Arrow diagram for ƒ
y
y
{ x, ƒ}
y ⫽ ƒ(x)
range
ƒ f (2) f (1) 0
1
2
x
x
x
0
domain FIGURE 4
FIGURE 5
y
EXAMPLE 1 The graph of a function f is shown in Figure 6. (a) Find the values of f 共1兲 and f 共5兲. (b) What are the domain and range of f ?
1
SOLUTION
0
1
FIGURE 6
The notation for intervals is given in Appendix A.
x
(a) We see from Figure 6 that the point 共1, 3兲 lies on the graph of f, so the value of f at 1 is f 共1兲 苷 3. (In other words, the point on the graph that lies above x 苷 1 is 3 units above the xaxis.) When x 苷 5, the graph lies about 0.7 unit below the xaxis, so we estimate that f 共5兲 ⬇ ⫺0.7. (b) We see that f 共x兲 is defined when 0 艋 x 艋 7, so the domain of f is the closed interval 关0, 7兴. Notice that f takes on all values from ⫺2 to 4, so the range of f is
ⱍ
兵y ⫺2 艋 y 艋 4其 苷 关⫺2, 4兴
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 1
Page 12
EXAMPLE 2 Sketch the graph and find the domain and range of each function. (a) f共x兲 苷 2x ⫺ 1 (b) t共x兲 苷 x 2 SOLUTION
y=2x1 x
1 2
FIGURE 7 y (2, 4)
y=≈ (_1, 1)
4:11 PM
FUNCTIONS AND MODELS
y
0 1
9/20/10
(a) The equation of the graph is y 苷 2x ⫺ 1, and we recognize this as being the equation of a line with slope 2 and yintercept ⫺1. (Recall the slopeintercept form of the equation of a line: y 苷 mx ⫹ b. See Appendix B.) This enables us to sketch a portion of the graph of f in Figure 7. The expression 2x ⫺ 1 is defined for all real numbers, so the domain of f is the set of all real numbers, which we denote by ⺢. The graph shows that the range is also ⺢. (b) Since t共2兲 苷 2 2 苷 4 and t共⫺1兲 苷 共⫺1兲2 苷 1, we could plot the points 共2, 4兲 and 共⫺1, 1兲, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y 苷 x 2, which represents a parabola (see Appendix C). The domain of t is ⺢. The range of t consists of all values of t共x兲, that is, all numbers of the form x 2. But x 2 艌 0 for all numbers x and any positive number y is a square. So the range of t is 兵 y y 艌 0其 苷 关0, ⬁兲. This can also be seen from Figure 8.
ⱍ
1 0
1
x
EXAMPLE 3 If f 共x兲 苷 2x 2 ⫺ 5x ⫹ 1 and h 苷 0, evaluate
f 共a ⫹ h兲 ⫺ f 共a兲 . h
SOLUTION We first evaluate f 共a ⫹ h兲 by replacing x by a ⫹ h in the expression for f 共x兲:
FIGURE 8
f 共a ⫹ h兲 苷 2共a ⫹ h兲2 ⫺ 5共a ⫹ h兲 ⫹ 1 苷 2共a 2 ⫹ 2ah ⫹ h 2 兲 ⫺ 5共a ⫹ h兲 ⫹ 1 苷 2a 2 ⫹ 4ah ⫹ 2h 2 ⫺ 5a ⫺ 5h ⫹ 1 Then we substitute into the given expression and simplify: f 共a ⫹ h兲 ⫺ f 共a兲 共2a 2 ⫹ 4ah ⫹ 2h 2 ⫺ 5a ⫺ 5h ⫹ 1兲 ⫺ 共2a 2 ⫺ 5a ⫹ 1兲 苷 h h
The expression f 共a ⫹ h兲 ⫺ f 共a兲 h in Example 3 is called a difference quotient and occurs frequently in calculus. As we will see in Chapter 2, it represents the average rate of change of f 共x兲 between x 苷 a and x 苷 a ⫹ h.
苷
2a 2 ⫹ 4ah ⫹ 2h 2 ⫺ 5a ⫺ 5h ⫹ 1 ⫺ 2a 2 ⫹ 5a ⫺ 1 h
苷
4ah ⫹ 2h 2 ⫺ 5h 苷 4a ⫹ 2h ⫺ 5 h
Representations of Functions There are four possible ways to represent a function: ■ verbally (by a description in words) ■
numerically
(by a table of values)
■
visually
(by a graph)
■
algebraically
(by an explicit formula)
If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.1
FOUR WAYS TO REPRESENT A FUNCTION
13
A. The most useful representation of the area of a circle as a function of its radius is
probably the algebraic formula A共r兲 苷 r 2, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is 兵r r ⬎ 0其 苷 共0, ⬁兲, and the range is also 共0, ⬁兲.
ⱍ
t
Population (millions)
0 10 20 30 40 50 60 70 80 90 100 110
1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870
B. We are given a description of the function in words: P共t兲 is the human population of
the world at time t. Let’s measure t so that t 苷 0 corresponds to the year 1900. The table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population P共t兲 at any time t. But it is possible to find an expression for a function that approximates P共t兲. In fact, using methods explained in Section 1.2, we obtain the approximation P共t兲 ⬇ f 共t兲 苷 共1.43653 ⫻ 10 9 兲 ⭈ 共1.01395兲 t Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary.
P
P
5x10'
5x10'
0
20
40
60
FIGURE 9
w (ounces)
⭈ ⭈ ⭈
100
120
t
0
20
40
60
80
100
120
t
FIGURE 10
A function defined by a table of values is called a tabular function.
0⬍w艋 1⬍w艋 2⬍w艋 3⬍w艋 4⬍w艋
80
1 2 3 4 5
The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function.
C共w兲 (dollars)
C. Again the function is described in words: Let C共w兲 be the cost of mailing a large enve
0.88 1.05 1.22 1.39 1.56
D. The graph shown in Figure 1 is the most natural representation of the vertical acceler
⭈ ⭈ ⭈
lope with weight w. The rule that the US Postal Service used as of 2010 is as follows: The cost is 88 cents for up to 1 oz, plus 17 cents for each additional ounce (or less) up to 13 oz. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10). ation function a共t兲. It’s true that a table of values could be compiled, and it is even possible to devise an approximate formula. But everything a geologist needs to know—amplitudes and patterns—can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for liedetection.)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
In the next example we sketch the graph of a function that is defined verbally. T
EXAMPLE 4 When you turn on a hotwater faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on. SOLUTION The initial temperature of the running water is close to room temperature t
0
FIGURE 11
because the water has been sitting in the pipes. When the water from the hotwater tank starts flowing from the faucet, T increases quickly. In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 11. In the following example we start with a verbal description of a function in a physical situation and obtain an explicit algebraic formula. The ability to do this is a useful skill in solving calculus problems that ask for the maximum or minimum values of quantities.
v
EXAMPLE 5 A rectangular storage container with an open top has a volume of 10 m3.
The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base. SOLUTION We draw a diagram as in Figure 12 and introduce notation by letting w and 2w
be the width and length of the base, respectively, and h be the height. The area of the base is 共2w兲w 苷 2w 2, so the cost, in dollars, of the material for the base is 10共2w 2 兲. Two of the sides have area wh and the other two have area 2wh, so the cost of the material for the sides is 6关2共wh兲 ⫹ 2共2wh兲兴. The total cost is therefore
h w
C 苷 10共2w 2 兲 ⫹ 6关2共wh兲 ⫹ 2共2wh兲兴 苷 20 w 2 ⫹ 36 wh
2w
To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10 m3. Thus
FIGURE 12
w 共2w兲h 苷 10
10 5 苷 2 2w 2 w
h苷
which gives
Substituting this into the expression for C , we have
冉 冊
PS In setting up applied functions as in
Example 5, it may be useful to review the principles of problem solving as discussed on page 75, particularly Step 1: Understand the Problem.
C 苷 20w 2 ⫹ 36w
5
w2
苷 20w 2 ⫹
180 w
Therefore the equation C共w兲 苷 20w 2 ⫹
180 w
w⬎0
expresses C as a function of w. EXAMPLE 6 Find the domain of each function. Domain Convention If a function is given by a formula and the domain is not stated explicitly, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number.
(a) f 共x兲 苷 sx ⫹ 2
(b) t共x兲 苷
1 x ⫺x 2
SOLUTION
(a) Because the square root of a negative number is not defined (as a real number), the domain of f consists of all values of x such that x ⫹ 2 艌 0. This is equivalent to x 艌 ⫺2, so the domain is the interval 关⫺2, ⬁兲.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.1
15
FOUR WAYS TO REPRESENT A FUNCTION
(b) Since t共x兲 苷
1 1 苷 x2 ⫺ x x共x ⫺ 1兲
and division by 0 is not allowed, we see that t共x兲 is not defined when x 苷 0 or x 苷 1. Thus the domain of t is
ⱍ
兵x x 苷 0, x 苷 1其 which could also be written in interval notation as 共⫺⬁, 0兲 傼 共0, 1兲 傼 共1, ⬁兲 The graph of a function is a curve in the xyplane. But the question arises: Which curves in the xyplane are graphs of functions? This is answered by the following test. The Vertical Line Test A curve in the xyplane is the graph of a function of x if and only if no vertical line intersects the curve more than once.
The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each vertical line x 苷 a intersects a curve only once, at 共a, b兲, then exactly one functional value is defined by f 共a兲 苷 b. But if a line x 苷 a intersects the curve twice, at 共a, b兲 and 共a, c兲, then the curve can’t represent a function because a function can’t assign two different values to a. y
y
x=a
(a, c)
x=a
(a, b) (a, b) a
0
x
a
0
x
FIGURE 13
For example, the parabola x 苷 y 2 ⫺ 2 shown in Figure 14(a) is not the graph of a function of x because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x. Notice that the equation x 苷 y 2 ⫺ 2 implies y 2 苷 x ⫹ 2, so y 苷 ⫾sx ⫹ 2 . Thus the upper and lower halves of the parabola are the graphs of the functions f 共x兲 苷 s x ⫹ 2 [from Example 6(a)] and t共x兲 苷 ⫺s x ⫹ 2 . [See Figures 14(b) and (c).] We observe that if we reverse the roles of x and y, then the equation x 苷 h共y兲 苷 y 2 ⫺ 2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h. y
y
y
_2 (_2, 0)
FIGURE 14
0
(a) x=¥2
x
_2 0
(b) y=œ„„„„ x+2
x
0
(c) y=_ œ„„„„ x+2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
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FUNCTIONS AND MODELS
Piecewise Deﬁned Functions The functions in the following four examples are defined by different formulas in different parts of their domains. Such functions are called piecewise defined functions.
v
EXAMPLE 7 A function f is defined by
f 共x兲 苷
再
1⫺x x2
if x 艋 ⫺1 if x ⬎ ⫺1
Evaluate f 共⫺2兲, f 共⫺1兲, and f 共0兲 and sketch the graph. SOLUTION Remember that a function is a rule. For this particular function the rule is the
following: First look at the value of the input x. If it happens that x 艋 ⫺1, then the value of f 共x兲 is 1 ⫺ x. On the other hand, if x ⬎ ⫺1, then the value of f 共x兲 is x 2. Since ⫺2 艋 ⫺1, we have f 共⫺2兲 苷 1 ⫺ 共⫺2兲 苷 3. Since ⫺1 艋 ⫺1, we have f 共⫺1兲 苷 1 ⫺ 共⫺1兲 苷 2.
y
Since 0 ⬎ ⫺1, we have f 共0兲 苷 0 2 苷 0.
1
_1
0
1
x
FIGURE 15
How do we draw the graph of f ? We observe that if x 艋 ⫺1, then f 共x兲 苷 1 ⫺ x, so the part of the graph of f that lies to the left of the vertical line x 苷 ⫺1 must coincide with the line y 苷 1 ⫺ x, which has slope ⫺1 and yintercept 1. If x ⬎ ⫺1, then f 共x兲 苷 x 2, so the part of the graph of f that lies to the right of the line x 苷 ⫺1 must coincide with the graph of y 苷 x 2, which is a parabola. This enables us to sketch the graph in Figure 15. The solid dot indicates that the point 共⫺1, 2兲 is included on the graph; the open dot indicates that the point 共⫺1, 1兲 is excluded from the graph. The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by a , is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have
ⱍ ⱍ
ⱍaⱍ 艌 0
For a more extensive review of absolute values, see Appendix A.
For example,
ⱍ3ⱍ 苷 3
ⱍ ⫺3 ⱍ 苷 3
for every number a
ⱍ0ⱍ 苷 0
ⱍ s2 ⫺ 1 ⱍ 苷 s2 ⫺ 1
ⱍ3 ⫺ ⱍ 苷 ⫺ 3
In general, we have
ⱍaⱍ 苷 a ⱍ a ⱍ 苷 ⫺a
if a 艌 0 if a ⬍ 0
(Remember that if a is negative, then ⫺a is positive.)
ⱍ ⱍ
EXAMPLE 8 Sketch the graph of the absolute value function f 共x兲 苷 x .
y
SOLUTION From the preceding discussion we know that
y= x 
ⱍxⱍ 苷 0
FIGURE 16
x
再
x ⫺x
if x 艌 0 if x ⬍ 0
Using the same method as in Example 7, we see that the graph of f coincides with the line y 苷 x to the right of the yaxis and coincides with the line y 苷 ⫺x to the left of the yaxis (see Figure 16).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Page 17
SECTION 1.1
FOUR WAYS TO REPRESENT A FUNCTION
17
EXAMPLE 9 Find a formula for the function f graphed in Figure 17. y
1 0
x
1
FIGURE 17 SOLUTION The line through 共0, 0兲 and 共1, 1兲 has slope m 苷 1 and yintercept b 苷 0, so
its equation is y 苷 x. Thus, for the part of the graph of f that joins 共0, 0兲 to 共1, 1兲, we have f 共x兲 苷 x
if 0 艋 x 艋 1
The line through 共1, 1兲 and 共2, 0兲 has slope m 苷 ⫺1, so its pointslope form is
Pointslope form of the equation of a line:
y ⫺ y1 苷 m共x ⫺ x 1 兲
y ⫺ 0 苷 共⫺1兲共x ⫺ 2兲
See Appendix B.
So we have
f 共x兲 苷 2 ⫺ x
or
y苷2⫺x
if 1 ⬍ x 艋 2
We also see that the graph of f coincides with the xaxis for x ⬎ 2. Putting this information together, we have the following threepiece formula for f :
再
x f 共x兲 苷 2 ⫺ x 0
EXAMPLE 10 In Example C at the beginning of this section we considered the cost C共w兲 of mailing a large envelope with weight w. In effect, this is a piecewise defined function because, from the table of values on page 13, we have
C 1.50
1.00
C共w兲 苷 0.50
0
FIGURE 18
if 0 艋 x 艋 1 if 1 ⬍ x 艋 2 if x ⬎ 2
1
2
3
4
5
w
0.88 if 0 ⬍ w 艋 1 1.05 if 1 ⬍ w 艋 2 1.22 if 2 ⬍ w 艋 3 1.39 if 3 ⬍ w 艋 4 ⭈ ⭈ ⭈
The graph is shown in Figure 18. You can see why functions similar to this one are called step functions—they jump from one value to the next. Such functions will be studied in Chapter 2.
Symmetry If a function f satisfies f 共⫺x兲 苷 f 共x兲 for every number x in its domain, then f is called an even function. For instance, the function f 共x兲 苷 x 2 is even because f 共⫺x兲 苷 共⫺x兲2 苷 x 2 苷 f 共x兲 The geometric significance of an even function is that its graph is symmetric with respect Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
to the yaxis (see Figure 19). This means that if we have plotted the graph of f for x 艌 0, we obtain the entire graph simply by reflecting this portion about the yaxis. y
y
f(_x)
ƒ _x
0
_x
ƒ
0
x
x
x
x
FIGURE 20 An odd function
FIGURE 19 An even function
If f satisfies f 共⫺x兲 苷 ⫺f 共x兲 for every number x in its domain, then f is called an odd function. For example, the function f 共x兲 苷 x 3 is odd because f 共⫺x兲 苷 共⫺x兲3 苷 ⫺x 3 苷 ⫺f 共x兲 The graph of an odd function is symmetric about the origin (see Figure 20). If we already have the graph of f for x 艌 0, we can obtain the entire graph by rotating this portion through 180⬚ about the origin.
v EXAMPLE 11 Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f 共x兲 苷 x 5 ⫹ x (b) t共x兲 苷 1 ⫺ x 4 (c) h共x兲 苷 2x ⫺ x 2 SOLUTION
f 共⫺x兲 苷 共⫺x兲5 ⫹ 共⫺x兲 苷 共⫺1兲5x 5 ⫹ 共⫺x兲
(a)
苷 ⫺x 5 ⫺ x 苷 ⫺共x 5 ⫹ x兲 苷 ⫺f 共x兲 Therefore f is an odd function. t共⫺x兲 苷 1 ⫺ 共⫺x兲4 苷 1 ⫺ x 4 苷 t共x兲
(b) So t is even.
h共⫺x兲 苷 2共⫺x兲 ⫺ 共⫺x兲2 苷 ⫺2x ⫺ x 2
(c)
Since h共⫺x兲 苷 h共x兲 and h共⫺x兲 苷 ⫺h共x兲, we conclude that h is neither even nor odd. The graphs of the functions in Example 11 are shown in Figure 21. Notice that the graph of h is symmetric neither about the yaxis nor about the origin.
1
y
y
y
1
f
g
h
1 1
_1
1
x
x
1
x
_1
FIGURE 21
(a)
( b)
(c)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.1 y
D
The graph shown in Figure 22 rises from A to B, falls from B to C , and rises again from C to D. The function f is said to be increasing on the interval 关a, b兴, decreasing on 关b, c兴, and increasing again on 关c, d兴. Notice that if x 1 and x 2 are any two numbers between a and b with x 1 ⬍ x 2 , then f 共x 1 兲 ⬍ f 共x 2 兲. We use this as the defining property of an increasing function.
y=ƒ C f(x™) f(x¡)
0 a x¡
x™
19
Increasing and Decreasing Functions
B
A
FOUR WAYS TO REPRESENT A FUNCTION
b
c
A function f is called increasing on an interval I if
x
d
f 共x 1 兲 ⬍ f 共x 2 兲
FIGURE 22
whenever x 1 ⬍ x 2 in I
It is called decreasing on I if
y
y=≈
f 共x 1 兲 ⬎ f 共x 2 兲
In the definition of an increasing function it is important to realize that the inequality f 共x 1 兲 ⬍ f 共x 2 兲 must be satisfied for every pair of numbers x 1 and x 2 in I with x 1 ⬍ x 2. You can see from Figure 23 that the function f 共x兲 苷 x 2 is decreasing on the interval 共⫺⬁, 0兴 and increasing on the interval 关0, ⬁兲.
x
0
FIGURE 23
1.1
Exercises
1. If f 共x兲 苷 x ⫹ s2 ⫺ x and t共u兲 苷 u ⫹ s2 ⫺ u , is it true
that f 苷 t?
2. If
f 共x兲 苷
x2 ⫺ x x⫺1
and
(c) (d) (e) (f)
Estimate the solution of the equation f 共x兲 苷 ⫺1. On what interval is f decreasing? State the domain and range of f. State the domain and range of t.
t共x兲 苷 x
is it true that f 苷 t?
y
g f
3. The graph of a function f is given.
(a) (b) (c) (d) (e) (f)
whenever x 1 ⬍ x 2 in I
State the value of f 共1兲. Estimate the value of f 共⫺1兲. For what values of x is f 共x兲 苷 1? Estimate the value of x such that f 共x兲 苷 0. State the domain and range of f. On what interval is f increasing?
0
2
x
5. Figure 1 was recorded by an instrument operated by the Cali
y
fornia Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
1 0
2
1
x
4. The graphs of f and t are given.
(a) State the values of f 共⫺4兲 and t共3兲. (b) For what values of x is f 共x兲 苷 t共x兲?
6. In this section we discussed examples of ordinary, everyday
functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
1. Homework Hints available at stewartcalculus.com Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
7–10 Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 7.
4:21 PM
y
in words what the graph tells you about this race. Who won the race? Did each runner finish the race?
y
8.
y (m)
0
9.
A
1
1
0
x
1
y
C
x
1
y
10.
B
100
0
t (s)
20
1
1 0
1
0
x
x
1
11. The graph shown gives the weight of a certain person as a
function of age. Describe in words how this person’s weight varies over time. What do you think happened when this person was 30 years old?
15. The graph shows the power consumption for a day in Septem
ber in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 AM? At 6 PM? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable? P 800 600
200 weight (pounds)
400
150
200
100 0
50
3
6
9
12
15
18
21
t
Pacific Gas & Electric
0
10
20 30 40
50
60 70
age (years)
12. The graph shows the height of the water in a bathtub as a
function of time. Give a verbal description of what you think happened. height (inches)
16. Sketch a rough graph of the number of hours of daylight as a
function of the time of year. 17. Sketch a rough graph of the outdoor temperature as a function
of time during a typical spring day. 18. Sketch a rough graph of the market value of a new car as a
function of time for a period of 20 years. Assume the car is well maintained.
15
19. Sketch the graph of the amount of a particular brand of coffee
10
sold by a store as a function of the price of the coffee.
5
20. You place a frozen pie in an oven and bake it for an hour. Then 0
5
10
15
time (min)
13. You put some ice cubes in a glass, fill the glass with cold
water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time. 14. Three runners compete in a 100meter race. The graph depicts
the distance run as a function of time for each runner. Describe
you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time. 21. A homeowner mows the lawn every Wednesday afternoon.
Sketch a rough graph of the height of the grass as a function of time over the course of a fourweek period. 22. An airplane takes off from an airport and lands an hour later at
another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x共t兲 be
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.1
the horizontal distance traveled and y共t兲 be the altitude of the plane. (a) Sketch a possible graph of x共t兲. (b) Sketch a possible graph of y共t兲. (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity. 23. The number N (in millions) of US cellular phone subscribers is
1996
1998
2000
2002
2004
2006
N
44
69
109
141
182
233
(a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cellphone subscribers at midyear in 2001 and 2005. 24. Temperature readings T (in °F) were recorded every two hours
from midnight to 2:00 PM in Phoenix on September 10, 2008. The time t was measured in hours from midnight. t
0
2
4
6
8
10
12
14
T
82
75
74
75
84
90
93
94
(a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 9:00 AM. f 共a ⫹ 1兲, 2 f 共a兲, f 共2a兲, f 共a 2 兲, [ f 共a兲] 2, and f 共a ⫹ h兲.
26. A spherical balloon with radius r inches has volume
V共r兲 苷 43 r 3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r ⫹ 1 inches. 27–30 Evaluate the difference quotient for the given function.
Simplify your answer.
38. Find the domain and range and sketch the graph of the
39–50 Find the domain and sketch the graph of the function. 39. f 共x兲 苷 2 ⫺ 0.4x
40. F 共x兲 苷 x 2 ⫺ 2x ⫹ 1
41. f 共t兲 苷 2t ⫹ t 2
42. H共t兲 苷
43. t共x兲 苷 sx ⫺ 5
44. F共x兲 苷 2x ⫹ 1
45. G共x兲 苷 47. f 共x兲 苷 48. f 共x兲 苷 49. f 共x兲 苷
f 共x兲 ⫺ f 共a兲 x⫺a
30. f 共x兲 苷
x⫹3 , x⫹1
50. f 共x兲 苷
ⱍ
ⱍ ⱍ
3x ⫹ x x
再 再 再
x⫹2 1⫺x 3 ⫺ 12 x 2x ⫺ 5
再
4 ⫺ t2 2⫺t
ⱍ
ⱍ ⱍ
46. t共x兲 苷 x ⫺ x
if x ⬍ 0 if x 艌 0 if x 艋 2 if x ⬎ 2
x ⫹ 2 if x 艋 ⫺1 x2 if x ⬎ ⫺1
x⫹9 ⫺2x ⫺6
if x ⬍ ⫺3 if x 艋 3 if x ⬎ 3
ⱍ ⱍ
51–56 Find an expression for the function whose graph is the given curve. 51. The line segment joining the points 共1, ⫺3兲 and 共5, 7兲 52. The line segment joining the points 共⫺5, 10兲 and 共7, ⫺10兲
f 共3 ⫹ h兲 ⫺ f 共3兲 h
1 29. f 共x兲 苷 , x
u⫹1 1 1⫹ u⫹1
37. F共 p兲 苷 s2 ⫺ s p
25. If f 共x兲 苷 3x 2 ⫺ x ⫹ 2, find f 共2兲, f 共⫺2兲, f 共a兲, f 共⫺a兲,
f 共a ⫹ h兲 ⫺ f 共a兲 h
36. f 共u兲 苷
53. The bottom half of the parabola x ⫹ 共 y ⫺ 1兲2 苷 0 54. The top half of the circle x 2 ⫹ 共 y ⫺ 2兲 2 苷 4 55.
f 共x兲 ⫺ f 共1兲 x⫺1
56.
y
y
1
1 0
1
x
0
1
31–37 Find the domain of the function. 31. f 共x兲 苷
x⫹4 x2 ⫺ 9
32. f 共x兲 苷
2x 3 ⫺ 5 2 x ⫹x⫺6
57–61 Find a formula for the described function and state its domain. 57. A rectangle has perimeter 20 m. Express the area of the rect
3 2t ⫺ 1 33. f 共t兲 苷 s
21
function h共x兲 苷 s4 ⫺ x 2 .
t
28. f 共x兲 苷 x 3,
1 4 x 2 ⫺ 5x s
35. h共x兲 苷
shown in the table. (Midyear estimates are given.)
27. f 共x兲 苷 4 ⫹ 3x ⫺ x 2,
FOUR WAYS TO REPRESENT A FUNCTION
34. t共t兲 苷 s3 ⫺ t ⫺ s2 ⫹ t
angle as a function of the length of one of its sides.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
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FUNCTIONS AND MODELS
58. A rectangle has area 16 m2. Express the perimeter of the rect
67. In a certain country, income tax is assessed as follows. There is
no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $14,000? On $26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I.
angle as a function of the length of one of its sides. 59. Express the area of an equilateral triangle as a function of the
length of a side. 60. Express the surface area of a cube as a function of its volume. 61. An open rectangular box with volume 2 m3 has a square base.
Express the surface area of the box as a function of the length of a side of the base. 62. A Norman window has the shape of a rectangle surmounted by
68. The functions in Example 10 and Exercise 67 are called step
functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.
69–70 Graphs of f and t are shown. Decide whether each function
is even, odd, or neither. Explain your reasoning. 69.
70.
y
y
g f
f
x
x g x
63. A box with an open top is to be constructed from a rectangular
piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.
x
x
x
x
x
12 x
other point must also be on the graph? (b) If the point 共5, 3兲 is on the graph of an odd function, what other point must also be on the graph? 72. A function f has domain 关⫺5, 5兴 and a portion of its graph is
20 x
71. (a) If the point 共5, 3兲 is on the graph of an even function, what
shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd. y
x
64. A cell phone plan has a basic charge of $35 a month. The plan
includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 艋 x 艋 600. 65. In a certain state the maximum speed permitted on freeways is
65 mi兾h and the minimum speed is 40 mi兾h. The fine for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph F共x兲 for 0 艋 x 艋 100. 66. An electricity company charges its customers a base rate of
$10 a month, plus 6 cents per kilowatthour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 艋 x 艋 2000.
_5
0
x
5
73–78 Determine whether f is even, odd, or neither. If you have a
graphing calculator, use it to check your answer visually. x2 x ⫹1
73. f 共x兲 苷
x x ⫹1
74. f 共x兲 苷
75. f 共x兲 苷
x x⫹1
76. f 共x兲 苷 x x
2
77. f 共x兲 苷 1 ⫹ 3x 2 ⫺ x 4
4
ⱍ ⱍ
78. f 共x兲 苷 1 ⫹ 3x 3 ⫺ x 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
79. If f and t are both even functions, is f ⫹ t even? If f and t are
80. If f and t are both even functions, is the product ft even? If f
both odd functions, is f ⫹ t odd? What if f is even and t is odd? Justify your answers.
1.2
23
and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.
Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description (often by means of a function or an equation) of a realworld phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 1 illustrates the process of mathematical modeling. Given a realworld problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numerical representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases.
Realworld problem
Formulate
Mathematical model
Solve
Mathematical conclusions
Interpret
Realworld predictions
Test
FIGURE 1 The modeling process
The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original realworld phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.
Linear Models The coordinate geometry of lines is reviewed in Appendix B.
When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slopeintercept form of the equation of a line to write a formula for
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
the function as y 苷 f 共x兲 苷 mx ⫹ b where m is the slope of the line and b is the yintercept. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f 共x兲 苷 3x ⫺ 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f 共x兲 increases by 0.3. So f 共x兲 increases three times as fast as x. Thus the slope of the graph y 苷 3x ⫺ 2, namely 3, can be interpreted as the rate of change of y with respect to x. y
y=3x2
0
x
_2
x
f 共x兲 苷 3x ⫺ 2
1.0 1.1 1.2 1.3 1.4 1.5
1.0 1.3 1.6 1.9 2.2 2.5
FIGURE 2
v
EXAMPLE 1
(a) As dry air moves upward, it expands and cools. If the ground temperature is 20⬚C and the temperature at a height of 1 km is 10⬚C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION
(a) Because we are assuming that T is a linear function of h, we can write T 苷 mh ⫹ b We are given that T 苷 20 when h 苷 0, so 20 苷 m ⴢ 0 ⫹ b 苷 b In other words, the yintercept is b 苷 20. We are also given that T 苷 10 when h 苷 1, so 10 苷 m ⴢ 1 ⫹ 20
T
The slope of the line is therefore m 苷 10 ⫺ 20 苷 ⫺10 and the required linear function is
20
T=_10h+20
T 苷 ⫺10h ⫹ 20
10
0
1
FIGURE 3
3
h
(b) The graph is sketched in Figure 3. The slope is m 苷 ⫺10⬚C兾km, and this represents the rate of change of temperature with respect to height. (c) At a height of h 苷 2.5 km, the temperature is T 苷 ⫺10共2.5兲 ⫹ 20 苷 ⫺5⬚C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
25
If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points.
v EXAMPLE 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2008. Use the data in Table 1 to find a model for the carbon dioxide level. SOLUTION We use the data in Table 1 to make the scatter plot in Figure 4, where t repre
sents time (in years) and C represents the CO2 level (in parts per million, ppm). C TABLE 1
Year
CO 2 level (in ppm)
1980 1982 1984 1986 1988 1990 1992 1994
338.7 341.2 344.4 347.2 351.5 354.2 356.3 358.6
380
Year
CO 2 level (in ppm)
1996 1998 2000 2002 2004 2006 2008
362.4 366.5 369.4 373.2 377.5 381.9 385.6
370 360 350 340 1980
FIGURE 4
1985
1990
1995
2000
2005
2010 t
Scatter plot for the average CO™ level
Notice that the data points appear to lie close to a straight line, so it’s natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? One possibility is the line that passes through the first and last data points. The slope of this line is 385.6 ⫺ 338.7 46.9 苷 苷 1.675 2008 ⫺ 1980 28 and its equation is C ⫺ 338.7 苷 1.675共t ⫺ 1980兲 or C 苷 1.675t ⫺ 2977.8
1
Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C 380 370 360 350
FIGURE 5
Linear model through first and last data points
340 1980
1985
1990
1995
2000
2005
2010 t
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 14.7.
Notice that our model gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the fit[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and yintercept of the regression line as m 苷 1.65429
b 苷 ⫺2938.07
So our least squares model for the CO2 level is C 苷 1.65429t ⫺ 2938.07
2
In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better fit than our previous linear model. C 380 370 360 350 340
FIGURE 6
1980
The regression line
1985
1990
1995
2000
2005
2010 t
v EXAMPLE 3 Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2015. According to this model, when will the CO2 level exceed 420 parts per million? SOLUTION Using Equation 2 with t 苷 1987, we estimate that the average CO2 level in
1987 was
C共1987兲 苷 共1.65429兲共1987兲 ⫺ 2938.07 ⬇ 349.00 This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.) With t 苷 2015, we get C共2015兲 苷 共1.65429兲共2015兲 ⫺ 2938.07 ⬇ 395.32 So we predict that the average CO2 level in the year 2015 will be 395.3 ppm. This is an example of extrapolation because we have predicted a value outside the region of observations. Consequently, we are far less certain about the accuracy of our prediction. Using Equation 2, we see that the CO2 level exceeds 420 ppm when 1.65429t ⫺ 2938.07 ⬎ 420 Solving this inequality, we get t⬎
3358.07 ⬇ 2029.92 1.65429
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
27
We therefore predict that the CO2 level will exceed 420 ppm by the year 2030. This prediction is risky because it involves a time quite remote from our observations. In fact, we see from Figure 6 that the trend has been for CO2 levels to increase rather more rapidly in recent years, so the level might exceed 420 ppm well before 2030.
Polynomials A function P is called a polynomial if P共x兲 苷 a n x n ⫹ a n⫺1 x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a 2 x 2 ⫹ a 1 x ⫹ a 0 where n is a nonnegative integer and the numbers a 0 , a 1, a 2 , . . . , a n are constants called the coefficients of the polynomial. The domain of any polynomial is ⺢ 苷 共⫺⬁, ⬁兲. If the leading coefficient a n 苷 0, then the degree of the polynomial is n. For example, the function P共x兲 苷 2x 6 ⫺ x 4 ⫹ 25 x 3 ⫹ s2 is a polynomial of degree 6. A polynomial of degree 1 is of the form P共x兲 苷 mx ⫹ b and so it is a linear function. A polynomial of degree 2 is of the form P共x兲 苷 ax 2 ⫹ bx ⫹ c and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola y 苷 ax 2, as we will see in the next section. The parabola opens upward if a ⬎ 0 and downward if a ⬍ 0. (See Figure 7.) y
y
2 2
x
1 0
FIGURE 7
The graphs of quadratic functions are parabolas.
1
x
(b) y=_2≈+3x+1
(a) y=≈+x+1
A polynomial of degree 3 is of the form P共x兲 苷 ax 3 ⫹ bx 2 ⫹ cx ⫹ d
a苷0
and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. y
y
1
2
0
FIGURE 8
y 20 1
1
(a) y=˛x+1
x
x
(b) y=x$3≈+x
1
x
(c) y=3x%25˛+60x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.7 we will explain why economists often use a polynomial P共x兲 to represent the cost of producing x units of a commodity. In the following example we use a quadratic function to model the fall of a ball. TABLE 2
Time (seconds)
Height (meters)
0 1 2 3 4 5 6 7 8 9
450 445 431 408 375 332 279 216 143 61
EXAMPLE 4 A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground. SOLUTION We draw a scatter plot of the data in Figure 9 and observe that a linear model
is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model: h 苷 449.36 ⫹ 0.96t ⫺ 4.90t 2
3
h (meters)
h
400
400
200
200
0
2
4
6
8
t (seconds)
0
2
4
6
8
FIGURE 9
FIGURE 10
Scatter plot for a falling ball
Quadratic model for a falling ball
t
In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit. The ball hits the ground when h 苷 0, so we solve the quadratic equation ⫺4.90t 2 ⫹ 0.96t ⫹ 449.36 苷 0 The quadratic formula gives t苷
⫺0.96 ⫾ s共0.96兲2 ⫺ 4共⫺4.90兲共449.36兲 2共⫺4.90兲
The positive root is t ⬇ 9.67, so we predict that the ball will hit the ground after about 9.7 seconds.
Power Functions A function of the form f 共x兲 苷 x a, where a is a constant, is called a power function. We consider several cases.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
(i) a 苷 n, where n is a positive integer
The graphs of f 共x兲 苷 x n for n 苷 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y 苷 x (a line through the origin with slope 1) and y 苷 x 2 [a parabola, see Example 2(b) in Section 1.1]. y
y=x
y=≈
y 1
1
0
1
x
0
y=x #
y
y
x
0
1
x
0
y=x%
y
1
1
1
y=x$
1
1
x
0
x
1
FIGURE 11 Graphs of ƒ=x n for n=1, 2, 3, 4, 5
The general shape of the graph of f 共x兲 苷 x n depends on whether n is even or odd. If n is even, then f 共x兲 苷 x n is an even function and its graph is similar to the parabola y 苷 x 2. If n is odd, then f 共x兲 苷 x n is an odd function and its graph is similar to that of y 苷 x 3. Notice from Figure 12, however, that as n increases, the graph of y 苷 x n becomes flatter near 0 and steeper when x 艌 1. (If x is small, then x 2 is smaller, x 3 is even smaller, x 4 is smaller still, and so on.)
ⱍ ⱍ
y
y
y=x $ y=x ^
y=x # y=≈
(_1, 1)
FIGURE 12
Families of power functions
(1, 1) y=x %
(1, 1)
x
0
(_1, _1) x
0
(ii) a 苷 1兾n, where n is a positive integer n The function f 共x兲 苷 x 1兾n 苷 s x is a root function. For n 苷 2 it is the square root function f 共x兲 苷 sx , whose domain is 关0, ⬁兲 and whose graph is the upper half of the n parabola x 苷 y 2. [See Figure 13(a).] For other even values of n, the graph of y 苷 s x is 3 similar to that of y 苷 sx . For n 苷 3 we have the cube root function f 共x兲 苷 sx whose domain is ⺢ (recall that every real number has a cube root) and whose graph is shown n 3 in Figure 13(b). The graph of y 苷 s x for n odd 共n ⬎ 3兲 is similar to that of y 苷 s x. y
y
(1, 1) 0
(1, 1) x
0
x
FIGURE 13
Graphs of root functions
x (a) ƒ=œ„
x (b) ƒ=Œ„
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
(iii) a 苷 ⫺1
y
The graph of the reciprocal function f 共x兲 苷 x ⫺1 苷 1兾x is shown in Figure 14. Its graph has the equation y 苷 1兾x, or xy 苷 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P :
y=Δ 1 0
9/20/10
x
1
V苷
C P
FIGURE 14
where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14.
The reciprocal function
V
FIGURE 15
Volume as a function of pressure at constant temperature
0
P
Power functions are also used to model speciesarea relationships (Exercises 26–27), illumination as a function of a distance from a light source (Exercise 25), and the period of revolution of a planet as a function of its distance from the sun (Exercise 28).
Rational Functions A rational function f is a ratio of two polynomials: f 共x兲 苷
y
20 0
2
x
P共x兲 Q共x兲
where P and Q are polynomials. The domain consists of all values of x such that Q共x兲 苷 0. A simple example of a rational function is the function f 共x兲 苷 1兾x, whose domain is 兵x x 苷 0其; this is the reciprocal function graphed in Figure 14. The function
ⱍ
f 共x兲 苷
2x 4 ⫺ x 2 ⫹ 1 x2 ⫺ 4
ⱍ
is a rational function with domain 兵x x 苷 ⫾2其. Its graph is shown in Figure 16.
FIGURE 16
2x$≈+1 ƒ= ≈4
Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: f 共x兲 苷 sx 2 ⫹ 1
t共x兲 苷
x 4 ⫺ 16x 2 3 ⫹ 共x ⫺ 2兲s x⫹1 x ⫹ sx
When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
31
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
y
y
y
1
1
2
1
_3
x
0
FIGURE 17
(a) ƒ=xœ„„„„ x+3
x
5
0
(b) ©=$œ„„„„„„ ≈25
x
1
(c) h(x)
[email protected]?#(x2)@
An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is m0 m 苷 f 共v兲 苷 s1 ⫺ v 2兾c 2 where m 0 is the rest mass of the particle and c 苷 3.0 ⫻ 10 5 km兾s is the speed of light in a vacuum.
Trigonometric Functions Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f 共x兲 苷 sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 18.
The Reference Pages are located at the front and back of the book.
y _ _π
π 2
y 3π 2
1 0 _1
π 2
π
_π 2π
5π 2
3π
_
π 2
π 0
x _1
(a) ƒ=sin x FIGURE 18
1 π 2
3π 3π 2
2π
5π 2
x
(b) ©=cos x
Notice that for both the sine and cosine functions the domain is 共⫺⬁, ⬁兲 and the range is the closed interval 关⫺1, 1兴. Thus, for all values of x, we have ⫺1 艋 sin x 艋 1
⫺1 艋 cos x 艋 1
or, in terms of absolute values,
ⱍ sin x ⱍ 艋 1
ⱍ cos x ⱍ 艋 1
Also, the zeros of the sine function occur at the integer multiples of ; that is, sin x 苷 0
when
x 苷 n n an integer
An important property of the sine and cosine functions is that they are periodic functions and have period 2. This means that, for all values of x, sin共x ⫹ 2兲 苷 sin x
cos共x ⫹ 2兲 苷 cos x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 4 in Section 1.3 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function
冋
L共t兲 苷 12 ⫹ 2.8 sin y
册
2 共t ⫺ 80兲 365
The tangent function is related to the sine and cosine functions by the equation tan x 苷
1 _
0
3π _π π _ 2 2
π 2
3π 2
π
sin x cos x
x
and its graph is shown in Figure 19. It is undefined whenever cos x 苷 0, that is, when x 苷 ⫾兾2, ⫾3兾2, . . . . Its range is 共⫺⬁, ⬁兲. Notice that the tangent function has period : tan共x ⫹ 兲 苷 tan x
for all x
FIGURE 19
The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D.
y=tan x
y
y
1 0
1 0
x
1
(a) y=2®
Exponential Functions
1
x
(b) y=(0.5)®
The exponential functions are the functions of the form f 共x兲 苷 a x , where the base a is a positive constant. The graphs of y 苷 2 x and y 苷 共0.5兲 x are shown in Figure 20. In both cases the domain is 共⫺⬁, ⬁兲 and the range is 共0, ⬁兲. Exponential functions will be studied in detail in Section 1.5, and we will see that they are useful for modeling many natural phenomena, such as population growth ( if a ⬎ 1) and radioactive decay ( if a ⬍ 1兲.
FIGURE 20
Logarithmic Functions y
The logarithmic functions f 共x兲 苷 log a x, where the base a is a positive constant, are the inverse functions of the exponential functions. They will be studied in Section 1.6. Figure 21 shows the graphs of four logarithmic functions with various bases. In each case the domain is 共0, ⬁兲, the range is 共⫺⬁, ⬁兲, and the function increases slowly when x ⬎ 1.
y=log™ x y=log£ x
1
0
1
x
y=log∞ x y=log¡¸ x
EXAMPLE 5 Classify the following functions as one of the types of functions that we have discussed. (a) f 共x兲 苷 5 x (b) t共x兲 苷 x 5
(c) h共x兲 苷 FIGURE 21
1⫹x 1 ⫺ sx
(d) u共t兲 苷 1 ⫺ t ⫹ 5t 4
SOLUTION
(a) f 共x兲 苷 5 x is an exponential function. (The x is the exponent.) (b) t共x兲 苷 x 5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5. 1⫹x (c) h共x兲 苷 is an algebraic function. 1 ⫺ sx (d) u共t兲 苷 1 ⫺ t ⫹ 5t 4 is a polynomial of degree 4. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
1.2
Page 33
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
33
Exercises
1–2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 1. (a) f 共x兲 苷 log 2 x
4 (b) t共x兲 苷 s x
2x 1 ⫺ x2
(e) v共t兲 苷 5
(d) u共t兲 苷 1 ⫺ 1.1t ⫹ 2.54t 2 (f) w 共 兲 苷 sin cos
t
2
(c) y 苷 x 2 共2 ⫺ x 3 兲 (e) y 苷
(f) y 苷
sx 3 ⫺ 1 3 1⫹s x
3– 4 Match each equation with its graph. Explain your choices.
(Don’t use a computer or graphing calculator.) 3. (a) y 苷 x
8. Find expressions for the quadratic functions whose graphs are
shown.
(b) y 苷 x
5
(c) y 苷 x
y
(0, 1) (4, 2)
0
x
g 0
3
x
(1, _2.5)
9. Find an expression for a cubic function f if f 共1兲 苷 6 and
f 共⫺1兲 苷 f 共0兲 苷 f 共2兲 苷 0.
10. Recent studies indicate that the average surface tempera
8
g h
0
y (_2, 2)
f
(d) y 苷 tan t ⫺ cos t
s 1⫹s
2
7. What do all members of the family of linear functions
y
(b) y 苷 x
2. (a) y 苷 x
f 共x兲 苷 1 ⫹ m共x ⫹ 3兲 have in common? Sketch several members of the family. f 共x兲 苷 c ⫺ x have in common? Sketch several members of the family.
3
(c) h共x兲 苷
6. What do all members of the family of linear functions
x
ture of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T 苷 0.02t ⫹ 8.50, where T is temperature in ⬚C and t represents years since 1900. (a) What do the slope and T intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. 11. If the recommended adult dosage for a drug is D ( in mg), then
to determine the appropriate dosage c for a child of age a, pharmacists use the equation c 苷 0.0417D共a ⫹ 1兲. Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn?
f
4. (a) y 苷 3x
(c) y 苷 x
(b) y 苷 3 x 3 (d) y 苷 s x
3
12. The manager of a weekend flea market knows from past expe
y
F
g f x
rience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y 苷 200 ⫺ 4x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative quantities.) (b) What do the slope, the yintercept, and the xintercept of the graph represent? 13. The relationship between the Fahrenheit 共F兲 and Celsius 共C兲
G
5. (a) Find an equation for the family of linear functions with
slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f 共2兲 苷 1 and sketch several members of the family. (c) Which function belongs to both families?
;
Graphing calculator or computer required
temperature scales is given by the linear function F 苷 95 C ⫹ 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the Fintercept and what does it represent?
14. Jason leaves Detroit at 2:00 PM and drives at a constant speed
west along I96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 PM. (a) Express the distance traveled in terms of the time elapsed.
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
(b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?
20. (a)
(b)
y
y
15. Biologists have noticed that the chirping rate of crickets of a
certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70⬚F and 173 chirps per minute at 80⬚F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 16. The manager of a furniture factory finds that it costs $2200
0
x
lation) for various family incomes as reported by the National Health Interview Survey.
17. At the surface of the ocean, the water pressure is the same as
18. The monthly cost of driving a car depends on the number of
0
; 21. The table shows (lifetime) peptic ulcer rates (per 100 popu
to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the yintercept of the graph and what does it represent?
the air pressure above the water, 15 lb兾in2. Below the surface, the water pressure increases by 4.34 lb兾in2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lb兾in2 ?
x
Income
Ulcer rate (per 100 population)
$4,000 $6,000 $8,000 $12,000 $16,000 $20,000 $30,000 $45,000 $60,000
14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2
(a) Make a scatter plot of these data and decide whether a linear model is appropriate. (b) Find and graph a linear model using the first and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of $25,000. (e) According to the model, how likely is someone with an income of $80,000 to suffer from peptic ulcers? (f) Do you think it would be reasonable to apply the model to someone with an income of $200,000?
miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? ; 22. Biologists have observed that the chirping rate of crickets of a (d) What does the Cintercept represent? certain species appears to be related to temperature. The table (e) Why does a linear function give a suitable model in this shows the chirping rates for various temperatures. situation? 19–20 For each scatter plot, decide what type of function you
might choose as a model for the data. Explain your choices. 19. (a)
(b)
y
0
y
x
0
x
Temperature (°F)
Chirping rate (chirps兾min)
Temperature (°F)
Chirping rate (chirps兾min)
50 55 60 65 70
20 46 79 91 113
75 80 85 90
140 173 198 211
(a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100⬚F.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.2
; 23. The table gives the winning heights for the men’s Olympic pole vault competitions up to the year 2004. Year
Height (m)
Year
Height (m)
1896 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956
3.30 3.30 3.50 3.71 3.95 4.09 3.95 4.20 4.31 4.35 4.30 4.55 4.56
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004
4.70 5.10 5.40 5.64 5.64 5.78 5.75 5.90 5.87 5.92 5.90 5.95
(a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to predict the height of the winning pole vault at the 2008 Olympics and compare with the actual winning height of 5.96 meters. (d) Is it reasonable to use the model to predict the winning height at the 2100 Olympics?
; 24. The table shows the percentage of the population of Argentina that has lived in rural areas from 1955 to 2000. Find a model for the data and use it to estimate the rural percentage in 1988 and 2002.
Year
Percentage rural
Year
Percentage rural
1955 1960 1965 1970 1975
30.4 26.4 23.6 21.1 19.0
1980 1985 1990 1995 2000
17.1 15.0 13.0 11.7 10.5
25. Many physical quantities are connected by inverse square
laws, that is, by power functions of the form f 共x兲 苷 kx ⫺2. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light? 26. It makes sense that the larger the area of a region, the larger
the number of species that inhabit the region. Many
Page 35
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
35
ecologists have modeled the speciesarea relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S 苷 0.7A0.3. (a) The cave called Misión Imposible near Puebla, Mexico, has a surface area of A 苷 60 m2. How many species of bats would you expect to find in that cave? (b) If you discover that four species of bats live in a cave, estimate the area of the cave.
; 27. The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles. Island
A
N
Saba Monserrat Puerto Rico Jamaica Hispaniola Cuba
4 40 3,459 4,411 29,418 44,218
5 9 40 39 84 76
(a) Use a power function to model N as a function of A. (b) The Caribbean island of Dominica has area 291 m2. How many species of reptiles and amphibians would you expect to find on Dominica?
; 28. The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods T (time of revolution in years). Planet
d
T
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune
0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086
0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784
(a) Fit a power model to the data. (b) Kepler’s Third Law of Planetary Motion states that “The square of the period of revolution of a planet is proportional to the cube of its mean distance from the sun.” Does your model corroborate Kepler’s Third Law?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
New Functions from Old Functions
1.3
In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition.
Transformations of Functions By applying certain transformations to the graph of a given function we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Let’s first consider translations. If c is a positive number, then the graph of y 苷 f 共x兲 ⫹ c is just the graph of y 苷 f 共x兲 shifted upward a distance of c units (because each ycoordinate is increased by the same number c). Likewise, if t共x兲 苷 f 共x ⫺ c兲, where c ⬎ 0, then the value of t at x is the same as the value of f at x ⫺ c (c units to the left of x). Therefore the graph of y 苷 f 共x ⫺ c兲 is just the graph of y 苷 f 共x兲 shifted c units to the right (see Figure 1). Vertical and Horizontal Shifts Suppose c ⬎ 0. To obtain the graph of
y 苷 f 共x兲 ⫹ c, shift the graph of y 苷 f 共x兲 a distance c units upward y 苷 f 共x兲 ⫺ c, shift the graph of y 苷 f 共x兲 a distance c units downward y 苷 f 共x ⫺ c兲, shift the graph of y 苷 f 共x兲 a distance c units to the right y 苷 f 共x ⫹ c兲, shift the graph of y 苷 f 共x兲 a distance c units to the left y
y
y=ƒ+c
y=f(x+c)
c
y =ƒ
y=cƒ (c>1)
y=f(_x)
y=f(xc)
y=ƒ c 0
y= 1c ƒ
c x
c
x
0
y=ƒc y=_ƒ
FIGURE 1
FIGURE 2
Translating the graph of ƒ
Stretching and reflecting the graph of ƒ
Now let’s consider the stretching and reflecting transformations. If c ⬎ 1, then the graph of y 苷 cf 共x兲 is the graph of y 苷 f 共x兲 stretched by a factor of c in the vertical direction (because each ycoordinate is multiplied by the same number c). The graph of y 苷 ⫺f 共x兲 is the graph of y 苷 f 共x兲 reflected about the xaxis because the point 共x, y兲 is
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.3
NEW FUNCTIONS FROM OLD FUNCTIONS
37
replaced by the point 共x, ⫺y兲. (See Figure 2 and the following chart, where the results of other stretching, shrinking, and reflecting transformations are also given.) Vertical and Horizontal Stretching and Reﬂecting Suppose c ⬎ 1. To obtain the
graph of y 苷 cf 共x兲, stretch the graph of y 苷 f 共x兲 vertically by a factor of c y 苷 共1兾c兲 f 共x兲, shrink the graph of y 苷 f 共x兲 vertically by a factor of c y 苷 f 共cx兲, shrink the graph of y 苷 f 共x兲 horizontally by a factor of c y 苷 f 共x兾c兲, stretch the graph of y 苷 f 共x兲 horizontally by a factor of c y 苷 ⫺f 共x兲, reflect the graph of y 苷 f 共x兲 about the xaxis y 苷 f 共⫺x兲, reflect the graph of y 苷 f 共x兲 about the yaxis
Figure 3 illustrates these stretching transformations when applied to the cosine function with c 苷 2. For instance, in order to get the graph of y 苷 2 cos x we multiply the ycoordinate of each point on the graph of y 苷 cos x by 2. This means that the graph of y 苷 cos x gets stretched vertically by a factor of 2. y
y=2 cos x
y
2
y=cos x
2
1 0
y=cos 1 x 2
1
1 y= cos x 2
x
1
0
x
y=cos x y=cos 2x
FIGURE 3
v EXAMPLE 1 Given the graph of y 苷 sx , use transformations to graph y 苷 sx ⫺ 2, y 苷 sx ⫺ 2 , y 苷 ⫺sx , y 苷 2sx , and y 苷 s⫺x . SOLUTION The graph of the square root function y 苷 sx , obtained from Figure 13(a)
in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch y 苷 sx ⫺ 2 by shifting 2 units downward, y 苷 sx ⫺ 2 by shifting 2 units to the right, y 苷 ⫺sx by reflecting about the xaxis, y 苷 2sx by stretching vertically by a factor of 2, and y 苷 s⫺x by reflecting about the yaxis. y
y
y
y
y
y
1 0
1
x
x
0
0
2
x
x
0
0
x
0
_2
(a) y=œ„x
(b) y=œ„2 x
(c) y=œ„„„„ x2
(d) y=_ œ„x
(e) y=2 œ„x
(f ) y=œ„„ _x
FIGURE 4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
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FUNCTIONS AND MODELS
EXAMPLE 2 Sketch the graph of the function f (x) 苷 x 2 ⫹ 6x ⫹ 10. SOLUTION Completing the square, we write the equation of the graph as
y 苷 x 2 ⫹ 6x ⫹ 10 苷 共x ⫹ 3兲2 ⫹ 1 This means we obtain the desired graph by starting with the parabola y 苷 x 2 and shifting 3 units to the left and then 1 unit upward (see Figure 5). y
y
1
(_3, 1) x
0
FIGURE 5
_3
(a) y=≈
_1
0
x
(b) y=(x+3)@+1
EXAMPLE 3 Sketch the graphs of the following functions. (a) y 苷 sin 2x (b) y 苷 1 ⫺ sin x SOLUTION
(a) We obtain the graph of y 苷 sin 2x from that of y 苷 sin x by compressing horizontally by a factor of 2. (See Figures 6 and 7.) Thus, whereas the period of y 苷 sin x is 2, the period of y 苷 sin 2x is 2兾2 苷 . y
y
y=sin x
1 0
π 2
π
FIGURE 6
y=sin 2x
1 x
0 π π 4
x
π
2
FIGURE 7
(b) To obtain the graph of y 苷 1 ⫺ sin x, we again start with y 苷 sin x. We reflect about the xaxis to get the graph of y 苷 ⫺sin x and then we shift 1 unit upward to get y 苷 1 ⫺ sin x. (See Figure 8.) y
y=1sin x
2 1 0
FIGURE 8
π 2
π
3π 2
2π
x
EXAMPLE 4 Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Philadelphia is located at approximately 40⬚N latitude, find a function that models the length of daylight at Philadelphia. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.3
NEW FUNCTIONS FROM OLD FUNCTIONS
39
20 18 16 14 12
20° N 30° N 40° N 50° N
Hours 10 8 6
FIGURE 9
Graph of the length of daylight from March 21 through December 21 at various latitudes
4
Lucia C. Harrison, Daylight, Twilight, Darkness and Time (New York, 1935) page 40.
0
60° N
2 Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
SOLUTION Notice that each curve resembles a shifted and stretched sine function. By
looking at the blue curve we see that, at the latitude of Philadelphia, daylight lasts about 14.8 hours on June 21 and 9.2 hours on December 21, so the amplitude of the curve (the 1 factor by which we have to stretch the sine curve vertically) is 2 共14.8 ⫺ 9.2兲 苷 2.8. By what factor do we need to stretch the sine curve horizontally if we measure the time t in days? Because there are about 365 days in a year, the period of our model should be 365. But the period of y 苷 sin t is 2, so the horizontal stretching factor is c 苷 2兾365. We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore we model the length of daylight in Philadelphia on the tth day of the year by the function
冋
L共t兲 苷 12 ⫹ 2.8 sin
Another transformation of some interest is taking the absolute value of a function. If y 苷 f 共x兲 , then according to the definition of absolute value, y 苷 f 共x兲 when f 共x兲 艌 0 and y 苷 ⫺f 共x兲 when f 共x兲 ⬍ 0. This tells us how to get the graph of y 苷 f 共x兲 from the graph of y 苷 f 共x兲: The part of the graph that lies above the xaxis remains the same; the part that lies below the xaxis is reflected about the xaxis.
ⱍ
y
0
_1
1
ⱍ
ⱍ
ⱍ
EXAMPLE 5 Sketch the graph of the function y 苷 x 2 ⫺ 1 .
SOLUTION We first graph the parabola y 苷 x ⫺ 1 in Figure 10(a) by shifting the parabola
y 苷 x 2 downward 1 unit. We see that the graph lies below the xaxis when ⫺1 ⬍ x ⬍ 1, so we reflect that part of the graph about the xaxis to obtain the graph of y 苷 x 2 ⫺ 1 in Figure 10(b).
y
ⱍ
1
(b) y= ≈1  FIGURE 10
ⱍ
2
(a) y=≈1
0
ⱍ
x
v
_1
册
2 共t ⫺ 80兲 365
x
ⱍ
Combinations of Functions Two functions f and t can be combined to form new functions f ⫹ t, f ⫺ t, ft, and f兾t in a manner similar to the way we add, subtract, multiply, and divide real numbers. The sum and difference functions are defined by 共 f ⫹ t兲共x兲 苷 f 共x兲 ⫹ t共x兲
共 f ⫺ t兲共x兲 苷 f 共x兲 ⫺ t共x兲
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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If the domain of f is A and the domain of t is B, then the domain of f ⫹ t is the intersection A 傽 B because both f 共x兲 and t共x兲 have to be defined. For example, the domain of f 共x兲 苷 sx is A 苷 关0, ⬁兲 and the domain of t共x兲 苷 s2 ⫺ x is B 苷 共⫺⬁, 2兴, so the domain of 共 f ⫹ t兲共x兲 苷 sx ⫹ s2 ⫺ x is A 傽 B 苷 关0, 2兴. Similarly, the product and quotient functions are defined by 共 ft兲共x兲 苷 f 共x兲t共x兲
冉冊
f f 共x兲 共x兲 苷 t t共x兲
The domain of ft is A 傽 B, but we can’t divide by 0 and so the domain of f兾t is 兵x 僆 A 傽 B t共x兲 苷 0其. For instance, if f 共x兲 苷 x 2 and t共x兲 苷 x ⫺ 1, then the domain of the rational function 共 f兾t兲共x兲 苷 x 2兾共x ⫺ 1兲 is 兵x x 苷 1其, or 共⫺⬁, 1兲 傼 共1, ⬁兲. There is another way of combining two functions to obtain a new function. For example, suppose that y 苷 f 共u兲 苷 su and u 苷 t共x兲 苷 x 2 ⫹ 1. Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x. We compute this by substitution:
ⱍ
ⱍ
y 苷 f 共u兲 苷 f 共t共x兲兲 苷 f 共x 2 ⫹ 1兲 苷 sx 2 ⫹ 1 The procedure is called composition because the new function is composed of the two given functions f and t. In general, given any two functions f and t, we start with a number x in the domain of t and find its image t共x兲. If this number t共x兲 is in the domain of f , then we can calculate the value of f 共t共x兲兲. Notice that the output of one function is used as the input to the next function. The result is a new function h共x兲 苷 f 共t共x兲兲 obtained by substituting t into f . It is called the composition (or composite) of f and t and is denoted by f ⴰ t (“f circle t”).
x (input)
g
©
f•g
Definition Given two functions f and t, the composite function f ⴰ t (also called the composition of f and t) is defined by
f
共 f ⴰ t兲共x兲 苷 f 共t共x兲兲
f { ©} (output) FIGURE 11
The domain of f ⴰ t is the set of all x in the domain of t such that t共x兲 is in the domain of f . In other words, 共 f ⴰ t兲共x兲 is defined whenever both t共x兲 and f 共t共x兲兲 are defined. Figure 11 shows how to picture f ⴰ t in terms of machines.
The f • g machine is composed of the g machine (first) and then the f machine.
EXAMPLE 6 If f 共x兲 苷 x 2 and t共x兲 苷 x ⫺ 3, find the composite functions f ⴰ t and t ⴰ f . SOLUTION We have
共 f ⴰ t兲共x兲 苷 f 共 t共x兲兲 苷 f 共x ⫺ 3兲 苷 共x ⫺ 3兲2 共t ⴰ f 兲共x兲 苷 t共 f 共x兲兲 苷 t共x 2 兲 苷 x 2 ⫺ 3 
NOTE You can see from Example 6 that, in general, f ⴰ t 苷 t ⴰ f . Remember, the notation f ⴰ t means that the function t is applied first and then f is applied second. In Example 6, f ⴰ t is the function that first subtracts 3 and then squares; t ⴰ f is the function that first squares and then subtracts 3.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.3
v
NEW FUNCTIONS FROM OLD FUNCTIONS
41
EXAMPLE 7 If f 共x兲 苷 sx and t共x兲 苷 s2 ⫺ x , find each function and its domain.
(a) f ⴰ t
(b) t ⴰ f
(c) f ⴰ f
(d) t ⴰ t
SOLUTION
(a)
4 共 f ⴰ t兲共x兲 苷 f 共t共x兲兲 苷 f (s2 ⫺ x ) 苷 ss2 ⫺ x 苷 s 2⫺x
ⱍ
ⱍ
The domain of f ⴰ t is 兵x 2 ⫺ x 艌 0其 苷 兵x x 艋 2其 苷 共⫺⬁, 2兴. (b)
If 0 艋 a 艋 b, then a 2 艋 b 2.
共t ⴰ f 兲共x兲 苷 t共 f 共x兲兲 苷 t(sx ) 苷 s2 ⫺ sx
For sx to be defined we must have x 艌 0. For s2 ⫺ sx to be defined we must have 2 ⫺ sx 艌 0, that is, sx 艋 2, or x 艋 4. Thus we have 0 艋 x 艋 4, so the domain of t ⴰ f is the closed interval 关0, 4兴. (c)
4 共 f ⴰ f 兲共x兲 苷 f 共 f 共x兲兲 苷 f (sx ) 苷 ssx 苷 s x
The domain of f ⴰ f is 关0, ⬁兲. (d)
共t ⴰ t兲共x兲 苷 t共t共x兲兲 苷 t(s2 ⫺ x ) 苷 s2 ⫺ s2 ⫺ x
This expression is defined when both 2 ⫺ x 艌 0 and 2 ⫺ s2 ⫺ x 艌 0. The first inequality means x 艋 2, and the second is equivalent to s2 ⫺ x 艋 2, or 2 ⫺ x 艋 4, or x 艌 ⫺2. Thus ⫺2 艋 x 艋 2, so the domain of t ⴰ t is the closed interval 关⫺2, 2兴. It is possible to take the composition of three or more functions. For instance, the composite function f ⴰ t ⴰ h is found by first applying h, then t, and then f as follows: 共 f ⴰ t ⴰ h兲共x兲 苷 f 共 t共h共x兲兲兲 EXAMPLE 8 Find f ⴰ t ⴰ h if f 共x兲 苷 x兾共x ⫹ 1兲, t共x兲 苷 x 10, and h共x兲 苷 x ⫹ 3. SOLUTION
共 f ⴰ t ⴰ h兲共x兲 苷 f 共 t共h共x兲兲兲 苷 f 共t共x ⫹ 3兲兲 苷 f 共共x ⫹ 3兲10 兲 苷
共x ⫹ 3兲10 共x ⫹ 3兲10 ⫹ 1
So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following example. EXAMPLE 9 Given F共x兲 苷 cos2共x ⫹ 9兲, find functions f , t, and h such that F 苷 f ⴰ t ⴰ h. SOLUTION Since F共x兲 苷 关cos共x ⫹ 9兲兴 2, the formula for F says: First add 9, then take the
cosine of the result, and finally square. So we let h共x兲 苷 x ⫹ 9 Then
t共x兲 苷 cos x
f 共x兲 苷 x 2
共 f ⴰ t ⴰ h兲共x兲 苷 f 共t共h共x兲兲兲 苷 f 共t共x ⫹ 9兲兲 苷 f 共cos共x ⫹ 9兲兲 苷 关cos共x ⫹ 9兲兴 2 苷 F共x兲
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
Exercises
1. Suppose the graph of f is given. Write equations for the graphs
that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the xaxis. (f) Reflect about the yaxis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.
6–7 The graph of y 苷 s3x ⫺ x 2 is given. Use transformations to create a function whose graph is as shown. y
2. Explain how each graph is obtained from the graph of y 苷 f 共x兲.
(a) y 苷 f 共x兲 ⫹ 8 (c) y 苷 8 f 共x兲 (e) y 苷 ⫺f 共x兲 ⫺ 1
(b) y 苷 f 共x ⫹ 8兲 (d) y 苷 f 共8x兲 (f) y 苷 8 f ( 18 x)
0
6.
3. The graph of y 苷 f 共x兲 is given. Match each equation with its
graph and give reasons for your choices. (a) y 苷 f 共x ⫺ 4兲 (b) y 苷 f 共x兲 ⫹ 3 (c) y 苷 13 f 共x兲 (d) y 苷 ⫺f 共x ⫹ 4兲 (e) y 苷 2 f 共x ⫹ 6兲
y=œ„„„„„„ 3x≈
1.5
x
3
7.
y
y
3
_1 0
_4
x _1 _2.5
0
5
2
x
y
@
!
6
8. (a) How is the graph of y 苷 2 sin x related to the graph of
f
3
y 苷 sin x ? Use your answer and Figure 6 to sketch the graph of y 苷 2 sin x. (b) How is the graph of y 苷 1 ⫹ sx related to the graph of y 苷 sx ? Use your answer and Figure 4(a) to sketch the graph of y 苷 1 ⫹ sx .
#
$ _6
0
_3
3
6
x
_3
%
4. The graph of f is given. Draw the graphs of the following
functions. (a) y 苷 f 共x兲 ⫺ 2 (c) y 苷 ⫺2 f 共x兲
(b) y 苷 f 共x ⫺ 2兲 (d) y 苷 f ( 13 x) ⫹ 1 y 2
9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9. y 苷
1 x⫹2
10. y 苷 共x ⫺ 1兲 3
3 x 11. y 苷 ⫺s
12. y 苷 x 2 ⫹ 6x ⫹ 4
13. y 苷 sx ⫺ 2 ⫺ 1
14. y 苷 4 sin 3x
15. y 苷 sin( 2 x)
16. y 苷
1
17. y 苷 2 共1 ⫺ cos x兲
18. y 苷 1 ⫺ 2 sx ⫹ 3
19. y 苷 1 ⫺ 2x ⫺ x
20. y 苷 x ⫺ 2
1
0
1
x
5. The graph of f is given. Use it to graph the following
functions. (a) y 苷 f 共2x兲 (c) y 苷 f 共⫺x兲
(b) y 苷 f ( 12 x) (d) y 苷 ⫺f 共⫺x兲
2 ⫺2 x
ⱍ
21. y 苷 x ⫺ 2
ⱍ
ⱍ
23. y 苷 sx ⫺ 1
2
ⱍ ⱍ
22. y 苷
ⱍ
冉 冊
1 tan x ⫺ 4 4
ⱍ
24. y 苷 cos x
ⱍ
y
25. The city of New Orleans is located at latitude 30⬚N. Use Fig1 0
1
x
ure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans.
1. Homework Hints available at stewartcalculus.com Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.3
26. A variable star is one whose brightness alternately increases
NEW FUNCTIONS FROM OLD FUNCTIONS
41– 46 Express the function in the form f ⴰ t.
and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by ⫾0.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.
41. F共x兲 苷 共2 x ⫹ x 2 兲 4
ⱍ ⱍ) related to the graph of f ? (b) Sketch the graph of y 苷 sin ⱍ x ⱍ. (c) Sketch the graph of y 苷 sⱍ x ⱍ.
45. v共t兲 苷 sec共t 2 兲 tan共t 2 兲
27. (a) How is the graph of y 苷 f ( x
28. Use the given graph of f to sketch the graph of y 苷 1兾f 共x兲.
Which features of f are the most important in sketching y 苷 1兾f 共x兲? Explain how they are used.
43. F共x兲 苷
42. F共x兲 苷 cos2 x
3 x s 3 1⫹s x
44. G共x兲 苷
冑 3
x 1⫹x
tan t 1 ⫹ tan t
46. u共t兲 苷
47– 49 Express the function in the form f ⴰ t ⴰ h.
ⱍ ⱍ
47. R共x兲 苷 ssx ⫺ 1
8 48. H共x兲 苷 s 2⫹ x
49. H共x兲 苷 sec (sx ) 4
y
50. Use the table to evaluate each expression.
1 0
x
1
29–30 Find (a) f ⫹ t, (b) f ⫺ t, (c) f t, and (d) f兾t and state their domains. 29. f 共x兲 苷 x 3 ⫹ 2x 2,
t共x兲 苷 3x 2 ⫺ 1 t共x兲 苷 sx ⫺ 1
30. f 共x兲 苷 s3 ⫺ x ,
2
(a) f 共 t共1兲兲 (d) t共 t共1兲兲
(b) t共 f 共1兲兲 (e) 共 t ⴰ f 兲共3兲
(c) f 共 f 共1兲兲 (f) 共 f ⴰ t兲共6兲
x
1
2
3
4
5
6
f 共x兲
3
1
4
2
2
5
t共x兲
6
3
2
1
2
3
51. Use the given graphs of f and t to evaluate each expression,
or explain why it is undefined. (a) f 共 t共2兲兲 (b) t共 f 共0兲兲 (d) 共 t ⴰ f 兲共6兲 (e) 共 t ⴰ t兲共⫺2兲
31–36 Find the functions (a) f ⴰ t, (b) t ⴰ f , (c) f ⴰ f , and (d) t ⴰ t
(c) 共 f ⴰ t兲共0兲 (f) 共 f ⴰ f 兲共4兲
y
and their domains. 31. f 共x兲 苷 x 2 ⫺ 1,
t共x兲 苷 2x ⫹ 1
32. f 共x兲 苷 x ⫺ 2,
t共x兲 苷 x ⫹ 3x ⫹ 4
36. f 共x兲 苷
2
t共x兲 苷 cos x
3 t共x兲 苷 s 1⫺x
34. f 共x兲 苷 sx ,
1 , x
t共x兲 苷
x , 1⫹x
0
52. Use the given graphs of f and t to estimate the value of
t共x兲 苷 sin 2x
37. f 共x兲 苷 3x ⫺ 2,
f 共 t共x兲兲 for x 苷 ⫺5, ⫺4, ⫺3, . . . , 5. Use these estimates to sketch a rough graph of f ⴰ t. y
t共x兲 苷 sin x,
ⱍ
t共x兲 苷 2 x,
h共x兲 苷 sx
39. f 共x兲 苷 sx ⫺ 3 ,
t共x兲 苷 x 2 ,
h共x兲 苷 x 3 ⫹ 2
t共x兲 苷
g
h共x兲 苷 x 2
38. f 共x兲 苷 x ⫺ 4 ,
40. f 共x兲 苷 tan x,
x
2
x⫹1 x⫹2
37– 40 Find f ⴰ t ⴰ h.
ⱍ
f
2
33. f 共x兲 苷 1 ⫺ 3x,
35. f 共x兲 苷 x ⫹
g
x 3 , h共x兲 苷 s x x⫺1
1 0
1
x
f
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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53. A stone is dropped into a lake, creating a circular ripple that
travels outward at a speed of 60 cm兾s. (a) Express the radius r of this circle as a function of the time t ( in seconds). (b) If A is the area of this circle as a function of the radius, find A ⴰ r and interpret it. 54. A spherical balloon is being inflated and the radius of the bal
loon is increasing at a rate of 2 cm兾s. (a) Express the radius r of the balloon as a function of the time t ( in seconds). (b) If V is the volume of the balloon as a function of the radius, find V ⴰ r and interpret it. 55. A ship is moving at a speed of 30 km兾h parallel to a straight
shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d , the distance the ship has traveled since noon; that is, find f so that s 苷 f 共d兲. (b) Express d as a function of t, the time elapsed since noon; that is, find t so that d 苷 t共t兲. (c) Find f ⴰ t. What does this function represent? 56. An airplane is flying at a speed of 350 mi兾h at an altitude of
one mile and passes directly over a radar station at time t 苷 0. (a) Express the horizontal distance d ( in miles) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d . (c) Use composition to express s as a function of t. 57. The Heaviside function H is defined by
H共t兲 苷
再
0 1
if t ⬍ 0 if t 艌 0
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V共t兲 in a circuit if the switch is turned on at time t 苷 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V共t兲 in terms of H共t兲.
1.4
5:11 PM
(c) Sketch the graph of the voltage V共t兲 in a circuit if the switch is turned on at time t 苷 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V共t兲 in terms of H共t兲. (Note that starting at t 苷 5 corresponds to a translation.) 58. The Heaviside function defined in Exercise 57 can also be used
to define the ramp function y 苷 ctH共t兲, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y 苷 tH共t兲. (b) Sketch the graph of the voltage V共t兲 in a circuit if the switch is turned on at time t 苷 0 and the voltage is gradually increased to 120 volts over a 60second time interval. Write a formula for V共t兲 in terms of H共t兲 for t 艋 60. (c) Sketch the graph of the voltage V共t兲 in a circuit if the switch is turned on at time t 苷 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V共t兲 in terms of H共t兲 for t 艋 32. 59. Let f and t be linear functions with equations f 共x兲 苷 m1 x ⫹ b1
and t共x兲 苷 m 2 x ⫹ b 2. Is f ⴰ t also a linear function? If so, what is the slope of its graph?
60. If you invest x dollars at 4% interest compounded annually,
then the amount A共x兲 of the investment after one year is A共x兲 苷 1.04x. Find A ⴰ A, A ⴰ A ⴰ A, and A ⴰ A ⴰ A ⴰ A. What do these compositions represent? Find a formula for the composition of n copies of A. 61. (a) If t共x兲 苷 2x ⫹ 1 and h共x兲 苷 4x 2 ⫹ 4x ⫹ 7, find a function
f such that f ⴰ t 苷 h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.) (b) If f 共x兲 苷 3x ⫹ 5 and h共x兲 苷 3x 2 ⫹ 3x ⫹ 2, find a function t such that f ⴰ t 苷 h.
62. If f 共x兲 苷 x ⫹ 4 and h共x兲 苷 4x ⫺ 1, find a function t such that
t ⴰ f 苷 h.
63. Suppose t is an even function and let h 苷 f ⴰ t. Is h always an
even function? 64. Suppose t is an odd function and let h 苷 f ⴰ t. Is h always an
odd function? What if f is odd? What if f is even?
Graphing Calculators and Computers In this section we assume that you have access to a graphing calculator or a computer with graphing software. We will see that the use of such a device enables us to graph more complicated functions and to solve more complex problems than would otherwise be possible. We also point out some of the pitfalls that can occur with these machines. Graphing calculators and computers can give very accurate graphs of functions. But we will see in Chapter 4 that only through the use of calculus can we be sure that we have uncovered all the interesting aspects of a graph. A graphing calculator or computer displays a rectangular portion of the graph of a function in a display window or viewing screen, which we refer to as a viewing rectangle. The default screen often gives an incomplete or misleading picture, so it is important to Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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45
choose the viewing rectangle with care. If we choose the xvalues to range from a minimum value of Xmin 苷 a to a maximum value of Xmax 苷 b and the yvalues to range from a minimum of Ymin 苷 c to a maximum of Ymax 苷 d , then the visible portion of the graph lies in the rectangle
ⱍ
关a, b兴 ⫻ 关c, d兴 苷 兵共x, y兲 a 艋 x 艋 b, c 艋 y 艋 d其 shown in Figure 1. We refer to this rectangle as the 关a, b兴 by 关c, d兴 viewing rectangle. (a, d )
y=d
( b, d )
x=b
x=a
FIGURE 1
The viewing rectangle 关a, b兴 by 关c, d兴
(a, c )
y=c
( b, c )
The machine draws the graph of a function f much as you would. It plots points of the form 共x, f 共x兲兲 for a certain number of equally spaced values of x between a and b. If an xvalue is not in the domain of f , or if f 共x兲 lies outside the viewing rectangle, it moves on to the next xvalue. The machine connects each point to the preceding plotted point to form a representation of the graph of f. EXAMPLE 1 Draw the graph of the function f 共x兲 苷 x 2 ⫹ 3 in each of the following
viewing rectangles. (a) 关⫺2, 2兴 by 关⫺2, 2兴 (c) 关⫺10, 10兴 by 关⫺5, 30兴
2
(b) 关⫺4, 4兴 by 关⫺4, 4兴 (d) 关⫺50, 50兴 by 关⫺100, 1000兴
SOLUTION For part (a) we select the range by setting X min 苷 ⫺2, X max 苷 2, _2
2
_2
(a) 关_2, 2兴 by 关_2, 2兴
Y min 苷 ⫺2, and Y max 苷 2. The resulting graph is shown in Figure 2(a). The display window is blank! A moment’s thought provides the explanation: Notice that x 2 艌 0 for all x, so x 2 ⫹ 3 艌 3 for all x. Thus the range of the function f 共x兲 苷 x 2 ⫹ 3 is 关3, ⬁兲. This means that the graph of f lies entirely outside the viewing rectangle 关⫺2, 2兴 by 关⫺2, 2兴. The graphs for the viewing rectangles in parts (b), (c), and (d) are also shown in Figure 2. Observe that we get a more complete picture in parts (c) and (d), but in part (d) it is not clear that the yintercept is 3.
4
_4
1000
30
4 10
_10
_50
50
_4
_5
_100
(b) 关_4, 4兴 by 关_4, 4兴
(c) 关_10, 10兴 by 关_5, 30兴
(d) 关_50, 50兴 by 关_100, 1000兴
FIGURE 2 Graphs of ƒ=≈+3
We see from Example 1 that the choice of a viewing rectangle can make a big difference in the appearance of a graph. Often it’s necessary to change to a larger viewing rectangle to obtain a more complete picture, a more global view, of the graph. In the next example we see that knowledge of the domain and range of a function sometimes provides us with enough information to select a good viewing rectangle. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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EXAMPLE 2 Determine an appropriate viewing rectangle for the function f 共x兲 苷 s8 ⫺ 2x 2 and use it to graph f. SOLUTION The expression for f 共x兲 is defined when
8 ⫺ 2x 2 艌 0
4
&?
2x 2 艋 8
&?
x2 艋 4
&?
ⱍxⱍ 艋 2
&? ⫺2 艋 x 艋 2
Therefore the domain of f is the interval 关⫺2, 2兴. Also, _3
0 艋 s8 ⫺ 2x 2 艋 s8 苷 2s2 ⬇ 2.83
3
so the range of f is the interval [0, 2s2 ]. We choose the viewing rectangle so that the xinterval is somewhat larger than the domain and the yinterval is larger than the range. Taking the viewing rectangle to be 关⫺3, 3兴 by 关⫺1, 4兴, we get the graph shown in Figure 3.
_1
FIGURE 3 ƒ=œ„„„„„„ 82≈
EXAMPLE 3 Graph the function y 苷 x 3 ⫺ 150x. 5
_5
SOLUTION Here the domain is ⺢, the set of all real numbers. That doesn’t help us choose
a viewing rectangle. Let’s experiment. If we start with the viewing rectangle 关⫺5, 5兴 by 关⫺5, 5兴, we get the graph in Figure 4. It appears blank, but actually the graph is so nearly vertical that it blends in with the yaxis. If we change the viewing rectangle to 关⫺20, 20兴 by 关⫺20, 20兴, we get the picture shown in Figure 5(a). The graph appears to consist of vertical lines, but we know that can’t be correct. If we look carefully while the graph is being drawn, we see that the graph leaves the screen and reappears during the graphing process. This indicates that we need to see more in the vertical direction, so we change the viewing rectangle to 关⫺20, 20兴 by 关⫺500, 500兴. The resulting graph is shown in Figure 5(b). It still doesn’t quite reveal all the main features of the function, so we try 关⫺20, 20兴 by 关⫺1000, 1000兴 in Figure 5(c). Now we are more confident that we have arrived at an appropriate viewing rectangle. In Chapter 4 we will be able to see that the graph shown in Figure 5(c) does indeed reveal all the main features of the function.
5
_5
FIGURE 4
20
_20
500
20
_20
1000
20
20
_20
_20
_500
_1000
(a)
( b)
(c)
FIGURE 5 Graphs of y=˛150x
v
EXAMPLE 4 Graph the function f 共x兲 苷 sin 50x in an appropriate viewing rectangle.
SOLUTION Figure 6(a) shows the graph of f produced by a graphing calculator using the
viewing rectangle 关⫺12, 12兴 by 关⫺1.5, 1.5兴. At first glance the graph appears to be
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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47
reasonable. But if we change the viewing rectangle to the ones shown in the following parts of Figure 6, the graphs look very different. Something strange is happening. 1.5
_12
The appearance of the graphs in Figure 6 depends on the machine used. The graphs you get with your own graphing device might not look like these figures, but they will also be quite inaccurate.
1.5
12
_10
10
_1.5
_1.5
(a)
(b)
1.5
1.5
_9
9
_6
6
FIGURE 6
Graphs of ƒ=sin 50x in four viewing rectangles
.25
_1.5
(d)
2 苷 ⬇ 0.126 50 25 This suggests that we should deal only with small values of x in order to show just a few oscillations of the graph. If we choose the viewing rectangle 关⫺0.25, 0.25兴 by 关⫺1.5, 1.5兴, we get the graph shown in Figure 7. Now we see what went wrong in Figure 6. The oscillations of y 苷 sin 50x are so rapid that when the calculator plots points and joins them, it misses most of the maximum and minimum points and therefore gives a very misleading impression of the graph.
FIGURE 7
ƒ=sin 50x
We have seen that the use of an inappropriate viewing rectangle can give a misleading impression of the graph of a function. In Examples 1 and 3 we solved the problem by changing to a larger viewing rectangle. In Example 4 we had to make the viewing rectangle smaller. In the next example we look at a function for which there is no single viewing rectangle that reveals the true shape of the graph.
1.5
6.5
_6.5
v _1.5
FIGURE 8
_1.5
(c)
In order to explain the big differences in appearance of these graphs and to find an appropriate viewing rectangle, we need to find the period of the function y 苷 sin 50x. We know that the function y 苷 sin x has period 2 and the graph of y 苷 sin 50x is shrunk horizontally by a factor of 50, so the period of y 苷 sin 50x is
1.5
_.25
_1.5
EXAMPLE 5 Graph the function f 共x兲 苷 sin x ⫹
1 100
cos 100x.
SOLUTION Figure 8 shows the graph of f produced by a graphing calculator with viewing
rectangle 关⫺6.5, 6.5兴 by 关⫺1.5, 1.5兴. It looks much like the graph of y 苷 sin x, but perhaps with some bumps attached. If we zoom in to the viewing rectangle 关⫺0.1, 0.1兴 by
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS 0.1
_0.1
0.1
关⫺0.1, 0.1兴, we can see much more clearly the shape of these bumps in Figure 9. The 1 reason for this behavior is that the second term, 100 cos 100x, is very small in comparison with the first term, sin x. Thus we really need two graphs to see the true nature of this function. EXAMPLE 6 Draw the graph of the function y 苷
1 . 1⫺x
SOLUTION Figure 10(a) shows the graph produced by a graphing calculator with view
_0.1
ing rectangle 关⫺9, 9兴 by 关⫺9, 9兴. In connecting successive points on the graph, the calculator produced a steep line segment from the top to the bottom of the screen. That line segment is not truly part of the graph. Notice that the domain of the function y 苷 1兾共1 ⫺ x兲 is 兵x x 苷 1其. We can eliminate the extraneous nearvertical line by experimenting with a change of scale. When we change to the smaller viewing rectangle 关⫺4.7, 4.7兴 by 关⫺4.7, 4.7兴 on this particular calculator, we obtain the much better graph in Figure 10(b).
FIGURE 9
ⱍ
Another way to avoid the extraneous line is to change the graphing mode on the calculator so that the dots are not connected.
9
4.7
_9
9
FIGURE 10
_4.7
4.7
_9
_4.7
(a)
(b)
3 x. EXAMPLE 7 Graph the function y 苷 s
SOLUTION Some graphing devices display the graph shown in Figure 11, whereas others
produce a graph like that in Figure 12. We know from Section 1.2 (Figure 13) that the graph in Figure 12 is correct, so what happened in Figure 11? The explanation is that some machines compute the cube root of x using a logarithm, which is not defined if x is negative, so only the right half of the graph is produced. 2
_3
2
3
_3
_2
FIGURE 11 You can get the correct graph with Maple if you first type with(RealDomain);
3
_2
FIGURE 12
You should experiment with your own machine to see which of these two graphs is produced. If you get the graph in Figure 11, you can obtain the correct picture by graphing the function x f 共x兲 苷 ⴢ x 1兾3 x
ⱍ ⱍ ⱍ ⱍ
3 Notice that this function is equal to s x (except when x 苷 0).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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49
To understand how the expression for a function relates to its graph, it’s helpful to graph a family of functions, that is, a collection of functions whose equations are related. In the next example we graph members of a family of cubic polynomials.
v
EXAMPLE 8 Graph the function y 苷 x 3 ⫹ cx for various values of the number c. How
does the graph change when c is changed? SOLUTION Figure 13 shows the graphs of y 苷 x 3 ⫹ cx for c 苷 2, 1, 0, ⫺1, and ⫺2. We
TEC In Visual 1.4 you can see an animation of Figure 13.
(a) y=˛+2x
see that, for positive values of c, the graph increases from left to right with no maximum or minimum points (peaks or valleys). When c 苷 0, the curve is flat at the origin. When c is negative, the curve has a maximum point and a minimum point. As c decreases, the maximum point becomes higher and the minimum point lower.
(b) y=˛+x
(c) y=˛
(d) y=˛x
(e) y=˛2x
FIGURE 13
Several members of the family of functions y=˛+cx, all graphed in the viewing rectangle 关_2, 2兴 by 关_2.5, 2.5兴
EXAMPLE 9 Find the solution of the equation cos x 苷 x correct to two decimal places. SOLUTION The solutions of the equation cos x 苷 x are the xcoordinates of the points of
intersection of the curves y 苷 cos x and y 苷 x. From Figure 14(a) we see that there is only one solution and it lies between 0 and 1. Zooming in to the viewing rectangle 关0, 1兴 by 关0, 1兴, we see from Figure 14(b) that the root lies between 0.7 and 0.8. So we zoom in further to the viewing rectangle 关0.7, 0.8兴 by 关0.7, 0.8兴 in Figure 14(c). By moving the cursor to the intersection point of the two curves, or by inspection and the fact that the xscale is 0.01, we see that the solution of the equation is about 0.74. (Many calculators have a builtin intersection feature.)
1.5
1 y=x
0.8 y=cos x
y=cos x _5
y=x
5
y=x y=cos x
FIGURE 14
Locating the roots of cos x=x
_1.5
(a) 关_5, 5兴 by 关_1.5, 1.5兴 xscale=1
1
0
(b) 关0, 1兴 by 关0, 1兴 xscale=0.1
0.8
0.7
(c) 关0.7, 0.8兴 by 关0.7, 0.8兴 xscale=0.01
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
; Exercises
1. Use a graphing calculator or computer to determine which of
the given viewing rectangles produces the most appropriate graph of the function f 共x兲 苷 sx 3 ⫺ 5x 2 . (a) 关⫺5, 5兴 by 关⫺5, 5兴 (b) 关0, 10兴 by 关0, 2兴 (c) 关0, 10兴 by 关0, 10兴 2. Use a graphing calculator or computer to determine which of
the given viewing rectangles produces the most appropriate graph of the function f 共x兲 苷 x 4 ⫺ 16x 2 ⫹ 20. (a) 关⫺3, 3兴 by 关⫺3, 3兴 (b) 关⫺10, 10兴 by 关⫺10, 10兴 (c) 关⫺50, 50兴 by 关⫺50, 50兴 (d) 关⫺5, 5兴 by 关⫺50, 50兴 3–14 Determine an appropriate viewing rectangle for the given function and use it to draw the graph. 3. f 共x兲 苷 x ⫺ 36x ⫹ 32
4. f 共x兲 苷 x ⫹ 15x ⫹ 65x
5. f 共x兲 苷 s50 ⫺ 0.2 x
6. f 共x兲 苷 s15x ⫺ x 2
7. f 共x兲 苷 x 3 ⫺ 225x
x 8. f 共x兲 苷 2 x ⫹ 100
9. f 共x兲 苷 sin 2 共1000x兲
10. f 共x兲 苷 cos共0.001x兲
2
3
2
11. f 共x兲 苷 sin sx
12. f 共x兲 苷 sec共20 x兲
13. y 苷 10 sin x ⫹ sin 100x
14. y 苷 x 2 ⫹ 0.02 sin 50x
24. We saw in Example 9 that the equation cos x 苷 x has exactly
one solution. (a) Use a graph to show that the equation cos x 苷 0.3x has three solutions and find their values correct to two decimal places. (b) Find an approximate value of m such that the equation cos x 苷 mx has exactly two solutions. 25. Use graphs to determine which of the functions f 共x兲 苷 10x 2
and t共x兲 苷 x 3兾10 is eventually larger (that is, larger when x is very large).
26. Use graphs to determine which of the functions
f 共x兲 苷 x 4 ⫺ 100x 3 and t共x兲 苷 x 3 is eventually larger.
ⱍ
ⱍ
27. For what values of x is it true that tan x ⫺ x ⬍ 0.01 and
⫺兾2 ⬍ x ⬍ 兾2?
28. Graph the polynomials P共x兲 苷 3x 5 ⫺ 5x 3 ⫹ 2x and Q共x兲 苷 3x 5
on the same screen, first using the viewing rectangle 关⫺2, 2兴 by [⫺2, 2] and then changing to 关⫺10, 10兴 by 关⫺10,000, 10,000兴. What do you observe from these graphs?
29. In this exercise we consider the family of root functions 15. (a) Try to find an appropriate viewing rectangle for
f 共x兲 苷 共x ⫺ 10兲3 2⫺x. (b) Do you need more than one window? Why? 16. Graph the function f 共x兲 苷 x 2s30 ⫺ x in an appropriate
viewing rectangle. Why does part of the graph appear to be missing? 17. Graph the ellipse 4x 2 ⫹ 2y 2 苷 1 by graphing the functions
whose graphs are the upper and lower halves of the ellipse. 18. Graph the hyperbola y 2 ⫺ 9x 2 苷 1 by graphing the functions
whose graphs are the upper and lower branches of the hyperbola. 19–20 Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? 19. y 苷 3x 2 ⫺ 6x ⫹ 1, y 苷 0.23x ⫺ 2.25;
关⫺1, 3兴 by 关⫺2.5, 1.5兴
20. y 苷 6 ⫺ 4x ⫺ x 2 , y 苷 3x ⫹ 18; 关⫺6, 2兴 by 关⫺5, 20兴 21–23 Find all solutions of the equation correct to two decimal
places.
30. In this exercise we consider the family of functions
f 共x兲 苷 1兾x n, where n is a positive integer. (a) Graph the functions y 苷 1兾x and y 苷 1兾x 3 on the same screen using the viewing rectangle 关⫺3, 3兴 by 关⫺3, 3兴. (b) Graph the functions y 苷 1兾x 2 and y 苷 1兾x 4 on the same screen using the same viewing rectangle as in part (a). (c) Graph all of the functions in parts (a) and (b) on the same screen using the viewing rectangle 关⫺1, 3兴 by 关⫺1, 3兴. (d) What conclusions can you make from these graphs? 31. Graph the function f 共x兲 苷 x 4 ⫹ cx 2 ⫹ x for several values
of c. How does the graph change when c changes?
21. x 4 ⫺ x 苷 1
22. sx 苷 x 3 ⫺ 1
23. tan x 苷 s1 ⫺ x 2
;
n f 共x兲 苷 s x , where n is a positive integer. 4 6 (a) Graph the functions y 苷 sx , y 苷 s x , and y 苷 s x on the same screen using the viewing rectangle 关⫺1, 4兴 by 关⫺1, 3兴. 3 5 (b) Graph the functions y 苷 x, y 苷 s x , and y 苷 s x on the same screen using the viewing rectangle 关⫺3, 3兴 by 关⫺2, 2兴. (See Example 7.) 3 4 (c) Graph the functions y 苷 sx , y 苷 s x, y 苷 s x , and 5 y 苷 sx on the same screen using the viewing rectangle 关⫺1, 3兴 by 关⫺1, 2兴. (d) What conclusions can you make from these graphs?
Graphing calculator or computer required
32. Graph the function f 共x兲 苷 s1 ⫹ cx 2 for various values
of c. Describe how changing the value of c affects the graph.
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.5
33. Graph the function y 苷 x n 2 ⫺x, x 艌 0, for n 苷 1, 2, 3, 4, 5,
51
EXPONENTIAL FUNCTIONS
[Hint: The TI83’s graphing window is 95 pixels wide. What specific points does the calculator plot?]
and 6. How does the graph change as n increases? 34. The curves with equations
y苷
ⱍxⱍ
0
sc ⫺ x 2
are called bulletnose curves. Graph some of these curves to see why. What happens as c increases? 35. What happens to the graph of the equation y 2 苷 cx 3 ⫹ x 2 as
c varies?
0
y=sin 96x
2π
y=sin 2x
38. The first graph in the figure is that of y 苷 sin 45x as displayed
by a TI83 graphing calculator. It is inaccurate and so, to help explain its appearance, we replot the curve in dot mode in the second graph. What two sine curves does the calculator appear to be plotting? Show that each point on the graph of y 苷 sin 45x that the TI83 chooses to plot is in fact on one of these two curves. (The TI83’s graphing window is 95 pixels wide.)
36. This exercise explores the effect of the inner function t on a
composite function y 苷 f 共 t共x兲兲. (a) Graph the function y 苷 sin( sx ) using the viewing rectangle 关0, 400兴 by 关⫺1.5, 1.5兴. How does this graph differ from the graph of the sine function? (b) Graph the function y 苷 sin共x 2 兲 using the viewing rectangle 关⫺5, 5兴 by 关⫺1.5, 1.5兴. How does this graph differ from the graph of the sine function? 37. The figure shows the graphs of y 苷 sin 96x and y 苷 sin 2x as
2π
0
2π
0
2π
displayed by a TI83 graphing calculator. The first graph is inaccurate. Explain why the two graphs appear identical.
1.5
Exponential Functions
In Appendix G we present an alternative approach to the exponential and logarithmic functions using integral calculus.
The function f 共x兲 苷 2 x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function t共x兲 苷 x 2, in which the variable is the base. In general, an exponential function is a function of the form f 共x兲 苷 a x where a is a positive constant. Let’s recall what this means. If x 苷 n, a positive integer, then an 苷 a ⴢ a ⴢ ⭈ ⭈ ⭈ ⴢ a n factors
If x 苷 0, then a 0 苷 1, and if x 苷 ⫺n, where n is a positive integer, then y
a ⫺n 苷
1 an
If x is a rational number, x 苷 p兾q, where p and q are integers and q ⬎ 0, then q p q a x 苷 a p兾q 苷 sa 苷 (sa )
p
1 0
1
x
FIGURE 1
Representation of y=2®, x rational
But what is the meaning of a x if x is an irrational number? For instance, what is meant by 2 s3 or 5 ? To help us answer this question we first look at the graph of the function y 苷 2 x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y 苷 2 x to include both rational and irrational numbers.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
There are holes in the graph in Figure 1 corresponding to irrational values of x. We want to fill in the holes by defining f 共x兲 苷 2 x, where x 僆 ⺢, so that f is an increasing function. In particular, since the irrational number s3 satisfies 1.7 ⬍ s3 ⬍ 1.8 we must have 2 1.7 ⬍ 2 s3 ⬍ 2 1.8 and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. Similarly, if we use better approximations for s3 , we obtain better approximations for 2 s3:
A proof of this fact is given in J. Marsden and A. Weinstein, Calculus Unlimited (Menlo Park, CA, 1981). For an online version, see caltechbook.library.caltech.edu/197/
1.73 ⬍ s3 ⬍ 1.74
?
2 1.73 ⬍ 2 s3 ⬍ 2 1.74
1.732 ⬍ s3 ⬍ 1.733
?
2 1.732 ⬍ 2 s3 ⬍ 2 1.733
1.7320 ⬍ s3 ⬍ 1.7321
?
2 1.7320 ⬍ 2 s3 ⬍ 2 1.7321
1.73205 ⬍ s3 ⬍ 1.73206 . . . . . .
?
2 1.73205 ⬍ 2 s3 ⬍ 2 1.73206 . . . . . .
It can be shown that there is exactly one number that is greater than all of the numbers 2 1.7,
2 1.73,
2 1.7320,
2 1.73205,
...
2 1.733,
2 1.7321,
2 1.73206,
...
and less than all of the numbers 2 1.8,
y
2 1.732,
2 1.74,
We define 2 s3 to be this number. Using the preceding approximation process we can compute it correct to six decimal places: 2 s3 ⬇ 3.321997 Similarly, we can define 2 x (or a x, if a ⬎ 0) where x is any irrational number. Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function f 共x兲 苷 2 x, x 僆 ⺢. The graphs of members of the family of functions y 苷 a x are shown in Figure 3 for various values of the base a. Notice that all of these graphs pass through the same point 共0, 1兲 because a 0 苷 1 for a 苷 0. Notice also that as the base a gets larger, the exponential function grows more rapidly (for x ⬎ 0).
1 0
1
x
FIGURE 2
y=2®, x real
® ” ’ 2 1
® ” ’ 4 1
y
10®
4®
2®
If 0 ⬍ a ⬍ 1, then a x approaches 0 as x becomes large. If a ⬎ 1, then a x approaches 0 as x decreases through negative values. In both cases the xaxis is a horizontal asymptote. These matters are discussed in Section 2.6.
FIGURE 3
1.5®
1®
0
1
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.5
53
EXPONENTIAL FUNCTIONS
You can see from Figure 3 that there are basically three kinds of exponential functions y 苷 a x. If 0 ⬍ a ⬍ 1, the exponential function decreases; if a 苷 1, it is a constant; and if a ⬎ 1, it increases. These three cases are illustrated in Figure 4. Observe that if a 苷 1 , then the exponential function y 苷 a x has domain ⺢ and range 共0, ⬁兲. Notice also that, since 共1兾a兲 x 苷 1兾a x 苷 a ⫺x, the graph of y 苷 共1兾a兲 x is just the reflection of the graph of y 苷 a x about the yaxis. y
y
y
1
(0, 1)
(0, 1) 0
FIGURE 4
0
x
(a) y=a®, 01 0
x
The logarithmic function log a has domain 共0, ⬁兲 and range ⺢. Its graph is the reflection of the graph of y 苷 a x about the line y 苷 x. Figure 11 shows the case where a ⬎ 1. (The most important logarithmic functions have base a ⬎ 1.) The fact that y 苷 a x is a very rapidly increasing function for x ⬎ 0 is reflected in the fact that y 苷 log a x is a very slowly increasing function for x ⬎ 1. Figure 12 shows the graphs of y 苷 log a x with various values of the base a ⬎ 1. Since log a 1 苷 0, the graphs of all logarithmic functions pass through the point 共1, 0兲.
y=log a x, a>1 y
FIGURE 11
y=log™ x y=log£ x
1
0
1
x
y=log∞ x y=log¡¸ x
FIGURE 12
The following properties of logarithmic functions follow from the corresponding properties of exponential functions given in Section 1.5.
Laws of Logarithms If x and y are positive numbers, then 1. log a 共xy兲 苷 log a x ⫹ log a y
冉冊
2. log a
x y
苷 log a x ⫺ log a y
3. log a 共x r 兲 苷 r log a x
(where r is any real number)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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FUNCTIONS AND MODELS
EXAMPLE 6 Use the laws of logarithms to evaluate log 2 80 ⫺ log 2 5. SOLUTION Using Law 2, we have
冉 冊
log 2 80 ⫺ log 2 5 苷 log 2
80 5
苷 log 2 16 苷 4
because 2 4 苷 16.
Natural Logarithms Notation for Logarithms Most textbooks in calculus and the sciences, as well as calculators, use the notation ln x for the natural logarithm and log x for the “common logarithm,” log10 x. In the more advanced mathematical and scientific literature and in computer languages, however, the notation log x usually denotes the natural logarithm.
Of all possible bases a for logarithms, we will see in Chapter 3 that the most convenient choice of a base is the number e, which was defined in Section 1.5. The logarithm with base e is called the natural logarithm and has a special notation: log e x 苷 ln x If we put a 苷 e and replace log e with “ln” in 6 and 7 , then the defining properties of the natural logarithm function become ln x 苷 y
8
9
ey 苷 x
&?
ln共e x 兲 苷 x
x僆⺢
e ln x 苷 x
x⬎0
In particular, if we set x 苷 1, we get ln e 苷 1 EXAMPLE 7 Find x if ln x 苷 5. SOLUTION 1 From 8 we see that
ln x 苷 5
means
e5 苷 x
Therefore x 苷 e 5. (If you have trouble working with the “ln” notation, just replace it by log e . Then the equation becomes log e x 苷 5; so, by the definition of logarithm, e 5 苷 x.) SOLUTION 2 Start with the equation
ln x 苷 5 and apply the exponential function to both sides of the equation: e ln x 苷 e 5 But the second cancellation equation in 9 says that e ln x 苷 x. Therefore x 苷 e 5.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.6
INVERSE FUNCTIONS AND LOGARITHMS
65
EXAMPLE 8 Solve the equation e 5⫺3x 苷 10. SOLUTION We take natural logarithms of both sides of the equation and use 9 :
ln共e 5⫺3x 兲 苷 ln 10 5 ⫺ 3x 苷 ln 10 3x 苷 5 ⫺ ln 10 x 苷 13 共5 ⫺ ln 10兲 Since the natural logarithm is found on scientific calculators, we can approximate the solution: to four decimal places, x ⬇ 0.8991.
v
EXAMPLE 9 Express ln a ⫹ 2 ln b as a single logarithm. 1
SOLUTION Using Laws 3 and 1 of logarithms, we have
ln a ⫹ 12 ln b 苷 ln a ⫹ ln b 1兾2 苷 ln a ⫹ ln sb 苷 ln(asb ) The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm.
10 Change of Base Formula For any positive number a 共a 苷 1兲, we have
log a x 苷
ln x ln a
PROOF Let y 苷 log a x. Then, from 6 , we have a y 苷 x. Taking natural logarithms of both
sides of this equation, we get y ln a 苷 ln x. Therefore y苷
ln x ln a
Scientific calculators have a key for natural logarithms, so Formula 10 enables us to use a calculator to compute a logarithm with any base (as shown in the following example). Similarly, Formula 10 allows us to graph any logarithmic function on a graphing calculator or computer (see Exercises 43 and 44). EXAMPLE 10 Evaluate log 8 5 correct to six decimal places. SOLUTION Formula 10 gives
log 8 5 苷
ln 5 ⬇ 0.773976 ln 8
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FUNCTIONS AND MODELS
y
Graph and Growth of the Natural Logarithm y=´ y=x
1
y=ln x
0 x
1
The graphs of the exponential function y 苷 e x and its inverse function, the natural logarithm function, are shown in Figure 13. Because the curve y 苷 e x crosses the yaxis with a slope of 1, it follows that the reflected curve y 苷 ln x crosses the xaxis with a slope of 1. In common with all other logarithmic functions with base greater than 1, the natural logarithm is an increasing function defined on 共0, ⬁兲 and the yaxis is a vertical asymptote. (This means that the values of ln x become very large negative as x approaches 0.) EXAMPLE 11 Sketch the graph of the function y 苷 ln共x ⫺ 2兲 ⫺ 1. SOLUTION We start with the graph of y 苷 ln x as given in Figure 13. Using the transfor
mations of Section 1.3, we shift it 2 units to the right to get the graph of y 苷 ln共x ⫺ 2兲 and then we shift it 1 unit downward to get the graph of y 苷 ln共x ⫺ 2兲 ⫺ 1. (See Figure 14.)
FIGURE 13 The graph of y=ln x is the reflection of the graph of y=´ about the line y=x y
y
y
x=2
y=ln x 0
(1, 0)
x=2 y=ln(x2)1
y=ln(x2) 0
x
2
x
(3, 0)
2
0
x (3, _1)
FIGURE 14
Although ln x is an increasing function, it grows very slowly when x ⬎ 1. In fact, ln x grows more slowly than any positive power of x. To illustrate this fact, we compare approximate values of the functions y 苷 ln x and y 苷 x 1兾2 苷 sx in the following table and we graph them in Figures 15 and 16. You can see that initially the graphs of y 苷 sx and y 苷 ln x grow at comparable rates, but eventually the root function far surpasses the logarithm. x
1
2
5
10
50
100
500
1000
10,000
100,000
ln x
0
0.69
1.61
2.30
3.91
4.6
6.2
6.9
9.2
11.5
sx
1
1.41
2.24
3.16
7.07
10.0
22.4
31.6
100
316
ln x sx
0
0.49
0.72
0.73
0.55
0.46
0.28
0.22
0.09
0.04
y
y
x y=œ„ 20
x y=œ„ 1
0
y=ln x
y=ln x 1
FIGURE 15
x
0
1000 x
FIGURE 16
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 1.6
INVERSE FUNCTIONS AND LOGARITHMS
67
Inverse Trigonometric Functions When we try to find the inverse trigonometric functions, we have a slight difficulty: Because the trigonometric functions are not onetoone, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become onetoone. You can see from Figure 17 that the sine function y 苷 sin x is not onetoone (use the Horizontal Line Test). But the function f 共x兲 苷 sin x, ⫺兾2 艋 x 艋 兾2, is onetoone (see Figure 18). The inverse function of this restricted sine function f exists and is denoted by sin ⫺1 or arcsin. It is called the inverse sine function or the arcsine function. y
y
y=sin x _ π2 _π
0
π 2
0
x
π
π 2
x
π
π
FIGURE 18 y=sin x, _ 2 ¯x¯ 2
FIGURE 17
Since the definition of an inverse function says that f ⫺1共x兲 苷 y
f 共y兲 苷 x
&?
we have sin⫺1x 苷 y
 sin⫺1x 苷 1
&?
sin y 苷 x
and ⫺
艋y艋 2 2
Thus, if ⫺1 艋 x 艋 1, sin ⫺1x is the number between ⫺兾2 and 兾2 whose sine is x.
sin x
( 12) and (b) tan(arcsin 13 ).
EXAMPLE 12 Evaluate (a) sin⫺1 SOLUTION
(a) We have sin⫺1( 12) 苷
3 1 ¨
6
because sin共兾6兲 苷 12 and 兾6 lies between ⫺兾2 and 兾2. (b) Let 苷 arcsin 13 , so sin 苷 13. Then we can draw a right triangle with angle as in Figure 19 and deduce from the Pythagorean Theorem that the third side has length s9 ⫺ 1 苷 2s2 . This enables us to read from the triangle that
2 2 œ„ FIGURE 19
tan(arcsin 13 ) 苷 tan 苷
1 2s2
The cancellation equations for inverse functions become, in this case,
sin⫺1共sin x兲 苷 x sin共sin⫺1x兲 苷 x
for ⫺
艋x艋 2 2
for ⫺1 艋 x 艋 1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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π 2
0
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FUNCTIONS AND MODELS
y
_1
9/20/10
x
1
The inverse sine function, sin⫺1, has domain 关⫺1, 1兴 and range 关⫺兾2, 兾2兴 , and its graph, shown in Figure 20, is obtained from that of the restricted sine function (Figure 18) by reflection about the line y 苷 x. The inverse cosine function is handled similarly. The restricted cosine function f 共x兲 苷 cos x, 0 艋 x 艋 , is onetoone (see Figure 21) and so it has an inverse function denoted by cos ⫺1 or arccos.
_ π2
cos⫺1x 苷 y
&?
cos y 苷 x
0艋y艋
and
FIGURE 20
y=sin–! x=arcsin x y
y
π 1 π 2
0
π 2
x
π
0
_1
1
FIGURE 21
FIGURE 22
y=cos x, 0¯x¯π
y=cos–! x=arccos x
x
The cancellation equations are cos ⫺1共cos x兲 苷 x cos共cos⫺1x兲 苷 x
for ⫺1 艋 x 艋 1
The inverse cosine function, cos⫺1, has domain 关⫺1, 1兴 and range 关0, 兴. Its graph is shown in Figure 22. The tangent function can be made onetoone by restricting it to the interval 共⫺兾2, 兾2兲. Thus the inverse tangent function is defined as the inverse of the function f 共x兲 苷 tan x, ⫺兾2 ⬍ x ⬍ 兾2. (See Figure 23.) It is denoted by tan⫺1 or arctan.
y
_ π2
for 0 艋 x 艋
0
tan⫺1x 苷 y
π 2
&?
tan y 苷 x
x
and
⫺
⬍y⬍ 2 2
EXAMPLE 13 Simplify the expression cos共tan⫺1x兲. SOLUTION 1 Let y 苷 tan⫺1x. Then tan y 苷 x and ⫺兾2 ⬍ y ⬍ FIGURE 23
兾2. We want to find
cos y but, since tan y is known, it is easier to find sec y first: π
π
y=tan x, _ 2 <x0, f is concave upward
v
EXAMPLE 6 Discuss the curve y 苷 x 4 ⫺ 4x 3 with respect to concavity, points of
inflection, and local maxima and minima. Use this information to sketch the curve. SOLUTION If f 共x兲 苷 x 4 ⫺ 4x 3, then
f ⬘共x兲 苷 4x 3 ⫺ 12x 2 苷 4x 2共x ⫺ 3兲 f ⬙共x兲 苷 12x 2 ⫺ 24x 苷 12x共x ⫺ 2兲 To find the critical numbers we set f ⬘共x兲 苷 0 and obtain x 苷 0 and x 苷 3. To use the Second Derivative Test we evaluate f ⬙ at these critical numbers: f ⬙共0兲 苷 0
Since f ⬘共3兲 苷 0 and f ⬙共3兲 ⬎ 0, f 共3兲 苷 ⫺27 is a local minimum. Since f ⬙共0兲 苷 0, the Second Derivative Test gives no information about the critical number 0. But since f ⬘共x兲 ⬍ 0 for x ⬍ 0 and also for 0 ⬍ x ⬍ 3, the First Derivative Test tells us that f does not have a local maximum or minimum at 0. [In fact, the expression for f ⬘共x兲 shows that f decreases to the left of 3 and increases to the right of 3.] Since f ⬙共x兲 苷 0 when x 苷 0 or 2, we divide the real line into intervals with these numbers as endpoints and complete the following chart.
y
y=x$4˛ (0, 0)
inflection points
2
3
(2, _16)
(3, _27)
FIGURE 11
f ⬙共3兲 苷 36 ⬎ 0
Interval
f ⬙共x兲 苷 12x共x ⫺ 2兲
Concavity
(⫺⬁, 0) (0, 2) (2, ⬁)
⫹ ⫺ ⫹
upward downward upward
x
The point 共0, 0兲 is an inflection point since the curve changes from concave upward to concave downward there. Also 共2, ⫺16兲 is an inflection point since the curve changes from concave downward to concave upward there. Using the local minimum, the intervals of concavity, and the inflection points, we sketch the curve in Figure 11. NOTE The Second Derivative Test is inconclusive when f ⬙共c兲 苷 0. In other words, at such a point there might be a maximum, there might be a minimum, or there might be neither (as in Example 6). This test also fails when f ⬙共c兲 does not exist. In such cases the First Derivative Test must be used. In fact, even when both tests apply, the First Derivative Test is often the easier one to use.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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EXAMPLE 7 Sketch the graph of the function f 共x兲 苷 x 2兾3共6 ⫺ x兲1兾3. SOLUTION Calculation of the first two derivatives gives Use the differentiation rules to check these calculations.
f ⬘共x兲 苷
Try reproducing the graph in Figure 12 with a graphing calculator or computer. Some machines produce the complete graph, some produce only the portion to the right of the yaxis, and some produce only the portion between x 苷 0 and x 苷 6. For an explanation and cure, see Example 7 in Section 1.4. An equivalent expression that gives the correct graph is y 苷 共x 2 兲1兾3 ⴢ
ⱍ
6⫺x 6⫺x
ⱍⱍ
6⫺x
ⱍ
(4, 2%?# )
3 2
0
1
2
3
4
f ⬙共x兲 苷
⫺8 x 4兾3共6 ⫺ x兲5兾3
Since f ⬘共x兲 苷 0 when x 苷 4 and f ⬘共x兲 does not exist when x 苷 0 or x 苷 6, the critical numbers are 0, 4, and 6. Interval
4⫺x
x 1兾3
共6 ⫺ x兲2兾3
f ⬘共x兲
f
x⬍0 0⬍x⬍4 4⬍x⬍6 x⬎6
⫹ ⫹ ⫺ ⫺
⫺ ⫹ ⫹ ⫹
⫹ ⫹ ⫹ ⫹
⫺ ⫹ ⫺ ⫺
decreasing on (⫺⬁, 0) increasing on (0, 4) decreasing on (4, 6) decreasing on (6, ⬁)
1兾3
y 4
4⫺x x 1兾3共6 ⫺ x兲2兾3
5
7 x
To find the local extreme values we use the First Derivative Test. Since f ⬘ changes from negative to positive at 0, f 共0兲 苷 0 is a local minimum. Since f ⬘ changes from positive to negative at 4, f 共4兲 苷 2 5兾3 is a local maximum. The sign of f ⬘ does not change at 6, so there is no minimum or maximum there. (The Second Derivative Test could be used at 4 but not at 0 or 6 since f ⬙ does not exist at either of these numbers.) Looking at the expression for f ⬙共x兲 and noting that x 4兾3 艌 0 for all x, we have f ⬙共x兲 ⬍ 0 for x ⬍ 0 and for 0 ⬍ x ⬍ 6 and f ⬙共x兲 ⬎ 0 for x ⬎ 6. So f is concave downward on 共⫺⬁, 0兲 and 共0, 6兲 and concave upward on 共6, ⬁兲, and the only inflection point is 共6, 0兲. The graph is sketched in Figure 12. Note that the curve has vertical tangents at 共0, 0兲 and 共6, 0兲 because f ⬘共x兲 l ⬁ as x l 0 and as x l 6.
ⱍ
ⱍ
[email protected]?#(6x)!?#
EXAMPLE 8 Use the first and second derivatives of f 共x兲 苷 e 1兾x, together with asympFIGURE 12
totes, to sketch its graph. SOLUTION Notice that the domain of f is 兵x
ⱍ
x 苷 0其, so we check for vertical asymptotes by computing the left and right limits as x l 0. As x l 0⫹, we know that t 苷 1兾x l ⬁, so lim⫹ e 1兾x 苷 lim e t 苷 ⬁ tl⬁
xl0
and this shows that x 苷 0 is a vertical asymptote. As x l 0⫺, we have t 苷 1兾x l ⫺⬁, so lim⫺ e 1兾x 苷 lim e t 苷 0 t l ⫺⬁
xl0
TEC In Module 4.3 you can practice using information about f ⬘, f ⬙, and asymptotes to determine the shape of the graph of f .
As x l ⫾⬁, we have 1兾x l 0 and so lim e 1兾x 苷 e 0 苷 1
x l ⫾⬁
This shows that y 苷 1 is a horizontal asymptote. Now let’s compute the derivative. The Chain Rule gives f ⬘共x兲 苷 ⫺
e 1兾x x2
Since e 1兾x ⬎ 0 and x 2 ⬎ 0 for all x 苷 0, we have f ⬘共x兲 ⬍ 0 for all x 苷 0. Thus f is decreasing on 共⫺⬁, 0兲 and on 共0, ⬁兲. There is no critical number, so the function has no Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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297
HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
local maximum or minimum. The second derivative is f ⬙共x兲 苷 ⫺
x 2e 1兾x共⫺1兾x 2 兲 ⫺ e 1兾x共2x兲 e 1兾x共2x ⫹ 1兲 苷 x4 x4
Since e 1兾x ⬎ 0 and x 4 ⬎ 0, we have f ⬙共x兲 ⬎ 0 when x ⬎ ⫺12 共x 苷 0兲 and f ⬙共x兲 ⬍ 0 1 1 when x ⬍ ⫺2 . So the curve is concave downward on (⫺⬁, ⫺2 ) and concave upward 1 1 ⫺2 on (⫺2 , 0) and on 共0, ⬁兲. The inflection point is (⫺2 , e ). To sketch the graph of f we first draw the horizontal asymptote y 苷 1 (as a dashed line), together with the parts of the curve near the asymptotes in a preliminary sketch [Figure 13(a)]. These parts reflect the information concerning limits and the fact that f is decreasing on both 共⫺⬁, 0兲 and 共0, ⬁兲. Notice that we have indicated that f 共x兲 l 0 as x l 0⫺ even though f 共0兲 does not exist. In Figure 13(b) we finish the sketch by incorporating the information concerning concavity and the inflection point. In Figure 13(c) we check our work with a graphing device. y
y
y=‰ 4
inflection point y=1 0
y=1 0
x
(a) Preliminary sketch
_3
x
3 0
(b) Finished sketch
(c) Computer confirmation
FIGURE 13
Exercises
4.3
(b) How do you determine where the graph of f is concave upward or concave downward? (c) How do you locate inflection points?
1–2 Use the given graph of f to find the following.
(a) (b) (c) (d) (e)
The open intervals on which f is increasing. The open intervals on which f is decreasing. The open intervals on which f is concave upward. The open intervals on which f is concave downward. The coordinates of the points of inflection.
1.
2.
y
4. (a) State the First Derivative Test.
(b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
y
5–6 The graph of the derivative f ⬘ of a function f is shown. (a) On what intervals is f increasing or decreasing? (b) At what values of x does f have a local maximum or minimum? 1
1 0
1
x
0
5.
6.
y
x
1
3. Suppose you are given a formula for a function f .
y
y=fª(x)
y=fª(x) 0
2
4
6
x
0
2
4
6
x
(a) How do you determine where f is increasing or decreasing?
;
Graphing calculator or computer required
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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7. In each part state the xcoordinates of the inflection points
23. Suppose f ⬙ is continuous on 共⫺⬁, ⬁兲.
(a) If f ⬘共2兲 苷 0 and f ⬙共2兲 苷 ⫺5, what can you say about f ? (b) If f ⬘共6兲 苷 0 and f ⬙共6兲 苷 0, what can you say about f ?
of f . Give reasons for your answers. (a) The curve is the graph of f . (b) The curve is the graph of f ⬘. (c) The curve is the graph of f ⬙.
24–29 Sketch the graph of a function that satisfies all of the given
conditions.
y
24. Vertical asymptote x 苷 0, 0
2
4
6
f ⬘共x兲 ⬎ 0 if x ⬍ ⫺2, f ⬘共x兲 ⬍ 0 if x ⬎ ⫺2 共x 苷 0兲, f ⬙共x兲 ⬍ 0 if x ⬍ 0, f ⬙共x兲 ⬎ 0 if x ⬎ 0
x
8
25. f ⬘共0兲 苷 f ⬘共2兲 苷 f ⬘共4兲 苷 0,
f ⬘共x兲 ⬎ 0 if x ⬍ 0 or 2 ⬍ x ⬍ 4, f ⬘共x兲 ⬍ 0 if 0 ⬍ x ⬍ 2 or x ⬎ 4, f ⬙共x兲 ⬎ 0 if 1 ⬍ x ⬍ 3, f ⬙共x兲 ⬍ 0 if x ⬍ 1 or x ⬎ 3
8. The graph of the first derivative f ⬘ of a function f is shown.
(a) On what intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or minimum? Explain. (c) On what intervals is f concave upward or concave downward? Explain. (d) What are the xcoordinates of the inflection points of f ? Why? y
f ⬘共x兲 ⬍ 0 if x ⬍ 1, f ⬘共x兲 ⬎ 0 if 1 ⬍ x ⬍ 2, f ⬘共x兲 苷 ⫺1 if x ⬎ 2, f ⬙共x兲 ⬍ 0 if ⫺2 ⬍ x ⬍ 0, inflection point 共0, 1兲
ⱍ ⱍ
1
3
f ⬘共⫺2兲 苷 0,
5
7
9
x
ⱍ ⱍ
27. f ⬘共x兲 ⬎ 0 if x ⬍ 2,
y=fª(x)
0
ⱍ ⱍ
26. f ⬘共1兲 苷 f ⬘共⫺1兲 苷 0,
ⱍ
ⱍ ⱍ
28. f ⬘共x兲 ⬎ 0 if x ⬍ 2,
f ⬘共2兲 苷 0,
ⱍ ⱍ
f ⬘共x兲 ⬍ 0 if x ⬎ 2,
ⱍ
lim f ⬘共x兲 苷 ⬁,
xl2
ⱍ ⱍ
f ⬙共x兲 ⬎ 0 if x 苷 2
ⱍ ⱍ
f ⬘共x兲 ⬍ 0 if x ⬎ 2,
lim f 共x兲 苷 1,
xl⬁
f ⬙共x兲 ⬍ 0 if 0 ⬍ x ⬍ 3,
f 共⫺x兲 苷 ⫺f 共x兲, f ⬙共x兲 ⬎ 0 if x ⬎ 3
9–18
(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f . (c) Find the intervals of concavity and the inflection points. 9. f 共x兲 苷 2x 3 ⫹ 3x 2 ⫺ 36x
29. f ⬘共x兲 ⬍ 0 and f ⬙共x兲 ⬍ 0 for all x 30. Suppose f 共3兲 苷 2, f ⬘共3兲 苷 2, and f ⬘共x兲 ⬎ 0 and f ⬙共x兲 ⬍ 0 for 1
all x. (a) Sketch a possible graph for f. (b) How many solutions does the equation f 共x兲 苷 0 have? Why? (c) Is it possible that f ⬘共2兲 苷 13 ? Why?
10. f 共x兲 苷 4x 3 ⫹ 3x 2 ⫺ 6x ⫹ 1 11. f 共x兲 苷 x 4 ⫺ 2x 2 ⫹ 3 13. f 共x兲 苷 sin x ⫹ cos x, 14. f 共x兲 苷 cos x ⫺ 2 sin x, 2
12. f 共x兲 苷
x x2 ⫹ 1
0 艋 x 艋 2
31–32 The graph of the derivative f ⬘ of a continuous function f
0 艋 x 艋 2
15. f 共x兲 苷 e 2x ⫹ e⫺x
16. f 共x兲 苷 x 2 ln x
17. f 共x兲 苷 x 2 ⫺ x ⫺ ln x
18. f 共x兲 苷 x 4e⫺x
19–21 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? 19. f 共x兲 苷 1 ⫹ 3x 2 ⫺ 2x 3
20. f 共x兲 苷
x2 x⫺1
4 x 21. f 共x兲 苷 sx ⫺ s
22. (a) Find the critical numbers of f 共x兲 苷 x 4共x ⫺ 1兲3.
(b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you?
is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of x does f have a local maximum? Local minimum? (c) On what intervals is f concave upward? Concave downward? (d) State the xcoordinate(s) of the point(s) of inflection. (e) Assuming that f 共0兲 苷 0, sketch a graph of f. 31.
y
y=fª(x) 2
0
2
4
6
8 x
_2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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32.
y
HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
299
; 55–56 (a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value.
y=fª(x) 2
0
2
4
8 x
6
55. f 共x兲 苷
_2
x⫹1 sx 2 ⫹ 1
56. f 共 x兲 苷 x 2 e⫺x
; 57–58 (a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of f ⬙ to give better estimates.
33– 44
(a) (b) (c) (d)
Find the intervals of increase or decrease. Find the local maximum and minimum values. Find the intervals of concavity and the inflection points. Use the information from parts (a)–(c) to sketch the graph. Check your work with a graphing device if you have one.
57. f 共x兲 苷 cos x ⫹
cos 2x, 0 艋 x 艋 2
1 2
58. f 共x兲 苷 x 共x ⫺ 2兲 3
4
33. f 共x兲 苷 x 3 ⫺ 12x ⫹ 2
34. f 共x兲 苷 36x ⫹ 3x 2 ⫺ 2x 3
35. f 共x兲 苷 2 ⫹ 2x 2 ⫺ x 4
36. t共x兲 苷 200 ⫹ 8x 3 ⫹ x 4
59–60 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f ⬙.
37. h共x兲 苷 共x ⫹ 1兲5 ⫺ 5x ⫺ 2
38. h共x兲 苷 5x 3 ⫺ 3x 5
59. f 共x兲 苷
39. F共x兲 苷 x s6 ⫺ x
40. G共x兲 苷 5x 2兾3 ⫺ 2x 5兾3
41. C共x兲 苷 x 1兾3共x ⫹ 4兲
42. f 共x兲 苷 ln共x 4 ⫹ 27兲
43. f 共 兲 苷 2 cos ⫹ cos2, 44. S共x兲 苷 x ⫺ sin x,
0 艋 艋 2
0 艋 x 艋 4
45–52
(a) (b) (c) (d) (e)
Find the vertical and horizontal asymptotes. Find the intervals of increase or decrease. Find the local maximum and minimum values. Find the intervals of concavity and the inflection points. Use the information from parts (a)–(d) to sketch the graph of f .
45. f 共x兲 苷 1 ⫹
1 1 ⫺ 2 x x
47. f 共x兲 苷 sx ⫹ 1 ⫺ x 2
49. f 共x兲 苷 e⫺x
2
51. f 共x兲 苷 ln共1 ⫺ ln x兲
46. f 共x兲 苷
x2 ⫺ 4 x2 ⫹ 4
x4 ⫹ x3 ⫹ 1 sx 2 ⫹ x ⫹ 1
60. f 共x兲 苷
x 2 tan⫺1 x 1 ⫹ x3
61. A graph of a population of yeast cells in a new laboratory cul
ture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.
700 600 500 Number 400 of yeast cells 300 200 100
ex 48. f 共x兲 苷 1 ⫺ ex
0
50. f 共x兲 苷 x ⫺ 6 x 2 ⫺ 3 ln x 1
CAS
2
52. f 共x兲 苷 e arctan x
53. Suppose the derivative of a function f is
f ⬘共x兲 苷 共x ⫹ 1兲2共x ⫺ 3兲5共x ⫺ 6兲 4. On what interval is f increasing? 54. Use the methods of this section to sketch the curve
y 苷 x 3 ⫺ 3a 2x ⫹ 2a 3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
2
4
6
8
10 12 14 16 18
Time (in hours)
62. Let f 共t兲 be the temperature at time t where you live and sup
pose that at time t 苷 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) f ⬘共3兲 苷 2, f ⬙共3兲 苷 4 (b) f ⬘共3兲 苷 2, f ⬙共3兲 苷 ⫺4 (c) f ⬘共3兲 苷 ⫺2, f ⬙共3兲 苷 4 (d) f ⬘共3兲 苷 ⫺2, f ⬙共3兲 苷 ⫺4
63. Let K共t兲 be a measure of the knowledge you gain by studying
for a test for t hours. Which do you think is larger, K共8兲 ⫺ K共7兲 or K共3兲 ⫺ K共2兲? Is the graph of K concave upward or concave downward? Why?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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64. Coffee is being poured into the mug shown in the figure at a
constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?
69. (a) If the function f 共x兲 苷 x 3 ⫹ ax 2 ⫹ bx has the local mini
mum value ⫺ 29 s3 at x 苷 1兾s3 , what are the values of a and b? (b) Which of the tangent lines to the curve in part (a) has the smallest slope?
70. For what values of a and b is 共2, 2.5兲 an inflection point of
the curve x 2 y ⫹ ax ⫹ by 苷 0? What additional inflection points does the curve have?
71. Show that the curve y 苷 共1 ⫹ x兲兾共1 ⫹ x 2兲 has three points
of inflection and they all lie on one straight line. 72. Show that the curves y 苷 e ⫺x and y 苷 ⫺e⫺x touch the curve
y 苷 e⫺x sin x at its inflection points.
73. Show that the inflection points of the curve y 苷 x sin x lie on 65. A drug response curve describes the level of medication in
the bloodstream after a drug is administered. A surge function S共t兲 苷 At pe⫺kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A 苷 0.01, p 苷 4, k 苷 0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.
the curve y 2共x 2 ⫹ 4兲 苷 4x 2.
74–76 Assume that all of the functions are twice differentiable and the second derivatives are never 0. 74. (a) If f and t are concave upward on I , show that f ⫹ t is
concave upward on I . (b) If f is positive and concave upward on I , show that the function t共x兲 苷 关 f 共x兲兴 2 is concave upward on I . 75. (a) If f and t are positive, increasing, concave upward func
66. The family of bellshaped curves
y苷
1 2 2 e⫺共x⫺兲 兾共2 兲 s2
occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, let’s scale the function so as to remove the factor 1兾( s2 ) and let’s analyze the special case where 苷 0. So we study the function f 共x兲 苷 e⫺x
;
兾共2 2 兲
2
(a) Find the asymptote, maximum value, and inflection points of f . (b) What role does play in the shape of the curve? (c) Illustrate by graphing four members of this family on the same screen. 67. Find a cubic function f 共x兲 苷 ax 3 ⫹ bx 2 ⫹ cx ⫹ d that has a
local maximum value of 3 at x 苷 ⫺2 and a local minimum value of 0 at x 苷 1.
68. For what values of the numbers a and b does the function
f 共x兲 苷 axe have the maximum value f 共2兲 苷 1?
bx 2
tions on I , show that the product function f t is concave upward on I . (b) Show that part (a) remains true if f and t are both decreasing. (c) Suppose f is increasing and t is decreasing. Show, by giving three examples, that f t may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case? 76. Suppose f and t are both concave upward on 共⫺⬁, ⬁兲.
Under what condition on f will the composite function h共x兲 苷 f 共 t共x兲兲 be concave upward? 77. Show that tan x ⬎ x for 0 ⬍ x ⬍ 兾2. [Hint: Show that
f 共x兲 苷 tan x ⫺ x is increasing on 共0, 兾2兲.]
78. (a) Show that e x 艌 1 ⫹ x for x 艌 0.
(b) Deduce that e x 艌 1 ⫹ x ⫹ 12 x 2 for x 艌 0. (c) Use mathematical induction to prove that for x 艌 0 and any positive integer n, ex 艌 1 ⫹ x ⫹
x2 xn ⫹ ⭈⭈⭈ ⫹ 2! n!
79. Show that a cubic function (a thirddegree polynomial)
always has exactly one point of inflection. If its graph has three xintercepts x 1, x 2, and x 3, show that the xcoordinate of the inflection point is 共x 1 ⫹ x 2 ⫹ x 3 兲兾3.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.4
P共x兲 苷 x ⫹ cx ⫹ x have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases? 3
2
81. Prove that if 共c, f 共c兲兲 is a point of inflection of the graph
f 共x兲 苷 cx ⫹ increasing on 共⫺⬁, ⬁兲?
situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, t, and h whose values at 0 are all 0 and, for x 苷 0,
82. Show that if f 共x兲 苷 x 4, then f ⬙共0兲 苷 0, but 共0, 0兲 is not an
inflection point of the graph of f .
f 共x兲 苷 x 4 sin
ⱍ ⱍ
83. Show that the function t共x兲 苷 x x has an inflection point at
1 x
f 共c兲 ⬎ 0. Does f have a local maximum or minimum at c ? Does f have a point of inflection at c ?
85. Suppose f is differentiable on an interval I and f ⬘共x兲 ⬎ 0 for
all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I .
冉 冊
t共x兲 苷 x 4 2 ⫹ sin
冉
h共x兲 苷 x 4 ⫺2 ⫹ sin
84. Suppose that f is continuous and f ⬘共c兲 苷 f ⬙共c兲 苷 0, but
4.4
1 x2 ⫹ 3
87. The three cases in the First Derivative Test cover the
of f and f ⬙ exists in an open interval that contains c, then f ⬙共c兲 苷 0. [Hint: Apply the First Derivative Test and Fermat’s Theorem to the function t 苷 f ⬘.]
共0, 0兲 but t⬙共0兲 does not exist.
301
86. For what values of c is the function
; 80. For what values of c does the polynomial 4
INDETERMINATE FORMS AND L’HOSPITAL’S RULE
1 x
冊
1 x
(a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local minimum at 0, t has a local minimum, and h has a local maximum.
Indeterminate Forms and l’Hospital’s Rule Suppose we are trying to analyze the behavior of the function F共x兲 苷
ln x x⫺1
Although F is not defined when x 苷 1, we need to know how F behaves near 1. In particular, we would like to know the value of the limit lim
1
x l1
ln x x⫺1
In computing this limit we can’t apply Law 5 of limits (the limit of a quotient is the quotient of the limits, see Section 2.3) because the limit of the denominator is 0. In fact, although the limit in 1 exists, its value is not obvious because both numerator and denominator approach 0 and 00 is not defined. In general, if we have a limit of the form lim
xla
f 共x兲 t共x兲
where both f 共x兲 l 0 and t共x兲 l 0 as x l a, then this limit may or may not exist and is called an indeterminate form of type 00 . We met some limits of this type in Chapter 2. For rational functions, we can cancel common factors: lim x l1
x2 ⫺ x x共x ⫺ 1兲 x 1 苷 lim 苷 lim 苷 x l1 共x ⫹ 1兲共x ⫺ 1兲 x l1 x ⫹ 1 x2 ⫺ 1 2
We used a geometric argument to show that lim
xl0
sin x 苷1 x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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But these methods do not work for limits such as 1 , so in this section we introduce a systematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms. Another situation in which a limit is not obvious occurs when we look for a horizontal asymptote of F and need to evaluate the limit ln x x⫺1
lim
2
xl⬁
It isn’t obvious how to evaluate this limit because both numerator and denominator become large as x l ⬁. There is a struggle between numerator and denominator. If the numerator wins, the limit will be ⬁; if the denominator wins, the answer will be 0. Or there may be some compromise, in which case the answer will be some finite positive number. In general, if we have a limit of the form lim
xla
f 共x兲 t共x兲
where both f 共x兲 l ⬁ (or ⫺⬁) and t共x兲 l ⬁ (or ⫺⬁), then the limit may or may not exist and is called an indeterminate form of type ⴥ兾ⴥ. We saw in Section 2.6 that this type of limit can be evaluated for certain functions, including rational functions, by dividing numerator and denominator by the highest power of x that occurs in the denominator. For instance, 1 x ⫺1 x2 1⫺0 1 lim 苷 lim 苷 苷 2 x l ⬁ 2x ⫹ 1 xl⬁ 1 2⫹0 2 2⫹ 2 x
y
f g
0
1⫺
2
a
y
x
This method does not work for limits such as 2 , but l’Hospital’s Rule also applies to this type of indeterminate form.
y=m¡(xa)
L’Hospital’s Rule Suppose f and t are differentiable and t⬘共x兲 苷 0 on an open
interval I that contains a (except possibly at a). Suppose that y=m™(xa) 0
a
or that
FIGURE 1 Figure 1 suggests visually why l’Hospital’s Rule might be true. The first graph shows two differentiable functions f and t, each of which approaches 0 as x l a. If we were to zoom in toward the point 共a, 0兲, the graphs would start to look almost linear. But if the functions actually were linear, as in the second graph, then their ratio would be m1共x ⫺ a兲 m1 苷 m2共x ⫺ a兲 m2 which is the ratio of their derivatives. This suggests that f 共x兲 f ⬘共x兲 lim 苷 lim x l a t共x兲 x l a t⬘共x兲
lim f 共x兲 苷 0
and
lim f 共x兲 苷 ⫾⬁
and
xla
x
xla
lim t共x兲 苷 0
xla
lim t共x兲 苷 ⫾⬁
xla
(In other words, we have an indeterminate form of type 00 or ⬁兾⬁.) Then lim
xla
f 共x兲 f ⬘共x兲 苷 lim x l a t⬘共x兲 t共x兲
if the limit on the right side exists (or is ⬁ or ⫺⬁).
NOTE 1 L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. It is especially important to verify the conditions regarding the limits of f and t before using l’Hospital’s Rule.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.4 L’Hospital L’Hospital’s Rule is named after a French nobleman, the Marquis de l’Hospital (1661– 1704), but was discovered by a Swiss mathematician, John Bernoulli (1667–1748). You might sometimes see l’Hospital spelled as l’Hôpital, but he spelled his own name l’Hospital, as was common in the 17th century. See Exercise 81 for the example that the Marquis used to illustrate his rule. See the project on page 310 for further historical details.
INDETERMINATE FORMS AND L’HOSPITAL’S RULE
303
NOTE 2 L’Hospital’s Rule is also valid for onesided limits and for limits at infinity or negative infinity; that is, “ x l a ” can be replaced by any of the symbols x l a⫹, x l a⫺, x l ⬁, or x l ⫺⬁. NOTE 3 For the special case in which f 共a兲 苷 t共a兲 苷 0, f ⬘ and t⬘ are continuous, and t⬘共a兲 苷 0, it is easy to see why l’Hospital’s Rule is true. In fact, using the alternative form of the definition of a derivative, we have
f ⬘共x兲 f ⬘共a兲 lim 苷 苷 x l a t⬘共x兲 t⬘共a兲
f 共x兲 ⫺ f 共a兲 f 共x兲 ⫺ f 共a兲 x⫺a x⫺a 苷 lim x l a t共x兲 ⫺ t共a兲 t共x兲 ⫺ t共a兲 lim xla x⫺a x⫺a
lim
xla
f 共x兲 ⫺ f 共a兲 f 共x兲 苷 lim x l a t共x兲 t共x兲 ⫺ t共a兲
苷 lim xla
It is more difficult to prove the general version of l’Hospital’s Rule. See Appendix F.
v
EXAMPLE 1 Find lim
xl1
ln x . x⫺1
SOLUTION Since
lim ln x 苷 ln 1 苷 0 x l1
and
lim 共x ⫺ 1兲 苷 0 x l1
we can apply l’Hospital’s Rule: d 共ln x兲 ln x dx 1兾x lim 苷 lim 苷 lim xl1 x ⫺ 1 xl1 d xl1 1 共x ⫺ 1兲 dx
 Notice that when using l’Hospital’s Rule we differentiate the numerator and denominator separately. We do not use the Quotient Rule.
苷 lim
xl1
v The graph of the function of Example 2 is shown in Figure 2. We have noticed previously that exponential functions grow far more rapidly than power functions, so the result of Example 2 is not unexpected. See also Exercise 71. 20
FIGURE 2
xl⬁
ex . x2
SOLUTION We have lim x l ⬁ e x 苷 ⬁ and lim x l ⬁ x 2 苷 ⬁, so l’Hospital’s Rule gives
d 共e x 兲 ex dx ex lim 2 苷 lim 苷 lim xl⬁ x xl⬁ d x l ⬁ 2x 共x 2 兲 dx Since e x l ⬁ and 2x l ⬁ as x l ⬁, the limit on the right side is also indeterminate, but a second application of l’Hospital’s Rule gives
y= ´ ≈ 0
EXAMPLE 2 Calculate lim
1 苷1 x
10
lim
xl⬁
ex ex ex 苷 lim 苷 lim 苷⬁ x l ⬁ 2x xl⬁ 2 x2
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v
The graph of the function of Example 3 is shown in Figure 3. We have discussed previously the slow growth of logarithms, so it isn’t surprising that this ratio approaches 0 as x l ⬁. See also Exercise 72.
EXAMPLE 3 Calculate lim
xl⬁
ln x . 3 x s
3 SOLUTION Since ln x l ⬁ and s x l ⬁ as x l ⬁, l’Hospital’s Rule applies:
lim
2
xl⬁
y= ln x Œ„ x 0
10,000
ln x 1兾x 苷 lim 1 ⫺2兾3 3 xl⬁ x sx 3
Notice that the limit on the right side is now indeterminate of type 00 . But instead of applying l’Hospital’s Rule a second time as we did in Example 2, we simplify the expression and see that a second application is unnecessary: ln x 1兾x 3 苷 lim 1 ⫺2兾3 苷 lim 3 苷 0 3 xl⬁ x x l ⬁ sx x s 3
lim
xl⬁
_1
FIGURE 3
EXAMPLE 4 Find lim
xl0
tan x ⫺ x . (See Exercise 44 in Section 2.2.) x3
SOLUTION Noting that both tan x ⫺ x l 0 and x 3 l 0 as x l 0, we use l’Hospital’s
Rule: lim
xl0
tan x ⫺ x sec2x ⫺ 1 苷 lim xl0 x3 3x 2
Since the limit on the right side is still indeterminate of type 00 , we apply l’Hospital’s Rule again: sec2x ⫺ 1 2 sec2x tan x lim 苷 lim 2 xl0 xl0 3x 6x
The graph in Figure 4 gives visual confirmation of the result of Example 4. If we were to zoom in too far, however, we would get an inaccurate graph because tan x is close to x when x is small. See Exercise 44(d) in Section 2.2.
Because lim x l 0 sec2 x 苷 1, we simplify the calculation by writing lim
xl0
1
2 sec2x tan x 1 tan x 1 tan x 苷 lim sec2 x ⴢ lim 苷 lim xl0 6x 3 xl0 x 3 xl0 x
We can evaluate this last limit either by using l’Hospital’s Rule a third time or by writing tan x as 共sin x兲兾共cos x兲 and making use of our knowledge of trigonometric limits. Putting together all the steps, we get y= _1
tan x x ˛
lim
xl0
tan x ⫺ x sec 2 x ⫺ 1 2 sec 2 x tan x 苷 lim 苷 lim xl0 xl0 x3 3x 2 6x
1 0
苷
FIGURE 4
v
EXAMPLE 5 Find lim
xl⫺
1 tan x 1 sec 2 x 1 lim 苷 lim 苷 3 xl0 x 3 xl0 1 3
sin x . 1 ⫺ cos x
SOLUTION If we blindly attempted to use l’Hospital’s Rule, we would get

lim
xl⫺
sin x cos x 苷 lim ⫺ 苷 ⫺⬁ x l 1 ⫺ cos x sin x
This is wrong! Although the numerator sin x l 0 as x l ⫺, notice that the denominator 共1 ⫺ cos x兲 does not approach 0, so l’Hospital’s Rule can’t be applied here.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INDETERMINATE FORMS AND L’HOSPITAL’S RULE
305
The required limit is, in fact, easy to find because the function is continuous at and the denominator is nonzero there: lim
xl⫺
sin x sin 0 苷 苷 苷0 1 ⫺ cos x 1 ⫺ cos 1 ⫺ 共⫺1兲
Example 5 shows what can go wrong if you use l’Hospital’s Rule without thinking. Other limits can be found using l’Hospital’s Rule but are more easily found by other methods. (See Examples 3 and 5 in Section 2.3, Example 3 in Section 2.6, and the discussion at the beginning of this section.) So when evaluating any limit, you should consider other methods before using l’Hospital’s Rule.
Indeterminate Products If lim x l a f 共x兲 苷 0 and lim x l a t共x兲 苷 ⬁ (or ⫺⬁), then it isn’t clear what the value of lim x l a 关 f 共x兲 t共x兲兴, if any, will be. There is a struggle between f and t. If f wins, the answer will be 0; if t wins, the answer will be ⬁ (or ⫺⬁). Or there may be a compromise where the answer is a finite nonzero number. This kind of limit is called an indeterminate form of type 0 ⴢ ⴥ. We can deal with it by writing the product ft as a quotient: ft 苷
f 1兾t
or
ft 苷
t 1兾f
This converts the given limit into an indeterminate form of type 00 or ⬁兾⬁ so that we can use l’Hospital’s Rule.
v Figure 5 shows the graph of the function in Example 6. Notice that the function is undefined at x 苷 0; the graph approaches the origin but never quite reaches it. y
EXAMPLE 6 Evaluate lim x ln x . x l0⫹
SOLUTION The given limit is indeterminate because, as x l 0 ⫹, the first factor 共x兲
approaches 0 while the second factor 共ln x兲 approaches ⫺⬁. Writing x 苷 1兾共1兾x兲, we have 1兾x l ⬁ as x l 0 ⫹, so l’Hospital’s Rule gives lim x ln x 苷 lim⫹
x l 0⫹
y=x ln x
xl0
ln x 1兾x 苷 lim⫹ 苷 lim⫹ 共⫺x兲 苷 0 x l 0 ⫺1兾x 2 xl0 1兾x
NOTE In solving Example 6 another possible option would have been to write
lim x ln x 苷 lim⫹
x l 0⫹
0
FIGURE 5
1
x
xl0
x 1兾ln x
This gives an indeterminate form of the type 0兾0, but if we apply l’Hospital’s Rule we get a more complicated expression than the one we started with. In general, when we rewrite an indeterminate product, we try to choose the option that leads to the simpler limit.
Indeterminate Differences If lim x l a f 共x兲 苷 ⬁ and lim x l a t共x兲 苷 ⬁, then the limit lim 关 f 共x兲 ⫺ t共x兲兴
xla
is called an indeterminate form of type ⴥ ⴚ ⴥ. Again there is a contest between f and t. Will the answer be ⬁ ( f wins) or will it be ⫺⬁ ( t wins) or will they compromise on a finite
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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number? To find out, we try to convert the difference into a quotient (for instance, by using a common denominator, or rationalization, or factoring out a common factor) so that we have an indeterminate form of type 00 or ⬁兾⬁. EXAMPLE 7 Compute
lim 共sec x ⫺ tan x兲.
x l共兾2兲⫺
SOLUTION First notice that sec x l ⬁ and tan x l ⬁ as x l 共兾2兲⫺, so the limit is
indeterminate. Here we use a common denominator: lim 共sec x ⫺ tan x兲 苷
x l共兾2兲⫺
苷
lim
x l共兾2兲⫺
lim
x l共兾2兲⫺
冉
1 sin x ⫺ cos x cos x
冊
1 ⫺ sin x ⫺cos x 苷 lim ⫺ 苷0 x l共兾2兲 cos x ⫺sin x
Note that the use of l’Hospital’s Rule is justified because 1 ⫺ sin x l 0 and cos x l 0 as x l 共兾2兲⫺.
Indeterminate Powers Several indeterminate forms arise from the limit lim 关 f 共x兲兴 t共x兲
xla
1. lim f 共x兲 苷 0
and
2. lim f 共x兲 苷 ⬁
and
3. lim f 共x兲 苷 1
and
xla
xla
xla
Although forms of the type 0 0, ⬁0, and 1⬁ are indeterminate, the form 0 ⬁ is not indeterminate. (See Exercise 84.)
lim t共x兲 苷 0
type 0 0
lim t共x兲 苷 0
type ⬁ 0
lim t共x兲 苷 ⫾⬁
type 1⬁
xla
xla
xla
Each of these three cases can be treated either by taking the natural logarithm: let
y 苷 关 f 共x兲兴 t共x兲,
then ln y 苷 t共x兲 ln f 共x兲
or by writing the function as an exponential: 关 f 共x兲兴 t共x兲 苷 e t共x兲 ln f 共x兲 (Recall that both of these methods were used in differentiating such functions.) In either method we are led to the indeterminate product t共x兲 ln f 共x兲, which is of type 0 ⴢ ⬁. EXAMPLE 8 Calculate lim⫹ 共1 ⫹ sin 4x兲cot x. xl0
SOLUTION First notice that as x l 0 ⫹, we have 1 ⫹ sin 4x l 1 and cot x l ⬁, so the
given limit is indeterminate. Let y 苷 共1 ⫹ sin 4x兲cot x Then
ln y 苷 ln关共1 ⫹ sin 4x兲cot x 兴 苷 cot x ln共1 ⫹ sin 4x兲
so l’Hospital’s Rule gives 4 cos 4x ln共1 ⫹ sin 4x兲 1 ⫹ sin 4x lim ln y 苷 lim⫹ 苷 lim⫹ 苷4 x l 0⫹ xl0 xl0 tan x sec2x Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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307
INDETERMINATE FORMS AND L’HOSPITAL’S RULE
So far we have computed the limit of ln y, but what we want is the limit of y. To find this we use the fact that y 苷 e ln y : lim 共1 ⫹ sin 4x兲cot x 苷 lim⫹ y 苷 lim⫹ e ln y 苷 e 4
x l 0⫹
v
The graph of the function y 苷 x x, x ⬎ 0, is shown in Figure 6. Notice that although 0 0 is not defined, the values of the function approach 1 as x l 0⫹. This confirms the result of Example 9.
xl0
xl0
EXAMPLE 9 Find lim⫹ x x. xl0
SOLUTION Notice that this limit is indeterminate since 0 x 苷 0 for any x ⬎ 0 but x 0 苷 1
for any x 苷 0. We could proceed as in Example 8 or by writing the function as an exponential:
2
x x 苷 共e ln x 兲 x 苷 e x ln x In Example 6 we used l’Hospital’s Rule to show that lim x ln x 苷 0
_1
x l 0⫹
2
0
4.4
lim x x 苷 lim⫹ e x ln x 苷 e 0 苷 1
Therefore
FIGURE 6
x l 0⫹
xl0
Exercises 5–6 Use the graphs of f and t and their tangent lines at 共2, 0兲 to
1– 4 Given that
lim f 共x兲 苷 0
lim t共x兲 苷 0
x la
x la
lim p共x兲 苷 ⬁ x la
lim h共x兲 苷 1 x la
lim q共 x兲 苷 ⬁
find lim x l2
5. y
f 共x兲 . t共x兲
y=1.8(x2)
x la
which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. 1. (a) lim
xla
(d) lim
xla
f 共x兲 t共x兲 p共x兲 f 共x兲
(b) lim
xla
(e) lim
xla
2. (a) lim 关 f 共x兲p共x兲兴
f 共x兲 p共x兲
(c) lim
xla
7. lim x l1
(b) lim 关 p共x兲 ⫺ q共x兲兴 xla
x2 ⫺ 1 x2 ⫺ x
4. (a) lim 关 f 共x兲兴 t共x兲
(b) lim 关 f 共x兲兴 p共x兲
(c) lim 关h共x兲兴 p共x兲
(d) lim 关 p共x兲兴 f 共x兲
(e) lim 关 p共x兲兴 q共x兲
(f) lim sp共x兲
xla
xla
xla
11.
lim
x l共兾2兲⫹
13. lim tl0
Graphing calculator or computer required
g
xl2
x ⫺ 2x ⫹ 1 x l1 x3 ⫺ 1
q共x兲
xla
x
0
8. lim
y=2x
x2 ⫹ x ⫺ 6 x⫺2
2
9. lim
xla
;
4
3
(c) lim 关 p共x兲 ⫹ q共x兲兴
xla
2 x
y=5 (x2)
xla
xla
f
7–66 Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
(c) lim 关 p共x兲q共x兲兴
xla
y=1.5(x2)
2 0
p共x兲 q共x兲
xla
3. (a) lim 关 f 共x兲 ⫺ p共x兲兴
y
g
h共x兲 p共x兲
(b) lim 关h共x兲p共x兲兴
xla
6.
f
cos x 1 ⫺ sin x
e 2t ⫺ 1 sin t
10. lim
x l 1兾2
6x 2 ⫹ 5x ⫺ 4 4x 2 ⫹ 16x ⫺ 9
12. lim
sin 4 x tan 5x
14. lim
x2 1 ⫺ cos x
xl 0
xl0
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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1 ⫺ sin 15. lim l 兾2 1 ⫹ cos 2
1 ⫺ sin 16. lim l 兾2 csc
59. lim⫹ x 1兾共1⫺x兲
60. lim x 共ln 2兲兾共1 ⫹ ln x兲
61. lim x 1兾x
62. lim 共e x ⫹ x兲1兾x
ln x 17. lim x l ⬁ sx
x ⫹ x2 18. lim x l ⬁ 1 ⫺ 2x 2
63. lim⫹ 共4x ⫹ 1兲
19. lim⫹ xl0
ln x x
20. lim
ln sx x2
22. lim
8 ⫺5 t
xl⬁
t ⫺1 t5 ⫺ 1 8
21. lim tl1
s1 ⫹ 2x ⫺ s1 ⫺ 4x 23. lim xl0 x
27. lim
tanh x tan x
28. lim
x ⫺ sin x x ⫺ tan x
67. lim
xl⬁
1⫹
2 x
x
30. lim
共ln x兲 x
31. lim
x3x 3x ⫺ 1
32. lim
cos mx ⫺ cos nx x2
70. f 共x兲 苷 2x sin x,
33. lim
x ⫹ sin x x ⫹ cos x
34. lim
x tan 共4x兲
71. Prove that
xl0
xl0
1 ⫺ x ⫹ ln x 35. lim x l 1 1 ⫹ cos x x a ⫺ ax ⫹ a ⫺ 1 共x ⫺ 1兲2
39. lim
cos x ⫺ 1 ⫹ x x4
xl1
1 2
xl0
69. f 共x兲 苷 e x ⫺ 1,
e x ⫺ e⫺x ⫺ 2 x x ⫺ sin x
38. lim
xl0
2
40. lim⫹ x la
cos x ln共x ⫺ a兲 ln共e x ⫺ e a 兲
41. lim x sin共兾x兲
42. lim sx e⫺x兾2
43. lim cot 2x sin 6x
44. lim⫹ sin x ln x
xl⬁
xl⬁
xl0
45. lim x e
47. lim⫹ ln x tan共 x兾2兲 x l1
49. lim
xl1
冉 冉
51. lim⫹ x l0
冊 冊
x 1 ⫺ x⫺1 ln x 1 1 ⫺ x x e ⫺1
5x ⫺ 4x 3x ⫺ 2x
t共x兲 苷 x 3 ⫹ 4x t共x兲 苷 sec x ⫺ 1
xl⬁
ex 苷⬁ xn
for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x. 72. Prove that
lim
xl⬁
ln x 苷0 xp
for any number p ⬎ 0. This shows that the logarithmic function approaches ⬁ more slowly than any power of x.
73. lim
xl⬁
48.
2x⫹1
limit? Evaluate the limit using another method.
46. lim x tan共1兾x兲
xl⬁
冊
73–74 What happens if you try to use l’Hospital’s Rule to find the
xl0
3 ⫺x 2
xl0
lim
xx ⫺ 1 36. lim⫹ x l0 ln x ⫹ x ⫺ 1
37. lim
68. lim
⫺1
xl0
2x ⫺ 3 2x ⫹ 5
f ⬘共x兲兾t⬘共x兲 near x 苷 0 to see that these ratios have the same limit as x l 0. Also, calculate the exact value of the limit.
sin x x
xl0
冉
; 69–70 Illustrate l’Hospital’s Rule by graphing both f 共x兲兾t共x兲 and
2
xl⬁
xl⬁
冉 冊
29. lim
xl0
66. lim
l’Hospital’s Rule to find the exact value.
sinh x ⫺ x x3
⫺1
2
; 67–68 Use a graph to estimate the value of the limit. Then use
26. lim
xl0
xl1
xl0
e ⫺1⫺x x2
xl0
64. lim 共2 ⫺ x兲tan共 x兾2兲
t
25. lim
xl0
xl⬁
cot x
x l0
e u兾10 24. lim 3 ul ⬁ u
x
xl0
xl⬁
65. lim⫹ 共cos x兲1兾x
t
tl0
xl⬁
x l1
xl⬁
x sx ⫹ 1
74.
2
lim
x l共 兾2兲⫺
sec x tan x
lim cos x sec 5x
x l共 兾2兲⫺
x ; 75. Investigate the family of curves f 共x兲 苷 e ⫺ cx. In particular,
50. lim 共csc x ⫺ cot x兲 xl0
冉
52. lim cot x ⫺ xl0
1 x
53. lim 共x ⫺ ln x兲 xl⬁
冊
find the limits as x l ⫾⬁ and determine the values of c for which f has an absolute minimum. What happens to the minimum points as c increases?
76. If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is v苷
54. lim⫹ 关ln共x 7 ⫺ 1兲 ⫺ ln共x 5 ⫺ 1兲兴
mt 共1 ⫺ e ⫺ct兾m 兲 c
x l1
55. lim x sx
56. lim⫹ 共tan 2 x兲 x
57. lim 共1 ⫺ 2x兲1兾x
58. lim
x l0⫹
xl0
xl0
xl⬁
冉 冊 1⫹
a x
bx
where t is the acceleration due to gravity and c is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; c is the proportionality constant.) (a) Calculate lim t l ⬁ v. What is the meaning of this limit?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.4
INDETERMINATE FORMS AND L’HOSPITAL’S RULE
(b) For fixed t, use l’Hospital’s Rule to calculate lim c l 0⫹ v. What can you conclude about the velocity of a falling object in a vacuum?
309
P A(¨ )
77. If an initial amount A0 of money is invested at an interest rate
r compounded n times a year, the value of the investment after t years is
B(¨ )
冉 冊
A 苷 A0 1 ⫹
r n
¨
nt
If we let n l ⬁, we refer to the continuous compounding of interest. Use l’Hospital’s Rule to show that if interest is compounded continuously, then the amount after t years is
O
冋
冉 冊册
83. Evaluate lim x ⫺ x 2 ln xl⬁
of resistance is proportional to the square of the velocity, then the distance the ball travels in time t is
冑
.
84. Suppose f is a positive function. If lim xla f 共x兲 苷 0 and
lim 关 f 共x兲兴 t共x兲 苷 0
78. If a metal ball with mass m is projected in water and the force
m ln cosh c
1⫹x x
lim xla t共x兲 苷 ⬁, show that
A 苷 A0 e rt
s共t兲 苷
R
Q
tc mt
xla
This shows that 0 ⬁ is not an indeterminate form. 85. If f ⬘ is continuous, f 共2兲 苷 0, and f ⬘共2兲 苷 7, evaluate
lim
xl0
where c is a positive constant. Find lim c l 0⫹ s共t兲.
f 共2 ⫹ 3x兲 ⫹ f 共2 ⫹ 5x兲 x
86. For what values of a and b is the following equation true? 79. If an electrostatic field E acts on a liquid or a gaseous polar
dielectric, the net dipole moment P per unit volume is e E ⫹ e ⫺E 1 P共E兲 苷 E ⫺E ⫺ e ⫺e E
lim
xl0
lim
hl0
the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is
冉冊 冉冊
v 苷 ⫺c
r R
2
ln
R lr
r l0
81. The first appearance in print of l’Hospital’s Rule was in the
book Analyse des Infiniment Petits published by the Marquis de l’Hospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function 3 aax s2a 3x ⫺ x 4 ⫺ a s y苷 4 3 a ⫺ sax
as x approaches a, where a ⬎ 0. (At that time it was common to write aa instead of a 2.) Solve this problem. 82. The figure shows a sector of a circle with central angle . Let
A共 兲 be the area of the segment between the chord PR and the arc PR. Let B共 兲 be the area of the triangle PQR. Find lim l 0⫹ 〈共 兲兾〉 共 兲.
冊
苷0
f 共x ⫹ h兲 ⫺ f 共x ⫺ h兲 苷 f ⬘共x兲 2h
Explain the meaning of this equation with the aid of a diagram. 88. If f ⬙ is continuous, show that
r R
where c is a positive constant. Find the following limits and interpret your answers. (a) lim⫹ v (b) lim⫹ v
sin 2x b ⫹a⫹ 2 x3 x
87. If f ⬘ is continuous, use l’Hospital’s Rule to show that
Show that lim E l 0⫹ P共E兲 苷 0. 80. A metal cable has radius r and is covered by insulation, so that
冉
lim
hl0
f 共x ⫹ h兲 ⫺ 2 f 共x兲 ⫹ f 共x ⫺ h兲 苷 f ⬙共x兲 h2
再
89. Let
e⫺1兾x f 共x兲 苷 0
2
if x 苷 0 if x 苷 0
(a) Use the definition of derivative to compute f ⬘共0兲. (b) Show that f has derivatives of all orders that are defined on ⺢. [Hint: First show by induction that there is a polynomial pn共x兲 and a nonnegative integer k n such that f 共n兲共x兲 苷 pn共x兲f 共x兲兾x k n for x 苷 0.]
; 90. Let f 共x兲 苷
再ⱍ ⱍ x 1
x
if x 苷 0 if x 苷 0
(a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several times toward the point 共0, 1兲 on the graph of f . (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
WRITING PROJECT
THE ORIGINS OF L’HOSPITAL’S RULE
Thomas Fisher Rare Book Library
L’Hospital’s Rule was first published in 1696 in the Marquis de l’Hospital’s calculus textbook Analyse des Infiniment Petits, but the rule was discovered in 1694 by the Swiss mathematician John (Johann) Bernoulli. The explanation is that these two mathematicians had entered into a curious business arrangement whereby the Marquis de l’Hospital bought the rights to Bernoulli’s mathematical discoveries. The details, including a translation of l’Hospital’s letter to Bernoulli proposing the arrangement, can be found in the book by Eves [1]. Write a report on the historical and mathematical origins of l’Hospital’s Rule. Start by providing brief biographical details of both men (the dictionary edited by Gillispie [2] is a good source) and outline the business deal between them. Then give l’Hospital’s statement of his rule, which is found in Struik’s sourcebook [4] and more briefly in the book of Katz [3]. Notice that l’Hospital and Bernoulli formulated the rule geometrically and gave the answer in terms of differentials. Compare their statement with the version of l’Hospital’s Rule given in Section 4.4 and show that the two statements are essentially the same. 1. Howard Eves, In Mathematical Circles (Volume 2: Quadrants III and IV) (Boston: Prindle,
www.stewartcalculus.com The Internet is another source of information for this project. Click on History of Mathematics for a list of reliable websites.
Weber and Schmidt, 1969), pp. 20–22. 2. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974). See the article on Johann Bernoulli by E. A. Fellmann and J. O. Fleckenstein in Volume II and the article on the Marquis de l’Hospital by Abraham Robinson in Volume VIII. 3. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993), p. 484. 4. D. J. Struik, ed., A Sourcebook in Mathematics, 1200–1800 (Princeton, NJ: Princeton University Press, 1969), pp. 315–316.
Summary of Curve Sketching
4.5
30
y=8˛21≈+18x+2
_2
4 _10
FIGURE 1 8
y=8˛21≈+18x+2 0
2 6
FIGURE 2
So far we have been concerned with some particular aspects of curve sketching: domain, range, and symmetry in Chapter 1; limits, continuity, and asymptotes in Chapter 2; derivatives and tangents in Chapters 2 and 3; and extreme values, intervals of increase and decrease, concavity, points of inflection, and l’Hospital’s Rule in this chapter. It is now time to put all of this information together to sketch graphs that reveal the important features of functions. You might ask: Why don’t we just use a graphing calculator or computer to graph a curve? Why do we need to use calculus? It’s true that modern technology is capable of producing very accurate graphs. But even the best graphing devices have to be used intelligently. We saw in Section 1.4 that it is extremely important to choose an appropriate viewing rectangle to avoid getting a misleading graph. (See especially Examples 1, 3, 4, and 5 in that section.) The use of calculus enables us to discover the most interesting aspects of graphs and in many cases to calculate maximum and minimum points and inflection points exactly instead of approximately. For instance, Figure 1 shows the graph of f 共x兲 苷 8x 3 ⫺ 21x 2 ⫹ 18x ⫹ 2. At first glance it seems reasonable: It has the same shape as cubic curves like y 苷 x 3, and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x 苷 0.75 and a minimum when x 苷 1. Indeed, if we zoom in to this portion of the graph, we see that behavior exhibited in Figure 2. Without calculus, we could easily have overlooked it. In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the following
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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311
SUMMARY OF CURVE SKETCHING
information. We don’t assume that you have a graphing device, but if you do have one you should use it as a check on your work.
Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a curve y 苷 f 共x兲 by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It’s often useful to start by determining the domain D of f , that is, the set of values of x for which f 共x兲 is defined. B. Intercepts The yintercept is f 共0兲 and this tells us where the curve intersects the yaxis. To find the xintercepts, we set y 苷 0 and solve for x. (You can omit this step if the equation is difficult to solve.)
y
C. Symmetry
0
x
(a) Even function: reflectional symmetry y
x
0
(b) Odd function: rotational symmetry FIGURE 3
( i) If f 共⫺x兲 苷 f 共x兲 for all x in D, that is, the equation of the curve is unchanged when x is replaced by ⫺x, then f is an even function and the curve is symmetric about the yaxis. This means that our work is cut in half. If we know what the curve looks like for x 艌 0, then we need only reflect about the yaxis to obtain the complete curve [see Figure 3(a)]. Here are some examples: y 苷 x 2, y 苷 x 4, y 苷 x , and y 苷 cos x. ( ii) If f 共⫺x兲 苷 ⫺f 共x兲 for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x 艌 0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y 苷 x, y 苷 x 3, y 苷 x 5, and y 苷 sin x. ( iii) If f 共x ⫹ p兲 苷 f 共x兲 for all x in D, where p is a positive constant, then f is called a periodic function and the smallest such number p is called the period. For instance, y 苷 sin x has period 2 and y 苷 tan x has period . If we know what the graph looks like in an interval of length p, then we can use translation to sketch the entire graph (see Figure 4).
ⱍ ⱍ
y
FIGURE 4
Periodic function: translational symmetry
ap
0
a
a+p
a+2p
x
D. Asymptotes
( i) Horizontal Asymptotes. Recall from Section 2.6 that if either lim x l ⬁ f 共x兲 苷 L or lim x l⫺ ⬁ f 共x兲 苷 L, then the line y 苷 L is a horizontal asymptote of the curve y 苷 f 共x兲. If it turns out that lim x l ⬁ f 共x兲 苷 ⬁ (or ⫺⬁), then we do not have an asymptote to the right, but that is still useful information for sketching the curve. ( ii) Vertical Asymptotes. Recall from Section 2.2 that the line x 苷 a is a vertical asymptote if at least one of the following statements is true:
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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(For rational functions you can locate the vertical asymptotes by equating the denominator to 0 after canceling any common factors. But for other functions this method does not apply.) Furthermore, in sketching the curve it is very useful to know exactly which of the statements in 1 is true. If f 共a兲 is not defined but a is an endpoint of the domain of f , then you should compute lim x l a⫺ f 共x兲 or lim x l a⫹ f 共x兲, whether or not this limit is infinite. ( iii) Slant Asymptotes. These are discussed at the end of this section. E. Intervals of Increase or Decrease Use the I/D Test. Compute f ⬘共x兲 and find the intervals on which f ⬘共x兲 is positive ( f is increasing) and the intervals on which f ⬘共x兲 is negative ( f is decreasing). F. Local Maximum and Minimum Values Find the critical numbers of f [the numbers c where f ⬘共c兲 苷 0 or f ⬘共c兲 does not exist]. Then use the First Derivative Test. If f ⬘ changes from positive to negative at a critical number c, then f 共c兲 is a local maximum. If f ⬘ changes from negative to positive at c, then f 共c兲 is a local minimum. Although it is usually preferable to use the First Derivative Test, you can use the Second Derivative Test if f ⬘共c兲 苷 0 and f ⬙共c兲 苷 0. Then f ⬙共c兲 ⬎ 0 implies that f 共c兲 is a local minimum, whereas f ⬙共c兲 ⬍ 0 implies that f 共c兲 is a local maximum. G. Concavity and Points of Inflection Compute f ⬙共x兲 and use the Concavity Test. The curve is concave upward where f ⬙共x兲 ⬎ 0 and concave downward where f ⬙共x兲 ⬍ 0. Inflection points occur where the direction of concavity changes. H. Sketch the Curve Using the information in items A–G, draw the graph. Sketch the asymptotes as dashed lines. Plot the intercepts, maximum and minimum points, and inflection points. Then make the curve pass through these points, rising and falling according to E, with concavity according to G, and approaching the asymptotes. If additional accuracy is desired near any point, you can compute the value of the derivative there. The tangent indicates the direction in which the curve proceeds.
v
EXAMPLE 1 Use the guidelines to sketch the curve y 苷
A. The domain is
兵x
ⱍx
2
2x 2 . x2 ⫺ 1
ⱍ x 苷 ⫾1其 苷 共⫺⬁, ⫺1兲 傼 共⫺1, 1兲 傼 共1, ⬁兲
⫺ 1 苷 0其 苷 兵x
B. The x and yintercepts are both 0. C. Since f 共⫺x兲 苷 f 共x兲, the function f is even. The curve is symmetric about the yaxis. y
lim
D.
x l⫾⬁
y=2 0
x=_1
x
x=1
FIGURE 5
Preliminary sketch We have shown the curve approaching its horizontal asymptote from above in Figure 5. This is confirmed by the intervals of increase and decrease.
2x 2 2 苷 lim 苷2 x l⫾⬁ 1 ⫺ 1兾x 2 x ⫺1 2
Therefore the line y 苷 2 is a horizontal asymptote. Since the denominator is 0 when x 苷 ⫾1, we compute the following limits: lim⫹
2x 2 苷⬁ x2 ⫺ 1
lim⫹
2x 2 苷 ⫺⬁ x2 ⫺ 1
x l1
x l⫺1
lim⫺
2x 2 苷 ⫺⬁ x2 ⫺ 1
lim⫺
2x 2 苷⬁ x2 ⫺ 1
x l1
x l⫺1
Therefore the lines x 苷 1 and x 苷 ⫺1 are vertical asymptotes. This information about limits and asymptotes enables us to draw the preliminary sketch in Figure 5, showing the parts of the curve near the asymptotes.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.5
f ⬘共x兲 苷
E.
313
4x共x 2 ⫺ 1兲 ⫺ 2x 2 ⴢ 2x ⫺4x 苷 2 2 2 共x ⫺ 1兲 共x ⫺ 1兲2
Since f ⬘共x兲 ⬎ 0 when x ⬍ 0 共x 苷 ⫺1兲 and f ⬘共x兲 ⬍ 0 when x ⬎ 0 共x 苷 1兲, f is increasing on 共⫺⬁, ⫺1兲 and 共⫺1, 0兲 and decreasing on 共0, 1兲 and 共1, ⬁兲. F. The only critical number is x 苷 0. Since f ⬘ changes from positive to negative at 0, f 共0兲 苷 0 is a local maximum by the First Derivative Test.
y
f ⬙共x兲 苷
G. y=2
⫺4共x 2 ⫺ 1兲2 ⫹ 4x ⴢ 2共x 2 ⫺ 1兲2x 12x 2 ⫹ 4 苷 2 2 4 共x ⫺ 1兲 共x ⫺ 1兲3
Since 12x 2 ⫹ 4 ⬎ 0 for all x, we have
0 x
x=_1
SUMMARY OF CURVE SKETCHING
Finished sketch of y=
x2 ⫺ 1 ⬎ 0
&?
ⱍ ⱍ
ⱍxⱍ ⬎ 1
and f ⬙共x兲 ⬍ 0 &? x ⬍ 1. Thus the curve is concave upward on the intervals 共⫺⬁, ⫺1兲 and 共1, ⬁兲 and concave downward on 共⫺1, 1兲. It has no point of inflection since 1 and ⫺1 are not in the domain of f . H. Using the information in E–G, we finish the sketch in Figure 6.
x=1
FIGURE 6
f ⬙共x兲 ⬎ 0 &?
2≈ ≈1
x2 . sx ⫹ 1 x ⬎ ⫺1其 苷 共⫺1, ⬁兲
EXAMPLE 2 Sketch the graph of f 共x兲 苷 A. B. C. D.
ⱍ
Domain 苷 兵x x ⫹ 1 ⬎ 0其 苷 兵x The x and yintercepts are both 0. Symmetry: None Since
ⱍ
lim
xl⬁
x2 苷⬁ sx ⫹ 1
there is no horizontal asymptote. Since sx ⫹ 1 l 0 as x l ⫺1⫹ and f 共x兲 is always positive, we have x2 lim⫹ 苷⬁ x l⫺1 sx ⫹ 1 and so the line x 苷 ⫺1 is a vertical asymptote. f ⬘共x兲 苷
E.
We see that f ⬘共x兲 苷 0 when x 苷 0 (notice that ⫺43 is not in the domain of f ), so the only critical number is 0. Since f ⬘共x兲 ⬍ 0 when ⫺1 ⬍ x ⬍ 0 and f ⬘共x兲 ⬎ 0 when x ⬎ 0, f is decreasing on 共⫺1, 0兲 and increasing on 共0, ⬁兲. F. Since f ⬘共0兲 苷 0 and f ⬘ changes from negative to positive at 0, f 共0兲 苷 0 is a local (and absolute) minimum by the First Derivative Test.
y
G.
y=
x=_1 FIGURE 7
0
2xsx ⫹ 1 ⫺ x 2 ⴢ 1兾(2sx ⫹ 1 ) x共3x ⫹ 4兲 苷 x⫹1 2共x ⫹ 1兲3兾2
≈ œ„„„„ x+1 x
f ⬙共x兲 苷
2共x ⫹ 1兲3兾2共6x ⫹ 4兲 ⫺ 共3x 2 ⫹ 4x兲3共x ⫹ 1兲1兾2 3x 2 ⫹ 8x ⫹ 8 苷 3 4共x ⫹ 1兲 4共x ⫹ 1兲5兾2
Note that the denominator is always positive. The numerator is the quadratic 3x 2 ⫹ 8x ⫹ 8, which is always positive because its discriminant is b 2 ⫺ 4ac 苷 ⫺32, which is negative, and the coefficient of x 2 is positive. Thus f ⬙共x兲 ⬎ 0 for all x in the domain of f , which means that f is concave upward on 共⫺1, ⬁兲 and there is no point of inflection. H. The curve is sketched in Figure 7.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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v A. B. C. D.
EXAMPLE 3 Sketch the graph of f 共x兲 苷 xe x.
The domain is ⺢. The x and yintercepts are both 0. Symmetry: None Because both x and e x become large as x l , we have lim x l xe x 苷 . As x l , however, e x l 0 and so we have an indeterminate product that requires the use of l’Hospital’s Rule: lim xe x 苷 lim
x l
x l
x 1 苷 lim 苷 lim 共e x 兲 苷 0 x l ex x l ex
Thus the xaxis is a horizontal asymptote. y
y=x´
Since e x is always positive, we see that f 共x兲 0 when x 1 0, and f 共x兲 0 when x 1 0. So f is increasing on 共1, 兲 and decreasing on 共, 1兲. F. Because f 共1兲 苷 0 and f changes from negative to positive at x 苷 1, f 共1兲 苷 e1 is a local (and absolute) minimum.
1 _2
_1
FIGURE 8
f 共x兲 苷 共x 1兲e x e x 苷 共x 2兲e x
G. x
(_1, _1/e)
f 共x兲 苷 xe x e x 苷 共x 1兲e x
E.
Since f 共x兲 0 if x 2 and f 共x兲 0 if x 2, f is concave upward on 共2, 兲 and concave downward on 共, 2兲. The inflection point is 共2, 2e2 兲. H. We use this information to sketch the curve in Figure 8. EXAMPLE 4 Sketch the graph of f 共x兲 苷
cos x . 2 sin x
A. The domain is ⺢. 1 B. The y intercept is f 共0兲 苷 2 . The x intercepts occur when cos x 苷 0, that is,
x 苷 共2n 1兲兾2, where n is an integer.
C. f is neither even nor odd, but f 共x 2兲 苷 f 共x兲 for all x and so f is periodic and
has period 2. Thus, in what follows, we need to consider only 0 x 2 and then extend the curve by translation in part H. D. Asymptotes: None E.
f 共x兲 苷
共2 sin x兲共sin x兲 cos x 共cos x兲 2 sin x 1 苷 共2 sin x兲 2 共2 sin x兲 2
Thus f 共x兲 0 when 2 sin x 1 0 &? sin x 12 &? 7兾6 x 11兾6. So f is increasing on 共7兾6, 11兾6兲 and decreasing on 共0, 7兾6兲 and 共11兾6, 2兲. F. From part E and the First Derivative Test, we see that the local minimum value is f 共7兾6兲 苷 1兾s3 and the local maximum value is f 共11兾6兲 苷 1兾s3 . G. If we use the Quotient Rule again and simplify, we get f 共x兲 苷
2 cos x 共1 sin x兲 共2 sin x兲 3
Because 共2 sin x兲 3 0 and 1 sin x 0 for all x , we know that f 共x兲 0 when cos x 0, that is, 兾2 x 3兾2. So f is concave upward on 共兾2, 3兾2兲 and concave downward on 共0, 兾2兲 and 共3兾2, 2兲. The inflection points are 共兾2, 0兲 and 共3兾2, 0兲.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.5
SUMMARY OF CURVE SKETCHING
315
H. The graph of the function restricted to 0 x 2 is shown in Figure 9. Then we
extend it, using periodicity, to the complete graph in Figure 10. y
”
1 2
11π 1 6 , œ„3
π
π 2
y
’
3π 2
1 2
2π x
_π
π
2π
3π
x
1  ’ ” 7π 6 , œ„3
FIGURE 9
FIGURE 10
EXAMPLE 5 Sketch the graph of y 苷 ln共4 x 2 兲. A. The domain is
兵x
ⱍ 4x
2
0其 苷 兵x
ⱍx
2
4其 苷 兵x
ⱍ ⱍ x ⱍ 2其 苷 共2, 2兲
B. The yintercept is f 共0兲 苷 ln 4. To find the xintercept we set
y 苷 ln共4 x 2 兲 苷 0 We know that ln 1 苷 0, so we have 4 x 2 苷 1 ? x 2 苷 3 and therefore the xintercepts are s3 . C. Since f 共x兲 苷 f 共x兲, f is even and the curve is symmetric about the yaxis. D. We look for vertical asymptotes at the endpoints of the domain. Since 4 x 2 l 0 as x l 2 and also as x l 2, we have lim ln共4 x 2 兲 苷
x l2
lim ln共4 x 2 兲 苷
x l2
Thus the lines x 苷 2 and x 苷 2 are vertical asymptotes. f 共x兲 苷
E. y (0, ln 4)
x=_2
x=2 0 {_ œ„3, 0}
x
Since f 共x兲 0 when 2 x 0 and f 共x兲 0 when 0 x 2, f is ncreasing i on 共2, 0兲 and decreasing on 共0, 2兲. F. The only critical number is x 苷 0. Since f changes from positive to negative at 0, f 共0兲 苷 ln 4 is a local maximum by the First Derivative Test.
{œ„3, 0}
G.
FIGURE 11 y=ln(4 ≈)
2x 4 x2
f 共x兲 苷
共4 x 2 兲共2兲 2x共2x兲 8 2x 2 苷 共4 x 2 兲2 共4 x 2 兲2
Since f 共x兲 0 for all x, the curve is concave downward on 共2, 2兲 and has no inflection point. H. Using this information, we sketch the curve in Figure 11.
Slant Asymptotes Some curves have asymptotes that are oblique, that is, neither horizontal nor vertical. If lim 关 f 共x兲 共mx b兲兴 苷 0
xl
then the line y 苷 mx b is called a slant asymptote because the vertical distance
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
between the curve y 苷 f 共x兲 and the line y 苷 mx b approaches 0, as in Figure 12. (A similar situation exists if we let x l .) For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following example.
y
y=ƒ ƒ(mx+b) y=mx+b
0
v
x
A. B. C. D.
FIGURE 12
EXAMPLE 6 Sketch the graph of f 共x兲 苷
x3 . x 1 2
The domain is ⺢ 苷 共, 兲. The x and yintercepts are both 0. Since f 共x兲 苷 f 共x兲, f is odd and its graph is symmetric about the origin. Since x 2 1 is never 0, there is no vertical asymptote. Since f 共x兲 l as x l and f 共x兲 l as x l , there is no horizontal asymptote. But long division gives f 共x兲 苷
x3 x 苷x 2 x 1 x 1 2
f 共x兲 x 苷
x 苷 x2 1
1 x 1 1 2 x
l 0 as
x l
So the line y 苷 x is a slant asymptote. f 共x兲 苷
E.
3x 2共x 2 1兲 x 3 ⴢ 2x x 2共x 2 3兲 苷 2 2 共x 1兲 共x 2 1兲2
Since f 共x兲 0 for all x (except 0), f is increasing on 共, 兲.
F. Although f 共0兲 苷 0, f does not change sign at 0, so there is no local maximum or
minimum. G.
y
y=
˛ ≈+1
f 共x兲 苷
共4x 3 6x兲共x 2 1兲2 共x 4 3x 2 兲 ⴢ 2共x 2 1兲2x 2x共3 x 2 兲 苷 共x 2 1兲4 共x 2 1兲3
Since f 共x兲 苷 0 when x 苷 0 or x 苷 s3 , we set up the following chart: Interval
”œ„3,
3œ„ 3 ’ 4
0
”_œ„3,
3œ„ 3 _ 4 ’
x
inflection points y=x FIGURE 13
x s3
x
3 x2
共x 2 1兲3
f 共x兲
f
CU on (, s3 )
CD on (s3 , 0)
0 x s3
CU on (0, s3 )
x s3
CD on (s3 , )
s3 x 0
The points of inflection are (s3 , 34 s3 ), 共0, 0兲, and (s3 , 34 s3 ). H. The graph of f is sketched in Figure 13.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.5
1–54 Use the guidelines of this section to sketch the curve. 1. y 苷 x 3 12x 2 36x
2. y 苷 2 3x 2 x 3
3. y 苷 x 4 4x
4. y 苷 x 4 8x 2 8
5. y 苷 x共x 4兲3
6. y 苷 x 5 5x
7. y 苷 5 x 5 3 x 3 16x
8. y 苷 共4 x 2 兲 5
1
8
x x1
10. y 苷
1 x 9
m苷
x 4 x 2 2x
18. y 苷
E 苷 sm 02 c 4 h 2 c 2兾 2
x x3 1
21. y 苷 共x 3兲sx
22. y 苷 2sx x
23. y 苷 sx 2 x 2
24. y 苷 sx 2 x x
p共t兲 苷
25. y 苷
x sx 1
26. y 苷 x s2 x 2
27. y 苷
s1 x 2 x
28. y 苷
30. y 苷 x 5兾3 5x 2兾3
31. y 苷 sx 1
32. y 苷 sx 1
33. y 苷 sin x
34. y 苷 x cos x
35. y 苷 x tan x,
兾2 x 兾2
37. y 苷 2 x sin x,
0 x 3
1
38. y 苷 sec x tan x, 39. y 苷
sin x 1 cos x x
where a, b, and K are positive constants and b a. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes? walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve y苷
sin x 2 cos x
44. y 苷 ex sin x,
e
0 x 2
45. y 苷 x ln x
46. y 苷 e
47. y 苷 共1 e x 兲2
48. y 苷 e x兾x 2
49. y 苷 ln共sin x兲
50. y 苷 ln共x 2 3x 2兲
51. y 苷 xe1兾x
52. y 苷
2x
ln x x2
W WL 3 WL 2 2 x4 x x 24EI 12EI 24EI
where E and I are positive constants. (E is Young’s modulus of elasticity and I is the moment of inertia of a crosssection of the beam.) Sketch the graph of the deflection curve.
42. y 苷 共1 x兲e x
兲
C共t兲 苷 K共eat ebt 兲
59. The figure shows a beam of length L embedded in concrete
0 x 兾2 40. y 苷
41. y 苷 arctan共e x 兲 43. y 苷 1兾共1 e
3
兾2 x 兾2
36. y 苷 2x tan x,
where p共t兲 is the proportion of the population that knows the rumor at time t and a and k are positive constants. (a) When will half the population have heard the rumor? (b) When is the rate of spread of the rumor greatest? (c) Sketch the graph of p. the bloodstream is
29. y 苷 x 3x 1兾3
3
1 1 aekt
58. A model for the concentration at time t of a drug injected into
x sx 2 1
3
where m 0 is the rest mass of the particle, is its wave length, and h is Planck’s constant. Sketch the graph of E as a function of . What does the graph say about the energy? 57. A model for the spread of a rumor is given by the equation
x3 20. y 苷 x2
2
m0 s1 v 2兾c 2
56. In the theory of relativity, the energy of a particle is
x2 x 9 2
x2 19. y 苷 2 x 3
2
x1 x1
where m 0 is the rest mass of the particle, m is the mass when the particle moves with speed v relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v.
1 1 16. y 苷 1 2 x x
x1 x2
3
54. y 苷 tan1
55. In the theory of relativity, the mass of a particle is
x 12. y 苷 2 x 9 14. y 苷
2
x 15. y 苷 2 x 9 17. y 苷
冉 冊
53. y 苷 e 3x e2x
2
x x2 11. y 苷 2 3x x 2 13. y 苷
317
Exercises
4.5
9. y 苷
SUMMARY OF CURVE SKETCHING
x
y
W
0 L
1. Homework Hints available at stewartcalculus.com Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
60. Coulomb’s Law states that the force of attraction between two
charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge 1 at a position x between them. It follows from Coulomb’s Law that the net force acting on the middle particle is k k F共x兲 苷 2 x 共x 2兲2
0x2
67. y 苷
x3 4 x2
68. y 苷
69. y 苷 1 2 x ex
x3 共x 1兲2
70. y 苷 1 x e 1x兾3
1
71. Show that the curve y 苷 x tan1x has two slant asymptotes:
y 苷 x 兾2 and y 苷 x 兾2. Use this fact to help sketch the curve.
72. Show that the curve y 苷 sx 2 4x has two slant asymptotes:
where k is a positive constant. Sketch the graph of the net force function. What does the graph say about the force? +1
_1
+1
0
x
2
y 苷 x 2 and y 苷 x 2. Use this fact to help sketch the curve.
73. Show that the lines y 苷 共b兾a兲x and y 苷 共b兾a兲x are slant
asymptotes of the hyperbola 共x 2兾a 2 兲 共 y 2兾b 2 兲 苷 1.
x
74. Let f 共x兲 苷 共x 3 1兲兾x. Show that 61–64 Find an equation of the slant asymptote. Do not sketch the
curve. 61. y 苷
x2 1 x1
62. y 苷
2x 3 x 2 x 3 x 2 2x
63. y 苷
4x 3 2x 2 5 2x 2 x 3
64. y 苷
5x 4 x 2 x x3 x2 2
65–70 Use the guidelines of this section to sketch the curve. In
guideline D find an equation of the slant asymptote. 2
65. y 苷
4.6
lim 关 f 共x兲 x 2 兴 苷 0
x l
x x1
66. y 苷
1 5x 2x x2
2
This shows that the graph of f approaches the graph of y 苷 x 2, and we say that the curve y 苷 f 共x兲 is asymptotic to the parabola y 苷 x 2. Use this fact to help sketch the graph of f . 75. Discuss the asymptotic behavior of f 共x兲 苷 共x 4 1兲兾x in the
same manner as in Exercise 74. Then use your results to help sketch the graph of f . 76. Use the asymptotic behavior of f 共x兲 苷 cos x 1兾x 2 to sketch
its graph without going through the curvesketching procedure of this section.
Graphing with Calculus and Calculators
If you have not already read Section 1.4, you should do so now. In particular, it explains how to avoid some of the pitfalls of graphing devices by choosing appropriate viewing rectangles.
The method we used to sketch curves in the preceding section was a culmination of much of our study of differential calculus. The graph was the final object that we produced. In this section our point of view is completely different. Here we start with a graph produced by a graphing calculator or computer and then we refine it. We use calculus to make sure that we reveal all the important aspects of the curve. And with the use of graphing devices we can tackle curves that would be far too complicated to consider without technology. The theme is the interaction between calculus and calculators. EXAMPLE 1 Graph the polynomial f 共x兲 苷 2x 6 3x 5 3x 3 2x 2. Use the graphs of f
and f to estimate all maximum and minimum points and intervals of concavity.
SOLUTION If we specify a domain but not a range, many graphing devices will deduce a
suitable range from the values computed. Figure 1 shows the plot from one such device if we specify that 5 x 5. Although this viewing rectangle is useful for showing that the asymptotic behavior (or end behavior) is the same as for y 苷 2x 6, it is obviously hiding some finer detail. So we change to the viewing rectangle 关3, 2兴 by 关50, 100兴 shown in Figure 2.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.6
GRAPHING WITH CALCULUS AND CALCULATORS
41,000
319
100 y=ƒ y=ƒ _3
_5
2
5 _1000
_50
FIGURE 1
FIGURE 2
From this graph it appears that there is an absolute minimum value of about 15.33 when x ⬇ 1.62 (by using the cursor) and f is decreasing on 共, 1.62兲 and increasing on 共1.62, 兲. Also there appears to be a horizontal tangent at the origin and inflection points when x 苷 0 and when x is somewhere between 2 and 1. Now let’s try to confirm these impressions using calculus. We differentiate and get f 共x兲 苷 12x 5 15x 4 9x 2 4x f 共x兲 苷 60x 4 60x 3 18x 4 When we graph f in Figure 3 we see that f 共x兲 changes from negative to positive when x ⬇ 1.62; this confirms (by the First Derivative Test) the minimum value that we found earlier. But, perhaps to our surprise, we also notice that f 共x兲 changes from positive to negative when x 苷 0 and from negative to positive when x ⬇ 0.35. This means that f has a local maximum at 0 and a local minimum when x ⬇ 0.35, but these were hidden in Figure 2. Indeed, if we now zoom in toward the origin in Figure 4, we see what we missed before: a local maximum value of 0 when x 苷 0 and a local minimum value of about 0.1 when x ⬇ 0.35. 20
1 y=ƒ
y=fª(x) _1 _3
2 _5
FIGURE 3
10 _3
2 y=f ·(x)
_30
1
_1
FIGURE 4
What about concavity and inflection points? From Figures 2 and 4 there appear to be inflection points when x is a little to the left of 1 and when x is a little to the right of 0. But it’s difficult to determine inflection points from the graph of f , so we graph the second derivative f in Figure 5. We see that f changes from positive to negative when x ⬇ 1.23 and from negative to positive when x ⬇ 0.19. So, correct to two decimal places, f is concave upward on 共, 1.23兲 and 共0.19, 兲 and concave downward on 共1.23, 0.19兲. The inflection points are 共1.23, 10.18兲 and 共0.19, 0.05兲. We have discovered that no single graph reveals all the important features of this polynomial. But Figures 2 and 4, when taken together, do provide an accurate picture.
FIGURE 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
v
EXAMPLE 2 Draw the graph of the function
f 共x兲 苷
x 2 7x 3 x2
in a viewing rectangle that contains all the important features of the function. Estimate the maximum and minimum values and the intervals of concavity. Then use calculus to find these quantities exactly. SOLUTION Figure 6, produced by a computer with automatic scaling, is a disaster. Some
graphing calculators use 关10, 10兴 by 关10, 10兴 as the default viewing rectangle, so let’s try it. We get the graph shown in Figure 7; it’s a major improvement.
3 10!*
10
10 y=ƒ
y=ƒ
_10
y=ƒ
y=1
10 _20
_5
20
5 _5
_10
FIGURE 7
FIGURE 6
FIGURE 8
The yaxis appears to be a vertical asymptote and indeed it is because lim
xl0
x 2 7x 3 苷 x2
Figure 7 also allows us to estimate the xintercepts: about 0.5 and 6.5. The exact values are obtained by using the quadratic formula to solve the equation x 2 7x 3 苷 0; we get x 苷 (7 s37 )兾2. To get a better look at horizontal asymptotes, we change to the viewing rectangle 关20, 20兴 by 关5, 10兴 in Figure 8. It appears that y 苷 1 is the horizontal asymptote and this is easily confirmed: lim
x l
2
_3
0
y=ƒ
冊
苷1
To estimate the minimum value we zoom in to the viewing rectangle 关3, 0兴 by 关4, 2兴 in Figure 9. The cursor indicates that the absolute minimum value is about 3.1 when x ⬇ 0.9, and we see that the function decreases on 共, 0.9兲 and 共0, 兲 and increases on 共0.9, 0兲. The exact values are obtained by differentiating: f 共x兲 苷
_4
FIGURE 9
冉
x 2 7x 3 7 3 苷 lim 1 2 x l x2 x x
7 6 7x 6 3 苷 x2 x x3
This shows that f 共x兲 0 when 67 x 0 and f 共x兲 0 when x 67 and when x 0. The exact minimum value is f ( 67 ) 苷 37 12 ⬇ 3.08. Figure 9 also shows that an inflection point occurs somewhere between x 苷 1 and x 苷 2. We could estimate it much more accurately using the graph of the second derivative, but in this case it’s just as easy to find exact values. Since f 共x兲 苷
14 18 2(7x 9兲 4 苷 x3 x x4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.6
GRAPHING WITH CALCULUS AND CALCULATORS
321
we see that f 共x兲 0 when x 97 共x 苷 0兲. So f is concave upward on (97 , 0) and 共0, 兲 and concave downward on (, 97 ). The inflection point is (97 , 71 27 ). The analysis using the first two derivatives shows that Figure 8 displays all the major aspects of the curve.
v
EXAMPLE 3 Graph the function f 共x兲 苷
x 2共x 1兲3 . 共x 2兲2共x 4兲4
SOLUTION Drawing on our experience with a rational function in Example 2, let’s start
10
y=ƒ _10
10
by graphing f in the viewing rectangle 关10, 10兴 by 关10, 10兴. From Figure 10 we have the feeling that we are going to have to zoom in to see some finer detail and also zoom out to see the larger picture. But, as a guide to intelligent zooming, let’s first take a close look at the expression for f 共x兲. Because of the factors 共x 2兲2 and 共x 4兲4 in the denominator, we expect x 苷 2 and x 苷 4 to be the vertical asymptotes. Indeed
_10
lim x l2
FIGURE 10
x 2共x 1兲3 苷 共x 2兲2共x 4兲4
and
lim
xl4
x 2共x 1兲3 苷 共x 2兲2共x 4兲4
To find the horizontal asymptotes, we divide numerator and denominator by x 6 : x 2 共x 1兲3 ⴢ x 共x 1兲 x3 x3 苷 苷 共x 2兲2共x 4兲4 共x 2兲2 共x 4兲4 ⴢ x2 x4 2
y
_1
1
2
3
4
3
冉 冊 冉 冊冉 冊 1 1 1 x x
1
2
1
4 x
4
This shows that f 共x兲 l 0 as x l , so the xaxis is a horizontal asymptote. It is also very useful to consider the behavior of the graph near the xintercepts using an analysis like that in Example 12 in Section 2.6. Since x 2 is positive, f 共x兲 does not change sign at 0 and so its graph doesn’t cross the xaxis at 0. But, because of the factor 共x 1兲3, the graph does cross the xaxis at 1 and has a horizontal tangent there. Putting all this information together, but without using derivatives, we see that the curve has to look something like the one in Figure 11. Now that we know what to look for, we zoom in (several times) to produce the graphs in Figures 12 and 13 and zoom out (several times) to get Figure 14.
x
FIGURE 11
0.05
0.0001
500 y=ƒ
y=ƒ _100
2 x
3
1
_1.5
0.5
y=ƒ _1 _0.05
FIGURE 12
_0.0001
FIGURE 13
10 _10
FIGURE 14
We can read from these graphs that the absolute minimum is about 0.02 and occurs when x ⬇ 20. There is also a local maximum ⬇0.00002 when x ⬇ 0.3 and a local minimum ⬇211 when x ⬇ 2.5. These graphs also show three inflection points near 35, 5, and 1 and two between 1 and 0. To estimate the inflection points closely we would need to graph f , but to compute f by hand is an unreasonable chore. If you have a computer algebra system, then it’s easy to do (see Exercise 15). Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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We have seen that, for this particular function, three graphs (Figures 12, 13, and 14) are necessary to convey all the useful information. The only way to display all these features of the function on a single graph is to draw it by hand. Despite the exaggerations and distortions, Figure 11 does manage to summarize the essential nature of the function. The family of functions f 共x兲 苷 sin共x sin cx兲 where c is a constant, occurs in applications to frequency modulation (FM) synthesis. A sine wave is modulated by a wave with a different frequency 共sin cx兲. The case where c 苷 2 is studied in Example 4. Exercise 27 explores another special case.
EXAMPLE 4 Graph the function f 共x兲 苷 sin共x sin 2x兲. For 0 x , estimate all maximum and minimum values, intervals of increase and decrease, and inflection points.
ⱍ
ⱍ
SOLUTION We first note that f is periodic with period 2. Also, f is odd and f 共x兲 1
for all x. So the choice of a viewing rectangle is not a problem for this function: We start with 关0, 兴 by 关1.1, 1.1兴. (See Figure 15.) It appears that there are three local maximum values and two local minimum values in that window. To confirm this and locate them more accurately, we calculate that
1.1
f 共x兲 苷 cos共x sin 2x兲 ⴢ 共1 2 cos 2x兲
π
0
and graph both f and f in Figure 16. Using zoomin and the First Derivative Test, we find the following approximate values: Intervals of increase: 共0, 0.6兲, 共1.0, 1.6兲, 共2.1, 2.5兲 共0.6, 1.0兲, 共1.6, 2.1兲, 共2.5, 兲
_1.1
Intervals of decrease:
FIGURE 15
Local maximum values: f 共0.6兲 ⬇ 1, f 共1.6兲 ⬇ 1, f 共2.5兲 ⬇ 1 Local minimum values:
1.2
f 共1.0兲 ⬇ 0.94, f 共2.1兲 ⬇ 0.94
The second derivative is
y=ƒ
f 共x兲 苷 共1 2 cos 2x兲2 sin共x sin 2x兲 4 sin 2x cos共x sin 2x兲 0
π
Graphing both f and f in Figure 17, we obtain the following approximate values:
y=f ª(x)
共0.8, 1.3兲, 共1.8, 2.3兲
Concave upward on:
_1.2
Concave downward on: 共0, 0.8兲, 共1.3, 1.8兲, 共2.3, 兲
FIGURE 16
共0, 0兲, 共0.8, 0.97兲, 共1.3, 0.97兲, 共1.8, 0.97兲, 共2.3, 0.97兲
Inflection points: 1.2
1.2 f
0
π
_2π
2π
f· _1.2
FIGURE 17
_1.2
FIGURE 18
Having checked that Figure 15 does indeed represent f accurately for 0 x , we can state that the extended graph in Figure 18 represents f accurately for 2 x 2.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.6
GRAPHING WITH CALCULUS AND CALCULATORS
323
Our final example is concerned with families of functions. As discussed in Section 1.4, this means that the functions in the family are related to each other by a formula that contains one or more arbitrary constants. Each value of the constant gives rise to a member of the family and the idea is to see how the graph of the function changes as the constant changes.
v
EXAMPLE 5 How does the graph of f 共x兲 苷 1兾共x 2 2x c兲 vary as c varies?
SOLUTION The graphs in Figures 19 and 20 (the special cases c 苷 2 and c 苷 2) show
2
two very differentlooking curves. Before drawing any more graphs, let’s see what members of this family have in common. Since _5
4 y=
lim
1 ≈+2x+2
x l
for any value of c, they all have the xaxis as a horizontal asymptote. A vertical asymptote will occur when x 2 2x c 苷 0. Solving this quadratic equation, we get x 苷 1 s1 c . When c 1, there is no vertical asymptote (as in Figure 19). When c 苷 1, the graph has a single vertical asymptote x 苷 1 because
_2
FIGURE 19
c=2 y= 2
1 苷0 x 2x c 2
1 ≈+2x2
_5
lim
x l1
4
1 1 苷 lim 苷 x l1 共x 1兲2 x 2 2x 1
When c 1, there are two vertical asymptotes: x 苷 1 s1 c (as in Figure 20). Now we compute the derivative: f 共x兲 苷
_2
FIGURE 20
c=_2
TEC See an animation of Figure 21 in Visual 4.6.
c=_1 FIGURE 21
2x 2 共x 2 2x c兲2
This shows that f 共x兲 苷 0 when x 苷 1 ( if c 苷 1), f 共x兲 0 when x 1, and f 共x兲 0 when x 1. For c 1, this means that f increases on 共, 1兲 and decreases on 共1, 兲. For c 1, there is an absolute maximum value f 共1兲 苷 1兾共c 1兲. For c 1, f 共1兲 苷 1兾共c 1兲 is a local maximum value and the intervals of increase and decrease are interrupted at the vertical asymptotes. Figure 21 is a “slide show” displaying five members of the family, all graphed in the viewing rectangle 关5, 4兴 by 关2, 2兴. As predicted, c 苷 1 is the value at which a transition takes place from two vertical asymptotes to one, and then to none. As c increases from 1, we see that the maximum point becomes lower; this is explained by the fact that 1兾共c 1兲 l 0 as c l . As c decreases from 1, the vertical asymptotes become more widely separated because the distance between them is 2s1 c , which becomes large as c l . Again, the maximum point approaches the xaxis because 1兾共c 1兲 l 0 as c l .
c=0
c=1
c=2
c=3
The family of functions ƒ=1/(≈+2x+c)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
There is clearly no inflection point when c 艋 1. For c ⬎ 1 we calculate that f ⬙共x兲 苷
2共3x 2 ⫹ 6x ⫹ 4 ⫺ c兲 共x 2 ⫹ 2x ⫹ c兲3
and deduce that inflection points occur when x 苷 ⫺1 ⫾ s3共c ⫺ 1兲兾3. So the inflection points become more spread out as c increases and this seems plausible from the last two parts of Figure 21.
4.6
; Exercises
1–8 Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ⬘ and f ⬙ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
all the maximum and minimum values are as given in the example. Calculate f ⬙ and use it to estimate the intervals of concavity and inflection points. CAS
1. f 共x兲 苷 4x 4 ⫺ 32x 3 ⫹ 89x 2 ⫺ 95x ⫹ 29
graphs to estimate the intervals of increase and decrease and concavity of f.
2. f 共x兲 苷 x ⫺ 15x ⫹ 75x ⫺ 125x ⫺ x 6
5
4
3
3. f 共x兲 苷 x 6 ⫺ 10x 5 ⫺ 400x 4 ⫹ 2500x 3
x ⫺1 40x 3 ⫹ x ⫹ 1 2
4. f 共x兲 苷
6. f 共x兲 苷 6 sin x ⫺ x 2,
5. f 共x兲 苷
16. If f is the function of Exercise 14, find f ⬘ and f ⬙ and use their
CAS
17–22 Use a computer algebra system to graph f and to find f ⬘
and f ⬙. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f .
x x3 ⫹ x2 ⫹ 1
⫺5 艋 x 艋 3
7. f 共x兲 苷 6 sin x ⫹ cot x,
17. f 共x兲 苷
⫺ 艋 x 艋
8. f 共x兲 苷 e x ⫺ 0.186x 4
x 3 ⫹ 5x 2 ⫹ 1 x ⫹ x3 ⫺ x2 ⫹ 2 4
19. f 共x兲 苷 sx ⫹ 5 sin x ,
9–10 Produce graphs of f that reveal all the important aspects of
20. f 共x兲 苷 共x 2 ⫺ 1兲 e arctan x
the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.
21. f 共x兲 苷
9. f 共x兲 苷 1 ⫹
1 8 1 ⫹ 2 ⫹ 3 x x x
10. f 共x兲 苷
2 ⫻ 10 8 1 8 ⫺ x x4 CAS
11–12
(a) Graph the function. (b) Use l’Hospital’s Rule to explain the behavior as x l 0. (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. 11. f 共x兲 苷 x 2 ln x
13–14 Sketch the graph by hand using asymptotes and intercepts,
but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values. 13. f 共x兲 苷
共x ⫹ 4兲共x ⫺ 3兲2 x 4共x ⫺ 1兲
14. f 共x兲 苷
共2 x ⫹ 3兲 2 共x ⫺ 2兲 5 x 3 共x ⫺ 5兲 2
x 2兾3 1 ⫹ x ⫹ x4
22. f 共x兲 苷
1 1 ⫹ e tan x
x 艋 20
23–24 Graph the function using as many viewing rectangles as you need to depict the true nature of the function. 23. f 共x兲 苷
CAS
12. f 共x兲 苷 xe 1兾x
1 ⫺ e 1兾x 1 ⫹ e 1兾x
18. f 共x兲 苷
1 ⫺ cos共x 4 兲 x8
ⱍ
24. f 共x兲 苷 e x ⫹ ln x ⫺ 4
ⱍ
25–26
(a) Graph the function. (b) Explain the shape of the graph by computing the limit as x l 0⫹ or as x l ⬁. (c) Estimate the maximum and minimum values and then use calculus to find the exact values. (d) Use a graph of f ⬙ to estimate the xcoordinates of the inflection points. 25. f 共x兲 苷 x 1兾x
26. f 共x兲 苷 共sin x兲sin x
27. In Example 4 we considered a member of the family of funcCAS
15. If f is the function considered in Example 3, use a computer
algebra system to calculate f ⬘ and then graph it to confirm that
;
Graphing calculator or computer required
tions f 共x兲 苷 sin共x ⫹ sin cx兲 that occur in FM synthesis. Here we investigate the function with c 苷 3. Start by graphing f in
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.7
the viewing rectangle 关0, 兴 by 关⫺1.2, 1.2兴. How many local maximum points do you see? The graph has more than are visible to the naked eye. To discover the hidden maximum and minimum points you will need to examine the graph of f ⬘ very carefully. In fact, it helps to look at the graph of f ⬙ at the same time. Find all the maximum and minimum values and inflection points. Then graph f in the viewing rectangle 关⫺2, 2兴 by 关⫺1.2, 1.2兴 and comment on symmetry. 28–35 Describe how the graph of f varies as c varies. Graph
several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes. 28. f 共x兲 苷 x 3 ⫹ cx 29. f 共x兲 苷 sx 4 ⫹ cx 2 ⫺x
30. f 共x兲 苷 x sc 2 ⫺ x 2
31. f 共x兲 苷 e ⫹ ce
32. f 共x兲 苷 ln共x ⫹ c兲
cx 33. f 共x兲 苷 1 ⫹ c 2x 2
34. f 共x兲 苷 x 2 ⫹ ce⫺x
x
2
35. f 共x兲 苷 cx ⫹ sin x
OPTIMIZATION PROBLEMS
325
discover graphically what happens as b increases. Then use calculus to prove what you have discovered. 37. Investigate the family of curves given by f 共x兲 苷 xe⫺cx,
where c is a real number. Start by computing the limits as x l ⫾⬁. Identify any transitional values of c where the basic shape changes. What happens to the maximum or minimum points and inflection points as c changes? Illustrate by graphing several members of the family. 38. Investigate the family of curves given by the equation
f 共x兲 苷 x 4 ⫹ cx 2 ⫹ x. Start by determining the transitional value of c at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of c at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered. 39. (a) Investigate the family of polynomials given by the equa
tion f 共x兲 苷 cx 4 ⫺ 2 x 2 ⫹ 1. For what values of c does the curve have minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the parabola y 苷 1 ⫺ x 2. Illustrate by graphing this parabola and several members of the family. 40. (a) Investigate the family of polynomials given by the equa
36. The family of functions f 共t兲 苷 C共e⫺at ⫺ e⫺bt 兲, where a, b,
and C are positive numbers and b ⬎ a, has been used to model the concentration of a drug injected into the bloodstream at time t 苷 0. Graph several members of this family. What do they have in common? For fixed values of C and a,
4.7
tion f 共x兲 苷 2x 3 ⫹ cx 2 ⫹ 2 x. For what values of c does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve y 苷 x ⫺ x 3. Illustrate by graphing this curve and several members of the family.
Optimization Problems
PS
The methods we have learned in this chapter for finding extreme values have practical applications in many areas of life. A businessperson wants to minimize costs and maximize profits. A traveler wants to minimize transportation time. Fermat’s Principle in optics states that light follows the path that takes the least time. In this section we solve such problems as maximizing areas, volumes, and profits and minimizing distances, times, and costs. In solving such practical problems the greatest challenge is often to convert the word problem into a mathematical optimization problem by setting up the function that is to be maximized or minimized. Let’s recall the problemsolving principles discussed on page 75 and adapt them to this situation: Steps in Solving Optimization Problems 1. Understand the Problem The first step is to read the problem carefully until it is
clearly understood. Ask yourself: What is the unknown? What are the given quantities? What are the given conditions? 2. Draw a Diagram In most problems it is useful to draw a diagram and identify the given and required quantities on the diagram. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
3. Introduce Notation Assign a symbol to the quantity that is to be maximized or
minimized (let’s call it Q for now). Also select symbols 共a, b, c, . . . , x, y兲 for other unknown quantities and label the diagram with these symbols. It may help to use initials as suggestive symbols—for example, A for area, h for height, t for time. 4. Express Q in terms of some of the other symbols from Step 3. 5. If Q has been expressed as a function of more than one variable in Step 4, use the given information to find relationships ( in the form of equations) among these variables. Then use these equations to eliminate all but one of the variables in the expression for Q. Thus Q will be expressed as a function of one variable x, say, Q 苷 f 共x兲. Write the domain of this function. 6. Use the methods of Sections 4.1 and 4.3 to find the absolute maximum or minimum value of f . In particular, if the domain of f is a closed interval, then the Closed Interval Method in Section 4.1 can be used. EXAMPLE 1 A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? PS Understand the problem
SOLUTION In order to get a feeling for what is happening in this problem, let’s experiment
PS Analogy: Try special cases
with some special cases. Figure 1 (not to scale) shows three possible ways of laying out the 2400 ft of fencing.
PS Draw diagrams
400
1000 2200
700
100
1000
700
1000
100
Area=100 · 2200=220,000
[email protected] Area=700 · 1000=700,000
[email protected] Area=1000 · 400=400,000
[email protected] FIGURE 1
We see that when we try shallow, wide fields or deep, narrow fields, we get relatively small areas. It seems plausible that there is some intermediate configuration that produces the largest area. Figure 2 illustrates the general case. We wish to maximize the area A of the rectangle. Let x and y be the depth and width of the rectangle ( in feet). Then we express A in terms of x and y: A 苷 xy
PS Introduce notation
y x
A
x
We want to express A as a function of just one variable, so we eliminate y by expressing it in terms of x. To do this we use the given information that the total length of the fencing is 2400 ft. Thus 2x ⫹ y 苷 2400
FIGURE 2
From this equation we have y 苷 2400 ⫺ 2x, which gives A 苷 x共2400 ⫺ 2x兲 苷 2400x ⫺ 2x 2 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.7
OPTIMIZATION PROBLEMS
327
Note that x 艌 0 and x 艋 1200 (otherwise A ⬍ 0). So the function that we wish to maximize is A共x兲 苷 2400x ⫺ 2x 2
0 艋 x 艋 1200
The derivative is A⬘共x兲 苷 2400 ⫺ 4x, so to find the critical numbers we solve the equation 2400 ⫺ 4x 苷 0 which gives x 苷 600. The maximum value of A must occur either at this critical number or at an endpoint of the interval. Since A共0兲 苷 0, A共600兲 苷 720,000, and A共1200兲 苷 0, the Closed Interval Method gives the maximum value as A共600兲 苷 720,000. [Alternatively, we could have observed that A⬙共x兲 苷 ⫺4 ⬍ 0 for all x, so A is always concave downward and the local maximum at x 苷 600 must be an absolute maximum.] Thus the rectangular field should be 600 ft deep and 1200 ft wide.
v EXAMPLE 2 A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. SOLUTION Draw the diagram as in Figure 3, where r is the radius and h the height (both
h
in centimeters). In order to minimize the cost of the metal, we minimize the total surface area of the cylinder (top, bottom, and sides). From Figure 4 we see that the sides are made from a rectangular sheet with dimensions 2 r and h. So the surface area is
r
A 苷 2 r 2 ⫹ 2 rh
FIGURE 3
To eliminate h we use the fact that the volume is given as 1 L, which we take to be 1000 cm3. Thus r 2h 苷 1000
2πr r
h
which gives h 苷 1000兾共 r 2 兲. Substitution of this into the expression for A gives
冉 冊
A 苷 2 r 2 ⫹ 2 r Area 2{π
[email protected]}
Area (2πr)h
FIGURE 4
1000 r 2
苷 2 r 2 ⫹
2000 r
Therefore the function that we want to minimize is A共r兲 苷 2 r 2 ⫹
2000 r
r⬎0
To find the critical numbers, we differentiate: y
A⬘共r兲 苷 4 r ⫺ y=A(r)
1000
0
FIGURE 5
10
r
2000 4共 r 3 ⫺ 500兲 苷 r2 r2
3 Then A⬘共r兲 苷 0 when r 3 苷 500, so the only critical number is r 苷 s 500兾 . Since the domain of A is 共0, ⬁兲, we can’t use the argument of Example 1 concerning 3 endpoints. But we can observe that A⬘共r兲 ⬍ 0 for r ⬍ s 500兾 and A⬘共r兲 ⬎ 0 for 3 r ⬎ s500兾 , so A is decreasing for all r to the left of the critical number and increas3 ing for all r to the right. Thus r 苷 s 500兾 must give rise to an absolute minimum. [Alternatively, we could argue that A共r兲 l ⬁ as r l 0 ⫹ and A共r兲 l ⬁ as r l ⬁, so there must be a minimum value of A共r兲, which must occur at the critical number. See Figure 5.]
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION 3 The value of h corresponding to r 苷 s 500兾 is
In the Applied Project on page 337 we investigate the most economical shape for a can by taking into account other manufacturing costs.
h苷
冑
1000 1000 苷 苷2 r 2 共500兾兲2兾3
3
500 苷 2r
3 Thus, to minimize the cost of the can, the radius should be s 500兾 cm and the height should be equal to twice the radius, namely, the diameter.
NOTE 1 The argument used in Example 2 to justify the absolute minimum is a variant of the First Derivative Test (which applies only to local maximum or minimum values) and is stated here for future reference.
TEC Module 4.7 takes you through six additional optimization problems, including animations of the physical situations.
First Derivative Test for Absolute Extreme Values Suppose that c is a critical number of a continuous function f defined on an interval. (a) If f ⬘共x兲 ⬎ 0 for all x ⬍ c and f ⬘共x兲 ⬍ 0 for all x ⬎ c, then f 共c兲 is the absolute maximum value of f . (b) If f ⬘共x兲 ⬍ 0 for all x ⬍ c and f ⬘共x兲 ⬎ 0 for all x ⬎ c, then f 共c兲 is the absolute minimum value of f .
NOTE 2 An alternative method for solving optimization problems is to use implicit differentiation. Let’s look at Example 2 again to illustrate the method. We work with the same equations
A 苷 2 r 2 ⫹ 2 rh
r 2h 苷 1000
but instead of eliminating h, we differentiate both equations implicitly with respect to r: A⬘ 苷 4 r ⫹ 2 h ⫹ 2 rh⬘
2 rh ⫹ r 2h⬘ 苷 0
The minimum occurs at a critical number, so we set A⬘ 苷 0, simplify, and arrive at the equations 2r ⫹ h ⫹ rh⬘ 苷 0 2h ⫹ rh⬘ 苷 0 and subtraction gives 2r ⫺ h 苷 0, or h 苷 2r.
v
EXAMPLE 3 Find the point on the parabola y 2 苷 2x that is closest to the point 共1, 4兲.
SOLUTION The distance between the point 共1, 4兲 and the point 共x, y兲 is
d 苷 s共x ⫺ 1兲2 ⫹ 共 y ⫺ 4兲2
y (1, 4)
(x, y)
1 0
(See Figure 6.) But if 共x, y兲 lies on the parabola, then x 苷 12 y 2, so the expression for d becomes d 苷 s( 12 y 2 ⫺ 1) 2 ⫹ 共y ⫺ 4兲2
¥=2x
1 2 3 4
x
(Alternatively, we could have substituted y 苷 s2x to get d in terms of x alone.) Instead of minimizing d , we minimize its square: d 2 苷 f 共y兲 苷 ( 12 y 2 ⫺ 1) 2 ⫹ 共y ⫺ 4兲2
FIGURE 6
(You should convince yourself that the minimum of d occurs at the same point as the Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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329
minimum of d 2, but d 2 is easier to work with.) Differentiating, we obtain f ⬘共y兲 苷 2( 12 y 2 ⫺ 1) y ⫹ 2共 y ⫺ 4兲 苷 y 3 ⫺ 8 so f ⬘共y兲 苷 0 when y 苷 2. Observe that f ⬘共y兲 ⬍ 0 when y ⬍ 2 and f ⬘共y兲 ⬎ 0 when y ⬎ 2, so by the First Derivative Test for Absolute Extreme Values, the absolute minimum occurs when y 苷 2. (Or we could simply say that because of the geometric nature of the problem, it’s obvious that there is a closest point but not a farthest point.) The corresponding value of x is x 苷 12 y 2 苷 2. Thus the point on y 2 苷 2x closest to 共1, 4兲 is 共2, 2兲. EXAMPLE 4 A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible (see Figure 7). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km兾h and run 8 km兾h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared with the speed at which the man rows.)
3 km A
C x D
SOLUTION If we let x be the distance from C to D, then the running distance is 8 km
ⱍ DB ⱍ 苷 8 ⫺ x and the Pythagorean Theorem gives the rowing distance as ⱍ AD ⱍ 苷 sx ⫹ 9 . We use the equation 2
time 苷
distance rate
B
Then the rowing time is sx 2 ⫹ 9兾6 and the running time is 共8 ⫺ x兲兾8, so the total time T as a function of x is FIGURE 7
T共x兲 苷
8⫺x sx 2 ⫹ 9 ⫹ 6 8
The domain of this function T is 关0, 8兴. Notice that if x 苷 0, he rows to C and if x 苷 8, he rows directly to B. The derivative of T is T⬘共x兲 苷
x 1 ⫺ 8 6sx 2 ⫹ 9
Thus, using the fact that x 艌 0, we have T⬘共x兲 苷 0 &?
x 1 苷 8 6sx 2 ⫹ 9
&? 4x 苷 3sx 2 ⫹ 9
&?
16x 2 苷 9共x 2 ⫹ 9兲 &? 7x 2 苷 81
&?
x苷
9 s7
The only critical number is x 苷 9兾s7 . To see whether the minimum occurs at this critical number or at an endpoint of the domain 关0, 8兴, we evaluate T at all three points: T共0兲 苷 1.5
T
冉 冊 9 s7
苷1⫹
s7 ⬇ 1.33 8
T共8兲 苷
s73 ⬇ 1.42 6
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Since the smallest of these values of T occurs when x 苷 9兾s7 , the absolute minimum value of T must occur there. Figure 8 illustrates this calculation by showing the graph of T. Thus the man should land the boat at a point 9兾s7 km (⬇3.4 km) downstream from his starting point.
y=T(x) 1
v 2
11:09 AM
APPLICATIONS OF DIFFERENTIATION
T
0
9/21/10
4
6
x
EXAMPLE 5 Find the area of the largest rectangle that can be inscribed in a semicircle
of radius r. SOLUTION 1 Let’s take the semicircle to be the upper half of the circle x 2 ⫹ y 2 苷 r 2 with
FIGURE 8 y
(x, y)
2x
y r x
0
_r
center the origin. Then the word inscribed means that the rectangle has two vertices on the semicircle and two vertices on the xaxis as shown in Figure 9. Let 共x, y兲 be the vertex that lies in the first quadrant. Then the rectangle has sides of lengths 2x and y, so its area is A 苷 2xy To eliminate y we use the fact that 共x, y兲 lies on the circle x 2 ⫹ y 2 苷 r 2 and so y 苷 sr 2 ⫺ x 2 . Thus A 苷 2xsr 2 ⫺ x 2
FIGURE 9
The domain of this function is 0 艋 x 艋 r . Its derivative is A⬘ 苷 2sr 2 ⫺ x 2 ⫺
2x 2 2共r 2 ⫺ 2x 2 兲 苷 sr 2 ⫺ x 2 sr 2 ⫺ x 2
which is 0 when 2x 2 苷 r 2, that is, x 苷 r兾s2 (since x 艌 0). This value of x gives a maximum value of A since A共0兲 苷 0 and A共r兲 苷 0. Therefore the area of the largest inscribed rectangle is
冉 冊
A
r s2
苷2
r s2
冑
r2 ⫺
r2 苷 r2 2
SOLUTION 2 A simpler solution is possible if we think of using an angle as a variable. Let
be the angle shown in Figure 10. Then the area of the rectangle is
r ¨ r cos ¨ FIGURE 10
r sin ¨
A共 兲 苷 共2r cos 兲共r sin 兲 苷 r 2共2 sin cos 兲 苷 r 2 sin 2 We know that sin 2 has a maximum value of 1 and it occurs when 2 苷 兾2. So A共 兲 has a maximum value of r 2 and it occurs when 苷 兾4. Notice that this trigonometric solution doesn’t involve differentiation. In fact, we didn’t need to use calculus at all.
Applications to Business and Economics In Section 3.7 we introduced the idea of marginal cost. Recall that if C共x兲, the cost function, is the cost of producing x units of a certain product, then the marginal cost is the rate of change of C with respect to x. In other words, the marginal cost function is the derivative, C⬘共x兲, of the cost function. Now let’s consider marketing. Let p共x兲 be the price per unit that the company can charge if it sells x units. Then p is called the demand function (or price function) and we would expect it to be a decreasing function of x. If x units are sold and the price per unit is p共x兲, then the total revenue is R共x兲 苷 xp共x兲 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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331
and R is called the revenue function. The derivative R⬘ of the revenue function is called the marginal revenue function and is the rate of change of revenue with respect to the number of units sold. If x units are sold, then the total profit is P共x兲 苷 R共x兲 ⫺ C共x兲 and P is called the profit function. The marginal profit function is P⬘, the derivative of the profit function. In Exercises 57– 62 you are asked to use the marginal cost, revenue, and profit functions to minimize costs and maximize revenues and profits.
v EXAMPLE 6 A store has been selling 200 Bluray disc players a week at $350 each. A market survey indicates that for each $10 rebate offered to buyers, the number of units sold will increase by 20 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue? SOLUTION If x is the number of Bluray players sold per week, then the weekly increase in
sales is x ⫺ 200. For each increase of 20 units sold, the price is decreased by $10. So for 1 each additional unit sold, the decrease in price will be 20 ⫻ 10 and the demand function is 1 p共x兲 苷 350 ⫺ 10 20 共x ⫺ 200兲 苷 450 ⫺ 2 x
The revenue function is R共x兲 苷 xp共x兲 苷 450x ⫺ 12 x 2 Since R⬘共x兲 苷 450 ⫺ x, we see that R⬘共x兲 苷 0 when x 苷 450. This value of x gives an absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward). The corresponding price is p共450兲 苷 450 ⫺ 12 共450兲 苷 225 and the rebate is 350 ⫺ 225 苷 125. Therefore, to maximize revenue, the store should offer a rebate of $125.
4.7
Exercises
1. Consider the following problem: Find two numbers whose sum
is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem.
(b) Use calculus to solve the problem and compare with your answer to part (a). 2. Find two numbers whose difference is 100 and whose product
is a minimum. 3. Find two positive numbers whose product is 100 and whose
sum is a minimum.
;
First number
Second number
Product
1 2 3 . . .
22 21 20 . . .
22 42 60 . . .
Graphing calculator or computer required
4. The sum of two positive numbers is 16. What is the smallest
possible value of the sum of their squares? 5. What is the maximum vertical distance between the line
y 苷 x ⫹ 2 and the parabola y 苷 x 2 for ⫺1 艋 x 艋 2? 6. What is the minimum vertical distance between the parabolas
y 苷 x 2 ⫹ 1 and y 苷 x ⫺ x 2 ?
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
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APPLICATIONS OF DIFFERENTIATION
7. Find the dimensions of a rectangle with perimeter 100 m
whose area is as large as possible. 8. Find the dimensions of a rectangle with area 1000 m2 whose
perimeter is as small as possible. 9. A model used for the yield Y of an agricultural crop as a func
tion of the nitrogen level N in the soil (measured in appropriate units) is kN Y苷 1 ⫹ N2 where k is a positive constant. What nitrogen level gives the best yield? 10. The rate 共in mg carbon兾m 3兾h兲 at which photosynthesis takes
place for a species of phytoplankton is modeled by the function P苷
100 I I2 ⫹ I ⫹ 4
where I is the light intensity (measured in thousands of footcandles). For what light intensity is P a maximum? 11. Consider the following problem: A farmer with 750 ft of
fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a). 12. Consider the following problem: A box with an open top is to
be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a). 13. A farmer wants to fence an area of 1.5 million square feet in a
rectangular field and then divide it in half with a fence parallel
to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? 14. A box with a square base and open top must have a volume of
32,000 cm3. Find the dimensions of the box that minimize the amount of material used. 15. If 1200 cm2 of material is available to make a box with a
square base and an open top, find the largest possible volume of the box. 16. A rectangular storage container with an open top is to have a
volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. 17. Do Exercise 16 assuming the container has a lid that is made
from the same material as the sides. 18. (a) Show that of all the rectangles with a given area, the one
with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square. 19. Find the point on the line y 苷 2x ⫹ 3 that is closest to the
origin. 20. Find the point on the curve y 苷 sx that is closest to the
point 共3, 0兲.
21. Find the points on the ellipse 4x 2 ⫹ y 2 苷 4 that are farthest
away from the point 共1, 0兲.
; 22. Find, correct to two decimal places, the coordinates of the point on the curve y 苷 sin x that is closest to the point 共4, 2兲. 23. Find the dimensions of the rectangle of largest area that can be
inscribed in a circle of radius r . 24. Find the area of the largest rectangle that can be inscribed in
the ellipse x 2兾a 2 ⫹ y 2兾b 2 苷 1. 25. Find the dimensions of the rectangle of largest area that can be
inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle. 26. Find the area of the largest trapezoid that can be inscribed in a
circle of radius 1 and whose base is a diameter of the circle. 27. Find the dimensions of the isosceles triangle of largest area that
can be inscribed in a circle of radius r . 28. Find the area of the largest rectangle that can be inscribed in a
right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs. 29. A right circular cylinder is inscribed in a sphere of radius r .
Find the largest possible volume of such a cylinder. 30. A right circular cylinder is inscribed in a cone with height h
and base radius r. Find the largest possible volume of such a cylinder. 31. A right circular cylinder is inscribed in a sphere of radius r .
Find the largest possible surface area of such a cylinder.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.7
32. A Norman window has the shape of a rectangle surmounted
by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 62 on page 22.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted. 33. The top and bottom margins of a poster are each 6 cm and the
side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area. 2
34. A poster is to have an area of 180 in with 1inch margins at
the bottom and sides and a 2inch margin at the top. What dimensions will give the largest printed area? 35. A piece of wire 10 m long is cut into two pieces. One piece
is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum? 36. Answer Exercise 35 if one piece is bent into a square and the
other into a circle. 37. A cylindrical can without a top is made to contain V cm3 of
liquid. Find the dimensions that will minimize the cost of the metal to make the can. 38. A fence 8 ft tall runs parallel to a tall building at a distance of
4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 39. A coneshaped drinking cup is made from a circular piece
of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A
B R
C
40. A coneshaped paper drinking cup is to be made to hold 27 cm3
of water. Find the height and radius of the cup that will use the smallest amount of paper.
OPTIMIZATION PROBLEMS
43. If a resistor of R ohms is connected across a battery of E volts
with internal resistance r ohms, then the power ( in watts) in the external resistor is P苷
E 2R 共R ⫹ r兲 2
If E and r are fixed but R varies, what is the maximum value of the power? 44. For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3. It is
believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u 共u ⬍ v兲, then the time required to swim a distance L is L兾共v ⫺ u兲 and the total energy E required to swim the distance is given by E共v兲 苷 av 3 ⴢ
L v⫺u
where a is the proportionality constant. (a) Determine the value of v that minimizes E . (b) Sketch the graph of E. Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed. 45. In a beehive, each cell is a regular hexagonal prism, open at
one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S 苷 6sh ⫺ 32 s 2 cot ⫹ (3s 2s3兾2) csc where s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dS兾d. (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell ( in terms of s and h). Note: Actual measurements of the angle in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 2⬚.
41. A cone with height h is inscribed in a larger cone with
height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h 苷 13 H .
trihedral angle ¨
rear of cell
42. An object with weight W is dragged along a horizontal plane
by a force acting along a rope attached to the object. If the rope makes an angle with a plane, then the magnitude of the force is W F苷 sin ⫹ cos where is a constant called the coefficient of friction. For what value of is F smallest?
333
h
b
s
front of cell
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
(b) If C共x兲 苷 16,000 ⫹ 200x ⫹ 4x 3兾2, in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.
46. A boat leaves a dock at 2:00 PM and travels due south at a
speed of 20 km兾h. Another boat has been heading due east at 15 km兾h and reaches the same dock at 3:00 PM. At what time were the two boats closest together? 47. Solve the problem in Example 4 if the river is 5 km wide and
point B is only 5 km downstream from A.
58. (a) Show that if the profit P共x兲 is a maximum, then the mar
ginal revenue equals the marginal cost. (b) If C共x兲 苷 16,000 ⫹ 500x ⫺ 1.6x 2 ⫹ 0.004x 3 is the cost function and p共x兲 苷 1700 ⫺ 7x is the demand function, find the production level that will maximize profit.
48. A woman at a point A on the shore of a circular lake with
radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi兾h and row a boat at 2 mi兾h. How should she proceed?
59. A baseball team plays in a stadium that holds 55,000 specta
tors. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue?
B
A
¨ 2
2
C
60. During the summer months Terry makes and sells necklaces
on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?
49. An oil refinery is located on the north bank of a straight river
that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000兾km over land to a point P on the north bank and $800,000兾km under the river to the tanks. To minimize the cost of the pipeline, where should P be located?
61. A manufacturer has been selling 1000 flatscreen TVs a
week at $450 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is C共x兲 苷 68,000 ⫹ 150x, how should the manufacturer set the size of the rebate in order to maximize its profit?
; 50. Suppose the refinery in Exercise 49 is located 1 km north of the river. Where should P be located? 51. The illumination of an object by a light source is directly pro
portional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?
62. The manager of a 100unit apartment complex knows from
52. Find an equation of the line through the point 共3, 5兲 that cuts
experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue?
off the least area from the first quadrant. 53. Let a and b be positive numbers. Find the length of the short
est line segment that is cut off by the first quadrant and passes through the point 共a, b兲.
63. Show that of all the isosceles triangles with a given perime
54. At which points on the curve y 苷 1 ⫹ 40x 3 ⫺ 3x 5 does the
ter, the one with the greatest area is equilateral.
tangent line have the largest slope? 55. What is the shortest possible length of the line segment that
is cut off by the first quadrant and is tangent to the curve y 苷 3兾x at some point? 56. What is the smallest possible area of the triangle that is cut
off by the first quadrant and whose hypotenuse is tangent to the parabola y 苷 4 ⫺ x 2 at some point?
CAS
64. The frame for a kite is to be made from six pieces of wood.
The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? a
b
a
b
57. (a) If C共x兲 is the cost of producing x units of a commodity,
then the average cost per unit is c共x兲 苷 C共x兲兾x. Show that if the average cost is a minimum, then the marginal cost equals the average cost.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.7
; 65. A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B, and C is minimized (see the figure). Express L as a function of x 苷 AP and use the graphs of L and dL兾dx to estimate the minimum value of L.
OPTIMIZATION PROBLEMS
335
between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when 1 苷 2.
ⱍ ⱍ
P S
A P Q 2m B
3m
R
T
69. The upper righthand corner of a piece of paper, 12 in. by
C
D
¨™
¨¡
5m
66. The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At
very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that c共v兲 is minimized for this car when v ⬇ 30 mi兾h. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Let’s call this consumption G. Using the graph, estimate the speed at which G has its minimum value.
8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y ? 12 y
x
8
c
70. A steel pipe is being carried down a hallway 9 ft wide. At the
end of the hall there is a rightangled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? 0
20
40
60
√
6 67. Let v1 be the velocity of light in air and v2 the velocity of
light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin 1 v1 苷 sin 2 v2
¨
9
where 1 (the angle of incidence) and 2 (the angle of refraction) are as shown. This equation is known as Snell’s Law. A ¨¡
71. An observer stands at a point P, one unit away from a track.
Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight between the runners. [Hint: Maximize tan .]
C
P ¨
¨™
1
B 68. Two vertical poles PQ and ST are secured by a rope PRS
going from the top of the first pole to a point R on the ground
S
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
72. A rain gutter is to be constructed from a metal sheet of width
sel with radius r1 branching at an angle into a smaller vessel with radius r 2 .
30 cm by bending up onethird of the sheet on each side through an angle . How should be chosen so that the gutter will carry the maximum amount of water?
C
r™ ¨ 10 cm
10 cm
b
vascular branching
¨ 10 cm A
r¡
¨ B
73. Where should the point P be chosen on the line segment AB so
as to maximize the angle ? B
a
2
(a) Use Poiseuille’s Law to show that the total resistance of the blood along the path ABC is
¨
P
R苷C
3
冊
a ⫺ b cot b csc ⫹ r14 r24
where a and b are the distances shown in the figure. (b) Prove that this resistance is minimized when
5
A
冉
cos 苷
74. A painting in an art gallery has height h and is hung so that its
lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle subtended at his eye by the painting?)
r24 r14
(c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is twothirds the radius of the larger vessel.
¨
d
75. Find the maximum area of a rectangle that can be circum
scribed about a given rectangle with length L and width W. [Hint: Express the area as a function of an angle .] 76. The blood vascular system consists of blood vessels (arteries,
arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille’s Laws gives the resistance R of the blood as R苷C
L r4
where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood ves
© Manfred Kage / Peter Arnold Images / Photolibrary
h
77. Ornithologists have determined that some species of birds tend
to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point C should the bird
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLIED PROJECT
fly in order to minimize the total energy expended in returning to its nesting area? (b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio W兾L mean in terms of the bird’s flight? What would a small value mean? Determine the ratio W兾L corresponding to the minimum expenditure of energy. (c) What should the value of W兾L be in order for the bird to fly directly to its nesting area D? What should the value of W兾L be for the bird to fly to B and then along the shore to D ? (d) If the ornithologists observe that birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land?
island
THE SHAPE OF A CAN
; 78. Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ᐉ parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on ᐉ so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity I共x兲 at the point P. (b) If d 苷 5 m, use graphs of I共x兲 and I⬘共x兲 to show that the intensity is minimized when x 苷 5 m, that is, when P is at the midpoint of ᐉ. (c) If d 苷 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere between d 苷 5 m and d 苷 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d . P
ᐉ
x
5 km C
d
D
B
337
nest
13 km
10 m
APPLIED PROJECT
h r
THE SHAPE OF A CAN In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height h and radius r that minimize the cost of the metal to make the can (see the figure). If we disregard any waste metal in the manufacturing process, then the problem is to minimize the surface area of the cylinder. We solved this problem in Example 2 in Section 4.7 and we found that h 苷 2r ; that is, the height should be the same as the diameter. But if you go to your cupboard or your supermarket with a ruler, you will discover that the height is usually greater than the diameter and the ratio h兾r varies from 2 up to about 3.8. Let’s see if we can explain this phenomenon. 1. The material for the cans is cut from sheets of metal. The cylindrical sides are formed by
bending rectangles; these rectangles are cut from the sheet with little or no waste. But if the top and bottom discs are cut from squares of side 2r (as in the figure), this leaves considerable waste metal, which may be recycled but has little or no value to the can makers. If this is the case, show that the amount of metal used is minimized when h 8 苷 ⬇ 2.55 r
Discs cut from squares
;
Graphing calculator or computer required
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
2. A more efficient packing of the discs is obtained by dividing the metal sheet into hexagons
and cutting the circular lids and bases from the hexagons (see the figure). Show that if this strategy is adopted, then h 4 s3 ⬇ 2.21 苷 r 3. The values of h兾r that we found in Problems 1 and 2 are a little closer to the ones that
actually occur on supermarket shelves, but they still don’t account for everything. If we look more closely at some real cans, we see that the lid and the base are formed from discs with radius larger than r that are bent over the ends of the can. If we allow for this we would increase h兾r. More significantly, in addition to the cost of the metal we need to incorporate the manufacturing of the can into the cost. Let’s assume that most of the expense is incurred in joining the sides to the rims of the cans. If we cut the discs from hexagons as in Problem 2, then the total cost is proportional to
Discs cut from hexagons
4 s3 r 2 ⫹ 2 rh ⫹ k共4 r ⫹ h兲 where k is the reciprocal of the length that can be joined for the cost of one unit area of metal. Show that this expression is minimized when 3 V s 苷 k
冑 3
h 2 ⫺ h兾r ⴢ r h兾r ⫺ 4 s3
3 ; 4. Plot sV 兾k as a function of x 苷 h兾r and use your graph to argue that when a can is large
or joining is cheap, we should make h兾r approximately 2.21 (as in Problem 2). But when the can is small or joining is costly, h兾r should be substantially larger. 5. Our analysis shows that large cans should be almost square but small cans should be tall
and thin. Take a look at the relative shapes of the cans in a supermarket. Is our conclusion usually true in practice? Are there exceptions? Can you suggest reasons why small cans are not always tall and thin?
4.8
Newton’s Method Suppose that a car dealer offers to sell you a car for $18,000 or for payments of $375 per month for five years. You would like to know what monthly interest rate the dealer is, in effect, charging you. To find the answer, you have to solve the equation 1
0.15
0 _0.05
FIGURE 1
0.012
48x共1 ⫹ x兲60 ⫺ 共1 ⫹ x兲60 ⫹ 1 苷 0
(The details are explained in Exercise 41.) How would you solve such an equation? For a quadratic equation ax 2 ⫹ bx ⫹ c 苷 0 there is a wellknown formula for the roots. For third and fourthdegree equations there are also formulas for the roots, but they are extremely complicated. If f is a polynomial of degree 5 or higher, there is no such formula (see the note on page 212). Likewise, there is no formula that will enable us to find the exact roots of a transcendental equation such as cos x 苷 x. We can find an approximate solution to Equation 1 by plotting the left side of the equation. Using a graphing device, and after experimenting with viewing rectangles, we produce the graph in Figure 1. We see that in addition to the solution x 苷 0, which doesn’t interest us, there is a solution between 0.007 and 0.008. Zooming in shows that the root is approximately 0.0076. If
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.8 Try to solve Equation 1 using the numerical rootfinder on your calculator or computer. Some machines are not able to solve it. Others are successful but require you to specify a starting point for the search. y {x ¡, f(x¡)}
y=ƒ L 0
r
x™ x ¡
x
FIGURE 2
NEWTON’S METHOD
339
we need more accuracy we could zoom in repeatedly, but that becomes tiresome. A faster alternative is to use a numerical rootfinder on a calculator or computer algebra system. If we do so, we find that the root, correct to nine decimal places, is 0.007628603. How do those numerical rootfinders work? They use a variety of methods, but most of them make some use of Newton’s method, also called the NewtonRaphson method. We will explain how this method works, partly to show what happens inside a calculator or computer, and partly as an application of the idea of linear approximation. The geometry behind Newton’s method is shown in Figure 2, where the root that we are trying to find is labeled r. We start with a first approximation x 1, which is obtained by guessing, or from a rough sketch of the graph of f, or from a computergenerated graph of f. Consider the tangent line L to the curve y 苷 f 共x兲 at the point 共x 1, f 共x 1兲兲 and look at the xintercept of L, labeled x 2. The idea behind Newton’s method is that the tangent line is close to the curve and so its xintercept, x2 , is close to the xintercept of the curve (namely, the root r that we are seeking). Because the tangent is a line, we can easily find its xintercept. To find a formula for x2 in terms of x1 we use the fact that the slope of L is f ⬘共x1 兲, so its equation is y ⫺ f 共x 1 兲 苷 f ⬘共x 1 兲共x ⫺ x 1 兲 Since the xintercept of L is x 2 , we set y 苷 0 and obtain 0 ⫺ f 共x 1 兲 苷 f ⬘共x 1 兲共x 2 ⫺ x 1 兲 If f ⬘共x 1兲 苷 0, we can solve this equation for x 2 : x2 苷 x1 ⫺
We use x 2 as a second approximation to r. Next we repeat this procedure with x 1 replaced by the second approximation x 2, using the tangent line at 共x 2 , f 共x 2 兲兲. This gives a third approximation:
y {x¡, f(x¡)}
x3 苷 x2 ⫺ {x™, f(x™)}
r 0
x£
x™ x ¡
x¢
x
f 共x 2 兲 f ⬘共x 2 兲
If we keep repeating this process, we obtain a sequence of approximations x 1, x 2, x 3, x 4, . . . as shown in Figure 3. In general, if the nth approximation is x n and f ⬘共x n 兲 苷 0, then the next approximation is given by
FIGURE 3
2
Sequences were briefly introduced in A Preview of Calculus on page 5. A more thorough discussion starts in Section 11.1.
f 共x 1 兲 f ⬘共x 1 兲
x n⫹1 苷 x n ⫺
f 共x n 兲 f ⬘共x n 兲
If the numbers x n become closer and closer to r as n becomes large, then we say that the sequence converges to r and we write lim x n 苷 r
nl⬁
 Although the sequence of successive approximations converges to the desired root for functions of the type illustrated in Figure 3, in certain circumstances the sequence may not Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 4
Page 340
converge. For example, consider the situation shown in Figure 4. You can see that x 2 is a worse approximation than x 1. This is likely to be the case when f ⬘共x 1兲 is close to 0. It might even happen that an approximation (such as x 3 in Figure 4) falls outside the domain of f . Then Newton’s method fails and a better initial approximation x 1 should be chosen. See Exercises 31–34 for specific examples in which Newton’s method works very slowly or does not work at all.
x™
0
x¡
11:14 AM
APPLICATIONS OF DIFFERENTIATION
y
x£
9/21/10
r
x
FIGURE 4
v EXAMPLE 1 Starting with x 1 苷 2, find the third approximation x 3 to the root of the equation x 3 ⫺ 2x ⫺ 5 苷 0. SOLUTION We apply Newton’s method with
f 共x兲 苷 x 3 ⫺ 2x ⫺ 5 TEC In Module 4.8 you can investigate how Newton’s Method works for several functions and what happens when you change x 1.
Figure 5 shows the geometry behind the first step in Newton’s method in Example 1. Since f ⬘共2兲 苷 10, the tangent line to y 苷 x 3 ⫺ 2x ⫺ 5 at 共2, ⫺1兲 has equation y 苷 10x ⫺ 21 so its xintercept is x 2 苷 2.1.
With n 苷 1 we have x2 苷 x1 ⫺ 苷2⫺
y=˛2x5 2.2 x™
_2
x13 ⫺ 2x 1 ⫺ 5 3x12 ⫺ 2 2 3 ⫺ 2共2兲 ⫺ 5 苷 2.1 3共2兲2 ⫺ 2
Then with n 苷 2 we obtain x3 苷 x2 ⫺
y=10x21
FIGURE 5
f ⬘共x兲 苷 3x 2 ⫺ 2
Newton himself used this equation to illustrate his method and he chose x 1 苷 2 after some experimentation because f 共1兲 苷 ⫺6, f 共2兲 苷 ⫺1, and f 共3兲 苷 16. Equation 2 becomes x n3 ⫺ 2x n ⫺ 5 x n⫹1 苷 x n ⫺ 3x n2 ⫺ 2
1
1.8
and
x 23 ⫺ 2x 2 ⫺ 5 共2.1兲3 ⫺ 2共2.1兲 ⫺ 5 苷 2.1 ⫺ ⬇ 2.0946 2 3x 2 ⫺ 2 3共2.1兲2 ⫺ 2
It turns out that this third approximation x 3 ⬇ 2.0946 is accurate to four decimal places. Suppose that we want to achieve a given accuracy, say to eight decimal places, using Newton’s method. How do we know when to stop? The rule of thumb that is generally used is that we can stop when successive approximations x n and x n⫹1 agree to eight decimal places. (A precise statement concerning accuracy in Newton’s method will be given in Exercise 39 in Section 11.11.) Notice that the procedure in going from n to n ⫹ 1 is the same for all values of n. (It is called an iterative process.) This means that Newton’s method is particularly convenient for use with a programmable calculator or a computer.
v
6 2 correct to eight decimal places. EXAMPLE 2 Use Newton’s method to find s
6 SOLUTION First we observe that finding s 2 is equivalent to finding the positive root of
the equation x6 ⫺ 2 苷 0 so we take f 共x兲 苷 x 6 ⫺ 2. Then f ⬘共x兲 苷 6x 5 and Formula 2 (Newton’s method) becomes x n6 ⫺ 2 x n⫹1 苷 x n ⫺ 6x n5 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.8
NEWTON’S METHOD
341
If we choose x 1 苷 1 as the initial approximation, then we obtain x 2 ⬇ 1.16666667 x 3 ⬇ 1.12644368 x 4 ⬇ 1.12249707 x 5 ⬇ 1.12246205 x 6 ⬇ 1.12246205 Since x 5 and x 6 agree to eight decimal places, we conclude that 6 2 ⬇ 1.12246205 s
to eight decimal places.
v
EXAMPLE 3 Find, correct to six decimal places, the root of the equation cos x 苷 x.
SOLUTION We first rewrite the equation in standard form:
cos x ⫺ x 苷 0 Therefore we let f 共x兲 苷 cos x ⫺ x. Then f ⬘共x兲 苷 ⫺sin x ⫺ 1, so Formula 2 becomes x n⫹1 苷 x n ⫺ y
y=x
y=cos x 1
π 2
x
π
cos x n ⫺ x n cos x n ⫺ x n 苷 xn ⫹ ⫺sin x n ⫺ 1 sin x n ⫹ 1
In order to guess a suitable value for x 1 we sketch the graphs of y 苷 cos x and y 苷 x in Figure 6. It appears that they intersect at a point whose xcoordinate is somewhat less than 1, so let’s take x 1 苷 1 as a convenient first approximation. Then, remembering to put our calculator in radian mode, we get x 2 ⬇ 0.75036387 x 3 ⬇ 0.73911289
FIGURE 6
x 4 ⬇ 0.73908513 x 5 ⬇ 0.73908513 Since x 4 and x 5 agree to six decimal places (eight, in fact), we conclude that the root of the equation, correct to six decimal places, is 0.739085. Instead of using the rough sketch in Figure 6 to get a starting approximation for Newton’s method in Example 3, we could have used the more accurate graph that a calculator or computer provides. Figure 7 suggests that we use x1 苷 0.75 as the initial approximation. Then Newton’s method gives
1
y=cos x
x 2 ⬇ 0.73911114
y=x
0
FIGURE 7
1
x 3 ⬇ 0.73908513
x 4 ⬇ 0.73908513
and so we obtain the same answer as before, but with one fewer step. You might wonder why we bother at all with Newton’s method if a graphing device is available. Isn’t it easier to zoom in repeatedly and find the roots as we did in Section 1.4? If only one or two decimal places of accuracy are required, then indeed Newton’s method is inappropriate and a graphing device suffices. But if six or eight decimal places are required, then repeated zooming becomes tiresome. It is usually faster and more efficient to use a computer and Newton’s method in tandem—the graphing device to get started and Newton’s method to finish.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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4.8
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APPLICATIONS OF DIFFERENTIATION
Exercises
1. The figure shows the graph of a function f . Suppose that New
ton’s method is used to approximate the root r of the equation f 共x兲 苷 0 with initial approximation x 1 苷 1. (a) Draw the tangent lines that are used to find x 2 and x 3, and estimate the numerical values of x 2 and x 3. (b) Would x 1 苷 5 be a better first approximation? Explain. y
7. x 5 ⫺ x ⫺ 1 苷 0,
x1 苷 1
8. x 7 ⫹ 4 苷 0,
x1 苷 ⫺1
; 9. Use Newton’s method with initial approximation x1 苷 ⫺1 to find x 2 , the second approximation to the root of the equation x 3 ⫹ x ⫹ 3 苷 0. Explain how the method works by first graphing the function and its tangent line at 共⫺1, 1兲.
; 10. Use Newton’s method with initial approximation x1 苷 1 to find x 2 , the second approximation to the root of the equation x 4 ⫺ x ⫺ 1 苷 0. Explain how the method works by first graphing the function and its tangent line at 共1, ⫺1兲.
1 0
r
1
s
11–12 Use Newton’s method to approximate the given number correct to eight decimal places.
x
5 11. s 20
2. Follow the instructions for Exercise 1(a) but use x 1 苷 9 as the
starting approximation for finding the root s. 共2, 5兲 has the equation y 苷 9 ⫺ 2x. If Newton’s method is used to locate a root of the equation f 共x兲 苷 0 and the initial approximation is x1 苷 2, find the second approximation x 2.
4. For each initial approximation, determine graphically what
happens if Newton’s method is used for the function whose graph is shown. (a) x1 苷 0 (b) x1 苷 1 (c) x1 苷 3 (d) x1 苷 4 (e) x1 苷 5 y
3
1
100
s100
13–16 Use Newton’s method to approximate the indicated root of
the equation correct to six decimal places.
3. Suppose the tangent line to the curve y 苷 f 共x兲 at the point
0
12.
x
5
13. The root of x 4 ⫺ 2 x 3 ⫹ 5x 2 ⫺ 6 苷 0 in the interval 关1, 2兴 14. The root of 2.2 x 5 ⫺ 4.4 x 3 ⫹ 1.3x 2 ⫺ 0.9x ⫺ 4.0 苷 0 in the
interval 关⫺2, ⫺1兴
15. The negative root of e x 苷 4 ⫺ x 2 16. The positive root of 3 sin x 苷 x 17–22 Use Newton’s method to find all roots of the equation
correct to six decimal places. 17. 3 cos x 苷 x ⫹ 1
18. sx ⫹ 1 苷 x 2 ⫺ x
19. 共x ⫺ 2兲 2 苷 ln x
20.
21. x 3 苷 tan⫺1x
22. sin x 苷 x 2 ⫺ 2
1 苷 1 ⫹ x3 x
; 23–28 Use Newton’s method to find all the roots of the equation 5. For which of the initial approximations x1 苷 a, b, c, and d do
you think Newton’s method will work and lead to the root of the equation f 共x兲 苷 0?
correct to eight decimal places. Start by drawing a graph to find initial approximations. 23. x 6 ⫺ x 5 ⫺ 6x 4 ⫺ x 2 ⫹ x ⫹ 10 苷 0 24. x 5 ⫺ 3x 4 ⫹ x 3 ⫺ x 2 ⫺ x ⫹ 6 苷 0
y
25.
x 苷 s1 ⫺ x x ⫹1
26. cos共x 2 ⫺ x兲 苷 x 4
2
27. 4e ⫺x sin x 苷 x 2 ⫺ x ⫹ 1
28. e arctan x 苷 sx 3 ⫹ 1
2
a
0
b
c
d
x
29. (a) Apply Newton’s method to the equation x 2 ⫺ a 苷 0 to 6–8 Use Newton’s method with the specified initial approximation
x 1 to find x 3 , the third approximation to the root of the given equation. (Give your answer to four decimal places.) 6. 3 x 3 ⫹ 2 x 2 ⫹ 3 苷 0, 1
;
1
derive the following squareroot algorithm (used by the ancient Babylonians to compute sa ) : x n⫹1 苷
x 1 苷 ⫺3
Graphing calculator or computer required
冉
1 a xn ⫹ 2 xn
冊
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.8
(b) Use part (a) to compute s1000 correct to six decimal places. 30. (a) Apply Newton’s method to the equation 1兾x ⫺ a 苷 0 to
343
40. In the figure, the length of the chord AB is 4 cm and the
length of the arc AB is 5 cm. Find the central angle , in radians, correct to four decimal places. Then give the answer to the nearest degree.
derive the following reciprocal algorithm: x n⫹1 苷 2x n ⫺ ax
NEWTON’S METHOD
5 cm
2 n
4 cm
A
(This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1兾1.6984 correct to six decimal places.
B
¨
31. Explain why Newton’s method doesn’t work for finding the
root of the equation x 3 ⫺ 3x ⫹ 6 苷 0 if the initial approximation is chosen to be x 1 苷 1. 32. (a) Use Newton’s method with x 1 苷 1 to find the root of the
;
equation x 3 ⫺ x 苷 1 correct to six decimal places. (b) Solve the equation in part (a) using x 1 苷 0.6 as the initial approximation. (c) Solve the equation in part (a) using x 1 苷 0.57. (You definitely need a programmable calculator for this part.) (d) Graph f 共x兲 苷 x 3 ⫺ x ⫺ 1 and its tangent lines at x1 苷 1, 0.6, and 0.57 to explain why Newton’s method is so sensitive to the value of the initial approximation.
33. Explain why Newton’s method fails when applied to the 3 equation s x 苷 0 with any initial approximation x 1 苷 0. Illustrate your explanation with a sketch.
34. If
f 共x兲 苷
再
sx ⫺s⫺x
if x 艌 0 if x ⬍ 0
then the root of the equation f 共x兲 苷 0 is x 苷 0. Explain why Newton’s method fails to find the root no matter which initial approximation x 1 苷 0 is used. Illustrate your explanation with a sketch. 35. (a) Use Newton’s method to find the critical numbers of the
function f 共x兲 苷 x 6 ⫺ x 4 ⫹ 3x 3 ⫺ 2x correct to six decimal places. (b) Find the absolute minimum value of f correct to four decimal places.
41. A car dealer sells a new car for $18,000. He also offers to sell
the same car for payments of $375 per month for five years. What monthly interest rate is this dealer charging? To solve this problem you will need to use the formula for the present value A of an annuity consisting of n equal payments of size R with interest rate i per time period: A苷
Replacing i by x, show that 48x共1 ⫹ x兲60 ⫺ 共1 ⫹ x兲60 ⫹ 1 苷 0 Use Newton’s method to solve this equation. 42. The figure shows the sun located at the origin and the earth
at the point 共1, 0兲. (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ⬇ 1.496 ⫻ 10 8 km.) There are five locations L 1 , L 2 , L 3 , L 4 , and L 5 in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If m1 is the mass of the sun, m 2 is the mass of the earth, and r 苷 m 2兾共m1 ⫹ m 2 兲, it turns out that the xcoordinate of L 1 is the unique root of the fifthdegree equation p共x兲 苷 x 5 ⫺ 共2 ⫹ r兲x 4 ⫹ 共1 ⫹ 2r兲x 3 ⫺ 共1 ⫺ r兲x 2 ⫹ 2共1 ⫺ r兲x ⫹ r ⫺ 1 苷 0
36. Use Newton’s method to find the absolute maximum value
of the function f 共x兲 苷 x cos x, 0 艋 x 艋 , correct to six decimal places. 37. Use Newton’s method to find the coordinates of the inflection
point of the curve y 苷 x 2 sin x, 0 艋 x 艋 , correct to six decimal places.
R 关1 ⫺ 共1 ⫹ i 兲⫺n 兴 i
and the xcoordinate of L 2 is the root of the equation p共x兲 ⫺ 2rx 2 苷 0 Using the value r ⬇ 3.04042 ⫻ 10 ⫺6, find the locations of the libration points (a) L 1 and (b) L 2. y
38. Of the infinitely many lines that are tangent to the curve
y 苷 ⫺sin x and pass through the origin, there is one that has the largest slope. Use Newton’s method to find the slope of that line correct to six decimal places.
L¢ sun
earth
L∞
L¡
L™
x
39. Use Newton’s method to find the coordinates, correct to six
decimal places, of the point on the parabola y 苷 共x ⫺ 1兲 2 that is closest to the origin.
L£
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
Antiderivatives
4.9
A physicist who knows the velocity of a particle might wish to know its position at a given time. An engineer who can measure the variable rate at which water is leaking from a tank wants to know the amount leaked over a certain time period. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. In each case, the problem is to find a function F whose derivative is a known function f. If such a function F exists, it is called an antiderivative of f. Deﬁnition A function F is called an antiderivative of f on an interval I if F⬘共x兲 苷 f 共x兲 for all x in I .
For instance, let f 共x兲 苷 x 2. It isn’t difficult to discover an antiderivative of f if we keep the Power Rule in mind. In fact, if F共x兲 苷 13 x 3, then F⬘共x兲 苷 x 2 苷 f 共x兲. But the function G共x兲 苷 13 x 3 ⫹ 100 also satisfies G⬘共x兲 苷 x 2. Therefore both F and G are antiderivatives of f . Indeed, any function of the form H共x兲 苷 13 x 3 ⫹ C, where C is a constant, is an antiderivative of f . The question arises: Are there any others? To answer this question, recall that in Section 4.2 we used the Mean Value Theorem to prove that if two functions have identical derivatives on an interval, then they must differ by a constant (Corollary 4.2.7). Thus if F and G are any two antiderivatives of f , then y
˛
F⬘共x兲 苷 f 共x兲 苷 G⬘共x兲
˛
so G共x兲 ⫺ F共x兲 苷 C, where C is a constant. We can write this as G共x兲 苷 F共x兲 ⫹ C, so we have the following result.
y= 3 +3 y= 3 +2 ˛
y= 3 +1 y= ˛ 0
x
1 Theorem If F is an antiderivative of f on an interval I , then the most general antiderivative of f on I is F共x兲 ⫹ C
3
˛
y= 3 1 ˛
y= 3 2
where C is an arbitrary constant. Going back to the function f 共x兲 苷 x 2, we see that the general antiderivative of f is x ⫹ C. By assigning specific values to the constant C, we obtain a family of functions whose graphs are vertical translates of one another (see Figure 1). This makes sense because each curve must have the same slope at any given value of x.
1 3
FIGURE 1
Members of the family of antiderivatives of ƒ=≈
3
EXAMPLE 1 Find the most general antiderivative of each of the following functions. (a) f 共x兲 苷 sin x (b) f 共x兲 苷 1兾x (c) f 共x兲 苷 x n, n 苷 ⫺1 SOLUTION
(a) If F共x兲 苷 ⫺cos x, then F⬘共x兲 苷 sin x, so an antiderivative of sin x is ⫺cos x. By Theorem 1, the most general antiderivative is G共x兲 苷 ⫺cos x ⫹ C. (b) Recall from Section 3.6 that d 1 共ln x兲 苷 dx x So on the interval 共0, ⬁兲 the general antiderivative of 1兾x is ln x ⫹ C . We also learned that d 1 共ln x 兲 苷 dx x
ⱍ ⱍ
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.9
ANTIDERIVATIVES
345
for all x 苷 0. Theorem 1 then tells us that the general antiderivative of f 共x兲 苷 1兾x is ln x ⫹ C on any interval that doesn’t contain 0. In particular, this is true on each of the intervals 共⫺⬁, 0兲 and 共0, ⬁兲. So the general antiderivative of f is
ⱍ ⱍ
F共x兲 苷
再
ln x ⫹ C1 ln共⫺x兲 ⫹ C2
if x ⬎ 0 if x ⬍ 0
(c) We use the Power Rule to discover an antiderivative of x n. In fact, if n 苷 ⫺1, then d dx
冉 冊 x n⫹1 n⫹1
苷
共n ⫹ 1兲x n 苷 xn n⫹1
Thus the general antiderivative of f 共x兲 苷 x n is F共x兲 苷
x n⫹1 ⫹C n⫹1
This is valid for n 艌 0 since then f 共x兲 苷 x n is defined on an interval. If n is negative (but n 苷 ⫺1), it is valid on any interval that doesn’t contain 0. As in Example 1, every differentiation formula, when read from right to left, gives rise to an antidifferentiation formula. In Table 2 we list some particular antiderivatives. Each formula in the table is true because the derivative of the function in the right column appears in the left column. In particular, the first formula says that the antiderivative of a constant times a function is the constant times the antiderivative of the function. The second formula says that the antiderivative of a sum is the sum of the antiderivatives. (We use the notation F⬘苷 f , G⬘ 苷 t.) 2
Table of Antidifferentiation Formulas
To obtain the most general antiderivative from the particular ones in Table 2, we have to add a constant (or constants), as in Example 1.
Function
Particular antiderivative
Function
Particular antiderivative
c f 共x兲
cF共x兲
sec x
tan x
f 共x兲 ⫹ t共x兲
F共x兲 ⫹ G共x兲
sec x tan x
sec x
1 s1 ⫺ x 2
sin⫺1x
2
n⫹1
x n 共n 苷 ⫺1兲
x n⫹1
1 x
ln x
1 1 ⫹ x2
tan⫺1x
ex
ex
cosh x
sinh x
cos x
sin x
sinh x
cosh x
sin x
⫺cos x
ⱍ ⱍ
EXAMPLE 2 Find all functions t such that
t⬘共x兲 苷 4 sin x ⫹
2x 5 ⫺ sx x
SOLUTION We first rewrite the given function as follows:
t⬘共x兲 苷 4 sin x ⫹
2x 5 1 sx ⫺ 苷 4 sin x ⫹ 2x 4 ⫺ x x sx
Thus we want to find an antiderivative of t⬘共x兲 苷 4 sin x ⫹ 2x 4 ⫺ x⫺1兾2 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
Using the formulas in Table 2 together with Theorem 1, we obtain t共x兲 苷 4共⫺cos x兲 ⫹ 2
x5 x1兾2 ⫺ 1 ⫹C 5 2
苷 ⫺4 cos x ⫹ 25 x 5 ⫺ 2sx ⫹ C In applications of calculus it is very common to have a situation as in Example 2, where it is required to find a function, given knowledge about its derivatives. An equation that involves the derivatives of a function is called a differential equation. These will be studied in some detail in Chapter 9, but for the present we can solve some elementary differential equations. The general solution of a differential equation involves an arbitrary constant (or constants) as in Example 2. However, there may be some extra conditions given that will determine the constants and therefore uniquely specify the solution. Figure 2 shows the graphs of the function f ⬘ in Example 3 and its antiderivative f. Notice that f ⬘共x兲 ⬎ 0, so f is always increasing. Also notice that when f ⬘ has a maximum or minimum, f appears to have an inflection point. So the graph serves as a check on our calculation.
EXAMPLE 3 Find f if f ⬘共x兲 苷 e x ⫹ 20共1 ⫹ x 2 兲⫺1 and f 共0兲 苷 ⫺2. SOLUTION The general antiderivative of
f ⬘共x兲 苷 e x ⫹
f 共x兲 苷 e x ⫹ 20 tan⫺1 x ⫹ C
is
40
20 1 ⫹ x2
To determine C we use the fact that f 共0兲 苷 ⫺2: fª _2
3 f
f 共0兲 苷 e 0 ⫹ 20 tan⫺1 0 ⫹ C 苷 ⫺2 Thus we have C 苷 ⫺2 ⫺ 1 苷 ⫺3, so the particular solution is f 共x兲 苷 e x ⫹ 20 tan⫺1 x ⫺ 3
_25
FIGURE 2
v
EXAMPLE 4 Find f if f ⬙共x兲 苷 12x 2 ⫹ 6x ⫺ 4, f 共0兲 苷 4, and f 共1兲 苷 1.
SOLUTION The general antiderivative of f ⬙共x兲 苷 12x 2 ⫹ 6x ⫺ 4 is
f ⬘共x兲 苷 12
x3 x2 ⫹6 ⫺ 4x ⫹ C 苷 4x 3 ⫹ 3x 2 ⫺ 4x ⫹ C 3 2
Using the antidifferentiation rules once more, we find that f 共x兲 苷 4
x4 x3 x2 ⫹3 ⫺4 ⫹ Cx ⫹ D 苷 x 4 ⫹ x 3 ⫺ 2x 2 ⫹ Cx ⫹ D 4 3 2
To determine C and D we use the given conditions that f 共0兲 苷 4 and f 共1兲 苷 1. Since f 共0兲 苷 0 ⫹ D 苷 4, we have D 苷 4. Since f 共1兲 苷 1 ⫹ 1 ⫺ 2 ⫹ C ⫹ 4 苷 1 we have C 苷 ⫺3. Therefore the required function is f 共x兲 苷 x 4 ⫹ x 3 ⫺ 2x 2 ⫺ 3x ⫹ 4 If we are given the graph of a function f , it seems reasonable that we should be able to sketch the graph of an antiderivative F. Suppose, for instance, that we are given that F共0兲 苷 1. Then we have a place to start, the point 共0, 1兲, and the direction in which we move our pencil is given at each stage by the derivative F⬘共x兲 苷 f 共x兲. In the next example we use the principles of this chapter to show how to graph F even when we don’t have a formula for f . This would be the case, for instance, when f 共x兲 is determined by experimental data. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 4.9
y=ƒ
1
2
3
4
x
FIGURE 3 y
347
v EXAMPLE 5 The graph of a function f is given in Figure 3. Make a rough sketch of an antiderivative F, given that F共0兲 苷 2.
y
0
ANTIDERIVATIVES
y=F(x)
SOLUTION We are guided by the fact that the slope of y 苷 F共x兲 is f 共x兲. We start at the
point 共0, 2兲 and draw F as an initially decreasing function since f 共x兲 is negative when 0 ⬍ x ⬍ 1. Notice that f 共1兲 苷 f 共3兲 苷 0, so F has horizontal tangents when x 苷 1 and x 苷 3. For 1 ⬍ x ⬍ 3, f 共x兲 is positive and so F is increasing. We see that F has a local minimum when x 苷 1 and a local maximum when x 苷 3. For x ⬎ 3, f 共x兲 is negative and so F is decreasing on 共3, ⬁兲. Since f 共x兲 l 0 as x l ⬁, the graph of F becomes flatter as x l ⬁. Also notice that F⬙共x兲 苷 f ⬘共x兲 changes from positive to negative at x 苷 2 and from negative to positive at x 苷 4, so F has inflection points when x 苷 2 and x 苷 4. We use this information to sketch the graph of the antiderivative in Figure 4.
2
Rectilinear Motion
1 0
1
FIGURE 4
x
Antidifferentiation is particularly useful in analyzing the motion of an object moving in a straight line. Recall that if the object has position function s 苷 f 共t兲, then the velocity function is v共t兲 苷 s⬘共t兲. This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is a共t兲 苷 v⬘共t兲, so the velocity function is an antiderivative of the acceleration. If the acceleration and the initial values s共0兲 and v共0兲 are known, then the position function can be found by antidifferentiating twice.
v
EXAMPLE 6 A particle moves in a straight line and has acceleration given by a共t兲 苷 6t ⫹ 4. Its initial velocity is v共0兲 苷 ⫺6 cm兾s and its initial displacement is s共0兲 苷 9 cm. Find its position function s共t兲. SOLUTION Since v⬘共t兲 苷 a共t兲 苷 6t ⫹ 4, antidifferentiation gives
v共t兲 苷 6
t2 ⫹ 4t ⫹ C 苷 3t 2 ⫹ 4t ⫹ C 2
Note that v共0兲 苷 C. But we are given that v共0兲 苷 ⫺6, so C 苷 ⫺6 and v共t兲 苷 3t 2 ⫹ 4t ⫺ 6
Since v共t兲 苷 s⬘共t兲, s is the antiderivative of v : s共t兲 苷 3
t3 t2 ⫹4 ⫺ 6t ⫹ D 苷 t 3 ⫹ 2t 2 ⫺ 6t ⫹ D 3 2
This gives s共0兲 苷 D. We are given that s共0兲 苷 9, so D 苷 9 and the required position function is s共t兲 苷 t 3 ⫹ 2t 2 ⫺ 6t ⫹ 9 An object near the surface of the earth is subject to a gravitational force that produces a downward acceleration denoted by t. For motion close to the ground we may assume that t is constant, its value being about 9.8 m兾s2 (or 32 ft兾s2 ). EXAMPLE 7 A ball is thrown upward with a speed of 48 ft兾s from the edge of a cliff 432 ft above the ground. Find its height above the ground t seconds later. When does it reach its maximum height? When does it hit the ground? SOLUTION The motion is vertical and we choose the positive direction to be upward. At
time t the distance above the ground is s共t兲 and the velocity v共t兲 is decreasing. Therefore
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
the acceleration must be negative and we have a共t兲 苷
dv 苷 ⫺32 dt
Taking antiderivatives, we have v共t兲 苷 ⫺32t ⫹ C
To determine C we use the given information that v共0兲 苷 48. This gives 48 苷 0 ⫹ C, so v共t兲 苷 ⫺32t ⫹ 48
The maximum height is reached when v共t兲 苷 0, that is, after 1.5 s. Since s⬘共t兲 苷 v共t兲, we antidifferentiate again and obtain s共t兲 苷 ⫺16t 2 ⫹ 48t ⫹ D Using the fact that s共0兲 苷 432, we have 432 苷 0 ⫹ D and so Figure 5 shows the position function of the ball in Example 7. The graph corroborates the conclusions we reached: The ball reaches its maximum height after 1.5 s and hits the ground after 6.9 s. 500
s共t兲 苷 ⫺16t 2 ⫹ 48t ⫹ 432 The expression for s共t兲 is valid until the ball hits the ground. This happens when s共t兲 苷 0, that is, when ⫺16t 2 ⫹ 48t ⫹ 432 苷 0 t 2 ⫺ 3t ⫺ 27 苷 0
or, equivalently,
Using the quadratic formula to solve this equation, we get t苷 8
0
FIGURE 5
4.9
We reject the solution with the minus sign since it gives a negative value for t . Therefore the ball hits the ground after 3(1 ⫹ s13 )兾2 ⬇ 6.9 s.
Exercises
1–22 Find the most general antiderivative of the function. (Check your answer by differentiation.) 1. f 共x兲 苷 x ⫺ 3
19. f 共x兲 苷 5e x ⫺ 3 cosh x 21. f 共x兲 苷
2. f 共x兲 苷 2 x 2 ⫺ 2x ⫹ 6 1
3. f 共x兲 苷 2 ⫹ 4 x 2 ⫺ 5 x 3
4. f 共x兲 苷 8x 9 ⫺ 3x 6 ⫹ 12x 3
5. f 共x兲 苷 共x ⫹ 1兲共2 x ⫺ 1兲
6. f 共x兲 苷 x 共2 ⫺ x兲 2
7. f 共x兲 苷 7x 2兾5 ⫹ 8x ⫺4兾5
8. f 共x兲 苷 x 3.4 ⫺ 2x s2⫺1
1
3
4
9. f 共x兲 苷 s2 3 11. f 共x兲 苷 3sx ⫺ 2 s x
13. f 共x兲 苷
1 2 ⫺ 5 x
15. t共t兲 苷
1⫹t⫹t st
24. f 共x兲 苷 4 ⫺ 3共1 ⫹ x 2 兲⫺1,
3t ⫺ t ⫹ 6t t4
2
18. f 共t兲 苷 sin t ⫹ 2 sinh t
Graphing calculator or computer required
F共0兲 苷 4 F共1兲 苷 0
2
16. r共 兲 苷 sec tan ⫺ 2e
17. h共 兲 苷 2 sin ⫺ sec
2 ⫹ x2 1 ⫹ x2
tion. Check your answer by comparing the graphs of f and F .
3 12. f 共x兲 苷 s x 2 ⫹ x sx
14. f 共t兲 苷
22. f 共x兲 苷
; 23–24 Find the antiderivative F of f that satisfies the given condi23. f 共x兲 苷 5x 4 ⫺ 2x 5,
3
20. f 共x兲 苷 2sx ⫹ 6 cos x
x 5 ⫺ x 3 ⫹ 2x x4
10. f 共x兲 苷 e 2
4
2
;
3 ⫾ 3s13 2
25– 48 Find f . 25. f ⬙共x兲 苷 20x 3 ⫺ 12x 2 ⫹ 6x 26. f ⬙共x兲 苷 x 6 ⫺ 4x 4 ⫹ x ⫹ 1 27. f ⬙共x兲 苷 3 x 2兾3 2
28. f ⬙共x兲 苷 6x ⫹ sin x
1. Homework Hints available at stewartcalculus.com
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SECTION 4.9
30. f 共t兲 苷 e t ⫹ t ⫺4
29. f 共t兲 苷 cos t 31. f ⬘共x兲 苷 1 ⫹ 3sx , 33. f ⬘共t兲 苷 4兾共1 ⫹ t 2 兲, 34. f ⬘共t兲 苷 t ⫹ 1兾t ,
√
f 共⫺1兲 苷 2
f 共1兲 苷 0 0
⫺兾2 ⬍ t ⬍ 兾2,
35. f ⬘共t兲 苷 2 cos t ⫹ sec 2 t, 36. f ⬘共x兲 苷 共x ⫺ 1兲兾x,
f 共1兲 苷 , 1 2
2
37. f ⬘共x兲 苷 x ⫺1兾3,
figure. Sketch the graph of a position function.
t ⬎ 0, f 共1兲 苷 6
3
f 共1兲 苷 1,
f 共兾3兲 苷 4
f is continuous and f 共0兲 苷 ⫺1.
f 共⫺1兲 苷 ⫺1
39. f ⬙共x兲 苷 ⫺2 ⫹ 12x ⫺ 12x 2,
y 2
f 共0兲 苷 4, f ⬘共0兲 苷 12
f 共0兲 苷 3,
f 共4兲 苷 20,
43. f ⬙共x兲 苷 4 ⫹ 6x ⫹ 24x 2, 44. f ⬙共x兲 苷 x 3 ⫹ sinh x, 45. f ⬙共x兲 苷 2 ⫹ cos x,
f ⬘共0兲 苷 4
f 共0兲 苷 3, f 共1兲 苷 10
f 共0兲 苷 ⫺1,
47. f ⬙共x兲 苷 x ⫺2,
f 共1兲 苷 0,
48. f 共x兲 苷 cos x,
f 共0兲 苷 1,
f 共2兲 苷 0
f ⬘共0兲 苷 2,
x
2
(b) Starting with the graph in part (a), sketch a rough graph of the antiderivative F that satisfies F共0兲 苷 1. (c) Use the rules of this section to find an expression for F共x兲. (d) Graph F using the expression in part (c). Compare with your sketch in part (b).
f 共兾2兲 苷 0
f 共0兲 苷 0, f 共 兲 苷 0
1
; 56. (a) Use a graphing device to graph f 共x兲 苷 2x ⫺ 3 sx .
f 共2兲 苷 2.6
46. f ⬙共t兲 苷 2e t ⫹ 3 sin t,
x ⬎ 0,
0 _1
f ⬘共4兲 苷 7
f 共0兲 苷 1,
y=fª(x)
1
40. f ⬙共x兲 苷 8x 3 ⫹ 5, f 共1兲 苷 0, f ⬘共1兲 苷 8 41. f ⬙共 兲 苷 sin ⫹ cos ,
t
55. The graph of f ⬘ is shown in the figure. Sketch the graph of f if
f 共⫺1兲 苷 0
f ( 12 ) 苷 1
38. f ⬘共x兲 苷 4兾s1 ⫺ x 2 ,
42. f ⬙共t兲 苷 3兾st ,
349
54. The graph of the velocity function of a particle is shown in the
f 共4兲 苷 25
32. f ⬘共x兲 苷 5x 4 ⫺ 3x 2 ⫹ 4,
ANTIDERIVATIVES
f ⬙共0兲 苷 3
; 57–58 Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin.
49. Given that the graph of f passes through the point 共1, 6兲
57. f 共x兲 苷
and that the slope of its tangent line at 共x, f 共x兲兲 is 2x ⫹ 1, find f 共2兲.
50. Find a function f such that f ⬘共x兲 苷 x 3 and the line x ⫹ y 苷 0
sin x , ⫺2 艋 x 艋 2 1 ⫹ x2
58. f 共x兲 苷 sx 4 ⫺ 2 x 2 ⫹ 2 ⫺ 2,
⫺3 艋 x 艋 3
is tangent to the graph of f . 51–52 The graph of a function f is shown. Which graph is an anti
derivative of f and why? 51.
y
52. f
59. v共t兲 苷 sin t ⫺ cos t, f
b
a
y
59–64 A particle is moving with the given data. Find the position of the particle.
a x
x
b c c
60. v共t兲 苷 1.5 st ,
s共4兲 苷 10
61. a共t兲 苷 2t ⫹ 1,
s共0兲 苷 3,
62. a共t兲 苷 3 cos t ⫺ 2 sin t, 63. a共t兲 苷 10 sin t ⫹ 3 cos t, 64. a共t兲 苷 t ⫺ 4t ⫹ 6, 2
53. The graph of a function is shown in the figure. Make a rough
sketch of an antiderivative F, given that F共0兲 苷 1. y
y=ƒ 0
1
x
s共0兲 苷 0 v 共0兲 苷 ⫺2
s共0兲 苷 0, s共0兲 苷 0,
s共0兲 苷 0,
v 共0兲 苷 4
s共2兲 苷 12
s共1兲 苷 20
65. A stone is dropped from the upper observation deck (the Space
Deck) of the CN Tower, 450 m above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 m兾s, how long does it take to reach the ground?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
66. Show that for motion in a straight line with constant acceleration a, initial velocity v 0 , and initial displacement s 0 , the dis
placement after time t is
a苷
s 苷 at 2 ⫹ v 0 t ⫹ s 0 1 2
67. An object is projected upward with initial velocity v 0 meters
per second from a point s0 meters above the ground. Show that
68. Two balls are thrown upward from the edge of the cliff in
Example 7. The first is thrown with a speed of 48 ft兾s and the other is thrown a second later with a speed of 24 ft兾s. Do the balls ever pass each other? 69. A stone was dropped off a cliff and hit the ground with a speed
of 120 ft兾s. What is the height of the cliff? 70. If a diver of mass m stands at the end of a diving board with
length L and linear density , then the board takes on the shape of a curve y 苷 f 共x兲, where EI y ⬙ 苷 mt共L ⫺ x兲 ⫹ t共L ⫺ x兲 1 2
再
9 ⫺ 0.9t 0
if 0 艋 t 艋 10 if t ⬎ 10
If the raindrop is initially 500 m above the ground, how long does it take to fall? 74. A car is traveling at 50 mi兾h when the brakes are fully applied,
关v共t兲兴 2 苷 v02 ⫺ 19.6关s共t兲 ⫺ s0 兴
2
E and I are positive constants that depend on the material of the board and t 共⬍ 0兲 is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use f 共L兲 to estimate the distance below the horizontal at the end of the board. y
0
downward acceleration is
x
producing a constant deceleration of 22 ft兾s2. What is the distance traveled before the car comes to a stop? 75. What constant acceleration is required to increase the speed of
a car from 30 mi兾h to 50 mi兾h in 5 s? 76. A car braked with a constant deceleration of 16 ft兾s2, pro
ducing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied? 77. A car is traveling at 100 km兾h when the driver sees an accident
80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup? 78. A model rocket is fired vertically upward from rest. Its acceler
ation for the first three seconds is a共t兲 苷 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to ⫺18 ft兾s in 5 s. The rocket then “floats” to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of s and v. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land? 79. A highspeed bullet train accelerates and decelerates at the rate
71. A company estimates that the marginal cost ( in dollars per
item) of producing x items is 1.92 ⫺ 0.002x. If the cost of producing one item is $562, find the cost of producing 100 items. 72. The linear density of a rod of length 1 m is given by
共x兲 苷 1兾sx , in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod. 73. Since raindrops grow as they fall, their surface area increases
and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m兾s and its
of 4 ft兾s2. Its maximum cruising speed is 90 mi兾h. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 4
4
REVIEW
351
Review
Concept Check 1. Explain the difference between an absolute maximum and a
local maximum. Illustrate with a sketch. 2. (a) What does the Extreme Value Theorem say?
(b) Explain how the Closed Interval Method works. 3. (a) State Fermat’s Theorem.
(b) Define a critical number of f . 4. (a) State Rolle’s Theorem.
(b) State the Mean Value Theorem and give a geometric interpretation. 5. (a) State the Increasing /Decreasing Test.
(b) What does it mean to say that f is concave upward on an interval I ? (c) State the Concavity Test. (d) What are inflection points? How do you find them? 6. (a) State the First Derivative Test.
(b) State the Second Derivative Test. (c) What are the relative advantages and disadvantages of these tests? 7. (a) What does l’Hospital’s Rule say?
(b) How can you use l’Hospital’s Rule if you have a product f 共x兲 t共x兲 where f 共x兲 l 0 and t共x兲 l ⬁ as x l a ?
(c) How can you use l’Hospital’s Rule if you have a difference f 共x兲 ⫺ t共x兲 where f 共x兲 l ⬁ and t共x兲 l ⬁ as x l a? (d) How can you use l’Hospital’s Rule if you have a power 关 f 共x兲兴 t共x兲 where f 共x兲 l 0 and t共x兲 l 0 as x l a ? 8. If you have a graphing calculator or computer, why do you
need calculus to graph a function? 9. (a) Given an initial approximation x1 to a root of the equa
tion f 共x兲 苷 0, explain geometrically, with a diagram, how the second approximation x 2 in Newton’s method is obtained. (b) Write an expression for x 2 in terms of x1, f 共x 1 兲, and f ⬘共x 1兲. (c) Write an expression for x n⫹1 in terms of x n , f 共x n 兲, and f ⬘共x n 兲. (d) Under what circumstances is Newton’s method likely to fail or to work very slowly? 10. (a) What is an antiderivative of a function f ?
(b) Suppose F1 and F2 are both antiderivatives of f on an interval I . How are F1 and F2 related?
TrueFalse Quiz Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f ⬘共c兲 苷 0, then f has a local maximum or minimum at c. 2. If f has an absolute minimum value at c, then f ⬘共c兲 苷 0. 3. If f is continuous on 共a, b兲, then f attains an absolute maxi
mum value f 共c兲 and an absolute minimum value f 共d 兲 at some numbers c and d in 共a, b兲.
4. If f is differentiable and f 共⫺1兲 苷 f 共1兲, then there is a num
ⱍ ⱍ
ber c such that c ⬍ 1 and f ⬘共c兲 苷 0.
10. There exists a function f such that f 共x兲 ⬍ 0, f ⬘共x兲 ⬍ 0,
and f ⬙ 共x兲 ⬎ 0 for all x.
11. If f and t are increasing on an interval I , then f ⫹ t is
increasing on I . 12. If f and t are increasing on an interval I , then f ⫺ t is
increasing on I . 13. If f and t are increasing on an interval I , then f t is
increasing on I . 14. If f and t are positive increasing functions on an interval I ,
then f t is increasing on I .
5. If f ⬘共x兲 ⬍ 0 for 1 ⬍ x ⬍ 6, then f is decreasing on (1, 6).
15. If f is increasing and f 共x兲 ⬎ 0 on I , then t共x兲 苷 1兾f 共x兲 is
6. If f ⬙共2兲 苷 0, then 共2, f 共2兲兲 is an inflection point of the
16. If f is even, then f ⬘ is even.
curve y 苷 f 共x兲.
7. If f ⬘共x兲 苷 t⬘共x兲 for 0 ⬍ x ⬍ 1, then f 共x兲 苷 t共x兲 for
0 ⬍ x ⬍ 1.
decreasing on I . 17. If f is periodic, then f ⬘ is periodic. 18. The most general antiderivative of f 共x兲 苷 x ⫺2 is
F共x兲 苷 ⫺
8. There exists a function f such that f 共1兲 苷 ⫺2, f 共3兲 苷 0, and
f ⬘共x兲 ⬎ 1 for all x.
9. There exists a function f such that f 共x兲 ⬎ 0, f ⬘共x兲 ⬍ 0, and
f ⬙ 共x兲 ⬎ 0 for all x.
1 ⫹C x
19. If f ⬘共x兲 exists and is nonzero for all x, then f 共1兲 苷 f 共0兲. 20. lim
xl0
x 苷1 ex
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
Exercises (c) Sketch the graph of f ⬙. (d) Sketch a possible graph of f .
1–6 Find the local and absolute extreme values of the function on the given interval. 1. f 共x兲 苷 x 3 ⫺ 6x 2 ⫹ 9x ⫹ 1, 2. f 共x兲 苷 xs1 ⫺ x ,
3x ⫺ 4 , x2 ⫹ 1
3. f 共x兲 苷
y=f ª(x) _2
关⫺2, 2兴
_1
关⫺1, 3兴
9. lim
xl0
e ⫺1 tan x
8. lim
xl0
e 4x ⫺ 1 ⫺ 4x x2
10. lim
xl⬁
13. lim⫹ xl1
冉
x 1 ⫺ x⫺1 ln x
tan 4x x ⫹ sin 2x
3
4
5
6
7
x
e 4x ⫺ 1 ⫺ 4x x2
x l
冊
14.
19. y 苷 2 ⫺ 2x ⫺ x 3
20. y 苷 x 3 ⫺ 6x 2 ⫺ 15x ⫹ 4
21. y 苷 x 4 ⫺ 3x 3 ⫹ 3x 2 ⫺ x
22. y 苷
x 1 ⫺ x2
24. y 苷
1 1 ⫺ x2 共x ⫺ 2兲 2
23. y 苷
12. lim⫺ 共x ⫺ 兲 csc x
11. lim 共x 2 ⫺ x 3 兲e 2x x l ⫺⬁
2
19–34 Use the guidelines of Section 4.5 to sketch the curve.
x
xl0
1
关⫺2, 1兴
7–14 Evaluate the limit. 7. lim
0
关⫺, 兴
5. f 共x兲 苷 x ⫹ 2 cos x, 6. f 共x兲 苷 x e ,
y
关⫺1, 1兴
4. f 共x兲 苷 sx 2 ⫹ x ⫹ 1 ,
2 ⫺x
关2, 4兴
1 x共x ⫺ 3兲2
25. y 苷 x 2兾共x ⫹ 8兲
26. y 苷 s1 ⫺ x ⫹ s1 ⫹ x
27. y 苷 x s2 ⫹ x
3 x2 ⫹ 1 28. y 苷 s
29. y 苷 e x sin x,
⫺ 艋 x 艋
30. y 苷 4x ⫺ tan x,
lim 共tan x兲cos x
x l共兾2兲 ⫺
⫺兾2 ⬍ x ⬍ 兾2
⫺1
15–17 Sketch the graph of a function that satisfies the given
2
31. y 苷 sin 共1兾x兲
32. y 苷 e 2x⫺x
33. y 苷 共x ⫺ 2兲e ⫺x
34. y 苷 x ⫹ ln共x 2 ⫹ 1兲
conditions.
; 35–38 Produce graphs of f that reveal all the important aspects
15. f 共0兲 苷 0,
f ⬘共⫺2兲 苷 f ⬘共1兲 苷 f ⬘共9兲 苷 0, lim f 共x兲 苷 0, lim f 共x兲 苷 ⫺⬁,
xl⬁
of the curve. Use graphs of f ⬘ and f ⬙ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly.
x l6
f ⬘共x兲 ⬍ 0 on 共⫺⬁, ⫺2兲, 共1, 6兲, and 共9, ⬁兲, f ⬘共x兲 ⬎ 0 on 共⫺2, 1兲 and 共6, 9兲, f ⬙共x兲 ⬎ 0 on 共⫺⬁, 0兲 and 共12, ⬁兲,
35. f 共x兲 苷
f ⬙共x兲 ⬍ 0 on 共0, 6兲 and 共6, 12兲 16. f 共0兲 苷 0,
f ⬙共x兲 ⬍ 0 for x ⬎ 3,
⫺5 艋 x 艋 5
2
main aspects of this function. Estimate the inflection points. Then use calculus to find them exactly.
f ⬙共x兲 ⬎ 0 for 0 ⬍ x ⬍ 3, lim f 共x兲 苷 ⫺2
CAS
xl⬁
18. The figure shows the graph of the derivative f ⬘ of a
function f . (a) On what intervals is f increasing or decreasing? (b) For what values of x does f have a local maximum or minimum?
;
x3 ⫺ x x ⫹x⫹3 2
⫺1兾x in a viewing rectangle that shows all the ; 39. Graph f 共x兲 苷 e
f ⬘共x兲 ⬍ 0 for 0 ⬍ x ⬍ 2,
f ⬘共x兲 ⬎ 0 for x ⬎ 2,
38. f 共x兲 苷 x 2 ⫹ 6.5 sin x,
f ⬘共x兲 苷 ⫺1 if 1 ⬍ x ⬍ 3,
f ⬘共x兲 苷 1 if x ⬎ 3 17. f is odd,
36. f 共x兲 苷
37. f 共x兲 苷 3x 6 ⫺ 5x 5 ⫹ x 4 ⫺ 5x 3 ⫺ 2x 2 ⫹ 2
f is continuous and even,
f ⬘共x兲 苷 2x if 0 ⬍ x ⬍ 1,
x2 ⫺ 1 x3
Graphing calculator or computer required
40. (a) Graph the function f 共x兲 苷 1兾共1 ⫹ e 1兾x 兲.
(b) Explain the shape of the graph by computing the limits of f 共x兲 as x approaches ⬁, ⫺⬁, 0⫹, and 0⫺. (c) Use the graph of f to estimate the coordinates of the inflection points. (d) Use your CAS to compute and graph f ⬙. (e) Use the graph in part (d) to estimate the inflection points more accurately.
CAS Computer algebra system required
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 4 CAS
ⱍ
REVIEW
ⱍ
41– 42 Use the graphs of f, f ⬘, and f ⬙ to estimate the
56. Solve Exercise 55 when CD 苷 2 cm.
xcoordinates of the maximum and minimum points and inflection points of f.
57. The velocity of a wave of length L in deep water is
41. f 共x兲 苷
cos 2 x sx 2 ⫹ x ⫹ 1
; 43. Investigate the family of functions f 共x兲 苷 ln共sin x ⫹ C 兲. What features do the members of this family have in common? How do they differ? For which values of C is f continuous on 共⫺⬁, ⬁兲? For which values of C does f have no graph at all? What happens as C l ⬁? 2
⫺cx ; 44. Investigate the family of functions f 共x兲 苷 cxe . What
happens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of the family. 45. Show that the equation 3x ⫹ 2 cos x ⫹ 5 苷 0 has exactly
one real root.
58. A metal storage tank with volume V is to be constructed in
the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal? 59. A hockey team plays in an arena with a seating capacity of
15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?
; 60. A manufacturer determines that the cost of making x units of
46. Suppose that f is continuous on 关0, 4兴, f 共0兲 苷 1, and
2 艋 f ⬘共x兲 艋 5 for all x in 共0, 4兲. Show that 9 艋 f 共4兲 艋 21.
47. By applying the Mean Value Theorem to the function
f 共x兲 苷 x 1兾5 on the interval 关32, 33兴, show that 5 33 ⬍ 2.0125 2⬍s
48. For what values of the constants a and b is 共1, 3兲 a point of
inflection of the curve y 苷 ax 3 ⫹ bx 2 ?
49. Let t共x兲 苷 f 共x 2 兲, where f is twice differentiable for all x,
f ⬘共x兲 ⬎ 0 for all x 苷 0, and f is concave downward on 共⫺⬁, 0兲 and concave upward on 共0, ⬁兲. (a) At what numbers does t have an extreme value? (b) Discuss the concavity of t.
50. Find two positive integers such that the sum of the first num
ber and four times the second number is 1000 and the product of the numbers is as large as possible. 51. Show that the shortest distance from the point 共x 1, y1 兲 to the
straight line Ax ⫹ By ⫹ C 苷 0 is
ⱍ Ax
⫹ By1 ⫹ C sA2 ⫹ B 2 1
53. Find the smallest possible area of an isosceles triangle that is 54. Find the volume of the largest circular cone that can be
inscribed in a sphere of radius r.
ⱍ ⱍ
ⱍ
55. In ⌬ ABC, D lies on AB, CD ⬜ AB, AD 苷 BD 苷 4 cm,
and CD 苷 5 cm. Where should a point P be chosen on CD so that the sum PA ⫹ PB ⫹ PC is a minimum?
ⱍ
x 5 ⫺ x 4 ⫹ 3x 2 ⫺ 3x ⫺ 2 苷 0 in the interval 关1, 2兴 correct to six decimal places. 62. Use Newton’s method to find all roots of the equation
sin x 苷 x 2 ⫺ 3x ⫹ 1 correct to six decimal places. 63. Use Newton’s method to find the absolute maximum value of
the function f 共t兲 苷 cos t ⫹ t ⫺ t 2 correct to eight decimal places. 64. Use the guidelines in Section 4.5 to sketch the curve
y 苷 x sin x, 0 艋 x 艋 2. Use Newton’s method when necessary.
66. f ⬘共x兲 苷 2e x ⫹ sec x tan x 3 67. f ⬘共x兲 苷 sx 3 ⫹ s x2
68. f ⬘共x兲 苷 sinh x ⫹ 2 cosh x, 69. f ⬘共t兲 苷 2t ⫺ 3 sin t,
circumscribed about a circle of radius r .
ⱍ ⱍ ⱍ ⱍ ⱍ
61. Use Newton’s method to find the root of the equation
65. f ⬘共x兲 苷 cos x ⫺ 共1 ⫺ x 2兲⫺1兾2
point 共3, 0兲.
ⱍ
a commodity is C共x兲 苷 1800 ⫹ 25x ⫺ 0.2x 2 ⫹ 0.001x 3 and the demand function is p共x兲 苷 48.2 ⫺ 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.
65–72 Find f .
ⱍ
52. Find the point on the hyperbola x y 苷 8 that is closest to the
ⱍ
L C ⫹ C L
where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?
42. f 共x兲 苷 e⫺0.1x ln共x 2 ⫺ 1兲
ⱍ
冑
v苷K
⫺ 艋 x 艋
,
353
u ⫹ su , u
f 共0兲 苷 2
f 共0兲 苷 5
2
70. f ⬘共u兲 苷
f 共1兲 苷 3
71. f ⬙共x兲 苷 1 ⫺ 6x ⫹ 48x 2,
f 共0兲 苷 1,
72. f ⬙共x兲 苷 2x ⫹ 3x ⫺ 4x ⫹ 5, 3
2
f ⬘共0兲 苷 2
f 共0兲 苷 2,
f 共1兲 苷 0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLICATIONS OF DIFFERENTIATION
73–74 A particle is moving with the given data. Find the
position of the particle. 73. v共t兲 苷 2t ⫺ 1兾共1 ⫹ t 2 兲, 74. a共t兲 苷 sin t ⫹ 3 cos t,
s共0兲 苷 1
as shown in the figure. Show that the range of the projectile, measured up the slope, is given by R共 兲 苷
s共0兲 苷 0, v 共0兲 苷 2
x ; 75. (a) If f 共x兲 苷 0.1e ⫹ sin x, ⫺4 艋 x 艋 4, use a graph of f
to sketch a rough graph of the antiderivative F of f that satisfies F共0兲 苷 0. (b) Find an expression for F共x兲. (c) Graph F using the expression in part (b). Compare with your sketch in part (a).
2v 2 cos sin共 ⫺ ␣兲 t cos2␣
(b) Determine so that R is a maximum. (c) Suppose the plane is at an angle ␣ below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R. y
; 76. Investigate the family of curves given by f 共x兲 苷 x 4 ⫹ x 3 ⫹ cx 2
¨
In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs. 77. A canister is dropped from a helicopter 500 m above the
ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m兾s. Will it burst? 78. In an automobile race along a straight road, car A passed
car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make. 79. A rectangular beam will be cut from a cylindrical log of
radius 10 inches. (a) Show that the beam of maximal crosssectional area is a square. (b) Four rectangular planks will be cut from the four sections of the log that remain after cutting the square beam. Determine the dimensions of the planks that will have maximal crosssectional area. (c) Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.
å
R x
0
81. Show that, for x ⬎ 0,
x ⬍ tan⫺1x ⬍ x 1 ⫹ x2 82. Sketch the graph of a function f such that f ⬘共x兲 ⬍ 0 for
ⱍ ⱍ
ⱍ ⱍ
all x, f ⬙共x兲 ⬎ 0 for x ⬎ 1, f ⬙共x兲 ⬍ 0 for x ⬍ 1, and lim x l⫾⬁ 关 f 共x兲 ⫹ x兴 苷 0.
83. A light is to be placed atop a pole of height h feet to
illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle (see the figure) and inversely proportional to the square of the distance d from the source. (a) How tall should the light pole be to maximize I ? (b) Suppose that the light pole is h feet tall and that a woman is walking away from the base of the pole at the rate of 4 ft兾s. At what rate is the intensity of the light at the point on her back 4 ft above the ground decreasing when she reaches the outer edge of the traffic circle? ¨ h
depth
10
d 40
84. Water is flowing at a constant rate into a spherical tank. Let
width 80. If a projectile is fired with an initial velocity v at an angle of
inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola y 苷 共tan 兲x ⫺
t x2 2v 2 cos 2
P
0⬍⬍
2
(a) Suppose the projectile is fired from the base of a plane that is inclined at an angle ␣, ␣ ⬎ 0, from the horizontal,
V共t兲 be the volume of water in the tank and H共t兲 be the height of the water in the tank at time t. (a) What are the meanings of V⬘共t兲 and H⬘共t兲? Are these derivatives positive, negative, or zero? (b) Is V ⬙共t兲 positive, negative, or zero? Explain. (c) Let t1, t 2, and t 3 be the times when the tank is onequarter full, half full, and threequarters full, respectively. Are the values H ⬙共t1兲, H ⬙共t 2 兲, and H ⬙共t 3 兲 positive, negative, or zero? Why?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Problems Plus One of the most important principles of problem solving is analogy (see page 75). If you are having trouble getting started on a problem, it is sometimes helpful to start by solving a similar, but simpler, problem. The following example illustrates the principle. Cover up the solution and try solving it yourself first. EXAMPLE 1 If x, y, and z are positive numbers, prove that
共x 2 ⫹ 1兲共 y 2 ⫹ 1兲共z 2 ⫹ 1兲 艌8 xyz SOLUTION It may be difficult to get started on this problem. (Some students have tackled
it by multiplying out the numerator, but that just creates a mess.) Let’s try to think of a similar, simpler problem. When several variables are involved, it’s often helpful to think of an analogous problem with fewer variables. In the present case we can reduce the number of variables from three to one and prove the analogous inequality x2 ⫹ 1 艌2 x
1
for x ⬎ 0
In fact, if we are able to prove 1 , then the desired inequality follows because 共x 2 ⫹ 1兲共 y 2 ⫹ 1兲共z 2 ⫹ 1兲 苷 xyz
冉 冊冉 冊冉 冊 x2 ⫹ 1 x
y2 ⫹ 1 y
z2 ⫹ 1 z
艌2ⴢ2ⴢ2苷8
The key to proving 1 is to recognize that it is a disguised version of a minimum problem. If we let f 共x兲 苷
x2 ⫹ 1 1 苷x⫹ x x
x⬎0
then f ⬘共x兲 苷 1 ⫺ 共1兾x 2 兲, so f ⬘共x兲 苷 0 when x 苷 1. Also, f ⬘共x兲 ⬍ 0 for 0 ⬍ x ⬍ 1 and f ⬘共x兲 ⬎ 0 for x ⬎ 1. Therefore the absolute minimum value of f is f 共1兲 苷 2. This means that x2 ⫹ 1 艌2 x
PS
LOOK BACK
What have we learned from the solution to this example? N
N
To solve a problem involving several variables, it might help to solve a similar problem with just one variable. When trying to prove an inequality, it might help to think of it as a maximum or minimum problem.
for all positive values of x
and, as previously mentioned, the given inequality follows by multiplication. The inequality in 1 could also be proved without calculus. In fact, if x ⬎ 0, we have x2 ⫹ 1 艌2 x
&? &?
x 2 ⫹ 1 艌 2x
&?
x 2 ⫺ 2x ⫹ 1 艌 0
共x ⫺ 1兲2 艌 0
Because the last inequality is obviously true, the first one is true too. 355 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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2
1. If a rectangle has its base on the xaxis and two vertices on the curve y 苷 e ⫺x , show that the
Problems
rectangle has the largest possible area when the two vertices are at the points of inflection of the curve.
ⱍ
ⱍ
2. Show that sin x ⫺ cos x 艋 s2 for all x. 3. Does the function f 共x兲 苷 e 10 ⱍ x⫺2 ⱍ⫺x have an absolute maximum? If so, find it. What about an 2
absolute minimum?
ⱍ ⱍ
4. Show that x 2 y 2共4 ⫺ x 2 兲共4 ⫺ y 2 兲 艋 16 for all numbers x and y such that x 艋 2 and
ⱍ y ⱍ 艋 2.
5. Show that the inflection points of the curve y 苷 共sin x兲兾x lie on the curve y 2共x 4 ⫹ 4兲 苷 4. 6. Find the point on the parabola y 苷 1 ⫺ x 2 at which the tangent line cuts from the first quad
rant the triangle with the smallest area. 7. If a, b, c, and d are constants such that
lim
xl0
ax 2 ⫹ sin bx ⫹ sin cx ⫹ sin dx 苷8 3x 2 ⫹ 5x 4 ⫹ 7x 6
find the value of the sum a ⫹ b ⫹ c ⫹ d.
ⱍ
ⱍ
8. Sketch the set of all points 共x, y兲 such that x ⫹ y 艋 e x. 9. Find the highest and lowest points on the curve x ⫹ x y ⫹ y 2 苷 12. 2
10. For what values of c does the curve y 苷 cx 3 ⫹ e x have inflection points? 11. If P共a, a 2 兲 is any point on the parabola y 苷 x 2, except for the origin, let Q be the point where
y
the normal line intersects the parabola again (see the figure). Show that the line segment PQ has the shortest possible length when a 苷 1兾s2 . Q
12. Sketch the region in the plane consisting of all points 共x, y兲 such that
ⱍ
ⱍ
2xy 艋 x ⫺ y 艋 x 2 ⫹ y 2 P 0
13. The line y 苷 mx ⫹ b intersects the parabola y 苷 x 2 in points A and B. (See the figure.) Find x
the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB. y
FIGURE FOR PROBLEM 11
y=≈ B A
y=mx+b O
P
x
14. ABCD is a square piece of paper with sides of length 1 m. A quartercircle is drawn from B
to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quartercircle. Determine the maximum and minimum areas that the triangle AEF can have. 15. For which positive numbers a does the curve y 苷 a x intersect the line y 苷 x ?
356 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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16. For what value of a is the following equation true?
lim
xl⬁
冉 冊 x⫹a x⫺a
x
苷e
17. Let f 共x兲 苷 a 1 sin x ⫹ a 2 sin 2x ⫹ ⭈ ⭈ ⭈ ⫹ a n sin nx, where a 1 , a 2 , . . . , a n are real numbers
ⱍ
ⱍ ⱍ
ⱍ
and n is a positive integer. If it is given that f 共x兲 艋 sin x for all x, show that
ⱍa
1
ⱍ
⫹ 2a 2 ⫹ ⭈ ⭈ ⭈ ⫹ na n 艋 1
18. An arc PQ of a circle subtends a central angle as in the figure. Let A共 兲 be the area between
the chord PQ and the arc PQ. Let B共 兲 be the area between the tangent lines PR, QR, and the arc. Find A共 兲 lim l 0⫹ B共 兲 P
A(¨ )
¨
B(¨ )
R
Q 19. The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h
of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at a point P and the transmitted signals are recorded at a point Q, which is a distance D from P. The first signal to arrive at Q travels along the surface and takes T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (a) Express T1, T2, and T3 in terms of D, h, c1, c2, and . (b) Show that T2 is a minimum when sin 苷 c1兾c2. (c) Suppose that D 苷 1 km, T1 苷 0.26 s, T2 苷 0.32 s, and T3 苷 0.34 s. Find c1, c2, and h. P
Q
D speed of sound=c¡
h
¨
¨ R
O
S
speed of sound=c™
Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines. 20. For what values of c is there a straight line that intersects the curve
y 苷 x 4 ⫹ cx 3 ⫹ 12x 2 ⫺ 5x ⫹ 2 in four distinct points?
357 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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21. One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des
d B
9/21/10
E
C
x r F
D FIGURE FOR PROBLEM 21
Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d ⬎ r ), a rope of length is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. As l’Hospital argued, this happens when the distance ED is maximized. Show that when the system reaches equilibrium, the value of x is r (r ⫹ sr 2 ⫹ 8d 2 ) 4d
ⱍ
ⱍ
Notice that this expression is independent of both W and . 22. Given a sphere with radius r , find the height of a pyramid of minimum volume whose base is
a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular ngon? (A regular ngon is a polygon with n equal sides and angles.) (Use the fact that the volume of a pyramid is 13 Ah, where A is the area of the base.) 23. Assume that a snowball melts so that its volume decreases at a rate proportional to its surface
area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? 24. A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical
bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n 苷 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1 ⫹ sn .
358 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Integrals
In Example 7 in Section 5.4 you will see how to use power consumption data and an integral to compute the amount of energy used in one day in San Francisco.
© Nathan Jaskowiak / Shutterstock
In Chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. In much the same way, this chapter starts with the area and distance problems and uses them to formulate the idea of a definite integral, which is the basic concept of integral calculus. We will see in Chapters 6 and 8 how to use the integral to solve problems concerning volumes, lengths of curves, population predictions, cardiac output, forces on a dam, work, consumer surplus, and baseball, among many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this chapter that it greatly simplifies the solution of many problems.
359 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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360
CHAPTER 5
5.1
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INTEGRALS
Areas and Distances
Now is a good time to read (or reread) A Preview of Calculus (see page 1). It discusses the unifying ideas of calculus and helps put in perspective where we have been and where we are going. y
y=ƒ x=a S
x=b
a
0
x
b
FIGURE 1
In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit.
The Area Problem We begin by attempting to solve the area problem: Find the area of the region S that lies under the curve y 苷 f 共x兲 from a to b. This means that S, illustrated in Figure 1, is bounded by the graph of a continuous function f [where f 共x兲 艌 0], the vertical lines x 苷 a and x 苷 b, and the xaxis. In trying to solve the area problem we have to ask ourselves: What is the meaning of the word area ? This question is easy to answer for regions with straight sides. For a rectangle, the area is defined as the product of the length and the width. The area of a triangle is half the base times the height. The area of a polygon is found by dividing it into triangles (as in Figure 2) and adding the areas of the triangles.
S=s(x, y)  a¯x¯b, 0¯y¯ƒd A™ w
h
A¢
A¡
b
l FIGURE 2
A£
A= 21 bh
A=lw
A=A¡+A™+A£+A¢
However, it isn’t so easy to find the area of a region with curved sides. We all have an intuitive idea of what the area of a region is. But part of the area problem is to make this intuitive idea precise by giving an exact definition of area. Recall that in defining a tangent we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We pursue a similar idea for areas. We first approximate the region S by rectangles and then we take the limit of the areas of these rectangles as we increase the number of rectangles. The following example illustrates the procedure.
y (1, 1)
y=≈
v
S
EXAMPLE 1 Use rectangles to estimate the area under the parabola y 苷 x 2 from 0 to 1
(the parabolic region S illustrated in Figure 3). SOLUTION We first notice that the area of S must be somewhere between 0 and 1 because
0
1
x
FIGURE 3
S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips S1, S2 , S3 , and S4 by drawing the vertical lines x 苷 14 , x 苷 12 , and x 苷 34 as in Figure 4(a). y
y
(1, 1)
(1, 1)
y=≈
S¢ S™
S£
S¡ 0
FIGURE 4
1 4
1 2
(a)
3 4
1
x
0
1 4
1 2
3 4
1
x
(b)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.1
AREAS AND DISTANCES
361
We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip [see Figure 4(b)]. In other words, the heights of these rectangles are the values of the function f 共x兲 苷 x 2 at the right endpoints of the subintervals [0, 14 ], [ 14 , 12 ], [ 12 , 34 ], and [ 34 , 1]. Each rectangle has width 41 and the heights are ( 14 )2, ( 12 )2, ( 34 )2, and 12. If we let R 4 be the sum of the areas of these approximating rectangles, we get R 4 苷 14 ⴢ ( 14 )2 ⫹ 14 ⴢ ( 12 )2 ⫹ 14 ⴢ ( 34 )2 ⫹ 14 ⴢ 12 苷 15 32 苷 0.46875 From Figure 4(b) we see that the area A of S is less than R 4 , so A ⬍ 0.46875 y
Instead of using the rectangles in Figure 4(b) we could use the smaller rectangles in Figure 5 whose heights are the values of f at the left endpoints of the subintervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is
(1, 1)
y=≈
L 4 苷 14 ⴢ 0 2 ⫹ 14 ⴢ ( 14 )2 ⫹ 14 ⴢ ( 12 )2 ⫹ 14 ⴢ ( 34 )2 苷 327 苷 0.21875 We see that the area of S is larger than L 4 , so we have lower and upper estimates for A: 0
1 4
1 2
3 4
1
x
FIGURE 5
0.21875 ⬍ A ⬍ 0.46875 We can repeat this procedure with a larger number of strips. Figure 6 shows what happens when we divide the region S into eight strips of equal width. y
y (1, 1)
(1, 1)
y=≈
0
FIGURE 6
Approximating S with eight rectangles
1 8
1
(a) Using left endpoints
x
0
1 8
1
x
(b) Using right endpoints
By computing the sum of the areas of the smaller rectangles 共L 8 兲 and the sum of the areas of the larger rectangles 共R 8 兲, we obtain better lower and upper estimates for A: 0.2734375 ⬍ A ⬍ 0.3984375 n
Ln
Rn
10 20 30 50 100 1000
0.2850000 0.3087500 0.3168519 0.3234000 0.3283500 0.3328335
0.3850000 0.3587500 0.3501852 0.3434000 0.3383500 0.3338335
So one possible answer to the question is to say that the true area of S lies somewhere between 0.2734375 and 0.3984375. We could obtain better estimates by increasing the number of strips. The table at the left shows the results of similar calculations (with a computer) using n rectangles whose heights are found with left endpoints 共L n 兲 or right endpoints 共R n 兲. In particular, we see by using 50 strips that the area lies between 0.3234 and 0.3434. With 1000 strips we narrow it down even more: A lies between 0.3328335 and 0.3338335. A good estimate is obtained by averaging these numbers: A ⬇ 0.3333335.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
From the values in the table in Example 1, it looks as if R n is approaching increases. We confirm this in the next example.
1 3
as n
v EXAMPLE 2 For the region S in Example 1, show that the sum of the areas of the upper approximating rectangles approaches 31 , that is, lim R n 苷 13
nl⬁
y
SOLUTION R n is the sum of the areas of the n rectangles in Figure 7. Each rectangle
(1, 1)
has width 1兾n and the heights are the values of the function f 共x兲 苷 x 2 at the points 1兾n, 2兾n, 3兾n, . . . , n兾n; that is, the heights are 共1兾n兲2, 共2兾n兲2, 共3兾n兲2, . . . , 共n兾n兲2. Thus
y=≈
Rn 苷
0
1
x
冉冊 冉冊 冉冊 1 n
2
⫹
1 n
2
2 n
⫹
1 n
3 n
2
⫹ ⭈⭈⭈ ⫹
苷
1 1 2 ⭈ 共1 ⫹ 2 2 ⫹ 3 2 ⫹ ⭈ ⭈ ⭈ ⫹ n 2 兲 n n2
苷
1 2 共1 ⫹ 2 2 ⫹ 3 2 ⫹ ⭈ ⭈ ⭈ ⫹ n 2 兲 n3
1 n
FIGURE 7
1 n
1 n
冉冊 n n
2
Here we need the formula for the sum of the squares of the first n positive integers:
1
12 ⫹ 2 2 ⫹ 3 2 ⫹ ⭈ ⭈ ⭈ ⫹ n 2 苷
n共n ⫹ 1兲共2n ⫹ 1兲 6
Perhaps you have seen this formula before. It is proved in Example 5 in Appendix E. Putting Formula 1 into our expression for R n , we get Rn 苷 Here we are computing the limit of the sequence 兵R n 其. Sequences and their limits were discussed in A Preview of Calculus and will be studied in detail in Section 11.1. The idea is very similar to a limit at infinity (Section 2.6) except that in writing lim n l ⬁ we restrict n to be a positive integer. In particular, we know that 1 lim 苷 0 nl ⬁ n When we write lim n l ⬁ Rn 苷 13 we mean that we can make Rn as close to 13 as we like by taking n sufficiently large.
1 n共n ⫹ 1兲共2n ⫹ 1兲 共n ⫹ 1兲共2n ⫹ 1兲 ⭈ 苷 n3 6 6n 2
Thus we have lim R n 苷 lim
nl⬁
nl⬁
共n ⫹ 1兲共2n ⫹ 1兲 6n 2
苷 lim
1 6
苷 lim
1 6
nl⬁
nl⬁
苷
冉 冊冉 冊 冉 冊冉 冊 n⫹1 n
1⫹
1 n
2n ⫹ 1 n
2⫹
1 n
1 1 ⴢ1ⴢ2苷 6 3 1
It can be shown that the lower approximating sums also approach 3 , that is, lim L n 苷 13
nl⬁
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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363
AREAS AND DISTANCES
From Figures 8 and 9 it appears that, as n increases, both L n and R n become better and better approximations to the area of S. Therefore we define the area A to be the limit of the sums of the areas of the approximating rectangles, that is, TEC In Visual 5.1 you can create pictures like those in Figures 8 and 9 for other values of n.
A 苷 lim R n 苷 lim L n 苷 13 nl⬁
nl⬁
y
y
n=10 R¡¸=0.385
0
y
n=50 R∞¸=0.3434
n=30 R£¸Å0.3502
1
x
0
1
x
0
1
x
1
x
FIGURE 8 Right endpoints produce upper sums because ƒ
[email protected] is increasing y
y
n=10 L¡¸=0.285
0
y
n=50 L∞¸=0.3234
n=30 L£¸Å0.3169
1
x
0
1
x
0
FIGURE 9 Left endpoints produce lower sums because ƒ
[email protected] is increasing
Let’s apply the idea of Examples 1 and 2 to the more general region S of Figure 1. We start by subdividing S into n strips S1, S2 , . . . , Sn of equal width as in Figure 10. y
y=ƒ
S¡
0
a
S™
⁄
S£
¤
Si
‹
. . . xi1
Sn
xi
. . . xn1
b
x
FIGURE 10
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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The width of the interval 关a, b兴 is b ⫺ a, so the width of each of the n strips is ⌬x 苷
b⫺a n
These strips divide the interval [a, b] into n subintervals 关x 0 , x 1 兴,
关x 1, x 2 兴,
关x 2 , x 3 兴,
...,
关x n⫺1, x n 兴
where x 0 苷 a and x n 苷 b. The right endpoints of the subintervals are x 1 苷 a ⫹ ⌬x, x 2 苷 a ⫹ 2 ⌬x, x 3 苷 a ⫹ 3 ⌬x, ⭈ ⭈ ⭈ Let’s approximate the ith strip Si by a rectangle with width ⌬x and height f 共x i 兲, which is the value of f at the right endpoint (see Figure 11). Then the area of the ith rectangle is f 共x i 兲 ⌬x . What we think of intuitively as the area of S is approximated by the sum of the areas of these rectangles, which is R n 苷 f 共x 1 兲 ⌬x ⫹ f 共x 2 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n 兲 ⌬x y
Îx
f(xi)
0
a
⁄
¤
‹
xi1
b
xi
x
FIGURE 11
Figure 12 shows this approximation for n 苷 2, 4, 8, and 12. Notice that this approximation appears to become better and better as the number of strips increases, that is, as n l ⬁. Therefore we define the area A of the region S in the following way. y
y
0
a
⁄
(a) n=2
b x
0
y
a
⁄
¤
(b) n=4
‹
b
x
0
y
b
a
(c) n=8
x
0
b
a
x
(d) n=12
FIGURE 12
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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365
2 Deﬁnition The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:
A 苷 lim R n 苷 lim 关 f 共x 1 兲 ⌬x ⫹ f 共x 2 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n 兲 ⌬x兴 nl⬁
nl⬁
It can be proved that the limit in Definition 2 always exists, since we are assuming that f is continuous. It can also be shown that we get the same value if we use left endpoints: 3
A 苷 lim L n 苷 lim 关 f 共x 0 兲 ⌬x ⫹ f 共x 1 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n⫺1 兲 ⌬x兴 nl⬁
nl⬁
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number x*i in the ith subinterval 关x i⫺1, x i 兴. We call the numbers x1*, x2*, . . . , x n* the sample points. Figure 13 shows approximating rectangles when the sample points are not chosen to be endpoints. So a more general expression for the area of S is A 苷 lim 关 f 共x*1 兲 ⌬x ⫹ f 共x2* 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x*n 兲 ⌬x兴
4
nl⬁
y
Îx
f(x *) i
0
FIGURE 13
a
⁄
x*¡
¤ x™*
‹ x£*
xi1
xi
b
xn1
x *i
x
x n*
NOTE It can be shown that an equivalent definition of area is the following: A is the unique number that is smaller than all the upper sums and bigger than all the lower sums. We saw in Examples 1 and 2, for instance, that the area ( A 苷 13 ) is trapped between all the left approximating sums L n and all the right approximating sums Rn. The function in those examples, f 共x兲 苷 x 2, happens to be increasing on 关0, 1兴 and so the lower sums arise from left endpoints and the upper sums from right endpoints. (See Figures 8 and 9.) In general, we form lower (and upper) sums by choosing the sample points x*i so that f 共x*i 兲 is the minimum (and maximum) value of f on the ith subinterval. (See Figure 14 and Exercises 7–8). y
FIGURE 14
Lower sums (short rectangles) and upper sums (tall rectangles)
0
a
b
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
We often use sigma notation to write sums with many terms more compactly. For instance,
This tells us to end with i=n. This tells us to add.
9/21/10
n
n
μ
兺 f 共x 兲 ⌬x 苷 f 共x 兲 ⌬x ⫹ f 共x 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x 兲 ⌬x
f(xi) Îx
i
This tells us to start with i=m.
1
2
n
i苷1
i=m
So the expressions for area in Equations 2, 3, and 4 can be written as follows: n
If you need practice with sigma notation, look at the examples and try some of the exercises in Appendix E.
A 苷 lim
兺 f 共x 兲 ⌬x
A 苷 lim
兺 f 共x
A 苷 lim
兺 f 共x*兲 ⌬x
i
n l ⬁ i苷1 n
n l ⬁ i苷1
i⫺1
兲 ⌬x
n
i
n l ⬁ i苷1
We can also rewrite Formula 1 in the following way: n
兺i
2
苷
i苷1
n共n ⫹ 1兲共2n ⫹ 1兲 6
EXAMPLE 3 Let A be the area of the region that lies under the graph of f 共x兲 苷 e⫺x
between x 苷 0 and x 苷 2. (a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. (b) Estimate the area by taking the sample points to be midpoints and using four subintervals and then ten subintervals. SOLUTION
(a) Since a 苷 0 and b 苷 2, the width of a subinterval is ⌬x 苷
2⫺0 2 苷 n n
So x 1 苷 2兾n, x 2 苷 4兾n, x 3 苷 6兾n, x i 苷 2i兾n, and x n 苷 2n兾n. The sum of the areas of the approximating rectangles is Rn 苷 f 共x 1 兲 ⌬x ⫹ f 共x 2 兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n 兲 ⌬x 苷 e⫺x1 ⌬x ⫹ e⫺x 2 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ e⫺xn ⌬x 苷 e⫺2兾n
冉冊 2 n
冉冊
⫹ e⫺4兾n
2 n
冉冊
⫹ ⭈ ⭈ ⭈ ⫹ e⫺2n兾n
2 n
According to Definition 2, the area is A 苷 lim Rn 苷 lim nl⬁
nl⬁
2 ⫺2兾n 共e ⫹ e⫺4兾n ⫹ e⫺6兾n ⫹ ⭈ ⭈ ⭈ ⫹ e⫺2n兾n 兲 n
Using sigma notation we could write A 苷 lim
nl⬁
2 n
n
兺e
⫺2i兾n
i苷1
It is difficult to evaluate this limit directly by hand, but with the aid of a computer algebra system it isn’t hard (see Exercise 28). In Section 5.3 we will be able to find A more easily using a different method. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.1 y 1
AREAS AND DISTANCES
367
(b) With n 苷 4 the subintervals of equal width ⌬x 苷 0.5 are 关0, 0.5兴, 关0.5, 1兴, 关1, 1.5兴, and 关1.5, 2兴. The midpoints of these subintervals are x1* 苷 0.25, x2* 苷 0.75, x3* 苷 1.25, and x4* 苷 1.75, and the sum of the areas of the four approximating rectangles (see Figure 15) is
y=e–®
4
M4 苷 0
1
2
x
兺 f 共x*兲 ⌬x i
i苷1
苷 f 共0.25兲 ⌬x ⫹ f 共0.75兲 ⌬x ⫹ f 共1.25兲 ⌬x ⫹ f 共1.75兲 ⌬x
FIGURE 15
苷 e⫺0.25共0.5兲 ⫹ e⫺0.75共0.5兲 ⫹ e⫺1.25共0.5兲 ⫹ e⫺1.75共0.5兲 苷 12 共e⫺0.25 ⫹ e⫺0.75 ⫹ e⫺1.25 ⫹ e⫺1.75 兲 ⬇ 0.8557
y
So an estimate for the area is 1
A ⬇ 0.8557
y=e–®
With n 苷 10 the subintervals are 关0, 0.2兴, 关0.2, 0.4兴, . . . , 关1.8, 2兴 and the midpoints are * 苷 1.9. Thus x1* 苷 0.1, x2* 苷 0.3, x3* 苷 0.5, . . . , x10 0
FIGURE 16
1
2
x
A ⬇ M10 苷 f 共0.1兲 ⌬x ⫹ f 共0.3兲 ⌬x ⫹ f 共0.5兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共1.9兲 ⌬x 苷 0.2共e⫺0.1 ⫹ e⫺0.3 ⫹ e⫺0.5 ⫹ ⭈ ⭈ ⭈ ⫹ e⫺1.9 兲 ⬇ 0.8632 From Figure 16 it appears that this estimate is better than the estimate with n 苷 4.
The Distance Problem Now let’s consider the distance problem: Find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. (In a sense this is the inverse problem of the velocity problem that we discussed in Section 2.1.) If the velocity remains constant, then the distance problem is easy to solve by means of the formula distance 苷 velocity ⫻ time But if the velocity varies, it’s not so easy to find the distance traveled. We investigate the problem in the following example.
v EXAMPLE 4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30second time interval. We take speedometer readings every five seconds and record them in the following table: Time (s) Velocity (mi兾h)
0
5
10
15
20
25
30
17
21
24
29
32
31
28
In order to have the time and the velocity in consistent units, let’s convert the velocity readings to feet per second (1 mi兾h 苷 5280兾3600 ft兾s): Time (s) Velocity (ft兾s)
0
5
10
15
20
25
30
25
31
35
43
47
46
41
During the first five seconds the velocity doesn’t change very much, so we can estimate the distance traveled during that time by assuming that the velocity is constant. If we Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
take the velocity during that time interval to be the initial velocity (25 ft兾s), then we obtain the approximate distance traveled during the first five seconds: 25 ft兾s ⫻ 5 s 苷 125 ft Similarly, during the second time interval the velocity is approximately constant and we take it to be the velocity when t 苷 5 s. So our estimate for the distance traveled from t 苷 5 s to t 苷 10 s is 31 ft兾s ⫻ 5 s 苷 155 ft If we add similar estimates for the other time intervals, we obtain an estimate for the total distance traveled: 共25 ⫻ 5兲 ⫹ 共31 ⫻ 5兲 ⫹ 共35 ⫻ 5兲 ⫹ 共43 ⫻ 5兲 ⫹ 共47 ⫻ 5兲 ⫹ 共46 ⫻ 5兲 苷 1135 ft We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity. Then our estimate becomes 共31 ⫻ 5兲 ⫹ 共35 ⫻ 5兲 ⫹ 共43 ⫻ 5兲 ⫹ 共47 ⫻ 5兲 ⫹ 共46 ⫻ 5兲 ⫹ 共41 ⫻ 5兲 苷 1215 ft If we had wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second.
√ 40
20
0
10
FIGURE 17
20
30
t
Perhaps the calculations in Example 4 remind you of the sums we used earlier to estimate areas. The similarity is explained when we sketch a graph of the velocity function of the car in Figure 17 and draw rectangles whose heights are the initial velocities for each time interval. The area of the first rectangle is 25 ⫻ 5 苷 125, which is also our estimate for the distance traveled in the first five seconds. In fact, the area of each rectangle can be interpreted as a distance because the height represents velocity and the width represents time. The sum of the areas of the rectangles in Figure 17 is L 6 苷 1135, which is our initial estimate for the total distance traveled. In general, suppose an object moves with velocity v 苷 f 共t兲, where a 艋 t 艋 b and f 共t兲 艌 0 (so the object always moves in the positive direction). We take velocity readings at times t0 共苷 a兲, t1, t2 , . . . , tn 共苷 b兲 so that the velocity is approximately constant on each subinterval. If these times are equally spaced, then the time between consecutive readings is ⌬t 苷 共b ⫺ a兲兾n. During the first time interval the velocity is approximately f 共t0 兲 and so the distance traveled is approximately f 共t0 兲 ⌬t. Similarly, the distance traveled during the second time interval is about f 共t1 兲 ⌬t and the total distance traveled during the time interval 关a, b兴 is approximately n
f 共t0 兲 ⌬t ⫹ f 共t1 兲 ⌬t ⫹ ⭈ ⭈ ⭈ ⫹ f 共tn⫺1 兲 ⌬t 苷
兺 f 共t
i⫺1
兲 ⌬t
i苷1
If we use the velocity at right endpoints instead of left endpoints, our estimate for the total distance becomes n
f 共t1 兲 ⌬t ⫹ f 共t2 兲 ⌬t ⫹ ⭈ ⭈ ⭈ ⫹ f 共tn 兲 ⌬t 苷
兺 f 共t 兲 ⌬t i
i苷1
The more frequently we measure the velocity, the more accurate our estimates become, so Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.1
AREAS AND DISTANCES
369
it seems plausible that the exact distance d traveled is the limit of such expressions: n
d 苷 lim
5
兺 f 共t
nl ⬁ i苷1
i⫺1
n
兲 ⌬t 苷 lim
兺 f 共t 兲 ⌬t
nl ⬁ i苷1
i
We will see in Section 5.4 that this is indeed true. Because Equation 5 has the same form as our expressions for area in Equations 2 and 3, it follows that the distance traveled is equal to the area under the graph of the velocity function. In Chapter 6 we will see that other quantities of interest in the natural and social sciences—such as the work done by a variable force or the cardiac output of the heart—can also be interpreted as the area under a curve. So when we compute areas in this chapter, bear in mind that they can be interpreted in a variety of practical ways.
5.1
Exercises
1. (a) By reading values from the given graph of f , use four rect
angles to find a lower estimate and an upper estimate for the area under the given graph of f from x 苷 0 to x 苷 8. In each case sketch the rectangles that you use. (b) Find new estimates using eight rectangles in each case. y
3. (a) Estimate the area under the graph of f 共x兲 苷 cos x from
x 苷 0 to x 苷 兾2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.
4. (a) Estimate the area under the graph of f 共x兲 苷 sx from x 苷 0
to x 苷 4 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.
4 2
5. (a) Estimate the area under the graph of f 共x兲 苷 1 ⫹ x 2 from
0
8 x
4
2. (a) Use six rectangles to find estimates of each type for the
area under the given graph of f from x 苷 0 to x 苷 12. (i) L 6 (sample points are left endpoints) (ii) R 6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L 6 an underestimate or overestimate of the true area? (c) Is R 6 an underestimate or overestimate of the true area? (d) Which of the numbers L 6 , R 6 , or M6 gives the best estimate? Explain. y 8
y=ƒ
x 苷 ⫺1 to x 苷 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)–(c), which appears to be the best estimate?
; 6. (a) Graph the function f 共x兲 苷 x ⫺ 2 ln x, 1 艋 x 艋 5 (b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles. 7. Evaluate the upper and lower sums for f 共x兲 苷 2 ⫹ sin x,
0 艋 x 艋 , with n 苷 2, 4, and 8. Illustrate with diagrams like Figure 14.
4
8. Evaluate the upper and lower sums for f 共x兲 苷 1 ⫹ x 2, 0
;
4
8
Graphing calculator or computer required
12 x
⫺1 艋 x 艋 1, with n 苷 3 and 4. Illustrate with diagrams like Figure 14.
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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9. The region under y 苷 x from 0 to 1 4
10. The region under y 苷 cos x from 0 to 兾2
11. Some computer algebra systems have commands that will
draw approximating rectangles and evaluate the sums of their areas, at least if x*i is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.) (a) If f 共x兲 苷 1兾共x 2 ⫹ 1兲, 0 艋 x 艋 1, find the left and right sums for n 苷 10, 30, and 50. (b) Illustrate by graphing the rectangles in part (a). (c) Show that the exact area under f lies between 0.780 and 0.791. CAS
Page 370
INTEGRALS
9–10 With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use the Is⬎ command or a ForEndFor loop, on a Casio use Isz, on an HP or in BASIC use a FORNEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for n 苷 10, 30, 50, and 100. Then guess the value of the exact area.
CAS
12:05 PM
12. (a) If f 共x兲 苷 ln x, 1 艋 x 艋 4, use the commands discussed
in Exercise 11 to find the left and right sums for n 苷 10, 30, and 50. (b) Illustrate by graphing the rectangles in part (a). (c) Show that the exact area under f lies between 2.50 and 2.59.
13. The speed of a runner increased steadily during the first
three seconds of a race. Her speed at halfsecond intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.
hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out. t 共h兲 r共t兲 (L兾h)
0
2
4
6
8
10
8.7
7.6
6.8
6.2
5.7
5.3
16. When we estimate distances from velocity data, it is some
times necessary to use times t0 , t1, t2 , t3 , . . . that are not equally spaced. We can still estimate distances using the time periods ⌬t i 苷 t i ⫺ t i⫺1. For example, on May 7, 1992, the space shuttle Endeavour was launched on mission STS49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the earth’s surface of the Endeavour, 62 seconds after liftoff. Event
Velocity (ft兾s)
0 10 15 20 32 59 62 125
0 185 319 447 742 1325 1445 4151
Launch Begin roll maneuver End roll maneuver Throttle to 89% Throttle to 67% Throttle to 104% Maximum dynamic pressure Solid rocket booster separation
17. The velocity graph of a braking car is shown. Use it to esti
mate the distance traveled by the car while the brakes are applied. √ (ft /s) 60
t (s)
0
0.5
1.0
1.5
2.0
2.5
3.0
40
v (ft兾s)
0
6.2
10.8
14.9
18.1
19.4
20.2
20 0
14. Speedometer readings for a motorcycle at 12second
intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
Time (s)
2
4
t 6 (seconds)
18. The velocity graph of a car accelerating from rest to a speed
of 120 km兾h over a period of 30 seconds is shown. Estimate the distance traveled during this period. √ (km / h) 80
t (s)
0
12
24
36
48
60
v (ft兾s)
30
28
25
22
24
27
15. Oil leaked from a tank at a rate of r共t兲 liters per hour. The
rate decreased as time passed and values of the rate at two
40
0
10
20
t 30 (seconds)
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SECTION 5.2
2x , 1艋x艋3 x2 ⫹ 1
20. f 共x兲 苷 x 2 ⫹ s1 ⫹ 2x , 21. f 共x兲 苷 ssin x ,
371
that the n rectangles representing R n ⫺ L n can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that
19–21 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 19. f 共x兲 苷
THE DEFINITE INTEGRAL
Rn ⫺ A ⬍
4艋x艋7
0艋x艋
b⫺a 关 f 共b兲 ⫺ f 共a兲兴 n
26. If A is the area under the curve y 苷 e x from 1 to 3, use
Exercise 25 to find a value of n such that Rn ⫺ A ⬍ 0.0001.
22–23 Determine a region whose area is equal to the given limit.
CAS
Do not evaluate the limit. n
22. lim
兺
n l ⬁ i苷1
2 n
冉 冊 5⫹
2i n
10
n
23. lim
兺
n l ⬁ i苷1
a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).
i tan 4n 4n CAS
24. (a) Use Definition 2 to find an expression for the area under
冋
n共n ⫹ 1兲 2
CAS
册
30. (a) Let A n be the area of a polygon with n equal sides
uous function f from a to b, and let L n and Rn be the approximations to A with n subintervals using left and right endpoints, respectively. (a) How are A, L n, and Rn related? (b) Show that
inscribed in a circle with radius r . By dividing the polygon into n congruent triangles with central angle 2兾n, show that A n 苷 12 nr 2 sin
b⫺a 关 f 共b兲 ⫺ f 共a兲兴 n
冉 冊 2 n
(b) Show that lim n l ⬁ A n 苷 r 2. [Hint: Use Equation 3.3.2 on page 192.]
Then draw a diagram to illustrate this equation by showing
5.2
29. Find the exact area under the cosine curve y 苷 cos x from
x 苷 0 to x 苷 b, where 0 艋 b 艋 兾2. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if b 苷 兾2?
2
25. Let A be the area under the graph of an increasing contin
Rn ⫺ L n 苷
28. Find the exact area of the region under the graph of y 苷 e⫺x
from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b).
the curve y 苷 x 3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 ⫹ 2 3 ⫹ 3 3 ⫹ ⭈ ⭈ ⭈ ⫹ n 3 苷
27. (a) Express the area under the curve y 苷 x 5 from 0 to 2 as
The Definite Integral We saw in Section 5.1 that a limit of the form n
1
lim
兺 f 共x*兲 ⌬x 苷 lim 关 f 共x *兲 ⌬x ⫹ f 共x *兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x *兲 ⌬x兴
n l ⬁ i苷1
i
nl⬁
1
2
n
arises when we compute an area. We also saw that it arises when we try to find the distance traveled by an object. It turns out that this same type of limit occurs in a wide variety of situations even when f is not necessarily a positive function. In Chapters 6 and 8 we will see that limits of the form 1 also arise in finding lengths of curves, volumes of solids, centers of mass, force due to water pressure, and work, as well as other quantities. We therefore give this type of limit a special name and notation.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
2 Deﬁnition of a Deﬁnite Integral If f is a function defined for a 艋 x 艋 b, we divide the interval 关a, b兴 into n subintervals of equal width ⌬x 苷 共b ⫺ a兲兾n. We let x 0 共苷 a兲, x 1, x 2 , . . . , x n (苷 b) be the endpoints of these subintervals and we let x1*, x2*, . . . , x n* be any sample points in these subintervals, so x*i lies in the ith subinterval 关x i⫺1, x i 兴. Then the definite integral of f from a to b is
y
b
n
f 共x兲 dx 苷 lim
兺 f 共x*兲 ⌬x i
n l ⬁ i苷1
a
provided that this limit exists and gives the same value for all possible choices of sample points. If it does exist, we say that f is integrable on 关a, b兴.
The precise meaning of the limit that defines the integral is as follows: For every number ⬎ 0 there is an integer N such that
冟y
b
a
n
f 共x兲 dx ⫺
兺 f 共x* 兲 ⌬x i
i苷1
冟
⬍
for every integer n ⬎ N and for every choice of x*i in 关x i⫺1, x i 兴. NOTE 1 The symbol x was introduced by Leibniz and is called an integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation xab f 共x兲 dx, f 共x兲 is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. For now, the symbol dx has no meaning by itself; xab f 共x兲 dx is all one symbol. The dx simply indicates that the independent variable is x. The procedure of calculating an integral is called integration. NOTE 2 The definite integral
xab f 共x兲 dx is a number; it does not depend on x. In fact,
we could use any letter in place of x without changing the value of the integral:
Riemann Bernhard Riemann received his Ph.D. under the direction of the legendary Gauss at the University of Göttingen and remained there to teach. Gauss, who was not in the habit of praising other mathematicians, spoke of Riemann’s “creative, active, truly mathematical mind and gloriously fertile originality.” The definition 2 of an integral that we use is due to Riemann. He also made major contributions to the theory of functions of a complex variable, mathematical physics, number theory, and the foundations of geometry. Riemann’s broad concept of space and geometry turned out to be the right setting, 50 years later, for Einstein’s general relativity theory. Riemann’s health was poor throughout his life, and he died of tuberculosis at the age of 39.
y
b
a
b
b
f 共x兲 dx 苷 y f 共t兲 dt 苷 y f 共r兲 dr a
a
NOTE 3 The sum n
兺 f 共x*兲 ⌬x i
i苷1
that occurs in Definition 2 is called a Riemann sum after the German mathematician Bernhard Riemann (1826–1866). So Definition 2 says that the definite integral of an integrable function can be approximated to within any desired degree of accuracy by a Riemann sum. We know that if f happens to be positive, then the Riemann sum can be interpreted as a sum of areas of approximating rectangles (see Figure 1). By comparing Definition 2 with the definition of area in Section 5.1, we see that the definite integral xab f 共x兲 dx can be interpreted as the area under the curve y 苷 f 共x兲 from a to b. (See Figure 2.)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.2 y
y
y=ƒ
0 a
b
x
a
y=ƒ
x *i
y=ƒ
b
a
ƒ dx is the net area.
b
x
FIGURE 1
FIGURE 2
If ƒ˘0, the integral ja ƒ dx is the area under the curve y=ƒ from a to b.
b
If f takes on both positive and negative values, as in Figure 3, then the Riemann sum is the sum of the areas of the rectangles that lie above the xaxis and the negatives of the areas of the rectangles that lie below the xaxis (the areas of the blue rectangles minus the areas of the gold rectangles). When we take the limit of such Riemann sums, we get the situation illustrated in Figure 4. A definite integral can be interpreted as a net area, that is, a difference of areas:
y
b
a
f 共x兲 dx 苷 A 1 ⫺ A 2
where A 1 is the area of the region above the xaxis and below the graph of f , and A 2 is the area of the region below the xaxis and above the graph of f .
y
j
a
If ƒ˘0, the Riemann sum μ f(x*i ) Îx is the sum of areas of rectangles.
μ f(x* i ) Î x is an approximation to the net area.
FIGURE 4
0
x
b
FIGURE 3
0 a
373
y
Îx
0
THE DEFINITE INTEGRAL
b x
NOTE 4 Although we have defined
xab f 共x兲 dx by dividing 关a, b兴 into subintervals of
equal width, there are situations in which it is advantageous to work with subintervals of unequal width. For instance, in Exercise 16 in Section 5.1 NASA provided velocity data at times that were not equally spaced, but we were still able to estimate the distance traveled. And there are methods for numerical integration that take advantage of unequal subintervals. If the subinterval widths are ⌬x 1, ⌬x 2 , . . . , ⌬x n , we have to ensure that all these widths approach 0 in the limiting process. This happens if the largest width, max ⌬x i , approaches 0. So in this case the definition of a definite integral becomes
y
b
a
n
f 共x兲 dx 苷
lim
兺 f 共x* 兲 ⌬x
max ⌬x i l 0 i苷1
i
i
NOTE 5 We have defined the definite integral for an integrable function, but not all functions are integrable (see Exercises 69–70). The following theorem shows that the most commonly occurring functions are in fact integrable. The theorem is proved in more advanced courses.
3 Theorem If f is continuous on 关a, b兴, or if f has only a finite number of jump discontinuities, then f is integrable on 关a, b兴; that is, the definite integral xab f 共x兲 dx exists.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
If f is integrable on 关a, b兴, then the limit in Definition 2 exists and gives the same value no matter how we choose the sample points x*i . To simplify the calculation of the integral we often take the sample points to be right endpoints. Then x*i 苷 x i and the definition of an integral simplifies as follows.
4
Theorem If f is integrable on 关a, b兴, then
y
b
b⫺a n
⌬x 苷
兺 f 共x 兲 ⌬x i
n l ⬁ i苷1
a
where
n
f 共x兲 dx 苷 lim
and
x i 苷 a ⫹ i ⌬x
EXAMPLE 1 Express n
兺 共x
lim
n l ⬁ i苷1
3 i
⫹ x i sin x i 兲 ⌬x
as an integral on the interval 关0, 兴. SOLUTION Comparing the given limit with the limit in Theorem 4, we see that they will
be identical if we choose f 共x兲 苷 x 3 ⫹ x sin x. We are given that a 苷 0 and b 苷 . Therefore, by Theorem 4, we have n
lim
兺 共x
n l ⬁ i苷1
3 i
⫹ x i sin x i 兲 ⌬x 苷 y 共x 3 ⫹ x sin x兲 dx 0
Later, when we apply the definite integral to physical situations, it will be important to recognize limits of sums as integrals, as we did in Example 1. When Leibniz chose the notation for an integral, he chose the ingredients as reminders of the limiting process. In general, when we write n
lim
兺 f 共x *兲 ⌬x 苷 y
n l ⬁ i苷1
i
b
a
f 共x兲 dx
we replace lim 冘 by x, x*i by x, and ⌬x by dx.
Evaluating Integrals When we use a limit to evaluate a definite integral, we need to know how to work with sums. The following three equations give formulas for sums of powers of positive integers. Equation 5 may be familiar to you from a course in algebra. Equations 6 and 7 were discussed in Section 5.1 and are proved in Appendix E. n
5
兺i苷
i苷1 n
6
兺i
2
苷
3
苷
i苷1 n
7
兺i
i苷1
n共n ⫹ 1兲 2 n共n ⫹ 1兲共2n ⫹ 1兲 6
冋
n共n ⫹ 1兲 2
册
2
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SECTION 5.2
THE DEFINITE INTEGRAL
375
The remaining formulas are simple rules for working with sigma notation: n
兺 c 苷 nc
8 Formulas 8–11 are proved by writing out each side in expanded form. The left side of Equation 9 is ca 1 ⫹ ca 2 ⫹ ⭈ ⭈ ⭈ ⫹ ca n The right side is c共a 1 ⫹ a 2 ⫹ ⭈ ⭈ ⭈ ⫹ a n 兲
i苷1 n
兺 ca
9
n
i
苷c
i苷1 n
兺 共a
10
These are equal by the distributive property. The other formulas are discussed in Appendix E.
⫹ bi 兲 苷
兺a
n
i
⫹
i苷1
n
兺 共a
i
i苷1 n
i
i苷1
11
兺a
i苷1
⫺ bi 兲 苷
兺a
i
i苷1
n
i
兺b n
i
⫺
i苷1
兺b
i
i苷1
EXAMPLE 2
(a) Evaluate the Riemann sum for f 共x兲 苷 x 3 ⫺ 6x, taking the sample points to be right endpoints and a 苷 0, b 苷 3, and n 苷 6. 3
(b) Evaluate y 共x 3 ⫺ 6x兲 dx. 0
SOLUTION
(a) With n 苷 6 the interval width is ⌬x 苷
b⫺a 3⫺0 1 苷 苷 n 6 2
and the right endpoints are x 1 苷 0.5, x 2 苷 1.0, x 3 苷 1.5, x 4 苷 2.0, x 5 苷 2.5, and x 6 苷 3.0. So the Riemann sum is 6
R6 苷
兺 f 共x 兲 ⌬x i
i苷1
苷 f 共0.5兲 ⌬x ⫹ f 共1.0兲 ⌬x ⫹ f 共1.5兲 ⌬x ⫹ f 共2.0兲 ⌬x ⫹ f 共2.5兲 ⌬x ⫹ f 共3.0兲 ⌬x 苷 12 共⫺2.875 ⫺ 5 ⫺ 5.625 ⫺ 4 ⫹ 0.625 ⫹ 9兲 苷 ⫺3.9375 Notice that f is not a positive function and so the Riemann sum does not represent a sum of areas of rectangles. But it does represent the sum of the areas of the blue rectangles (above the xaxis) minus the sum of the areas of the gold rectangles (below the xaxis) in Figure 5. y
5
0
y=˛6x
3
x
FIGURE 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
(b) With n subintervals we have ⌬x 苷
b⫺a 3 苷 n n
Thus x 0 苷 0, x 1 苷 3兾n, x 2 苷 6兾n, x 3 苷 9兾n, and, in general, x i 苷 3i兾n. Since we are using right endpoints, we can use Theorem 4:
y
3
0
In the sum, n is a constant (unlike i), so we can move 3兾n in front of the 冘 sign.
n
共x 3 ⫺ 6x兲 dx 苷 lim
nl⬁
苷 lim
nl⬁
苷 lim
nl⬁
苷 lim
y
nl⬁
5
y=˛6x
苷 lim
nl⬁
A¡ 0
A™
3
x
FIGURE 6
j
3
0
(˛6x) dx=A¡A™=_6.75
i
n l ⬁ i苷1
苷 lim
苷
冉冊 冊 冉 冊册 册 兺 册 册 冎 冊 冉 冊册 n
3i n
兺 f 共x 兲 ⌬x 苷 lim 兺 f 3 n 3 n
n l ⬁ i苷1
冋冉 兺冋 n
兺
i苷1
n
3
3i n
(Equation 9 with c 苷 3兾n)
i
(Equations 11 and 9)
⫺6
27 3 18 i ⫺ i n3 n
i苷1
冋 兺 再 冋 冋 冉 81 n4
3i n
3 n
n
i3 ⫺
i苷1
54 n2
81 n4
n共n ⫹ 1兲 2
81 4
1⫹
1 n
n
i苷1
2
⫺
54 n共n ⫹ 1兲 n2 2
2
⫺ 27 1 ⫹
(Equations 7 and 5)
1 n
81 27 ⫺ 27 苷 ⫺ 苷 ⫺6.75 4 4
This integral can’t be interpreted as an area because f takes on both positive and negative values. But it can be interpreted as the difference of areas A 1 ⫺ A 2 , where A 1 and A 2 are shown in Figure 6. Figure 7 illustrates the calculation by showing the positive and negative terms in the right Riemann sum R n for n 苷 40. The values in the table show the Riemann sums approaching the exact value of the integral, ⫺6.75, as n l ⬁. y
5
0
y=˛6x
3
x
n
Rn
40 100 500 1000 5000
⫺6.3998 ⫺6.6130 ⫺6.7229 ⫺6.7365 ⫺6.7473
FIGURE 7
R¢¸Å_6.3998
A much simpler method for evaluating the integral in Example 2 will be given in Section 5.4. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.2
THE DEFINITE INTEGRAL
377
EXAMPLE 3 Because f 共x兲 苷 e x is positive, the integral in Example 3 represents the area shown in Figure 8.
(a) Set up an expression for x13 e x dx as a limit of sums. (b) Use a computer algebra system to evaluate the expression. SOLUTION
y
(a) Here we have f 共x兲 苷 e x, a 苷 1, b 苷 3, and ⌬x 苷
y=´ 10
b⫺a 2 苷 n n
So x0 苷 1, x1 苷 1 ⫹ 2兾n, x2 苷 1 ⫹ 4兾n, x 3 苷 1 ⫹ 6兾n, and 2i n
xi 苷 1 ⫹ 0
1
3
x
From Theorem 4, we get
FIGURE 8
y
3
1
n
e x dx 苷 lim
兺 f 共x 兲 ⌬x i
n l ⬁ i苷1 n
苷 lim
兺f
苷 lim
2 n
n l ⬁ i苷1
nl⬁
冉 冊 1⫹
2i n
2 n
n
兺e
1⫹2i兾n
i苷1
(b) If we ask a computer algebra system to evaluate the sum and simplify, we obtain A computer algebra system is able to find an explicit expression for this sum because it is a geometric series. The limit could be found using l’Hospital’s Rule.
n
兺e
1⫹2i兾n
苷
i苷1
e 共3n⫹2兲兾n ⫺ e 共n⫹2兲兾n e 2兾n ⫺ 1
Now we ask the computer algebra system to evaluate the limit:
y
3
1
e x dx 苷 lim
nl⬁
2 e 共3n⫹2兲兾n ⫺ e 共n⫹2兲兾n ⴢ 苷 e3 ⫺ e n e 2兾n ⫺ 1
We will learn a much easier method for the evaluation of integrals in the next section.
y
1
v
EXAMPLE 4 Evaluate the following integrals by interpreting each in terms of areas.
(a)
y
1
0
s1 ⫺ x 2 dx
(b)
FIGURE 9
1
3
0
共x ⫺ 1兲 dx
SOLUTION
y= œ„„„„„ 1≈ or ≈+¥=1
(a) Since f 共x兲 苷 s1 ⫺ x 2 艌 0, we can interpret this integral as the area under the curve y 苷 s1 ⫺ x 2 from 0 to 1. But, since y 2 苷 1 ⫺ x 2, we get x 2 ⫹ y 2 苷 1, which shows that the graph of f is the quartercircle with radius 1 in Figure 9. Therefore 1
0
y
x
y s1 ⫺ x 0
2
dx 苷 14 共1兲2 苷
4
(In Section 7.3 we will be able to prove that the area of a circle of radius r is r 2.)
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INTEGRALS
(b) The graph of y 苷 x ⫺ 1 is the line with slope 1 shown in Figure 10. We compute the integral as the difference of the areas of the two triangles:
y
3
0
共x ⫺ 1兲 dx 苷 A 1 ⫺ A 2 苷 12 共2 ⭈ 2兲 ⫺ 12 共1 ⭈ 1兲 苷 1.5 y
(3, 2)
y=x1 A¡ 0 A™
1
3
x
_1
FIGURE 10
The Midpoint Rule We often choose the sample point x*i to be the right endpoint of the i th subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose x*i to be the midpoint of the interval, which we denote by x i . Any Riemann sum is an approximation to an integral, but if we use midpoints we get the following approximation. TEC Module 5.2/7.7 shows how the Midpoint Rule estimates improve as n increases.
Midpoint Rule
y
b
a
n
f 共x兲 dx ⬇ ⌬x 苷
where
i
1
n
b⫺a n
x i 苷 12 共x i⫺1 ⫹ x i 兲 苷 midpoint of 关x i⫺1, x i 兴
and
v
兺 f 共 x 兲 ⌬x 苷 ⌬x 关 f 共x 兲 ⫹ ⭈ ⭈ ⭈ ⫹ f 共x 兲兴
i苷1
EXAMPLE 5 Use the Midpoint Rule with n 苷 5 to approximate
y
2
1
1 dx. x
SOLUTION The endpoints of the five subintervals are 1, 1.2, 1.4, 1.6, 1.8, and 2.0,
so the midpoints are 1.1, 1.3, 1.5, 1.7, and 1.9. The width of the subintervals is ⌬x 苷 共2 ⫺ 1兲兾5 苷 15 , so the Midpoint Rule gives y
1 y= x
y
2
1
1 dx ⬇ ⌬x 关 f 共1.1兲 ⫹ f 共1.3兲 ⫹ f 共1.5兲 ⫹ f 共1.7兲 ⫹ f 共1.9兲兴 x 苷
1 5
冉
1 1 1 1 1 ⫹ ⫹ ⫹ ⫹ 1.1 1.3 1.5 1.7 1.9
冊
⬇ 0.691908 0
1
FIGURE 11
2
x
Since f 共x兲 苷 1兾x ⬎ 0 for 1 艋 x 艋 2, the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in Figure 11.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.2
THE DEFINITE INTEGRAL
379
At the moment we don’t know how accurate the approximation in Example 5 is, but in Section 7.7 we will learn a method for estimating the error involved in using the Midpoint Rule. At that time we will discuss other methods for approximating definite integrals. If we apply the Midpoint Rule to the integral in Example 2, we get the picture in Figure 12. The approximation M40 ⬇ ⫺6.7563 is much closer to the true value ⫺6.75 than the right endpoint approximation, R 40 ⬇ ⫺6.3998, shown in Figure 7. y
TEC In Visual 5.2 you can compare left, right, and midpoint approximations to the integral in Example 2 for different values of n.
5
y=˛6x
0
3
x
FIGURE 12
M¢¸Å_6.7563
Properties of the Definite Integral When we defined the definite integral xab f 共x兲 dx, we implicitly assumed that a ⬍ b. But the definition as a limit of Riemann sums makes sense even if a ⬎ b. Notice that if we reverse a and b, then ⌬x changes from 共b ⫺ a兲兾n to 共a ⫺ b兲兾n. Therefore
y
a
b
b
f 共x兲 dx 苷 ⫺y f 共x兲 dx a
If a 苷 b, then ⌬x 苷 0 and so
y
a
a
f 共x兲 dx 苷 0
We now develop some basic properties of integrals that will help us to evaluate integrals in a simple manner. We assume that f and t are continuous functions. Properties of the Integral 1.
y
b
a
2.
y
y
b
a
y=c
c
3.
y
b
a
area=c(ba)
4.
y
b
a
0
a
FIGURE 13
j
b
a
c dx=c(ba)
b
c dx 苷 c共b ⫺ a兲,
where c is any constant b
b
关 f 共x兲 ⫹ t共x兲兴 dx 苷 y f 共x兲 dx ⫹ y t共x兲 dx a
b
cf 共x兲 dx 苷 c y f 共x兲 dx, a
b
a
where c is any constant b
关 f 共x兲 ⫺ t共x兲兴 dx 苷 y f 共x兲 dx ⫺ y t共x兲 dx a
a
x
Property 1 says that the integral of a constant function f 共x兲 苷 c is the constant times the length of the interval. If c ⬎ 0 and a ⬍ b, this is to be expected because c共b ⫺ a兲 is the area of the shaded rectangle in Figure 13.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
y
Property 2 says that the integral of a sum is the sum of the integrals. For positive functions it says that the area under f ⫹ t is the area under f plus the area under t. Figure 14 helps us understand why this is true: In view of how graphical addition works, the corresponding vertical line segments have equal height. In general, Property 2 follows from Theorem 4 and the fact that the limit of a sum is the sum of the limits:
f+g
g
f
y
b
a
0
12:08 PM
n
关 f 共x兲 ⫹ t共x兲兴 dx 苷 lim
nl⬁
FIGURE 14 b
a
冋兺
i
n
苷 lim
j
i
n l ⬁ i苷1
b x
a
兺 关 f 共x 兲 ⫹ t共x 兲兴 ⌬x i苷1
兺 t共x 兲 ⌬x i
i苷1
n
苷 lim
[ƒ+©] dx=
j
b
a
n
兺
f 共x i 兲 ⌬x ⫹ lim
n l ⬁ i苷1
b
ƒ dx+j © dx a
册
n
f 共x i 兲 ⌬x ⫹
兺 t共x 兲 ⌬x
n l ⬁ i苷1
b
i
b
苷 y f 共x兲 dx ⫹ y t共x兲 dx a
Property 3 seems intuitively reasonable because we know that multiplying a function by a positive number c stretches or shrinks its graph vertically by a factor of c. So it stretches or shrinks each approximating rectangle by a factor c and therefore it has the effect of multiplying the area by c.
a
Property 3 can be proved in a similar manner and says that the integral of a constant times a function is the constant times the integral of the function. In other words, a constant (but only a constant) can be taken in front of an integral sign. Property 4 is proved by writing f ⫺ t 苷 f ⫹ 共⫺t兲 and using Properties 2 and 3 with c 苷 ⫺1.
y
EXAMPLE 6 Use the properties of integrals to evaluate
1
共4 ⫹ 3x 2 兲 dx.
0
SOLUTION Using Properties 2 and 3 of integrals, we have
y
1
0
1
1
1
1
共4 ⫹ 3x 2 兲 dx 苷 y 4 dx ⫹ y 3x 2 dx 苷 y 4 dx ⫹ 3 y x 2 dx 0
0
0
0
We know from Property 1 that
y
1
0
4 dx 苷 4共1 ⫺ 0兲 苷 4 1
and we found in Example 2 in Section 5.1 that y x 2 dx 苷 13 . So 0
y
1
0
1
1
共4 ⫹ 3x 2 兲 dx 苷 y 4 dx ⫹ 3 y x 2 dx 0
0
苷4⫹3⭈ 苷5 1 3
The next property tells us how to combine integrals of the same function over adjacent intervals:
y
y=ƒ 5.
y
c
a
0
a
FIGURE 15
c
b
x
b
b
f 共x兲 dx ⫹ y f 共x兲 dx 苷 y f 共x兲 dx c
a
This is not easy to prove in general, but for the case where f 共x兲 艌 0 and a ⬍ c ⬍ b Property 5 can be seen from the geometric interpretation in Figure 15: The area under y 苷 f 共x兲 from a to c plus the area from c to b is equal to the total area from a to b.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.2
v
THE DEFINITE INTEGRAL
381
EXAMPLE 7 If it is known that x0 f 共x兲 dx 苷 17 and x0 f 共x兲 dx 苷 12, find x8 f 共x兲 dx. 10
8
10
SOLUTION By Property 5, we have
y
8
0
so
y
10
8
10
10
f 共x兲 dx ⫹ y f 共x兲 dx 苷 y f 共x兲 dx 8
0
10
8
f 共x兲 dx 苷 y f 共x兲 dx ⫺ y f 共x兲 dx 苷 17 ⫺ 12 苷 5 0
0
Properties 1–5 are true whether a ⬍ b, a 苷 b, or a ⬎ b. The following properties, in which we compare sizes of functions and sizes of integrals, are true only if a 艋 b. Comparison Properties of the Integral 6. If f 共x兲 艌 0 for a 艋 x 艋 b, then
y
b
a
7. If f 共x兲 艌 t共x兲 for a 艋 x 艋 b, then
f 共x兲 dx 艌 0.
y
b
a
b
f 共x兲 dx 艌 y t共x兲 dx. a
8. If m 艋 f 共x兲 艋 M for a 艋 x 艋 b, then b
m共b ⫺ a兲 艋 y f 共x兲 dx 艋 M共b ⫺ a兲 a
y M
y=ƒ m 0
a
FIGURE 16
b
x
If f 共x兲 艌 0, then xab f 共x兲 dx represents the area under the graph of f , so the geometric interpretation of Property 6 is simply that areas are positive. (It also follows directly from the definition because all the quantities involved are positive.) Property 7 says that a bigger function has a bigger integral. It follows from Properties 6 and 4 because f ⫺ t 艌 0. Property 8 is illustrated by Figure 16 for the case where f 共x兲 艌 0. If f is continuous we could take m and M to be the absolute minimum and maximum values of f on the interval 关a, b兴. In this case Property 8 says that the area under the graph of f is greater than the area of the rectangle with height m and less than the area of the rectangle with height M. PROOF OF PROPERTY 8 Since m 艋 f 共x兲 艋 M , Property 7 gives
y
b
a
b
b
m dx 艋 y f 共x兲 dx 艋 y M dx a
a
Using Property 1 to evaluate the integrals on the left and right sides, we obtain b
m共b ⫺ a兲 艋 y f 共x兲 dx 艋 M共b ⫺ a兲 a
Property 8 is useful when all we want is a rough estimate of the size of an integral without going to the bother of using the Midpoint Rule. EXAMPLE 8 Use Property 8 to estimate
y
1
0
2
e⫺x dx.
2
SOLUTION Because f 共x兲 苷 e⫺x is a decreasing function on 关0, 1兴, its absolute maximum
value is M 苷 f 共0兲 苷 1 and its absolute minimum value is m 苷 f 共1兲 苷 e⫺1. Thus, by
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
y
Property 8,
y=1
1
9/21/10
1
2
e⫺1共1 ⫺ 0兲 艋 y e⫺x dx 艋 1共1 ⫺ 0兲 y=e–x
0
2
1
2
e⫺1 艋 y e⫺x dx 艋 1
or
0
y=1/e
Since e⫺1 ⬇ 0.3679, we can write 1
2
0.367 艋 y e⫺x dx 艋 1 0
1
x
The result of Example 8 is illustrated in Figure 17. The integral is greater than the area of the lower rectangle and less than the area of the square.
FIGURE 17
5.2
0
Exercises
1. Evaluate the Riemann sum for f 共x兲 苷 3 ⫺ 2 x, 2 艋 x 艋 14, 1
with six subintervals, taking the sample points to be left endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.
4 t共x兲 dx with six sub6. The graph of t is shown. Estimate x⫺2
intervals using (a) right endpoints, (b) left endpoints, and (c) midpoints. y
2. If f 共x兲 苷 x 2 ⫺ 2x, 0 艋 x 艋 3, evaluate the Riemann sum with
n 苷 6, taking the sample points to be right endpoints. What does the Riemann sum represent? Illustrate with a diagram.
1
3. If f 共x兲 苷 e x ⫺ 2, 0 艋 x 艋 2, find the Riemann sum with
x
1
n 苷 4 correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.
4. (a) Find the Riemann sum for f 共x兲 苷 sin x, 0 艋 x 艋 3兾2,
with six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points. 5. The graph of a function f is given. Estimate x010 f 共x兲 dx using
five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. y
1 0
1
CAS Computer algebra system required
x
7. A table of values of an increasing function f is shown. Use the
table to find lower and upper estimates for x1030 f 共x兲 dx. x
10
14
18
22
26
30
f 共x兲
⫺12
⫺6
⫺2
1
3
8
8. The table gives the values of a function obtained from an
experiment. Use them to estimate x39 f 共x兲 dx using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral? x
3
4
5
6
7
8
9
f 共x兲
⫺3.4
⫺2.1
⫺0.6
0.3
0.9
1.4
1.8
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.2
using a Riemann sum with right endpoints and n 苷 8. (b) Draw a diagram like Figure 3 to illustrate the approximation in part (a). (c) Use Theorem 4 to evaluate x04 共x 2 ⫺ 3x兲 dx. (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4.
approximate the integral. Round the answer to four decimal places.
y
11.
y
8
0
sin sx dx,
n苷4
10.
y
x dx, x⫹1
n苷5
12.
y
2
0
兾2
0
5
1
cos 4 x dx, n 苷 4
x 2e⫺x dx,
n苷4
b2 ⫺ a2 . 2
b
CAS
27. Prove that y x dx 苷
13. If you have a CAS that evaluates midpoint approximations
a
and graphs the corresponding rectangles (use RiemannSum or middlesum and middlebox commands in Maple), check the answer to Exercise 11 and illustrate with a graph. Then repeat with n 苷 10 and n 苷 20.
b
28. Prove that y x 2 dx 苷 a
instructions for Exercise 9 in Section 5.1), compute the left and right Riemann sums for the function f 共x兲 苷 x兾共x ⫹ 1兲 on the interval 关0, 2兴 with n 苷 100. Explain why these estimates show that 2
0
x dx ⬍ 0.9081 x⫹1
29.
CAS
y
16. Use a calculator or computer to make a table of values of
left and right Riemann sums L n and R n for the integral 2 x02 e⫺x dx with n 苷 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a 2 2 similar statement for the integral x⫺1 e⫺x dx ? Explain.
x dx 1 ⫹ x5
6
2
31.
y
0
sin 5x dx
18. lim
兺
19. lim
兺 关5共x*兲
20. lim
兺
n
n l ⬁ i苷1
10
2
x 6 dx
preting it in terms of areas. (a)
y
(c)
y
2
0 7
5
f 共x兲 dx
(b)
y
f 共x兲 dx
(d)
y
5
0 9
0
f 共x兲 dx f 共x兲 dx
y
y=ƒ
2 i
共x ⫺ 4 ln x兲 dx
33. The graph of f is shown. Evaluate each integral by inter
n
兺x
y
32.
interval. n l ⬁ i苷1
10
1
using a computer algebra system to find both the sum and the limit.
17–20 Express the limit as a definite integral on the given
17. lim
y
30.
31–32 Express the integral as a limit of sums. Then evaluate,
15. Use a calculator or computer to make a table of values of
right Riemann sums R n for the integral x0 sin x dx with n 苷 5, 10, 50, and 100. What value do these numbers appear to be approaching?
b3 ⫺ a3 . 3
29–30 Express the integral as a limit of Riemann sums. Do not evaluate the limit.
14. With a programmable calculator or computer (see the
0.8946 ⬍ y
ln共1 ⫹ x i2 兲 ⌬x, 关2, 6兴
cos x i ⌬x, xi
0
关, 2兴
2
4
6
x
8
n
n l ⬁ i苷1 n
n l ⬁ i苷1
i
3
⫺ 4 x*i 兴 ⌬x,
x *i ⌬x, 共x *i 兲2 ⫹ 4
关2, 7] 34. The graph of t consists of two straight lines and a semi
关1, 3兴
circle. Use it to evaluate each integral. (a)
21–25 Use the form of the definition of the integral given in
Theorem 4 to evaluate the integral. 21.
y
23.
y
25.
5
2
共4 ⫺ 2x兲 dx
22.
y
共x 2 ⫹ x 兲 dx
24.
y
0
⫺2
y
1
0
4
1 2
0
共x 2 ⫺ 4x ⫹ 2 兲 dx 共2x ⫺ x 3 兲 dx
y
2
0
(b)
t共x兲 dx
y
6
2
t共x兲 dx
(c)
y
7
0
t共x兲 dx
y 4 2
0
y=©
4
7 x
共x ⫺ 3x 兲 dx 3
383
26. (a) Find an approximation to the integral x04 共x 2 ⫺ 3x兲 dx
9–12 Use the Midpoint Rule with the given value of n to
9.
THE DEFINITE INTEGRAL
2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_05_ch05_p380389.qk:97909_05_ch05_p380389
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CHAPTER 5
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y
2
⫺1
共1 ⫺ x兲 dx
0
Page 384
INTEGRALS
35– 40 Evaluate the integral by interpreting it in terms of areas. 35.
12:08 PM
9
36.
y ( 0
1 3
5
y (1 ⫹ s9 ⫺ x ) dx
38.
y ( x ⫺ s25 ⫺ x ) dx
39.
y ⱍ x ⱍ dx
40.
y ⱍ x ⫺ 5 ⱍ dx
2
2
⫺1
x2x f 共t兲 dt, where f is the function whose graph is
given, which of the following values is largest? (A) F共0兲 (B) F共1兲 (C) F共2兲 (D) F共3兲 (E) F共4兲
x ⫺ 2) dx
37.
⫺3
52. If F共x兲 苷
2
⫺5
y
10
0
y=f(t)
41. Evaluate y sin 2 x cos 4 x dx.
0
1
2
3
t
4
1
42. Given that y 3x sx 2 ⫹ 4 dx 苷 5s5 ⫺ 8, what is 0
y
0
1
3usu 2 ⫹ 4 du ? 53. Each of the regions A, B, and C bounded by the graph of f and
43. In Example 2 in Section 5.1 we showed that x x dx 苷 .
the xaxis has area 3. Find the value of
Use this fact and the properties of integrals to evaluate x01 共5 ⫺ 6x 2 兲 dx.
y
1 0
2
1 3
2
⫺4
关 f 共x兲 ⫹ 2x ⫹ 5兴 dx
44. Use the properties of integrals and the result of Example 3 to
y
evaluate x13 共2e x ⫺ 1兲 dx. 45. Use the result of Example 3 to evaluate x13 e x⫹2 dx.
B
兾2 0
cos x dx 苷 1 (from Exercise 29 in Section 5.1), together with the properties of integrals, to evaluate x0兾2 共2 cos x ⫺ 5x兲 dx.
46. Use the result of Exercise 27 and the fact that x
47. Write as a single integral in the form xab f 共x兲 dx :
y
2
⫺2
f 共x兲 dx ⫹
y
5
2
f 共x兲 dx ⫺ y
⫺1
⫺2
49. If x09 f 共x兲 dx 苷 37 and x09 t共x兲 dx 苷 16, find
x09 关2 f 共x兲 ⫹ 3t共x兲兴 dx.
再
3 for x ⬍ 3 f 共x兲 苷 x for x 艌 3
0
C
2
x
55–58 Use the properties of integrals to verify the inequality without evaluating the integrals. 55.
y
56.
y
4
0 1
0
共x 2 ⫺ 4x ⫹ 4兲 dx 艌 0 s1 ⫹ x 2 dx 艋
51. For the function f whose graph is shown, list the following
(C) x38 f 共x兲 dx (E) f ⬘共1兲
_2
imum value M . Between what two values must x02 f 共x兲 dx lie? Which property of integrals allows you to make your conclusion?
57. 2 艋
quantities in increasing order, from smallest to largest, and explain your reasoning. (A) x08 f 共x兲 dx (B) x03 f 共x兲 dx
A
54. Suppose f has absolute minimum value m and absolute max
f 共x兲 dx
48. If x15 f 共x兲 dx 苷 12 and x45 f 共x兲 dx 苷 3.6, find x14 f 共x兲 dx.
50. Find x05 f 共x兲 dx if
_4
58.
y
1
⫺1
y
1
0
s1 ⫹ x dx
s1 ⫹ x 2 dx 艋 2 s2
s2 艋 24
兾4
y
兾6
cos x dx 艋
s3 24
(D) x48 f 共x兲 dx 59–64 Use Property 8 to estimate the value of the integral. y
2
0
5
x
4
59.
y
61.
y
63.
y
1
sx dx
兾3 兾4
2
0
tan x dx
xe⫺x dx
2
60.
y
62.
y
64.
y
0 2
0
1 dx 1 ⫹ x2 共x 3 ⫺ 3x ⫹ 3兲 dx
2
共x ⫺ 2 sin x兲 dx
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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DISCOVERY PROJECT
65–66 Use properties of integrals, together with Exercises 27 and 28, to prove the inequality. 65.
y
66.
y
3
1
sx 4 ⫹ 1 dx 艌
兾2
0
69. Let f 共x兲 苷 0 if x is any rational number and f 共x兲 苷 1 if x is
any irrational number. Show that f is not integrable on 关0, 1兴. not integrable on 关0, 1兴. [Hint: Show that the first term in the Riemann sum, f 共x1* 兲 ⌬x, can be made arbitrarily large.]
71–72 Express the limit as a definite integral. n
67. Prove Property 3 of integrals.
b
a
ⱍ
ⱍ
f 共x兲 dx 艋 y
ⱍ
ⱍ
2
0
b
a
ⱍ f 共x兲 ⱍ dx
ⱍ
ⱍ
i4 n5
72. lim
1 n
兺
2
0
n
i苷1
[Hint: Consider f 共x兲 苷 x 4.] 1 1 ⫹ 共i兾n兲2
73. Find x12 x ⫺2 dx. Hint: Choose x * i to be the geometric mean
ⱍ
f 共x兲 sin 2x dx 艋 y
DISCOVERY PROJECT
兺
nl⬁
[Hint: ⫺ f 共x兲 艋 f 共x兲 艋 f 共x兲 .] (b) Use the result of part (a) to show that
y
71. lim
n l ⬁ i苷1
68. (a) If f is continuous on 关a, b兴, show that
ⱍy
385
70. Let f 共0兲 苷 0 and f 共x兲 苷 1兾x if 0 ⬍ x 艋 1. Show that f is
26 3
2 8
x sin x dx 艋
AREA FUNCTIONS
of x i⫺1 and x i (that is, x *i 苷 sx i⫺1 x i ) and use the identity 1 1 1 苷 ⫺ m共m ⫹ 1兲 m m⫹1
ⱍ f 共x兲 ⱍ dx
AREA FUNCTIONS 1. (a) Draw the line y 苷 2t ⫹ 1 and use geometry to find the area under this line, above the
taxis, and between the vertical lines t 苷 1 and t 苷 3. (b) If x ⬎ 1, let A共x兲 be the area of the region that lies under the line y 苷 2t ⫹ 1 between t 苷 1 and t 苷 x. Sketch this region and use geometry to find an expression for A共x兲. (c) Differentiate the area function A共x兲. What do you notice?
2. (a) If x 艌 ⫺1, let x
A共x兲 苷 y 共1 ⫹ t 2 兲 dt ⫺1
A共x兲 represents the area of a region. Sketch that region. (b) Use the result of Exercise 28 in Section 5.2 to find an expression for A共x兲. (c) Find A⬘共x兲. What do you notice? (d) If x 艌 ⫺1 and h is a small positive number, then A共x ⫹ h兲 ⫺ A共x兲 represents the area of a region. Describe and sketch the region. (e) Draw a rectangle that approximates the region in part (d). By comparing the areas of these two regions, show that A共x ⫹ h兲 ⫺ A共x兲 ⬇ 1 ⫹ x2 h (f ) Use part (e) to give an intuitive explanation for the result of part (c). 2 ; 3. (a) Draw the graph of the function f 共x兲 苷 cos共x 兲 in the viewing rectangle 关0, 2兴
by 关⫺1.25, 1.25兴. (b) If we define a new function t by
x
t共x兲 苷 y cos共t 2 兲 dt 0
then t共x兲 is the area under the graph of f from 0 to x [until f 共x兲 becomes negative, at which point t共x兲 becomes a difference of areas]. Use part (a) to determine the value of
;
Graphing calculator or computer required
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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x at which t共x兲 starts to decrease. [Unlike the integral in Problem 2, it is impossible to evaluate the integral defining t to obtain an explicit expression for t共x兲.] (c) Use the integration command on your calculator or computer to estimate t共0.2兲, t共0.4兲, t共0.6兲, . . . , t共1.8兲, t共2兲. Then use these values to sketch a graph of t. (d) Use your graph of t from part (c) to sketch the graph of t⬘ using the interpretation of t⬘共x兲 as the slope of a tangent line. How does the graph of t⬘ compare with the graph of f ? 4. Suppose f is a continuous function on the interval 关a, b兴 and we define a new function t
by the equation t共x兲 苷
y
x
a
f 共t兲 dt
Based on your results in Problems 1–3, conjecture an expression for t⬘共x兲.
5.3
The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches of calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem, the area problem. Newton’s mentor at Cambridge, Isaac Barrow (1630–1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical method. In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums as we did in Sections 5.1 and 5.2. The first part of the Fundamental Theorem deals with functions defined by an equation of the form t共x兲 苷
1
y
x
a
f 共t兲 dt
where f is a continuous function on 关a, b兴 and x varies between a and b. Observe that t depends only on x, which appears as the variable upper limit in the integral. If x is a fixed number, then the integral xax f 共t兲 dt is a definite number. If we then let x vary, the number xax f 共t兲 dt also varies and defines a function of x denoted by t共x兲. If f happens to be a positive function, then t共x兲 can be interpreted as the area under the graph of f from a to x, where x can vary from a to b. (Think of t as the “area so far” function; see Figure 1.) y
y=f(t ) area=©
FIGURE 1
0
a
x
b
t
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.3
y=f(t) 1 1
2
387
v EXAMPLE 1 If f is the function whose graph is shown in Figure 2 and t共x兲 苷 x0x f 共t兲 dt, find the values of t共0兲, t共1兲, t共2兲, t共3兲, t共4兲, and t共5兲. Then sketch a rough graph of t.
y 2
0
THE FUNDAMENTAL THEOREM OF CALCULUS
SOLUTION First we notice that t共0兲 苷
t
4
x00 f 共t兲 dt 苷 0. From Figure 3 we see that t共1兲 is the
area of a triangle: 1
t共1兲 苷 y f 共t兲 dt 苷 12 共1 ⴢ 2兲 苷 1 0
FIGURE 2
To find t共2兲 we add to t共1兲 the area of a rectangle: 2
1
2
t共2兲 苷 y f 共t兲 dt 苷 y f 共t兲 dt ⫹ y f 共t兲 dt 苷 1 ⫹ 共1 ⴢ 2兲 苷 3 0
0
1
We estimate that the area under f from 2 to 3 is about 1.3, so t共3兲 苷 t共2兲 ⫹
y
3
2
f 共t兲 dt ⬇ 3 ⫹ 1.3 苷 4.3
y 2
y 2
y 2
y 2
y 2
1
1
1
1
1
0
1
t
0
1
g(1)=1
g(2)=3
t
0
1
2
3
t
0
1
2
4
t
0
1
2
4
t
g(3)Å4.3
g(4)Å3
FIGURE 3
g(5)Å1.7
For t ⬎ 3, f 共t兲 is negative and so we start subtracting areas:
y 4
g
4
t共4兲 苷 t共3兲 ⫹ y f 共t兲 dt ⬇ 4.3 ⫹ 共⫺1.3兲 苷 3.0
3
3
2
5
t共5兲 苷 t共4兲 ⫹ y f 共t兲 dt ⬇ 3 ⫹ 共⫺1.3兲 苷 1.7
1 0
2
4
1
2
FIGURE 4 x
©=j f(t) dt a
3
4
5 x
We use these values to sketch the graph of t in Figure 4. Notice that, because f 共t兲 is positive for t ⬍ 3, we keep adding area for t ⬍ 3 and so t is increasing up to x 苷 3, where it attains a maximum value. For x ⬎ 3, t decreases because f 共t兲 is negative. If we take f 共t兲 苷 t and a 苷 0, then, using Exercise 27 in Section 5.2, we have x
t共x兲 苷 y t dt 苷 0
x2 2
Notice that t⬘共x兲 苷 x, that is, t⬘ 苷 f . In other words, if t is defined as the integral of f by Equation 1, then t turns out to be an antiderivative of f , at least in this case. And if we sketch the derivative of the function t shown in Figure 4 by estimating slopes of tangents, we get a graph like that of f in Figure 2. So we suspect that t⬘苷 f in Example 1 too. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
To see why this might be generally true we consider any continuous function f with f 共x兲 艌 0. Then t共x兲 苷 xax f 共t兲 dt can be interpreted as the area under the graph of f from a to x, as in Figure 1. In order to compute t⬘共x兲 from the definition of a derivative we first observe that, for h ⬎ 0, t共x ⫹ h兲 ⫺ t共x兲 is obtained by subtracting areas, so it is the area under the graph of f from x to x ⫹ h (the blue area in Figure 5). For small h you can see from the figure that this area is approximately equal to the area of the rectangle with height f 共x兲 and width h :
y
h ƒ 0
a
Page 388
t共x ⫹ h兲 ⫺ t共x兲 ⬇ hf 共x兲 x
b
x+h
t
t共x ⫹ h兲 ⫺ t共x兲 ⬇ f 共x兲 h
so
FIGURE 5
Intuitively, we therefore expect that t⬘共x兲 苷 lim
hl0
t共x ⫹ h兲 ⫺ t共x兲 苷 f 共x兲 h
The fact that this is true, even when f is not necessarily positive, is the first part of the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus, Part 1 If f is continuous on 关a, b兴, then the We abbreviate the name of this theorem as FTC1. In words, it says that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit.
function t defined by x
t共x兲 苷 y f 共t兲 dt
a艋x艋b
a
is continuous on 关a, b兴 and differentiable on 共a, b兲, and t⬘共x兲 苷 f 共x兲. PROOF If x and x ⫹ h are in 共a, b兲, then
t共x ⫹ h兲 ⫺ t共x兲 苷 y
x⫹h
a
苷
a
冉y
x
a
苷y
x⫹h
x
x
f 共t兲 dt ⫺ y f 共t兲 dt f 共t兲 dt ⫹ y
x⫹h
x
冊
x
f 共t兲 dt ⫺ y f 共t兲 dt
(by Property 5)
a
f 共t兲 dt
and so, for h 苷 0,
y
y=ƒ
2
FIGURE 6
x u
√=x+h
y
x⫹h
x
f 共t兲 dt
For now let’s assume that h ⬎ 0. Since f is continuous on 关x, x ⫹ h兴, the Extreme Value Theorem says that there are numbers u and v in 关x, x ⫹ h兴 such that f 共u兲 苷 m and f 共v兲 苷 M, where m and M are the absolute minimum and maximum values of f on 关x, x ⫹ h兴. (See Figure 6.) By Property 8 of integrals, we have
M m
0
t共x ⫹ h兲 ⫺ t共x兲 1 苷 h h
x
mh 艋 y
x⫹h
x
f 共t兲 dt 艋 Mh
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.3
THE FUNDAMENTAL THEOREM OF CALCULUS
f 共u兲h 艋 y
that is,
x⫹h
x
389
f 共t兲 dt 艋 f 共v兲h
Since h ⬎ 0, we can divide this inequality by h : f 共u兲 艋
1 h
x⫹h
y
x
f 共t兲 dt 艋 f 共v兲
Now we use Equation 2 to replace the middle part of this inequality: f 共u兲 艋
3 TEC Module 5.3 provides visual evidence for FTC1.
t共x ⫹ h兲 ⫺ t共x兲 艋 f 共v兲 h
Inequality 3 can be proved in a similar manner for the case where h ⬍ 0. (See Exercise 71.) Now we let h l 0. Then u l x and v l x, since u and v lie between x and x ⫹ h. Therefore lim f 共u兲 苷 lim f 共u兲 苷 f 共x兲
hl0
and
ulx
lim f 共v兲 苷 lim f 共v兲 苷 f 共x兲
hl0
vlx
because f is continuous at x. We conclude, from 3 and the Squeeze Theorem, that 4
t⬘共x兲 苷 lim
hl0
t共x ⫹ h兲 ⫺ t共x兲 苷 f 共x兲 h
If x 苷 a or b, then Equation 4 can be interpreted as a onesided limit. Then Theorem 2.8.4 (modified for onesided limits) shows that t is continuous on 关a, b兴. Using Leibniz notation for derivatives, we can write FTC1 as 5
d dx
y
x
a
f 共t兲 dt 苷 f 共x兲
when f is continuous. Roughly speaking, Equation 5 says that if we first integrate f and then differentiate the result, we get back to the original function f .
v
EXAMPLE 2 Find the derivative of the function t共x兲 苷
y
x
0
s1 ⫹ t 2 dt.
SOLUTION Since f 共t兲 苷 s1 ⫹ t 2 is continuous, Part 1 of the Fundamental Theorem of
Calculus gives t⬘共x兲 苷 s1 ⫹ x 2 EXAMPLE 3 Although a formula of the form t共x兲 苷
xax f 共t兲 dt may seem like a strange
way of defining a function, books on physics, chemistry, and statistics are full of such functions. For instance, the Fresnel function x
S共x兲 苷 y sin共 t 2兾2兲 dt 0
is named after the French physicist Augustin Fresnel (1788–1827), who is famous for his works in optics. This function first appeared in Fresnel’s theory of the diffraction of light waves, but more recently it has been applied to the design of highways.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Part 1 of the Fundamental Theorem tells us how to differentiate the Fresnel function: S⬘共x兲 苷 sin共 x 2兾2兲 This means that we can apply all the methods of differential calculus to analyze S (see Exercise 65). Figure 7 shows the graphs of f 共x兲 苷 sin共 x 2兾2兲 and the Fresnel function S共x兲 苷 x0x f 共t兲 dt. A computer was used to graph S by computing the value of this integral for many values of x. It does indeed look as if S共x兲 is the area under the graph of f from 0 to x [until x ⬇ 1.4 when S共x兲 becomes a difference of areas]. Figure 8 shows a larger part of the graph of S. y
y 1
f
0.5
S 0
x
1
x
1
FIGURE 7
FIGURE 8
ƒ=sin(π≈/2)
The Fresnel function S(x)=j sin(π
[email protected]/2) dt
x
0
x
S(x)= j sin(π
[email protected]/2) dt 0
If we now start with the graph of S in Figure 7 and think about what its derivative should look like, it seems reasonable that S⬘共x兲 苷 f 共x兲. [For instance, S is increasing when f 共x兲 ⬎ 0 and decreasing when f 共x兲 ⬍ 0.] So this gives a visual confirmation of Part 1 of the Fundamental Theorem of Calculus. d x y sec t dt. dx 1 SOLUTION Here we have to be careful to use the Chain Rule in conjunction with FTC1. Let u 苷 x 4. Then 4
EXAMPLE 4 Find
d dx
y
x4
1
sec t dt 苷
d dx
苷
d du
y
u
1
sec t dt
冋y
苷 sec u
u
1
册
sec t dt
du dx
du dx
(by the Chain Rule)
(by FTC1)
苷 sec共x 4 兲 ⴢ 4x 3 In Section 5.2 we computed integrals from the definition as a limit of Riemann sums and we saw that this procedure is sometimes long and difficult. The second part of the Fundamental Theorem of Calculus, which follows easily from the first part, provides us with a much simpler method for the evaluation of integrals. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.3
THE FUNDAMENTAL THEOREM OF CALCULUS
391
The Fundamental Theorem of Calculus, Part 2 If f is continuous on 关a, b兴, then
y
We abbreviate this theorem as FTC2.
b
f 共x兲 dx 苷 F共b兲 ⫺ F共a兲
a
where F is any antiderivative of f, that is, a function such that F⬘ 苷 f. PROOF Let t共x兲 苷
xax f 共t兲 dt. We know from Part 1 that t⬘共x兲 苷 f 共x兲; that is, t is an anti
derivative of f . If F is any other antiderivative of f on 关a, b兴, then we know from Corollary 4.2.7 that F and t differ by a constant: F共x兲 苷 t共x兲 ⫹ C
6
for a ⬍ x ⬍ b. But both F and t are continuous on 关a, b兴 and so, by taking limits of both sides of Equation 6 (as x l a⫹ and x l b⫺ ), we see that it also holds when x 苷 a and x 苷 b. If we put x 苷 a in the formula for t共x兲, we get a
t共a兲 苷 y f 共t兲 dt 苷 0 a
So, using Equation 6 with x 苷 b and x 苷 a, we have F共b兲 ⫺ F共a兲 苷 关t共b兲 ⫹ C兴 ⫺ 关t共a兲 ⫹ C兴 b
苷 t共b兲 ⫺ t共a兲 苷 t共b兲 苷 y f 共t兲 dt a
Part 2 of the Fundamental Theorem states that if we know an antiderivative F of f , then we can evaluate xab f 共x兲 dx simply by subtracting the values of F at the endpoints of the interval 关a, b兴. It’s very surprising that xab f 共x兲 dx, which was defined by a complicated procedure involving all of the values of f 共x兲 for a 艋 x 艋 b, can be found by knowing the values of F共x兲 at only two points, a and b. Although the theorem may be surprising at first glance, it becomes plausible if we interpret it in physical terms. If v共t兲 is the velocity of an object and s共t兲 is its position at time t, then v共t兲 苷 s⬘共t兲, so s is an antiderivative of v. In Section 5.1 we considered an object that always moves in the positive direction and made the guess that the area under the velocity curve is equal to the distance traveled. In symbols:
y
b
a
v共t兲 dt 苷 s共b兲 ⫺ s共a兲
That is exactly what FTC2 says in this context.
v
EXAMPLE 5 Evaluate the integral
y
3
1
e x dx.
SOLUTION The function f 共x兲 苷 e x is continuous everywhere and we know that an anti
derivative is F共x兲 苷 e x, so Part 2 of the Fundamental Theorem gives
Compare the calculation in Example 5 with the much harder one in Example 3 in Section 5.2.
y
3
1
e x dx 苷 F共3兲 ⫺ F共1兲 苷 e 3 ⫺ e
Notice that FTC2 says we can use any antiderivative F of f. So we may as well use the simplest one, namely F共x兲 苷 e x, instead of e x ⫹ 7 or e x ⫹ C . Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
We often use the notation
]
F共x兲
b
苷 F共b兲 ⫺ F共a兲
a
So the equation of FTC2 can be written as
y
b
a
]
f 共x兲 dx 苷 F共x兲
Other common notations are F共x兲
ⱍ
b a
b
where
a
F⬘苷 f
and 关F共x兲兴 ab .
EXAMPLE 6 Find the area under the parabola y 苷 x 2 from 0 to 1. SOLUTION An antiderivative of f 共x兲 苷 x 2 is F共x兲 苷 3 x 3. The required area A is found 1
using Part 2 of the Fundamental Theorem: 1
A 苷 y x 2 dx 苷 0
3
In applying the Fundamental Theorem we use a particular antiderivative F of f. It is not necessary to use the most general antiderivative.
苷
x3 3
册
1
0
3
1 0 1 ⫺ 苷 3 3 3
If you compare the calculation in Example 6 with the one in Example 2 in Section 5.1, you will see that the Fundamental Theorem gives a much shorter method. EXAMPLE 7 Evaluate
y
6
3
dx . x
SOLUTION The given integral is an abbreviation for
y
1 dx x
6
3
ⱍ ⱍ
An antiderivative of f 共x兲 苷 1兾x is F共x兲 苷 ln x and, because 3 艋 x 艋 6, we can write F共x兲 苷 ln x. So
y
6
3
1 6 dx 苷 ln x 3 苷 ln 6 ⫺ ln 3 x
]
苷 ln
6 苷 ln 2 3
y 1
EXAMPLE 8 Find the area under the cosine curve from 0 to b, where 0 艋 b 艋 兾2.
y=cos x
SOLUTION Since an antiderivative of f 共x兲 苷 cos x is F共x兲 苷 sin x, we have area=1 0
FIGURE 9
π 2
x
b
A 苷 y cos x dx 苷 sin x 0
]
b 0
苷 sin b ⫺ sin 0 苷 sin b
In particular, taking b 苷 兾2, we have proved that the area under the cosine curve from 0 to 兾2 is sin共兾2兲 苷 1. (See Figure 9.)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.3
THE FUNDAMENTAL THEOREM OF CALCULUS
393
When the French mathematician Gilles de Roberval first found the area under the sine and cosine curves in 1635, this was a very challenging problem that required a great deal of ingenuity. If we didn’t have the benefit of the Fundamental Theorem, we would have to compute a difficult limit of sums using obscure trigonometric identities (or a computer algebra system as in Exercise 29 in Section 5.1). It was even more difficult for Roberval because the apparatus of limits had not been invented in 1635. But in the 1660s and 1670s, when the Fundamental Theorem was discovered by Barrow and exploited by Newton and Leibniz, such problems became very easy, as you can see from Example 8. EXAMPLE 9 What is wrong with the following calculation?
y

3
⫺1
1 x⫺1 2 dx 苷 x ⫺1
册
3
苷⫺
⫺1
1 4 ⫺1苷⫺ 3 3
SOLUTION To start, we notice that this calculation must be wrong because the answer is
negative but f 共x兲 苷 1兾x 2 艌 0 and Property 6 of integrals says that xab f 共x兲 dx 艌 0 when f 艌 0. The Fundamental Theorem of Calculus applies to continuous functions. It can’t be applied here because f 共x兲 苷 1兾x 2 is not continuous on 关⫺1, 3兴. In fact, f has an infinite discontinuity at x 苷 0, so
y
3
⫺1
1 dx x2
does not exist
Differentiation and Integration as Inverse Processes We end this section by bringing together the two parts of the Fundamental Theorem. The Fundamental Theorem of Calculus Suppose f is continuous on 关a, b兴. 1. If t共x兲 苷 2.
xax f 共t兲 dt, then t⬘共x兲 苷 f 共x兲.
xab f 共x兲 dx 苷 F共b兲 ⫺ F共a兲, where F is any antiderivative of
f, that is, F⬘苷 f.
We noted that Part 1 can be rewritten as d dx
y
x
a
f 共t兲 dt 苷 f 共x兲
which says that if f is integrated and then the result is differentiated, we arrive back at the original function f . Since F⬘共x兲 苷 f 共x兲, Part 2 can be rewritten as
y
b
a
F⬘共x兲 dx 苷 F共b兲 ⫺ F共a兲
This version says that if we take a function F, first differentiate it, and then integrate the result, we arrive back at the original function F, but in the form F共b兲 ⫺ F共a兲. Taken together, the two parts of the Fundamental Theorem of Calculus say that differentiation and integration are inverse processes. Each undoes what the other does. The Fundamental Theorem of Calculus is unquestionably the most important theorem in calculus and, indeed, it ranks as one of the great accomplishments of the human mind.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
Before it was discovered, from the time of Eudoxus and Archimedes to the time of Galileo and Fermat, problems of finding areas, volumes, and lengths of curves were so difficult that only a genius could meet the challenge. But now, armed with the systematic method that Newton and Leibniz fashioned out of the Fundamental Theorem, we will see in the chapters to come that these challenging problems are accessible to all of us.
5.3
Exercises
1. Explain exactly what is meant by the statement that “differenti
ation and integration are inverse processes.”
(e) Sketch a rough graph of t. (f ) Use the graph in part (e) to sketch the graph of t⬘共x兲. Compare with the graph of f.
x0x f 共t兲 dt, where f is the function whose graph is shown. (a) Evaluate t共x兲 for x 苷 0, 1, 2, 3, 4, 5, and 6. (b) Estimate t共7兲. (c) Where does t have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of t.
2. Let t共x兲 苷
y 2
0
2
5
x
y
5–6 Sketch the area represented by t共x兲. Then find t⬘共x兲 in two
ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.
1 0
4
1
6
t
5. t共x兲 苷 3. Let t共x兲 苷
x0x f 共t兲 dt, where
f is the function whose graph is shown. (a) Evaluate t共0兲, t共1兲, t共2兲, t共3兲, and t共6兲. (b) On what interval is t increasing? (c) Where does t have a maximum value? (d) Sketch a rough graph of t. y
0
4. Let t共x兲 苷
1
x0x f 共t兲 dt, where
x
共2 ⫹ sin t兲 dt
0
7. t共x兲 苷
y
9. t共s兲 苷
y
1 dt t3 ⫹ 1
8. t共x兲 苷
y
共t ⫺ t 2 兲8 dt
10. t共r兲 苷
y
x
1 s
5
y
x
r
0
s1 ⫹ sec t dt
x
3
2
e t ⫺t dt sx 2 ⫹ 4 dx
册
x
Hint: y s1 ⫹ sec t dt 苷 ⫺y s1 ⫹ sec t dt
5
t
Graphing calculator or computer required
x
12. G共x兲 苷
y
13. h共x兲 苷
y
f is the function whose graph is
shown. (a) Evaluate t共0兲 and t共6兲. (b) Estimate t共x兲 for x 苷 1, 2, 3, 4, and 5. (c) On what interval is t increasing? (d) Where does t have a maximum value?
;
y
7–18 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
冋
f
6. t共x兲 苷
t 2 dt
1
11. F共x兲 苷 1
x
y
15. y 苷
y
17. y 苷
y
1
x
1
tan x
0 1
1⫺3x
CAS Computer algebra system required
ex
cos st dt ln t dt
14. h共x兲 苷
st ⫹ s t dt
16. y 苷
y
u3 du 1 ⫹ u2
18. y 苷
y
x
y
4
0 1
sin x
sx
1
z2 dz z ⫹1 4
cos2 d s1 ⫹ t 2 dt
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.3
19– 44 Evaluate the integral. 19.
y
2
4
21.
y
23.
y
25.
y
27. 29. 31. 33.
35.
1 9
1
1
0
y
24.
y
26.
y
sin d
共u ⫹ 2兲共u ⫺ 3兲 du
兾4
1
y y
39.
y
v 3 ⫹ 3v 6
2
1
43.
44.
y y y y y
y
2
3
1
2
1
sec tan d
共2 sin x ⫺ e x 兲 dx
冑
18
0
再 再
3 dz z
region that lies beneath the given curve. Then find the exact area. 3 49. y 苷 s x,
51. y 苷 sin x, 0 艋 x 艋
53.
y
2
⫺1
if ⫺2 艋 x 艋 0 if 0 ⬍ x 艋 2
cos x dx
兾6
55–59 Find the derivative of the function. 55. t共x兲 苷
冋
y
3x
2x
u2 ⫺ 1 du u2 ⫹ 1
3x
0
册
3x
Hint: y f 共u兲 du 苷 y f 共u兲 du ⫹ y f 共u兲 du 2x
1⫹2x
56. t共x兲 苷
y
57. F共x兲 苷
y
y
2x
1⫺2x x2
x
sin x
cos x
0
t sin t dt 2
58. F共x兲 苷
e t dt
y
2x
arctan t dt
sx
ln共1 ⫹ 2v兲 dv
x0x 共1 ⫺ t 2 兲e t
2
dt, on what interval is f increasing?
61. On what interval is the curve
y苷y
x
0
2 4 ⫺ x2
2
y
54.
60. If f 共x兲 苷
1兾2
52. y 苷 sec x, 0 艋 x 艋 兾3
x 3 dx
4 ⫹ u2 du u3 4 dx s1 ⫺ x 2
1艋x艋6
2
53–54 Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
59. y 苷
1兾s2
50. y 苷 x ⫺4,
0 艋 x 艋 27
cosh t dt
sin x if 0 艋 x ⬍ 兾2 cos x if 兾2 艋 x 艋
where f 共x兲 苷
f 共x兲 dx where f 共x兲 苷
⫺2
共 y ⫺ 1兲共2y ⫹ 1兲 dy
1
42.
f 共x兲 dx
共4 ⫺ t兲 st dt
兾4
0
y
2
u 4 ⫺ 25 u 9) du
e dx
0
40.
e u⫹1 du
4
0
8 dx 1 ⫹ x2
1
5
0
1 2
x ⫺2兾3 dx
⫺5
y
s3
8
1
38.
0
y
36.
0
共x e ⫹ e x 兲 dx
⫺1
y
34.
dv
v4
0
y
32.
共1 ⫹ 2y兲 2 dy
1兾s3
41.
30.
sec 2 t dt
1
37.
28.
x⫺1 dx sx
0
y
1
sx dx
9
2
x 100 dx
⫺1
y (1 ⫹
1
y
y
22.
兾6
y
20.
1
共5 ⫺ 2t ⫹ 3t 2兲 dt
395
; 49–52 Use a graph to give a rough estimate of the area of the
共x 3 ⫺ 2x兲 dx
⫺1
THE FUNDAMENTAL THEOREM OF CALCULUS
t2 dt t ⫹t⫹2 2
concave downward? 62. If f 共x兲 苷
x0sin x s1 ⫹ t 2
find t ⬙共兾6兲.
dt and t共 y兲 苷 x3y f 共x兲 dx,
63. If f 共1兲 苷 12, f ⬘ is continuous, and x14 f ⬘共x兲 dx 苷 17, what is
the value of f 共4兲?
; 45– 48 What is wrong with the equation? 45.
y
1
⫺2
y
47.
y
⫺1
兾3
y
0
x⫺3 ⫺3
册 册
3 8
苷⫺
⫺2
3 苷 2 ⫺1
sec tan d 苷 sec
]
sec 2 x dx 苷 tan x
0
兾3
]
苷0
64. The error function
erf共x兲 苷
2
4 2 dx 苷 ⫺ 2 x3 x
2
46.
48.
x ⫺4 dx 苷
1
苷 ⫺3
2 s
y
x
0
2
e⫺t dt
is used in probability, statistics, and engineering. 2 1 (a) Show that xab e⫺t dt 苷 2 s 关erf共b兲 ⫺ erf共a兲兴. 2 (b) Show that the function y 苷 e x erf共x兲 satisfies the differential equation y⬘ 苷 2xy ⫹ 2兾s . 65. The Fresnel function S was defined in Example 3 and
graphed in Figures 7 and 8. (a) At what values of x does this function have local maximum values?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
CAS
(b) On what intervals is the function concave upward? (c) Use a graph to solve the following equation correct to two decimal places:
Page 396
INTEGRALS
y
x
0
CAS
2:29 PM
69–70 Evaluate the limit by first recognizing the sum as a
Riemann sum for a function defined on 关0, 1兴. n
69. lim
兺
70. lim
1 n
n l ⬁ i苷1
sin共 t 2兾2兲 dt 苷 0.2
66. The sine integral function
nl⬁
Si共x兲 苷 y
x
0
sin t dt t
is important in electrical engineering. [The integrand f 共t兲 苷 共sin t兲兾t is not defined when t 苷 0, but we know that its limit is 1 when t l 0. So we define f 共0兲 苷 1 and this makes f a continuous function everywhere.] (a) Draw the graph of Si. (b) At what values of x does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation correct to one decimal place:
y
x
0
i3 n4
冉冑 冑 冑 1 ⫹ n
2 ⫹ n
72. If f is continuous and t and h are differentiable functions,
find a formula for d dx
y 3
74. (a) Show that cos共x 2兲 艌 cos x for 0 艋 x 艋 1. 兾6
(b) Deduce that x0
cos共x 2兲 dx 艌 12.
10
x2 dx 艋 0.1 x ⫹ x2 ⫹ 1 4
by comparing the integrand to a simpler function. 76. Let
0 x 2⫺x 0
f 共x兲 苷
if if if if
x⬍0 0艋x艋1 1⬍x艋2 x⬎2
x
t共x兲 苷 y f 共t兲 dt 0
4
6
t
8
77. Find a function f and a number a such that
_2
6⫹
y
f 共t兲 dt 苷 2 sx t2
x
a
for all x ⬎ 0
78. The area labeled B is three times the area labeled A. Express
y
b in terms of a.
f 0.4
y
y
y=´
0.2 0
f 共t兲 dt
(a) Find an expression for t共x兲 similar to the one for f 共x兲. (b) Sketch the graphs of f and t. (c) Where is f differentiable? Where is t differentiable?
f
1
68.
h共x兲
t共x兲
(b) Show that 1 艋 x01 s1 ⫹ x 3 dx 艋 1.25.
and
2
y
73. (a) Show that 1 艋 s1 ⫹ x 3 艋 1 ⫹ x 3 for x 艌 0.
0艋y
67–68 Let t共x兲 苷 x f 共t兲 dt, where f is the function whose graph is shown. (a) At what values of x do the local maximum and minimum values of t occur? (b) Where does t attain its absolute maximum value? (c) On what intervals is t concave downward? (d) Sketch the graph of t.
0 _1
n n
71. Justify 3 for the case h ⬍ 0.
5
2
冑冊
75. Show that
sin t dt 苷 1 t
x 0
67.
3 ⫹ ⭈⭈⭈ ⫹ n
1
3
5
7
9
y=´
t
_0.2
B
A 0
a
x
0
b
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.4
that depreciates at the (continuous) rate f 苷 f 共t兲, where t is the time measured in months since its last overhaul. Because a fixed cost A is incurred each time the machine is overhauled, the company wants to determine the optimal time T (in months) between overhauls. (a) Explain why x0t f 共s兲 ds represents the loss in value of the machine over the period of time t since the last overhaul. (b) Let C 苷 C共t兲 be given by
冋
t
Show that the critical numbers of C occur at the numbers t where C共t兲 苷 f 共t兲 ⫹ t共t兲. (b) Suppose that
册
f 共t兲 苷
A ⫹ y f 共s兲 ds 0
What does C represent and why would the company want to minimize C ? (c) Show that C has a minimum value at the numbers t 苷 T where C共T 兲 苷 f 共T 兲.
再
V V ⫺ t 15 450 0
t共t兲 苷
and
if 0 ⬍ t 艋 30 if t ⬎ 30
Vt 2 12,900
t⬎0
Determine the length of time T for the total depreciation D共t兲 苷 x0t f 共s兲 ds to equal the initial value V. (c) Determine the absolute minimum of C on 共0, T 兴. (d) Sketch the graphs of C and f ⫹ t in the same coordinate system, and verify the result in part (a) in this case.
80. A hightech company purchases a new computing system
whose initial value is V. The system will depreciate at the rate f 苷 f 共t兲 and will accumulate maintenance costs at the
5.4
397
rate t 苷 t共t兲, where t is the time measured in months. The company wants to determine the optimal time to replace the system. (a) Let 1 t C共t兲 苷 y 关 f 共s兲 ⫹ t共s兲兴 ds t 0
79. A manufacturing company owns a major piece of equipment
1 C共t兲 苷 t
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
Indefinite Integrals and the Net Change Theorem We saw in Section 5.3 that the second part of the Fundamental Theorem of Calculus provides a very powerful method for evaluating the definite integral of a function, assuming that we can find an antiderivative of the function. In this section we introduce a notation for antiderivatives, review the formulas for antiderivatives, and use them to evaluate definite integrals. We also reformulate FTC2 in a way that makes it easier to apply to science and engineering problems.
Indefinite Integrals Both parts of the Fundamental Theorem establish connections between antiderivatives and definite integrals. Part 1 says that if f is continuous, then xax f 共t兲 dt is an antiderivative of f. Part 2 says that xab f 共x兲 dx can be found by evaluating F共b兲 ⫺ F共a兲, where F is an antiderivative of f. We need a convenient notation for antiderivatives that makes them easy to work with. Because of the relation given by the Fundamental Theorem between antiderivatives and integrals, the notation x f 共x兲 dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus
y f 共x兲 dx 苷 F共x兲
means
F⬘共x兲 苷 f 共x兲
For example, we can write
yx
2
dx 苷
x3 ⫹C 3
because
d dx
冉
冊
x3 ⫹ C 苷 x2 3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
So we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).  You should distinguish carefully between definite and indefinite integrals. A definite integral xab f 共x兲 dx is a number, whereas an indefinite integral x f 共x兲 dx is a function (or family of functions). The connection between them is given by Part 2 of the Fundamental Theorem: If f is continuous on 关a, b兴, then
y
b
a
册
f 共x兲 dx 苷 y f 共x兲 dx
b a
The effectiveness of the Fundamental Theorem depends on having a supply of antiderivatives of functions. We therefore restate the Table of Antidifferentiation Formulas from Section 4.9, together with a few others, in the notation of indefinite integrals. Any formula can be verified by differentiating the function on the right side and obtaining the integrand. For instance,
y sec x dx 苷 tan x ⫹ C 2
d 共tan x ⫹ C兲 苷 sec2x dx
because
Table of Indefinite Integrals
1
y cf 共x兲 dx 苷 c y f 共x兲 dx
y 关 f 共x兲 ⫹ t共x兲兴 dx 苷 y f 共x兲 dx ⫹ y t共x兲 dx
y k dx 苷 kx ⫹ C x n⫹1 ⫹C n⫹1
yx
n
dx 苷
ye
x
dx 苷 e x ⫹ C
共n 苷 ⫺1兲
y
1 dx 苷 ln x ⫹ C x
ⱍ ⱍ
ya
x
dx 苷
ax ⫹C ln a
y sin x dx 苷 ⫺cos x ⫹ C
y cos x dx 苷 sin x ⫹ C
y sec x dx 苷 tan x ⫹ C
y csc x dx 苷 ⫺cot x ⫹ C
y sec x tan x dx 苷 sec x ⫹ C
y csc x cot x dx 苷 ⫺csc x ⫹ C
2
yx
2
1 dx 苷 tan⫺1x ⫹ C ⫹1
y sinh x dx 苷 cosh x ⫹ C
2
y
1 dx 苷 sin⫺1x ⫹ C s1 ⫺ x 2
y cosh x dx 苷 sinh x ⫹ C
Recall from Theorem 4.9.1 that the most general antiderivative on a given interval is obtained by adding a constant to a particular antiderivative. We adopt the convention that when a formula for a general indefinite integral is given, it is valid only on an interval. Thus we write 1 1 y x 2 dx 苷 ⫺ x ⫹ C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.4
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
399
with the understanding that it is valid on the interval 共0, ⬁兲 or on the interval 共⫺⬁, 0兲. This is true despite the fact that the general antiderivative of the function f 共x兲 苷 1兾x 2, x 苷 0, is 1 ⫹ C1 if x ⬍ 0 x F共x兲 苷 1 ⫺ ⫹ C2 if x ⬎ 0 x ⫺
EXAMPLE 1 Find the general indefinite integral
y 共10x
4
4
⫺ 2 sec 2x兲 dx
SOLUTION Using our convention and Table 1, we have _1.5
1.5
y 共10x
4
⫺ 2 sec2x兲 dx 苷 10 y x 4 dx ⫺ 2 y sec2x dx
_4
苷 10
FIGURE 1 The indefinite integral in Example 1 is graphed in Figure 1 for several values of C. Here the value of C is the yintercept.
x5 ⫺ 2 tan x ⫹ C 5
苷 2x 5 ⫺ 2 tan x ⫹ C You should check this answer by differentiating it.
v
EXAMPLE 2 Evaluate
y
cos d. sin2
SOLUTION This indefinite integral isn’t immediately apparent in Table 1, so we use trigo
nometric identities to rewrite the function before integrating: cos d 苷 y sin2
y
冉 冊冉 冊 cos sin
1 sin
d
苷 y csc cot d 苷 ⫺csc ⫹ C
EXAMPLE 3 Evaluate
y
3
0
共x 3 ⫺ 6x兲 dx.
SOLUTION Using FTC2 and Table 1, we have
y
3
0
共x 3 ⫺ 6x兲 dx 苷
x4 x2 ⫺6 4 2
册
3
0
苷 ( ⴢ 3 ⫺ 3 ⴢ 3 2 ) ⫺ ( 14 ⴢ 0 4 ⫺ 3 ⴢ 0 2 ) 1 4
4
苷 814 ⫺ 27 ⫺ 0 ⫹ 0 苷 ⫺6.75 Compare this calculation with Example 2(b) in Section 5.2.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
v
EXAMPLE 4 Find
y
2x 3 ⫺ 6x ⫹
0
areas. Figure 2 shows the graph of the integrand in Example 4. We know from Section 5.2 that the value of the integral can be interpreted as a net area: the sum of the areas labeled with a plus sign minus the area labeled with a minus sign.
冉
2
3 x ⫹1 2
冊
dx and interpret the result in terms of
SOLUTION The Fundamental Theorem gives
y
2
0
冉
2x 3 ⫺ 6x ⫹
3 x2 ⫹ 1
冊
dx 苷 2
册
x4 x2 ⫺6 ⫹ 3 tan⫺1x 4 2
2
0
2
]
y
苷 12 x 4 ⫺ 3x 2 ⫹ 3 tan⫺1x
0
苷 共2 4 兲 ⫺ 3共2 2 兲 ⫹ 3 tan⫺1 2 ⫺ 0 1 2
3
苷 ⫺4 ⫹ 3 tan⫺1 2 0
2 x
This is the exact value of the integral. If a decimal approximation is desired, we can use a calculator to approximate tan⫺1 2. Doing so, we get
y
FIGURE 2
2
0
EXAMPLE 5 Evaluate
y
9
1
冉
2x 3 ⫺ 6x ⫹
3 x ⫹1 2
冊
dx ⬇ ⫺0.67855
2t 2 ⫹ t 2 st ⫺ 1 dt. t2
SOLUTION First we need to write the integrand in a simpler form by carrying out the
division:
y
9
1
2t 2 ⫹ t 2 st ⫺ 1 9 dt 苷 y 共2 ⫹ t 1兾2 ⫺ t⫺2 兲 dt 1 t2 苷 2t ⫹
t 3兾2 3 2
⫺
t⫺1 ⫺1
册
9
苷 2t ⫹ 23 t 3兾2 ⫹ 1
1 t
册
9
1
苷 (2 ⴢ 9 ⫹ 23 ⴢ 9 3兾2 ⫹ 19 ) ⫺ (2 ⴢ 1 ⫹ 23 ⴢ 13兾2 ⫹ 11 ) 苷 18 ⫹ 18 ⫹ 19 ⫺ 2 ⫺ 23 ⫺ 1 苷 32 49
Applications Part 2 of the Fundamental Theorem says that if f is continuous on 关a, b兴, then
y
b
a
f 共x兲 dx 苷 F共b兲 ⫺ F共a兲
where F is any antiderivative of f. This means that F⬘ 苷 f , so the equation can be rewritten as b y F⬘共x兲 dx 苷 F共b兲 ⫺ F共a兲 a
We know that F⬘共x兲 represents the rate of change of y 苷 F共x兲 with respect to x and F共b兲 ⫺ F共a兲 is the change in y when x changes from a to b. [Note that y could, for instance, increase, then decrease, then increase again. Although y might change in both directions, F共b兲 ⫺ F共a兲 represents the net change in y.] So we can reformulate FTC2 in words as follows.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.4
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
401
Net Change Theorem The integral of a rate of change is the net change:
y
b
F⬘共x兲 dx 苷 F共b兲 ⫺ F共a兲
a
This principle can be applied to all of the rates of change in the natural and social sciences that we discussed in Section 3.7. Here are a few instances of this idea: If V共t兲 is the volume of water in a reservoir at time t, then its derivative V⬘共t兲 is the rate at which water flows into the reservoir at time t. So
■
y
t2
V⬘共t兲 dt 苷 V共t2 兲 ⫺ V共t1 兲
t1
is the change in the amount of water in the reservoir between time t1 and time t2 . If 关C兴共t兲 is the concentration of the product of a chemical reaction at time t, then the rate of reaction is the derivative d关C兴兾dt. So
■
y
d关C兴 dt 苷 关C兴共t2 兲 ⫺ 关C兴共t1 兲 dt
t2
t1
is the change in the concentration of C from time t1 to time t2 . If the mass of a rod measured from the left end to a point x is m共x兲, then the linear density is 共x兲 苷 m⬘共x兲. So
■
y
b
a
共x兲 dx 苷 m共b兲 ⫺ m共a兲
is the mass of the segment of the rod that lies between x 苷 a and x 苷 b. If the rate of growth of a population is dn兾dt, then
■
y
t2
t1
dn dt 苷 n共t 2 兲 ⫺ n共t1 兲 dt
is the net change in population during the time period from t1 to t2 . (The population increases when births happen and decreases when deaths occur. The net change takes into account both births and deaths.) If C共x兲 is the cost of producing x units of a commodity, then the marginal cost is the derivative C⬘共x兲. So
■
y
x2
x1
C⬘共x兲 dx 苷 C共x 2 兲 ⫺ C共x 1 兲
is the increase in cost when production is increased from x1 units to x2 units. If an object moves along a straight line with position function s共t兲, then its velocity is v共t兲 苷 s⬘共t兲, so
■
2
y
t2
t1
v共t兲 dt 苷 s共t2 兲 ⫺ s共t1 兲
is the net change of position, or displacement, of the particle during the time period from t1 to t2 . In Section 5.1 we guessed that this was true for the case where the object moves in the positive direction, but now we have proved that it is always true.
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INTEGRALS
If we want to calculate the distance the object travels during the time interval, we have to consider the intervals when v共t兲 艌 0 (the particle moves to the right) and also the intervals when v共t兲 艋 0 (the particle moves to the left). In both cases the distance is computed by integrating v共t兲 , the speed. Therefore
■
ⱍ
ⱍ
y ⱍ v共t兲 ⱍ dt 苷 total distance traveled t2
3
t1
Figure 3 shows how both displacement and distance traveled can be interpreted in terms of areas under a velocity curve. √
√(t)
t™
displacement=j √(t) dt=A¡A™+A£ t¡
A¡
t™
A£ 0
t¡
t™
A™
distance=j  √(t) dt=A¡+A™+A£ t¡
t
FIGURE 3 ■
The acceleration of the object is a共t兲 苷 v⬘共t兲, so
y
t2
t1
a共t兲 dt 苷 v共t2 兲 ⫺ v共t1 兲
is the change in velocity from time t1 to time t2 .
v EXAMPLE 6 A particle moves along a line so that its velocity at time t is v共t兲 苷 t 2 ⫺ t ⫺ 6 (measured in meters per second).
(a) Find the displacement of the particle during the time period 1 艋 t 艋 4. (b) Find the distance traveled during this time period. SOLUTION
(a) By Equation 2, the displacement is 4
s共4兲 ⫺ s共1兲 苷
y
苷
冋
1
v共t兲 dt 苷
y
4
1
共t 2 ⫺ t ⫺ 6兲 dt
册
t3 t2 ⫺ ⫺ 6t 3 2
4
苷⫺
1
9 2
This means that the particle moved 4.5 m toward the left. (b) Note that v共t兲 苷 t 2 ⫺ t ⫺ 6 苷 共t ⫺ 3兲共t ⫹ 2兲 and so v共t兲 艋 0 on the interval 关1, 3兴 and v共t兲 艌 0 on 关3, 4兴. Thus, from Equation 3, the distance traveled is To integrate the absolute value of v共t兲, we use Property 5 of integrals from Section 5.2 to split the integral into two parts, one where v共t兲 艋 0 and one where v共t兲 艌 0.
y ⱍ v共t兲 ⱍ dt 苷 y 4
1
3
1
4
关⫺v共t兲兴 dt ⫹ y v共t兲 dt 3
3
4
苷 y 共⫺t 2 ⫹ t ⫹ 6兲 dt ⫹ y 共t 2 ⫺ t ⫺ 6兲 dt 1
冋
苷 ⫺ 苷
3
册 冋
t3 t2 ⫹ ⫹ 6t 3 2
3
⫹
1
册
t3 t2 ⫺ ⫺ 6t 3 2
4
3
61 ⬇ 10.17 m 6
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.4
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
403
EXAMPLE 7 Figure 4 shows the power consumption in the city of San Francisco for a day in September (P is measured in megawatts; t is measured in hours starting at midnight). Estimate the energy used on that day. P 800 600 400 200
0
FIGURE 4
3
6
9
12
15
18
t
21
Pacific Gas & Electric
SOLUTION Power is the rate of change of energy: P共t兲 苷 E⬘共t兲. So, by the Net Change
Theorem,
y
24
0
24
P共t兲 dt 苷 y E⬘共t兲 dt 苷 E共24兲 ⫺ E共0兲 0
is the total amount of energy used on that day. We approximate the value of the integral using the Midpoint Rule with 12 subintervals and ⌬t 苷 2:
y
24
0
P共t兲 dt ⬇ 关P共1兲 ⫹ P共3兲 ⫹ P共5兲 ⫹ ⭈ ⭈ ⭈ ⫹ P共21兲 ⫹ P共23兲兴 ⌬t ⬇ 共440 ⫹ 400 ⫹ 420 ⫹ 620 ⫹ 790 ⫹ 840 ⫹ 850 ⫹ 840 ⫹ 810 ⫹ 690 ⫹ 670 ⫹ 550兲共2兲 苷 15,840
The energy used was approximately 15,840 megawatthours. How did we know what units to use for energy in Example 7? The integral x024 P共t兲 dt is defined as the limit of sums of terms of the form P共ti*兲 ⌬t. Now P共ti*兲 is measured in megawatts and ⌬t is measured in hours, so their product is measured in megawatthours. The same is true of the limit. In general, the unit of measurement for xab f 共x兲 dx is the product of the unit for f 共x兲 and the unit for x.
A note on units
Exercises
5.4
1– 4 Verify by differentiation that the formula is correct.
1 s1 ⫹ x 2 dx 苷 ⫺ ⫹C 2 x s1 ⫹ x x
1.
y
2.
y cos x dx 苷
3.
y cos x dx 苷 sin x ⫺
y
x 2 dx 苷 2 共bx ⫺ 2a兲 sa ⫹ bx ⫹ C 3b sa ⫹ bx
2
2
;
4.
3
1 2
x ⫹ 14 sin 2x ⫹ C 1 3
sin3 x ⫹ C
Graphing calculator or computer required
5–18 Find the general indefinite integral. 5.
y 共x
2
⫹ x ⫺2 兲 dx
6.
y (sx
3
3 ⫹s x 2 ) dx
1. Homework Hints available at stewartcalculus.com
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404
CHAPTER 5
7. 9.
y (x
4
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INTEGRALS
⫺ 12 x 3 ⫹ 14 x ⫺ 2) dx
y 共u ⫹ 4兲共2u ⫹ 1兲 du
8. 10.
y 共y
⫹ 1.8y 2 ⫺ 2.4y兲 dy
3
y v共v
2
⫹ 2兲2 dv
冉
冊
11.
y
x 3 ⫺ 2 sx dx x
12.
y
13.
y 共sin x ⫹ sinh x兲 dx
14.
y 共csc t ⫺ 2e 兲 dt
15.
y 共 ⫺ csc cot 兲 d
16.
y sec t 共sec t ⫹ tan t兲 dt
y 共1 ⫹ tan ␣兲 d␣
18.
1 x2 ⫹ 1 ⫹ 2 x ⫹1
1兾s3
t2 ⫺ 1 dt t4 ⫺ 1
43.
y
45.
y ( x ⫺ 2 ⱍ x ⱍ) dx
0 2
⫺1
44.
y ⱍ 2x ⫺ 1 ⱍ dx
46.
y
2
0
3兾2
0
ⱍ sin x ⱍ dx
dx
; 47. Use a graph to estimate the xintercepts of the curve
17.
2
2
t
sin 2x dx sin x
y
several members of the family on the same screen.
y (cos x ⫹ x) dx 1 2
20.
y 共e
x
2 ⫺1 4 ; 48. Repeat Exercise 47 for the curve y 苷 共x ⫹ 1兲 ⫺ x .
49. The area of the region that lies to the right of the yaxis and
; 19–20 Find the general indefinite integral. Illustrate by graphing
19.
y 苷 1 ⫺ 2x ⫺ 5x 4. Then use this information to estimate the area of the region that lies under the curve and above the xaxis.
to the left of the parabola x 苷 2y ⫺ y 2 (the shaded region in the figure) is given by the integral x02 共2y ⫺ y 2 兲 dy. (Turn your head clockwise and think of the region as lying below the curve x 苷 2y ⫺ y 2 from y 苷 0 to y 苷 2.) Find the area of the region.
⫺ 2x 2 兲 dx
y 2
x=2y¥ 21– 46 Evaluate the integral. 21.
y
3
⫺2
共x 2 ⫺ 3兲 dx
22.
0
23.
y⫺2 ( 12 t 4 ⫹ 14 t 3 ⫺ t) dt
25.
y
27.
2
0
y
0
y
共2x ⫺ 3兲共4x 2 ⫹ 1兲 dx
26.
y
共5e x ⫹ 3 sin x兲 dx
28.
4 ⫹ 6u su
冊
y
31.
y x (sx ⫹ sx ) dx
33.
y
35.
37.
1
0
y
1
0
y
2 x ⫺ 2 x
y
39.
y
兾3
64
0
共1 ⫹ 6w 2 ⫺ 10w 4 兲 dw
1
⫺1
y
2
1
t共1 ⫺ t兲 2 dt
冉
4 1 ⫺ 3 x2 x
冊
4
30.
y0 (3st
32.
y
dx
34.
y
36.
1
0
共5x ⫺ 5 x 兲 dx
兾3
y
⫺ 2e t) dt
兾4
y=1
1
y=$œ„ x
0
1
x
csc 2 d 51. If w⬘共t兲 is the rate of growth of a child in pounds per year, what does x510 w⬘共t兲 dt represent? 52. The current in a wire is defined as the derivative of the
charge: I共t兲 苷 Q⬘共t兲. (See Example 3 in Section 3.7.) What does xab I共t兲 dt represent?
sin ⫹ sin tan d sec2
dr s1 ⫺ r 2
4 y 苷 1, and the curve y 苷 s x . Find the area of this region by writing x as a function of y and integrating with respect to y (as in Exercise 49).
sy ⫺ y dy y2
4
1
x
1
50. The boundaries of the shaded region are the yaxis, the line
2
3 1⫹s x dx sx
s3兾2
0
0
3
1 ⫹ cos2 d cos2
0
y
dx
共x 10 ⫹ 10 x 兲 dx
兾4
1
du
4
冉 冊
0
38.
41.
2
1
3
共4x 3 ⫺ 3x 2 ⫹ 2x兲 dx
y
29.
1
2
1
24.
冉
4
y
40.
42.
y
10
⫺10
y
2
1
2e x dx sinh x ⫹ cosh x
共x ⫺ 1兲3 dx x2
53. If oil leaks from a tank at a rate of r共t兲 gallons per minute at
time t, what does x0120 r共t兲 dt represent? 54. A honeybee population starts with 100 bees and increases
at a rate of n⬘共t兲 bees per week. What does 100 ⫹ x015 n⬘共t兲 dt represent?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.4
55. In Section 4.7 we defined the marginal revenue function R⬘共x兲
as the derivative of the revenue function R共x兲, where x is the 5000 number of units sold. What does x1000 R⬘共x兲 dx represent? 56. If f 共x兲 is the slope of a trail at a distance of x miles from the
start of the trail, what does x f 共x兲 dx represent? 5 3
57. If x is measured in meters and f 共x兲 is measured in newtons,
what are the units for x0100 f 共x兲 dx ?
58. If the units for x are feet and the units for a共x兲 are pounds
per foot, what are the units for da兾dx ? What units does x28 a共x兲 dx have? 59–60 The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. 59. v共t兲 苷 3t ⫺ 5,
1艋t艋6
61–62 The acceleration function (in m兾s2 ) and the initial velocity
are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. 61. a共t兲 苷 t ⫹ 4,
v 共0兲 苷 5,
62. a共t兲 苷 2t ⫹ 3,
405
given in the table. The time t is measured in seconds and the units for r共t兲 are tonnes (metric tons) per second. t
0
1
2
3
4
5
6
r共t兲
2
10
24
36
46
54
60
(a) Give upper and lower estimates for the total quantity Q共6兲 of erupted materials after 6 seconds. (b) Use the Midpoint Rule to estimate Q共6兲. 67. The marginal cost of manufacturing x yards of a certain
fabric is C⬘共x兲 苷 3 ⫺ 0.01x ⫹ 0.000006x 2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards. 68. Water flows into and out of a storage tank. A graph of the rate
of change r共t兲 of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t 苷 0 is 25,000 L, use the Midpoint Rule to estimate the amount of water in the tank four days later.
0艋t艋3
60. v共t兲 苷 t 2 ⫺ 2t ⫺ 8,
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
r 2000 1000
0 艋 t 艋 10
v 共0兲 苷 ⫺4,
0
0艋t艋3
1
2
3
4 t
_1000
63. The linear density of a rod of length 4 m is given by
共x兲 苷 9 ⫹ 2 sx measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.
69. A bacteria population is 4000 at time t 苷 0 and its rate of
growth is 1000 ⴢ 2 t bacteria per hour after t hours. What is the population after one hour?
64. Water flows from the bottom of a storage tank at a rate of
r共t兲 苷 200 ⫺ 4t liters per minute, where 0 艋 t 艋 50. Find the amount of water that flows from the tank during the first 10 minutes. 65. The velocity of a car was read from its speedometer at
10second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t (s)
v (mi兾h)
t (s)
v (mi兾h)
0 10 20 30 40 50
0 38 52 58 55 51
60 70 80 90 100
56 53 50 47 45
66. Suppose that a volcano is erupting and readings of the rate r共t兲
at which solid materials are spewed into the atmosphere are
70. Shown is the graph of traffic on an Internet service provider’s
T1 data line from midnight to 8:00 AM. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period. D 0.8
0.4
0
2
4
6
8 t (hours)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
71. Shown is the power consumption in the province of Ontario,
Canada, for December 9, 2004 (P is measured in megawatts; t is measured in hours starting at midnight). Using the fact that power is the rate of change of energy, estimate the energy used on that day.
P
; 72. On May 7, 1992, the space shuttle Endeavour was launched on mission STS49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to model these data by a thirddegree polynomial. (b) Use the model in part (a) to estimate the height reached by the Endeavour, 125 seconds after liftoff.
22,000
Event 20,000 18,000 16,000 0
3
6
9
12
15
18
21
t
Launch Begin roll maneuver End roll maneuver Throttle to 89% Throttle to 67% Throttle to 104% Maximum dynamic pressure Solid rocket booster separation
Time (s)
Velocity (ft兾s)
0 10 15 20 32 59 62 125
0 185 319 447 742 1325 1445 4151
Independent Electricity Market Operator
WRITING PROJECT
NEWTON, LEIBNIZ, AND THE INVENTION OF CALCULUS We sometimes read that the inventors of calculus were Sir Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). But we know that the basic ideas behind integration were investigated 2500 years ago by ancient Greeks such as Eudoxus and Archimedes, and methods for finding tangents were pioneered by Pierre Fermat (1601–1665), Isaac Barrow (1630–1677), and others. Barrow––who taught at Cambridge and was a major influence on Newton––was the first to understand the inverse relationship between differentiation and integration. What Newton and Leibniz did was to use this relationship, in the form of the Fundamental Theorem of Calculus, in order to develop calculus into a systematic mathematical discipline. It is in this sense that Newton and Leibniz are credited with the invention of calculus. Read about the contributions of these men in one or more of the given references and write a report on one of the following three topics. You can include biographical details, but the main thrust of your report should be a description, in some detail, of their methods and notations. In particular, you should consult one of the sourcebooks, which give excerpts from the original publications of Newton and Leibniz, translated from Latin to English. N
The Role of Newton in the Development of Calculus
N
The Role of Leibniz in the Development of Calculus
N
The Controversy between the Followers of Newton and Leibniz over Priority in the Invention of Calculus
References 1. Carl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1987),
Chapter 19. 2. Carl Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover,
1959), Chapter V. 3. C. H. Edwards, The Historical Development of the Calculus (New York: SpringerVerlag,
1979), Chapters 8 and 9.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.5
THE SUBSTITUTION RULE
407
4. Howard Eves, An Introduction to the History of Mathematics, 6th ed. (New York: Saunders,
1990), Chapter 11. 5. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974).
See the article on Leibniz by Joseph Hofmann in Volume VIII and the article on Newton by I. B. Cohen in Volume X. 6. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993),
Chapter 12. 7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford
University Press, 1972), Chapter 17. Sourcebooks 1. John Fauvel and Jeremy Gray, eds., The History of Mathematics: A Reader (London:
MacMillan Press, 1987), Chapters 12 and 13. 2. D. E. Smith, ed., A Sourcebook in Mathematics (New York: Dover, 1959), Chapter V. 3. D. J. Struik, ed., A Sourcebook in Mathematics, 1200–1800 (Princeton, NJ: Princeton
University Press, 1969), Chapter V.
5.5
The Substitution Rule Because of the Fundamental Theorem, it’s important to be able to find antiderivatives. But our antidifferentiation formulas don’t tell us how to evaluate integrals such as
y 2xs1 ⫹ x
1 PS
Differentials were defined in Section 3.10. If u 苷 f 共x兲, then du 苷 f ⬘共x兲 dx
2
dx
To find this integral we use the problemsolving strategy of introducing something extra. Here the “something extra” is a new variable; we change from the variable x to a new variable u. Suppose that we let u be the quantity under the root sign in 1 , u 苷 1 ⫹ x 2. Then the differential of u is du 苷 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in 1 and so, formally, without justifying our calculation, we could write
y 2xs1 ⫹ x
2
2
dx 苷 y s1 ⫹ x 2 2x dx 苷 y su du 苷 23 u 3兾2 ⫹ C 苷 23 共x 2 ⫹ 1兲3兾2 ⫹ C
But now we can check that we have the correct answer by using the Chain Rule to differentiate the final function of Equation 2: d dx
[
2 3
]
共x 2 ⫹ 1兲3兾2 ⫹ C 苷 23 ⴢ 32 共x 2 ⫹ 1兲1兾2 ⴢ 2x 苷 2xsx 2 ⫹ 1
In general, this method works whenever we have an integral that we can write in the form
x f 共t共x兲兲 t⬘共x兲 dx. Observe that if F⬘苷 f , then 3
y F⬘共t共x兲兲 t⬘共x兲 dx 苷 F共t共x兲兲 ⫹ C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
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INTEGRALS
because, by the Chain Rule, d 关F共t共x兲兲兴 苷 F⬘共t共x兲兲 t⬘共x兲 dx If we make the “change of variable” or “substitution” u 苷 t共x兲, then from Equation 3 we have
y F⬘共t共x兲兲 t⬘共x兲 dx 苷 F共t共x兲兲 ⫹ C 苷 F共u兲 ⫹ C 苷 y F⬘共u兲 du or, writing F⬘ 苷 f , we get
y f 共t共x兲兲 t⬘共x兲 dx 苷 y f 共u兲 du Thus we have proved the following rule. 4 The Substitution Rule If u 苷 t共x兲 is a differentiable function whose range is an interval I and f is continuous on I , then
y f 共t共x兲兲 t⬘共x兲 dx 苷 y f 共u兲 du Notice that the Substitution Rule for integration was proved using the Chain Rule for differentiation. Notice also that if u 苷 t共x兲, then du 苷 t⬘共x兲 dx, so a way to remember the Substitution Rule is to think of dx and du in 4 as differentials. Thus the Substitution Rule says: It is permissible to operate with dx and du after integral signs as if they were differentials. EXAMPLE 1 Find
yx
3
cos共x 4 ⫹ 2兲 dx.
SOLUTION We make the substitution u 苷 x 4 ⫹ 2 because its differential is du 苷 4x 3 dx,
which, apart from the constant factor 4, occurs in the integral. Thus, using x 3 dx 苷 14 du and the Substitution Rule, we have
yx
3
cos共x 4 ⫹ 2兲 dx 苷 y cos u ⴢ 14 du 苷 14 y cos u du 苷 14 sin u ⫹ C 苷 14 sin共x 4 ⫹ 2兲 ⫹ C
Check the answer by differentiating it.
Notice that at the final stage we had to return to the original variable x. The idea behind the Substitution Rule is to replace a relatively complicated integral by a simpler integral. This is accomplished by changing from the original variable x to a new variable u that is a function of x. Thus in Example 1 we replaced the integral x x 3 cos共x 4 ⫹ 2兲 dx by the simpler integral 14 x cos u du. The main challenge in using the Substitution Rule is to think of an appropriate substitution. You should try to choose u to be some function in the integrand whose differential also occurs (except for a constant factor). This was the case in Example 1. If that is not possible, try choosing u to be some complicated part of the integrand (perhaps the inner function in a composite function). Finding the right substitution is a bit of an art. It’s not unusual to guess wrong; if your first guess doesn’t work, try another substitution.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.5
EXAMPLE 2 Evaluate
THE SUBSTITUTION RULE
409
y s2x ⫹ 1 dx.
SOLUTION 1 Let u 苷 2x ⫹ 1. Then du 苷 2 dx, so dx 苷
1 2
du. Thus the Substitution Rule
gives
y s2x ⫹ 1 dx 苷 y su 苷
ⴢ 12 du 苷 12 y u 12 du
1 u 32 ⴢ ⫹ C 苷 13 u 32 ⫹ C 2 32
苷 13 2x ⫹ 132 ⫹ C SOLUTION 2 Another possible substitution is u 苷 s2x ⫹ 1 . Then
dx s2x ⫹ 1
du 苷
dx 苷 s2x ⫹ 1 du 苷 u du
so
(Or observe that u 2 苷 2x ⫹ 1, so 2u du 苷 2 dx.) Therefore
y s2x ⫹ 1 dx 苷 y u ⴢ u du 苷 y u 苷
v
EXAMPLE 3 Find
du
u3 ⫹ C 苷 13 2x ⫹ 132 ⫹ C 3
x dx. s1 ⫺ 4x 2
y
SOLUTION Let u 苷 1 ⫺ 4x 2. Then du 苷 ⫺8x dx, so x dx 苷 ⫺ 8 du and 1
1
x
f _1
1
y s1 ⫺ 4x
2
dx 苷 ⫺ 18 y
_1
FIGURE 1
x 14≈ œ„„„„„„
©=j ƒ dx=_ 41 œ„„„„„„ 14≈
1 du 苷 ⫺ 18 y u ⫺12 du su
苷 ⫺ 18 (2su ) ⫹ C 苷 ⫺ 14 s1 ⫺ 4x 2 ⫹ C
©= ƒ dx
ƒ=
2
The answer to Example 3 could be checked by differentiation, but instead let’s check it with a graph. In Figure 1 we have used a computer to graph both the integrand f x 苷 xs1 ⫺ 4x 2 and its indefinite integral tx 苷 ⫺ 14 s1 ⫺ 4x 2 (we take the case C 苷 0). Notice that tx decreases when f x is negative, increases when f x is positive, and has its minimum value when f x 苷 0. So it seems reasonable, from the graphical evidence, that t is an antiderivative of f . EXAMPLE 4 Calculate
ye
5x
dx.
SOLUTION If we let u 苷 5x, then du 苷 5 dx, so dx 苷
ye
5x
1 5
du. Therefore
dx 苷 15 y e u du 苷 15 e u ⫹ C 苷 15 e 5x ⫹ C
NOTE With some experience, you might be able to evaluate integrals like those in Examples 1–4 without going to the trouble of making an explicit substitution. By recognizing the pattern in Equation 3, where the integrand on the left side is the product of the derivative of an outer function and the derivative of the inner function, we could work Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
Example 1 as follows:
yx
3
cos共x 4 ⫹ 2兲 dx 苷 y cos共x 4 ⫹ 2兲 ⴢ x 3 dx 苷 14 y cos共x 4 ⫹ 2兲 ⴢ 共4x 3 兲 dx 苷 14 y cos共x 4 ⫹ 2兲 ⴢ
d 共x 4 ⫹ 2兲 dx 苷 14 sin共x 4 ⫹ 2兲 ⫹ C dx
Similarly, the solution to Example 4 could be written like this:
ye
5x
dx 苷 15 y 5e 5x dx 苷 15 y
d 5x 共e 兲 dx 苷 15 e 5x ⫹ C dx
The following example, however, is more complicated and so an explicit substitution is advisable. EXAMPLE 5 Find
y s1 ⫹ x
2
x 5 dx.
SOLUTION An appropriate substitution becomes more obvious if we factor x 5 as x 4 ⴢ x.
Let u 苷 1 ⫹ x 2. Then du 苷 2x dx, so x dx 苷 12 du. Also x 2 苷 u ⫺ 1, so x 4 苷 共u ⫺ 1兲2:
y s1 ⫹ x
2
x 5 dx 苷 y s1 ⫹ x 2 x 4 ⭈ x dx 苷 y su 共u ⫺ 1兲2 ⴢ 12 du 苷 12 y su 共u 2 ⫺ 2u ⫹ 1兲 du 苷 12 y 共u 5兾2 ⫺ 2u 3兾2 ⫹ u 1兾2 兲 du 苷 12 ( 27 u 7兾2 ⫺ 2 ⭈ 25 u 5兾2 ⫹ 23 u 3兾2 ) ⫹ C 苷 17 共1 ⫹ x 2 兲7兾2 ⫺ 25 共1 ⫹ x 2 兲5兾2 ⫹ 13 共1 ⫹ x 2 兲3兾2 ⫹ C
v
EXAMPLE 6 Calculate
y tan x dx.
SOLUTION First we write tangent in terms of sine and cosine:
y tan x dx 苷 y
sin x dx cos x
This suggests that we should substitute u 苷 cos x, since then du 苷 ⫺sin x dx and so sin x dx 苷 ⫺du: sin x 1 y tan x dx 苷 y cos x dx 苷 ⫺y u du
ⱍ ⱍ
ⱍ
ⱍ
苷 ⫺ln u ⫹ C 苷 ⫺ln cos x ⫹ C
ⱍ
ⱍ
ⱍ
Since ⫺ln cos x 苷 ln共 cos x 6 can also be written as 5
ⱍ
⫺1
ⱍ
ⱍ
ⱍ
ⱍ
兲 苷 ln共1兾 cos x 兲 苷 ln sec x , the result of Example
y tan x dx 苷 ln ⱍ sec x ⱍ ⫹ C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 5.5
THE SUBSTITUTION RULE
411
Definite Integrals When evaluating a definite integral by substitution, two methods are possible. One method is to evaluate the indefinite integral first and then use the Fundamental Theorem. For instance, using the result of Example 2, we have
y
4
4
]
s2x ⫹ 1 dx 苷 y s2x ⫹ 1 dx
0
]
0
4
苷 13 共2x ⫹ 1兲3兾2 0 苷 13 共9兲3兾2 ⫺ 13 共1兲3兾2 苷 13 共27 ⫺ 1兲 苷 263 Another method, which is usually preferable, is to change the limits of integration when the variable is changed.
This rule says that when using a substitution in a definite integral, we must put everything in terms of the new variable u, not only x and dx but also the limits of integration. The new limits of integration are the values of u that correspond to x 苷 a and x 苷 b.
6
The Substitution Rule for Definite Integrals If t⬘ is continuous on 关a, b兴 and f is
continuous on the range of u 苷 t共x兲, then
y
b
a
f 共t共x兲兲 t⬘共x兲 dx 苷 y
t共b兲
t共a兲
f 共u兲 du
PROOF Let F be an antiderivative of f . Then, by 3 , F共t共x兲兲 is an antiderivative of
f 共t共x兲兲 t⬘共x兲, so by Part 2 of the Fundamental Theorem, we have
y
b
a
]
b
f 共t共x兲兲 t⬘共x兲 dx 苷 F共t共x兲兲 a 苷 F共t共b兲兲 ⫺ F共t共a兲兲
But, applying FTC2 a second time, we also have
y
t共b兲
t共a兲
EXAMPLE 7 Evaluate
y
4
0
]
f 共u兲 du 苷 F共u兲
t共b兲 t共a兲
苷 F共t共b兲兲 ⫺ F共t共a兲兲
s2x ⫹ 1 dx using 6 .
SOLUTION Using the substitution from Solution 1 of Example 2, we have u 苷 2x ⫹ 1
and dx 苷 12 du. To find the new limits of integration we note that when x 苷 0, u 苷 2共0兲 ⫹ 1 苷 1 Therefore
y
4
0
when x 苷 4, u 苷 2共4兲 ⫹ 1 苷 9
and 9 1 2 1
s2x ⫹ 1 dx 苷 y
su du
]
苷 12 ⴢ 23 u 3兾2 苷 共9 1 3
3兾2
9 1
⫺ 13兾2 兲 苷 263
Observe that when using 6 we do not return to the variable x after integrating. We simply evaluate the expression in u between the appropriate values of u.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
2
1
4:44 PM
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INTEGRALS
EXAMPLE 8 Evaluate
The integral given in Example 8 is an abbreviation for
y
9/21/10
y
2
1
dx . 共3 ⫺ 5x兲2
SOLUTION Let u 苷 3 ⫺ 5x. Then du 苷 ⫺5 dx, so dx 苷 ⫺5 du . When x 苷 1, u 苷 ⫺2 1
1 dx 共3 ⫺ 5x兲2
and when x 苷 2, u 苷 ⫺7. Thus
y
dx 1 苷⫺ 共3 ⫺ 5x兲2 5
2
1
苷⫺ 1 5
苷
Since the function f 共x兲 苷 共ln x兲兾x in Example 9 is positive for x ⬎ 1, the integral represents the area of the shaded region in Figure 2. y 0.5
y=
v
EXAMPLE 9 Calculate
y
y
⫺7
⫺2
1 5
冋 册 ⫺
冉
du u2
⫺
1 u
⫺7
1 5u
苷 ⫺2
1 1 ⫹ 7 2
冊
苷
册
⫺7
⫺2
1 14
ln x dx. x
e
1
SOLUTION We let u 苷 ln x because its differential du 苷 dx兾x occurs in the integral.
When x 苷 1, u 苷 ln 1 苷 0; when x 苷 e, u 苷 ln e 苷 1. Thus
ln x x
y
e
1
ln x 1 u2 dx 苷 y u du 苷 0 x 2
册
1
苷
0
1 2
Symmetry 0
FIGURE 2
1
e
x
The next theorem uses the Substitution Rule for Definite Integrals 6 to simplify the calculation of integrals of functions that possess symmetry properties. 7
Integrals of Symmetric Functions Suppose f is continuous on 关⫺a, a兴.
a (a) If f is even 关 f 共⫺x兲 苷 f 共x兲兴, then x⫺a f 共x兲 dx 苷 2 x0a f 共x兲 dx. a (b) If f is odd 关 f 共⫺x兲 苷 ⫺f 共x兲兴, then x⫺a f 共x兲 dx 苷 0.
PROOF We split the integral in two:
8
y
a
⫺a
0
a
f 共x兲 dx 苷 y f 共x兲 dx ⫹ y f 共x兲 dx 苷 ⫺y ⫺a
0
⫺a
0
a
f 共x兲 dx ⫹ y f 共x兲 dx 0
In the first integral on the far right side we make the substitution u 苷 ⫺x. Then du 苷 ⫺dx and when x 苷 ⫺a, u 苷 a. Therefore ⫺y
⫺a
0
a
a
f 共x兲 dx 苷 ⫺y f 共⫺u兲 共⫺du兲 苷 y f 共⫺u兲 du 0
0
and so Equation 8 becomes 9
y
a
⫺a
a
a
f 共x兲 dx 苷 y f 共⫺u兲 du ⫹ y f 共x兲 dx 0
0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Page 413
SECTION 5.5
y
a
a
⫺a
0
a
a
0
y
a
0
a
f 共x兲 dx 苷 2 y f 共x兲 dx 0
(b) If f is odd, then f 共⫺u兲 苷 ⫺f 共u兲 and so Equation 9 gives
(a) ƒ even, j ƒ dx=2 j ƒ dx _a
f 共x兲 dx 苷 y f 共u兲 du ⫹
x
a
0
y
y
_a
a
⫺a
0 a
x
a
(b) ƒ odd, j ƒ dx=0 _a
FIGURE 3
a
f 共x兲 dx 苷 ⫺y f 共u兲 du ⫹ 0
y
a
0
f 共x兲 dx 苷 0
Theorem 7 is illustrated by Figure 3. For the case where f is positive and even, part (a) says that the area under y 苷 f 共x兲 from ⫺a to a is twice the area from 0 to a because of symmetry. Recall that an integral xab f 共x兲 dx can be expressed as the area above the xaxis and below y 苷 f 共x兲 minus the area below the axis and above the curve. Thus part (b) says the integral is 0 because the areas cancel.
v
EXAMPLE 10 Since f 共x兲 苷 x 6 ⫹ 1 satisfies f 共⫺x兲 苷 f 共x兲, it is even and so
y
2
⫺2
2
共x 6 ⫹ 1兲 dx 苷 2 y 共x 6 ⫹ 1兲 dx 0
2
[
]
128 284 苷 2 17 x 7 ⫹ x 0 苷 2( 7 ⫹ 2) 苷 7
EXAMPLE 11 Since f 共x兲 苷 共tan x兲兾共1 ⫹ x 2 ⫹ x 4 兲 satisfies f 共⫺x兲 苷 ⫺f 共x兲, it is odd
and so
y
1
⫺1
1–6 Evaluate the integral by making the given substitution. ⫺x
ye
2.
y x 共2 ⫹ x
dx,
兲 dx,
u苷2⫹x
x sx ⫹ 1 dx,
u苷x ⫹1
3
2
4 5
3.
y
4.
y 共1 ⫺ 6t兲 ,
5.
y cos sin d,
6.
y
3
dt
4
sec 2共1兾x兲 dx, x2
3
u 苷 cos u 苷 1兾x
7– 48 Evaluate the indefinite integral.
;
2
兲 dx
y 共1 ⫺ 2x兲
9
dx
10.
y 共3t ⫹ 2兲
12.
y sec
2.4
dt
2 d
11.
y 共x ⫹ 1兲s2x ⫹ x
13.
y 5 ⫺ 3x
14.
y u s1 ⫺ u
15.
y sin t dt
16.
ye
17.
y
18.
y
19.
y
20.
yz
21.
y
22.
y cos sin d
23.
y sec tan d
24.
y sx
2
dx
2
4
u 苷 1 ⫺ 6t
3
y x sin共 x
9.
u 苷 ⫺x
1.
7.
tan x dx 苷 0 1 ⫹ x2 ⫹ x4
Exercises
5.5
413
(a) If f is even, then f 共⫺u兲 苷 f 共u兲 so Equation 9 gives
y
_a
THE SUBSTITUTION RULE
8.
yx
Graphing calculator or computer required
2
3
e x dx
dx
eu du 共1 ⫺ e u 兲2 a ⫹ bx 2 s3ax ⫹ bx
3
dx
共ln x兲2 dx x 2
3
x
2
du
cos共e x 兲 dx
sin sx dx sx
3
z2 dz ⫹1 4
sin共1 ⫹ x 3兾2 兲 dx
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
x
25.
ye
27.
y 共x
29.
y5
31.
33.
ye
t
⫹ 1兲共x 3 ⫹ 3x兲 4 dx
sin共5 t 兲 dt
tan x
2
sec x dx
cos x dx sin 2x
y
4:46 PM
dx
26.
y ax ⫹ b
28.
ye
30.
y 1⫹x
32.
34.
cos t
共a 苷 0兲
sin t dt
tan⫺1 x
dx
2
sin共ln x兲 dx x
y
cos共兾x兲 dx x2
y
2
y scot x csc x dx 2
37.
y sinh x cosh x dx
39.
sin 2x dx 1 ⫹ cos2x
41.
y
y cot x dx
36.
38.
40.
42.
2 dt 2t ⫹ 3
y
dt
y
cos2 t s1 ⫹ tan t sin x dx 1 ⫹ cos2x
y
y sin t sec 共cos t兲 dt 2
兾4
共x 3 ⫹ x 4 tan x兲 dx
61.
y
63.
y
65.
y
67.
y
69.
y
71.
y
73.
y (1 ⫹ sx )
t
35.
Page 414
INTEGRALS
s1 ⫹ e x dx 2
9/21/10
y
64.
y
x sx 2 ⫹ a 2 dx 共a ⬎ 0兲
66.
y
x sx ⫺ 1 dx
68.
y
dx x sln x
70.
y
ez ⫹ 1 dz ez ⫹ z
72.
y
13
0 a
0 2
1 e4
e 1
0
1
兾2
62.
⫺ 兾4
dx 3 共1 ⫹ 2x兲2 s
0 a
0
cos x sin共sin x兲 dx
x sa 2 ⫺ x 2 dx
兾3
⫺ 兾3 4
0
x 4 sin x dx
x dx s1 ⫹ 2x
1兾2
0 T兾2
0
sin⫺1 x dx s1 ⫺ x 2 sin共2 t兾T ⫺ ␣兲 dt
dx
0
4
3 74. Verify that f 共x兲 苷 sin s x is an odd function and use that fact
to show that 3
3 0 艋 y sin s x dx 艋 1
⫺2
; 75–76 Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area.
43.
45.
47.
y
dx s1 ⫺ x 2 sin⫺1x
44.
y
1⫹x dx 1 ⫹ x2
46.
y x共2x ⫹ 5兲
8
x dx 1 ⫹ x4
y
2
yx
75. y 苷 s2x ⫹ 1 , 0 艋 x 艋 1 76. y 苷 2 sin x ⫺ sin 2x, 0 艋 x 艋
s2 ⫹ x dx 2 77. Evaluate x⫺2 共x ⫹ 3兲s4 ⫺ x 2 dx by writing it as a sum of
dx
48.
3
y x sx
2
⫹ 1 dx
two integrals and interpreting one of those integrals in terms of an area.
78. Evaluate x01 x s1 ⫺ x 4 dx by making a substitution and inter
; 49–52 Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C 苷 0).
preting the resulting integral in terms of an area. 79. Which of the following areas are equal? Why? y
49.
y x共x 2 ⫺ 1兲3 dx
50.
y
y tan2 sec2 d
y=2x´ y=e
51.
ye
cos x
sin x dx
52.
œ„ x
4
y sin x cos x dx 0
0
1 x
1 x
y
53–73 Evaluate the definite integral. 53.
y
55.
y
57.
y
59.
1
0
54.
y
s1 ⫹ 7x dx
56.
y
sec 2共t兾4兲 dt
58.
y
1 3
0
0
y
cos共 t兾2兲 dt
2
1
e 1兾x dx x2
60.
1
0 3
0
dx 5x ⫹ 1
1兾2
1兾6
y
1
0
y=e sin x sin 2x
共3t ⫺ 1兲50 dt
csc t cot t dt
xe
⫺x 2
dx
0
1
π x 2
80. A model for the basal metabolism rate, in kcal兾h, of a young
man is R共t兲 苷 85 ⫺ 0.18 cos共 t兾12兲, where t is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, x024 R共t兲 dt, over a 24hour time period?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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81. An oil storage tank ruptures at time t 苷 0 and oil leaks from
the tank at a rate of r共t兲 苷 100e⫺0.01t liters per minute. How much oil leaks out during the first hour?
82. A bacteria population starts with 400 bacteria and grows at a
rate of r共t兲 苷 共450.268兲e1.12567t bacteria per hour. How many bacteria will there be after three hours?
y
冉
100 dx 苷 5000 1 ⫺ dt 共t ⫹ 10兲2
冊
⫺a
⫺b
f 共x兲 dx
88. If f is continuous on ⺢, prove that
y
b
a
f 共x ⫹ c兲 dx 苷
y
b⫹c
a⫹c
f 共x兲 dx
For the case where f 共x兲 艌 0, draw a diagram to interpret this equation geometrically as an equality of areas. 89. If a and b are positive numbers, show that
y
1
x a 共1 ⫺ x兲 b dx 苷
0
y
1
0
x b 共1 ⫺ x兲 a dx
90. If f is continuous on 关0, 兴, use the substitution u 苷 ⫺ x to
y
0
y
y
兾2
0
3
0
2
y
0
f 共sin x兲 dx
x sin x dx 1 ⫹ cos2x
92. (a) If f is continuous, prove that
0
86. If f is continuous and y f 共x兲 dx 苷 4, find y x f 共x 2 兲 dx.
0
2
0
x f 共sin x兲 dx 苷
91. Use Exercise 90 to evaluate the integral
85. If f is continuous and y f 共x兲 dx 苷 10, find y f 共2x兲 dx.
5
y
For the case where f 共x兲 艌 0 and 0 ⬍ a ⬍ b, draw a diagram to interpret this equation geometrically as an equality of areas.
calculators兾week
4
9
f 共⫺x兲 dx 苷
show that
(Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers’ unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.
0
b
a
84. Alabama Instruments Company has set up a production line to
manufacture a new calculator. The rate of production of these calculators after t weeks is
415
87. If f is continuous on ⺢, prove that
83. Breathing is cyclic and a full respiratory cycle from the begin
ning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L兾s. This explains, in part, why the function f 共t兲 苷 12 sin共2 t兾5兲 has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time t.
REVIEW
f 共cos x兲 dx 苷 y
兾2
0
f 共sin x兲 dx
(b) Use part (a) to evaluate x0兾2 cos 2 x dx and x0兾2 sin 2 x dx.
Review
Concept Check 1. (a) Write an expression for a Riemann sum of a function f.
Explain the meaning of the notation that you use. (b) If f 共x兲 艌 0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If f 共x兲 takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. 2. (a) Write the definition of the definite integral of a continuous
function from a to b. (b) What is the geometric interpretation of xab f 共x兲 dx if f 共x兲 艌 0? (c) What is the geometric interpretation of xab f 共x兲 dx if f 共x兲 takes on both positive and negative values? Illustrate with a diagram. 3. State both parts of the Fundamental Theorem of Calculus. 4. (a) State the Net Change Theorem.
(b) If r共t兲 is the rate at which water flows into a reservoir, what does xtt r共t兲 dt represent? 2
1
5. Suppose a particle moves back and forth along a straight line with velocity v共t兲, measured in feet per second, and accelera
tion a共t兲. (a) What is the meaning of x60120 v共t兲 dt ?
ⱍ
ⱍ
(b) What is the meaning of x60120 v共t兲 dt ? 120 60
(c) What is the meaning of x
a共t兲 dt ?
6. (a) Explain the meaning of the indefinite integral x f 共x兲 dx.
(b) What is the connection between the definite integral xab f 共x兲 dx and the indefinite integral x f 共x兲 dx ? 7. Explain exactly what is meant by the statement that “differen
tiation and integration are inverse processes.” 8. State the Substitution Rule. In practice, how do you use it?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
TrueFalse Quiz Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
8. If f and t are differentiable and f 共x兲 艌 t共x兲 for a ⬍ x ⬍ b,
then f ⬘共x兲 艌 t⬘共x兲 for a ⬍ x ⬍ b.
1. If f and t are continuous on 关a, b兴, then
y
b
a
b
关 f 共x兲 ⫹ t共x兲兴 dx 苷 y f 共x兲 dx ⫹
y
a
b
a
t共x兲 dx
2. If f and t are continuous on 关a, b兴, then
y
b
a
关 f 共x兲 t共x兲兴 dx 苷
冉y
b
a
冊冉y
f 共x兲 dx
b
a
冊
t共x兲 dx
b
b
y
b
5
冉
x 5 ⫺ 6x 9 ⫹
b
a
b
dx 苷 0
共ax 2 ⫹ bx ⫹ c兲 dx 苷 2 y 共ax 2 ⫹ c兲 dx 0
y
3
0
2
5
3
2
2
e x dx 苷 y e x dx ⫹ y e x dx 0
5
15. If f is continuous on 关a, b兴, then
a
sf 共x兲 dx 苷
冊 5
⫺5
d dx
5. If f is continuous on 关a, b兴 and f 共x兲 艌 0, then
y
sin x 共1 ⫹ x 4 兲2
14. If x01 f 共x兲 dx 苷 0, then f 共x兲 苷 0 for 0 艋 x 艋 1.
x f 共x兲 dx 苷 x y f 共x兲 dx
a
1
⫺1
11. All continuous functions have derivatives.
a
4. If f is continuous on 关a, b兴, then
y
10.
13.
5f 共x兲 dx 苷 5 y f 共x兲 dx
a
y
12. All continuous functions have antiderivatives.
3. If f is continuous on 关a, b兴, then
y
9.
冑y
b
a
f 共x兲 dx
16.
冉y
b
a
冊
f 共x兲 dx 苷 f 共x兲
x02 共x ⫺ x 3 兲 dx represents the area under the curve y 苷 x ⫺ x 3 from 0 to 2.
3
6. If f ⬘ is continuous on 关1, 3兴, then y f ⬘共v兲 dv 苷 f 共3兲 ⫺ f 共1兲. 1
7. If f and t are continuous and f 共x兲 艌 t共x兲 for a 艋 x 艋 b, then
y
b
a
b
17.
y
1
⫺2
1 3 dx 苷 ⫺ x4 8
18. If f has a discontinuity at 0, then y
f 共x兲 dx 艌 y t共x兲 dx
1
⫺1
a
f 共x兲 dx does not exist.
Exercises 1. Use the given graph of f to find the Riemann sum with six
subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. y
y=ƒ
2
2. (a) Evaluate the Riemann sum for
f 共x兲 苷 x 2 ⫺ x
with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. (b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral
y
2
0
0
;
2
Graphing calculator or computer required
6
x
0艋x艋2
共x 2 ⫺ x兲 dx
(c) Use the Fundamental Theorem to check your answer to part (b). (d) Draw a diagram to explain the geometric meaning of the integral in part (b).
CAS Computer algebra system required
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 5
3. Evaluate 1
y ( x ⫹ s1 ⫺ x ) dx
y
23.
y
25.
y
27.
y sin t cos t dt
29.
y
31.
y tan x ln共cos x兲 dx
33.
y 1⫹x
35.
y
37.
y ⱍx
2
⫺ 兾4
冉 冊
0
by interpreting it in terms of areas.
lim
兺 sin x
n l ⬁ i苷1
i
⌬x
as a definite integral on the interval 关0, 兴 and then evaluate the integral. 5. If x f 共x兲 dx 苷 10 and x f 共x兲 dx 苷 7, find x f 共x兲 dx. 6 0
CAS
4 0
6 4
6. (a) Write x15 共x ⫹ 2x 5 兲 dx as a limit of Riemann sums,
taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b) Use the Fundamental Theorem to check your answer to part (a). 7. The following figure shows the graphs of f, f ⬘, and x f 共t兲 dt. x 0
Identify each graph, and explain your choices. y
y
dx
24.
y
dx
26.
y 1 ⫹ cot x dx
28.
y sin x cos共cos x兲 dx
30.
y
32.
y
34.
y sinh共1 ⫹ 4x兲 dx
36.
y
38.
y ⱍ sx
x⫹2 sx ⫹ 4x 2
e sx dx sx
x3
dx
4
sec tan d 1 ⫹ sec 3
ⱍ
⫺ 4 dx
2
0
b c
417
ex dx 1 ⫹ e 2x
1
22.
0
2
1⫺x x
4. Express n
t 4 tan t dt 2 ⫹ cos t
兾4
21.
REVIEW
x dx x2 ⫺ 4
10
1
csc 2x
cos共ln x兲 dx x x
dx
s1 ⫺ x 4
兾4
0
共1 ⫹ tan t兲3 sec2t dt
ⱍ
4
⫺ 1 dx
0
; 39– 40 Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C 苷 0).
x
a
39.
cos x
y
s1 ⫹ sin x
dx
40.
x3
y
sx ⫹ 1 2
dx
; 41. Use a graph to give a rough estimate of the area of the region 8. Evaluate:
d arctan x 共e 兲 dx dx
1
(a)
y
(c)
d dx
0
y
x
0
d (b) dx
y
1
0
e
arctan x
dx
2 ; 42. Graph the function f 共x兲 苷 cos x sin x and use the graph to
e arctan t dt
guess the value of the integral x02 f 共x兲 dx. Then evaluate the integral to confirm your guess.
9–38 Evaluate the integral. 9.
11. 13.
y
2
1
y
1
0
y
9
1
15.
y
17.
y
1
0 5
1
19.
y
1
0
共8x ⫹ 3x 兲 dx 3
2
共1 ⫺ x 9 兲 dx su ⫺ 2u 2 du u
that lies under the curve y 苷 x sx , 0 艋 x 艋 4. Then find the exact area.
43– 48 Find the derivative of the function. 10.
12. 14.
y
T
0
y
1
0
共x ⫺ 8x ⫹ 7兲 dx 4
共1 ⫺ x兲9 dx
1
y (su 4
0
y共 y 2 ⫹ 1兲5 dy
16.
y
dt 共t ⫺ 4兲2
18.
y
v 2 cos共 v 3兲 dv
20.
2
0 1
0
y
⫹ 1兲 du
43. F共x兲 苷
y
45. t共x兲 苷
y
2
47. y 苷
y
x
sx
x
0
x4
0
t2 dt 1 ⫹ t3
44. F共x兲 苷
y
cos共t 2 兲 dt
46. t共x兲 苷
y
et dt t
48. y 苷
y
1
x
sin x
1
3x⫹1
2x
st ⫹ sin t dt 1 ⫺ t2 dt 1 ⫹ t4
sin共t 4 兲 dt
y 2s1 ⫹ y 3 dy sin共3 t兲 dt
1
⫺1
sin x dx 1 ⫹ x2
49–50 Use Property 8 of integrals to estimate the value of the integral. 49.
y
3
1
sx 2 ⫹ 3 dx
50.
y
5
3
1 dx x⫹1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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INTEGRALS
62. The Fresnel function S共x兲 苷
51–54 Use the properties of integrals to verify the inequality. 51. 53.
y
1
0
y
1
0
x 2 cos x dx 艋
Page 418
1 3
52.
e x cos x dx 艋 e ⫺ 1
54.
兾2
y
兾4
y
1
0
in Section 5.3. Fresnel also used the function
sin x s2 dx 艋 2 x
C共x兲 苷
x sin⫺1x dx 艋 兾4
55. Use the Midpoint Rule with n 苷 6 to approximate x03 sin共x 3 兲 dx.
CAS
56. A particle moves along a line with velocity function v共t兲 苷 t 2 ⫺ t, where v is measured in meters per second.
Find (a) the displacement and (b) the distance traveled by the particle during the time interval 关0, 5兴. 57. Let r共t兲 be the rate at which the world’s oil is consumed, where
t is measured in years starting at t 苷 0 on January 1, 2000, and r共t兲 is measured in barrels per year. What does x08 r共t兲 dt represent? 58. A radar gun was used to record the speed of a runner at the
times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds. t (s)
v (m兾s)
0 0.5 1.0 1.5 2.0 2.5
t (s)
0 4.67 7.34 8.86 9.73 10.22
3.0 3.5 4.0 4.5 5.0
y
x
0
CAS
y
x
0
cos ( 12 t 2) dt
in his theory of the diffraction of light waves. (a) On what intervals is C increasing? (b) On what intervals is C concave upward? (c) Use a graph to solve the following equation correct to two decimal places: cos ( 12 t 2) dt 苷 0.7
(d) Plot the graphs of C and S on the same screen. How are these graphs related?
; 63. Estimate the value of the number c such that the area under the curve y 苷 sinh cx between x 苷 0 and x 苷 1 is equal to 1. 64. Suppose that the temperature in a long, thin rod placed along
ⱍ ⱍ
ⱍ ⱍ
the xaxis is initially C兾共2a兲 if x 艋 a and 0 if x ⬎ a. It can be shown that if the heat diffusivity of the rod is k , then the temperature of the rod at the point x at time t is T共x, t兲 苷
v (m兾s)
10.51 10.67 10.76 10.81 10.81
x0x sin ( 12 t 2) dt was introduced
C a s4 kt
y
a
0
2
e ⫺共x⫺u兲 兾共4kt兲 du
To find the temperature distribution that results from an initial hot spot concentrated at the origin, we need to compute lim T共x, t兲
al0
Use l’Hospital’s Rule to find this limit.
59. A population of honeybees increased at a rate of r共t兲 bees
per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. r
65. If f is a continuous function such that
y
x
1
x
f 共t兲 dt 苷 共x ⫺ 1兲e 2x ⫹ y e ⫺t f 共t兲 dt 1
for all x, find an explicit formula for f 共x兲. 66. Suppose h is a function such that h共1兲 苷 ⫺2, h⬘共1兲 苷 2,
12000
h⬙共1兲 苷 3, h共2兲 苷 6, h⬘共2兲 苷 5, h⬙共2兲 苷 13, and h ⬙ is continuous everywhere. Evaluate x12 h⬙共u兲 du.
8000
67. If f ⬘ is continuous on 关a, b兴, show that b
2 y f 共x兲 f ⬘共x兲 dx 苷 关 f 共b兲兴 2 ⫺ 关 f 共a兲兴 2
4000
a
0
60. Let
f 共x兲 苷
4
再
8
⫺x ⫺ 1 ⫺s1 ⫺ x 2
12
16
20
t 24 (weeks)
hl0
1 h
y
2⫹h
2
s1 ⫹ t 3 dt.
69. If f is continuous on 关0, 1兴 , prove that
y if ⫺3 艋 x 艋 0 if 0 艋 x 艋 1
1 f 共x兲 dx by interpreting the integral as a difference Evaluate x⫺3 of areas.
61. If f is continuous and x02 f 共x兲 dx 苷 6, evaluate
x0兾2 f 共2 sin 兲 cos d.
68. Find lim
1
0
1
f 共x兲 dx 苷 y f 共1 ⫺ x兲 dx 0
70. Evaluate
lim
nl⬁
1 n
冋冉 冊 冉 冊 冉 冊 1 n
9
⫹
2 n
9
⫹
3 n
9
⫹ ⭈⭈⭈ ⫹
冉 冊册 n n
9
71. Suppose f is continuous, f 共0兲 苷 0, f 共1兲 苷 1, f ⬘共x兲 ⬎ 0, and
x01 f 共x兲 dx 苷 13. Find the value of the integral x01 f ⫺1共 y兲 dy.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_05_ch05_p410419.qk:97909_05_ch05_p410419
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Page 419
Problems Plus Before you look at the solution of the following example, cover it up and first try to solve the problem yourself. EXAMPLE 1 Evaluate lim x l3
冉
x x⫺3
y
x
3
冊
sin t dt . t
SOLUTION Let’s start by having a preliminary look at the ingredients of the function.
What happens to the first factor, x兾共x ⫺ 3兲, when x approaches 3? The numerator approaches 3 and the denominator approaches 0, so we have x l ⬁ as x⫺3
PS The principles of problem solving are discussed on page 75.
x l 3⫹
x l ⫺⬁ as x⫺3
and
x l 3⫺
The second factor approaches x33 共sin t兲兾t dt, which is 0. It’s not clear what happens to the function as a whole. (One factor is becoming large while the other is becoming small.) So how do we proceed? One of the principles of problem solving is recognizing something familiar. Is there a part of the function that reminds us of something we’ve seen before? Well, the integral
y
x
3
sin t dt t
has x as its upper limit of integration and that type of integral occurs in Part 1 of the Fundamental Theorem of Calculus: d dx
y
x
a
f 共t兲 dt 苷 f 共x兲
This suggests that differentiation might be involved. Once we start thinking about differentiation, the denominator 共x ⫺ 3兲 reminds us of something else that should be familiar: One of the forms of the definition of the derivative in Chapter 2 is F共x兲 ⫺ F共a兲 F⬘共a兲 苷 lim xla x⫺a and with a 苷 3 this becomes F⬘共3兲 苷 lim
xl3
F共x兲 ⫺ F共3兲 x⫺3
So what is the function F in our situation? Notice that if we define F共x兲 苷 y
x
3
sin t dt t
then F共3兲 苷 0. What about the factor x in the numerator? That’s just a red herring, so let’s factor it out and put together the calculation:
lim x l3
Another approach is to use l’Hospital’s Rule.
冉
x x⫺3
y
x
3
冊
sin t dt 苷 lim x ⴢ lim x l3 x l3 t 苷 3 lim x l3
y
x
3
sin t dt t x⫺3
F共x兲 ⫺ F共3兲 x⫺3
苷 3F⬘共3兲 苷 3
sin 3 3
(FTC1)
苷 sin 3 419 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_05_ch05_p420.qk:97909_05_ch05_p420
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1. If x sin x 苷
Problems
y
x2
Page 420
f 共t兲 dt, where f is a continuous function, find f 共4兲.
0
2. Find the minimum value of the area of the region under the curve y 苷 x ⫹ 1兾x from x 苷 a
to x 苷 a ⫹ 1.5, for all a ⬎ 0. 4
4
3. If x04 e 共x⫺2兲 dx 苷 k, find the value of x04 xe 共x⫺2兲 dx. 2 3 ; 4. (a) Graph several members of the family of functions f 共x兲 苷 共2cx ⫺ x 兲兾c for c ⬎ 0 and
look at the regions enclosed by these curves and the xaxis. Make a conjecture about how the areas of these regions are related. (b) Prove your conjecture in part (a). (c) Take another look at the graphs in part (a) and use them to sketch the curve traced out by the vertices (highest points) of the family of functions. Can you guess what kind of curve this is? (d) Find an equation of the curve you sketched in part (c). t共x兲
1 cos x dt, where t共x兲 苷 y 关1 ⫹ sin共t 2 兲兴 dt, find f ⬘共兾2兲. 0 s1 ⫹ t 3
5. If f 共x兲 苷
y
6. If f 共x兲 苷
x0x x 2 sin共t 2 兲 dt, find
0
1 x
7. Evaluate lim
xl0
y
x
0
f ⬘共x兲.
共1 ⫺ tan 2t兲1兾t dt.
8. The figure shows two regions in the first quadrant: A共t兲 is the area under the curve y 苷 sin共x 2 兲
y
P { t, sin( t @ ) }
from 0 to t, and B共t兲 is the area of the triangle with vertices O, P, and 共t, 0兲. Find lim⫹ A共t兲兾B共t兲. tl0
9. Find the interval 关a, b兴 for which the value of the integral xab 共2 ⫹ x ⫺ x 2 兲 dx is a maximum.
y=sin{≈}
10000
兺
10. Use an integral to estimate the sum
A(t)
si .
i苷1
O
t
11. (a) Evaluate x0n 冀x冁 dx, where n is a positive integer.
x
(b) Evaluate xab 冀x冁 dx, where a and b are real numbers with 0 艋 a ⬍ b.
y
P { t, sin( t @ ) }
12. Find
d2 dx 2
x
冉
y y 0
sin t
1
冊
s1 ⫹ u 4 du dt .
13. Suppose the coefficients of the cubic polynomial P共x兲 苷 a ⫹ bx ⫹ cx 2 ⫹ dx 3 satisfy the
B(t)
equation O
t
x
a⫹
c d b ⫹ ⫹ 苷0 2 3 4
Show that the equation P共x兲 苷 0 has a root between 0 and 1. Can you generalize this result for an nthdegree polynomial?
FIGURE FOR PROBLEM 8
14. A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to
be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a height r兾s1 ⫹ 2 above the surface of the liquid.
2
x
15. Prove that if f is continuous, then y f 共u兲共x ⫺ u兲 du 苷 0
x
冉
y y 0
u
0
冊
f 共t兲 dt du.
16. The figure shows a region consisting of all points inside a square that are closer to the center 2
than to the sides of the square. Find the area of the region.
2
17. Evaluate lim
nl⬁
2
冊
1 1 1 . ⫹ ⫹ ⭈⭈⭈ ⫹ sn sn ⫹ 1 sn sn ⫹ 2 sn sn ⫹ n
18. For any number c, we let fc 共x兲 be the smaller of the two numbers 共x ⫺ c兲 2 and 共x ⫺ c ⫺ 2兲 2.
Then we define t共c兲 苷 x01 fc 共x兲 dx. Find the maximum and minimum values of t共c兲 if ⫺2 艋 c 艋 2.
FIGURE FOR PROBLEM 16
420
冉
;
Graphing calculator or computer required
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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6
Applications of Integration
The Great Pyramid of King Khufu was built in Egypt from 2580 BC to 2560 BC and was the tallest manmade structure in the world for more than 3800 years. The techniques of this chapter will enable us to estimate the total work done in building this pyramid and therefore to make an educated guess as to how many laborers were needed to construct it.
© Ziga Camernik / Shutterstock
In this chapter we explore some of the applications of the definite integral by using it to compute areas between curves, volumes of solids, and the work done by a varying force. The common theme is the following general method, which is similar to the one we used to find areas under curves: We break up a quantity Q into a large number of small parts. We next approximate each small part by a quantity of the form f 共x i*兲 ⌬x and thus approximate Q by a Riemann sum. Then we take the limit and express Q as an integral. Finally we evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint Rule.
421 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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422
CHAPTER 6
APPLICATIONS OF INTEGRATION
Areas Between Curves
6.1 y
y=ƒ
S 0
a
b
x
y=©
In Chapter 5 we defined and calculated areas of regions that lie under the graphs of functions. Here we use integrals to find areas of regions that lie between the graphs of two functions. Consider the region S that lies between two curves y 苷 f 共x兲 and y 苷 t共x兲 and between the vertical lines x 苷 a and x 苷 b, where f and t are continuous functions and f 共x兲 艌 t共x兲 for all x in 关a, b兴. (See Figure 1.) Just as we did for areas under curves in Section 5.1, we divide S into n strips of equal width and then we approximate the ith strip by a rectangle with base ⌬x and height f 共x*i 兲 ⫺ t共x*i 兲. (See Figure 2. If we like, we could take all of the sample points to be right endpoints, in which case x*i 苷 x i .) The Riemann sum
FIGURE 1
n
兺 关 f 共x*兲 ⫺ t共x*兲兴 ⌬x
S=s(x, y)  a¯x¯b, ©¯y¯ƒd
i
i
i苷1
is therefore an approximation to what we intuitively think of as the area of S. y
y
f (x *i )
0
a
f (x *i )g(x *i )
Îx FIGURE 2
x
b
_g(x *i )
0
a
b
x
x *i
(a) Typical rectangle
(b) Approximating rectangles
This approximation appears to become better and better as n l ⬁. Therefore we define the area A of the region S as the limiting value of the sum of the areas of these approximating rectangles. n
1
A 苷 lim
兺 关 f 共x*兲 ⫺ t共x*兲兴 ⌬x
n l ⬁ i苷1
i
i
We recognize the limit in 1 as the definite integral of f ⫺ t. Therefore we have the following formula for area. 2 The area A of the region bounded by the curves y 苷 f 共x兲, y 苷 t共x兲, and the lines x 苷 a, x 苷 b, where f and t are continuous and f 共x兲 艌 t共x兲 for all x in 关a, b兴, is b
A 苷 y 关 f 共x兲 ⫺ t共x兲兴 dx a
Notice that in the special case where t共x兲 苷 0, S is the region under the graph of f and our general definition of area 1 reduces to our previous definition (Definition 2 in Section 5.1).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.1 y
AREAS BETWEEN CURVES
In the case where both f and t are positive, you can see from Figure 3 why 2 is true: y=ƒ
A 苷 关area under y 苷 f 共x兲兴 ⫺ 关area under y 苷 t共x兲兴
S
b
0
b
b
苷 y f 共x兲 dx ⫺ y t共x兲 dx 苷 y 关 f 共x兲 ⫺ t共x兲兴 dx
y=©
a
a
b
x
a
a
EXAMPLE 1 Find the area of the region bounded above by y 苷 e x, bounded below by
y 苷 x, and bounded on the sides by x 苷 0 and x 苷 1.
FIGURE 3 b
b
SOLUTION The region is shown in Figure 4. The upper boundary curve is y 苷 e x and the
A=j ƒ dxj © dx a
a
lower boundary curve is y 苷 x. So we use the area formula 2 with f 共x兲 苷 e x, t共x兲 苷 x, a 苷 0, and b 苷 1:
y
1
]
A 苷 y 共e x ⫺ x兲 dx 苷 e x ⫺ 12 x 2 0
y=´
y=x Îx 0
1
x
In Figure 4 we drew a typical approximating rectangle with width ⌬x as a reminder of the procedure by which the area is defined in 1 . In general, when we set up an integral for an area, it’s helpful to sketch the region to identify the top curve yT , the bottom curve yB , and a typical approximating rectangle as in Figure 5. Then the area of a typical rectangle is 共yT ⫺ yB兲 ⌬x and the equation
FIGURE 4
n
A 苷 lim
兺 共y
n l ⬁ i苷1
y
yT yTyB yB
Îx
a
b
x
1 0
苷 e ⫺ 12 ⫺ 1 苷 e ⫺ 1.5
x=1
1
0
423
T
b
⫺ yB兲 ⌬x 苷 y 共yT ⫺ yB兲 dx a
summarizes the procedure of adding (in a limiting sense) the areas of all the typical rectangles. Notice that in Figure 5 the lefthand boundary reduces to a point, whereas in Figure 3 the righthand boundary reduces to a point. In the next example both of the side boundaries reduce to a point, so the first step is to find a and b.
v
EXAMPLE 2 Find the area of the region enclosed by the parabolas y 苷 x 2 and
y 苷 2x ⫺ x 2.
FIGURE 5
SOLUTION We first find the points of intersection of the parabolas by solving their equa
tions simultaneously. This gives x 2 苷 2x ⫺ x 2, or 2x 2 ⫺ 2x 苷 0. Thus 2x共x ⫺ 1兲 苷 0, so x 苷 0 or 1. The points of intersection are 共0, 0兲 and 共1, 1兲. We see from Figure 6 that the top and bottom boundaries are
yT=2x≈ y
yT 苷 2x ⫺ x 2
(1, 1)
yB 苷 x 2
and
The area of a typical rectangle is yB=≈
共yT ⫺ yB兲 ⌬x 苷 共2x ⫺ x 2 ⫺ x 2 兲 ⌬x
Îx (0, 0)
FIGURE 6
x
and the region lies between x 苷 0 and x 苷 1. So the total area is 1
1
A 苷 y 共2x ⫺ 2x 2 兲 dx 苷 2 y 共x ⫺ x 2 兲 dx 0
冋
苷2
0
x2 x3 ⫺ 2 3
册 冉 冊 1
苷2
0
1 1 ⫺ 2 3
苷
1 3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_06_ch06_p421431.qk_97909_06_ch06_p421431 9/21/10 3:49 PM Page 424
424
CHAPTER 6
APPLICATIONS OF INTEGRATION
Sometimes it’s difficult, or even impossible, to find the points of intersection of two curves exactly. As shown in the following example, we can use a graphing calculator or computer to find approximate values for the intersection points and then proceed as before. EXAMPLE 3 Find the approximate area of the region bounded by the curves y 苷 x兾sx 2 ⫹ 1 and y 苷 x 4 ⫺ x. SOLUTION If we were to try to find the exact intersection points, we would have to solve
the equation x 苷 x4 ⫺ x sx ⫹ 1 2
1.5 x y= œ„„„„„ ≈+1
_1
2 y=x $x
This looks like a very difficult equation to solve exactly (in fact, it’s impossible), so instead we use a graphing device to draw the graphs of the two curves in Figure 7. One intersection point is the origin. We zoom in toward the other point of intersection and find that x ⬇ 1.18. (If greater accuracy is required, we could use Newton’s method or a rootfinder, if available on our graphing device.) Thus an approximation to the area between the curves is
_1
A⬇
y
冋
1.18
0
FIGURE 7
册
x ⫺ 共x 4 ⫺ x兲 dx sx ⫹ 1 2
To integrate the first term we use the substitution u 苷 x 2 ⫹ 1. Then du 苷 2x dx, and when x 苷 1.18, we have u ⬇ 2.39. So A ⬇ 12 y
2.39
1
苷 su
du 1.18 ⫺ y 共x 4 ⫺ x兲 dx 0 su
2.39
]
1
⫺
冋
x5 x2 ⫺ 5 2
册
1.18
0
共1.18兲 共1.18兲2 ⫹ 5 2 5
苷 s2.39 ⫺ 1 ⫺ ⬇ 0.785 √ (mi/h)
EXAMPLE 4 Figure 8 shows velocity curves for two cars, A and B, that start side by side and move along the same road. What does the area between the curves represent? Use the Midpoint Rule to estimate it.
60
A
50
SOLUTION We know from Section 5.4 that the area under the velocity curve A represents
40 30
B
20 10 0
2
FIGURE 8
4
6
8 10 12 14 16 t (seconds)
the distance traveled by car A during the first 16 seconds. Similarly, the area under curve B is the distance traveled by car B during that time period. So the area between these curves, which is the difference of the areas under the curves, is the distance between the cars after 16 seconds. We read the velocities from the graph and convert them to feet per second 共1 mi兾h 苷 5280 3600 ft兾s兲. t
0
2
4
6
8
10
12
14
16
vA
0
34
54
67
76
84
89
92
95
vB
0
21
34
44
51
56
60
63
65
vA ⫺ vB
0
13
20
23
25
28
29
29
30
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97909_06_ch06_p421431.qk_97909_06_ch06_p421431 9/21/10 3:50 PM Page 425
SECTION 6.1
AREAS BETWEEN CURVES
425
We use the Midpoint Rule with n 苷 4 intervals, so that ⌬t 苷 4. The midpoints of the intervals are t1 苷 2, t2 苷 6, t3 苷 10, and t4 苷 14. We estimate the distance between the cars after 16 seconds as follows:
y
16
0
共vA ⫺ vB 兲 dt ⬇ ⌬t 关13 ⫹ 23 ⫹ 28 ⫹ 29兴 苷 4共93兲 苷 372 ft
If we are asked to find the area between the curves y 苷 f 共x兲 and y 苷 t共x兲 where f 共x兲 艌 t共x兲 for some values of x but t共x兲 艌 f 共x兲 for other values of x, then we split the given region S into several regions S1 , S2 , . . . with areas A1 , A2 , . . . as shown in Figure 9. We then define the area of the region S to be the sum of the areas of the smaller regions S1 , S2 , . . . , that is, A 苷 A1 ⫹ A2 ⫹ ⭈ ⭈ ⭈. Since
y
y=© S¡
S£
S™ y=ƒ
0
a
b
x
ⱍ f 共x兲 ⫺ t共x兲 ⱍ 苷
FIGURE 9
再
f 共x兲 ⫺ t共x兲 t共x兲 ⫺ f 共x兲
when f 共x兲 艌 t共x兲 when t共x兲 艌 f 共x兲
we have the following expression for A. 3 The area between the curves y 苷 f 共x兲 and y 苷 t共x兲 and between x 苷 a and x 苷 b is
A苷y
b
a
ⱍ f 共x兲 ⫺ t共x兲 ⱍ dx
When evaluating the integral in 3 , however, we must still split it into integrals corresponding to A1 , A2 , . . . .
v EXAMPLE 5 Find the area of the region bounded by the curves y 苷 sin x, y 苷 cos x, x 苷 0, and x 苷 兾2. SOLUTION The points of intersection occur when sin x 苷 cos x, that is, when x 苷 兾4 (since 0 艋 x 艋 兾2). The region is sketched in Figure 10. Observe that cos x 艌 sin x when 0 艋 x 艋 兾4 but sin x 艌 cos x when 兾4 艋 x 艋 兾2. Therefore the required area is
y y =cos x A¡
y=sin x A™ π 2
x=0
x=
A苷y
兾2
0
FIGURE 10
π 4
π 2
x
苷y
兾4
0
ⱍ cos x ⫺ sin x ⱍ dx 苷 A
1
0
共cos x ⫺ sin x兲 dx ⫹ y
[
苷
冉
兾2
兾4
兾4
]
苷 sin x ⫹ cos x
0
[
⫹ A2 共sin x ⫺ cos x兲 dx
冊 冉
兾2
]
⫹ ⫺cos x ⫺ sin x
兾4
1 1 1 1 ⫹ ⫺ 0 ⫺ 1 ⫹ ⫺0 ⫺ 1 ⫹ ⫹ s2 s2 s2 s2
冊
苷 2s2 ⫺ 2 In this particular example we could have saved some work by noticing that the region is symmetric about x 苷 兾4 and so A 苷 2A1 苷 2 y
兾4
0
共cos x ⫺ sin x兲 dx
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97909_06_ch06_p421431.qk_97909_06_ch06_p421431 9/21/10 3:50 PM Page 426
426
CHAPTER 6
APPLICATIONS OF INTEGRATION
Some regions are best treated by regarding x as a function of y. If a region is bounded by curves with equations x 苷 f 共y兲, x 苷 t共y兲, y 苷 c, and y 苷 d, where f and t are continuous and f 共y兲 艌 t共 y兲 for c 艋 y 艋 d (see Figure 11), then its area is d
A 苷 y 关 f 共y兲 ⫺ t共y兲兴 dy c
y
y
y=d
d
d
xR
xL Îy x=g(y)
Îy
x=f(y) xR x L
c
y=c
c
0
0
x
FIGURE 11
x
FIGURE 12
If we write x R for the right boundary and x L for the left boundary, then, as Figure 12 illustrates, we have d
A 苷 y 共x R ⫺ x L 兲 dy c
Here a typical approximating rectangle has dimensions x R ⫺ x L and ⌬y.
y
(5, 4)
4
v
1 x L=2 ¥3
EXAMPLE 6 Find the area enclosed by the line y 苷 x ⫺ 1 and the parabola
y 2 苷 2x ⫹ 6. xR=y+1
SOLUTION By solving the two equations we find that the points of intersection are
共⫺1, ⫺2兲 and 共5, 4兲. We solve the equation of the parabola for x and notice from Figure 13 that the left and right boundary curves are
x
0 _2
(_1, _2)
x L 苷 12 y 2 ⫺ 3
xR 苷 y ⫹ 1
and
We must integrate between the appropriate yvalues, y 苷 ⫺2 and y 苷 4. Thus FIGURE 13
4
A 苷 y 共x R ⫺ x L 兲 dy 苷 ⫺2
y
苷y
y= œ„„„„„ 2x+6
(5, 4)
苷⫺
A™ y=x1 ⫺3
A¡
0
x (_1, _2)
y=_ œ„„„„„ 2x+6 FIGURE 14
4
⫺2
4
y [共y ⫹ 1兲 ⫺ ( ⫺2
(⫺12 y 2 ⫹ y ⫹ 4) dy
1 2
冉 冊 y3 3
⫹
1 2
]
y 2 ⫺ 3) dy
册
y2 ⫹ 4y 2
4
⫺2
苷 ⫺ 共64兲 ⫹ 8 ⫹ 16 ⫺ ( ⫹ 2 ⫺ 8) 苷 18 1 6
4 3
NOTE We could have found the area in Example 6 by integrating with respect to x instead of y, but the calculation is much more involved. It would have meant splitting the region in two and computing the areas labeled A1 and A2 in Figure 14. The method we used in Example 6 is much easier.
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97909_06_ch06_p421431.qk_97909_06_ch06_p421431 9/21/10 3:50 PM Page 427
SECTION 6.1
427
Exercises
6.1
21. y 苷 tan x,
1– 4 Find the area of the shaded region. y
1.
AREAS BETWEEN CURVES
x+2 y=œ„„„„ (4, 4)
x=2
y 苷 sin 2x,
24. y 苷 cos x,
y 苷 1 ⫺ cos x, 0 艋 x 艋
y=x y=
y
3.
ⱍ ⱍ
x
1 x+1
x苷9
26. y 苷 x ,
y 苷 x2 ⫺ 2
27. y 苷 1兾x,
y 苷 x,
28. y 苷 x ,
y 苷 2x ,
1 4
y
4.
y 苷 12 x,
2
x 苷 0,
x 苷 兾2
23. y 苷 cos x,
25. y 苷 sx ,
x
⫺兾3 艋 x 艋 兾3
22. y 苷 x , y 苷 x
y
2.
y=5x≈
y 苷 2 sin x,
3
y 苷 14 x, 2
x⬎0
x ⫹ y 苷 3,
x艌0
x=¥4y x=¥2
y=1
29–30 Use calculus to find the area of the triangle with the given vertices.
(_3, 3) x
x=e y
x
y=_1
29. 共0, 0兲,
共3, 1兲,
共1, 2兲
30. 共2, 0兲,
共0, 2兲,
共⫺1, 1兲
x=2y¥ 31–32 Evaluate the integral and interpret it as the area of a region. 5–12 Sketch the region enclosed by the given curves. Decide
Sketch the region.
whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
31.
5. y 苷 e x,
y 苷 x 2 ⫺ 1,
6. y 苷 sin x, y 苷 x,
x 苷 ⫺1,
x 苷 兾2,
7. y 苷 共x ⫺ 2兲 ,
y苷x
8. y 苷 x ⫺ 2x,
y苷x⫹4
2
2
9. y 苷 1兾x,
y 苷 1兾x 2,
x苷
12. 4x ⫹ y 2 苷 12,
32.
y ⱍ3 1
x
⫺1
ⱍ
⫺ 2 x dx
; 33–36 Use a graph to find approximate xcoordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. 33. y 苷 x sin共x 2 兲, 34. y 苷
x艌0
x 苷 y2 ⫺ 1
11. x 苷 1 ⫺ y 2,
0
x苷1
x苷2
y 苷 2x兾,
10. y 苷 sin x,
兾2
y ⱍ sin x ⫺ cos 2x ⱍ dx
x , y 苷 x 5 ⫺ x, x 艌 0 共x 2 ⫹ 1兲2
35. y 苷 3x 2 ⫺ 2 x,
x苷y
y 苷 x4
36. y 苷 e x,
y 苷 x 3 ⫺ 3x ⫹ 4
y 苷 2 ⫺ x2
13–28 Sketch the region enclosed by the given curves and find its
area. 13. y 苷 12 ⫺ x 2,
y 苷 x2 ⫺ 6
14. y 苷 x 2,
y 苷 4x ⫺ x 2
15. y 苷 e x,
y 苷 xe x,
17. x 苷 2y ,
x苷4⫹y
2
x⫺y苷1
19. y 苷 cos x,
y 苷 4x 2 ⫺ 1
;
37. y 苷
x苷0
18. y 苷 sx ⫺ 1 , 20. x 苷 y 4,
to compute the area correct to five decimal places.
y 苷 2 ⫺ cos x, 0 艋 x 艋 2
16. y 苷 cos x, 2
; 37– 40 Graph the region between the curves and use your calculator
y 苷 s2 ⫺ x ,
y苷0
Graphing calculator or computer required
CAS
2 , 1 ⫹ x4
y 苷 x2
39. y 苷 tan 2 x,
y 苷 sx
40. y 苷 cos x,
y 苷 x ⫹ 2 sin 4 x
2
38. y 苷 e1⫺x ,
y 苷 x4
41. Use a computer algebra system to find the exact area enclosed
by the curves y 苷 x 5 ⫺ 6x 3 ⫹ 4x and y 苷 x.
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
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CHAPTER 6
APPLICATIONS OF INTEGRATION
42. Sketch the region in the xyplane defined by the inequalities
ⱍ ⱍ
x ⫺ 2y 2 艌 0, 1 ⫺ x ⫺ y 艌 0 and find its area.
(d) Estimate the time at which the cars are again side by side. √
43. Racing cars driven by Chris and Kelly are side by side at the
start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.
A B
t
vC
vK
t
vC
vK
0 1 2 3 4 5
0 20 32 46 54 62
0 22 37 52 61 71
6 7 8 9 10
69 75 81 86 90
80 86 93 98 102
44. The widths (in meters) of a kidneyshaped swimming pool
were measured at 2meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.
0
1
t (min)
2
48. The figure shows graphs of the marginal revenue function R⬘
and the marginal cost function C⬘ for a manufacturer. [Recall from Section 4.7 that R共x兲 and C共x兲 represent the revenue and cost when x units are manufactured. Assume that R and C are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity. y
Rª(x) 3 2
6.2
7.2
6.8
5.6 5.0 4.8
4.8
1 0
C ª(x)
50
100
x
2 2 ; 49. The curve with equation y 苷 x 共x ⫹ 3兲 is called Tschirn
45. A crosssection of an airplane wing is shown. Measurements of
the thickness of the wing, in centimeters, at 20centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing’s crosssection.
hausen’s cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop. 50. Find the area of the region bounded by the parabola y 苷 x 2, the
tangent line to this parabola at 共1, 1兲, and the xaxis.
51. Find the number b such that the line y 苷 b divides the region
bounded by the curves y 苷 x 2 and y 苷 4 into two regions with equal area.
200 cm 46. If the birth rate of a population is b共t兲 苷 2200e 0.024t people per
year and the death rate is d共t兲 苷 1460e0.018t people per year, find the area between these curves for 0 艋 t 艋 10. What does this area represent?
47. Two cars, A and B, start side by side and accelerate from rest.
The figure shows the graphs of their velocity functions. (a) Which car is ahead after one minute? Explain. (b) What is the meaning of the area of the shaded region? (c) Which car is ahead after two minutes? Explain.
52. (a) Find the number a such that the line x 苷 a bisects the area
under the curve y 苷 1兾x 2, 1 艋 x 艋 4. (b) Find the number b such that the line y 苷 b bisects the area in part (a).
53. Find the values of c such that the area of the region bounded by
the parabolas y 苷 x 2 ⫺ c 2 and y 苷 c 2 ⫺ x 2 is 576.
54. Suppose that 0 ⬍ c ⬍ 兾2. For what value of c is the area of
the region enclosed by the curves y 苷 cos x, y 苷 cos共x ⫺ c兲, and x 苷 0 equal to the area of the region enclosed by the curves y 苷 cos共x ⫺ c兲, x 苷 , and y 苷 0 ?
55. For what values of m do the line y 苷 mx and the curve
y 苷 x兾共x 2 ⫹ 1兲 enclose a region? Find the area of the region.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPLIED PROJECT
APPLIED PROJECT
y
(0.8, 0.5)
(0.4, 0.12) 0.2
0.4
0.6
0.8
429
THE GINI INDEX
1
0
THE GINI INDEX
1
x
How is it possible to measure the distribution of income among the inhabitants of a given country? One such measure is the Gini index, named after the Italian economist Corrado Gini who first devised it in 1912. We first rank all households in a country by income and then we compute the percentage of households whose income is at most a given percentage of the country’s total income. We define a Lorenz curve y 苷 L共x兲 on the interval 关0, 1兴 by plotting the point 共a兾100, b兾100兲 on the curve if the bottom a% of households receive at most b% of the total income. For instance, in Figure 1 the point 共0.4, 0.12兲 is on the Lorenz curve for the United States in 2008 because the poorest 40% of the population received just 12% of the total income. Likewise, the bottom 80% of the population received 50% of the total income, so the point 共0.8, 0.5兲 lies on the Lorenz curve. (The Lorenz curve is named after the American economist Max Lorenz.) Figure 2 shows some typical Lorenz curves. They all pass through the points 共0, 0兲 and 共1, 1兲 and are concave upward. In the extreme case L共x兲 苷 x, society is perfectly egalitarian: The poorest a% of the population receives a% of the total income and so everybody receives the same income. The area between a Lorenz curve y 苷 L共x兲 and the line y 苷 x measures how much the income distribution differs from absolute equality. The Gini index (sometimes called the Gini coefficient or the coefficient of inequality) is the area between the Lorenz curve and the line y 苷 x (shaded in Figure 3) divided by the area under y 苷 x. 1. (a) Show that the Gini index G is twice the area between the Lorenz curve and the line
y 苷 x, that is,
FIGURE 1
Lorenz curve for the US in 2008
1
G 苷 2 y 关x ⫺ L共x兲兴 dx 0
y 1
(b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? What is the value of G for a perfectly totalitarian society (a single person receives all the income?)
(1, 1)
2. The following table (derived from data supplied by the US Census Bureau) shows values
of the Lorenz function for income distribution in the United States for the year 2008.
y=x income fraction
x L共x兲
0
0.2
0.4
0.6
0.8
1.0
0.000 0.034 0.120 0.267 0.500 1.000
(a) What percentage of the total US income was received by the richest 20% of the population in 2008? (b) Use a calculator or computer to fit a quadratic function to the data in the table. Graph the data points and the quadratic function. Is the quadratic model a reasonable fit? (c) Use the quadratic model for the Lorenz function to estimate the Gini index for the United States in 2008.
1 x
population fraction
0.0
FIGURE 2
3. The following table gives values for the Lorenz function in the years 1970, 1980, 1990,
y
and 2000. Use the method of Problem 2 to estimate the Gini index for the United States for those years and compare with your answer to Problem 2(c). Do you notice a trend? x
y=x
y= L (x)
0
FIGURE 3
1
x
CAS
0.0
0.2
0.4
0.6
0.8
1.0
1970
0.000 0.041 0.149 0.323 0.568 1.000
1980
0.000 0.042 0.144 0.312 0.559 1.000
1990
0.000 0.038 0.134 0.293 0.530 1.000
2000
0.000 0.036 0.125 0.273 0.503 1.000
4. A power model often provides a more accurate fit than a quadratic model for a Lorenz
function. If you have a computer with Maple or Mathematica, fit a power function 共 y 苷 ax k 兲 to the data in Problem 2 and use it to estimate the Gini index for the United States in 2008. Compare with your answer to parts (b) and (c) of Problem 2.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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6.2
CHAPTER 6
APPLICATIONS OF INTEGRATION
Volumes In trying to find the volume of a solid we face the same type of problem as in finding areas. We have an intuitive idea of what volume means, but we must make this idea precise by using calculus to give an exact definition of volume. We start with a simple type of solid called a cylinder (or, more precisely, a right cylinder). As illustrated in Figure 1(a), a cylinder is bounded by a plane region B1, called the base, and a congruent region B2 in a parallel plane. The cylinder consists of all points on line segments that are perpendicular to the base and join B1 to B2. If the area of the base is A and the height of the cylinder (the distance from B1 to B2 ) is h, then the volume V of the cylinder is defined as V 苷 Ah In particular, if the base is a circle with radius r , then the cylinder is a circular cylinder with volume V 苷 r 2h [see Figure 1(b)], and if the base is a rectangle with length l and width w, then the cylinder is a rectangular box (also called a rectangular parallelepiped ) with volume V 苷 lwh [see Figure 1(c)].
B™ h h
h B¡ FIGURE 1
w
r
(a) Cylinder V=Ah
(b) Circular cylinder V=π
[email protected] l (c) Rectangular box V=lwh
For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate each piece by a cylinder. We estimate the volume of S by adding the volumes of the cylinders. We arrive at the exact volume of S through a limiting process in which the number of pieces becomes large. We start by intersecting S with a plane and obtaining a plane region that is called a crosssection of S. Let A共x兲 be the area of the crosssection of S in a plane Px perpendicular to the xaxis and passing through the point x, where a 艋 x 艋 b. (See Figure 2. Think of slicing S with a knife through x and computing the area of this slice.) The crosssectional area A共x兲 will vary as x increases from a to b. y
Px
A(a) A(b)
0
FIGURE 2
a
x
b
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.2
VOLUMES
431
Let’s divide S into n “slabs” of equal width ⌬x by using the planes Px1 , Px 2 , . . . to slice the solid. (Think of slicing a loaf of bread.) If we choose sample points x*i in 关x i⫺1, x i 兴, we can approximate the ith slab Si (the part of S that lies between the planes Px i⫺1 and Px i ) by a cylinder with base area A共x*i 兲 and “height” ⌬x. (See Figure 3.) y
y
Îx
S
0
a
xi1 x*i xi
b
x
0
⁄
a=x¸
¤
‹
x x¢
x∞
xß
x¶=b
x
FIGURE 3
The volume of this cylinder is A共x*i 兲 ⌬x, so an approximation to our intuitive conception of the volume of the i th slab Si is V共Si 兲 ⬇ A共x*i 兲 ⌬x Adding the volumes of these slabs, we get an approximation to the total volume (that is, what we think of intuitively as the volume): n
V⬇
兺 A共x*兲 ⌬x i
i苷1
This approximation appears to become better and better as n l ⬁. (Think of the slices as becoming thinner and thinner.) Therefore we define the volume as the limit of these sums as n l ⬁. But we recognize the limit of Riemann sums as a definite integral and so we have the following definition. It can be proved that this definition is independent of how S is situated with respect to the xaxis. In other words, no matter how we slice S with parallel planes, we always get the same answer for V.
Definition of Volume Let S be a solid that lies between x 苷 a and x 苷 b. If the crosssectional area of S in the plane Px , through x and perpendicular to the xaxis, is A共x兲, where A is a continuous function, then the volume of S is n
V 苷 lim
y
兺 A共x*兲 ⌬x 苷 y
n l ⬁ i苷1
y
_r
r x
x
i
b
a
A共x兲 dx
When we use the volume formula V 苷 xab A共x兲 dx, it is important to remember that A共x兲 is the area of a moving crosssection obtained by slicing through x perpendicular to the xaxis. Notice that, for a cylinder, the crosssectional area is constant: A共x兲 苷 A for all x. So our definition of volume gives V 苷 xab A dx 苷 A共b ⫺ a兲; this agrees with the formula V 苷 Ah. EXAMPLE 1 Show that the volume of a sphere of radius r is V 苷 3 r 3. 4
SOLUTION If we place the sphere so that its center is at the origin (see Figure 4), then the FIGURE 4
plane Px intersects the sphere in a circle whose radius (from the Pythagorean Theorem)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 6
APPLICATIONS OF INTEGRATION
is y 苷 sr 2 ⫺ x 2 . So the crosssectional area is A共x兲 苷 y 2 苷 共r 2 ⫺ x 2 兲 Using the definition of volume with a 苷 ⫺r and b 苷 r , we have r
r
⫺r
⫺r
V 苷 y A共x兲 dx 苷 y 共r 2 ⫺ x 2 兲 dx r
苷 2 y 共r 2 ⫺ x 2 兲 dx 0
冋
苷 2 r 2x ⫺
x3 3
(The integrand is even.)
册 冉 r
苷 2 r 3 ⫺
0
r3 3
冊
苷 43 r 3 Figure 5 illustrates the definition of volume when the solid is a sphere with radius r 苷 1. From the result of Example 1, we know that the volume of the sphere is 43 , which is approximately 4.18879. Here the slabs are circular cylinders, or disks, and the three parts of Figure 5 show the geometric interpretations of the Riemann sums n
n
兺 A共x 兲 ⌬x 苷 兺 共1
2
i
i苷1
TEC Visual 6.2A shows an animation
of Figure 5.
(a) Using 5 disks, VÅ4.2726
⫺ x i2 兲 ⌬x
i苷1
when n 苷 5, 10, and 20 if we choose the sample points x*i to be the midpoints xi . Notice that as we increase the number of approximating cylinders, the corresponding Riemann sums become closer to the true volume.
(b) Using 10 disks, VÅ4.2097
(c) Using 20 disks, VÅ4.1940
FIGURE 5 Approximating the volume of a sphere with radius 1
v EXAMPLE 2 Find the volume of the solid obtained by rotating about the xaxis the region under the curve y 苷 sx from 0 to 1. Illustrate the definition of volume by sketching a typical approximating cylinder. SOLUTION The region is shown in Figure 6(a). If we rotate about the xaxis, we get the
solid shown in Figure 6(b). When we slice through the point x, we get a disk with radius sx . The area of this crosssection is A共x兲 苷 (sx ) 2 苷 x and the volume of the approximating cylinder (a disk with thickness ⌬x) is A共x兲 ⌬x 苷 x ⌬x Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.2
VOLUMES
433
The solid lies between x 苷 0 and x 苷 1, so its volume is 1
1
V 苷 y A共x兲 dx 苷 y x dx 苷 0
0
y
x2 2
册
1
苷
0
2
y
y=œ„
Did we get a reasonable answer in Example 2? As a check on our work, let’s replace the given region by a square with base 关0, 1兴 and height 1. If we rotate this square, we get a cylinder with radius 1, height 1, and volume ⴢ 12 ⴢ 1 苷 . We computed that the given solid has half this volume. That seems about right.
œ„
0
x
1
x
0
x
1
Îx
FIGURE 6
(a)
(b)
v EXAMPLE 3 Find the volume of the solid obtained by rotating the region bounded by y 苷 x 3, y 苷 8, and x 苷 0 about the yaxis. SOLUTION The region is shown in Figure 7(a) and the resulting solid is shown in Fig
ure 7(b). Because the region is rotated about the yaxis, it makes sense to slice the solid perpendicular to the yaxis and therefore to integrate with respect to y. If we slice at 3 height y, we get a circular disk with radius x, where x 苷 s y . So the area of a crosssection through y is 3 A共y兲 苷 x 2 苷 (s y )2 苷 y 2兾3 and the volume of the approximating cylinder pictured in Figure 7(b) is A共y兲 ⌬y 苷 y 2兾3 ⌬y Since the solid lies between y 苷 0 and y 苷 8, its volume is 8
8
V 苷 y A共y兲 dy 苷 y y 2兾3 dy 苷 0
0
[
3 5
y 5兾3
y
]
8 0
苷
96 5
y
y=8
8 x (x, y)
Îy x=0 y=˛ or 3 x=œ„ œ y 0
FIGURE 7
(a)
x
0
x
(b)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 6
APPLICATIONS OF INTEGRATION
EXAMPLE 4 The region enclosed by the curves y 苷 x and y 苷 x 2 is rotated about the
xaxis. Find the volume of the resulting solid. SOLUTION The curves y 苷 x and y 苷 x 2 intersect at the points 共0, 0兲 and 共1, 1兲. The
TEC Visual 6.2B shows how solids of revolution are formed.
region between them, the solid of rotation, and a crosssection perpendicular to the xaxis are shown in Figure 8. A crosssection in the plane Px has the shape of a washer (an annular ring) with inner radius x 2 and outer radius x, so we find the crosssectional area by subtracting the area of the inner circle from the area of the outer circle: A共x兲 苷 x 2 ⫺ 共x 2 兲2 苷 共x 2 ⫺ x 4 兲 Therefore we have 1
1
V 苷 y A共x兲 dx 苷 y 共x 2 ⫺ x 4 兲 dx 0
冋
苷 y
x3 x5 ⫺ 3 5
册
0
1
苷
0
2 15
y (1, 1)
A(x)
y=x y=≈ ≈ x
(0, 0)
FIGURE 8
(a)
x
x
0
(c)
( b)
EXAMPLE 5 Find the volume of the solid obtained by rotating the region in Example 4 about the line y 苷 2. SOLUTION The solid and a crosssection are shown in Figure 9. Again the crosssection is
a washer, but this time the inner radius is 2 ⫺ x and the outer radius is 2 ⫺ x 2. y 4
y=2
y=2
2x 2≈ y=≈
y=x 0
FIGURE 9
x
≈
x
1
x
x
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.2
VOLUMES
435
The crosssectional area is A共x兲 苷 共2 ⫺ x 2 兲2 ⫺ 共2 ⫺ x兲2 and so the volume of S is 1
V 苷 y A共x兲 dx 0
1
苷 y 关共2 ⫺ x 2 兲2 ⫺ 共2 ⫺ x兲2 兴 dx 0
1
苷 y 共x 4 ⫺ 5x 2 ⫹ 4x兲 dx 0
冋
苷 苷
x5 x3 x2 ⫺5 ⫹4 5 3 2
册
1
0
8 15
The solids in Examples 1–5 are all called solids of revolution because they are obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using the basic defining formula b
V 苷 y A共x兲 dx a
or
d
V 苷 y A共y兲 dy c
and we find the crosssectional area A共x兲 or A共y兲 in one of the following ways: N
If the crosssection is a disk (as in Examples 1–3), we find the radius of the disk (in terms of x or y) and use A 苷 共radius兲2
N
If the crosssection is a washer (as in Examples 4 and 5), we find the inner radius r in and outer radius rout from a sketch (as in Figures 8, 9, and 10) and compute the area of the washer by subtracting the area of the inner disk from the area of the outer disk: A 苷 共outer radius兲2 ⫺ 共inner radius兲2
rin rout
FIGURE 10
The next example gives a further illustration of the procedure.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 6
APPLICATIONS OF INTEGRATION
EXAMPLE 6 Find the volume of the solid obtained by rotating the region in Example 4 about the line x 苷 ⫺1. SOLUTION Figure 11 shows a horizontal crosssection. It is a washer with inner radius
1 ⫹ y and outer radius 1 ⫹ sy , so the crosssectional area is A共y兲 苷 共outer radius兲2 ⫺ 共inner radius兲2 苷 (1 ⫹ sy ) 2 ⫺ 共1 ⫹ y兲2 The volume is 1
V 苷 y A共y兲 dy 苷 y 0
1
0
[(1 ⫹ sy )
1
2
苷 y (2sy ⫺ y ⫺ y 2 ) dy 苷 0
]
⫺ 共1 ⫹ y兲2 dy
冋
4y 3兾2 y2 y3 ⫺ ⫺ 3 2 3
册
1
苷
0
2
y
1+œ„y 1+y 1 x=œ„y y x=y x
0
x=_1
FIGURE 11
We now find the volumes of two solids that are not solids of revolution. EXAMPLE 7 Figure 12 shows a solid with a circular base of radius 1. Parallel crosssections perpendicular to the base are equilateral triangles. Find the volume of the solid.
TEC Visual 6.2C shows how the solid in Figure 12 is generated.
SOLUTION Let’s take the circle to be x 2 ⫹ y 2 苷 1. The solid, its base, and a typical cross
section at a distance x from the origin are shown in Figure 13. y
y
B(x, y)
≈ y=œ„„„„„„
C
C
y B y
_1
0
0 1
x
A (a) The solid
FIGURE 12
Computergenerated picture of the solid in Example 7
œ3y œ„
x
A
x
x
(b) Its base
A
60° y
60° y
B
(c) A crosssection
FIGURE 13
Since B lies on the circle, we have y 苷 s1 ⫺ x 2 and so the base of the triangle ABC is AB 苷 2s1 ⫺ x 2 . Since the triangle is equilateral, we see from Figure 13(c) that its
ⱍ ⱍ
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.2
VOLUMES
437
height is s3 y 苷 s3 s1 ⫺ x 2 . The crosssectional area is therefore A共x兲 苷 12 ⴢ 2s1 ⫺ x 2 ⴢ s3 s1 ⫺ x 2 苷 s3 共1 ⫺ x 2 兲 and the volume of the solid is 1
1
⫺1
⫺1
V 苷 y A共x兲 dx 苷 y s3 共1 ⫺ x 2 兲 dx
冋 册
1
苷 2 y s3 共1 ⫺ x 2 兲 dx 苷 2s3 x ⫺ 0
x3 3
1
苷 0
4s3 3
v EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h. SOLUTION We place the origin O at the vertex of the pyramid and the xaxis along its cen
tral axis as in Figure 14. Any plane Px that passes through x and is perpendicular to the xaxis intersects the pyramid in a square with side of length s, say. We can express s in terms of x by observing from the similar triangles in Figure 15 that x s兾2 s 苷 苷 h L兾2 L and so s 苷 Lx兾h. [Another method is to observe that the line OP has slope L兾共2h兲 and so its equation is y 苷 Lx兾共2h兲.] Thus the crosssectional area is A共x兲 苷 s 2 苷
L2 2 x h2
y
y
x
P
h
O
O
x
s
L
x
x
h
FIGURE 14
FIGURE 15
The pyramid lies between x 苷 0 and x 苷 h, so its volume is y h
V 苷 y A共x兲 dx 苷 y
h
0
FIGURE 16
0
L2 2 L2 x 3 x dx 苷 h2 h2 3
册
h
0
苷
L2 h 3
NOTE We didn’t need to place the vertex of the pyramid at the origin in Example 8. We did so merely to make the equations simple. If, instead, we had placed the center of the base at the origin and the vertex on the positive yaxis, as in Figure 16, you can verify that we would have obtained the integral
y
0
h
x
V苷y
h
0
L2 L2h 共h ⫺ y兲2 dy 苷 2 h 3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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438
CHAPTER 6
APPLICATIONS OF INTEGRATION
EXAMPLE 9 A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of 30⬚ along a diameter of the cylinder. Find the volume of the wedge. SOLUTION If we place the xaxis along the diameter where the planes meet, then the
base of the solid is a semicircle with equation y 苷 s16 ⫺ x 2 , ⫺4 艋 x 艋 4. A crosssection perpendicular to the xaxis at a distance x from the origin is a triangle ABC , as shown in Figure 17, whose base is y 苷 s16 ⫺ x 2 and whose height is BC 苷 y tan 30⬚ 苷 s16 ⫺ x 2 兾s3 . Thus the crosssectional area is
C 0
ⱍ ⱍ
y
A
y=œ„„„„„„ 16 ≈
B
4
1 16 ⫺ x 2 s16 ⫺ x 2 苷 2s3 s3
A共x兲 苷 12 s16 ⫺ x 2 ⴢ
x
C
and the volume is 4
4
⫺4
⫺4
V 苷 y A共x兲 dx 苷 y 30° A
y
1 苷 s3
B
FIGURE 17
苷
y
4
0
16 ⫺ x 2 dx 2s3
冋
1 x3 共16 ⫺ x 兲 dx 苷 16x ⫺ 3 s3 2
册
4
0
128 3s3
For another method see Exercise 62.
Exercises
6.2
1–18 Find the volume of the solid obtained by rotating the region
14. y 苷 sin x, y 苷 cos x, 0 艋 x 艋 兾4;
bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
15. y 苷 x , y 苷 0, x 苷 1;
1. y 苷 2 ⫺ 2 x, y 苷 0, x 苷 1, x 苷 2; 1
about the xaxis
16. xy 苷 1, y 苷 0, x 苷 1, x 苷 2; 17. x 苷 y , x 苷 1 ⫺ y ; 2
2. y 苷 1 ⫺ x 2, y 苷 0;
about the xaxis
3. y 苷 sx ⫺ 1 , y 苷 0, x 苷 5;
about the xaxis
4. y 苷 s25 ⫺ x 2 , y 苷 0, x 苷 2, x 苷 4; 5. x 苷 2sy , x 苷 0, y 苷 9;
about the yaxis
6. y 苷 ln x, y 苷 1, y 苷 2, x 苷 0; 7. y 苷 x 3, y 苷 x, x 艌 0; 8. y 苷 4 x 2, y 苷 5 ⫺ x 2; 1
9. y 2 苷 x, x 苷 2y ; 2
11. y 苷 x 2, x 苷 y 2;
13. y 苷 1 ⫹ sec x, y 苷 3;
;
about x 苷 1
19–30 Refer to the figure and find the volume generated by rotating the given region about the specified line. y
about the yaxis
B(1, 1)
C(0, 1)
about the xaxis
T™
about the xaxis
y=œ„ $x T£
T¡
about the yaxis O
about y 苷 1
12. y 苷 e ⫺x, y 苷 1, x 苷 2;
about x 苷 ⫺1
about x 苷 3
18. y 苷 x, y 苷 0, x 苷 2, x 苷 4;
about the yaxis
10. y 苷 x , x 苷 2, y 苷 0; 1 4
about the xaxis
2
about y 苷 ⫺1
about x 苷 2
3
about y 苷 2 about y 苷 1
Graphing calculator or computer required
A(1, 0)
x
19. 1 about OA
20. 1 about OC
21. 1 about AB
22. 1 about BC
CAS Computer algebra system required
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.2
23. 2 about OA
24. 2 about OC
25. 2 about AB
26. 2 about BC
27. 3 about OA
28. 3 about OC
29. 3 about AB
30. 3 about BC
sectional areas A (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with n 苷 5 to estimate the volume of the log.
rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. 2
31. y 苷 e⫺x , y 苷 0, x 苷 ⫺1, x 苷 1
(b) About y 苷 ⫺1
(a) About y 苷 2
(b) About x 苷 2
y 4
(b) About the yaxis
2
34. y 苷 x 2, x 2 ⫹ y 2 苷 1, y 艌 0
(a) About the xaxis
0
; 35–36 Use a graph to find approximate xcoordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the xaxis the region bounded by these curves. 35. y 苷 2 ⫹ x 2 cos x,
CAS
of the solid obtained by rotating the region bounded by the given curves about the specified line. 38. y 苷 x, y 苷 xe
0 1 2 3 4 5
0.68 0.65 0.64 0.61 0.58 0.59
6 7 8 9 10
0.53 0.55 0.52 0.50 0.48
2
4
6
10 x
8
46. (a) A model for the shape of a bird’s egg is obtained by
f 共x兲 苷 共ax 3 ⫹ bx 2 ⫹ cx ⫹ d兲s1 ⫺ x 2
37–38 Use a computer algebra system to find the exact volume
37. y 苷 sin2 x, y 苷 0, 0 艋 x 艋 ;
A (m2 )
rotating about the xaxis the region under the graph of
y 苷 e x兾2 ⫹ e⫺2x
36. y 苷 3 sin共x 2 兲,
x (m)
(b) Estimate the volume if the region is rotated about the yaxis. Again use the Midpoint Rule with n 苷 4. CAS
y 苷 x4 ⫹ x ⫹ 1
A (m2 )
xaxis to form a solid, use the Midpoint Rule with n 苷 4 to estimate the volume of the solid.
(b) About y 苷 1
33. x 2 ⫹ 4y 2 苷 4
x (m)
45. (a) If the region shown in the figure is rotated about the
32. y 苷 0, y 苷 cos x, ⫺兾2 艋 x 艋 兾2 2
(a) About the xaxis
439
44. A log 10 m long is cut at 1meter intervals and its cross
31–34 Set up an integral for the volume of the solid obtained by
(a) About the xaxis
VOLUMES
about y 苷 ⫺1
; about y 苷 3
Use a CAS to find the volume of such an egg. (b) For a Redthroated Loon, a 苷 ⫺0.06, b 苷 0.04, c 苷 0.1, and d 苷 0.54. Graph f and find the volume of an egg of this species. 47–59 Find the volume of the described solid S.
1⫺x兾2
47. A right circular cone with height h and base radius r
39– 42 Each integral represents the volume of a solid. Describe
48. A frustum of a right circular cone with height h, lower base
radius R, and top radius r
the solid.
39. y sin x dx 0
1
40. y 共1 ⫺ y 2 兲2 dy
r
⫺1
h
1
41. y 共 y 4 ⫺ y 8 兲 dy 0
42. y
兾2
0
关共1 ⫹ cos x兲2 ⫺ 12 兴 dx
R 49. A cap of a sphere with radius r and height h
43. A CAT scan produces equally spaced crosssectional views
of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows crosssections spaced 1.5 cm apart. The liver is 15 cm long and the crosssectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.
h r
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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440
CHAPTER 6
APPLICATIONS OF INTEGRATION
50. A frustum of a pyramid with square base of side b, square top
of side a, and height h
61. (a) Set up an integral for the volume of a solid torus (the
donutshaped solid shown in the figure) with radii r and R. (b) By interpreting the integral as an area, find the volume of the torus.
a
R r
b
What happens if a 苷 b ? What happens if a 苷 0 ? 51. A pyramid with height h and rectangular base with dimensions
b and 2b
62. Solve Example 9 taking crosssections to be parallel to the line
of intersection of the two planes.
52. A pyramid with height h and base an equilateral triangle with
63. (a) Cavalieri’s Principle states that if a family of parallel planes
gives equal crosssectional areas for two solids S1 and S2 , then the volumes of S1 and S2 are equal. Prove this principle. (b) Use Cavalieri’s Principle to find the volume of the oblique cylinder shown in the figure.
side a (a tetrahedron)
a a
a h
53. A tetrahedron with three mutually perpendicular faces and
three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm 54. The base of S is a circular disk with radius r . Parallel cross
sections perpendicular to the base are squares.
r
64. Find the volume common to two circular cylinders, each with
radius r, if the axes of the cylinders intersect at right angles.
55. The base of S is an elliptical region with boundary curve
9x 2 ⫹ 4y 2 苷 36. Crosssections perpendicular to the xaxis are isosceles right triangles with hypotenuse in the base.
56. The base of S is the triangular region with vertices 共0, 0兲,
共1, 0兲, and 共0, 1兲. Crosssections perpendicular to the yaxis are equilateral triangles.
57. The base of S is the same base as in Exercise 56, but cross
sections perpendicular to the xaxis are squares. 58. The base of S is the region enclosed by the parabola
y 苷 1 ⫺ x and the xaxis. Crosssections perpendicular to the yaxis are squares. 2
59. The base of S is the same base as in Exercise 58, but cross
sections perpendicular to the xaxis are isosceles triangles with height equal to the base. 60. The base of S is a circular disk with radius r . Parallel cross
sections perpendicular to the base are isosceles triangles with height h and unequal side in the base. (a) Set up an integral for the volume of S. (b) By interpreting the integral as an area, find the volume of S.
65. Find the volume common to two spheres, each with radius r , if
the center of each sphere lies on the surface of the other sphere. 66. A bowl is shaped like a hemisphere with diameter 30 cm. A
heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl. 67. A hole of radius r is bored through the middle of a cylinder of
radius R ⬎ r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.
68. A hole of radius r is bored through the center of a sphere of
radius R ⬎ r. Find the volume of the remaining portion of the sphere.
69. Some of the pioneers of calculus, such as Kepler and Newton,
were inspired by the problem of finding the volumes of wine
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.3
441
VOLUMES BY CYLINDRICAL SHELLS
(b) Show that the volume enclosed by the barrel is
barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas. (a) A barrel with height h and maximum radius R is constructed by rotating about the xaxis the parabola y 苷 R ⫺ cx 2, ⫺h兾2 艋 x 艋 h兾2, where c is a positive constant. Show that the radius of each end of the barrel is r 苷 R ⫺ d, where d 苷 ch 2兾4.
V 苷 13 h (2R 2 ⫹ r 2 ⫺ 25 d 2 ) 70. Suppose that a region has area A and lies above the xaxis.
When is rotated about the xaxis, it sweeps out a solid with volume V1. When is rotated about the line y 苷 ⫺k (where k is a positive number), it sweeps out a solid with volume V2 . Express V2 in terms of V1, k, and A.
Volumes by Cylindrical Shells
6.3 y
y=2≈˛ 1
xL=?
xR=?
0
2
x
Some volume problems are very difficult to handle by the methods of the preceding section. For instance, let’s consider the problem of finding the volume of the solid obtained by rotating about the yaxis the region bounded by y 苷 2x 2 ⫺ x 3 and y 苷 0. (See Figure 1.) If we slice perpendicular to the yaxis, we get a washer. But to compute the inner radius and the outer radius of the washer, we’d have to solve the cubic equation y 苷 2x 2 ⫺ x 3 for x in terms of y; that’s not easy. Fortunately, there is a method, called the method of cylindrical shells, that is easier to use in such a case. Figure 2 shows a cylindrical shell with inner radius r1, outer radius r2 , and height h. Its volume V is calculated by subtracting the volume V1 of the inner cylinder from the volume V2 of the outer cylinder:
FIGURE 1
r
V 苷 V2 ⫺ V1 苷 r22 h ⫺ r12 h 苷 共r22 ⫺ r12 兲h
Îr
r¡
苷 共r2 ⫹ r1 兲共r2 ⫺ r1 兲h
r™
苷 2 h
r2 ⫹ r1 h共r2 ⫺ r1 兲 2
If we let ⌬r 苷 r2 ⫺ r1 (the thickness of the shell) and r 苷 12 共r2 ⫹ r1 兲 (the average radius of the shell), then this formula for the volume of a cylindrical shell becomes V 苷 2 rh ⌬r
1 FIGURE 2
and it can be remembered as V 苷 [circumference][height][thickness] Now let S be the solid obtained by rotating about the yaxis the region bounded by y 苷 f 共x兲 [where f 共x兲 艌 0], y 苷 0, x 苷 a, and x 苷 b, where b ⬎ a 艌 0. (See Figure 3.) y
y
y=ƒ
y=ƒ
0
a
b
x
0
a
b
FIGURE 3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
97909_06_ch06_p442451.qk_97909_06_ch06_p442451 9/21/10 3:57 PM Page 442
442
CHAPTER 6
APPLICATIONS OF INTEGRATION
We divide the interval 关a, b兴 into n subintervals 关x i⫺1, x i 兴 of equal width ⌬x and let x i be the midpoint of the ith subinterval. If the rectangle with base 关x i⫺1, x i 兴 and height f 共x i 兲 is rotated about the yaxis, then the result is a cylindrical shell with average radius x i , height f 共x i 兲, and thickness ⌬x (see Figure 4), so by Formula 1 its volume is Vi 苷 共2 x i 兲关 f 共x i 兲兴 ⌬x y
y
y
y=ƒ
0
a
b x i1 x–i
y=ƒ
b
x
y=ƒ
x
x
b
xi
FIGURE 4
Therefore an approximation to the volume V of S is given by the sum of the volumes of these shells: n
V⬇
n
兺
Vi 苷
i苷1
兺 2 x
f 共x i 兲 ⌬x
i
i苷1
This approximation appears to become better as n l ⬁. But, from the definition of an integral, we know that n
lim
兺 2 x
n l ⬁ i苷1
b
i
f 共 x i 兲 ⌬x 苷 y 2 x f 共x兲 dx a
Thus the following appears plausible: 2 The volume of the solid in Figure 3, obtained by rotating about the yaxis the region under the curve y 苷 f 共x兲 from a to b, is b
V 苷 y 2 x f 共x兲 dx a
where 0 艋 a ⬍ b
The argument using cylindrical shells makes Formula 2 seem reasonable, but later we will be able to prove it (see Exercise 71 in Section 7.1). The best way to remember Formula 2 is to think of a typical shell, cut and flattened as in Figure 5, with radius x, circumference 2 x, height f 共x兲, and thickness ⌬x or dx:
y
b
共2 x兲
关 f 共x兲兴
dx
circumference
height
thickness
a
y
ƒ
ƒ x
x
2πx
Îx
FIGURE 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.3
VOLUMES BY CYLINDRICAL SHELLS
443
This type of reasoning will be helpful in other situations, such as when we rotate about lines other than the yaxis. EXAMPLE 1 Find the volume of the solid obtained by rotating about the yaxis the region bounded by y 苷 2x 2 x 3 and y 苷 0.
y
SOLUTION From the sketch in Figure 6 we see that a typical shell has radius x, circumfer
ence 2 x, and height f 共x兲 苷 2x 2 x 3. So, by the shell method, the volume is 2
2≈˛
2
V 苷 y 共2 x兲共2x 2 x 3 兲 dx 苷 2 y 共2x 3 x 4 兲 dx 0
0
2 x
x
苷 2
[
1 2
x 4 15 x 5
]
2 0
苷 2 (8 325 ) 苷 165
It can be verified that the shell method gives the same answer as slicing.
FIGURE 6
y
Figure 7 shows a computergenerated picture of the solid whose volume we computed in Example 1.
x
FIGURE 7 NOTE Comparing the solution of Example 1 with the remarks at the beginning of this section, we see that the method of cylindrical shells is much easier than the washer method for this problem. We did not have to find the coordinates of the local maximum and we did not have to solve the equation of the curve for x in terms of y. However, in other examples the methods of the preceding section may be easier.
v EXAMPLE 2 Find the volume of the solid obtained by rotating about the yaxis the region between y 苷 x and y 苷 x 2.
y
y=x y=≈
SOLUTION The region and a typical shell are shown in Figure 8. We see that the shell has shell radius x, circumference 2 x, and height x x 2. So the volume is height=x≈ 0
FIGURE 8
x
x
1
1
V 苷 y 共2 x兲共x x 2 兲 dx 苷 2 y 共x 2 x 3 兲 dx 0
苷 2
0
冋
x3 x4 3 4
册
1
0
苷
6
As the following example shows, the shell method works just as well if we rotate about the xaxis. We simply have to draw a diagram to identify the radius and height of a shell.
v EXAMPLE 3 Use cylindrical shells to find the volume of the solid obtained by rotating about the xaxis the region under the curve y 苷 sx from 0 to 1.
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97909_06_ch06_p442451.qk_97909_06_ch06_p442451 9/21/10 3:57 PM Page 444
444
CHAPTER 6
y
APPLICATIONS OF INTEGRATION
SOLUTION This problem was solved using disks in Example 2 in Section 6.2. To use
shell height=1¥
shells we relabel the curve y 苷 sx (in the figure in that example) as x 苷 y 2 in Figure 9. For rotation about the xaxis we see that a typical shell has radius y, circumference 2 y, and height 1 ⫺ y 2. So the volume is
1 y
x= =¥
x=1
shell radius=y
0
1
1
V 苷 y 共2 y兲共1 ⫺ y 2 兲 dy 苷 2 y 共y ⫺ y 3 兲 dy 0
x
1
FIGURE 9
苷 2
冋
y2 y4 ⫺ 2 4
0
册
1
苷 0
2
In this problem the disk method was simpler.
v EXAMPLE 4 Find the volume of the solid obtained by rotating the region bounded by y 苷 x ⫺ x 2 and y 苷 0 about the line x 苷 2. SOLUTION Figure 10 shows the region and a cylindrical shell formed by rotation about the
line x 苷 2. It has radius 2 ⫺ x, circumference 2 共2 ⫺ x兲, and height x ⫺ x 2. y
y
x=2
y=x≈
x
0
0
1
x
FIGURE 10
2
3
4
x
2x
The volume of the given solid is 1
V 苷 y 2 共2 ⫺ x兲共x ⫺ x 2 兲 dx 0
1
苷 2 y 共x 3 ⫺ 3x 2 ⫹ 2x兲 dx 0
苷 2
6.3
冋
册
x4 ⫺ x3 ⫹ x2 4
1
苷 0
2
Exercises
1. Let S be the solid obtained by rotating the region shown in
the figure about the yaxis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What are its circumference and height? Use shells to find V.
2. Let S be the solid obtained by rotating the region shown in the
figure about the yaxis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Explain. y
y
y=sin{ ≈}
y=x(x1)@
0
;
Graphing calculator or computer required
1
x
CAS Computer algebra system required
0
π œ„
x
1. Homework Hints available at stewartcalculus.com
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97909_06_ch06_p442451.qk_97909_06_ch06_p442451 9/21/10 3:58 PM Page 445
SECTION 6.3
26. x 2 y 2 苷 7, x 苷 4;
3–7 Use the method of cylindrical shells to find the volume gener
ated by rotating the region bounded by the given curves about the yaxis. 3 3. y 苷 s x,
4. y 苷 x 3,
y 苷 0, y 苷 0,
2
5. y 苷 ex ,
y 苷 0,
6. y 苷 4x x 2, 7. y 苷 x 2,
28. If the region shown in the figure is rotated about the yaxis to
x苷1
form a solid, use the Midpoint Rule with n 苷 5 to estimate the volume of the solid.
y苷x
y 苷 6x 2x 2
y
8. Let V be the volume of the solid obtained by rotating about the
4
yaxis the region bounded by y 苷 sx and y 苷 x 2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.
2
9–14 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the xaxis. 9. xy 苷 1,
0
x 苷 0, y 苷 1, y 苷 3
10. y 苷 sx , 11. y 苷 x 3,
y 苷 8,
12. x 苷 4y y , 3
14. x y 苷 3,
x苷0 x苷2
29.
y
31.
y
32.
y
x 苷 4 共 y 1兲
15. y 苷 x 4, y 苷 0, x 苷 1;
10 x
8
3
0 1
0
2 x 5 dx
30. 2 y
2
0
y dy 1 y2
2 共3 y兲共1 y 2 兲 dy
intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the yaxis the region enclosed by these curves.
about x 苷 1
y 苷 sx 1
33. y 苷 e x,
about x 苷 1
34. y 苷 x x 1,
19. y 苷 x 3, y 苷 0, x 苷 1; 20. x 苷 y 1, x 苷 2;
2 共 x兲共cos x sin x兲 dx
about x 苷 1
17. y 苷 4x x , y 苷 3; 2
18. y 苷 x 2, y 苷 2 x 2;
兾4
0
; 33–34 Use a graph to estimate the xcoordinates of the points of
about x 苷 2
16. y 苷 sx , y 苷 0, x 苷 1;
3
CAS
about y 苷 2
about the yaxis
22. y 苷 tan x, y 苷 0, x 苷 兾4;
about x 苷 兾2
23. y 苷 cos x, y 苷 cos x, 兾2 x 兾2; 4
24. y 苷 x, y 苷 2x兾共1 x 兲; 3
about x 苷
about x 苷 1
25. x 苷 ssin y , 0 y , x 苷 0;
35–36 Use a computer algebra system to find the exact volume of
the solid obtained by rotating the region bounded by the given curves about the specified line.
(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. 21. y 苷 xe x, y 苷 0, x 苷 2;
y 苷 x 4 4x 1
about y 苷 1
21–26
4
6
2
15–20 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
2
4
solid.
x苷0
13. x 苷 1 共 y 2兲2,
2
29–32 Each integral represents the volume of a solid. Describe the
x 苷 0, y 苷 2
2
about y 苷 5
obtained by rotating about the yaxis the region under the curve y 苷 s1 x 3 , 0 x 1.
x苷2
x 苷 0,
445
27. Use the Midpoint Rule with n 苷 5 to estimate the volume
x苷1 x 苷 1,
VOLUMES BY CYLINDRICAL SHELLS
about y 苷 4
35. y 苷 sin 2 x, y 苷 sin 4 x, 0 x ; 36. y 苷 x sin x, y 苷 0, 0 x ; 3
about x 苷 兾2 about x 苷 1
37– 43 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. 37. y 苷 x 2 6x 8, y 苷 0;
about the yaxis
38. y 苷 x 6x 8, y 苷 0;
about the xaxis
2
39. y x 苷 1, y 苷 2;
about the xaxis
40. y x 苷 1, y 苷 2;
about the yaxis
2 2
2 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 6
APPLICATIONS OF INTEGRATION
41. x 2 ⫹ 共 y ⫺ 1兲2 苷 1; 42. x 苷 共 y ⫺ 3兲 , x 苷 4; 2
48. Suppose you make napkin rings by drilling holes with different
about the yaxis about y 苷 1
43. x 苷 共 y ⫺ 1兲2, x ⫺ y 苷 1;
about x 苷 ⫺1
44. Let T be the triangular region with vertices 共0, 0兲, 共1, 0兲, and
共1, 2兲, and let V be the volume of the solid generated when T is rotated about the line x 苷 a, where a ⬎ 1. Express a in terms of V.
diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure. (a) Guess which ring has more wood in it. (b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius r through the center of a sphere of radius R and express the answer in terms of h.
45– 47 Use cylindrical shells to find the volume of the solid. 45. A sphere of radius r
h
46. The solid torus of Exercise 61 in Section 6.2 47. A right circular cone with height h and base radius r
6.4
Work The term work is used in everyday language to mean the total amount of effort required to perform a task. In physics it has a technical meaning that depends on the idea of a force. Intuitively, you can think of a force as describing a push or pull on an object—for example, a horizontal push of a book across a table or the downward pull of the earth’s gravity on a ball. In general, if an object moves along a straight line with position function s共t兲, then the force F on the object (in the same direction) is given by Newton’s Second Law of Motion as the product of its mass m and its acceleration: F苷m
1
d 2s dt 2
In the SI metric system, the mass is measured in kilograms (kg), the displacement in meters (m), the time in seconds (s), and the force in newtons ( N 苷 kg⭈m兾s2 ). Thus a force of 1 N acting on a mass of 1 kg produces an acceleration of 1 m兾s2. In the US Customary system the fundamental unit is chosen to be the unit of force, which is the pound. In the case of constant acceleration, the force F is also constant and the work done is defined to be the product of the force F and the distance d that the object moves: 2
W 苷 Fd
work 苷 force ⫻ distance
If F is measured in newtons and d in meters, then the unit for W is a newtonmeter, which is called a joule (J). If F is measured in pounds and d in feet, then the unit for W is a footpound (ftlb), which is about 1.36 J.
v
EXAMPLE 1
(a) How much work is done in lifting a 1.2kg book off the floor to put it on a desk that is 0.7 m high? Use the fact that the acceleration due to gravity is t 苷 9.8 m兾s2. (b) How much work is done in lifting a 20lb weight 6 ft off the ground? SOLUTION
(a) The force exerted is equal and opposite to that exerted by gravity, so Equation 1 gives F 苷 mt 苷 共1.2兲共9.8兲 苷 11.76 N Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.4
WORK
447
and then Equation 2 gives the work done as W 苷 Fd 苷 共11.76兲共0.7兲 ⬇ 8.2 J (b) Here the force is given as F 苷 20 lb, so the work done is W 苷 Fd 苷 20 ⴢ 6 苷 120 ftlb Notice that in part (b), unlike part (a), we did not have to multiply by t because we were given the weight (which is a force) and not the mass of the object. Equation 2 defines work as long as the force is constant, but what happens if the force is variable? Let’s suppose that the object moves along the xaxis in the positive direction, from x 苷 a to x 苷 b, and at each point x between a and b a force f 共x兲 acts on the object, where f is a continuous function. We divide the interval 关a, b兴 into n subintervals with endpoints x 0 , x 1, . . . , x n and equal width ⌬x. We choose a sample point x*i in the i th subinterval 关x i⫺1, x i 兴. Then the force at that point is f 共x*i 兲. If n is large, then ⌬x is small, and since f is continuous, the values of f don’t change very much over the interval 关x i⫺1, x i 兴. In other words, f is almost constant on the interval and so the work Wi that is done in moving the particle from x i⫺1 to x i is approximately given by Equation 2: Wi ⬇ f 共x*i 兲 ⌬x Thus we can approximate the total work by n
W⬇
3
兺 f 共x*兲 ⌬x i
i苷1
It seems that this approximation becomes better as we make n larger. Therefore we define the work done in moving the object from a to b as the limit of this quantity as n l ⬁. Since the right side of 3 is a Riemann sum, we recognize its limit as being a definite integral and so n
W 苷 lim
4
frictionless surface
x
0
兺 f 共x*兲 ⌬x 苷 y
n l ⬁ i苷1
i
b
a
f 共x兲 dx
EXAMPLE 2 When a particle is located a distance x feet from the origin, a force of x 2 ⫹ 2x pounds acts on it. How much work is done in moving it from x 苷 1 to x 苷 3?
(a) Natural position of spring SOLUTION
3
W 苷 y 共x 2 ⫹ 2x兲 dx 苷 1
册
x3 ⫹ x2 3
3
1
苷
50 3
ƒ=kx
The work done is 16 23 ftlb.
0
x
x
(b) Stretched position of spring FIGURE 1
Hooke’s Law
In the next example we use a law from physics: Hooke’s Law states that the force required to maintain a spring stretched x units beyond its natural length is proportional to x : f 共x兲 苷 kx where k is a positive constant (called the spring constant). Hooke’s Law holds provided that x is not too large (see Figure 1).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 6
APPLICATIONS OF INTEGRATION
v EXAMPLE 3 A force of 40 N is required to hold a spring that has been stretched from its natural length of 10 cm to a length of 15 cm. How much work is done in stretching the spring from 15 cm to 18 cm? SOLUTION According to Hooke’s Law, the force required to hold the spring stretched
x meters beyond its natural length is f 共x兲 苷 kx. When the spring is stretched from 10 cm to 15 cm, the amount stretched is 5 cm 苷 0.05 m. This means that f 共0.05兲 苷 40, so 0.05k 苷 40
40 k 苷 0.05 苷 800
Thus f 共x兲 苷 800x and the work done in stretching the spring from 15 cm to 18 cm is W苷y
0.08
0.05
x2 800x dx 苷 800 2
册
0.08
0.05
苷 400关共0.08兲 ⫺ 共0.05兲 兴 苷 1.56 J 2
2
v EXAMPLE 4 A 200lb cable is 100 ft long and hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building?
0
SOLUTION Here we don’t have a formula for the force function, but we can use an argux*i
Îx
100 x
ment similar to the one that led to Definition 4. Let’s place the origin at the top of the building and the x axis pointing downward as in Figure 2. We divide the cable into small parts with length ⌬x . If x*i is a point in the ith such interval, then all points in the interval are lifted by approximately the same amount, namely x*i . The cable weighs 2 pounds per foot, so the weight of the ith part is 2⌬x . Thus the work done on the ith part, in footpounds, is 共2⌬x兲 ⴢ x*i
FIGURE 2
force
If we had placed the origin at the bottom of the cable and the xaxis upward, we would have gotten W苷y
100
0
2共100 ⫺ x兲 dx
which gives the same answer.
苷 2x*i ⌬x
distance
We get the total work done by adding all these approximations and letting the number of parts become large (so ⌬x l 0 ): n
W 苷 lim
兺 2x* ⌬x 苷 y
n l ⬁ i苷1
]
苷 x2
100 0
i
100
0
2x dx
苷 10,000 ftlb
EXAMPLE 5 A tank has the shape of an inverted circular cone with height 10 m and base radius 4 m. It is filled with water to a height of 8 m. Find the work required to empty the tank by pumping all of the water to the top of the tank. (The density of water is 1000 kg兾m3.) SOLUTION Let’s measure depths from the top of the tank by introducing a vertical coordi
nate line as in Figure 3. The water extends from a depth of 2 m to a depth of 10 m and so we divide the interval 关2, 10兴 into n subintervals with endpoints x 0 , x 1, . . . , x n and choose x*i in the i th subinterval. This divides the water into n layers. The ith layer is approximated by a circular cylinder with radius ri and height ⌬x. We can compute ri from similar triangles, using Figure 4, as follows: ri 4 苷 10 ⫺ x*i 10
ri 苷 25 共10 ⫺ x*i 兲
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.4
449
WORK
Thus an approximation to the volume of the ith layer of water is
4m 0
Vi ⬇ ri2 ⌬x 苷
2m xi*
4 共10 ⫺ x*i 兲2 ⌬x 25
and so its mass is 10 m
Îx
mi 苷 density ⫻ volume
ri
⬇ 1000 ⴢ x
4 共10 ⫺ x*i 兲2 ⌬x 苷 160 共10 ⫺ x*i 兲2 ⌬x 25
The force required to raise this layer must overcome the force of gravity and so
FIGURE 3
Fi 苷 mi t ⬇ 共9.8兲160 共10 ⫺ x*i 兲2 ⌬x 苷 1568 共10 ⫺ x*i 兲2 ⌬x
4
Each particle in the layer must travel a distance of approximately x*i . The work Wi done to raise this layer to the top is approximately the product of the force Fi and the distance x*i :
ri
10 10xi*
Wi ⬇ Fi x*i ⬇ 1568 x*i 共10 ⫺ x*i 兲2 ⌬x To find the total work done in emptying the entire tank, we add the contributions of each of the n layers and then take the limit as n l ⬁: n
FIGURE 4
W 苷 lim
兺 1568 x*共10 ⫺ x*兲 i
n l ⬁ i苷1
i
2
10
⌬x 苷 y 1568 x共10 ⫺ x兲2 dx 2
冋
10
苷 1568 y 共100x ⫺ 20x 2 ⫹ x 3 兲 dx 苷 1568 50x 2 ⫺ 2
苷 1568 (
6.4
2048 3
册
10
2
) ⬇ 3.4 ⫻ 10 6 J
Exercises
1. A 360lb gorilla climbs a tree to a height of 20 ft. Find the
How much work is done by the force in moving an object a distance of 8 m?
work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds
F (N)
2. How much work is done when a hoist lifts a 200kg rock to a
30 20 10
height of 3 m? 3. A variable force of 5x ⫺2 pounds moves an object along a
straight line when it is x feet from the origin. Calculate the work done in moving the object from x 苷 1 ft to x 苷 10 ft. 4. When a particle is located a distance x meters from the origin,
a force of cos共 x兾3兲 newtons acts on it. How much work is done in moving the particle from x 苷 1 to x 苷 2? Interpret your answer by considering the work done from x 苷 1 to x 苷 1.5 and from x 苷 1.5 to x 苷 2.
5. Shown is the graph of a force function (in newtons) that
increases to its maximum value and then remains constant.
;
20x 3 x4 ⫹ 3 4
Graphing calculator or computer required
0
1
2 3 4 5 6 7 8
x (m)
6. The table shows values of a force function f 共x兲, where x is
measured in meters and f 共x兲 in newtons. Use the Midpoint Rule to estimate the work done by the force in moving an object from x 苷 4 to x 苷 20.
x
4
6
8
10
12
14
16
18
20
f 共x兲
5
5.8
7.0
8.8
9.6
8.2
6.7
5.2
4.1
1. Homework Hints available at stewartcalculus.com
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CHAPTER 6
APPLICATIONS OF INTEGRATION
7. A force of 10 lb is required to hold a spring stretched 4 in.
18. A 10ft chain weighs 25 lb and hangs from a ceiling. Find the
beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length? 8. A spring has a natural length of 20 cm. If a 25N force is
required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 25 cm?
work done in lifting the lower end of the chain to the ceiling so that it’s level with the upper end. 19. An aquarium 2 m long, 1 m wide, and 1 m deep is full of
water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg兾m3.)
9. Suppose that 2 J of work is needed to stretch a spring from its
natural length of 30 cm to a length of 42 cm. (a) How much work is needed to stretch the spring from 35 cm to 40 cm? (b) How far beyond its natural length will a force of 30 N keep the spring stretched?
20. A circular swimming pool has a diameter of 24 ft, the sides are
5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5 lb兾ft 3.)
10. If the work required to stretch a spring 1 ft beyond its natural
length is 12 ftlb, how much work is needed to stretch it 9 in. beyond its natural length? 11. A spring has natural length 20 cm. Compare the work W1
done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related? 12. If 6 J of work is needed to stretch a spring from 10 cm to
21–24 A tank is full of water. Find the work required to pump the
water out of the spout. In Exercises 23 and 24 use the fact that water weighs 62.5 lb兾ft3. 21.
1m
2m 3m 3m
12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring? 13–20 Show how to approximate the required work by a Riemann
22.
3m
8m 23.
24.
6 ft
12 ft
sum. Then express the work as an integral and evaluate it. 13. A heavy rope, 50 ft long, weighs 0.5 lb兾ft and hangs over the
edge of a building 120 ft high. (a) How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling half the rope to the top of the building? 14. A chain lying on the ground is 10 m long and its mass is
80 kg. How much work is required to raise one end of the chain to a height of 6 m?
8 ft
6 ft
3 ft 10 ft frustum of a cone
; 25. Suppose that for the tank in Exercise 21 the pump breaks down after 4.7 ⫻ 10 5 J of work has been done. What is the depth of the water remaining in the tank?
26. Solve Exercise 22 if the tank is half full of oil that has a 15. A cable that weighs 2 lb兾ft is used to lift 800 lb of coal up a
mine shaft 500 ft deep. Find the work done. 16. A bucket that weighs 4 lb and a rope of negligible weight are
used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft兾s, but water leaks out of a hole in the bucket at a rate of 0.2 lb兾s. Find the work done in pulling the bucket to the top of the well.
density of 900 kg兾m3. 27. When gas expands in a cylinder with radius r , the pressure at
any given time is a function of the volume: P 苷 P共V 兲. The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F 苷 r 2P. Show that the work done by the gas when the volume expands from volume V1 to volume V2 is V2
W 苷 y P dV V1
17. A leaky 10kg bucket is lifted from the ground to a height of
12 m at a constant speed with a rope that weighs 0.8 kg兾m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12m level. How much work is done?
piston head
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.5
28. In a steam engine the pressure P and volume V of steam satisfy
the equation PV 1.4 苷 k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use Exercise 27 to calculate the work done by the engine during a cycle when the steam starts at a pressure of 160 lb兾in2 and a volume of 100 in3 and expands to a volume of 800 in3.
29. (a) Newton’s Law of Gravitation states that two bodies with
masses m1 and m2 attract each other with a force F苷G
AVERAGE VALUE OF A FUNCTION
451
base is a square with side length 756 ft and its height when built was 481 ft. (It was the tallest manmade structure in the world for more than 3800 years.) The density of the limestone is about 150 lb兾ft 3. (a) Estimate the total work done in building the pyramid. (b) If each laborer worked 10 hours a day for 20 years, for 340 days a year, and did 200 ftlb兾h of work in lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?
m1 m2 r2
where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r 苷 a to r 苷 b. (b) Compute the work required to launch a 1000kg satellite vertically to a height of 1000 km. You may assume that the earth’s mass is 5.98 ⫻ 10 24 kg and is concentrated at its center. Take the radius of the earth to be 6.37 ⫻ 10 6 m and G 苷 6.67 ⫻ 10 ⫺11 N⭈m2兾 kg 2. 30. The Great Pyramid of King Khufu was built of limestone in
Egypt over a 20year time period from 2580 BC to 2560 BC. Its
6.5
© Vladimir Korostyshevskiy / Shutterstock
Average Value of a Function It is easy to calculate the average value of finitely many numbers y1 , y2 , . . . , yn : yave 苷
T 15 10 5
Tave
6 0
FIGURE 1
12
18
24
t
y1 ⫹ y2 ⫹ ⭈ ⭈ ⭈ ⫹ yn n
But how do we compute the average temperature during a day if infinitely many temperature readings are possible? Figure 1 shows the graph of a temperature function T共t兲, where t is measured in hours and T in ⬚C, and a guess at the average temperature, Tave. In general, let’s try to compute the average value of a function y 苷 f 共x兲, a 艋 x 艋 b. We start by dividing the interval 关a, b兴 into n equal subintervals, each with length ⌬x 苷 共b ⫺ a兲兾n. Then we choose points x1*, . . . , x *n in successive subintervals and calculate the average of the numbers f 共x1*兲, . . . , f 共x *兲 n : f 共x1*兲 ⫹ ⭈ ⭈ ⭈ ⫹ f 共x *n 兲 n (For example, if f represents a temperature function and n 苷 24, this means that we take temperature readings every hour and then average them.) Since ⌬x 苷 共b ⫺ a兲兾n, we can write n 苷 共b ⫺ a兲兾⌬x and the average value becomes f 共x 1*兲 ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n*兲 1 苷 关 f 共x1*兲 ⌬x ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n*兲 ⌬x兴 b⫺a b⫺a ⌬x n 1 苷 f 共x i*兲 ⌬x 兺 b ⫺ a i苷1
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CHAPTER 6
APPLICATIONS OF INTEGRATION
If we let n increase, we would be computing the average value of a large number of closely spaced values. (For example, we would be averaging temperature readings taken every minute or even every second.) The limiting value is lim
nl⬁
n
1 b⫺a
1
兺 f 共x *兲 ⌬x 苷 b ⫺ a y i
b
a
i苷1
f 共x兲 dx
by the definition of a definite integral. Therefore we define the average value of f on the interval 关a, b兴 as For a positive function, we can think of this definition as saying
fave 苷
area 苷 average height width
v
1 b⫺a
y
b
a
f 共x兲 dx
EXAMPLE 1 Find the average value of the function f 共x兲 苷 1 ⫹ x 2 on the
interval 关⫺1, 2兴.
SOLUTION With a 苷 ⫺1 and b 苷 2 we have
fave 苷
1 b⫺a
1 苷 3
y
b
a
f 共x兲 dx 苷
冋 册 x3 x⫹ 3
1 2 ⫺ 共⫺1兲
y
2
⫺1
共1 ⫹ x 2 兲 dx
2
苷2 ⫺1
If T共t兲 is the temperature at time t , we might wonder if there is a specific time when the temperature is the same as the average temperature. For the temperature function graphed in Figure 1, we see that there are two such times––just before noon and just before midnight. In general, is there a number c at which the value of a function f is exactly equal to the average value of the function, that is, f 共c兲 苷 fave ? The following theorem says that this is true for continuous functions. The Mean Value Theorem for Integrals If f is continuous on 关a, b兴, then there exists a
number c in 关a, b兴 such that
y
f 共c兲 苷 fave 苷
y=ƒ
y
that is,
b
a
1 b⫺a
y
b
a
f 共x兲 dx
f 共x兲 dx 苷 f 共c兲共b ⫺ a兲
f(c)=fave 0 a
c
b
x
FIGURE 2 You can always chop off the top of a (twodimensional) mountain at a certain height and use it to fill in the valleys so that the mountain becomes completely flat.
The Mean Value Theorem for Integrals is a consequence of the Mean Value Theorem for derivatives and the Fundamental Theorem of Calculus. The proof is outlined in Exercise 25. The geometric interpretation of the Mean Value Theorem for Integrals is that, for positive functions f, there is a number c such that the rectangle with base 关a, b兴 and height f 共c兲 has the same area as the region under the graph of f from a to b. (See Figure 2 and the more picturesque interpretation in the margin note.)
v
EXAMPLE 2 Since f 共x兲 苷 1 ⫹ x 2 is continuous on the interval 关⫺1, 2兴, the Mean
Value Theorem for Integrals says there is a number c in 关⫺1, 2兴 such that
y
2
⫺1
共1 ⫹ x 2 兲 dx 苷 f 共c兲关2 ⫺ 共⫺1兲兴
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 6.5 y
AVERAGE VALUE OF A FUNCTION
453
In this particular case we can find c explicitly. From Example 1 we know that fave 苷 2, so the value of c satisfies
(2, 5)
y=1+≈
f 共c兲 苷 fave 苷 2 1 ⫹ c2 苷 2
Therefore (_1, 2)
fave=2
c2 苷 1
so
So in this case there happen to be two numbers c 苷 ⫾1 in the interval 关⫺1, 2兴 that work in the Mean Value Theorem for Integrals. Examples 1 and 2 are illustrated by Figure 3.
0
_1
1
x
2
v EXAMPLE 3 Show that the average velocity of a car over a time interval 关t1, t2 兴 is the same as the average of its velocities during the trip.
FIGURE 3
SOLUTION If s共t兲 is the displacement of the car at time t , then, by definition, the average
velocity of the car over the interval is ⌬s s共t2 兲 ⫺ s共t1 兲 苷 ⌬t t2 ⫺ t1 On the other hand, the average value of the velocity function on the interval is vave 苷
6.5
1 t2 ⫺ t1
y
t2
t1
1 t2 ⫺ t1
v共t兲 dt 苷
y
t2
t1
s⬘共t兲 dt
苷
1 关s共t2 兲 ⫺ s共t1 兲兴 t2 ⫺ t1
苷
s共t2 兲 ⫺ s共t1 兲 苷 average velocity t2 ⫺ t1
(by the Net Change Theorem)
Exercises
1–8 Find the average value of the function on the given interval. 1. f 共x兲 苷 4x ⫺ x ,
关0, 4兴
2
2. f 共x兲 苷 sin 4 x, 3. t共x兲 苷 sx , 3
4. t共t兲 苷
10. f 共x兲 苷 1兾x,
关⫺, 兴
关1, 3兴
2 2 ; 12. f 共x兲 苷 2x兾共1 ⫹ x 兲 , 关0, 2兴
t , 关1, 3兴 s3 ⫹ t 2
5. f 共t兲 苷 e sin t cos t,
关0, 兾2兴
6. f 共 兲 苷 sec 共兾2兲,
关0, 兾2兴
7. h共x兲 苷 cos x sin x,
关0, 兴
8. h共u兲 苷 共3 ⫺ 2u兲⫺1,
关⫺1, 1兴
4
关2, 5兴
; 11. f 共x兲 苷 2 sin x ⫺ sin 2x, 关0, 兴
关1, 8兴
2
9. f 共x兲 苷 共x ⫺ 3兲2,
13. If f is continuous and x13 f 共x兲 dx 苷 8, show that f takes on the
value 4 at least once on the interval 关1, 3兴.
14. Find the numbers b such that the average value of
f 共x兲 苷 2 ⫹ 6x ⫺ 3x 2 on the interval 关0, b兴 is equal to 3.
15. Find the average value of f on 关0, 8兴. y
9–12
(a) Find the average value of f on the given interval. (b) Find c such that fave 苷 f 共c兲. (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f .
;
Graphing calculator or computer required
1 0
2
4
6
x
1. Homework Hints available at stewartcalculus.com
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454
CHAPTER 6
APPLICATIONS OF INTEGRATION
16. The velocity graph of an accelerating car is shown.
temperature of the coffee after t minutes is T共t兲 苷 20 ⫹ 75e⫺kt
√ (km/h) 60
where k ⬇ 0.02. (b) What is the average temperature of the coffee during the first half hour?
40 20
21. In Example 1 in Section 3.8 we modeled the world population
0
in the second half of the 20th century by the equation P共t兲 苷 2560e 0.017185t. Use this equation to estimate the average world population during this time period.
4
8
12 t (seconds)
(a) Use the Midpoint rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity? 17. In a certain city the temperature (in ⬚ F) t hours after 9 AM was
modeled by the function
t T共t兲 苷 50 ⫹ 14 sin 12 Find the average temperature during the period from 9 AM to 9 PM. 18. The velocity v of blood that flows in a blood vessel with radius
R and length l at a distance r from the central axis is
22. If a freely falling body starts from rest, then its displacement is given by s 苷 12 tt 2. Let the velocity after a time T be v T . Show
that if we compute the average of the velocities with respect to t we get vave 苷 12 v T , but if we compute the average of the velocities with respect to s we get vave 苷 23 v T . 23. Use the result of Exercise 83 in Section 5.5 to compute the
average volume of inhaled air in the lungs in one respiratory cycle. 24. Use the diagram to show that if f is concave upward on 关a, b兴,
then fave ⬎ f
冉 冊 a⫹b 2
y
f
P v共r兲 苷 共R 2 ⫺ r 2 兲 4 l where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 7 in Section 3.7). Find the average velocity (with respect to r ) over the interval 0 艋 r 艋 R. Compare the average velocity with the maximum velocity. 19. The linear density in a rod 8 m long is 12兾sx ⫹ 1 kg兾m,
where x is measured in meters from one end of the rod. Find the average density of the rod. 20. (a) A cup of coffee has temperature 95⬚ C and takes 30 minutes
to cool to 61⬚ C in a room with temperature 20⬚ C. Use Newton’s Law of Cooling (Section 3.8) to show that the
0
a
a+b 2
b
x
25. Prove the Mean Value Theorem for Integrals by applying the
Mean Value Theorem for derivatives (see Section 4.2) to the function F共x兲 苷 xax f 共t兲 dt. 26. If fave 关a, b兴 denotes the average value of f on the interval 关a, b兴
and a ⬍ c ⬍ b, show that fave 关a, b兴 苷
c⫺a b⫺c fave 关a, c兴 ⫹ fave 关c, b兴 b⫺a b⫺a
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APPLIED PROJECT
APPLIED PROJECT
CALCULUS AND BASEBALL
455
CALCULUS AND BASEBALL In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002). 1. It may surprise you to learn that the collision of baseball and bat lasts only about a thou
sandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum. The momentum p of an object is the product of its mass m and its velocity v, that is, p 苷 mv. Suppose an object, moving along a straight line, is acted on by a force F 苷 F共t兲 that is a continuous function of time. (a) Show that the change in momentum over a time interval 关t0 , t1 兴 is equal to the integral of F from t0 to t1; that is, show that t1
p共t1 兲 ⫺ p共t 0 兲 苷 y F共t兲 dt
Batter’s box
t0
This integral is called the impulse of the force over the time interval. (b) A pitcher throws a 90mi兾h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi兾h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m 苷 w兾t where t 苷 32 ft兾s 2. (i) Find the change in the ball’s momentum. (ii) Find the average force on the bat.
An overhead view of the position of a baseball bat, shown every fiftieth of a second during a typical swing. (Adapted from The Physics of Baseball)
2. In this problem we calculate the work required for a pitcher to throw a 90mi兾h fastball by
first considering kinetic energy. The kinetic energy K of an object of mass m and velocity v is given by K 苷 12 mv 2. Suppose an object of mass m, moving in a straight line, is acted on by a force F 苷 F共s兲 that depends on its position s. According to Newton’s Second Law F共s兲 苷 ma 苷 m
dv dt
where a and v denote the acceleration and velocity of the object. (a) Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that s1
W 苷 y F共s兲 ds 苷 12 mv12 ⫺ 12 mv 02 s0
where v0 苷 v共s0 兲 and v1 苷 v共s1 兲 are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule, m
dv dv ds dv 苷m 苷 mv dt ds dt ds
(b) How many footpounds of work does it take to throw a baseball at a speed of 90 mi兾h?
;
Graphing calculator or computer required
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456
CHAPTER 6
APPLICATIONS OF INTEGRATION
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft兾s. Assume that the velocity v共t兲 of the ball after t seconds satisfies the differential equation dv兾dt 苷 ⫺ 101 v because of air resist
;
APPLIED PROJECT
CAS
ance. How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.) (b) The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft兾s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft兾s? (c) For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?
WHERE TO SIT AT THE MOVIES
A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the seating area is inclined at an angle of ␣ 苷 20⬚ above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats, so 0 艋 x 艋 60. Suppose you decide that the best place to sit is in the row where the angle subtended by the screen at your eyes is a maximum. Let’s also suppose that your eyes are 4 ft above the floor, as shown in the figure. (In Exercise 74 in Section 4.7 we looked at a simpler version of this problem, where the floor is horizontal, but this project involves a more complicated situation and requires technology.) 25 ft
1. Show that ¨
冉
苷 arccos
4 ft
x
a 2 ⫹ b 2 ⫺ 625 2ab
冊
where
10 ft å
a 2 苷 共9 ⫹ x cos ␣兲2 ⫹ 共31 ⫺ x sin ␣兲2
9 ft
and b 2 苷 共9 ⫹ x cos ␣兲2 ⫹ 共x sin ␣ ⫺ 6兲2 2. Use a graph of as a function of x to estimate the value of x that maximizes . In which
row should you sit? What is the viewing angle in this row?
3. Use your computer algebra system to differentiate and find a numerical value for the root
of the equation d兾dx 苷 0. Does this value confirm your result in Problem 2?
4. Use the graph of to estimate the average value of on the interval 0 艋 x 艋 60. Then use
your CAS to compute the average value. Compare with the maximum and minimum values of .
CAS Computer algebra system required
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CHAPTER 6
6
REVIEW
457
Review
Concept Check 1. (a) Draw two typical curves y 苷 f 共x兲 and y 苷 t共x兲, where
f 共x兲 艌 t共x兲 for a 艋 x 艋 b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area. (b) Explain how the situation changes if the curves have equations x 苷 f 共 y兲 and x 苷 t共 y兲, where f 共 y兲 艌 t共 y兲 for c 艋 y 艋 d.
2. Suppose that Sue runs faster than Kathy throughout a
(b) If S is a solid of revolution, how do you find the crosssectional areas? 4. (a) What is the volume of a cylindrical shell?
(b) Explain how to use cylindrical shells to find the volume of a solid of revolution. (c) Why might you want to use the shell method instead of slicing? 5. Suppose that you push a book across a 6meterlong table
by exerting a force f 共x兲 at each point from x 苷 0 to x 苷 6. What does x06 f 共x兲 dx represent? If f 共x兲 is measured in newtons, what are the units for the integral?
1500meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race? 3. (a) Suppose S is a solid with known crosssectional areas.
6. (a) What is the average value of a function f on an
interval 关a, b兴 ? (b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation?
Explain how to approximate the volume of S by a Riemann sum. Then write an expression for the exact volume.
Exercises 1–6 Find the area of the region bounded by the given curves.
y 苷 4x ⫺ x
1. y 苷 x , 2
2. y 苷 1兾x,
y 苷 x 2,
6. y 苷 sx ,
y 苷 0,
x苷e
15. Find the volumes of the solids obtained by rotating the region
bounded by the curves y 苷 x and y 苷 x 2 about the following lines. (a) The xaxis (b) The yaxis (c) y 苷 2
ⱍ ⱍ
x 苷 y ⫹ 3y 2
5. y 苷 sin共 x兾2兲,
16. Let be the region in the first quadrant bounded by the
y 苷 x 2 ⫺ 2x
y 苷 x 2,
7. y 苷 2x, y 苷 x ;
curves y 苷 x 3 and y 苷 2x ⫺ x 2. Calculate the following quantities. (a) The area of (b) The volume obtained by rotating about the xaxis (c) The volume obtained by rotating about the yaxis
x苷2
7–11 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 2
17. Let be the region bounded by the curves y 苷 tan共x 2 兲,
x 苷 1, and y 苷 0. Use the Midpoint Rule with n 苷 4 to estimate the following quantities. (a) The area of (b) The volume obtained by rotating about the xaxis
about the xaxis
8. x 苷 1 ⫹ y , y 苷 x ⫺ 3; 2
9. x 苷 0, x 苷 9 ⫺ y 2;
about the yaxis
about x 苷 ⫺1
10. y 苷 x ⫹ 1, y 苷 9 ⫺ x 2; 2
about y 苷 ⫺1
2 ; 18. Let be the region bounded by the curves y 苷 1 ⫺ x and
y 苷 x 6 ⫺ x ⫹ 1. Estimate the following quantities. (a) The xcoordinates of the points of intersection of the curves (b) The area of (c) The volume generated when is rotated about the xaxis (d) The volume generated when is rotated about the yaxis
11. x 2 ⫺ y 2 苷 a 2, x 苷 a ⫹ h (where a ⬎ 0, h ⬎ 0);
about the yaxis 12–14 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 12. y 苷 tan x, y 苷 x, x 苷 兾3; 13. y 苷 cos x, 2
;
about y 苷 2
2
y苷 x
3. y 苷 1 ⫺ 2x , 2
4. x ⫹ y 苷 0,
14. y 苷 sx , y 苷 x 2;
ⱍ x ⱍ 艋 兾2,
about the yaxis
y苷 ; 1 4
about x 苷 兾2
19–22 Each integral represents the volume of a solid. Describe the solid. 19.
y
兾2
0
2 x cos x dx
20.
y
兾2
0
2 cos2x dx
Graphing calculator or computer required
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458 21.
CHAPTER 6
y
0
APPLICATIONS OF INTEGRATION
共2 ⫺ sin x兲2 dx
22.
y
4
0
29. A tank full of water has the shape of a paraboloid of revolution
2 共6 ⫺ y兲共4y ⫺ y 2 兲 dy
23. The base of a solid is a circular disk with radius 3. Find the
volume of the solid if parallel crosssections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
;
as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank. (b) After 4000 ftlb of work has been done, what is the depth of the water remaining in the tank? 4 ft
24. The base of a solid is the region bounded by the parabolas
y 苷 x and y 苷 2 ⫺ x . Find the volume of the solid if the crosssections perpendicular to the xaxis are squares with one side lying along the base. 2
2
4 ft
25. The height of a monument is 20 m. A horizontal crosssection
at a distance x meters from the top is an equilateral triangle with side 14 x meters. Find the volume of the monument. 26. (a) The base of a solid is a square with vertices located at
共1, 0兲, 共0, 1兲, 共⫺1, 0兲, and 共0, ⫺1兲. Each crosssection perpendicular to the xaxis is a semicircle. Find the volume of the solid. (b) Show that by cutting the solid of part (a), we can rearrange it to form a cone. Thus compute its volume more simply.
27. A force of 30 N is required to maintain a spring stretched from
its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm? 28. A 1600lb elevator is suspended by a 200ft cable that weighs
10 lb兾ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?
30. Find the average value of the function f 共t兲 苷 t sin共t 2 兲 on the
interval 关0, 10兴.
31. If f is a continuous function, what is the limit as h l 0 of the
average value of f on the interval 关x, x ⫹ h兴?
32. Let 1 be the region bounded by y 苷 x 2, y 苷 0, and x 苷 b,
where b ⬎ 0. Let 2 be the region bounded by y 苷 x 2, x 苷 0, and y 苷 b 2. (a) Is there a value of b such that 1 and 2 have the same area? (b) Is there a value of b such that 1 sweeps out the same volume when rotated about the xaxis and the yaxis? (c) Is there a value of b such that 1 and 2 sweep out the same volume when rotated about the xaxis? (d) Is there a value of b such that 1 and 2 sweep out the same volume when rotated about the yaxis?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Problems Plus 1. (a) Find a positive continuous function f such that the area under the graph of f from 0 to t
is A共t兲 苷 t 3 for all t ⬎ 0. (b) A solid is generated by rotating about the xaxis the region under the curve y 苷 f 共x兲, where f is a positive function and x 艌 0. The volume generated by the part of the curve from x 苷 0 to x 苷 b is b 2 for all b ⬎ 0. Find the function f .
2. There is a line through the origin that divides the region bounded by the parabola y 苷 x ⫺ x 2
and the xaxis into two regions with equal area. What is the slope of that line? y
3. The figure shows a horizontal line y 苷 c intersecting the curve y 苷 8x ⫺ 27x 3. Find the
y=8x27˛
number c such that the areas of the shaded regions are equal. y=c
4. A cylindrical glass of radius r and height L is filled with water and then tilted until the water
remaining in the glass exactly covers its base. (a) Determine a way to “slice” the water into parallel rectangular crosssections and then set up a definite integral for the volume of the water in the glass. (b) Determine a way to “slice” the water into parallel crosssections that are trapezoids and then set up a definite integral for the volume of the water. (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b). (d) Find the volume of the water in the glass from purely geometric considerations. (e) Suppose the glass is tilted until the water exactly covers half the base. In what direction can you “slice” the water into triangular crosssections? Rectangular crosssections? Crosssections that are segments of circles? Find the volume of water in the glass.
x
0
FIGURE FOR PROBLEM 3
L
r
L
r
5. (a) Show that the volume of a segment of height h of a sphere of radius r is 1 V 苷 3 h 2共3r ⫺ h兲
r
(See the figure.) (b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of one segment is twice the volume of the other, then x is a solution of the equation h
FIGURE FOR PROBLEM 5
3x 3 ⫺ 9x ⫹ 2 苷 0 where 0 ⬍ x ⬍ 1. Use Newton’s method to find x accurate to four decimal places. (c) Using the formula for the volume of a segment of a sphere, it can be shown that the depth x to which a floating sphere of radius r sinks in water is a root of the equation x 3 ⫺ 3rx 2 ⫹ 4r 3s 苷 0 where s is the specific gravity of the sphere. Suppose a wooden sphere of radius 0.5 m has specific gravity 0.75. Calculate, to fourdecimalplace accuracy, the depth to which the sphere will sink. CAS Computer algebra system required
459
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(d) A hemispherical bowl has radius 5 inches and water is running into the bowl at the rate of 0.2 in3兾s. (i) How fast is the water level in the bowl rising at the instant the water is 3 inches deep? (ii) At a certain instant, the water is 4 inches deep. How long will it take to fill the bowl? y=Lh y=0 L
h
6. Archimedes’ Principle states that the buoyant force on an object partially or fully submerged
in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density 0 floating partly submerged in a fluid of density f , the buoyant force is given by 0 A共 y兲 dy, where t is the acceleration due to gravity and A共 y兲 is the area of a F 苷 f t x⫺h typical crosssection of the object (see the figure). The weight of the object is given by
y=_h
W 苷 0t y
L⫺h
⫺h
A共 y兲 dy
FIGURE FOR PROBLEM 6
(a) Show that the percentage of the volume of the object above the surface of the liquid is 100
f ⫺ 0 f
(b) The density of ice is 917 kg兾m3 and the density of seawater is 1030 kg兾m3. What percentage of the volume of an iceberg is above water? (c) An ice cube floats in a glass filled to the brim with water. Does the water overflow when the ice melts? (d) A sphere of radius 0.4 m and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the water is 1000 kg兾m3. y
7. Water in an open bowl evaporates at a rate proportional to the area of the surface of the water.
y=2≈
(This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.
C y=≈ B
P
8. A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection
is a circle of radius r . (In other words, they intersect in a great circle of the small sphere.) Find r so that the volume inside the small sphere and outside the large sphere is as large as possible.
A
x
0
FIGURE FOR PROBLEM 9
9. The figure shows a curve C with the property that, for every point P on the middle curve
y 苷 2x 2, the areas A and B are equal. Find an equation for C .
10. A paper drinking cup filled with water has the shape of a cone with height h and semivertical
angle . (See the figure.) A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?
460
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11. A clepsydra, or water clock, is a glass container with a small hole in the bottom through
which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x 苷 f 共 y兲 be continuous on the interval 关0, b兴 and assume that the container is formed by rotating the graph of f about the yaxis. Let V denote the volume of water and h the height of the water level at time t. (a) Determine V as a function of h. (b) Show that dV dh 苷 关 f 共h兲兴 2 dt dt (c) Suppose that A is the area of the hole in the bottom of the container. It follows from Torricelli’s Law that the rate of change of the volume of the water is given by dV 苷 k A sh dt where k is a negative constant. Determine a formula for the function f such that dh兾dt is a constant C. What is the advantage in having dh兾dt 苷 C ? y b
x=f(y) h x
y
12. A cylindrical container of radius r and height L is partially filled with a liquid whose volume
is V. If the container is rotated about its axis of symmetry with constant angular speed , then the container will induce a rotational motion in the liquid around the same axis. Eventually, the liquid will be rotating at the same angular speed as the container. The surface of the liquid will be convex, as indicated in the figure, because the centrifugal force on the liquid particles increases with the distance from the axis of the container. It can be shown that the surface of the liquid is a paraboloid of revolution generated by rotating the parabola
v
L h r FIGURE FOR PROBLEM 12
x
y苷h⫹
2x 2 2t
about the yaxis, where t is the acceleration due to gravity. (a) Determine h as a function of . (b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top? (c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis. (i) Determine the angular speed of the container and the volume of the fluid. (ii) How far below the top of the tank is the liquid at the wall of the container? 13. Suppose the graph of a cubic polynomial intersects the parabola y 苷 x 2 when x 苷 0, x 苷 a,
and x 苷 b, where 0 ⬍ a ⬍ b. If the two regions between the curves have the same area, how is b related to a?
461
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CAS
14. Suppose we are planning to make a taco from a round tortilla with diameter 8 inches by
bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will fill the tortilla to the edge (but no more) with meat, cheese, and other ingredients. Our problem is to decide how to curve the tortilla in order to maximize the volume of food it can hold. (a) We start by placing a circular cylinder of radius r along a diameter of the tortilla and folding the tortilla around the cylinder. Let x represent the distance from the center of the tortilla to a point P on the diameter (see the figure). Show that the crosssectional area of the filled taco in the plane through P perpendicular to the axis of the cylinder is
冉
1 A共x兲 苷 r s16 ⫺ x 2 ⫺ 2 r 2 sin
2 s16 ⫺ x 2 r
冊
and write an expression for the volume of the filled taco. (b) Determine (approximately) the value of r that maximizes the volume of the taco. (Use a graphical approach with your CAS.)
x P 15. If the tangent at a point P on the curve y 苷 x 3 intersects the curve again at Q, let A be the
area of the region bounded by the curve and the line segment PQ. Let B be the area of the region defined in the same way starting with Q instead of P. What is the relationship between A and B ?
462
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Techniques of Integration
Shown is a photograph of Omega Centauri, which contains several million stars and is the largest globular cluster in our galaxy. Astronomers use stellar stereography to determine the actual density of stars in a star cluster from the (twodimensional) density that can be analyzed from a photograph. In Section 7.8 you are asked to evaluate an integral to calculate the perceived density from the actual density.
© 2010 Thomas V. Davis, www.tvdavisastropics.com
Because of the Fundamental Theorem of Calculus, we can integrate a function if we know an antiderivative, that is, an indefinite integral. We summarize here the most important integrals that we have learned so far. x n⫹1 ⫹C n⫹1
yx
n
dx 苷
ye
x
dx 苷 e x ⫹ C
共n 苷 ⫺1兲
ⱍ ⱍ
冉冊
1 1 x dx 苷 tan⫺1 ⫹ a2 a a
x
dx 苷
ax ⫹C ln a
y cos x dx 苷 sin x ⫹ C y csc x dx 苷 ⫺cot x ⫹ C y csc x cot x dx 苷 ⫺csc x ⫹ C y cosh x dx 苷 sinh x ⫹ C y cot x dx 苷 ln ⱍ sin x ⱍ ⫹ C
2
2
1 dx 苷 ln x ⫹ C x
ya
y sin x dx 苷 ⫺cos x ⫹ C y sec x dx 苷 tan x ⫹ C y sec x tan x dx 苷 sec x ⫹ C y sinh x dx 苷 cosh x ⫹ C y tan x dx 苷 ln ⱍ sec x ⱍ ⫹ C yx
y
2
⫹C
y
冉冊
x 1 dx 苷 sin⫺1 a sa 2 ⫺ x 2
⫹ C,
a⬎0
In this chapter we develop techniques for using these basic integration formulas to obtain indefinite integrals of more complicated functions. We learned the most important method of integration, the Substitution Rule, in Section 5.5. The other general technique, integration by parts, is presented in Section 7.1. Then we learn methods that are special to particular classes of functions, such as trigonometric functions and rational functions. Integration is not as straightforward as differentiation; there are no rules that absolutely guarantee obtaining an indefinite integral of a function. Therefore we discuss a strategy for integration in Section 7.5.
463 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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7.1
CHAPTER 7
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TECHNIQUES OF INTEGRATION
Integration by Parts Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration corresponds to the Chain Rule for differentiation. The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts. The Product Rule states that if f and t are differentiable functions, then d 关 f 共x兲t共x兲兴 苷 f 共x兲t⬘共x兲 ⫹ t共x兲 f ⬘共x兲 dx In the notation for indefinite integrals this equation becomes
y 关 f 共x兲t⬘共x兲 ⫹ t共x兲 f ⬘共x兲兴 dx 苷 f 共x兲t共x兲 y f 共x兲t⬘共x兲 dx ⫹ y t共x兲 f ⬘共x兲 dx 苷 f 共x兲t共x兲
or
We can rearrange this equation as
1
y f 共x兲t⬘共x兲 dx 苷 f 共x兲t共x兲 ⫺ y t共x兲 f ⬘共x兲 dx
Formula 1 is called the formula for integration by parts. It is perhaps easier to remember in the following notation. Let u 苷 f 共x兲 and v 苷 t共x兲. Then the differentials are du 苷 f ⬘共x兲 dx and dv 苷 t⬘共x兲 dx, so, by the Substitution Rule, the formula for integration by parts becomes
y u dv 苷 uv ⫺ y v du
2
EXAMPLE 1 Find
y x sin x dx.
SOLUTION USING FORMULA 1 Suppose we choose f 共x兲 苷 x and t⬘共x兲 苷 sin x. Then
f ⬘共x兲 苷 1 and t共x兲 苷 ⫺cos x. (For t we can choose any antiderivative of t⬘.) Thus, using Formula 1, we have
y x sin x dx 苷 f 共x兲t共x兲 ⫺ y t共x兲 f ⬘共x兲 dx 苷 x共⫺cos x兲 ⫺ y 共⫺cos x兲 dx 苷 ⫺x cos x ⫹ y cos x dx 苷 ⫺x cos x ⫹ sin x ⫹ C It’s wise to check the answer by differentiating it. If we do so, we get x sin x, as expected. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 7.1
INTEGRATION BY PARTS
465
SOLUTION USING FORMULA 2 Let It is helpful to use the pattern: u苷䊐 dv 苷 䊐 du 苷 䊐 v苷䊐
Then
u苷x
dv 苷 sin x dx
du 苷 dx
v 苷 ⫺cos x
and so
u
y x sin x dx 苷 y x
d√
u
√
√
du
sin x dx 苷 x 共⫺cos x兲 ⫺ y 共⫺cos x兲 dx
苷 ⫺x cos x ⫹ y cos x dx 苷 ⫺x cos x ⫹ sin x ⫹ C NOTE Our aim in using integration by parts is to obtain a simpler integral than the one we started with. Thus in Example 1 we started with x x sin x dx and expressed it in terms of the simpler integral x cos x dx. If we had instead chosen u 苷 sin x and dv 苷 x dx, then du 苷 cos x dx and v 苷 x 2兾2, so integration by parts gives
y x sin x dx 苷 共sin x兲
x2 1 ⫺ 2 2
yx
2
cos x dx
Although this is true, x x 2 cos x dx is a more difficult integral than the one we started with. In general, when deciding on a choice for u and dv, we usually try to choose u 苷 f 共x兲 to be a function that becomes simpler when differentiated (or at least not more complicated) as long as dv 苷 t⬘共x兲 dx can be readily integrated to give v.
v
EXAMPLE 2 Evaluate
y ln x dx.
SOLUTION Here we don’t have much choice for u and dv. Let
u 苷 ln x du 苷
Then
dv 苷 dx
1 dx x
v苷x
Integrating by parts, we get
y ln x dx 苷 x ln x ⫺ y x
dx x
It’s customary to write x 1 dx as x dx.
苷 x ln x ⫺ y dx
Check the answer by differentiating it.
苷 x ln x ⫺ x ⫹ C Integration by parts is effective in this example because the derivative of the function f 共x兲 苷 ln x is simpler than f .
v
EXAMPLE 3 Find
2 t
y t e dt.
SOLUTION Notice that t 2 becomes simpler when differentiated (whereas e t is unchanged
when differentiated or integrated), so we choose u 苷 t2 Then
du 苷 2t dt
dv 苷 e t dt v 苷 et
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
Integration by parts gives
y t e dt 苷 t e 2 t
3
2 t
⫺ 2 y te t dt
The integral that we obtained, x te t dt, is simpler than the original integral but is still not obvious. Therefore we use integration by parts a second time, this time with u 苷 t and dv 苷 e t dt. Then du 苷 dt, v 苷 e t, and
y te dt 苷 te t
t
⫺ y e t dt
苷 te t ⫺ e t ⫹ C Putting this in Equation 3, we get
yt
e dt 苷 t 2 e t ⫺ 2 y te t dt
2 t
苷 t 2 e t ⫺ 2共te t ⫺ e t ⫹ C兲 苷 t 2 e t ⫺ 2te t ⫹ 2e t ⫹ C1
v An easier method, using complex numbers, is given in Exercise 50 in Appendix H.
EXAMPLE 4 Evaluate
ye
x
where C1 苷 ⫺2C
sin x dx.
SOLUTION Neither e x nor sin x becomes simpler when differentiated, but we try choosing
u 苷 e x and dv 苷 sin x dx anyway. Then du 苷 e x dx and v 苷 ⫺cos x, so integration by parts gives
ye
4
x
sin x dx 苷 ⫺e x cos x ⫹ y e x cos x dx
The integral that we have obtained, x e x cos x dx, is no simpler than the original one, but at least it’s no more difficult. Having had success in the preceding example integrating by parts twice, we persevere and integrate by parts again. This time we use u 苷 e x and dv 苷 cos x dx. Then du 苷 e x dx, v 苷 sin x, and
ye
5
Figure 1 illustrates Example 4 by showing the graphs of f 共x兲 苷 e x sin x and F共x兲 苷 12 e x共sin x ⫺ cos x兲. As a visual check on our work, notice that f 共x兲 苷 0 when F has a maximum or minimum. 12
x
cos x dx 苷 e x sin x ⫺ y e x sin x dx
At first glance, it appears as if we have accomplished nothing because we have arrived at x e x sin x dx, which is where we started. However, if we put the expression for x e x cos x dx from Equation 5 into Equation 4 we get
ye
x
sin x dx 苷 ⫺e x cos x ⫹ e x sin x ⫺ y e x sin x dx
This can be regarded as an equation to be solved for the unknown integral. Adding x e x sin x dx to both sides, we obtain
F f
2 y e x sin x dx 苷 ⫺e x cos x ⫹ e x sin x 6
_3
Dividing by 2 and adding the constant of integration, we get _4
FIGURE 1
ye
x
sin x dx 苷 12 e x 共sin x ⫺ cos x兲 ⫹ C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 7.1
INTEGRATION BY PARTS
467
If we combine the formula for integration by parts with Part 2 of the Fundamental Theorem of Calculus, we can evaluate definite integrals by parts. Evaluating both sides of Formula 1 between a and b, assuming f ⬘ and t⬘ are continuous, and using the Fundamental Theorem, we obtain
y
6
b
a
EXAMPLE 5 Calculate
y
b
b
]
f 共x兲t⬘共x兲 dx 苷 f 共x兲t共x兲 a ⫺ y t共x兲 f ⬘共x兲 dx
1
0
a
tan⫺1x dx.
SOLUTION Let
u 苷 tan⫺1x du 苷
Then
dv 苷 dx
dx 1 ⫹ x2
v苷x
So Formula 6 gives
y
1
0
1
]
tan⫺1x dx 苷 x tan⫺1x 0 ⫺ y
x dx 1 ⫹ x2
1
0
苷 1 ⴢ tan⫺1 1 ⫺ 0 ⴢ tan⫺1 0 ⫺ y
1
0
苷
Since tan⫺1x 艌 0 for x 艌 0, the integral in Example 5 can be interpreted as the area of the region shown in Figure 2.
1 x ⫺y 2 dx 0 1 ⫹ x 4
To evaluate this integral we use the substitution t 苷 1 ⫹ x 2 (since u has another meaning in this example). Then dt 苷 2x dx, so x dx 苷 12 dt . When x 苷 0, t 苷 1; when x 苷 1, t 苷 2; so
y
y=tan–!x
0 1
x dx 1 ⫹ x2
y
x
1
0
x 1 2 dt 苷 12 ln t 2 dx 苷 2 y 1 t 1⫹x
ⱍ ⱍ]
2 1
苷 12 共ln 2 ⫺ ln 1兲 苷 12 ln 2
FIGURE 2
Therefore
y
1
0
tan⫺1x dx 苷
1 x ln 2 ⫺y ⫺ 2 dx 苷 0 1 ⫹ x 4 4 2
EXAMPLE 6 Prove the reduction formula Equation 7 is called a reduction formula because the exponent n has been reduced to n ⫺ 1 and n ⫺ 2.
7
1
y sin x dx 苷 ⫺ n cos x sin n
x⫹
n⫺1
n⫺1 n
y sin
n⫺2
x dx
where n 艌 2 is an integer. SOLUTION Let
u 苷 sin n⫺1x Then
du 苷 共n ⫺ 1兲 sin n⫺2x cos x dx
dv 苷 sin x dx v 苷 ⫺cos x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
so integration by parts gives
y sin x dx 苷 ⫺cos x sin n
x ⫹ 共n ⫺ 1兲 y sin n⫺2x cos 2x dx
n⫺1
Since cos 2x 苷 1 ⫺ sin 2x, we have
y sin x dx 苷 ⫺cos x sin n
x ⫹ 共n ⫺ 1兲 y sin n⫺2x dx ⫺ 共n ⫺ 1兲 y sin n x dx
n⫺1
As in Example 4, we solve this equation for the desired integral by taking the last term on the right side to the left side. Thus we have n y sin n x dx 苷 ⫺cos x sin n⫺1x ⫹ 共n ⫺ 1兲 y sin n⫺2x dx 1
y sin x dx 苷 ⫺ n cos x sin n
or
x⫹
n⫺1
n⫺1 n
y sin
n⫺2
x dx
The reduction formula 7 is useful because by using it repeatedly we could eventually express x sin n x dx in terms of x sin x dx ( if n is odd) or x 共sin x兲0 dx 苷 x dx ( if n is even).
Exercises
7.1
1–2 Evaluate the integral using integration by parts with the indicated choices of u and dv. 2
yx
2.
y cos d ;
y s2
y 共ln x兲 dx
16.
y t sinh mt dt
y x cos 5x dx ⫺3t
17.
y e sin 3 d
18.
ye
19.
yz e
20.
y x tan x dx
4.
y ye
21.
y
22.
y 共arcsin x兲
23.
y
24.
y
x dx
25.
y
t cosh t dt
26.
y
ln p dp
27.
r 3 ln r dr
28.
7.
y 共x
9.
y ln sx dx
10.
y sin
y arctan 4t dt
12.
yp
11.
2
⫹ 2x兲 cos x dx
3
;
6.
8.
yt
2
Graphing calculator or computer required
2
3 z
dz
⫺
ds
cos 2 d
2
dy
sin  t dt ⫺1
5
2t dt
2
xe 2x dx 共1 ⫹ 2x兲2
共x ⫺ 1兲 sin x dx
y
dt
y
0.2y
5.
te
15.
u 苷 , dv 苷 cos d
3–36 Evaluate the integral. 3.
y t sec
s
14.
ln x dx ; u 苷 ln x, dv 苷 x 2 dx
1.
2
13.
1兾2
0
1
0
y
3
1
x cos x dx
2
1
0
9
共x 2 ⫹ 1兲e⫺x dx ln y
4
y
2
0
dx
dy
sy t 2 sin 2t dt
1. Homework Hints available at stewartcalculus.com
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SECTION 7.1
29.
y
y dy e 2y
1
0
1兾2
30.
s3
1
31.
y
cos ⫺1x dx
32.
33.
y cos x ln共sin x兲 dx
34.
0
y y
1
y
r3
1
s4 ⫹ r 2
0
y
y
2
1
x 4共ln x兲2 dx
36.
y
t
0
兾2
0
sin 2n⫹1x dx 苷
2 ⴢ 4 ⴢ 6 ⴢ ⭈ ⭈ ⭈ ⴢ 2n 3 ⴢ 5 ⴢ 7 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫹ 1兲
50. Prove that, for even powers of sine,
dr
y 35.
469
(c) Use part (a) to show that, for odd powers of sine,
arctan共1兾x兲 dx
共ln x兲2 dx x3
2
INTEGRATION BY PARTS
兾2
0
e s sin共t ⫺ s兲 ds
sin 2nx dx 苷
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲 2 ⴢ 4 ⴢ 6 ⴢ ⭈ ⭈ ⭈ ⴢ 2n 2
51–54 Use integration by parts to prove the reduction formula. 37– 42 First make a substitution and then use integration by parts
51.
y 共ln x兲 dx 苷 x 共ln x兲
52.
yx e
53.
y tan x dx 苷
tan n⫺1 x ⫺ y tan n⫺2 x dx 共n 苷 1兲 n⫺1
54.
y sec x dx 苷
tan x sec n⫺2x n⫺2 ⫹ n⫺1 n⫺1
n
n
⫺ n y 共ln x兲n⫺1 dx
to evaluate the integral. 37.
y cos sx dx
39.
y
41.
y x ln共1 ⫹ x兲 dx
s
s 兾2
3 cos共 2 兲 d
3 ⫺t 2
38.
yt e
40.
y
42.
y sin共ln x兲 dx
0
dt
e cos t sin 2t dt
; 43– 46 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C 苷 0). 43.
y xe
⫺2x
dx
44.
yx
3兾2
n x
dx 苷 x ne x ⫺ n y x n⫺1e x dx
n
n
yx
3
s1 ⫹ x 2 dx
46.
yx
2
n⫺2
x dx
共n 苷 1兲
55. Use Exercise 51 to find x 共ln x兲3 dx. 56. Use Exercise 52 to find x x 4e x dx. 57–58 Find the area of the region bounded by the given curves.
ln x dx
57. y 苷 x 2 ln x, 45.
y sec
58. y 苷 x 2e ⫺x,
y 苷 4 ln x
y 苷 xe ⫺x
sin 2x dx
; 59–60 Use a graph to find approximate xcoordinates of the 47. (a) Use the reduction formula in Example 6 to show that
y sin x dx 苷 2
x sin 2x ⫺ ⫹C 2 4
59. y 苷 arcsin( 2 x), 1
(b) Use part (a) and the reduction formula to evaluate x sin 4x dx.
1 n⫺1 y cos x dx 苷 n cos n⫺1x sin x ⫹ n n
y cos
n⫺2
x dx
(b) Use part (a) to evaluate x cos 2x dx. (c) Use parts (a) and (b) to evaluate x cos 4x dx. 49. (a) Use the reduction formula in Example 6 to show that 兾2
0
sin n x dx 苷
n⫺1 n
60. y 苷 x ln共x ⫹ 1兲,
y 苷 2 ⫺ x2 y 苷 3x ⫺ x 2
61–63 Use the method of cylindrical shells to find the volume
48. (a) Prove the reduction formula
y
points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
y
兾2
0
sin n⫺2x dx
where n 艌 2 is an integer. (b) Use part (a) to evaluate x0兾2 sin 3x dx and x0兾2 sin 5x dx.
generated by rotating the region bounded by the given curves about the specified axis. 61. y 苷 cos共 x兾2兲, y 苷 0, 0 艋 x 艋 1; 62. y 苷 e x, y 苷 e⫺x, x 苷 1;
about the yaxis
about the yaxis
63. y 苷 e⫺x, y 苷 0, x 苷 ⫺1, x 苷 0;
about x 苷 1
64. Calculate the volume generated by rotating the region
bounded by the curves y 苷 ln x, y 苷 0, and x 苷 2 about each axis. (a) the yaxis (b) the xaxis
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
65. Calculate the average value of f 共x兲 苷 x sec2 x on the interval
Make the substitution y 苷 f 共x兲 and then use integration by parts on the resulting integral to prove that
66. A rocket accelerates by burning its onboard fuel, so its mass
V 苷 y 2 x f 共x兲 dx
关0, 兾4兴.
b
a
decreases with time. Suppose the initial mass of the rocket at liftoff ( including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ve (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation
y
m ⫺ rt m
v共t兲 苷 ⫺tt ⫺ ve ln
72. Let In 苷
0
is continuous. Find the value of x x f ⬙共x兲 dx.
70. (a) Use integration by parts to show that
and deduce that lim n l ⬁ I2n⫹1兾I2n 苷 1. (d) Use part (c) and Exercises 49 and 50 to show that lim
nl⬁
y f 共x兲 dx 苷 x f 共x兲 ⫺ y x f ⬘共x兲 dx (b) If f and t are inverse functions and f ⬘ is continuous, prove that
y
b
a
f 共x兲 dx 苷 bf 共b兲 ⫺ af 共a兲 ⫺ y
f 共b兲
f 共a兲
t共 y兲 dy
[Hint: Use part (a) and make the substitution y 苷 f 共x兲.] (c) In the case where f and t are positive functions and b ⬎ a ⬎ 0, draw a diagram to give a geometric interpretation of part (b). (d) Use part (b) to evaluate x1e ln x dx. 71. We arrived at Formula 6.3.2, V 苷
x0兾2 sin n x dx.
I2n⫹1 2n ⫹ 1 艋 艋1 2n ⫹ 2 I2n
a
4 1
x
(c) Use parts (a) and (b) to show that
f 共x兲 t ⬙共x兲 dx 苷 f 共a兲 t⬘共a兲 ⫺ f ⬘共a兲 t共a兲 ⫹ y f ⬙共x兲 t共x兲 dx
69. Suppose that f 共1兲 苷 2, f 共4兲 苷 7, f ⬘共1兲 苷 5, f ⬘共4兲 苷 3, and f ⬙
b
I2n⫹2 2n ⫹ 1 苷 I2n 2n ⫹ 2
it travel during the first t seconds? 68. If f 共0兲 苷 t共0兲 苷 0 and f ⬙ and t ⬙ are continuous, show that
a
(a) Show that I2n⫹2 艋 I2n⫹1 艋 I2n . (b) Use Exercise 50 to show that
67. A particle that moves along a straight line has velocity v共t兲 苷 t 2e⫺t meters per second after t seconds. How far will
0
x=b x=a
0
a
y=ƒ
c
where t is the acceleration due to gravity and t is not too large. If t 苷 9.8 m兾s 2, m 苷 30,000 kg, r 苷 160 kg兾s, and ve 苷 3000 m兾s, find the height of the rocket one minute after liftoff.
y
x=g(y)
d
2n 2n 2 2 4 4 6 6 ⴢ ⴢ ⴢ ⴢ ⴢ ⴢ ⭈⭈⭈ ⴢ ⴢ 苷 1 3 3 5 5 7 2n ⫺ 1 2n ⫹ 1 2
This formula is usually written as an infinite product:
2 2 4 4 6 6 苷 ⴢ ⴢ ⴢ ⴢ ⴢ ⴢ ⭈⭈⭈ 2 1 3 3 5 5 7 and is called the Wallis product. (e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.
xab 2 x f 共x兲 dx, by using
cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where f is onetoone and therefore has an inverse function t. Use the figure to show that d
V 苷 b 2d ⫺ a 2c ⫺ y 关 t共 y兲兴 2 dy c
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 7.2
7.2
TRIGONOMETRIC INTEGRALS
471
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. 3
y cos x dx.
EXAMPLE 1 Evaluate
SOLUTION Simply substituting u 苷 cos x isn’t helpful, since then du 苷 ⫺sin x dx. In
order to integrate powers of cosine, we would need an extra sin x factor. Similarly, a power of sine would require an extra cos x factor. Thus here we can separate one cosine factor and convert the remaining cos2x factor to an expression involving sine using the identity sin 2x ⫹ cos 2x 苷 1: cos 3x 苷 cos 2x ⴢ cos x 苷 共1 ⫺ sin 2x兲 cos x We can then evaluate the integral by substituting u 苷 sin x, so du 苷 cos x dx and
y cos x dx 苷 y cos x ⴢ cos x dx 苷 y 共1 ⫺ sin x兲 cos x dx 3
2
2
苷 y 共1 ⫺ u 2 兲 du 苷 u ⫺ 13 u 3 ⫹ C 苷 sin x ⫺ 13 sin 3x ⫹ C In general, we try to write an integrand involving powers of sine and cosine in a form where we have only one sine factor (and the remainder of the expression in terms of cosine) or only one cosine factor (and the remainder of the expression in terms of sine). The identity sin 2x ⫹ cos 2x 苷 1 enables us to convert back and forth between even powers of sine and cosine.
v
EXAMPLE 2 Find
5
2
y sin x cos x dx.
SOLUTION We could convert cos 2x to 1 ⫺ sin 2x, but we would be left with an expression
in terms of sin x with no extra cos x factor. Instead, we separate a single sine factor and rewrite the remaining sin 4x factor in terms of cos x : sin 5x cos 2x 苷 共sin2x兲2 cos 2x sin x 苷 共1 ⫺ cos 2x兲2 cos 2x sin x Figure 1 shows the graphs of the integrand sin 5x cos 2x in Example 2 and its indefinite integral (with C 苷 0). Which is which?
Substituting u 苷 cos x, we have du 苷 ⫺sin x dx and so
y sin x cos x dx 苷 y 共sin x兲 5
2
2
2
cos 2x sin x dx
0.2
苷 y 共1 ⫺ cos 2x兲2 cos 2x sin x dx _π
π
_ 0.2
FIGURE 1
苷 y 共1 ⫺ u 2 兲2 u 2 共⫺du兲 苷 ⫺y 共u 2 ⫺ 2u 4 ⫹ u 6 兲 du
冉
苷⫺
u3 u5 u7 ⫺2 ⫹ 3 5 7
冊
⫹C
苷 ⫺ 13 cos 3x ⫹ 25 cos 5x ⫺ 17 cos 7x ⫹ C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
In the preceding examples, an odd power of sine or cosine enabled us to separate a single factor and convert the remaining even power. If the integrand contains even powers of both sine and cosine, this strategy fails. In this case, we can take advantage of the following halfangle identities (see Equations 17b and 17a in Appendix D): sin 2x 苷 12 共1 ⫺ cos 2x兲
v Example 3 shows that the area of the region shown in Figure 2 is 兾2.
EXAMPLE 3 Evaluate
y
0
sin 2x dx.
SOLUTION If we write sin 2x 苷 1 ⫺ cos 2x, the integral is no simpler to evaluate. Using
the halfangle formula for sin 2x, however, we have
1.5
y
[email protected] x
0
sin 2x dx 苷 12 y 共1 ⫺ cos 2x兲 dx 0
苷 0 _0.5
FIGURE 2
cos 2x 苷 12 共1 ⫹ cos 2x兲
and
π
[ (x ⫺ 1 2
1 2
0
]
sin 2x)
苷 12 ( ⫺ 12 sin 2) ⫺ 12 (0 ⫺ 12 sin 0) 苷 12 Notice that we mentally made the substitution u 苷 2x when integrating cos 2x. Another method for evaluating this integral was given in Exercise 47 in Section 7.1. EXAMPLE 4 Find
4
y sin x dx.
SOLUTION We could evaluate this integral using the reduction formula for x sin n x dx
(Equation 7.1.7) together with Example 3 (as in Exercise 47 in Section 7.1), but a better method is to write sin 4x 苷 共sin 2x兲2 and use a halfangle formula:
y sin x dx 苷 y 共sin x兲 dx 4
2
苷
y
冉
2
1 ⫺ cos 2x 2
冊
2
dx
苷 14 y 共1 ⫺ 2 cos 2x ⫹ cos 2 2x兲 dx Since cos 2 2x occurs, we must use another halfangle formula cos 2 2x 苷 12 共1 ⫹ cos 4x兲 This gives
y sin x dx 苷 y 关1 ⫺ 2 cos 2x ⫹ 4
1 4
苷 14 y
1 2
共1 ⫹ cos 4x兲兴 dx
( 32 ⫺ 2 cos 2x ⫹ 12 cos 4x) dx
苷 14 ( 32 x ⫺ sin 2x ⫹ 18 sin 4x) ⫹ C To summarize, we list guidelines to follow when evaluating integrals of the form
x sin mx cos nx dx, where m 艌 0 and n 艌 0 are integers. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Page 473
SECTION 7.2
TRIGONOMETRIC INTEGRALS
473
Strategy for Evaluating y sin m x cos n x dx
(a) If the power of cosine is odd 共n 苷 2k ⫹ 1兲, save one cosine factor and use cos 2x 苷 1 ⫺ sin 2x to express the remaining factors in terms of sine:
y sin
m
x cos 2k⫹1x dx 苷 y sin m x 共cos 2x兲k cos x dx 苷 y sin m x 共1 ⫺ sin 2x兲k cos x dx
Then substitute u 苷 sin x. (b) If the power of sine is odd 共m 苷 2k ⫹ 1兲, save one sine factor and use sin 2x 苷 1 ⫺ cos 2x to express the remaining factors in terms of cosine:
y sin
2k⫹1
x cos n x dx 苷 y 共sin 2x兲k cos n x sin x dx 苷 y 共1 ⫺ cos 2x兲k cos n x sin x dx
Then substitute u 苷 cos x. [Note that if the powers of both sine and cosine are odd, either (a) or (b) can be used.] (c) If the powers of both sine and cosine are even, use the halfangle identities sin 2x 苷 12 共1 ⫺ cos 2x兲
cos 2x 苷 12 共1 ⫹ cos 2x兲
It is sometimes helpful to use the identity sin x cos x 苷 12 sin 2x We can use a similar strategy to evaluate integrals of the form x tan mx sec nx dx. Since 共d兾dx兲 tan x 苷 sec 2x, we can separate a sec 2x factor and convert the remaining (even) power of secant to an expression involving tangent using the identity sec 2x 苷 1 ⫹ tan 2x. Or, since 共d兾dx兲 sec x 苷 sec x tan x, we can separate a sec x tan x factor and convert the remaining (even) power of tangent to secant.
v
EXAMPLE 5 Evaluate
6
4
y tan x sec x dx.
SOLUTION If we separate one sec 2x factor, we can express the remaining sec 2x factor in
terms of tangent using the identity sec 2x 苷 1 ⫹ tan 2x. We can then evaluate the integral by substituting u 苷 tan x so that du 苷 sec 2x dx :
y tan x sec x dx 苷 y tan x sec x sec x dx 6
4
6
2
2
苷 y tan 6x 共1 ⫹ tan 2x兲 sec 2x dx 苷 y u 6共1 ⫹ u 2 兲 du 苷 y 共u 6 ⫹ u 8 兲 du 苷
u7 u9 ⫹ ⫹C 7 9
苷 17 tan 7x ⫹ 19 tan 9x ⫹ C Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
EXAMPLE 6 Find
y tan sec d. 5
7
SOLUTION If we separate a sec 2 factor, as in the preceding example, we are left with a
sec 5 factor, which isn’t easily converted to tangent. However, if we separate a sec tan factor, we can convert the remaining power of tangent to an expression involving only secant using the identity tan 2 苷 sec 2 ⫺ 1. We can then evaluate the integral by substituting u 苷 sec , so du 苷 sec tan d :
y tan 5
sec 7 d 苷 y tan 4 sec 6 sec tan d 苷
y 共sec ⫺ 1兲 sec sec 2
2
6
tan d
苷 y 共u 2 ⫺ 1兲2 u 6 du 苷 y 共u 10 ⫺ 2u 8 ⫹ u 6 兲 du 苷
u 11 u9 u7 ⫺2 ⫹ ⫹C 11 9 7
苷 111 sec 11 ⫺ 29 sec 9 ⫹ 17 sec 7 ⫹ C The preceding examples demonstrate strategies for evaluating integrals of the form
x tan mx sec nx dx for two cases, which we summarize here. Strategy for Evaluating y tan mx sec nx dx
(a) If the power of secant is even 共n 苷 2k, k 艌 2兲, save a factor of sec 2x and use sec 2x 苷 1 ⫹ tan 2x to express the remaining factors in terms of tan x :
y tan
m
x sec 2kx dx 苷 y tan m x 共sec 2x兲k⫺1 sec 2x dx 苷 y tan m x 共1 ⫹ tan 2x兲k⫺1 sec 2x dx
Then substitute u 苷 tan x. (b) If the power of tangent is odd 共m 苷 2k ⫹ 1兲, save a factor of sec x tan x and use tan 2x 苷 sec 2x ⫺ 1 to express the remaining factors in terms of sec x :
y tan
2k⫹1
x sec n x dx 苷 y 共tan 2x兲k sec n⫺1x sec x tan x dx 苷 y 共sec 2x ⫺ 1兲k sec n⫺1x sec x tan x dx
Then substitute u 苷 sec x. For other cases, the guidelines are not as clearcut. We may need to use identities, integration by parts, and occasionally a little ingenuity. We will sometimes need to be able to Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 7.2
TRIGONOMETRIC INTEGRALS
475
integrate tan x by using the formula established in (5.5.5):
y tan x dx 苷 ln ⱍ sec x ⱍ ⫹ C We will also need the indefinite integral of secant: Formula 1 was discovered by James Gregory in 1668. (See his biography on page 199.) Gregory used this formula to solve a problem in constructing nautical tables.
y sec x dx 苷 ln ⱍ sec x ⫹ tan x ⱍ ⫹ C
1
We could verify Formula 1 by differentiating the right side, or as follows. First we multiply numerator and denominator by sec x ⫹ tan x : sec x ⫹ tan x
y sec x dx 苷 y sec x sec x ⫹ tan x dx 苷y
sec 2x ⫹ sec x tan x dx sec x ⫹ tan x
If we substitute u 苷 sec x ⫹ tan x, then du 苷 共sec x tan x ⫹ sec 2x兲 dx, so the integral becomes x 共1兾u兲 du 苷 ln u ⫹ C. Thus we have
ⱍ ⱍ
y sec x dx 苷 ln ⱍ sec x ⫹ tan x ⱍ ⫹ C EXAMPLE 7 Find
3
y tan x dx.
SOLUTION Here only tan x occurs, so we use tan 2x 苷 sec 2x ⫺ 1 to rewrite a tan 2x factor
in terms of sec 2x :
y tan x dx 苷 y tan x tan x dx 苷 y tan x 共sec x ⫺ 1兲 dx 3
2
2
苷 y tan x sec 2x dx ⫺ y tan x dx 苷
tan 2x ⫺ ln sec x ⫹ C 2
ⱍ
ⱍ
In the first integral we mentally substituted u 苷 tan x so that du 苷 sec 2x dx. If an even power of tangent appears with an odd power of secant, it is helpful to express the integrand completely in terms of sec x. Powers of sec x may require integration by parts, as shown in the following example. EXAMPLE 8 Find
3
y sec x dx.
SOLUTION Here we integrate by parts with
u 苷 sec x du 苷 sec x tan x dx
dv 苷 sec 2x dx v 苷 tan x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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TECHNIQUES OF INTEGRATION
y sec x dx 苷 sec x tan x ⫺ y sec x tan x dx 3
Then
2
苷 sec x tan x ⫺ y sec x 共sec 2x ⫺ 1兲 dx 苷 sec x tan x ⫺ y sec 3x dx ⫹ y sec x dx Using Formula 1 and solving for the required integral, we get
y sec x dx 苷 (sec x tan x ⫹ ln ⱍ sec x ⫹ tan x ⱍ) ⫹ C 1 2
3
Integrals such as the one in the preceding example may seem very special but they occur frequently in applications of integration, as we will see in Chapter 8. Integrals of the form x cot m x csc n x dx can be found by similar methods because of the identity 1 ⫹ cot 2x 苷 csc 2x. Finally, we can make use of another set of trigonometric identities: 2 To evaluate the integrals (a) x sin mx cos nx dx, (b) x sin mx sin nx dx, or (c) x cos mx cos nx dx, use the corresponding identity: 1 (a) sin A cos B 苷 2 关sin共A ⫺ B兲 ⫹ sin共A ⫹ B兲兴
These product identities are discussed in Appendix D.
1 (b) sin A sin B 苷 2 关cos共A ⫺ B兲 ⫺ cos共A ⫹ B兲兴 1 (c) cos A cos B 苷 2 关cos共A ⫺ B兲 ⫹ cos共A ⫹ B兲兴
EXAMPLE 9 Evaluate
y sin 4x cos 5x dx.
SOLUTION This integral could be evaluated using integration by parts, but it’s easier to use
the identity in Equation 2(a) as follows:
y sin 4x cos 5x dx 苷 y 苷
1 2
1 2
关sin共⫺x兲 ⫹ sin 9x兴 dx
y 共⫺sin x ⫹ sin 9x兲 dx
苷 12 (cos x ⫺ 19 cos 9x兲 ⫹ C
Exercises
7.2
1–49 Evaluate the integral. 1.
2
3
y sin x cos x dx 兾2
sin7 cos 5 d
3.
y
5.
y sin 共 x兲 cos 共 x兲 dx
7.
y
0
2
;
兾2
0
5
cos2 d
2. 4.
y
兾2
0
6.
y
8.
y
11.
y
13.
y t sin t dt
15.
y
0
0
Graphing calculator or computer required
sin 2 t cos 4 t dt
10.
y
12.
y
14.
y cos cos 共sin 兲 d
16.
y x sin x dx
0
4
sin 5 x dx
sin3 (sx ) dx sx 2
cos 4 共2t兲 dt
y
y sin cos d 3
9.
sin 2 ( 13 ) d
兾2
0
sin 2x cos 2x dx 2
cos 5 ␣ d␣ ssin ␣
兾2
0
共2 ⫺ sin 兲 2 d 5
3
1. Homework Hints available at stewartcalculus.com
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 7.2 3
18.
y cot sin d
cos x ⫹ sin 2x dx sin x
20.
y cos x sin 2x dx
3
22.
y tan sec d
24.
y 共tan x ⫹ tan x兲 dx
26.
y
28.
y tan x sec x dx
2
17.
y cos x tan x dx
19.
y
21.
y tan x sec x dx
23.
y tan x dx
25.
y tan x sec x dx
27.
y
2
4
兾3
0
6
tan 5x sec 4x dx 3
29.
y tan x sec x dx
31.
y tan x dx
5
33.
y x sec x tan x dx
35.
y
37.
y
兾2 兾6
兾2 兾4
5
4
2
兾4
0
兾4
3
4
30.
y
32.
y tan x sec x dx
0
tan t dt
sin d cos3
y
cot 5 csc 3 d
38.
y csc
兾2 兾4
兾3
cot 3x dx
4
csc 3x dx
y
41.
y sin 8x cos 5x dx
42.
y cos x cos 4 x dx
44.
y
cos x ⫹ sin x dx sin 2x
46.
y
1 ⫺ tan 2x dx sec 2x
47.
y
49.
y x tan x dx
48.
兾6
兾4
0
y
dx cos x ⫺ 1
your answer is reasonable, by graphing both the integrand and its antiderivative (taking C 苷 0兲. 兲 dx
58. y 苷 sin3x,
y 苷 cos 3 x,
兾4 艋 x 艋 5兾4
; 59–60 Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. 59.
y
2
0
cos 3x dx
60.
y
2
0
sin 2 x cos 5 x dx
61. y 苷 sin x, y 苷 0, 兾2 艋 x 艋 ; 62. y 苷 sin 2 x, y 苷 0, 0 艋 x 艋 ;
about the xaxis
about the xaxis
63. y 苷 sin x, y 苷 cos x, 0 艋 x 艋 兾4;
about y 苷 1
64. y 苷 sec x, y 苷 cos x, 0 艋 x 艋 兾3;
about y 苷 ⫺1
65. A particle moves on a straight line with velocity function v共t兲 苷 sin t cos 2 t. Find its position function s 苷 f 共t兲
if f 共0兲 苷 0.
E共t兲 苷 155 sin共120 t兲
; 51–54 Evaluate the indefinite integral. Illustrate, and check that
2
y 苷 cos 2 x, ⫺兾4 艋 x 艋 兾4
current that varies from 155 V to ⫺155 V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the equation
x0兾4 tan 8 x sec x dx in terms of I.
2
57. y 苷 sin 2 x,
66. Household electricity is supplied in the form of alternating
2
y x sin 共x
57–58 Find the area of the region bounded by the given curves.
s1 ⫺ cos 4 d
50. If x0兾4 tan 6 x sec x dx 苷 I , express the value of
51.
(a) the substitution u 苷 cos x (b) the substitution u 苷 sin x (c) the identity sin 2x 苷 2 sin x cos x (d) integration by parts Explain the different appearances of the answers.
61–64 Find the volume obtained by rotating the region bounded by the given curves about the specified axis.
40.
s1 ⫹ cos 2x dx
56. Evaluate x sin x cos x dx by four methods:
x cot 6 x dx
y csc x dx
兾6
x dx 2
2
39.
0
4
the interval 关⫺, 兴.
4
5
36.
y
y sec
55. Find the average value of the function f 共x兲 苷 sin 2x cos 3x on
sec 4 tan 4 d
cot 2x dx
45.
54.
4
2
y
y sin 5 sin d
y sin 3x sin 6x dx
477
2
34.
43.
53.
TRIGONOMETRIC INTEGRALS
52.
5
3
y sin x cos x dx
where t is the time in seconds. Voltmeters read the RMS (rootmeansquare) voltage, which is the square root of the average value of 关E共t兲兴 2 over one cycle. (a) Calculate the RMS voltage of household current. (b) Many electric stoves require an RMS voltage of 220 V. Find the corresponding amplitude A needed for the voltage E共t兲 苷 A sin共120 t兲.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_07_ch07_p474483.qk:97909_07_ch07_p474483
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CHAPTER 7
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TECHNIQUES OF INTEGRATION
67–69 Prove the formula, where m and n are positive integers. 67.
5:26 PM
70. A finite Fourier series is given by the sum
N
y sin mx cos nx dx 苷 0 ⫺
再 再
0 68. y sin mx sin nx dx 苷 ⫺
0 69. y cos mx cos nx dx 苷 ⫺
7.3
f 共x兲 苷
兺a
n
sin nx
n苷1
if m 苷 n if m 苷 n
苷 a 1 sin x ⫹ a 2 sin 2x ⫹ ⭈ ⭈ ⭈ ⫹ a N sin Nx Show that the mth coefficient a m is given by the formula
if m 苷 n if m 苷 n
am 苷
1
y ⫺
f 共x兲 sin mx dx
Trigonometric Substitution In finding the area of a circle or an ellipse, an integral of the form x sa 2 ⫺ x 2 dx arises, where a ⬎ 0. If it were x xsa 2 ⫺ x 2 dx, the substitution u 苷 a 2 ⫺ x 2 would be effective but, as it stands, x sa 2 ⫺ x 2 dx is more difficult. If we change the variable from x to by the substitution x 苷 a sin , then the identity 1 ⫺ sin 2 苷 cos 2 allows us to get rid of the root sign because
ⱍ
sa 2 ⫺ x 2 苷 sa 2 ⫺ a 2 sin 2 苷 sa 2共1 ⫺ sin 2 兲 苷 sa 2 cos 2 苷 a cos
ⱍ
Notice the difference between the substitution u 苷 a 2 ⫺ x 2 (in which the new variable is a function of the old one) and the substitution x 苷 a sin (the old variable is a function of the new one). In general, we can make a substitution of the form x 苷 t共t兲 by using the Substitution Rule in reverse. To make our calculations simpler, we assume that t has an inverse function; that is, t is onetoone. In this case, if we replace u by x and x by t in the Substitution Rule (Equation 5.5.4), we obtain
y f 共x兲 dx 苷 y f 共 t共t兲兲t⬘共t兲 dt This kind of substitution is called inverse substitution. We can make the inverse substitution x 苷 a sin provided that it defines a onetoone function. This can be accomplished by restricting to lie in the interval 关⫺兾2, 兾2兴. In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities. In each case the restriction on is imposed to ensure that the function that defines the substitution is onetoone. (These are the same intervals used in Section 1.6 in defining the inverse functions.) Table of Trigonometric Substitutions Expression
Substitution
Identity
sa 2 ⫺ x 2
x 苷 a sin ,
⫺
艋艋 2 2
1 ⫺ sin 2 苷 cos 2
sa 2 ⫹ x 2
x 苷 a tan ,
⫺
⬍⬍ 2 2
1 ⫹ tan 2 苷 sec 2
sx 2 ⫺ a 2
x 苷 a sec ,
0艋⬍
3 or 艋 ⬍ 2 2
sec 2 ⫺ 1 苷 tan 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_07_ch07_p474483.qk:97909_07_ch07_p474483
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5:26 PM
Page 479
SECTION 7.3
v
EXAMPLE 1 Evaluate
y
TRIGONOMETRIC SUBSTITUTION
479
s9 ⫺ x 2 dx. x2
SOLUTION Let x 苷 3 sin , where ⫺兾2 艋
艋 兾2. Then dx 苷 3 cos d and
ⱍ
ⱍ
s9 ⫺ x 2 苷 s9 ⫺ 9 sin 2 苷 s9 cos 2 苷 3 cos 苷 3 cos (Note that cos 艌 0 because ⫺兾2 艋 艋 兾2.) Thus the Inverse Substitution Rule gives 3 cos s9 ⫺ x 2 y x 2 dx 苷 y 9 sin 2 3 cos d 苷y
cos 2 d 苷 y cot 2 d sin 2
苷 y 共csc 2 ⫺ 1兲 d 苷 ⫺cot ⫺ ⫹ C Since this is an indefinite integral, we must return to the original variable x. This can be done either by using trigonometric identities to express cot in terms of sin 苷 x兾3 or by drawing a diagram, as in Figure 1, where is interpreted as an angle of a right triangle. Since sin 苷 x兾3, we label the opposite side and the hypotenuse as having lengths x and 3. Then the Pythagorean Theorem gives the length of the adjacent side as s9 ⫺ x 2 , so we can simply read the value of cot from the figure:
3 x ¨ 9≈ œ„„„„„ FIGURE 1
cot 苷
x sin ¨= 3
s9 ⫺ x 2 x
(Although ⬎ 0 in the diagram, this expression for cot is valid even when ⬍ 0.) Since sin 苷 x兾3, we have 苷 sin⫺1共x兾3兲 and so
y v
冉冊
x s9 ⫺ x 2 s9 ⫺ x 2 dx 苷 ⫺ ⫺ sin⫺1 2 x x 3
⫹C
EXAMPLE 2 Find the area enclosed by the ellipse
x2 y2 苷1 2 ⫹ a b2 SOLUTION Solving the equation of the ellipse for y, we get
y
y2 x2 a2 ⫺ x2 苷 1 ⫺ 苷 b2 a2 a2
(0, b) (a, 0) 0
x
or
y苷⫾
b sa 2 ⫺ x 2 a
Because the ellipse is symmetric with respect to both axes, the total area A is four times the area in the first quadrant (see Figure 2). The part of the ellipse in the first quadrant is given by the function b sa 2 ⫺ x 2 a
y