Vasili P. Minorski
Aufgabensammlung der höheren Mathematik
15., aktualisierte Auflage
Vasili P. Minorski
Aufgabensa...
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Vasili P. Minorski
Aufgabensammlung der höheren Mathematik
15., aktualisierte Auflage
Vasili P. Minorski
Aufgabensammlung der höheren Mathematik Gute Studienergebnisse setzen in der Mathematik neben Kenntnissen auch Fertigkeiten voraus. Die Fertigkeiten kann man sich nur durch Üben aneignen. Mehr als 2500 Aufgaben wurden dafür in diesem Buch zusammengestellt. Ihre Lösungen, teils sogar mit Lösungsweg, sind am Ende der Sammlung zu finden. Diese moderne Aufgabensammlung, gedacht vor allem für Studenten ingenieurwissenschaftlicher Studiengänge an Hochschulen, ■ ist auf den Grundkurs Mathematik (Analysis, lineare Algebra) abgestimmt, ■ enthält viele Aufgaben mit technikorientierten Problemstellungen, ■ ermöglicht effektive Wiederholung und optimale Prüfungsvorbereitung. Aber auch Studenten der Mathematik und naturwissenschaftlicher Studiengänge können aus der Aufgabensammlung Nutzen ziehen.
www.hanser.de ISBN 978-3-446-41616-1
9
783446 416161
Vasili P. Minorski
Aufgabensammlung der höheren Mathematik Bearbeitet von Prof. Klaus Dibowski und Dr. Horst Schlegel
15., aktualisierte Auflage Mit 68 Bildern und 2670 Aufgaben mit Lösungen
Fachbuchverlag Leipzig im Carl Hanser Verlag
Aus dem Russischen übersetzt von Eberhardt Lacher, Schwarzenberg und Gerhard Liebold, Chemnitz Bearbeitung der deutschsprachigen Ausgabe von Heinz Birnbaum, Leipzig Titel der Originalausgabe: Сборник задач по высшей математике, 7. Auflage, Staatlicher Verlag für physikalisch-mathematische Literatur, Moskau 1962
Bearbeiter der 15. Auflage Prof. Dr. Klaus Dibowski Hochschule für Technik, Wirtschaft und Kultur Leipzig (FH) FB Informatik, Mathematik und Naturwissenschaften Dr. rer. nat. Horst Schlegel, Leipzig
Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. ISBN 978-3-446-41616-1
Dieses Werk ist urheberrechtlich geschützt. Alle Rechte, auch die der Übersetzung, des Nachdruckes und der Vervielfältigung des Buches, oder Teilen daraus, vorbehalten. Kein Teil des Werkes darf ohne schriftliche Genehmigung des Verlages in irgendeiner Form (Fotokopie, Mikrofilm oder ein anderes Verfahren), auch nicht für Zwecke der Unterrichtsgestaltung, reproduziert oder unter Verwendung elektronischer Systeme verarbeitet, vervielfältigt oder verbreitet werden.
Fachbuchverlag Leipzig im Carl Hanser Verlag © 2008 Carl Hanser Verlag München www.hanser.de Lektorat: Christine Fritzsch Herstellung: Renate Roßbach Satz: Klaus Dibowski, Leipzig Druck und Binden: Druckhaus „Thomas Müntzer“ GmbH, Bad Langensalza Printed in Germany
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A, B, . . .
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(A B)
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n ≥ n0
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tan(α + β) = 2 tan α " & ' ( a, b, c %) α, β, γ a2 − b 2 sin(α − β) = sin(α + β) c2 a 2 + c2 − b 2 tan α = 2 tan β b + c2 − a 2 * a sin(β − γ) + b sin(γ − α) + c sin(α − β) = 0
* " # $ α β sin(α + β) < sin α + sin β √ 7 lg 5 3x − 5 > −1 0 < x < ∞ 7x + 5 " a, b, c, d (a + b)2 +(c + d)2 ≤ a2 + c2 + b2 + d2 + ,- . ) n n(n + 1) i= 2 i=1
n i n+2 =2− n i 2 2 i=1 n−1
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n i=1
n
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n i=1
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A
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B A a > b a, b
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, A- a1 , a2 , a3 , . . .
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B := {x | x ∈ T ∧ H(x)} , B - x-
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A ∪ B := {x | x ∈ A ∨ x ∈ B} A ∩ B := {x | x ∈ A ∧ x ∈ B} A \ B := {x | x ∈ A ∧ x ∈ B} A × B := {(x, y) | x ∈ A ∧ y ∈ B ∧(x, y) } Komplement¨ armenge A := M \ A .M /" ( 0
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"
F # F F −1 F $ M N !%!& M ∼ N,
M ' N ( ! N %) * {x | x ∈ R ∧ 0 ≤ x ≤ 1} $ ! +
, ! '% A = {x | x ∈ R ∧ x3 + x2 − 2x = 0}
B = {x | x ∈ R ∧ x+
4 ≤ 4 ∧ x > 0} x
! C = {x | x ∈ N ∧ x2 − 4x − 5 ≤ 0} 1 ≤ 2x < 6}
D = {x | x ∈ Z ∧ 8 , x, y . / 0! 2
A = {(x, y) | (x, y) ∈ R ∧ x + y − 3 = 0}
B = {(x, y) | (x, y) ∈ R2 ∧ 4x2 − y 2 < 0} ! C = {(x, y) | (x, y) ∈ R2 ∧ (x2 − 4)(y + 1) = 0} 2
D = {(x, y) | (x, √ y) ∈ R ∧ y > x + 1 ∧ x ≥ −1}
1 2 A ∪ B A ∩ B A \ B B \ A A × B B × A
! '% A = {x | x ∈ R ∧ x2 − 2x = 15}
B = {x | x ∈ R ∧ x2 − 4x = 5}
3 A, B, C 4 05' ' 6 0 7 8 '5 ! ( 6 , A ∩ B ⊆ A A ∩ B ⊆ B
, A ⊆ A ∪ B B ⊆ A ∪ B ! (A) = A
A ⊆ B ' A ⊇ B A ∪ B = A ∩ B ' A ∩ B = A ∪ B (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) (A \ B) ∩ (A ∪ B) = A 9 6 '5 A, B & A ∩ B = B ⇐⇒ B ⊆ A
A ∪ B = B ⇐⇒ A ⊆ B : $ ! A = {m | m ∈ N ∧ m = n2 + 1 ∧ n ∈ N}
%
A = {2; 4; 6; 8; . . . } B = {1; 0, 1; 0, 01; 0, 001; . . .}
P1 (x) = 0 P2 (x) · P3 (x) = 0
# ,)
F = {(a, b), (c, b), (d, e)} P1 (x) = 0 G = {(c, f ), (c, b), (g, e)} P2 (x) = 0 ! " P3 (x) = 0 M = {a, c, h, d, g} N = {b, e, f, k} & + ' &) # $ " %& A × B x, y 2 ! ' # A = {x | x ∈ R ∧ (1 ≤ x ≤ 2 # ()& ∨ x = 3)} %&' B = {y | y ∈ R ∧ (2 ≤ y ≤ 3 * P1 (x), P2 (x), P3 (x) +', ∨ y = 4} ') - % # A = {x | x = i ∧ i ∈ {1, 2, 3, 4}} ' . &/ B = {y | y ∈ R ∧ 1 ≤ y < 3} Li = {x | Pi (x) = 0} (i = 1, 2, 3) ) 0 L1 , L2 M = {a, b, c} N = {α, β} 3 L3 1-)
" # P1 (x) · P2 (x) · P3 (x) = 0 # ,)
# ' M ! N
# ' ! N M
4 ' 5 Q
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− 2 = −1, 999 · · · = −1, 9
() 3 ' 5 x = 2, 34 3 5 2 &
−(
100x = 234, 444 . . . 10x = 23, 444 . . . ) 90x = 211
;
x=
211 90
[a, b] := {x | x ∈ R ∧ a ≤ x ≤ b}; [a, b) := {x | x ∈ R ∧ a ≤ x < b}; [a, ∞) := {x | x ∈ R ∧ a ≤ x}; (∞, b] := {x | x ∈ R ∧ x ≤ b};
(a, b] := {x | x ∈ R ∧ a < x ≤ b} (a, b) := {x | x ∈ R ∧ a < x < b} (a, ∞) := {x | x ∈ R ∧ a < x} (∞, b) := {x | x ∈ R ∧ x < b}
[a, b] ! " P0 P # ! $R1 %& ' $R2 % $R3 % ( ε $ε > 0% # ! P0 # ! P & ) | P − P0 |=| x − x0 |= (x − x0 )2 < ε $* % | P − P0 |= (x − x0 )2 + (y − y0 )2 < ε $* +,- % | P − P0 |= (x − x0 )2 + (y − y0 )2 + (z − z0 )2 < ε $* +% M . Ri (i = 1, 2, 3) / M # ! P ∈ Ri &
0 ε1( P # ! M ") - Ri
Ri
' H(M ) 2- ! M ⊆ R H(M ) • - !&
H(M ) M : sup H(M ) = lim sup M = lim M • - !&
3 H(M ) M : inf H(M ) = lim inf M = lim M
x √ 2+5 2 x = √ 2−1 √ √ 1, 47 − 2, 45 √ √ x = 5− 3 x = log9 5 · log25 27 √
√ a = 7 + 0, 2 b = 7 − 0, 2 ! a − b, a2 + b2 a3 − b3 " # # $ (a − b ∈ Q ∧ a2 + b2 ∈ Q) =⇒ a3 − b 3 ∈ Q % & # $ 0, 321 0, 132 2, 59 ' ( )
I1 = [−2, 3) I2 = [1, 5) I1 = (−5, 1) I2 = (−2, 0] I1 = [−1, 5; 3, 5)
I2 = (0, 5; 4, 5) $ * I1 ∪ I2 , I1 ∩ I2 , I1 \ I2 , I2 \ I1
+ ) , M
, -. / H(M )
lim M lim M !
0 n+1 ∧ n ∈ N} ⊂ R M = {x|x = n n M = {x|x = (−1)n+1 n+1 ∧ n ∈ N} ⊂ R 3n + (−1)n M = {x|x = 2n ∧ n ∈ N} ⊂ R 2n + 7 M = {x|x = (−1)n · 3n ∧ n ∈ N} ⊂ R 2
M = {(x, y, z)|x = 2 + n 4 ∧y = 4 + n 5 ∧z = 5 + ∧ n ∈ N} ⊂ R3 n
|a| =
a a ≥ 0 −a a < 0
1
2 . |a| 2 1 ) )
a )# 3 /
|a| ) 4 4 # $ . | − a| = |a| |x| = a ⇐⇒ x = ±a; |x| < a ⇐⇒ −a < x < a |x| > a ⇐⇒ ((x < −a) ∨ (x > a)) a |a| |a · b| = |a| · |b| ; = |b| = 0 b |b|
1 1% 1' 1+
|a + b| ≤ |a| + |b|
a < b ⇐⇒ a ± c < b ± c
a < b ⇐⇒ a · c < b · c c > 0
a < b ⇐⇒ a · c > b · c c < 0
((a < b) ∧ (c < d)) =⇒ a + c < b + d 1 > 1b a · b > 0 a < b ⇐⇒ a1 1 a · b < 0 a < b
! "
x# $ 2x + 6 ≤ 18 − 9x % −1, 5x − 3 < 3 − 4, 5x 3x + 6 ≤ 4x + 2 (1 − x)(x + 2) > 0 x2 + 2x − 8 < x − 2 x3 − x2 ≤ 4x − 4 5−x 8 4x − 3 3x + 2 ≤ −2 2x − 3 4x + 3 ≤6 & 2, 5 − x 2x2 + 12x + 8 ≤2 x2 + x − 6 √ 4x − 8 < 1 lg (2 + x) ≤ 1 !'$ ( %%
a, b |a + b| ≥ |a| − |b|
% |a + b| ≥ |a| − |b|
! "
x# ) *+ %,' $ x+3 ≥3 2x − 5 % |2x − 3| ≤ 6 |3 − 2x| > 5 7x − 3 1)
! z1 + z2 , z1 − z2 , z1 · z2 " z1 : z2 , z1 · z2 # z2 : z1 $! %& '' ! ' ! ( !' #" ) z1 = 9 − 7 , z2 = 3 + 2 4 1 4 1 % z1 = + , z2 = − 3 2 3 2 z1 = 2(cos 15◦ + sin 15◦ ), z2 = 3 e π/6 z1 = (1 + 2 )2 , z2 = (1 − )3 z1 = 2 e
5π/12
, z2 = 4 e
π/6
! ! * ! % x, y %) u, x, y, z #' (1 + 2 )x + (3 − 5 )y = 1 − 3 ⎧ ⎪ (1 + )u + (1 + 2 )x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ +(1 + 3 )y + (1 + 4 )z = % 1 + 5 ⎪ ⎪ ⎪ (3 − )u + (4 − 2 )x ⎪ ⎪ ⎪ ⎩ +(1 + )y + 4 z = 2 − ! $! %'' ! & ' ! ( !' # √ · x2 · · y 2 x, y ∈ R % (−)18 −17 √ √ 1 1 (2 3 − 3 2)2 4 + 7 i i 123 −99
+ ! ,# " ! - ! ! )!
5 17 − % 1 − 2√ 3 √ 3 3+ 2 √ √ √ 2 + 5 2 3− 2 . * % / & ' ! ' " !& !' # ( !' # # %
z = −5
% z = 9 π r = 3 , ϕ = 2 r = 8 , ϕ = −π π r = 2 , ϕ = 3 5π r = 7 , ϕ = 6 2π r = 1 , ϕ = − 3
r = 5 , ϕ = −127◦ Re z = 4 , Im z = −6 0 Re z = 0 , Im z = 2 / z = cos 60◦ + sin 30◦ Re z = −0, 5 , Im z = 8 √ Re z = 3 , Im z = −1 Re z = − Im z = −2 √ Re z = − 3 , Im z = −3 z = cos 30◦ − sin 30◦ 1 z = 9(− cos 270◦ + sin 270◦ ) ! z = e−3 π ' z = e2+3 π z = e4−11,5 π
Re z Im z 3+4 3
z = z = 2+ 1− 1+ 1− + −1 +1 z = (1 + )7 + (1 − )7 √ √ z = (1 − 3)10 − (1 + 3)10 √ (−1 + 3)15 z = (1 − )20 √ (−1 − 3)15 + (1 + )20
z =
!
e ϕ + e− ϕ
cos ϕ = 2 e ϕ − e− ϕ sin ϕ = 2 "
# $%√! z = −1 + 3 # $% z = 3 + 4
# $% √ z = 2 3 + 2 # $% z = −16 # $% & z = cos 225◦ − sin 225◦ '() * ) + ,- ) . ) /) # $% 0 '() * )1
x6 = 1
x2 = −3 + x4 = 1 −
x5 = −1 x3 = −2 − 3 x4 = −
/
2 f (x ) 34 # 0
' "
f (x) = Pn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ; an = 0
* Pn (x) : (x − x0 ) & 1 an
x0
? bn−1
an−1
an−2
...
x0 bn−1 x0 bn−2 . . . * * bn−2 bn−3 . . .
a2
a1
a0
x0 b2
x0 b1 *
x0 b0 *
b1
b0
f (x0 )
f (x0 ) ' + x − x0 5 f (x0 ) = 0 x0 6 34 2 , # Pn (x) $ 1 Pn (x) : (x − x0 ) = bn−1 xn−1 + bn−2 xn−2 + · · · + b1 x + b0 +
Pn (x) = (x − x0 )(bn−1 xn−1 + bn−2 xn−2 + · · · + b1 x + b0 )
' 7
n Pn (x) = 0 an , an−1 , . . . , a1 , a0 C n ! "# $% &
' ! ( ) * + x1 , x2 , . . . , xk , # ( α1 , α2 , . . . , αk α1 + α2 + · · · + αk = n% - "*..$
Pn (x) = an (x − x1 )α1 (x − x2 )α2 . . . (x − xk )αk
/ Pn (x) = 0 % • 0 x1 = α + iβ x2 = x1 = α − iβ # "β = 0$* • 1 n % ! *
1 Pn (x) = 0 an = 0 ,% ) pn (x) = xn + bn−1 xn−1 + · · · + b1 x + b0 = 0 an−i = bn−i (i = 1, 2, . . . , n)* an + x1 , x2 , . . . , xn n # , pn (x) = 0 "αi # αi 2 $% & # xi bi−1 (i = 1, 2, . . . , n)
x1 + x2 + · · · + xn = x1 x2 + x1 x3 + · · · + xn−1 xn =
n
xi = −bn−1
i=1 n
xi xj = bn−2
i,j=1 (i<j)
x1 x2 x3 + x1 x2 x4 + · · · + xn−2 xn−1 xn =
n
xi xj xk = −bn−3
i,j,k=1 (i<j 0 ⇔ a, b, c ! [a, b, c] < 0 ⇔ a, b, c 1(! [a, b, c] = 0 ⇔ a, b, c ( ⇔ c = λa + μb.
a = e1 + 2e2 + 3e3 , b = 6e1 + 4e2 − 2e3 a · b a × b
7 A(2; 2; 2) C(4; 5; 4) D(5; 5; 6)
B(4; 3; 3)
[(a − b), (b − c), (c − a)]
a = (2, 2, 1)Ì b = (6, 3, 2)Ì + 8 $ a ! a = b $ b ! a 3e1 + 4e2 + 7e3 b = (2, −5, 2)Ì ! $ "# $ λ a b 4a + 3b + 2c 7a + 6b + 5c,
a = λe1 +3e2 +4e3 |a| = 3, |b| = 2, |c| = 1, π b = 4e1 + λe2 − 7e3 % ∠(a, b) = ∠(a, c) = ∠(b, c) = 3 & (5a + 3b) · (2a − b) |a| = 3, |b| = 5 a b ' & ,, 7 $ s◦ ! a = (1, 1, 2)Ì b = (2, 1, 1)Ì ( ) "* + ,, ( !# λ Ì a = a = (−6, −3, 2) Ì (1, 1, λ)Ì , b = (1, 1, λ + 1)Ì c = b = (−3, 2, −6) (1, −1, λ)Ì , a + 5b 5a + b |a| = |b| = 3 ∠(a, b) = 30◦ % - . a, b, c $ 9* ' - "* ,, c . A B C a = e1 + e2 b = e2 + e3 A(4; 3; 2), B(2; 3; 4) C(1; 1; 1) / , :' ϕ a b
/ a = (3, 4, 5)Ì b = (−4, −5, 3)Ì
, 7 A(1; 1; −1), B(1; −1; 1), C(−1; 1; 1), D(1; 1; 1)
0 ' 0 a × (a × b) +
a = (2, −1, −1)Ì , (b × a) × a !# a = (2, 1, −3)Ì b = (−1, −3, 1)Ì c = e1 + e2 + 4e3 b = (1, −1, 1)Ì a = $ + (2, 5, 7)Ì , b = (1, 1, −1)Ì c = A(1; 2; −1), B(−1; 3; −4), C(0; 5; −7) (1, 2, 2)Ì , D(2; 4; −4) a , 1! b $ + c 7 23 4 a = λb + μc 5 , +6, ,
7 + ,,
!
"! #$ % & ' $ ( ) * ! +,-
. / !
0 1 (c2 = a2 + b2 −2ab cos γ) ,- (c2 = a2 + b2 )!
g g
r0 e3 O6 e2 e1
P0 (x0 ; y0 ; z0 ) w
v
1 P (x; y; z) r
g : r = r0 + λv
+!2
r 0 % - / P0 g v % % g λ / (−∞ < λ < +∞)
/ P1 (x1 ; y1 ; z1 ) P2 (x2 ; y2 ; z2 ) +!3
r = r1 + λ(r 2 − r1 )
r 1 = (x1 , y1 , z1 )Ì r2 = (x2 , y2 , z2 )Ì 4 5 F (xF ; yF ; zF ) 6 l % / P1 (x1 ; y1 ; z1 ) g g : r = r 0 + λv r F = (xF , yF , zF )Ì = r0 +
(r 1 − r0 ) · v ·v v2
+!
, S(xS ; yS ; zS ) -$ g1 g2 g1 : r = r 1 + λv 7 R3
g2 : r = r2 + μw
r 1 + λv = r2 + μw
( λ μ! #- +! 8 8! 69 =⇒ : ; , S !
+! 8
=⇒ S r S = (xS , yS , zS )Ì = r 1 + λv = r 2 + μw =⇒ g1 ≡ g2 ! |l| "# # $ g1 g2 g1 : r = r1 + λv g2 : r = r 2 + μw !" l : l = r 2 − r1 + μw − λv $% & l & g1 g2 $% $ "# $% λ μ' l·v =0
l·w =0
g1 g2 # $( & ) *
( * P1 P2 ( # P1 (−2; 3; −5) P2 (1; −4; −1) P1 (3; −2; 1) P2 (1; −2; 2) *%$( & * A(−5; 1; 2) B(3; −3; 1) $ r = (1, −2, 5)Ì + λ(2, −1, 1)Ì +& $& , "# # ", r = 5e1 + e2 − 2e3 + λ(4e1 − e2 −3e3 ) r = (7 − 3μ)e1 + (2μ − 2)e2 +(11 − 5μ)e3 √ 2 Ì 1 , 1) r = (1, 2, 1)Ì + λ( √ , − 2 3 r = (−4, 3, −1)Ì
√ 3 +μ( 3, − √ , 3)Ì 2
r = (2 + 4λ, −1 + 2λ, 3 − λ)Ì r = (1 + 2μ, 2 − μ, 2 + 3μ)Ì √ r = (2 + 2 3, −3, 7)Ì
√ +λ( 12, −2, 4)Ì √ r = (2 − 3, 0, 1)Ì √ +μ(− 3, 1, −2)Ì
r = (3, −1, 2)Ì + λ(2, 4, 10)Ì r = (−1, 5, 3)Ì + μ(−4, 4, 6)Ì
$ r = (3, −1, 2)Ì + λ(2, 4, 3)Ì r = (−1, 5, 10)Ì + μ(−4, 4, 6)Ì
−∞ < λ < ∞,
−∞ < μ < ∞)
+& # +
r = (−2, 5, 1)Ì + λ(−1, 2, 3)Ì r = (3, −1, 2)Ì + μ(1, −1, 1)Ì ? (−∞ < λ, μ < ∞)
. * P1 (3; −1; 2) P2 (1; 2; −1) *
g ( P1 P2 / 01 F & & 2
$ g / ! 2
, & g 3 / !
g1 g2 01 2 , 3# F 4 l - P1 (−2; 1; 1) g1 : r = (−1, 0, 1)Ì + λ(1, 1, 2)Ì P2 (0; 0; 1) P3 (1; −1; 0) g2 : r = μe1 +(3μ−1)e2 +(4μ+2)e3 4- l (−∞ < λ, μ < +∞) 4. |l| g 1
P1 (1; −2; 1) P2 (−2; 3; 5)! 00 5 g2 A(1; 2; 3) B(−2; 3; 1) Q1 (1; −5; −2) Q2 (10; −11; −5) C(2; −3; −1) " #$ " + 2 ,. g1 g2 % 4. 6 7 ABC. & ' '+ ' * P (2; −3; 4) y 89 :
( ' g : r = (1 + λ, −2 + 2λ, 5 − 2λ)Ì (−∞ < λ < +∞) P0 (1; −2; 5) g )* λ = 0+ ,* " - λ . / - g - P0 + ' '+ ' 0& ' %
" '
+ 5 6 # :# - 7 ; $ 9 - - E d = n · r 0 , ' r 0 = (x0 , y0 , z0 )Ì '$ P0 (x0 ; y0 ; z0 ) - E
⎛ ⎞ a n = ⎝b ⎠ c
z 6
n = (a, b, c)Ì E
P0 (x0 ; y0 ; z0 ) q P (x; y; z)
ax + by + cx = d
- y x
A P1 (x1 ; y1 ; z1 ) E E : n · r = d !" ax + by + cz = d
E E:
n·r−d =0 |n|
!"
ax + by + cz − d √ =0 a 2 + b 2 + c2
#
A=
|n · r 1 − d| |n|
!" A =
|ax1 + by1 + cz1 − d| √ a 2 + b 2 + c2
$
%" E & ' r = r 0 + λv + μw ( n = v × w d = n · r0 ) n · r = d %" & ' E
E : n · r = d n = (n1 , n2 , n3 )Ì v w ) n · v = 0 n · w = 0 ! * v = (0, n3 , −n2 )Ì w = (n2 , −n1 , 0)Ì + , r 0 * n · r 0 = d '+ Ì d , 0, 0 ' " ' n1 = 0 - r0 , v, w ! * r0 = n1 E : r = r0 + λv + μw ,. S(xS ; yS ; zS ) ) g E g : r = r0 + λv
E : n · r = d
r S = (xS , yS , zS )Ì = r 0 +
d − n · r0 ·v n·v
F (xF ; yF ; zF ) ! " P1 (x1 ; y1 ; z1 )Ì # $% E E : n·r =d r1 = (x1 , y1 , z1 )Ì r F = (xF , yF , zF )Ì = r1 +
d − n · r1 ·n n2
&
' g () $% E1 E2 E1 : n1 · r = d1 E2 : n2 · r = d2 ( #* g : r = r0 + λ · v % v = n1 × n2 + r 0 = (x0 , y0 , z0 )Ì () n1 · r0 = d1 n2 · r0 = d2 #* ,% x0 , y0 , z0 * (
% ,% - ) ( . () ,% %/ %
' P2 (x2 ; y2 ; z2 ) " P1 (x1 ; y1 ; z1 ) %(* $% E . % r 1 = (x1 , y1 , z1 )Ì E : n · r = d r 2 = r1 + 2 ·
d − n · r1 ·n n2
2 3 % 4
% " $% E1 ≡ E2 ( . % E1 E2 ( 5# *% ) E1 : r = (4, −2, −11)Ì +λ(−2, 5, 15)Ì + μ(2, 10, 21)Ì E2 : r = (2, 3, 4)Ì + λ(4, 5, 6)Ì +μ(0, 5, 12)Ì (−∞ < λ, μ < ∞)
6 . % " P1 ! $% E ) E : x + y − z = −1; P1 (2; 1; 1) % E : 6x − 3y + 2z = 28;
01
P1 (3; 5; −8)
3 $% E '# ) E " P (2; 3; 5) a = (4, 3, 2)Ì # E 7 % $% E : 2x + 3y + z = 6 '(( 8 ! E 2 9 : (x ≥ 0, y ≥ 0, z ≥ 0) 0 * % $% " ( 4
% E : (3, −2, 5) · r = 8
E : −y + 7z = 13
& & 0* E ) .*
E1 E2 +1 2* # P1 (1; 2; 1) . E1 : 2x + y − 2z − 4 = 0 E : x − 2y + z = 7 E2 : 3x + 6y − 2z − 12 = 0 3* ./ 4** E1 5 %
E2
+ ( 6 # P1 (2; 3; 4)
# P2 6 E1 : x − 2y + z = 1 . E E2 : (−2, 4, −2) · r = −1 E : x − 3y + 5z + 22 = 0
! " # $% + & # & g E A(−3; 2; 5), B(−2; 1; 6), C(1; 3; 2) g : x − 1 = 2y + 2 = z + 3 = λ
& −∞ < λ < +∞ g : r = (5 + 9λ)e1 − (5 + 6λ)e2 E : x+y−z+1=0 −(4 + 3λ)e3 g : r = (1, 2, 1)Ì + λ(2, −1, 2)Ì (−∞ < λ < +∞). −∞ < λ < +∞ ( E : 2x + y − z − 4 = 0 & E ' ( ) g # . E1 E2 *%
) 7 *% S & E1 : 2x − y + 3z = 1 g E E2 : (1, 1, −1) · r = 2 + & 6 E1 : 2x − 2y + 2z = 3 E : x − 2y − 3z = 0 E2 : −2x − 3y + 6z = 7 E2 # E1 : −x + 3y − 3z = 2 P1 (2; 2; −2) E2 : 3x + 2y + z = 5 + ) & +, &
# E . # A(2; 1; −2), B(0; 2; 1), C(1; 2; 0) (P1 (−1; −1; 2) . E1 E2 & E E1 : (1, −2, 1) · r = 4 A, B, C ) .* E2 : (1, 2, −2) · r = −4 & & g + ( ) S A . E E1 , E2 , E3 E1 : (2, −1, 3) · r − 9 = 0 ++ E / # E2 : (1, 2, 2) · r − 3 = 0 P (3; 2; −1) & E3 : (3, 1, −4) · r + 6 = 0 r = (−2, 0, 1)Ì + λ(−1, 3, −2)Ì (−∞ < λ < +∞).
+! &
E P1 (1; 2; 4) E1 E2 E1 : 2x − y + 3z − 6 = 0 E2 : x + 2y − z + 3 = 0
! " # E1 E2 $ % # E : 2x + 2y + z − 8 = 0 & A = 4 !
" + E g1 g2 , + P1 (4; −2; 5) & E
+ g3 E g1 P2 x = x2 = −1 # + $ S & g2 g3 !
'(! ) # E ''! - a = −e3 P1 (1; 2; 4) . M1 & # E1 E2 * / * E3 E1 : 4x − y + 3z − 6 = 0 E : x+y−z =2 E2 : x + 5y − z + 10 = 0 & , * . E3 : 2x − y + 5z − 5 = 0 M2 P2 (2; 3; −3)! '! $ + 0 A 1 / E Ì Ì g1 : r = (−2, 5, 2) + λ(1, 0, −1) + * & , * / . M2 g2 : r = (2, −4, 2)Ì + μ(1, 0, −1)Ì
+ ) 2 " & # (−∞ < λ, μ < +∞) sin α1 , n = sin α2
n ⎞ a11 a12 . . . a1n ⎜ a21 a22 . . . a2n ⎟ ⎟ A=⎜ ⎝ . . . . . . . . . . . . . . . . . . .⎠ an1 an2 . . . ann ⎛
n2 n ! " n # $ % aik i " k $ & " % % n ' Dn n a11 a12 . . . a1n a a22 . . . a2n Dn = det A = 21 . . . . . . . . . . . . . . . . . . . an1 an2 . . . ann
(
" % ) % ( ' D2 * & a D2 = 11 a21
a12 = a11 · a22 − a12 · a21 a22
+
Dn i " k $ , Aik ' n − 1 !# Dn i "Dn = (−1)i+1 ai1 · Ai1 + (−1)i+2 ai2 · Ai1 + . . . + (−1)i+n ain · Ain
!# Dn k $ Dn = (−1)1+k a1k · A1k + (−1)2+k a2k · A2k + . . . + (−1)n+k ank · Ank
/0 1
23 (−1)i+k 41
5 + − + · · · − + − · · · + − + · · · . . . . . . . . . . . . .
.
! "# a11 , a22 , . . . , ann # $ %& & & ! "
' !
λ & λ & & ! # λ() ! # * + ! # " Dn λ() ! # Dn = 0 ,
- ). / Dn = a11 · a22 · . . . · ann
0 −3 10 9 D3 = 1 −4 −2 5 −16 12 # 1 $ # 1 $ ' ! # &" () ") # 1(
$ 0 ( 3 −2 2 3 # # 6 −10 4 6 √ a −1 sin α cos α √ # # − cos α sin α a a ' 0 # 1
! # & 1(
2 3 4 5 −2 1 1 2 3 0 ) ( & 1 "( ) / a −a a # a a −a a −a −a 1 2 5 7 # 3 −4 −3 12 −15 2 x x 1 2 # y y 1 z 2 z 1 1 + cos α 1 + sin α 1 # 1 − sin α 1 + cos α 1 1 1 1 1 −2 −6 4 −3 1 2 −5 # 0 −4 3 4 6 0 1 8
−1 2
−3 1 2 −3 7 −9 1
0 2 −3 1 −3 2 2 1 −1 3 2 1
6 0 1 2 5 −1 2 −1 −1 3 −2 3 2 2 3 5 1 1 −1 −2
x
2 ! x 4 9 x 2 3 = 0 1 1 1 2 x 3 2 x −1 1 = 0 0 1 4 2 − x 1 −1 2 = 0 −2 4 − x x − 3 3 − x 3 − x
" A # $ %&' (m, n) (& # m · n )
* aik + + m * n (' , ⎞ ⎛ a11 a12 . . . a1n ⎜ a21 a22 . . . a2n ⎟ ⎟ A = A(m,n) = (aik )(m,n) = ⎜ ) - ⎝. . . . . . . . . . . . . . . . . . . . . ⎠ am1 am2 . . . amn ! aik i * k (' *. A = (aik ) B = (bik ) + . A B $ aik = bik / i k *. A = (aik ) B = (bik ) # $ .
+ ' ) , A(m,n) ± B (m,n) = S (m,n) aik ± bik = sik (i = 1, 2, . . . , m; k = 1, 2, . . . , n)
) 0
! , A + B = B + A; A + (B + C) = (A + B) + C
) 1
" A = (aik ) . λ + 2 aik # A λ ' , λA = Aλ = (λaik ) (−1) · A = −A; λ(A + B) = λA + λB (λ + μ)A = λA + μB
) ) 34 ) 33
! A · B A B + # 5 / ) 3 $5 A # %&' )m, n $5 B # %&' )n, r+
P m, r pik i ! "# $ A k % $ B ' A(m,n) · B (n,r) = P (m,r) ai1 · b1k + ai2 · b2k + . . . + ain · bnk = pik
#"&
(i = 1, 2, . . . , m; k = 1, 2, . . . , r)
' #"( #"
(A + B) · C = AC + BC; (AB)C = A(BC) AB = BA ) (λA)B = A(λB)
! * +, - +, . %+,
A·B a11 ## #
...
a1n ## #
ai1 ## #
...
ain ## #
am1
. . . amn
b11 ## #
...
bn1
. . . bnk
→
b1k ## #
...
'
b1r ## #
. . . bnr
↓ pik
0 / # 0 - 1 $ m, n# 2 3. -' #"
A + 0 = A; A − A = 0 ; 0 · A = 0
$ n, n ,4 a11 , a22 , . . . , ann # 6 +, ,4
#
5,
7 a = 0 .3 i > k# • 7 a = 0 .3 i < k # • 7 a = 0 .3 i = k # 8 a = 1 .3 i ,4 E , •
ik
ik
ik
. +,.'
ii
E (n,n) · A(n,n) = A(n,n) · E (n,n) = A(n,n) E (m,m) · A(m,n) = A(m,n) · E (n,n) = A(m,n)
$3 E (n,n) +, 3 - E n #
#"9 #":
A = (aik )(m,n)
AÌ = (aki )(n,m)
! "#$%" &! ' (AÌ )Ì = A; (A + B)Ì = AÌ + B Ì ; (AB)Ì = B Ì AÌ
(
) A *
+ $#
A = AÌ ! aik = aki (i, k = 1, 2, . . . , n).
, ! % 1 3 0 A= 2 5 −1 −1 4 0 B= 3 0 2 ⎛ ⎞ ⎛ ⎞ 4 3 −1 2 C = ⎝ 2 1 −2⎠ , d = ⎝−1⎠ −1 3 4 3 - $! .#" $ / ' 5A − 3B, 2B + 7C (A + B)C, 2AB, BA 2AÌ B, 3B Ì A, −BC, CB
% Ad, Bd, Cd, dA, dÌ C dÌ AÌ
AA, AÌ A, AAÌ , CC, CC Ì C Ì C Ì , dd, dÌ dÌ , ddÌ , dÌ d
-
⎛ $#
⎞ 1 −2 2 3 1 A = ⎝−2 3 ⎠ , B = 4 6 2 4 −5 AB % BA 0 - det (AAÌ ) + 3 2 1 2 A= 4 1 1 3 -
⎛ ABC 2 −1 A = ⎝−2 0 1 2
+ ⎞ 3 1 2 4 1 −3⎠ 3 1 0
⎛
3 ⎜7 ⎜ B=⎜ ⎜−1 ⎝2 −4 ⎛ 1 ⎜−3 C =⎜ ⎝2 −1
-
⎛ 1 A = ⎝0 0
⎞ −2 1 3 −4⎟ ⎟ 1 3⎟ ⎟ 0 2⎠ 2 1 ⎞ −1 1⎟ ⎟ 1⎠ 2 0 2 5 1 2
det A + ⎞ ⎛ 0 0 0 0 1 0⎠ · ⎝0 b 0 1 c ⎛ 0 0 × ⎝0 1
⎞ a 0⎠ 0 ⎞ 0 1 1 0⎠ 0 0
1 -
2 3 3 X 453 ! , ! 4 0 X · XÌ = 0 9 6 - + ' 7 A % %! )3 & 8 (n, n) ! det (λA) = λn · det A $# λ ∈ R A & 8 0 D & 8 0 + 83 # B
C AB − 2(C + D)
!( Tk * $%&' * Bi )( Bi * $%&' * Ej & $%&' * Tk (k = 1, 2, 3, 4) +, - & . / $%&' * E1 / $%&' * E2 *
T1 , T2 , T3 , T4 B1 , B2 , B3 E1 E2 ! " # $%&'( ./ ( 0 A 1 ! ) & - 2 - 2 T1 T2 T3 T4 E1 E2 - 2 S = A + AÌ 3 & B1 4 2 0 5 2 1 B2 1 2 1 3 3 2 3 4 1 1 1 B3 0
* n! 4 & n* 5'* ⎛ ⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ a = ⎜ ⎟ , ai ∈ R 6/7 ⎝ ⎠ an
ai 8 9** 5'* a - ' a 6n, 17 - 2 , , 6$'* 7 & ! * 61, n7 - 2 & 64'* 7( aÌ = (a1 , a2 , . . . , an )
:% *& n* 5'*
Ì λ '" ( Ì
6.7
0 a = (a1 , a2 , . . . , an ) b = (b1 , b2 , . . . , bn ) * ( aÌ + bÌ = (a1 + b1 , a2 + b2 , . . . , an + bn ) λaÌ = (λa1 , λa2 , . . . , λan )
67 6;7
Rn & 0 λi ai = o o +
i=1
i=1
o = (0, 0, . . . , 0)Ì .# $ T x ∈ Rn $ B = {b1 , b2 , . . . , bm } T % , • bi ∈ T, i = 1, 2, . . . , m • B ( • -* . x ∈ T )
α1 , . . . , αm
m x= αk bk k=1
α1 , α2 , . . . , αm / x 0* B $ ' 1 Rn 2
{e1 , e2 , . . . , en } ei = (0 , . . . , 0 , 1 , 0 , . . . , 0)Ì ↑ i 3
5
a1 = (2, 4, 4)Ì , a2 = (−3, 2, −2)Ì a3 = (2, −1, 4)Ì (
b = (2, 2, 8)Ì & a1 , a2 , a3 ! S1 = {a, b, c|a, b, c ∈ R3 }
!4#
(
S2 = {a, a + b, a + b + c}
(
S = {(3, −1, 1)Ì , (−1, 3, 1)Ì , (1, 1, 1)Ì }
(
a1 = (4, 1, 3, −2)Ì a2 = (1, 2, −3, 2)Ì a3 = (16, 9, 1, −3)Ì a4 = (0, 1, 2, 3)Ì a5 = (1, −1, 15, 0)Ì 3a1 + 5a2 − a3 , 2a1 + 4a3 − 2a5 1 1 a1 + 3a3 − a4 + a5 2 2 x 2x + a1 − 2a2 − a5 = o
" x = (1, −3, 5)Ì ! S = {(1, −1, 0)Ì , (3, 5, 0)Ì } # b1 = (1, 1, 1, 1, 1)Ì b2 = (0, 1, 1, 1, 1)Ì b3 = (0, 0, 1, 1, 1)Ì b4 = (0, 0, 0, 1, 1)Ì b5 = (0, 0, 0, 0, 1)Ì
2(a1 − x) + 5(a4 + x) = o 3(a3 + 2x) − 2(a5 − x) = o
x = (19, 1, 10)Ì ! S = {(1, 1, −1)Ì , (0, −1, 1)Ì , (5, 0, 3)Ì , (−2, 1 − 3)Ì }
B = {b1 , b2 , b3 , b4 , b5 } R5 $ % & x '( B ) x = (1, 1, 0, 1, 0)Ì x = (3, 5, 4, −1, −2)Ì x = (−5, 4, −3, 2, −1)Ì
m, n A r, r r, r r A ! A " r
# $ %& $ " r +1
# A r A' ( (A) = r(A) = r;
r ≤ min (m, n)
)*+
% • , ! • • ! -. . ! ( -. % A = (aik )(m,n) r(A) = r /
⎛ d11 d12 ⎜ 0 d22 ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0
. . . d1r . . . d2r . . . drr
d1,r+1 d2,r+1
. . . d1n . . . d2n
dr,r+1
. . . drn 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
r ! "
# $ ! % & "' A ( a11 = 0 ai1 ' & ( ) − a11
i* & i = 2, 3, . . . , m ( +
⎛ ⎞ a12 ... a1n a11 ⎜ 0 b11 ... b1,n−1 ⎟ a11 a12 . . . a1n ⎜ ⎟ ,- ⎜ ⎟= o B ⎝ ⎠ 0 bm−1,1 . . . bm−1,n−1 , B A "
$ .! !
+ r(A) r /*
D (r,r) = (dik )(r,r) ( m n* ./ aÌ i = (ai1 , ai2 , . . . , ain ), i = 1, 2, . . . , m * 0 ! (m, n)* ⎛ Ì⎞ ⎛ ⎞ a1 a11 a12 . . . a1n Ì ⎜ a2 ⎟ ⎜ a21 a22 . . . a2n ⎟ ⎜ ⎟ ⎟ A=⎜ ⎟=⎜ ,$ ⎝ ⎠ ⎝. . . . . . . . . . . . . . . . . . . . .⎠ am1 am2 . . . amn aÌ m
1 r(A) = r #
• 2 r = m m ./ ai 3 • 2 r < m m ./ ai 3 4 m ./ ai ) r ./ 3 1 (dik )(rr)
⎛ ⎞ 2 1 −4 1 3 ⎜−4 7 5 −2 0 ⎟ ⎟ A=⎜ ⎝ 5 6 9 −3 −3⎠ 0 3 −1 0 2
⎛ ⎞ 2 −1 3 −2 4 A = ⎝4 −2 5 1 7⎠ 2 −1 1 8 2 ⎛ ⎞ 1 3 5 −1 ⎜2 −1 −3 4 ⎟ ⎟
B=⎜ ⎝5 1 −1 7 ⎠ 7 7 9 1 ⎛ ⎞ 4 3 −5 2 3 ⎜8 6 −7 4 2 ⎟ ⎜ ⎟ ⎟ C =⎜ ⎜4 3 −8 2 7 ⎟ ⎝4 3 1 2 −5⎠ 8 6 −1 4 −6 ⎛ ⎞ 0 2 −4 ⎜3 1 7 ⎟ ⎜ ⎟ ⎜ 5 −10⎟ D=⎜ 0 ⎟ ⎝−1 −4 5 ⎠ 2 3 0 ⎛ ⎞ 2 2 −1 1 2 ⎜4 3 −1 2 1 ⎟ ⎟ F =⎜ ⎝8 5 −3 4 −1⎠ 3 3 −2 2 1 ⎛ ⎞ 25 31 17 43 ⎜75 94 53 132⎟ ⎟ G=⎜ ⎝75 94 54 134⎠ 25 32 20 48
! " #
$ % & λ ∈ R' ⎛ ⎞ 3 1 1 4 ⎜λ 4 10 1⎟ ⎟ A=⎜ ⎝ 1 7 17 3⎠ 2 2 4 3 ⎛ ⎞ 1 λ −1 2
B = ⎝2 −1 λ 5 ⎠ 1 10 −6 λ
# ( #
)% $ $ a1 a2 a3 a4
= (1, 1, 1, 1)Ì = (1, 1, −1, −1)Ì = (1, −1, −1, 1)Ì = (1, −1, 1, −1)Ì
a1 = (4, −1, 5, 6)Ì a2 = (4, −5, 2, 6)Ì a3 = (2, −2, 1, 3)Ì a4 = (6, −3, 3, 9)Ì
* & )%# T = {a1 , a2 , a3 , a4 } + B T ( B , )% & T % # & B ( + a1 = (−3, −6, 0, 0)Ì a2 = (1, 2, 3, 4)Ì a3 = (1, 2, 0, 0)Ì
a1 = (3, 4, −1, 2)Ì a2 = (1, 1, −1, −2)Ì a3 = (4, 1, −2, 3)Ì a4 = (5, 2, −3, 1)Ì
n n a11 x1 + a12 x2 + . . . + a1n xn = a1 a21 x1 + a22 x2 + . . . + a2n xn = a2 ................................. an1 x1 + an2 x2 + . . . + ann xn = an
!
" #$% aik & ' ! ai ( i, k = 1, 2, . . . , n! " ! ' ) *
a11 a12 . . . a1n a21 a22 . . . a2n = D= 0 . . . . . . . . . . . . . . . . . . . an1 an2 . . . ann
!
" Dk (k = 1, 2, . . . , n) a11 . . . a1,k−1 a1 a1,k+1 . . . a1n a21 . . . a2,k−1 a2 a2,k+1 . . . a2n Dk = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . an1 . . . an,k−1 an an,k+1 . . . ann
!
+, ) x1 =
D1 D2 Dn , x2 = , . . . , xn = D D D
!
' ! - ' . / 0. 1 ! 2ax1 − 3bx2 = 0 3ax1 − 6bx2 = ab (ab = 0) ! 3x1 − x2 + 2x3 = 3 −x1 + 3x2 − 2x3 = −1 2x1 + 2x2 + 3x3 = 14
! x1 − x2 + x3 − x4 = −2 x1 + x2 + x3 + x4 = 0 = 5 x1 + 2x2 3x3 + 4x4 = −10 ! 3x1 + 4x2 + x3 + 2x4 = −3 3x1 + 5x2 + 3x3 + 5x4 = −6 6x1 + 8x2 + x3 + 5x4 = −8 3x1 + 5x2 + 3x3 + 7x4 = −8
P2 (x) = ax2 + bx + c P2 (1) = −1, P2 (−1) = 9 P2 (2) = −3
P3 (x) = ax3 + bx2 + cx + d P3 (1) = 2, P3 (−1) = 8 P3 (2) = −10, P3 (−2) = 26
! " ! #$ " ! % ! R · i = u i = (I1 , I2 , I3 , I4 , I5 , I6 )Ì
u = (0, 0, 0, 0, 0, E)Ì R ⎛= ⎞ 0 −1 0 −1 1 0 ⎜ 0 0 −1 1 0 1⎟ ⎜ ⎟ ⎜ −1 ⎟ 1 0 0 0 −1 ⎜ ⎟ ⎜ 0 −R2 0 R4 0 −R6 ⎟ ⎜ ⎟ ⎝−R1 0 R3 0 0 R6 ⎠ R1 R2 0 0 0 0 E = 10 &' R1 = R2 = R3 = 2 Ω R4 = R5 = R6 = 4 Ω ! ( ) " I6 !
% ( m % ! n & * a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 ................................................. am1 x1 + am2 x2 + . . . + amn xn = bm
+,
- $ * ⎞ ⎛x ⎞ ⎛ b ⎞ 1 1 a11 a12 . . . a1n ⎟ ⎜ b2 ⎟ x ⎜ a21 a22 . . . a2n ⎟ ⎜ 2 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟=⎜ ⎟ ⎝. . . . . . . . . . . . . . . . . . . . .⎠ ⎜ ⎝ ⎠ ⎝ ⎠ am1 am2 . . . amn xn bm ⎛
+.
" $* A · x = b A = (aik )(m,n) ' x = (x1 , x2 , . . . , xn )Ì b = (b1 , b2 , . . . , bm )Ì ∈ Rn / (0 " 12$ 3 !4 B = (A|b) / ( +, !4 ' b = o ' !4 / ! ( $ 0 ' ) ! 5 x = o ∈ Rn 6)! ! ! ( m = n det A = 0 ! )"' ! ! 0 5 > 7 m = n 8 ! ! A
0 .+ det A < 0 & / " *& c = o * / -
( $ 0 1 A = A(n,n) 2
AÌ A = E n
3 A det A = ±14 / & $ 5# / 0 1 ! A−1 = AÌ " 6%& ')%& % A ) $# %& 0 + i = k ai1 ak1 + ai2 ak2 + . . . + ain akn = 1 + i = k k 0 + i = a1i a1k + a2i a2k + . . . + ani ank = 1 + i = k $ 2 0# 1 A 3 . det A = 1 & " -
(
y1 = x1 + 1, 5 y2 = x2 − 2, 5 4 3 x1 + x2 y1 = 5 5 3 4 y2 = − x1 + x2 5 5 4 3 y1 = − x1 + x2 + 1, 5 5 5 3 4 x1 + x2 − 2, 5 y2 = 5 5 1 5 4 A= 3 4 5
⎞ ⎛ 1 3 5 A = ⎝2 4 −1⎠ 1 7 3 * + ai + bi / ,$ (31 3 [b1 , b2 , b3 ] 1
4 [b1 , b2 , b3 ] = det A · [a1 , a2 , a3 ]
% & ' R3 ( {e1 , e2 , e3 } ) * + x , -* . E : x1 + x2 + x3 = 0 / 0 P 1 2
5 " ⎛ ⎞ 1 2 −2 1 2⎠ A = ⎝2 1 3 2 −2 −1 #$ A−1 " (6 3) $ + x1 = (4, 5, −2)Ì x2 = (1, −2, −3)Ì x3 = (19, −10, 13)Ì $ $ 1 y = Ax A ) (6 y 1 , y 2 , y 3 3) $1
+ 7 7$ )
& ' R3 ( {e1 , e2 , e3 } + Ì aÌ 1 = (2, 3, −1), a2 = (1, −1, −3)" Ì a3 = (1, 9, −11) ,$ (33 [a1 , a2 , a3 ] $ 1 bi = Aai
" / #$ / ! 1 1 1 A = √ 2 −1 1 ⎛ ⎞ 1 1 1 −1 1 ⎜−1 1 1 1 ⎟ ⎟ A = ⎜ 2 ⎝−1 −1 1 −1⎠ 1 −1 1 1
! det A = 1" A #$
& A = A(n,n) 8$ " det (A − λE n ) = 0
955
$ $ n1 λ ( $: A
n λ1 , λ2 , . . . , λn A xi ! " #$
(A − λi E)x = o
xi % & μxi (μ = 0) '( ) ri =
xi |xi |
" #$
*+ % ! (AÌ = A) , $ '- )$ . '- '( $ / k '- k 0 - '( $ . n '( r i ( -1 - ) , 1 + i = k 2 3( 0 ( (r i )Ì · rk = 0 + i = k 4 ) " #5$
R = (r 1 , r2 , . . . , r n )
det R = 1 6 R 1 A . , ⎛ ⎞ λ1 0 . . . 0 ⎜ ⎟ ⎜ 0 λ2 ⎟ ⎟ RÌ AR = D = ⎜ ⎜ 0 ⎟ ⎝ ⎠ 0 . . . 0 λn 2
) ! ⎛ ⎞ 6 2 2 A = ⎝2 3 −4⎠ 2 −4 3 ' $ '- A )$ '(
" #7$
8! ) $ 9 RÌ AR = D 4 : '( ) ! A 8! ) (
⎞ λ1 0 0 RÌ AR = ⎝ 0 λ2 0 ⎠ 0 0 λ3 λ1 , λ2 , λ3 A ⎛ ⎞ 15 6 0 1 ⎝ A = 6 22 6 ⎠ 11 0 6 29 ⎛ ⎞ −2 2 −4 1
A = ⎝ 2 −5 −2⎠ 3 −4 −2 −2 ⎛
⎛ ⎞ 0 1 0 0 1⎠ A = ⎝ 0 −6 −1 4 ⎛ ⎞ 2 −1 2
B = ⎝ 5 −3 3 ⎠ −1 0 −2 ⎛ ⎞ 4 −5 2 C = ⎝5 −7 3⎠ 6 −9 4
⎛
4 −5 D = ⎝ 1 −4 −4 0 ⎛ 5 1 3 ⎜2 3 1 F = ⎜ ⎝4 2 4 3 1 3 ⎛ 0 1 ⎜0 0 G = ⎜ ⎝0 0 48 −28
⎞ 7 9⎠ 5 ⎞ 2 3⎟ ⎟ 6⎠ 4 0 1 0 −8
⎞ 0 0⎟ ⎟ 1⎠ 7
! " #$ ⎛ ⎞ 11 −6 2 A = ⎝−6 10 −4⎠ 2 −4 6 ⎛ ⎞ 1 2 2
B = ⎝2 1 −2⎠ 2 −2 1
% &'! P ' ( ) ! x" *' +' xÌ Ax + aÌ x + a0 = 0
, -
$ A (AÌ = A) ." / •
,0 "
A= •
a11 a21
a12 , a= a22
a1 , x= a2
⎞" ⎛
x1 x2
⎛ ⎞ ⎛ ⎞ a1 x1 a11 a12 a13 A = ⎝a21 a22 a23 ⎠ , a = ⎝a2 ⎠ , x = ⎝x2 ⎠ a31 a32 a33 a3 x3 , 1 a12 = a21 , a13 = a31 , a23 = a32
A det A = 0
x = y − v v =
1 −1 A a 2
! 1 y Ì Ay + b0 = 0 b0 = a0 − aÌ A−1 a 4
#
"
$
y = Rz
" % z Ì Dz + b0 = 0
! #! D # & ' A( R ) & *+ ', det R = 1 - . %/ " 0 1 .2* ( / ! * λ1 > 0 3 3 ! )4, 5 *
. " 0 6 λ1 z12 + λ2 z22 + b0 = 0 λ1 >0 >0 >0
λ2 >0 0 ∧ a > 0)
lim log an = log lim an = log a (an > 0 ∧ a > 0)
{an } {sn } sn = a1 + a 2 + · · · + a n =
n
ai
i=1
s = a1 + a2 + · · · =
∞
ai
s = lim sn
n→∞
i=1
! " # $% lim sn = s & ' s ( ∞ ) ai
i=1
# an % " * $ ∞ • | ai | ! " i=1
•
+ sn = sn =
n
n
2a1 + (n − 1)d a1 + (i − 1)d = 2 i=1 n
a1 q i−1 = a1
i=1
s =
∞
a1 q i−1 =
i=1
∞ 1 i=1 ∞ i=1
i
"
qn − 1 , q = 1 q−1
a1 , | q |< 1 ! " 1−q
∞ (−1)i+1 i=1
i
1 ! " , a > 1 " , a ≤ 1 ia
! "
∞ ai =⇒ lim an = 0 n→∞
i=1
∞ an+1 n lim |an | lim ai n→∞ n→∞ an i=1 1 =1
! "
∞
(−1)i+1 ai ai > 0 ! i # {an } $
i=1
# # lim an = 0 an+1 ≤ an ! n ≥ n0 n→∞
%& ∞
ai
# %&
∞
i=1
i=1
% ∞ ai
∞
i=1
i=1
' # ( | an |≤ bn ! n ≥ n0 .
# %
' # ( an ≥ bn ! n0 ≥ n0 .
bi
bi
) " ! * + " ! ,- . ∞ ∞ / 0 ai 1 2 c = 0# cai i=1
1 2 0 3 (
i=1 ∞ i=1
c · ai = c
∞
ai
i=1
" ' ! ' # 4
3 2n − 5n2 + 8 7n3 + 2 n+7 ! 2 1− n−3 √ 2 { 4n + 3n − 2n} a √ 3n + 4 1+ n an = an = 2n + 1 n3
n0 = n0 (ε) | an − a |< ε n > n0 !" ε = 0, 001 # $ % ε & ' ( % ) a n 2 3n − 2 2 n 5n + 1 3n 2n n2 2 2 n +1 n +1 * +
an+1 = q < 1 $ lim n→∞ an lim an = 0 n→∞ # + n n lim n = 0 lim 2 = 0 n→∞ 2 n→∞ n! n! (n!)2 lim n = 0 lim =0 n→∞ n n→∞ (2n)!
, + 6n − 3 an = 6 − 5n
an =
2n(n − 1)2 (n + 2)3
(2n − 1)3 (4n − 1)2 (1 − 5n) 1 an = (−1)n · 2 n +1 1 n an = (−1) · 1 + n 10 3 an = − √ n n 3 5n − 2
an = 3n − 1 n − 10 an = 3 n−1 an = 3 8n + 10 n - an = √ 3 3 8n − n − n √ an = n 2 + n − n (−0, 3)n an = 3n − 2 n 1 an = 1 + 2n n +3 5 4 an = 1 − n √ n an = 3
an =
27log3 n 16log2 n n n+3 / an = n−5
. an =
0 . a = 7 a = 3 1 2 a = −6
Ko = −3 Ku = −12
" 3 -
s ** 0 2 = 2%% 1%+ 5% {sn } ! 1 1 1 *, > 1 2 2 + + + ... " 1·2 2·3 3·4 4 1 2 3 k(k 1+ 1) = k1 − k +1 1 + ... " + + + 2 5 8 11 1 1 1 1 2 3 + + + ... " 2 3 1 1·3 3·5 5·7 " + + 5 7 9 4 1 (2k − 1)(2k = 4 + 1) + + ... 11 1 1 1 2
2k − 1
−
2k + 1
# $ % & '
! " 0, 25 " 0, 49 " 0, 562 ( & ) * + , ) * " ) ) * " ) ) ./ */ 0 1 2 r 3 4 5 + 6 1 + 3 2 4 5 + 7 ) 8 " 1 9: " ; 9: 3 " 0 $
"- x ∈ E
f (−x) = f (x)
%/
"- x ∈ D(f ) #
f (−x) = −f (x))
%0
•
p p > 0 f (x + p) = f (x)
x ∈ D(f )
!
" y = f (u), u ∈ D(f ) u = g(x), x ∈ D(g) # " W (g) ⊆ D(f ) $ x%
y = f g(x) , x ∈ D(g) &
'( )*) + , ( D(f ) ⊆ R f (x) √ f (x) = x + 1 1 ( f (x) = 4 − x2 √ f (x) = x4 − 2x2 √ f (x) = x − x3 √ 1 f (x) = −x + √ 2+x x2 − 3x + 2 f (x) = lg x+1 2x
f (x) = arccos 1+x x f (x) = arcsin lg 10 √ 1 − lg(2x − 3) f (x) = x + 3 x−2 - f (x) = 21/(1−x) f (x) = x − arctan x x−3 − lg(4 − x)
f (x) = arcsin 2 1 f (x) = | x2 − 2x − 3 |
. '( )*) + , ( D(f ) ⊆ R " ),
/ ( W (f ) 0 , ( 3 y = f (x) = √ x−5
x−3 x2 − x − 6 1 1 − y = f (x) = x+2 x−2 y = f (x) =⎧ ⎨3 − x2 | x |≤ 1 2 ⎩ | x |> 1 |x|
( y = f (x) =
1 2"" % y =
10 x2 + 1
( y =
2x − 3 3x + 2
y = | sin x| 3 y = − 25 − x2 5 π y = 5 sin 2x − 2 y = |x|
y = x|x| 1 x2 − 1 y = arcsin y = 2 x x −4 ! 2
, (" 3 4 5 f (x) = −7 sin x cos x ( f (x) = | sin x| f (x) = 6 e−2x f (x) = 3x + 7 f (x) = 5x4 − 2x2 + 3 1+x f (x) = lg 1−x
1 + x2 f (x + 1) = x2 − 3x + 2 3 x −1 1 1 ' f x + = x2 + 2 ; x = 0 x x √ 1 1 − 1 + 4x √ y = f (x) = ; = x +
f 1 + x2 ; x > 0 x " 11 + 1+ 4x 2 3 4 * D(f ) = − , +∞ 4 [−a , a] & f (x) ! 5 $ " # $ + % & $ "'$ ( 22 ' $ f .% . & $ & ' D(f ) $ ' D(f ) ⊆ R 6 7 - 89 : −1 )' * f + 5 : ; $ & '
' 4 y = f (x) = (x − 5)3 ; ' D(f ) = [5 , +∞) ' + 3 5
' y = f (x) = x2 + 1; D(f ) = (−∞ , 0] x3 − 2x2 − 9x + 18 y = f (x) = x2 − 7x + 12
y = f (x) = (x − 3)2 ; D(f ) = (−∞ , 3] x2 − 4 ' y = f (x) = 2 1−x x − 2x + 1 ; y = f (x) = 1+x 2 f (x0 ) %& x ∈ D(f ) \ {x0 }! # x0
# ' ) '
" % *! ) ! f (x) x0 % # ' & x0 f (x0 ) = 0
'+
f (x0 ) = f (x0 ) = · · · = f (n−1) (x0 ) = 0 ∧ f (n) (x0 ) = 0 ∧ n ≥ 2
',-
' . n ! # x0 )
• ! % f (n) (x0 ) > 0/ • ! % f (n) (x0 ) < 0'
. n ! # x0 ) ' & x0 f (x0 ) = 0
',+
f (x0 ) = f (x0 ) = · · · = f (n−1) (x0 ) = 0 ∧ f (n) (x0 ) = 0 ∧ n ≥ 3 ∧ n ',, '
f (x) % . ) I • ! f (x) ≥ 0 %& x ∈ I, • ! f (x) ≤ 0 %& x ∈ I.
# ' % I ) %& f (x) > 0 # '
f (x) < 0 %& x ∈ I '
" ) 0! # # * ' # 1! ' ' % % 2% 3
! " # ! $ %! & %! ' ( ( ) * x → +∞ +,- * x → −∞ ! ) . / , ( f (x) g(x) ,0 [a, ∞) 1
lim (f (x) − g(x)) = 0
x→+∞
2 ! g 2 ! f ( 3 - ( 4! * 5 (−∞, a] x → −∞6 0 * + 4 6 7 898 :1 + ! - 5 - 6 f (x) = cosh2 x + 1 2x2 − 1 +6 f (x) = x4 6 f (x) = x 1 − x2 x 6 f (x) = ln x 6 f (x) = x − 2 sin x ; - 5 + 7 < 6 f (x) = x3 ; D(f ) = (−∞, +∞) +6 f (x) = ex ; D(f ) = (−∞, +∞) x 6 f (x) = ; 1 − x2 D(f ) = (−∞, −1) ∪ (−1, 1) ∪(1, +∞) " 0 ! 7 =
√ 3 6 f (x) = 2x + 3 x2 ; D(f ) = (−∞, +∞) +6 f (x) = ln 1 + x2 + arctan x; D(f ) = (−∞, +∞)
6 f (x) = 2x ex−2 +4 ex−2 −x2 − 6x; D(f ) = (−∞, +∞) 6 f (x) = sin3 x + cos3 x; D(f ) = [0, 2π) 1 6 f (x) = (x − 4) · cosh (2x + 3) 2 1 − sinh (2x + 3); 4 D(f ) = (−∞, +∞) 1 6 f (x) = (x2 − 6x + 5) · ln (x − 1) 2 5x x2 ; − + 4 2 D(f ) = (1, +∞) 6 f (x) = x(ln x)2 − x ln x + x; D(f ) = (0, +∞) 0 89 % + 7 :1 + = 6 f (x) = x(10 − x); D(f ) = [0, 10]
f (x) = x3 − 3x + 3; 3 5 D(f ) = − , 2 2
! " #$ %&" ' ( 2
y = f (x) = e−x ; −∞ < x < +∞. ) * ( y +,
x → ±∞ - ' ( &
7 8 9 ( :;< ! =
" 8 8 :< - 3 > " 0 1 7 " 4 a 60 h ?" 0 1
! ?"
. , +, / " ' (" # 01 0 2 3 :< = @ -," >& % 3 "
& - a 1 4 & " #" 1 A B , y = f (x) = 2x3 − 6x2 " ' 2 2
" % y = f (x) = x (9 − x ) - A 1 y = f (x) = x3 − x2 − 3x 3 ? 3 ' 0 2x2 + 1 1 C y = f (x) = 2 x −9 :: ! = x2 + x + 14 y = f (x) = / ! @ x+2 !D 3 * x+3 ( 32 3 3 +, = + y = f (x) = x−3 % ! 6 0 y = f (x) = 3 + x2 ( x+2 √ y = f (x) = :; -" " x 4 & 5 y = f (x) = x(x2 − 9) :< ! 01
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# * # a # # # # A(ϕ = α; r = a) # , # :
-
' # ; , M (ϕ = 0; r = a) 0) π 6 " ! ϕ = * (x0 ; y0 ) # 4 π ' % r = r(ϕ) = 5ϕ ϕ = 3 x2 y2 % + =1 & # a2 b2 " % y 2 = 2px " ' # % x = x(t) = 2 cos t − cos 2t, y = y(t) = −2 sin t + sin 2t * ' π t = 6 % y = x4 − 4x3 − 18x2 x = y = 0 % x = x(t) = sin 2t, % x2 + xy + y 2 = 3 x = y = 1 y = y(t) = sin2 t π x2 y2 t = % 2 + 2 = 1 " 8 a b A(a; 0) B(0; b) t % x = x(t) = arcsin √ , 1 + t2 % x = t2 , y = t3 x = y = 1 1 y = y(t) = arccos √ % r2 = 2a2 cos 2ϕ ϕ = 0 2 1 + t √ (a > 0) t = 5 ( ) * P1 (1; 0) + x y = e + # " , ' # 1 , % y = 1 + x2 - . + % x2 − y 2 = 4 /$, % y = sin x xy = a2 " 0 , # % 2y = x2 + 4x " % y = ex 1 2 2 % y = e−x , 3 % x = a(t − sin t), y = a(1 − cos t) % y = x2 y = x−1 % y = x e−x % y = sin x y = 0
! " r = r(ϕ) = a(1 − cos ϕ), a > 0 2
2
" r = a cos 2ϕ, a > 0 #" r2 =
2
a ,a>0 cos 2ϕ
x2 2 #" x = 2 cos t, y = sin t
" y = 1 −
" x = a cosh t, y = a sinh t " x = a(cos t + t sin t) y = a(sin t − t cos t) *" y 2 = 2x + 2
$ %#& y 2 = 8x #& 0, 128'
t3 3 3 &" x = a cos t, y = a sin3 t
( #&
) * * + !
" r = eϕ
" xy = 4 x = 2 " y = ln x +#&
) y ,#& x3 + 1 #" y = − +#&
)
3 x,#& " y = e +#&
) y ,#& x
" x = t2 , y =
. &3 & 1 #& * r = r(t) ' " r = ae1 cos t + ae2 sin t + cte3 " r = e1 cos t + e2 + e3 sin t #" r = t(1, 1, 1)Ì
" r = (1, 1, t)Ì
" r = (cosh t, 0, sinh t)Ì
4 / % #& #& x3 /#&% #& " y = x = −1 3 * 5 ) ! *" y 2 = x3 (1; 1) π π " r = (3 cos t, 3 sin t, 4t)Ì , t = " y = cos x x = 4 4 2 3 " r = te1 + t e2 + t e3 , t = 1 - . /#& 51 6 r r˙ * y = x2 − 6x + 10 P0 (3; 1)' #& & 1 %
r = (cos t, sin t, 1)Ì 0 %1 x
y = a cosh a + * #& y 2 /a /#& #& 2 #&
7 2 %#& x,#& (a > *! √ 0) π " r = (cos t, sin t, t 3)Ì , t = 2 + * 2 " r = (a sin t, b sin t cos t, c cos2 t)Ì , /#& *! π t= " y = x2 4
√ √ 2 t 2 t e sin t, 1, e cos t)Ì , r = ( 2 2 t=0 r = (t, t2 , t3 ), P (2; 4; 8) sin 2t π
r = R(cos2 t, , sin t), t = 2 4 Ì r = t(cos t, sin t, b) , P (0; 0; 0) x2 + y 2 = 10, y 2 + z 2 = 25, P (1; 3; 4)
x : y : z
x1 + 3x2 + 2x3 = 0 $
% & ' P (x, y, z)
r = r(t) = (t + 1, t2 − 1, t3 )Ì ( ! " #
x + 2y + z − 1 = 0$ )
*
√ r = (cos t, sin t, 2t 2)Ì π x, y # & t = $ 4
+ , (( ( - r = r(u) = (u + 1)e1 + u2 e2 + (2u − 1)e3
4 3 2 r = r(v) = 2v 2 e1 + (3v − 2)e2 + v 2 e3 t t t r = e1 + e2 + e3 ' ( 4 3 2 ! " #
* &
. ( & / ( ( .0" ( ( 1 " / (/ ( ( 2 " / ("( " # ( 2 3 ( (
• y
y
6
f 2 a b x f1
x = x(t) 6 y = y(t) x a b
#b A=
f2 (x) − f1 (x) x,
a
5 +
( f2 (x) ≥ f1 (x) 2 4 ( x ∈ [a, b].
#t2 y(t)x(t) ˙ t,
A= t1
( a = x(t1 ), b = x(t2 ) y(t) ≥ 0 2 t ∈ [t1 , t2 ].
5 6
y d6 g1 g2 -x c
#d (g2 (y) − g1 (y)) y,
A= c
g2 (y) ≥ g1 (y) y ∈ [c, d].
y 6 d x = x(t) y = y(t) -x c
#t2 x(t)y(t) ˙ t,
A=
t1
c = y(t1 ), d = y(t2 ) x(t) ≥ 0 t ∈ [t1 , t2 ].
• y
6
P2 k
1 A= 2
#ϕ2
1 A= 2
#t2 (xy˙ − y x) ˙ t
t1
P1 -x
k : x = x(t), y = y(t) t ∈ [t1 , t2 ] P1 (x(t1 ), y(t1 )), P2 (x(t2 ), y(t2 )).
r2 (ϕ) ϕ k : r = r(ϕ), ϕ ∈ [ϕ1 , ϕ2 ]
ϕ1
OP1 = r(ϕ1 ), OP2 = r(ϕ2 ).
!" " #$!" % & ' % &
( ) y = 4 − x2 , y = 0 & y = 6x − x2 , y = 0 !
2
2
x y + 2 =1 2 a b
y 2 = 2px, x = h (p > 0, h > 0)
y = 3 − 2x − x2 , y = 0 y = x3 , y = 8, x = 0 xy = 4, x = 1, x = 4, y = 0 " y 2 = 1 − x, x = −3 y = ln x, x = e, y = 0 * y 2 = 2x + 4, x = 0 + y 2 = x3 , y = 8, x = 0 y 2 = (4 − x)3 , x = 0
4(y 2 − x2 ) + x3 = 0 y 2 + x4 = x2 y = x2 + 4x + 5, x = 0, y = 0 ! " #! y = sin x y=0 $ y = x2 , y = 2 − x2 4y = x2 , y 2 = 4x y = x2 + 4x, y = x + 4 xy = 6, x + y − 7 = 0 a2 y 2 = x3 (2a − x) (y − x)2 = x3 , x = 1 % x3 + x2 − y 2 = 0 & ! y 2 (2a − x) = x(x − a)2 x ' y = a cosh a x = ±a, y = 0 ( ) * # #+ %
y = 0 ,' x = a(t − sin t), y = a(1 − cos t) # x = a cos3 t, y = a sin3 t -. / r2 = a2 cos 2ϕ -0 r = a(1 − cos ϕ) - r = 3 + sin 2ϕ ( r = 2 − cos 3ϕ 1% ## 23 4 r = a eϕ , ϕ = −π, ϕ = π r = |a cos 2ϕ| r = |a sin 3ϕ| r = a(sin ϕ + cos ϕ) a π ≤ ϕ ≤ 2π 1 r = , ϕ 4 r = a(1 + sin2 2ϕ), r = a 5 ϕ r = a sin3 3 6 4 x3 + y 3 − 3axy = 0
0 # 73+ 0 8 + + 9+ * + + + : 4 # +
s AB y = f (x), x1 ≤ x ≤ x2 #x2 s= 1 + (y )2 x x1
s = 1 + (y )2 x = x2 + y 2
x = x(t), y = y(t), t1 ≤ t ≤ t2 #t2 s= x˙ 2 + y˙ 2 t
s =
x˙ 2 + y˙ 2 t
t1
r = r(ϕ), ϕ1 ≤ ϕ ≤ ϕ2 #ϕ2 s= r2 + (r )2 ϕ
s =
r2 + (r )2 ϕ
ϕ1
s AB r = r(t) = (x(t), y(t), z(t))Ì , t1 ≤ t ≤ t2 #t2 #t2 2 2 2 s= x˙ + y˙ + z˙ t = t1
t1
r t
2
t
s = x˙ 2 + y˙ 2 + z˙ 2 t = x2 + y 2 + z 2
! " P0 P #" $ r = r(t)% !
t0 & t ' % ( ) #t 2 + (y(t)) 2 + (z(t)) 2 t. s= (x(t)) ˙ ˙ ˙
*
t0
( + P0 " ( ) s , $ '
+- ) +- #" $ r = r(s) = (x(s), y(s), z(s))Ì .
t t=
r = (x (s), y (s), z (s))Ì s
& 0) |t| = 1. 1 " + 2 $" t 2 r 2 n = s = 2s |n| = 1. t r s s2
/
n
t n
b
b = t × n |b| = 1
!
b, n, b " # $ % n b n t b
'( )"
t & "
b
6
P t+
sn
" #
κ ≥2 0 r = r(s) s
r * 2 + s 2 r 1 κ = 2 = (x (s))2 + (y (s))2 + (z (s))2 = s
( !
,
!
w +
w=
r 2 r 3 r , , s s2 s3 2 2 r s 2
-!
# t b 1 = k · n = · n, = −w · n ! s
s . r = r(t) " " / t 0 T =
r = r˙ * t
!
r 2 r × 2 = r˙ × r¨ t t N = B × T
B=
! " # t, b, n$ t=
B N T , b= , n= . |T | |B| |N |
"& '
% $
r 2 r t × t2 |r˙ × r¨| κ= = 3 r |r| ˙3 t w=
r 2 r d3 r , , t t2 t3 r 2 r 2 × t t2
=
[r, ˙ r¨, r¨] |r˙ × r¨|2
) *" ) & + + , # !* y 2 = x3 ) ' *" 4 - x = +*"
3 4 + y 2 = (2 − x)3 +*"
9 *" x = −1 * x2 + y 2 = a2
2
2
2
(
x 3 + y 3 = a 3
2π π ≤x≤ y = ln (sin x), 3 3 1 1 y = ln (1 − x2 ), − ≤ x < 2 2 y 2 = (x+1)3 +*"
*" x=4 x2 − 1 +*"
*" " y = 2 y=0 x y = a cosh , −a ≤ x ≤ a a
12 3 ≤x≤ 4 5 2 y = 2px +*"
*" p x= 2 y = ln (2 cos x) '*" + *"+ /*"
0
y x1*" 9y 2 = x(x − 3)2 '*" /*"
0 x1*"
. y = ln x,
e2y tanh x = 1, 1 ≤ x ≤ 2 ) 23 x = a(t − sin t), y = a(1 − cos t) t4 t6 0 x = , y = 2 − '*" 6 4 /*"
0 , *" 1 4 x = t2 , y = t(t2 − 3) ' 3 *" /*"
0 x1*" , r = a(1 − cos ϕ)
r = aϕ ϕ r = a sin3 3 ! " # $ %& ' A B () * ' A B #+ ) ) ,-) )# .#/ AB = 2b * '( )-) f 0 $ $ '/ # ) ( )/ f ) $ b 2 f2 s ≈ 2b(1 + ) 3 b2 # 1 2&/ 0 ) ( √ 1 1 + α ≈ 1 + α ( α 2 ) #/ 2 x = t, y = t2 , z = t3 , 3 0≤t≤3 # x = 3 cos t, y = 3 sin t, z = 4t, 0 ≤ t ≤ t0 x3 x2 , z= , 0≤x≤3 y = 2 6 x = t − sin t, y = 1 − cos t, t z = 4 sin , 0 ≤ t ≤ π 2 √ x = et , y = e−t , z = t 2, 0≤t≤1 x2 1 , 1≤x≤2 ( y = ln x, z = 2 2 x = t − sin t, y = 1 − cos t, t z = 4 cos %& ) %& 2 ) x, z /"#
, N & %)- ") t, b n ' t = 0 x = 1 − sin t, y = cos t, z = t 4 ) , / )# % x = t, y = t2 , z = t3 ' t = 1 ( 5 4 ) , / / x = et , y = e−t , z = t ' t = 0 ( 6 4# r = (t cos t, t sin t, t)Ì 1 ) )/ # 4 / ) , / 3
( 7 4) # # / r = r(t) ' t % ' s 8 .( t = 0 # )/ ( r = r(s) ") t, b, n & κ w #
$/ 8 ( .(/ # 9 √ r = r(t) = (2 cos t, 2 sin t, t 5)Ì # # # r = r(t) =
t (t − sin t, 1 − cos t, 4 sin )Ì 2 ' P (x = π, y = 2, z = 4)
) ( √ r = r(t) = (et , e−t , t 2)Ì b ' t = 0 / )
3 T B : 4 ) , /
21 2 23 1
y = x , z = y
x = 1
!" ,, ( t . t = 1 - - !" 2 r = r(t) = (t, t , t )Ì , 3 0 !" . 1 w , * ' r = r(t) = (t, t , t )Ì ,, . t = 0 √ ,' r = r(t) = (et , e−t , t 2)Ì ,, . t=0 x x , z = !' y = , 2 3 , . x = 1 ' r = r(t) = (2t, ln t, t )Ì ,, . t = 1 y ' x = , z = x ,, 2 . y = 1 ' r = r(t) = (et sin t, et cos t, et )Ì t = 0
!" #!" v = vt !" t $ % &' !" a v ˙ + n ( ") * a = vt
+ ,- !" , x = t, y = t − t -, t - , !" . " ( !" ( t = 0 , , (
,- !" ( x = 4 cos t, y = 3 sin t -, t - , !" . " ( !" π ( t = 4 / !" . "
κ
1 2!" !" , &r = r(t) = (a cos t, a sin t, bt)Ì ) ( 3 !" , &r = r(t) = (a cos t, −a sin t, bt)Ì ) (a > 0, b > 0)
" , !" , !" " !" 45" "
% - 6 . Δ
& !" - ' !" x, y, z 7 x x , # )!" !" B %. 8 x ∈ [x , x ] Ex 8 , (x; 0; 0) ") ( y, z , Q(x) " !" Ex B !"
VB B !
#x2 VB =
" #
%$Q(x) x x1
A V y = f (x) ≥ 0, x1 ≤ x ≤ x2 & x'( ) #x2 Ax = 2π y 1 + (y )2 x
#x2 Vx = π [f (x)]2 x
x1
" #%
x1
x = g(y) ≥ 0, y1 ≤ y ≤ y2 & y '( )
#y2 Ay = 2π
x 1+
x y
2
#y2 Vy = π [g(y)]2 y
y
y1
" #*%
y1
x = x(t), y = y(t), t1 ≤ t ≤ t2 & x'( ) #t2 Ax = 2π
|y(t)| x˙ 2 + y˙ 2 t
" #%
t1
#t2 Vx = π t1
y 2 (t) · x(t) ˙ t, + , & - x'( .
" ##%
r = r(ϕ), ϕ1 ≤ ϕ ≤ ϕ2 & / p)
#ϕ2 Ap = 2π
r(ϕ)
ϕ1 #ϕ2
Vp =
2 π 3
ϕ1
r2
+
r ϕ
2 sin ϕ ϕ
r3 (ϕ) sin ϕ ϕ, + , r(ϕ1 ), r(ϕ2 ) r = r(ϕ) ! & ' 0 p
" #%
" #1%
! " # $%&'( ' ) *+ , ) -. $/ . a b' 0 h ' $1 R' ' ! 2 3 x = 1 x = 32 Q(x) 4
-. , ! ! 4
" 3 5 ! Q(2) = 27 ' 6,2 ! ) ,-. 1 , / . A B # a b 2 #. 0 h
' ! 3 0 h2 ) -. ! 0 a b
7' 8+ 72 3 + $1 R' 2 # /
8+ , . 9 , α
: 1 , ! 2 1 3 ;, x" # ' y 2 = 2px, x = h ' xy = 4, x = 1, x = 4 x ' y = a cosh , x = ±a a π , ' y = cos x − 3 5π x = 0, x = 6 ' y = sin x, x = 0, x = π
7' y = 2
1 , x = ±1 1 + x2 2
2
' x 3 + y 3 = a 3 ' x2 − y 2 = a2 , x = ±2a ' (y − 3)2 + 3x = 0, x = −3
0
x
'
n > n0 (ε)
x∈I
/
I = [a, b] ! " # $ ci % |fi (x)| ≤ ci
&
&
x∈I
i
∞ &
i=1
ci
0
I = [a, b] fi (x) (i = 1, 2, 3, ...) I #b $ ∞ a
i=1
% fi (x)
∞ #
#b
b
x =
fi (x) x =
i=1 a
$ % &
( fi (x)
s(x) x
∞ &
fi (x) ' I = [a, b]
i=1
∞ &
fi (x)
i=1
∞ & i=1
fi (x) I
$∞ % ∞ fi (x) = fi (x) = s (x) ) x ∈ I x i=1 i=1 % ) |x| < 1 +
1 + x + x2 + x3 + . . . & ' [0 , 1/2] ) , n ) |rn (x)| < 0, 001 - x ' .
! "#
a
! *#
|rn (x)| < 0, 01 - x ' [1/2 , 1].
/
x3 x4 x x2 − + − ± ... 1 2 3 4 ' [0 , 1] ) , n x ' |rn (x)| < 0, 1.
$ /
1 1 1 − 2 + 2 2 x +1 x +4 x +9 1 ± ... − 2 x + 16 / ) , n ! x# |rn (x)| < 0, 0001.
2 /
x3 x3 x3 + + + ... 3 1+x (1 + x3 )2 ) x > 0 ) x ≥ 1 1 0 ) , n |rn | < 0, 001 ) x ≥ 1.
/
x + x(1 − x) + x(1 − x)2 +x(1 − x)3 + ... ' [0 , 1] 0 1 ' [1/2 , 1] ) , n )
3 /
1 1 + x(x + 1) (x + 1)(x + 2) 1 + ... + (x + 2)(x + 3) (0 , +∞) 1/x
n x > 0 |rn (x)| < 0, 1 ! " # $ ! $ % "$ [0 , +∞) &' " n |rn (x)| < 0, 01 1 1 $ √ + √ 1 + x 3 1 + 3x 1 1 + 2√ + 3√ + ... 3 1 + 5x 3 1 + 7x 1 1 1 √ +√ +√ 1+x 22 + 2x 24 + 3x 1 + ... +√ 26 + 4x ( ! " # $ !
$ %! "$ (−∞ , +∞) &' ! " 1 1 $ cos x + cos 2x + 2 cos 3x 2 2 1 + 3 cos 4x + . . . 2 1 1 x x sin x + 2 sin + 2 sin 2 2 3 3 1 x + 2 sin + ... 4 4 ) * " x " $ 1 + e−x + e−2x + e−3x + . . . 1 1 1 1 + x + x + x + . . . 2 3 4 1 1 1 + + + ... 1 + x2 1 + x4 1 + x6
+, ∞ n=0 ∞
an xn = a0 + a1 x + a2 x2 + a3 x3 + . . . an (x − x0 )n = a0 + a1 (x − x0 ) + a2 (x − x0 )2 + . . .
)
n=0
. / , ) , x0 = 00 $ ) |x| < r " # |x| > r 1 " .$ ' $ % "$ (−r , r) $ / ! , ) . / , $ x0 = 0 % r # , $ (x0 − r , x0 + r) $ 2 , 3 # r (r ≥ 0) an r = lim n→∞ an+1
r#
, 1 r = lim n→∞ n |a | n
4
r = 0 r = ∞ x ∈ R 0 < r < ∞ x |x − x0 | < r x |x − x0 | > r x = x0 ± r !" # $ % & !" ! " & $ ' !" ! " % (
!" % ) ∞ ∞ & & * ai xi = s1 (x) |x| < r1 bk xk = s2 (x) |x| < r2 # " i=0
k=0
|x| < r = min(r1 , r2 )) $∞
% $ ·
i
ai x
i=0
∞
% k
bk x
∞
=
cn xn = s1 (x) · s2 (x)
+
n=0
k=0
cn =
n
al · bn−l = a0 bn + a1 bn−1 + · · · + an b0
l=0
, !" )
$
s (x) =
∞
% n
an x
=
n=0
#b s(x) x = a
#b $ ∞ a
#x s(t) t = 0
nan xn−1 , |x| < r
n=0
n=0
%
n
an x
x =
n=0
#x $ ∞ 0
∞
!" !
an ·
n=0
bn+1 − an+1 , n+1
-
[a , b] ⊂ (−r , r)
% a n tn
∞
t =
∞ n=0
an ·
xn+1 , x ∈ (−r , r) n+1
.
* f (x) * ! I # x0 * /# $ $ " 0 $# 1 "$ ∞ f (k) (x0 ) k=0
k!
(x − x0 )k
2
f (x) =
f (x) x0
∞ f (k) (x0 )
k!
k=0
(x − x0 )k x ∈ I
lim Rn (x) = 0 x ∈ I,
n→∞
Rn (x) ! " # $ %& ' ( f (x) ) "! x0 = 0* ∞ f (k) (0) k=0
k!
xk
+ ( $ $ " ( , -. ( . / -. ( . ! 0 $ ! $ -. ( |x| = r x2 x4 x + 2 + 3 + ... 1 + 3·2 3 ·3 3 ·3 x3 x5 x 1 − √ + √ − √ ± . . . 5 2 52 3 53 4 4x2 2x $ 1 + 2 √ + √ 3 3 52 32 8x3 + √ + ... 72 33 2 4x 2x +√ 1 + √ 5·5 9 · 52 8x3 +√ + ... 13 · 53 x4 x2 √ + √ 1 − 3 · 2 2 32 · 3 3 x6 √ ± ... − 33 · 4 4 ∞ xn n! n=1
∞ (−x)n−1 n n=1
1
∞
3n xn (3n − 2) · 2n n=1 ∞ 10n xn √ n n=1 ∞
(−1)n−1 ·
n=1
∞
x2n−1 2n − 1
xn−1 n!
n=1
∞ n! n x nn n=1
(x + 1)3 (x + 1)2 + 2·4 3 · 42 (x + 1)4 + + ... 4 · 43 2x − 3 (2x − 3)2 − 1 3 (2x − 3)3 + ∓ ... 5
(x + 1) +
x − 1 (x − 1)2 + 1·2 3 · 22 +
(x − 1)3 + ... 5 · 23
2x + 1 (2x + 1)2 + 1 4 (2x + 1)3 + + ... 7 s(x) ! 1 + 2x + 3x2 + 4x3 + . . . ! 1 − 3x2 + 5x4 − 7x6 ± . . . x5 x7 x3 + − ± ... " x − 3 5 7 2 3 x x + + ... x + 2 3 1 + 3x + 5x2 + 7x3 + . . . # 1 − 4x + 7x2 − 10x3 ± . . .
s(x)x s(x) x s + s · x
#
s(t) t
0
s−s·x
$ #% ! & ' #( ! ! ) *!" $ + " #% n → ∞ f (x) = cosh x ! f (x) = sin2 x
#% & ' f (x) = ex/a , ' x0 = a #( ! "
- . ! #% # & ' " " /
. ! , ' x0 0 f (x) = ecos x , x0 = 0 √ ! f (x) = x3 , x0 = 1 1 " f (x) = , x0 = 2 x f (x) = ln cos x, x0 = 0 1 . ! #% # & ' / "' , ' x0 = 0
2 3/ !' 0 1 f (x) = (1 + x)3 1 ! f (x) = √ 1 + x2 1+x " f (x) = ln 1−x f (x) = ln (2 − 3x + x2 ) f (x) = ln (1 − x + x2 ) 1 # f (x) = √ 1 − x2 f (x) = e−x
2
f (x) = x e−2x 3x − 5 f (x) = 2 x − 4x + 3 x 4 f (x) = 9 + x2 ' f (x) = sin 3x + x cos 3x f (x) = cos2 x 1 f (x) = √ 3 + 2x . ! #% # & ' # , ' x0 / 2 3 / !' 0
f (x) = ln x, x0 = 1
f (x) =
x2
1 , x0 = −4 + 3x + 2
1 , x0 = −2 x x π f (x) = cos , x0 = 2 2
f (x) =
π f (x) = sin 3x, x0 = − 3 √ f (x) = 3 x, x0 = −1 f (x) = x4 − 4x2 , x0 = −2 πx , x0 = 1 f (x) = sin √ 3
f (x) = x, x0 = 4 f (x) = ex , x0 = −2 1 f (x) = 2 , x0 = −1 x 1 f (x) = 1 + x2 ! " " # $ % arctan x 1 & ! x = √ 3 $ % arctan x ' $ ! π π (
$ ) $ % '
* # ) $ + $ , # # x sin x e x x x x & #x 2 Φ(x) = e−t dt - ! 0
(x0 = 0) Φ( 13 ).
$ $ $ / . 0, 000001 ! & #x 3 Φ(x) = 1 + t2 t ' 0
Φ( 15 ) ) $ ! ' $ / . 0, 00001 ! 0 " $ #x t2 Φ(x) = cos t 4 0
Φ( 12 ) ' 0, 000001 1 2 " + x = a cos t, y = b sin t, a > b > 0
3 # I = [−l , l] f (x) ( . %$ . %$ . % x ∈ I , f (x) =
1 f (x − 0) + f (x + 0) . 2
5 ( f (x)
4 00
,
a0 nπx nπx an cos + + bn sin 2 l l n=1 ∞
f (x) =
4 01
1 an = l
#l −l
nπx f (x) cos x ; l
1 bn = l
#l f (x) sin −l
n = 0, 1, 2, . . .
nπx x l
n = 1, 2, 3, . . .
!" " "# (−∞, +∞) " !! $ #"!! % 2l "# I " &" #'!!
f (x)
%
f (−x) = f (x) (
∞
f (x) =
an =
2 l
f (x)
f (x) =
a0 nπx + an cos 2 l n=1 #l f (x) cos 0
∞ n=1
bn =
2 l
bn sin
nπx x, n = 0, 1, 2, . . . l
%
f (x) sin
+
nπx x, n = 1, 2, 3, . . . l
- ! #! % 2l = 2π
. !# "!
/ ! " ! #' 0 1 2" ! ( " f (x) = 1 #' x ∈ (0 , π) f (−x) = −f (x); π 1 1 1 1 − + − + ··· = 3 5 7 4
*
f (−x) = −f (x) (
nπx l
#l 0
)
,
f (x) = x #' x ∈ [0 , π] f (−x) = f (x); 1 1 1 π2 1 + 2 + 2 + 2 + ··· = 3 5 7 8 2 f (x) = x #' x ∈ [−π , π]; 1 1 π2 1 1 − 2 + 2 − 2 + ··· = 2 3 4 12 1 1 1 π2 1 + 2 + 2 + 2 + ··· = 2 3 4 6 π #' x ∈ (−π , 0) f (x) = π − x #' x ∈ [0 , π]
π−x x ∈ (0 , π] 2 f (−x) = f (x) f (x) = | sin x|; 1 1 1 1 + + + ··· = 1·3 3·5 5·7 2 x ∈ [0 , π/2] x f (x) = π − x x ∈ [π/2 , π]
f (x) =
f (x) =
1 x
x ∈ [−1 , 0) x ∈ (0 , 1]
l = 1 f (x) = ex x ∈ (−l , l) " f (x) # $ [0 , 2] % ! y 6
f (−x) = −f (x) 2l f (x) = 1 x ∈ (0 , l) f (−x) = −f (x) ! f (x) = 1 − x x ∈ [0 , 1], f (−x) = f (x) l = 1 0 x ∈ (−l , 0] f (x) = x x ∈ [0 , l) f (x) = x x ∈ [0 , l) f (−x) = f (x)
-x & '
! 2l = 4 !
f n x1 , x2 , . . . , xn n (x1 , x2 , . . . , xn ) ∈ D(f ) ⊆ Rn y ∈ W (f ) ⊆ R
y = f (x1 , x2 , . . . , xn ) = f (x) = f (P ), x ∈ D(f )
x = x1 e1 + x2 e2 + · · · + xn en = (x1 , x2 , . . . , xn )Ì
! "! P (x1 , x2 , . . . , xn ) #
$ % $ n = 2 z = f (x, y)& w = f (x, y, z); u = f (x, y, z) n = 3 ' d(P, Q) "! Q(q1 , q2 , . . . , qn ) d(P, Q) =
P (p1 , p2 , . . . , pn )
(p1 − q1 )2 + (p2 − q2 )2 + · · · + (pn − qn )2 = |p − q|
z = f (x, y), (x, y) ∈ D(f )
R3
P (x1 , x2 , . . . , xn ) ∈ Rn
! "
f (x1 , x2 , . . . , xn ) = c = const. # $
% f (x , x , . . . , x 1
R2
2
n ) &%
c
&% '( ) % w = f (x, y, z) R3
&% '( ) %
*%
!+ % ,
% + + .
-$ - .
+
.
z = f (x, y)
/ +
.
% &% &.
+ , 0
% 1
z = f (x, y) = −3x + 4y + 8 z = f (x, y) = 25 − x2 − y 2
z = f (x, y) = xy z = f (x, y) = x2 + y 2 4 z = f (x, y) = 2 x + y2 √ z = f (x, y) = xy ! " # " x, y "$ 1 f (x, y) = 4 − x2 − y 2 x √ f (x, y) = arcsin + xy 2
f (x, y) = ln (x + y) 1 f (x, y) = √ y− x f (x, y) = 1 − x2 + 1 − y 2 f (x, y) = x2 + y 2 − 1 + ln (x2 + y) f (x, y) = ln (1 − ex+y )
% &' " x, y "$ (c = −4, −1, 0, +1, +4 z = f (x, y) = x + y z = f (x, y) = x2 − y 2 ) *# &'+ ' w = f (x, y, z) w = x + y + z w = x2 + y 2 + z 2
w = x2 + y 2 − z 2 , f (x, y) = x4 + y 4 − 2xy - ## f (tx, ty) = t2 · f (x, y) x . , f (x, y) = - ## x−y f (a, b) + f (b, a) = 1 / *# !" # " w = f (x, y, z) √ √ √ w = x + y + z w = ln (xyz)
w = arcsin x + arcsin y + arcsin z w = 1 − x2 − y 2 − z 2
$ {xm } ' 0 Pm (x1m , x2m , . . . , xnm ), m = 1, 2, . . . - 0 A(a1 , a2 , . . . , an )- 1 lim xim = ai i = 1, 2, . . . , n
m→∞
(
.
(
/
2 # a = (a1 , a2 , . . . , an )Ì 3 lim xm = a = lim Pm = A
m→∞
m→∞
y = f (x) = f (x1 , x2 , . . . xn ) # 4 U ' x0 !" - ' 5# ' x0 $ ( , α ' f (x) x x0 - 1 {xm } xm ∈ D(f ), xm = x0 lim xm = x0 3 m→∞
lim f (xm ) = α
m→∞
(
6
lim f (x) = α
x→x0
! "! ε > 0 δ = δ(ε) > 0 !
#
! x
0 < |x − x0 | < δ $ |f (x) − α| < ε.
%
& '
$ f (x) ( ! x0 f ) U *$ x0 !+ !
lim f (x) = f (x0 )
x→x0
' z = f (x, y) $ ! , !+ Δz = f (x + Δx, y + Δy) − f (x, y) Δx z = f (x + Δx, y) − f (x, y) Δy z = f (x, y + Δy) − f (x, y) z = x2 − xy + y 2 Δz, Δx z, Δy z Δz, Δx z, Δy z ! ' ! x *$ 2 2, 1 ! y *$ 2 1, 9 .! / ! !
y lim (x;y)→(0;0) x − y #
! ! 0.1 ! 2 (x; y) ! 2 (0; 0) . ! ! y = mx ! 3. 4 1 ! ! 0. ! / 3 2 −2 1 !x2 − y 2 f (x, y) = 2 x + y2 x2 y 2 f (x, y) = 2 2 x y + (x − y)2 !
lim lim f (x, y) ! y→0 x→0
lim lim f (x, y)
x→0 y→0
/ ! #
lim
(x;y)→(0;0)
f (x, y)
5 ! $ ! #
-√ 2 − xy + 4 lim (x;y)→(0;0) xy sin (xy) lim xy (x;y)→(0;0) sin (xy) lim (x;y)→(0;0) x sin (xy) ! lim x (x;y)→(0;2) 1 lim (x2 + y 2 ) sin xy (x;y)→(0;0) x lim (x;y)→(0;0) x + y x2 + (y − 2)2 + 1 − 1 lim x→0 x2 + (y − 2)2 y→2 4−x y−x 2(2x + y) ln (x2 y 3 )
lim exp x→−0,5 y 2 − 4x2
lim
(x;y)→(4;4)
y→1
) ! $ ! '
$
z = f (x, y) (0; 0) f (x, y)⎧ ⎨ xy x2 + y 2 > 0 = x2 + y 2 ⎩0 x = y = 0
z =
x2 + y 2 > 0
f (x, y)⎧ 2 ⎨ x y 4 = x + y2 ⎩ 0
x2 + y 2 > 0
x = y = 0
x = y = 0
! """ # z = f (x, y)
1 + (y + 1)2
(x − 1 z = sin x sin y
z = ln (1 − x2 − y 2 ) z =
f (x, y)⎧ 2 ⎨ x y = x2 + y 2 ⎩ 0
1)2
x2 + y 2 (x + y)(y 2 − x)
$ % " w = f (x, y, z) "& 1 w = xyz 1 w = 2 x + y2 − z 2 1
w = 2 2 x + y − z2 + 1
' ( ) " " " ) ( * + , " " lim
h→0
f (x10 , . . . , xi−1,0 , xi0 + h, xi+1,0 , . . . , xn0 ) − f (x10 , . . . , xn0 ) h ∂f = ∂xi x=x0
+
-
. ' / f xi x0 (i = 1, 2, . . . , n) " " 0 ) 1" " 2 . ' " . ' "( " " 3 # ( 4 fx1 x2 = fx2 x1 , fx1 x1 x4 = fx1 x4 x1 = fx4 x1 x1 ") 5" y = f (x) x0 xi " ( "
∂f ∂f ∂f , ,..., ∂x1 ∂x2 ∂xn
/ f
Ì
x=x0
x0
= grad f (x)x=x0
+
6
∇=
∂ ∂ ∂ , ,..., ∂x1 ∂x2 ∂xn
Ì
grad f (x) = ∇f (x)
! "# f (x) $
x0 s = (s1 , s2 , . . . , sn )Ì %
∂f (x) s Ì = f (x ) = (grad f (x)) · s 0 ∂s x=x0 |s| x=x0
1 fxk (x0 ) · sk |s| n
=
&
k=1
' "# # (
)* f (x, y) = x3 + 3x2 y − y 3
f (x, y) = ln x2 + y 2 y f (x, y) = arctan x 1 1 √ − g(x, t) = ln √ 3 3 x t 2 2 c(a, b, γ) = a + b − 2ab cos γ z x y f (x, y, z) = + − x y z −yx f (x, y) = x e 2x − t g(x, t) = x + 2t √ α(x, t) = arcsin (t x) + f (x, y) = cos (ax − by) y f (x, y) = arcsin x x
f (x, y) = 3y − 2x h(x, t) = ln sin (x − 2t) g(x, y) = sin2 (x + y) − sin2 x − sin2 y
# f (x, y) = xy y ( f (x, y) = exp sin x
, f (x, y) = arcsin
x2 − y 2 x2 + y 2
f (x, y, z) = (xy)z f (x, y, z) = z xy f (x, y, z) = exyz cos y
f (x, y) = ln cos x - ' ! * # # (
*
z = ln (y − x2 ). % zxx , zxy , zyy u+v w = arctan . % 1 − uv w ,w ,w uu
uv
vv
z = x3 + x2 y + y 3 . % zxxx, zxxy , zxyy , zyyy / ! * # !
√ √
z = ln x + y
∂z 1 ∂z +y = ∂x ∂y 2 √ y z = x sin x ∂z ∂z z x +y = ∂x ∂y 2 x
u = exp 2 t ∂u ∂u +t =0 2x ∂x ∂t u = x2 + y 2 + z 2 2 2 2 ∂u ∂u ∂u + + =1 ∂x ∂y ∂z xy z = x−y ∂ 2z ∂2z ∂2z 2 + + 2 = 2 2 ∂x ∂x∂y ∂y x−y x
z = ex/y ∂z ∂z ∂2z = − y ∂x∂y ∂y ∂x !! y xf (x) +ϕ u= y x " # f ϕ $ xyuxy + y 2 uyy + xux + 2yuy = 0 % & ∂2z ∂2z = ∂x∂y ∂y∂x ' z = sin (ax − by) x2 z = 2 y
z = ln (x − 2y) & ! z = f (x, y) = x2 + y 2 ( grad z ) (3; 4) ) (3; 4) * +*
&
( , f - α = 30◦ $!* x#, ! ) (3; 4) . ( z = f (x, y) = x2 − y 2 & ) P (2; 1) ) P * +* & / ( , f (x, y) = x3 − 2x2 y + xy 2 + 1 ) M (1; 2) - * M N (4; 6) "! 0 ( grad u ) P (1; 2; 3) u = f (x, y, z) = xyz 1 ( 2 "! & y z = f (x, y) = ln x 1 1 ; ) A 2 4 B(1; 1) ( , # z = f (x, y) = x2 − xy − 2y 2 ) P (1; 2) - x#, ! 2 * 60◦ 3 ( , # z = f (x, y) = ln x2 + y 2 ) P (1; 1) - 2 ! 1. 4 # 5 x, y #6 7 ( , # w = f (x, y, z) = x2 − 3yz + 5 ) M (1; 2; −1) - #
! " # $% & !
w = f (x, y, z) = xy + yz + zx ' ! M (2; 1; 3) (
M N (5; 5; 15) )* # grad f (x) )* f (x1 , x2 , x3 , x4 ) = 6x1 x2 + 3x21 − cos x3 + x4 ex2 + # ," $ &- z 2 = xy (z ≥ 0) ' ! (4; 2)
y = f (x) . x0
/ h1 = Δx1 , h2 = Δx2 , . . . , hn = Δxn Δy = Δf = f (x10 + h1 , x20 + h2 , . . . , xn0 + hn ) − f (x10 , x20 , . . . , xn0 ) = f (x10 + Δx1 , x20 + Δx2 , . . . , xn0 + Δxn ) − f (x10 , x20 , . . . , xn0 )
0% 1 x0 ∈ D(f ), x0 + h ∈ D(f )
2
h = (h1 , h2 , . . . , hn )Ì = (Δx1 , Δx2 , . . . , Δxn )Ì .
3
f (x) " . x0 0
. x0 + h ∈ D(f ) ) 1 Δy = Δf = fx1 (x0 )h1 + fx2 (x0 )h2 + · · · + fxn (x0 )hn + η · ρ n = fxk (x0 ) · Δxk + η · ρ
45
k=1
(Δx1 )2 + (Δx2 )2 + · · · + (Δxn )2
4
η = η(x10 , . . . , xn0 ; Δx1 , . . . , Δxn )
44
lim η = 0.
46
ρ=
ρ→0
$
/ 45 " y & ! y = f (x) . x0 1 y =
n k=1
fxk (x0 ) · Δxk
4
Δy = y + η · ρ
! y = xi %
" # $
x1 = Δx1 , x2 = Δx2 , . . . , xn = Δxn
&
y = f (x1 , x2 , . . . , xn )% y =
∂y ∂y ∂y x1 + x2 + · · · + xn ∂x1 ∂x2 ∂xn
ρ ( ) ) y = f (x1 , x2 , . . . , xn )%
'
Δy ≈ y
f (x1 + Δx1 , . . . , xn + Δxn ) ≈ f (x1 , . . . , xn ) + f (x1 , . . . , xn )
*
z = f (x, y) + , ! % 2 z =
∂2z 2 ∂2z ∂2z 2 x + 2 x y + y ∂x2 ∂x∂y ∂y 2
-.
-
/ -. 0) 1 )+ % 2 z =
∂ ∂ x + y ∂x ∂y
2 z
2 f (x1 , x2 , . . . , xn ) # ) k $ 3 4 k %
k f (x1 , . . . , xn ) =
∂ ∂ x1 + · · · + xn ∂x1 ∂xn
2 ) $ z = f (x, y) 5 6 $ Δz $ ) 1 (x; y) ) 5$ ) 6 Δx Δy 7 ) 5 Δz z )
k f (x1 , . . . , xn )
-
z = xy 1 (5; 4) Δx = 0, 1; Δy = −0, 2 ) z = x2 y 1 (−3; 2) Δx = 0, 01; Δy = −0, 02
z = x2 − 3xy + y 2 1 (2; 1) Δx = −0, 1; Δy = 0, 2
y " (2; 1) f (x, y) = x Δx = 0, 1; Δy = 0, 2 g(x, y) = exy " (1; 2) Δx = −0, 1; Δy = 0, 1 y ϕ(x, y) = arctan " x (2; 3) Δx = 0, 1; Δy = −0, 5
xy f (x, y) = x−y g(s, t) = es/t f (x, y) = x2 + y 2 u(x, y, z) = x2 + y 2 + z 2 y f (x, y) = ln tan x z x h(x, y, z) = xy + y ϕ(x, y) = ex cos y + y sin 3x z ψ(x, y, z) = x2 + y 2 f (x1 , x2 , x3 , x4 ) = xx1 2 −x3 ln x4
!
# Δz z $ z = ln (x2 + y 2 )% &
x 2 2, 1 y 1 0, 9 ' ! 2 u y2 u(x, y) = 2 x y u(x, y) = x ln x u(x, y, z) = xy + yz + xz u(x, y) = cos (mx + ny)
( f (x1 , x2 , . . . , xn ) ) x1 , x2 , . . . , xn ' *+% ! , - fxi % . xi . . Δxi . (i = 1, 2, . . . , n)% $ Δf *+ f - ' n ∂f (x1 , . . . , xn ) · |Δxi | |Δf | ≈ |f | ≤ / ∂x i i=1 $ - ' Δf f f ≈ f
Δf f
.0 /
#
- & - ' / / # % |Δxi | %
' 1 |Δf | |f | ∂f 2 $ ∂x xi − |Δxi | ≤ xi ≤ xi + |Δxi | i (i = 1, 2, . . . , n) " " . / % ' . $ |Δf |
x z = f (x, y) = x = 2 ± 0, 1 y y = 4±0, 3 ! " f (1, 9; 3, 7), f (1, 9; 4, 3), f (2, 1; 3, 7), f (2, 1; 4, 3)# $ 1, 9 ≤ x ≤ 2, 1 3, 7 ≤ y ≤ 4, 3 %& '$ |Δz|max ( f (2; 4) ) ! " ' ( Δz * ∂z ∂z |Δz| ≤ · |Δx| + ∂x (2;4) ∂y (2;4) ×|Δy| ( |Δz|max
! " + ∂z ∂z " ∂x ∂y ( 1, 9 ≤ x ≤ 2, 1; 3, 7 ≤ y ≤ 4, 3 + |Δz| |Δz|max , $ R (
$ - R1 R2 R1 R2 R= R1 + R2 $ R1 = (550 ± 3)Ω R2 = (150 ± 1)Ω ! " ' ( ( R . //! ! " + ( / 0 $ 1 r = (5±0, 01) *% h = (12 ± 0, 04) 2 0
3 A0 ! " ' ( ( A0 ! " + ( 4 0 - $ 1 R = (400 ± 5), r = (300 ± 6) *% h = (500 ± 8) " ' ( 5 " & + a b γ 6 $ a = (92, 5 ± 0, 2)# b = (65, 6 ± 0, 1) γ = (57, 8◦ ± 0, 3◦ ) " ' ( # " 7 c + ( 8 9: # $ ; " 2 ( 6 p1 V1κ = p2 V2κ " ' ( " ( V1 # $ V2 , p1 , p2 ( #5 = $ #4 = $ # = $ (κ = 1, 4) ? 7 P '& $ Ra Ra Pa = E 2 · . (Ri + Ra )2 " ' ( " (
Pa |ΔRi | |ΔRa | = = 10 Ra Ri
Ra = 100Ri E = const.
P0 (x0 ; y0 ; z0 ) F (x, y, z) = 0
(x − x0 )Fx (x0 , y0 , z0 ) + (y − y0 )Fy (x0 , y0 , z0 ) + (z − z0 )Fz (x0 , y0 , z0 ) = 0
!
"
Ì n = Fx (x0 , y0 , z0 ), Fy (x0 , y0 , z0 ), Fz (x0 , y0 , z0 )
r = (x0 , y0 , z0 )Ì + λn, −∞ < λ < ∞
P (x0 ; y0 ; z0 ) F (x, y, z) = 0 #$ % z = f (x, y) &'( %) f (x, y) − z = 0 ' $ % F (x, y, z) = 0 * '+ Fx = Fy = Fz = 0 , #
- * ( ' z = 1 + x2 + y 2 - x=y=1 x2 + 2y 2 + 3z 2 − 21 = 0 P0 (1; 2; 2) z = ln (x2 + y 2 ) - x = 1, y = 0 z = sin x cos y - π x=y= 4 . - * ( ' xy = z 2 P0 (x0 ; y0 ; z0 )
xyz = a3
P0 (x0 ; y0 ; z0 )
x2 y2 z2 + 2 − 2 =1 2 a b c P1 (x1 ; y1 ; z1 ) P2 (a; b; c)
x2 + 4y 2 + z 2 = 36 x + y −
z=0
!"
x2 + y 2 = z 2
(3; 4; 5) #$ % & ' #
( !") * + !"
x = 0, y = 2 x2 + y 2 − xz − yz = 0 ! " xyz = a3 # $%
$ x0 = 4, z0 = 0 y0 > 0 0 1 ( # 2 ) " # 3 4 (2a2 − z 2 )x2 − a2 y 2 = 0 $ (a; a; a)
& ! '# ( ! ) " 2 2 2 2 x 3 + y 3 + z 3 = a 3 # * ! +,- a2
5 6 * $ " # z = 4 − x2 − y 2 2
. + / x2 + y 2 − (z − 5)2 = 0
+ " #
4 x, y #7 ! 4 7 2x + 2y + z = 08
6 y = f (x1 , x2 , . . . , xn ) 2 # x1 , x2 , . . . , xn ! * # t 9 x1 = ϕ1 (t), x2 = ϕ2 (t), . . . , xn = ϕn (t)
3
:.4
3
:04
( y = f (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))
t ; 3 49 ∂y x1 ∂y x2 ∂y xn y = + + ···+ = · · · t ∂x1 t ∂x2 t ∂xn t ∂y ∂y ∂y x˙ 1 + x˙ 2 + · · · + x˙ n ∂x1 ∂x2 ∂xn
3
0; i = 1, 2, . . . , n!" " #$ %&$$ a, b
c $ '$( & & )$&* A $ +, &$ "
##
# f (x, y) b = B
z=f(x,y)
z
f (x, y) x y B
Z
Z B x, y
z = f (x, y)
• f (x, y) ≥ 0 :
# V =
y B
x
f (x, y) b
Z
B
# • f (x, y) ≡ 1 :
A=
#
b
B
B
αf (x, y) + βg(x, y) b = α
B
#
# g(x, y) b , α, β ∈ R
f (x, y) b + β B
B
! B = B1 ∪ B2 " B1 B2 # $ % & #
#
f (x, y) b = B
f (x, y) b + B1
B2
& ' & %% & ( #b f (x, y) b = B
y
a≤x≤b f1 (x) ≤ y ≤ f2 (x)
#
B2
f (x, y) b
B x B:
B1
#
x=a
⎡ ⎢ ⎣
f# 2 (x)
y=f1 (x)
6
a
⎤
⎥ f (x, y) y ⎦ x =
#b
B
f# 2 (x)
x=a y=f1 (x)
f2
f1 -x b
f (x, y) y x ) ! *
B B:
y
y d
c≤y≤d ϕ1 (y) ≤ x ≤ ϕ2 (y)
#
#d
ϕ#2 (y)
f (x, y) b =
ϕ1
6
ϕ2
B
c
-x
f (x, y) x y y=c x=ϕ1 (y)
B
! B x, y " #
= (x, y) $ % &
'
% (! x y = (x, y) , rA = rA (x, y) $! ## % A x, y " !) ! ! ## % A ⊥ x, y " ! ! ## #
x b B & * B
+ * # , y = x2 y = x + 2 % & - & # x
y
% , % # . ! & * + * % % / 0 1%% y = x , xy = 4 , x = 4 y 2 = 4 + x , x + 3y = 0
(x − 2)(y − 2) = 4 (x + 3)(y + 3) = 4 y = ln x , x − y = 1 , y = −1 2 1%% & ! ' ' ! 3 4 ' #2
x+2 #
I =
y x x=−1 y=x2
#a
I =
√ 2 2 a −y # x y
y=0 x=a−y
#1 2−x # 2 I = y x x=0 y=x
#a
√
2a2 −y 2
#
I =
x y √
y=0 x= ay
##
xy 2 x y B
B
y 2 = 2px ! " x = p # (p > 0) $ % &" B ' # ( #
xy b , B : (x − 2)2 + y 2 = 1 B
#
(x2 + y 2 ) b
B
B : y = x2 , y = 1
) & & ## xy − y 2 x y B
B
* + ,+-+ O(0; 0), A(10; 1) B(1; 1) & ##
ex/y x y B B
y 2 = x (y ≥ 0) ! y = 1 x = 0 " # ## xy x y B B
x".& / " + & (x − 2)2 + y 2 = 1
#
& 0 &
### f (x, y, z) b =
B
1 /2- xy = 4 ! x + y = 5 # # ! y = x ( ≡ 1) # x 3 b B & x2 + y 2 B
& & & 1
y = x2 ! 2 y = x # 4 * + ,+-+ (0; 0) (0; 1) (1; 0) & - 0 & " # & 5-+&
≡ 1 & 6 & 7 0 " & Jx , Jy & -" 0 & Jp # & 5-+& 1 & +& & ' x = 0 x = a y = 0 y = b
# ≡ 1 & 8 -+ 1" % ' " # ≡ 1(
y = 0 / - 8&+ y = sin x
y = x2 x = 4 y = 0 # xy b B B
' & .&" x = a cos3 t , y = a sin3 t , 0 ≤ t ≤ π/2 (a > 0)
#
#
f (x, y, z) x y z B
B R3 # f (x, y, z) ≡ 1 V = b B B
! " ! #$ % &' (
z
x, y
B
B : (x, y) ∈ B g1 (x, y) ≤ z ≤ g2 (x, y)
g2 B
g1
' B )' x, y "' * x ⎡ ⎤ g2#(x,y) ## # ⎢ ⎥ f (x, y, z) b = f (x, y, z) z ⎦ x y ⎣ B
B
y
B' % +(
z=g1 (x,y)
, B '! )' '- x & . B = {(x, y, z)| a ≤ x ≤ b , f1 (x) ≤ y ≤ f2 (x) , g1 (x, y) ≤ z ≤ g2 (x, y)}
/ 0 ' !* #
#b
f# 2 (x)
g2#(x,y)
f (x, y, z) b = B
f (x, y, z) z y x
% + (
x=a y=f1 (x) z=g1 (x,y)
x, z
B
B = {(x, y, z) ∈ R3 | (x, z) ∈ B , g1 (x, z) ≤ y ≤ g2 (x, z)}
' B )' x, z "' * ⎡ ⎤ g2#(x,z) ## # ⎢ ⎥ f (x, y, z) b = f (x, y, z) y ⎦ x z ⎣ B
+ B
B
y=g1 (x,z)
y, z B = {(x, y, z) ∈ R3 | (y, z) ∈ B , g1 (y, z) ≤ x ≤ g2 (y, z)}
% (
B y, z ! ##
# f (x, y, z) b =
B
B
⎡ ⎢ ⎣
g2#(y,z)
⎤ ⎥ f (x, y, z) x⎦ y z
x=g1 (y,z)
= (x, y, z) " #$ P (x; y; z) %& ' B = const. ' (
)
! " * +, '$* ) %& ' B # m = b " B
+, '$ $ )
- . A
xS =
1 m #
JA =
# x b B 2 rA
b
B
/0 ( 0 1 %& ' * ( 0 2 - , !
y = x2 , x + y + z = 2, y = 1, z = 0
z = xy, x + y = 1, x = 0, y = 0, z = 0 # (x + y + z) b* , B B
( x = 0* x = 1* y = 0* y = 1* z = 0 z = 1 - , 3 ###
x y z * (x + y + z + 1)3
B
yS =
, B ( % x+y +z = 1 - ,
1 m
# y b B
zS =
1 m
# z b B
rA = rA (x, y, z) . ( P (x; y; z) ∈ B - . A
4 1 %& ' * - , ( 2 z = x2 + y 2 * x + y = 4* x = 0* y = 0* z = 0 #
x3 y 2 z b* , B 5
B
0 ≤ x ≤ 1* 0 ≤ y ≤ x * 0 ≤ z ≤ xy 6
7 " #8 *
x + y + z = a* x = 0* y = 0* z = 0
- , * , 9 #$ z : /& #$ " %& ' * 2 x = a* 2x+z = 2a* x + z = a* y 2 = ax* y = 0 - , 0 y > 0* ,
y
!" # ≡ 1$% &' () * $ x + y + z = a% x = 0% y = 0% z = 0 (a > 0) ($ az = a2 − x2 − y 2 % z = 0 (a > 0)
+ ,' ( )- z . - !"% &' () #/ ≡ 1$* $ x = 0% y = 0% y = a% z = 0 x + z = a (a > 0) √ ($ x + y + z = a 2% x2 + y 2 = a2 % z = 0 (a > 0) $ z 2 = 2ax, z = 0, x2 + y 2 = ax (a > 0)
. - ! T : x = x(u, v, w) y = y(u, v, w)
# $
z = z(u, v, w)
/( () u, v, w ! B 0 B ∗ 0 T * B = T (B ∗ ) 1- T 1 % % # #
∂(x, y, z) ∗ b # $ f (x, y, z) b = f x(u, v, w), y(u, v, w), z(u, v, w) ∂(u, v, w) B
B∗
∂x ∂u ∂y ∂(x, y, z) = ∂(u, v, w) ∂u ∂z ∂u
∂x ∂v ∂y ∂v ∂z ∂v
∂x ∂w ∂y . . . F unktionaldeterminante ∂w ∂z ∂w
. # $ ( #, b$* ∂(x, y, z) ∗ ∂(x, y, z) b b = x y z = ∂(u, v, w) u v w ∂(u, v, w)
# ($
# $
z w ! " ∂x ∂(x, y) ∂u = ∂(u, v) ∂y ∂u
∂x ∂v ∂y ∂v
# $ % $
b
(r, ϕ) r # $ % ϕ & " $ % $ x# ' $" % $
0 ≤ r < ∞, 0 ≤ ϕ < 2π
y
6 r Y ϕ
P
-x
( " T : x = r cos ϕ y = r sin ϕ
) *+
{(r, ϕ)| r = const. , ϕ ∈ [0, 2π)} ϕ, ) -% " +
(r, ϕ) , 0 ≤ r < ∞ , 0 ≤ ϕ < 2π ( . T : x = a r cos ϕ
y = b r sin ϕ , a, b % $
) /+
ϕ, % 0 r · a r · b 1 % $
z
(r, ϕ, z) P P x, y r, ϕ P z x, y
6
P z
ϕ
−∞ < z < ∞
x
!" #
> r
T : x = r cos ϕ y = r sin ϕ
-y
P
$
%
z=z
& #'( {(r, ϕ, z)| r = const. , ϕ ∈ [0, 2π) , z ∈ (−∞, ∞)} #)*"(
z
(r, ϑ, ϕ) r " # + ϑ , + z ) " - ϕ . /* 0 ≤ r < ∞ , 0 ≤ ϑ ≤ π , 0 ≤ ϕ < 2π
!" #
x
6
P U r ϑ ϕ P
-y
T : x = r sin ϑ cos ϕ
$ %
y = r sin ϑ sin ϕ z = r cos ϑ
& #'( {(r, ϑ, ϕ)| r = const. , ϑ ∈ [0, π] , ϕ ∈ [0, 2π)} # '(
(r, ϕ)
b = r r ϕ
+ # (r, ϕ) b = rab r ϕ /* (r, ϕ, z) b = r r ϕ z # (r, ϑ, ϕ) b = r2 sin ϑ r ϑ ϕ
## 1 − x2 − y 2 x y
xy = u
B (0, 0) ! r = 1 ## (x2 + y 2 ) x y " # 2
y = vx
2
B
B
2
B $ $ x + y = 2ax (a > 0) # % &
' ( #2 #x f x2 + y 2 y x x=0 y=0
)
* ## x y " # B & 2 a − x2 − y 2 B
+ $ ! a (0; 0) " ( , ## & xy x y " # B $ B
x- . (x − 2)2 + y 2 = 1 #
/ ' ( #1 #1 f (x, y) y x x=0 y=0
0 u v )1 u = x + y" v = x − y
2 34 " 5 $ # 1 a2 & xy = , xy = 2a2 , 2 x y = , y = 2x 2
& y = ax, y 2 = 16ax, ay 2 = x3 , 16ay 2 = x3 (a > 0)
y 2 = ux, vy 2 = x3 .
& x2/3 + y 2/3 = a2/3 (a > 0)
x = r cos3 ϕ , y = r sin3 ϕ .
. 5 6 34" $ $ ' y 2 = x" y 2 = 16x" y 2 = x3 " 16y 2 = x3 # 3) 0 √ 5 1 x = uv" √ 4 y = u3 v 7 +4 34" $ $ x2 + y 2 = 1" x = 0" y = 0 # " ) y -" # 34 ≡ 1 . 5 6 5 - # 0 5 x = r cos ϕ" y = 2r sin ϕ" z = z 0 8 " $ 34 y2 − z 2 = 1 ' x2 + 4 √ z = ± 3 # 9 : 8 # $ ; 0 x2 +y 2 = 2x % ### & x2 x y z B B
x2 y2 z2 ' () 2 + 2 + 2 = 1 a b c ### z x y z B & B
( z = 0 *$ () x2 y2 z2 + 2 + 2 = 1 2 a b c + , ' -) #$ . x2 z y2 & = 1 − 2 − 2 z = 0 c a b
x = ar cos ϕ y = br sin ϕ, z = z & x2/3 + y 2/3 + z 2/3 = a2/3 , a > 0 /0 x = r cos3 ϕ y = r sin3 ϕ, z = z &
& 4z = 16 − x2 − y 2 z = 0 x2 + y 2 = 4 / 1 !"& 2 & z = (x + a)2 x2 + y 2 = a2 4 & z = 2 z = 0 x2 + y 2 = 1 x + y2 x2 + y 2 = 4 & az = x2 + y 2 z = 0 x2 + y 2 ± ax = 0 (a > 0) & az = a2 − x2 − y 2 z = 0 x2 + y 2 ± ax = 0, a > 0 / !"&
2 '' * 3
x2 + y 2 + z 2 = a2 x2 + y 2 + z 2 = 4a2 '' 4 5' 6 ' ) ) ' 7 6 ' ) , 8 '' , ' -) #$ & (x2 + y 2 + z 2 )2 = a3 x & (x2 + y 2 + z 2 )2 = az(x2 + y 2 ) (a > 0)
!
! x2 + y 2 − z 2 = 0 , ' x2 + y 2 + z 2 = 2az ' , $ . (a > 0) 9 '' 3' Q #$ x2 + y 2 − z 2 = 0
z = h > 0 - ) :
(x, y, z) = z
n
F x, y, y , . . . , y (n) = 0 implizite
(n) (n−1) y = f x, y, y , . . . , y explizite
n y(x) x ∈ [a, b] [a, b]! • • •
! ! ! "# $ # % % x = a #& ' ( ) $ # # #* ! % ## % a, b, . . . +# ( ) $ # # !
y = f (x, y)
f (x, y) # , $ ## - P (x; y) ! . / - x, y0 ,
& # ! ! ! ! 1 2 # 2 ## (f (x, y) = d = const.) # + t & 3 & ! # 4 yt = y˙ ! 5! ' ) (y − y0 )2 = 2px 1 # /) y = c1 e2t +c2 e−t 2 ) y = cx ) y = ct ) y = c1 cos 2t + c2 sin 2t 2 ) y = 2cx ) x2 + y2 = c2 ) y = (c1 + c2 t) et +c3 ) y = c et ) t3 = c(t2 − y2 ) 6! + # ' %
) y2 + x1 = 2 + c e−y /2 ) 1 # x2 + y2 = 2cx x
) ln y = 1 + ay ) - y = x2 + 2cx 2
!
" xy = 2y, y = 5x2 1 " (y) ˙ 2 = t2 + y 2 , y = t c2 − x2
" (x+y) x+x y = 0, y = 2x " y¨ + y˙ = 0, y = 3 sin t − 4 cos t 2 x " 2 + ω 2 x = 0 t x = c1 cos ωt + c2 sin ωt " y¨ − 2y˙ + y = 0 y = t et y = t2 et " y − (λ1 + λ2 )y + λ1 λ2 y = 0 y = c1 eλ1 x +c2 eλ2 x
# $ y =
cx3 (c ∈ R) ! 3y − xy = 0 $ % & ' ( 1 1 " 1; " (1; 1) " 1; − 3 3
) * + ( y " y˙ = y − t " y˙ = t 2
" y˙ = y + t , y˙ = f (t, y) = t2 + y 2 * - . %
f (t, y) = d = const. / 1 d = ; 1; 2; 3 + 2 0 &1 2
3 % & 4
y˙ =
u(t) , v(y) = 0 v(y)
1 - 5 & y˙ =
v(y) y = u(t) t
- % / - ! ( # # v(y) y = u(t) t =⇒ V (y) = U (t) + C V (y) v(y) u(t)
y ( t
x2 y + y = 0 x + xy + y (y + xy) = 0 φ2 r + (r − a)φ = 0 2st2 s = (1 + t2 )t t 1 + y 2 + y 1 + t2 y˙ = 0 ty y˙ = 1 − t2 y − ty˙ = a + at2 y˙ y tan x = y y x3 = 2y ! (x2 + x)y = 2y + 1 " y a2 + x2 = y
(1 + x2 )y + 1 + y 2 = 0 # $ √ 2y x = y, y(4) = 1 π 1 = y = (2y + 1) cot x, y 4 2 x2 y + y 2 = 0, y(−1) = 1 y y t t − =0 1+y 1+t y(1) = 1 y(0) = 1 t t (1 + e )y y˙ = e , y(1) = 1 r + r tan φ φ = 0, r(π) = 2 √ y = 2 y ln x, y(e) = 1 (1 + x2 )y + y 1 + x2 = xy y(0) = 1 % $ & "' ( ) * + 100 ◦, $ +
& -* 20 ◦ , 25 ◦ , + $
10 . 60 ◦ , ) "' / 01 1$ 2) 2$ " #) "' * * -* ) 3 4 ( ( * 5 C $ 6 )
R 7 8 2 *
) 9
8*
U = const. 4 q = q(t) ( + $
q(0) = 0 : 6 8 5 " I
2 " U = 3 ;, R = 6 Ω, L = 0, 06 < & t = 0, 01 + $
8 & t = 0 $ / 1 $ & 8 5 " I $ / 2 R L
U
= 8 0 ) >" 0& 3# y x $! ! 4 # )
y (μ > 0& 3 x+μ ,
y = −λ
8# $ ' ! 7 * p * $- # 9 t ' ' : +! y(t)# ; : +! 9 k > 0 ! + ( < · · · > • ( * •
" k % - '
1 s 2 3 ' ( k : x = x(s) , s ∈ [0 , L]
+,
+, 4
k 5 ' L 6
k
7 • D ⊆ R3 • k : x = x(t) , t ∈ [α, β] % - ' • k⊂D • f : D −→ R 8 # • v : D −→ R3 # +P (x) Q(x) R(x)
,
#
#β f (x) s =
k
f x(t) |x(t)| ˙ t
+9,
α
#
#β v(x) · x =
k
α
v x(t) · x(t) ˙ t
+:,
x = (x, y, z)Ì v(x) · x = P (x) x + Q(x) y + R(x) z ! " #"
$ % & ! ' ( &
" k ) ! * +
' (
" k , x = x(s) , s ∈ [0 , L] -
" [0 , L] : 0 = s0 < s1 < · · · < sn = L (j = 0, . . . , n) xj := x(sj ) Δsi := si − si−1 , Δxi := xi − xi−1 (i = 1, . . . , n) . + -
, n
# f (x) s ;
f (xi−1 ) Δsi ≈
i=1
n
# v(x) · x
v(xi−1 ) · Δxi ≈
i=1
k
k
/ *
+) ,
• , k 0 $ (x) ! " 1
(x) s 1 " k k 1 • , F (x) + + ! " k F (x) · x * ! .
0 %
k + +
F (x0 ) x1 1 Δx1
x0
#
k
Δxn : xn Δx2 z
xn−1 x2
R F (xn−1 )
#
f (x) s = k
F (x1 ) 7
#
# f (x) s
−k
v(x) · x = −
v(x) · x
−k
k
α, β ∈ R # # # α f1 (x) + β f2 (x) s = α f1 (x) s + β f2 (x) s k
#
k
k
α v 1 (x) + β v 2 (x) · x = α
#
k
#
v 1 (x) · x + β k
v 2 (x) · x k
k = k1 ⊕ k2 ⊕ · · · ⊕ kn # # # f (x) s = f (x) s + · · · + f (x) s k
k1
#
v(x) · x = k
kn
#
#
v(x) · x + · · · + k1
v(x) · x kn
• M ⊆ R
x ∈ M ! " x n
# M $
• % M ∈ Rn
& # ' M ( "! M !
• G ∈ Rn # 1 • )* "! k v(x) · x ! + ( , - !
! "! k
"!
! " v
: G −→ Rn (n = 2 , 3) v
G ⊆#R v(x) · x G n
k
k . v ϕ !
v
(/ "! / !
A B ! ( "!0 # v(x) · x = ϕ(A) − ϕ(B) 12
AB
#$
, , 3 G ⊆ Rn - , 45 6 7- ( # * 3 R2 # R3 0 R2
R3 G G
v : G −→ Rn
G ⊆ Rn ! !
Py = Qx rot v = o
A(4; 2) B(2; 0) # [(x + y) x − x y] k
OA
OBA # !"
"# $
(y x + x y) %&
k
% $ '& ( ) $ ' ) * + A(a; 0; 0), B(a; a; 0) # C(a; a; a) $
(y x + z y + x z) k
OC
OABC - " " && F = (x − y , x)Ì '
" ' F . & -/ 0 & x = ±a y = ±a !( , ' ) "# 1 (
2 " & ' 3&"
0 4 - " " ) && F = (P , Q)Ì = (x + y , 2x)Ì ' ( " ' F & ! ( "
/ . 1 ' x = a cos t, y = a sin t ( ! , ' ) "# 1
2 ( " & &"
!"
n=2 n=3
P = x + y , Q = x %& ! *
5 - " " ) " F = (y , a)Ì && -& ! , ' ) "# (
1
& m
, '( ' 1 -/ &"
x = a cos t, y = b sin t ) 6 - 7 ) " F = (x , y , z)Ì && (
! , ' ) "# 1
2
OABCO, O(0; 0; 0), A(0; a; 0), B(a; a; 0) C(a; a; a) 8 "' 9 #
2xy x + x2 y
AB
#
(cos 2y x − 2x sin 2y y)
AB
# tan y x +
x y cos2 y
AB
'& π π A 1;
B 2; 6 4 A(0; 1), B(2; 5) C(0; 5) # [(x + y) x − 2y y] k
AB
AB y = x2 + 1 ACB
k : x= t y = t2 z = t3 , 0≤t≤1 k 0 - % & " -
A(−a; 0) B(0; a) ! ( ! " F = (y , y − x)Ì # cx # v(x)· x "' 2 v(x) = |x|3 k AB c > 0 3 ! ( k AOB Ì AB t , x = cos t , sin t , x2 2π y =a− a 0 ≤ t ≤ 2π 6 Ì x = (1 , 0 , t) , 0 ≤ t ≤ 1 $ [y x + (x + y) y] %
2
0 k - & - t % && ' %
" - ! (
) * '" + & 4 #$ ! ( x , ! ( · x ' & " % - y = x2 y = 4 |x|3 k & . && - % " v = v(x) = (y , z , x)Ì # 5 6 ! ( # ! ( v(x) · x v(x) · x ' & " % k
- ! ( k &
k : x = a cos t y = a sin t z = bt , 0 ≤ t ≤ 2π / k 0 - & - %
" -
1 " v(x) =
Ì 2 y − z 3 , 2yz , −x#2 ! (
v(x) · x k
-
! ( k %
&
k
&& " 7% - v(x) = (−y , x , z)Ì v(x) = (x , y , z)Ì v(x) = (y , x , 0)Ì v(x) = (z , x , y)Ì 8 6 , (z −y) x+(x−z) y +(y −x) z k
' 3 ABC & +* A(a; 0; 0) B(0; a; 0) C(0; 0; a) - (a > 0)
x = (x, y, z)Ì ∈ R3
Ì F : x = x(u, v) = x(u, v), y(u, v), z(u, v) , (u, v) ∈ B ⊆ R2
B !" #$ ◦
B %
$& B • •
◦
x(u, v) ' B x(u, v) ( )
xu =
∂x(u, v) ∂y(u, v) ∂z(u, v) , , ∂u ∂u ∂u
Ì
, xv =
◦
∂x(u, v) ∂y(u, v) ∂z(u, v) , , ∂v ∂v ∂v
Ì
' (u, v) ∈ B
xu × xv = o
F F &*** !' + $ " , ! $-
•
. ! / )" & . ! 0 * !
1 * 2! #0* . 3 • F : x = x(u, v) , (u, v) ∈ B . ! ◦
• x0 = x(u0 , v0 ) , (u0 , v0 ) ∈ B ! ' F
$& F
x0
n(x0 ) = xu (u0 , v0 )×xv (u0 , v0 ) & n(x0 ) = xv (u0 , v0 )×xu (u0 , v0 )
3 " 4 )" . * . 1 5! " 6 $& & 7,4 8 1 9& $!&
& ) " 74 8 )
• G ⊆ R3
• F : x = x(u, v) , (u, v) ∈ B
• F ⊂G
v : G −→ R3
• f : G −→ R •
#
f (x) σ =
F
#
f x(u, v) n x(u, v) b
B
#
v(x) · σ =
F
#
v x(u, v) · n x(u, v) b
B
Ì σ = (σ1 , σ2 , σ3 ) v(x) · σ = P (x) σ1 + Q(x) σ2 + R(x) σ3
! "# # $ % & # " σ $ % n 0 '# σ σ (" σ = σ = n σ
|n|
) *# + • + ! F , (x) & 1 Q = F (x) σ Q - F • + ! v(x) # . 1 / & U = F v(x) · σ - v F & F 0 ' / U > 0& / 1 2 - 3 " * f (x) ≡ 1 A - F + # A= F
σ
F : z = z(x, y) , (x, y) ∈ B ! " F # $ |n| = (−zx , −zy , 1)Ì = zx2 + zy2 + 1 ! Φ(x, y, z) = 0 ! F : z = z(x, y)# (x, y) ∈ B Ì Φ Φ Φ2x + Φ2y + Φ2z x y |n| = , ,1 = Φz Φz |Φz |
% ! 2
! &' 2z = x # x ($ ) y = # y = 2 √ 2x# x = 2 2 * ! + z 2 = 2xy # ($ ) x = a y = a , x ≥ 0 y ≥ 0 " * 2
2
2
+ y + z = x #
&' x2 + y 2 = a2 ! az = xy #
&' x2 + y 2 = a2 + x2 + y 2 = z 2 #
&' z 2 = 2px , ! &' x2 +y 2 = a2 #
&' x2 + y 2 = a2 ! + x2 + y 2 + z 2 = a2 #
&' " x2 + y 2 ± ax = 0 ! -$$ x2 + y 2 = 2az #
&'" x2 + y 2 = 3a2 % ., $ " ! /
)$ # 0" 0◦ β # 12" $ - α * % 3, α = 30◦ # β = 60◦ # % v(x) · σ F
Ì v = x3 , y 3 , z 3 # *
F 4" 5 -' # ) x + y + z = a# x = 0# y = 0# z = 0 (a > 0) *
# % v(x) · σ # *
F
Ì
v = (3x, 3y, 3z) F " + # 0" +$$ 6
7 % " # v(x) · σ , 8" F
cx # c > 0# *$ |x|3 2 2 y (z + 3)2 x + + = 1 F : 16 16 25 9$ ($ n(x) $ * # 5 *
$, v(x) =
x = 4r sin ϑ cos ϕ y = 4r sin ϑ cos ϕ z = −3 + 5r cos ϑ cx v(x) = (c > 0) |x|3 #
v(x) · σ ! F F
" # $ %
(z − 3)2 1 2 = 1 x + y2 + 16 25 −2 ≤ z ≤ 3 & ' ( ) n(x) * ! + ! ,& - . / ) , . F
0 - #
v(x) · σ ( v = (x, y, z)Ì
F
1 ,2 + 3 x + y + z = a ,a > 0 (
4 # -
v(x) · σ ( v = (x2 , y 2 , z 2 )Ì
F
,2+ 5 x2 + y 2 + 2az = a2 ,a > 0 ( *! , x < 0 y > 0 z > 0
3 • G ⊆ R3 / • v : G −→ R3 6 * • B ⊂ G ( ,n(x) + F
& # # v(x) · σ = div v(x) b F
,7
B
-( . v(x) , /! 8( . 2 ) ,79 F B
B
- * R3 R2 * * B ⊂ G ) ! : ) k (
k B $ % 6 # P v(x) · x = (Qx − Py ) b , v = !" Q k
6
B
Z = v(x) · x # k
% B &
v
k $ %
'( )% x0 ∈ G ' {Bn } * Bn ⊂ G + , 1 -% x0 Fn . / Bn n 4 π Vn = 0 " 3 n3 # 1 div v(x)x=x = lim v(x) · σ 0 n−→∞ Vn Fn
1 • G ⊆ R3 2 • v : G −→ R3 3 0% • F ⊂ G n(x) # " ' (% * k • k F 4% 5#
# 6
rot v(x) · σ v(x) · x = k
6"
F
6" 5 7 8$ % v k 9 : 8'# rot v F 9 : 82 % F 9
'( )% x0 ∈ G + n0 |n0 | = 1"
1 x0 n π n0 kn ! Fn An = 2 n " # kn $ n0 # % &' 6 1 rot v x=x · n0 = lim v(x) · x 0 n−→∞ An
{Fn } Fn ⊂ G
kn
#
(
v(x) · σ
) * $ v = 2 2 2 Ì x ,y ,z F + ," ./ " a 0 1 + F
( + ," 2 3% 3' ) * 3 ( 2 &% ' ) *
x + y + z = a 2 4 x2 + 2y 2 − z 2 = 1 $ 0 z = 0 z = 3 5 " + 6/ F * $ ( #
rot v(x) · σ / F
Ì x , 0 , ln (1 + z) 1 + y2 4 6*
v(x) =
7 . 6* v = u1 grad u2 %u1 u2 *$ 8 * 6 '
# # # (u1 Δ u2 B
6*
+ grad u1#· grad u2 ) x y z (u1 grad u2 ) · σ
= F
.
# # # 6* (u1 Δ u2 − u2 Δ u1 ) x y z B#
(u1 grad u2 − u2 grad u1 ) · σ ?
= F
( 9 v = (1 , x)Ì k = k1 ⊕ k2 ⊕ k3 k1 : y = x2 , x ∈ [0 , 1] 2 1 1 −k2 : x− + (y − 1)2 = , 2 4 x ∈ [0 , 1] , y ≥ 1 −k3 : x = (0 , t)Ì , t ∈ [0 , 1] .
& ) : 2ε # = |x| %ε # *" ' # = : * E x 9$ : |x| (* = ε div E
B a # Q = b B
! " # $ !% v = (2x − y, −yz 2 , −y 2 z)Ì & ' F ( x2 + y 2 + z 2 = 1
y = 0& x + y = a ( 4 % 5
6 ! ' 2( 6 2 y x + (x + y)2 y k
7 4 % ABC 2%% A(a; 0)& B(a; a) C(0; a)
) ! " # $ !% v = (4x, −2y 2 , z 2 )Ì & ' B * x2 + y 2 = 4& z = 0 z = 3 ( '
! " 2( Ì $ !% 1 1 ,− $( 7 v = y x 4 % ABC 2%% A(1; 1)& B(2; 1) C(2; 2)
+ , - *
8 #
U = F
x v(x) · σ # v = |x|3
. * & /% &
(, U & ' 0 x2 +y 2 +z 2 = r2 ' ' , 6-& - % Z = 1 v(x) · x # v = 2 x + y2 k
×(−y, x, 0)Ì . #& z 1 &
(, Z & ' x2 + y 2 = r2 # ' '
! " 2( $ !% v = (x + y, −2x)Ì & ' k # & # 3 x = 0&
v(x) · σ & '( v = (x, y, z)Ì &
F
$( 9( 0 x2 + y 2 + z 2 = a2 % '
: #
v(x)· σ & '( v = (x2 , y 2 , z 2 )Ì &
F
$( 5 * ( 9% ; % ' & * x2 + y 2 + 2az = a2 & x = 0& y = 0& z = 0 ( ' v = (x, y, z)Ì * $ #!< 1 V = (x σ1 + y σ2 + z σ3 ) 3
*$
F
* ! 2 x2 y2 z2 + + = 1 a2 b2 c2 - = # &
#
6 (yz x + xz y + xy z)
k
! " # $% & OAB % ' O(0; 0; 0) A(1; 1; 0) B(1; 1; 1) ( ) " * " # 3
x σ1 + y 3 σ2 + z 3 σ3 F
+ ,# x2 + y 2 + z 2 = a2 - ' ! + ,#
. ) " * "
x(z − y) x + y(x − z) y +z(y − x) z
k
- $% & % ' A(a; 0; 0) B(0; a; 0) C(0; 0; a) * ' ! / v = grad u % " * ### # ∂u Δ u x y z = σ ∂n B
F
0u *-% 1
* 2 3
4 - 5 %
u = x2 + y 2 + z 2 + ,# F : x2 + y 2 + z 2 = a2
2x2 − 5x 2
2x − 17x + 109 667 − x+6
−3x2 + 2x − 1 2x − 3 − 2 x + 2x + 8
1 1 x+ 3 2
0, 25
0, 375x + 0, 45 2x2 − 0, 1x + 3 3 ax − bx2 − c −
a = q ; p = −q 2 − 1
(x + 3)2 (2x − 5)2 (x − 1)2 (x + 1)2 (1 + a)2 (1 − a) 8x2 (x4 + 1)
144
x10 , ax = 0 2a13 5x + 2y n a + 5b
|a| · b3 · b≥0
√ a8 b 9
12
|2x − 5y| √ n xn · x2 , x ≥ 0 √ 12 a11 , a ≥ 0 x3 + 1 x > 0 − x3 + 1 x < 0
|a| = 5|b| 5|x| = 2|y| r5 u3 (s + t)5 ru = 0, s = −t
x − y , x ≥ y
|x|3 |x − y| 3 √ 5−2 2x − 1 x ≥ 1 1 x < 1 23 36
√
17 − 12 2 √ 36 + 11 10 √43 4 − 15 √ 15 a6 b 5 ab √ 28 − 4 2 √ √ 2 + 6
1 √ 5 25 −2 16 16
lg 17 = 2, 57890 . . . lg 3 √ w· u
√ 4 v3 6 a2 − b2
0, 8
4(3 − a) 3+a
3 lg a + 4 lg |b| − lg c a > 0, b = 0, c > 0
1
− lg (a2 + b2 ) 2 a2 + b2 = 0
1 lg b 2 +2 lg (a + b)
4 lg a +
7 5 − lg a − lg b 2 2 a > 0, b > 0
−7; 9 2 5
− ; 3 6
−3; 3; −5; 5 2 2 − ; ; −1; 1 3 3
x1 = x2 = 0 x3; 4 −7; −3 7; 2
λ=
1 5
λ = n2 + n n = 1, 2, 3, 4, . . . a ≥
1 16
9
−4
1 7 27
5; −
30 127
−2; 2
! "# k = 0, ±1, ±2, . . . # # 2π 2π 2π 5π 193, 22135◦+k·360◦ , +k· + k · 2π
k · 3 9 3 6 346, 77865◦+k·360◦ π 7
k · 2π, (2k + 1) · π + k · 2π k · π 6 6 k · π 11π π 3π + k · 2π + k ·2π, + k ·π
π 6 2 3 (2k + 1) · π 16 2π + k · 2π +k·π 2π 2 3 (2k + 1)π, k · ◦ ◦ 3 48, 47127 + k · 360 π + k · π ◦ ◦ 4 172, 64082 +k·360 4 π+k·π 15 π π π π π +k·π k · π, + k · , + k · 4 1, 24905 + k · π 6 5 9 3 3π π +k·π 0, 24498 + k · π k · 2π, + k · 2π 4 2 5π π π 3π +k·2π +k·2π, 5π π + k · π, +k ·π 3 3 +k·2π +k·2π, 4 3 6 6 ◦ ◦ 2π % 46, 77865 + k · 360 π π +k·π $ + k · π, + k · 2π 133, 22135◦+k·360◦ 3 2 6
B : a2 + b2 < 2ab
a, b a2 − 2ab + b2 < 0 A : (a − b)2 < 0
a, b. A
! "# B A :
(a − b)2 ≥ 0
a, b. a2 − 2ab + b2 ≥ 0 B : a2 + b2 ≥ 2ab
a, b
3
! α 0 < cos α < 1
"
× (k + 2) k+1 # i P (k + 1) :
$ % i=1 (k + 1)(k + 2) = 2 & '
* P (n) ( n ≥ 1 + , a > 0 b > 0 b a =⇒ + ≥ 2 ) P (n) : b a n n(n + 1) # i= 2 i=1 !
P (n0 ) = P (1) : 1·2 . 1= 2 P (k) : k k(k + 1)
i= 2 i=1 k ≥ 1. $ k % i + (k + 1) i=1
k(k + 1) +(k+1) = 2 k+1 (k + 1) i= 2 i=1
/ A = {−2, 0, 1} B = {2} C = {1, 2, 3, 4, 5} D = {−3, −2, −1,
0, 1, 2}
(2x − y)(2x
+y) < 0 =⇒ (y > 2x ∧ y > −2x) ∨ (y < 2x
∧ y < −2x)
0 1 2 3 1 4 5
y ¾ ½ ¹½ ¹½ ¹¾
6 -x
½
A ∪ B = {−3, −1, 5} A ∩ B = {5} A \ B = {−3} B \ A = {−1} 7 A × B = (−3, −1), 8 (−3, 5), (5, 7 −1), (5, 5) B × A = (−1, −3), 8 (−1, 5), (5, −3), (5, 5)
! !
4 ( x ∈ B ⇒ x∈ A∩B ⇒ x ∈ A ⇒ B ⊆ A; x∈B⇒x∈A⇒ (x ∈ A ∧ x ∈ B) ⇒ x∈ A∩B ⇒ (1) B ⊆ A ∩ B (2) A ∩ B ⊆ B &
((
0 &'( & ( % A∩B =B
! " X Y # 5 + ,% m ↔ n2 + 1 ! $ ! ∧n ∈ {1, 2, . . . } % &'( X ⊆ Y & ( Y ⊆ X 0
) &'($ & ( X = Y * + 6 A ∼ B $ ,% &'( ! - . 1 2n ↔ n−1 x / 10 (n = 1, 2, 3, . . . ) ,% X $ %
% ! / ,% Y 0 7 ( F 1 %$ G
% ! & ( 12 ( G−1 1 %$ ( &'( ! F −1
x∈A∪B ⇒ x ∈ A ∪ B 8 ( L1 ∪ L2 ∪ L3 ⇒ (x ∈ A ∧ x ∈ B) ( L1 ∩ (L2 ∪ L3 ) ⇒ (x ∈ A ∧ x ∈ B) ( L1 ∩ L2 ∩ L3 ⇒x∈A∩B ⇒ '' ( F1 = (1) A ∪ B ⊆ A ∩ B {(a, α), (b, α), (c, α)} 3 % F2 = (2) A ∩ B ⊆ A ∪ B
{(a, α), (b, α), (c, β)} F3 = {(a, α), (b, β), (c, α)} F4 = {(a, β), (b, α), (c, α)} F5 = {(a, α), (b, β), (c, β)} F6 = {(a, β), (b, α), (c, β)} F7 = {(a, β), (b, β), (c, α)} F8 = {(a, β), (b, β), (c, β)}
( G1 = {(α, a), (β, a)} G2 = {(α, a), (β, b)} G3 = {(α, a), (β, c)} G4 = {(α, b), (β, a)} G5 = {(α, b), (β, b)} G6 = {(α, b), (β, c)} G7 = {(α, c), (β, a)} G8 = {(α, c), (β, b)} G9 = {(α, c), (β, c)} G10 = {(α, a)} G11 = {(α, b)} G12 = {(α, c)} G13 = {(β, a)} G14 = {(β, b)} G15 = {(β, c)}
x = −0, 7
x = 0, 75 x = log9 5
1 =⇒ log25 3 = 4x
a3 − b3 = (a − b)(a2 +ab + b2 )
107 333
131 990
13 5 [−2, 5), [1, 3) [−2, 1), [3, 5) (−5, 1), (−2, 0] (−5, 2] ∪ (0, 1), ∅ [−1, 5; 3, 5), (0, 5; 3, 5) [−1, 5; 0, 5], [3, 5; 4, 5)
{1}
{−1, 1}
lim M = 1 lim M = −1
{3} − 32 , 23
2 3 2 lim M = − 3 {(2, 4, 5)} lim M =
L 12 5 −∞, 11 (−∞, 2)
[4, +∞) (−2, 1) (−3, 2) (−∞, −2] ∪ [1, 2] (−∞, −5) 3 ∪ −3 , +∞ 4 43 , 17 14 "4 3 7 , 2 (−∞; 1, 2] ∪ (2, 5; +∞)
! (−∞, −3) ∪ [−2, 2) " 2; 49 " (−2, 8] L # $
%& ' (! ## x+3 ≥3 2x − 5 |x + 3| ≥3 ⇔ |2x − 5| ⇔ (∗) |x + 3| ≥ 3|2x
5 −5| ∧ x = 2 ' x ≤ −3 (∗) ⇔ −(x + 3) ≥ −3(2x − 5) 18 ⇔x≥ 5 L1 = ∅ ' −3 < x < 52 (∗) ⇔ x + 3 ≥ −3(2x − 5) 12 ⇔x≥ 7 " 12 5 , L2 = 7 2 ' 52 < x (∗) ⇔ x + 3 ≥ 3(2x − 5)
18 ⇔x≤ 55 18 5 L3 = , 2 5
) " & "
L=L 1 ∪ L 2 ∪ L3 " 12 5 , L= 7 2 5 18 5 ∪ , 2 5 " 3 95 − 2 , 2
(−∞, −1)∪(4, +∞) −∞, 85 " 32 , +∞ 1 , − 11 2 2 38 0, 23 23 , +∞ [3, 5]
(−∞, −2) 9 ∪ − , +∞ 5
y
6
½ ½
(−7, −5] ∪ [1, 13)
-x
1 · 2 · . . . · n · (n + 1) ×(n + 2) · (n + 3) n + 1 · 2 · 3 3n + 1 · 2 · 3 · . . . · n 3 · 1 · 2 · . . . · n − 5 1 · 2 · . . . · (2n − 4) ×(2n − 3) 1 ·1·2·. . .·(n−1)·n 3
(n + 2)! n! (n − 2)!(n + 1) n + 1 (n + 2)! 2n · (2n + 1) (n − k − 1) · (n − k) 1 (2n − 1) · 2n n+2 (n + 1)! 4n2 + 2n + 1 (2n + 1)! n+1 n!
4n2 − 2n + 1 (2n)!
21 3921225 0 −36
2n = (1 + 1)n 0n = (1 − 1)n 495a4 x−2
0
60
1 8 1155 2048 14 − 81 (−1)n · (2n − 1)! n!(n − 1)!
x1 = 10−2,5 ; x2 = 10
−
9x2 y −4 − 42xy −2 z +49z 2 a3 + 6a2 b + 12ab2 +8b3 32x−5 − 240x−2 +720x − 1080x4 +810x7 − 243x10
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 25 25 25−k (−1)k x k k=0 n n 2(n−k) k b a k k=0 n k n (−1) k k=0
1 + 0, 3 + 0, 03 +0, 001 = 1, 331 !
2
×an
−2kn
P3 · P5 = 720
VW(5) = 3125 5
P4 = 24 P4 − P3 = 18 PW(2,3,1) = 60 PW(2,1,2,1) − 5 (1,2,1)
×PW6
(2) 10 · PW4
25
(k)
C25 = 225 − 1
k=1
= 33 554 431
4
6 6
VW(15) = 1 073 741 824 b a e d c 54. V5(2) = 20 V4(1) = 4
1 28.
= 120 = 120
C49(6) = 13 983 816 CW(3) = 56 VW(3) = 216 6
6
P10 = 3 628 800 V10(5) = 30 240 P5 = 120
5
(i)
C5 = 31
i=1
5
(i)
VW2 = 62
i=1
C25(k)
25 = k
C20(15) · C25(20)
= 823 727 520
VW(10) = 1024 PW(7,3) = 120 2
10
10
(i,10−i)
PW10
= 176
i=7
968 N2 = C97(3) · C3(2) S =
= 442 320
(5) C100
= 75 287 520 (5) S1 = S − C97 = 10 841 496
12 − 5 ; 6 − 9 41 − 3 ; 1 − 3 3 41 + 13+39 ; 130 130
− 2 + 14 ;
83 ; ;
73 55 48 ; + 36 73 73 55 4 − ; 1 36 3 4, 52993 + 2, 01764 − 0, 66622 −0, 98236 4, 24264 + 4, 24264 0, 64395 − 0, 17255 5, 79555 + 1, 55291 1, 06066 − 1, 06066
−5 + 2 ; −1 + 6 1 7 14 − 2 ; − − 4 4
2 14 + 25 25 3, 98174 + 3, 93185 − 2, 94646 −0, 06815 − 2, 07055 +7, 72741 0, 35355 + 0, 35355 − 0, 51764 −1, 93185
5 4 ;y= x = − 11 11
3 u = −2; x = 2 1 y = 2; z = − 2
−|xy|
−1
− 1 + √ −6 − 12 6 − e−π/2 1 + 2 − 31 − 17 3 √
15 + 2 5 6 √ 5 2 + 18
6
0 − 5 = 5 e (−π/2)
= 5 cos (−90◦ )
+ sin (−90)
9 + 0 = 9 e ·0 = 9(cos 0◦ + sin 0◦ ) π/2
0 + 3 = 3 e = 3(cos 90◦ + sin 90◦ ) −8 + 0 · = 8 e π = 8(cos 180◦ + sin 180◦ ) √ 1 + 3 = 2 e π/3 = 2(cos 60◦ + sin 60◦ ) 7√ 7 − 3+
2 2 = 7 e 5π/6 = 7(cos 150◦ + sin 150◦ ) √ 1 3 − −
2 2 (−2π/3) =e = cos (−120◦) + sin (−120◦ ) −3, 00908 −3, 99318
= 5 e (−2,21657) = 5 cos (−127◦)
+ sin (−127◦)
4 −√ 6
= 52 e√(−0,98279) = 52 cos (−56, 30993◦)+
sin (−56, 30993◦) 0 + 2 = 2 e π/2 = 2(cos 90◦ + sin 90◦ ) 0, 5 + √ · 0, 5 2 π/4 = e √2 2 (cos 45◦ = 2 + sin 45◦ ) −0, 5 + 8
=
·1,63322 64, 25 e = 64, 25( cos 93, 57633◦ + sin 93, 57633◦)
√ 3 − = 2 e (−π/6) = 2 cos (−30◦ )
+ sin (−30◦ ) −2 + 2 √ =√ 2 2 e 3π/4 = 2 2(cos 135◦ + sin 135◦) √ − 3− √ 3
=√ 2 3 e (−2π/3) = 2 3 cos (−120◦)
+ sin (−120◦) 1√ 1 3−
2 2 = e (−π/6) = cos (−30◦ ) + sin (−30◦ ) 0 − 9 = 9 e (−π/2) = 9 cos (−90◦ )
+ sin (−90◦ )
−1 + 0 = e π = cos 180◦ + sin 180◦ − e2 +0 = e2 e π = e2 (cos 180◦ + sin 180◦) π 0 + e4 · = e4 · e 2 = e4 (cos 90◦ + sin 90◦ ) Re z = 2 Im z = 1 3 Re z = − 2 3 Im z = 2
Re z = 0 Im z = −2 Re z = 16 Im z = 0
Re z = 0 √ Im z = 1024 3 Re z = −64 Im z = 0 √
1 √ 2+ 6 2 1 √ 2 z2 = − 2 √
+ 6
! z1 =
z1 = 1, 62894 +0, 52017
z2 = −1, 26495 +1, 15061
z3 = −0, 36398 −1, 67079
z1 = 1, 40211 +0, 18459
z2 = −0, 18459 +1, 40211
z3 = −1, 40211 −0, 18459
z4 = 0, 18459 −1, 40211
√ √ 2+ √ z1 = √ 2 z2 = −√2 + √2 z3 = − √ 2 −√ 2 z4 = 2 − 2 z1 = 0, 89101 +0, 45399
z2 = −0, 15643 +0, 98769
z3 = −0, 98769 +0, 15643
z4 = −0, 45399 −0, 89101
z5 = 0, 70711 −0, 70711
√
1 3 " 1; + 2 2
√3; −1 1 + 2 2 1 √3 − − 2 2 1 √3 − 2 2 −
0, 80902 + 0, 58779 − 0, 30902 +0, 95106 − 1; −0, 30902 −0, 95106
0, 80902 − 0, 58779
+0, 21275 − 0, 21275 −1, 06955 1, 06955 − 0, 21275
0, 28485 + 1, 75532 − 0, 28485 −1, 75532
0, 29863 + 1, 50405
− 1, 45186 −049340 1, 15323 − 1, 01065 0, 21275 + 1, 06955 − 1, 06955
0, 38268 + 0, 92388 − 0, 92388 +0, 38268 − 0, 38268 −0, 92388 0, 92388 − 0, 38268
x(2x3 +x2 +2x+1) = 0 x1 = 0 2x3 + x2 + 2x + 1 = 0
! " # $"" % & ' ! ( "
an−1 = a23 = 22 : n 3 3 2 x + 22 x2 + 22 · 2x +22 = 0
)
y = an x = a3 x = 2x : y 3 + y 2 + 4y + 4 = 0
$"" * " b0 = 4 = 22 "+ #
±1, ±2, ±4 , ' ,
y1 = −1& 1 1 4 4 −1 0 −4 −1 1 0 4 0 y2 + 4 = 0 y2 = 2 ; y3 = −2 xk = yk 2− 1 + k = 2, 3, 4 - " x2 , x3 , x4 : 1 x1 = 0; x2 = − ; 2 x3 = ; x4 = −
x1 x2 + x1 x3 +x2 x3 = −16 = b1 x1 x2 x3 = 80 = −b0
x1 = 2;
x2 = 3 x3,4 = 3 ± 2 x1 + x2 + x3 + x4 = 11 = −b3 x1 x2 + x1 x3 + x1 x4 +x2 x3 + x2 x4 +x3 x4 = 49 = b2 x1 x2 x3 + x1 x2 x4 +x1 x3 x4 + x2 x3 x4 = 101 = −b1 x1 x2 x3 x4 = 78 = b0
"(&
y 3 + y 2 + 4y + 4 = 0
(y 2 + 4)(y + 1) = 0
y1 = −1; y2,3 = ±2
x1 = −3 + x2 = −2 +
x1 + x2 = −5 + 2 = −b1 x1 x2 = 5 − 5 = b0 x1 = 2 ; x2 = −1 x1,2 = ±4 x3 = −5 x1 + x2 + x3 = −5 = −b2
x1 = x2 = x3 = 1 x1 = x2 = −1 x3 = x4 = 2
* x1,2 = 1 ± 2 x3,4 = −2 ± x1 = 0; x2,3 = 1; x4,5 = −2
x1 = −2; x2,3 = 3 x1 = 1; x2 = −2
x3 = −3; x4 = 7 √ x1,2 = 1 ± √ 3 x3,4 = ± 2
x1 = 1; x2√= 3 x3,4 = 1 ± 2
x1,2 = ±1 x3,4 = 1 ± 2
x1 = 1; x2 = x3 = x4 = −1 2x = y 1 1 x1 = − ; x2 = 2 2 3 5 x3 = ; x4 = 2 2 12x = y 1 x1 = 0; x2 = − 3 1 x3 = x4 = 2 x5,6 = −1 ±
y = 2x/3 x1 = 0; x2 = 3 y = log√ x5 5 x1 = 5; x2 = 5
3 2 −3 2 9 6 9 3 2 3 11 6 3 2 9
x2 (x − 2)2 (x2 + 2x +2) 2(x − 1)(x − 2)(x +3)(x2 + x + 1) (x − 3)(x + 2)2 (x2 +2x + 10) (x + √ 1)(x − 9)2 (x2 −2 2 + √ 4) · (x2 +2 2x + 4)
a1 = 0, a2 = 2, a3 = −2 λ2 + λ + 1 2 |a| = 70, cos α = 7 6 3 cos β = , cos γ = − 7 7 √ −−→ −−→ −−→ |AB | = |AC | = |BC | = 2 2 √ ! P (−4; 4; 4 2)
" D(9; −5; 6) −−→ AC = (−2, 6, −10)Ì −−→ BD = (14, −8, 8)Ì √ −−→ −−→ |AC | = 2 35, |BD | = 18
P3 (x) = 2(x − 3)3 +9(x − 3)2 +11(x − 3) + 9
−(x + 2)5 + 10(x +2)4 − 39(x + 2)3 +73(x + 2)2 − 64(x +2) + 21
2 −9 11 3 6 −9 6 3 2 −3
2x3 − 6x2 + 2x + 8 x = −1
2 9
3 2 1 Ì a0 = − √ , √ , − √ 14 14 14 5 3 2 Ì 0 b = √ , −√ , √ 38 38 38 a + b = (2, −1, 1)Ì b − a = (8, −5, 3)Ì − 2a + 3b = (21, −13, 8)Ì
3 2
x =
5 3
,−
35 10 Ì , 3 3
rA + rB + rC 3 S(3; 1; 3)
# rS =
5 11 13 1 13 17 ; ; ,D ; ; 3 3 3 3 3 3 c + λb v = 1+λ 17 P − ; 0; 0 10 √ √ √ a = 11, b = 5, c = 2 3 3 3 3 , Mb 1; ; 2 Ma ; ; 2 2 2 2 3 1 Mc ; 1; 2 2 1 1 1 Ì 1√ , , m = , |m| = 3 2 2 2 2
$ C
P (16; −5; 0)
8 (−16, 20, −8)Ì
ϕ ≈ 73, 39845◦ a · b = 0 λ = 4 15 49 108
[a, b, c] = 0
a = −b + 3c 76 0
!"
20 , 20 7 3 547 s0 = ± √1
√ 2 6
11
(1, −3, 1)Ì
r = (−2, 3, −5)Ì
+λ(3, −7, 4)Ì r = (3, −2, 1)Ì +λ(−2, 0, 1)Ì − ∞ < λ < +∞
18 23 1 ; ;− 22 √ 11 22 1826 22
F
√
33 A " &' S(4; −7; −3) (λ = −3) B ϕ ≈ 29, 62048◦ S(1; 2; 1) r = (2, −3, 4)Ì +λ(1, 0, 2)Ì !""" −∞ n2 + n
* n ≥ 1 0 1 / 1 n ≥ n2 + 1 2n n ≥ 1 0
an =
*
∞ i 2 i +1 i=1 ∞ 11 ≥ 2 i=1 i
! "! n |an | = √ n n 1 √ n 2 ( n) (1 + n12 ) n ⇒ lim n |an | n→∞ 1 = 2 0 = 1# 1 ·1 # 1 an = (3n − 1)2 1 < 2 $% n ≥ 1! n
)# $% a = 2# & # !
#
# ',! # )# ) !# # ) '(! #
# )# & '-! # # )#
# '! # )# #
# # '*! # )#
#
# ) # $# '.! # −1 ≤ x < 1 )# −∞ < x < +∞ # 0 < x < +∞
#
# |x| < 1
# #
# |x| < 2
'+! #
$# −1 ≤ x < 1 # −1 < x < 1
y
y
y
2 3 2 3 32
5
x
3 2
34Π
Π4
2 Π 4
Π 2
1
x
3Π 4
2
5
1
1
2
x
y
y Π 2
1 2
x
1 2
1
1
x
Π2
[−1; +∞) (−∞; −2)∪
(−2; 2) ∪ (2; +∞) √ (−∞; − √ 2] ∪ {0} ∪[ 2; +∞)
(−∞; −1] ∪ [0; 1] (−2; 0] (−1; 1) ∪ (2; +∞) − 31 ; 1 [1; 100] (2; +∞) (−∞; 1) ∪ (1; +∞) (−∞; +∞) [1; 4) (−∞; −1)∪
(−1; 3) ∪ (3; +∞)
D(f ) =
{x | x ∈ R ∧ x > 5} W (f ) = {y | y ∈ R ∧ y > 0}
D(f ) = {x | x ∈ R
∧x = −2 9 ∧ x = 3}
W (f ) = y | y ∈ R 1: ∧y = 0 ∧ y = 5 D(f ) = {x | x ∈ R ∧x = −2 ∧ x = 2} W (f ) = {y | y ∈ R ∧y ∈ (−∞; 0) ∪[1; +∞)}
D(f ) = {x | x ∈ R}
W (f ) = {y | y ∈ R ∧y ∈ (0; 3]}
f (−x) = 7 sin x cos x = −f (x)
! !
y
y
1 2 9 5 3
1 2 4
x
2 4
x
−1
y = f (x) 2 1 1−x −1 = 4 1+x D(f −1 ) = (−1; 1] W (f −1 ) " 1 = − ; +∞ 4
x =
8 +3 y−1
1 y − 7 ln x = 3 5
2 −1
+1
x = x
ey/5 +1 ey/5 −1
%&' x = 4; y = x + 5 f (x) = x + 5 26x − 42 + 2 x − 7x + 12 ( xN1 = −2; xN2 = 2 xP = 1 )* # y x=0 = −4
%&' x = 1; y = 1 f (x) = 1 2x − 5 + 2 x − 2x + 1 (
! x2 − 5x + 6 y = f −1 (x) = 5 √ x2 − 2 +x 3 x √ −1 D(f ) = [0; +∞) 1 + 1 + x2 x y = − x3 − 1 p0 = 10π [1; +∞) 1 p0 = 4π √ " f (x) + f (−x) y = 3 − x 2 p0 = 6π 1 [0; +∞) + f (x) − f (−x) π 2 p0 = 1−x 2 ; (−∞; −1) y = 1+x xN1 = 2; xN2 = −3 + , - ∪(−1; +∞) xP = 4; xL = 3 . 2 1 3 3 − 5 sin ey x = #$ x = 0 y = 2 2
(x − 5)(x + 3) x→5 lim (x − 5)(x + 5) = lim
x→5
=
3
4 8 = 10 5 lim
x→0
x→∞ lim
x+3 x+5
sin ax =a x
1+ 3+
1 x 2 x
x2
4 12
−1 1 − 56 72
=0 2(x − 1) lim x→1−0 −(x − 1) = −2
z = arcsin 7x x→0⇒z→0 1 x = sin z 7 z lim 1 z→0 7 sin z 1 = 7 × lim sin z z→0
=7·
z
52 12
√ 2 − 2 1 2 3 2 π 2
1 =7 1
0 e e2 −1 x → −0; + 1 x → +0 −∞ x → 2 − 0 + ∞ x → 2 + 0
+∞ x → 1 − 0 0 x → 1 + 0 2 x → −2 − 0 − 2 x → −2 + 0 +∞ x → −1 − 0 −∞ x → −1+0 −1 x → 1 − 0 + 1 x → 1 + 0 π2 x → 1 − 0 π − x → 1 + 0 2 1 − 36
x → −3±0
! x0 = 0 " # #
" ! x0 = 1 " # #
"
! x01 = −2 x02 = 2 # $ " " ! x01 = 1 " " x02 = 2
% & #
" ' # ! x0 = 0 " # ( # f (2) = 1
" ! x01 = 0 # ! x0 = 1 '# ' # ( f (0) = 2 f (0) = 1 ! f (0) = 0 x0 = kπ
f (0) = 1 (k = ±1, ±2, . . . ) " f (2) = 78 " # f (0) = n1 = n ! x0 = 2 " # # f (0) = 2
f (0) = 0
lim f (x)
lim f (x) x→2−0
x→1−0
= lim f (x)
= lim f (x) = 2
a = 0 f (0) = 1
x→1+0
x→2+0
x2 − 4x + 4 = (x − 2)2 y = 2 3 10x−3 − 6x 3 + 4x 4 y = −30x−4 − 4 · 3 −1 12 − 1 x 3+ x 4 3 4 30 4 y = − 4 − √ 3 x x 3 +√ 4 x y = 1 13 5x2 − 3x 2 + 3x 4 2 2x 3 5 4 3 1 = x 3 − x− 6 2 2 3 31 + x 12 2 5 4 1 y = · x 3 23 7 1 3 − · − x− 6 2 6 3 31 19 + · x 12 2 12 10 √ 3 y = x+ 3 √ 31 1 12 √ + ·x· x7 6 4x x 8 x y =π· 1 − x2 y = π×
1(1 − x2 ) − x(−2x) (1 − x2 )2 1 + x2 y = π · (1 − x2 )2
2 cos x sin x − 2 + x3 x tan x 1 √ −√ 2x x x cos2 x 1 # 1 − sin x
1 √ $ √ √ y= x2 · 3x 2x 2 x( x + 1)2 √ √ 4 = 3 2 x6 · x y = x ln x − lg 5 √ 4 −3 lg x = 3 2 x7/8 1 1 7 4 √ y = ln x + x · y = · 3 2 x− 8 x 8 1 √ −3 lg e 4 7· 3 2 x √ y = 8· 8x 3 lg e y = 1 + ln x − 4 3 2 x − 3x + 12x + 3x
3 ln x −42x − 40 / x % − 2
2 x −2x + 7 6x2 ex +2x3 ex −3x ln 3 y⎧= ln |x| = −7x2 + 20x − 6 & ⎨ln x ex ,x>0
u vw + uv w + uvw ⎩ln (−x) , x < 0 u vwz + uv wz +uvw z + uvwz y = ⎧ ⎧ ⎨1 ⎨−2x + 2 , x ≤ 2 ,x>0 x y = ⎩ (−1) ⎩−2 1 ,x>2 ,x0 ⎧ ⎨−1 , x < 0 y = ⎩1 ,x>0
y =
x ∈ (−∞, +∞)
x ∈ (−∞, 0) ∪(0, +∞) y⎧= ⎪ ⎪ ⎪ ⎨sin x , x ∈ [0, π] − sin x , ⎪ ⎪ ⎪ ⎩ x ∈ (π, 2π]
y=
y = ⎧ ⎪ ⎪ ⎪ ⎨cos x , x ∈ [0, π)
1 1 y = − · 2 1−x 1 1 − · 2 1+x 1 1+x+1−x = · 2 (1 − x)(1 + x)
x ∈ [0, 2π]
− 3 y = 5 1 + x2 2 3 y = 5 − 1 2 5
− +x2 2 · 2x
y = −
15x √ 2 (1 + x )2 1 + x2
1 ln (1 − x) 2 1 − ln (1 + x) 2
− cos x , ⎪ ⎪ ⎪ ⎩ x ∈ (π, 2π]
x ∈ [0, π) ∪(π, +2π]
√ x+1 √ √ 2 x+2 x· x 2 f (1) = √ 2 1 + 2√ 3 = 3 6 sin (6x + 4) 1 √ x2 + a2
2
" − x x+ 1 e + e− −3x # −6 (1 −e e−3x )2 y˙ = cos 2t x 2
y¨ = −2 sin 2t
x˙ = − sin 2t
1 −1 a ebx (b cos cx −c sin cx) y =
x2
y⎧=
⎪ ⎪ ⎪ ⎨ln (− ln x) , 0<x 1 y =
sin1 x
1 x ln x
x 2
x ¨ = −2 cos 2t v = − 1 s (s − 1)2 2 v= 2 s2 (s − 1)3 w = s3 (4 ln s + 1) s 2 w = s2 (12 ln s s2 +7) 2 (2x − 3) x y = (x − 1)2 y =2 x(x3 − 3x + 3) × (x − 1)3
cos x, − sin x,
− cos x, sin x 1 2 1 6 , − 2, 3, − 4 x x x x 1 √ , 2 x+1 1 , − 4 (x + 1)3 3 , 8 (x + 1)5 15 − 16 (x + 1)7
−3 sin 3x, − 9 cos 3x, 27 sin 3x, 81 cos 3x
f (x + p) = f (x) p ⇒
f (x + p) = f (x)
p
y (n) =
2 · n! (1 − x)n+1
y (n) = an · eax y (n) = (−1)n−1
×
(2n − 3)! 22n−2 (n − 2)!
arctan x 1 ×x 2 −n ln y = ln 10−3x ln x 1 √ y (n) = 2n−1 |x| x2 − 1 y n−1 = 0 − 3 ln x π × sin 2x + y y = ⎧ 2 1 ⎪ √ 2 −3x · ⎪ , |x| < √12 ⎪ ⎨ 1−x2 x = −3(ln x + 1) √ 2 − , 2 f (n) (x) $ = 1−x ⎪ ⎪ ⎪ ⎩ 1 −3x (−1)n 1 √ < |x| < 1 y = −30x (ln x 2 · n! 2 (1 + x)n+1 +1) 6x cosh2 x2 sinh x2 % y 1 a · tanh (ax + b) =0 ln y = 1, + y (1 − x)n+1 1 y = 0 x + sinh 2x 1 2 ! ! x ln x = e = const.
arsinh x 1 " # (tan x) cos x x (arctan x) 1 n × 2 × ln arctan x sin x cos x 3 i(i − 1)xi−2 sin x x i=2 + 2 ln tan x + n cos x (1 + x2 ) arctan x +(x − 1) · i(i − 1) x−1 2 x+2 e x+1 i=3 2 (x + 1) x2 + x + 1 ×(i − 2)xi−3 x 2 x (1 + ln x) 4x = (n + 1)n(n − 1)xn−2 4 $% x = 1 1 1 − ln x x −1 √ x x · n 2 x 3 x 3 i(i − 1) f (−x) = f (x) 2(1 + x3 ) i=2
⇒ 1 = (n + 1)n(n − 1) 2 − f (−x) = f (x) & k − 1 = 0 % (1 + x ) arctan x ⇒ k = 1 cos x sinh (sin x) n f (−x) = −f (x) 2 2 k(k − 1) 6x tanh x
2 2 k=1 cosh x f (−x) = −f (x) (n + 1)n(n − 1) 1 = ⇒ 3 cosh x $% ' − f (−x) = −f (x) 4 sinh 4x ( ) ⇒ √ n 4x − 1 f (−x) = f (x) k(k − 1)
2x k=1
=
n
k2 −
k=1
n
n
k
k=1
√ √ k
k=1
π6 , − 63 , 7363 f (k) (x) =
n
n k!xn−k k
n = (1 + 1)n k k=0
! v = t2 − 4t + 3 a = 2t − 4
" #$ v = 0 :
!
t1 = 1, t2 = 3 x = A cos ωt x˙ = −Aω sin ωt x ¨ = −Aω 2 cos ωt = −ω 2 A cos ωt = −ω 2 x x=0 π ⇒ ωt = + kπ 2 π x˙ + kπ = ±Aω 2 π x ¨ + kπ = 0 2
!
x=A ⇒ ωt = 2kπ x(2kπ) ˙ =0 x ¨(2kπ) = −Aω 2 x = −A ⇒ ωt = (2k + 1)π x˙ (2k + 1)π = 0 x ¨ (2k + 1)π = Aω 2 (k ∈ Z) V = πr · 2h · h t t 2 h −h t = v v h = t π(2r − h)h v = π 2
! v = x˙ = Ak e−kt =
k·A e−kt = k(A−x)
! x = 10+20t− 12 ·gt2 v = 20 − gt a = −g = −9, 81 2 v=0: t=
20 ≈ 2, 04 g
% ! y = nxn−1 x ! y = √ x 2 x 1+x
! r = 4 sin2 ϕ ϕ ! (sin2 t) = sin 2t t ! cos ϕ2 ϕ 1 = − sin ϕ 2 2 ! y = 0, 04 Δy = 0, 0401
! y = 0, 05
Δy ≈ 0, 049876
& ! VV !
3x2 x x3 = 0, 6% 3b s f = 8f =
! x = 5x2 √yx
0 0 -- "1 5 ,-- I = 2 , 1 ), i=1 i - - &54 . " " - 0 " n
∞ (−1)i+1 ai < an+1 i=n+1
" "& $ |rn (x)|
−lglg2ε 6 7 # -- |rn (x)| < 0, 01 0 n ≥ 7 /0 " ' " - " ' - |rn (x)| < ε
1 xn+1 ≤ n+1 n+1 x ∈ [0, 1] 1 |rn (x)| < 0, 1.
|rn (x)|
0 s(x) = ⎩0 x = 0 ⎧ 1 ⎪ ⎨ x > 0 3 )n−1 (1 + x r(x) = ⎪ ⎩0 x = 0 n rn (x) √ 0, 1 n−1 x3 < 10 − 1 x ≥ 0 ! " " # $ x ≥ 1
"# % x≥1: 1 |rn (x)| ≤ n−1 |rn (x)| < ε 2 lg ε n > 1 − & lg 2 ' ε = 0, 001 : n ≥ 11 1 1 − ( fi (x) = x+i−1 x+i 1 1 ) sn (x) = − x x+n 1 s(x) = lim sn (x) = n→∞ x 1 1 ≤ x ≥ 0 |rn (x)| = x+n n ⇒ |rn (x)| < 0, 1 n ≥ 10 * $ % x ≥ 0 + ! , "* + -
1 1 1 1 + + 2 + 3 + ... 3 3 3 . " / ! "# x ≥ 0 rn (x) " " ! - 1 |rn (x)| ≤ ) 2 · 3n−1 |rn (x)| < 0, 01 n ≥ 5
x ≥ 0
* $ x ≥ 0 + ! " " + -
1 1 1 1 + + + + ... 2 4 8 0 !
"# x ≥ 0; 1 |rn (x)| ≤ n−1 . 2 |rn (x)| < 0, 01 n ≥ 8 ∞ cos nx 1 1 1 * n−1 ≤ n−1 n−1 2 2 2 n=1 ∞ 1 1 x 1 * 2 sin ≤ 2 n n n n2 2 * x > 0 * x > 1 "* |x| > 1
" √ √ 5 * − 5, 5
√ √
√ √ % 3 3 5 5 , , "* − * − 2 2 2 2
3 * [−3, 3)
√ √
* [− 3, 3]
* (−∞, ∞)& #
n=1
* (−1, 1]
√ √ % 2 2 , * − 3 3
1 1 %* [−1, 1] * − , 10 10 *
* r = e
* [−5, 3)
* (1, 2]
* [−1, 3)
[−1, 0) s(x) = (1 −1 x)2 ,
sin2 x = |x| < 1
1 − 2x s(x) = , |x| < 1 (1 + x)2 ∞ x2n |x| < ∞ (2n)! n=0
x2n+1 sinh ϑx R2n (x) = (2n + 1)! ϑx e − e−ϑx ≤ e|x| | sinh ϑx| = 2 0 < ϑ < 1 |x|2n+1 |x| e |R2n (x)| ≤ (2n + 1)! ∞ |x|2n+1 ! (2n + 1)! n=0
" # $ % &#' (( ) 2n+1 |x| = 0 |x| < ∞ lim n→∞ (2n + 1)! (( ) n→∞ lim |R2n (x)| = 0 |x| < ∞$ ∞ x2n cosh x = |x| < ∞ (2n)! n=0
* 0 < ϑ < 1 R2n (x) = (−1)n 2n+1
x · 22n sin (2ϑx). (2n + 1)! |2x|2n+1 |R2n (x)| < (2n + 1)! lim |R2n (x)| = 0 +#( ,- ×
n→∞
# $ (
∞
(−1)n+1
n=1
|x| < ∞
2
s(x) = (11+−xx2 )2 , |x| < 1
s(x) = arctan x, |x| ≤ 1 s(x) = − ln (1 − x), x ∈ [−1, 1) s(x) = (11−+x)x 2 , |x| < 1
.
22n−1 2n x (2n)!
∞ (x − a)n e =e , |x| < ∞ n!an n=0 x (x − a)n+1 · e1+ϑ( a −1) Rn (x) = (n + 1)!an+1 lim |Rn (x)| = 0 x/a
n→∞
x4 x2 + − ... 1− 2 6 3 (x − 1)2 3 + ... 1 + (x − 1) + 2 4 2! 1 x − 2 (x − 2)2 − + − ... 2 4 8 x4 x6 x2 − − ... − − 2 12 45 ∞ −3 n f (x) = x n n=0 ∞ (n + 1)(n + 2) n x ; = (−1)n · 2 n=0 |x| < 1 ∞ 1 − 2 2n f (x) = x n n=0 ∞ (2n)! 2n = (−1)n · 2n x ; 2 (n!)2 n=0 |x| < 1 ∞ x2k−1 ; |x| < 1 f (x) = 2 2k − 1
e
/
k=1
f (x) = ln 2 + ln (1 − x)
x = ln 2 + ln 1 − 2 ∞ xn 1 + 2−n − ; |x| < 1 n n=1
f (x) = ln∞(1 + x3 ) − ln (1 + x) = −2
n=1
cos
nπ xn ; |x| < 1 3 n
∞
(2n)! 2n f (x) = x ; |x| < 1 2n (n!)2 2 n=0 ∞
x2n ; |x| < ∞ f (x) = (−1)n n! n=0
f (x) =
∞
(−1)n
n=0
2n n+1 x ; n!
|x| < ∞
2 1 1 − · 1 − x 3 1 − x3 ∞ 2 1 + n+1 xn ; |x| < 1 =− 3 n=0
f (x) = −
f (x) =
∞
(−1)n
n=0 ∞
f (x) = 2
x2n+1 ; |x| < 3 9n+1
(−1)n · 32n
n=0
(n + 2)x2n+1 ; |x| < ∞ (2n + 1)!x2n+1 ∞ (2x)2n 1 ; f (x) = 1 + (−1)n 2 n=1 (2n)! |x| < ∞ ∞ (2n)! n 1 √ f (x) = (−1)n n x ; 6 (n!)2 3 n=0 3 |x| < 2 f (x) = ln [1 + (x − 1)] = ∞ (x − 1)n ; 0<x 0) 5+ c = 0
4
4 2 0 5
5
z
0y x0 5
5
3
6
22
6
24 1
2 4
0
2 6
2
4 0
2
4
2 0
0 1
26 4 4
2 6
2 0
6 0 3 4 4 3 2 1 0 1
0
4
2
2 3
! " #
x = a, z 2 = ay y = b, z 2 = bx $% % & ' ( z = f (x, y)
7
8
( D(f ) = (x, y)| x2 + y2 < 4 ( D(f ) = {(x, y)| (−2 ≤ x ≤ 0
∧ y ≤ 0) ∨ (0 ≤ x ≤ 2 ∧ y ≥ 0)}
( D(f ) = {(x, y)| y > −x} 7 8 √ ( D(f ) = (x, y)| y > x ∧ x ≥ 0 ( D(f ) = {(x, y)| − 1 ≤ x ≤ 1
*
- * . -
( c > 0 / 0, - * z , 1 c < 0 !% / 0, - * z , 1 c = 0 2 f (tx,ty) = = t2
4 − 2tx · ty (tx)4 + (ty)
x2 + y 2 − 2xy = t2 · f (x, y)
3 a −a b + b −b a = 1
( 1. 4 5 6, ! (x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0) ( ( 1., 3., 6. 8. 4 6, ! (xyz > 0)
( 78 * ! ( D(f ) = {(x, y)| y < −x} ) x = ±1 y = ±1 ( # z = ±1 78 , ( # ( ) * * +*, ( * * / r = 1 $ n = (1, 1, 1)Ì * O(0; 0; 0) ( ! * ∧ −1 ≤ y ≤ 1} 7 D(f ) = (x, y)| x2 + y 2 ≥ 1 8 ∧ y > −x2
Δx z = (2x − y + Δx)Δx Δy z = (2y − x + Δy)Δy Δz = Δx z + Δy z − ΔxΔy
x = 2, Δx = 0, 1, y = 2, Δy = −0, 1 : Δx z = 0, 21; Δy z = −0, 19 Δz = 0, 03 m lim f (x, y) = y = mx x→0 1−m 3 2 y = x y = x 4 3 y = 2x
3 2 (−2) ! lim lim f (x, y) = −1 y→0 x→0 lim lim f (x, y) = 1 x→0 y→0
1 − m2 lim f (x, y) = x→0 1 + m2 y = mx " # $ % lim lim f (x, y) = 0 y→0 x→0 lim lim f (x, y) = 0 x→0 y→0 ⎧ ⎨1 y = x lim f (x, y) = x→0 ⎩0 y = 2x y→0
& −
1 4
1
%
0 1 2 1 4 '
2 $ % $ %
% ( x = 1, y = −1 ) * # x = mπ y = nπ (m, n ∈ Z) % ) * + x2 + y 2 = 1 ) * # x + y = 0 , - y 2 = x . ) * +,, / x = 0, y = 0, z = 0 ) * + z 2 = x2 + y 2 % ) * % 012 ,, z 2 − x2 − y 2 = 1
fx = 3x(x + 2y); fy = 3(x2 − y 2 ) 2x 2y ; fy = 2 2 +y x + y2 y x % fx = − 2 ; fy = 2 2 x +y x + y2 √ 3 t √ gx = √ 3 3x( √ x − 3 t) 3 x √ gy = √ 3 3t( t − 3 x)
fx =
x2
a − b cos γ c b − a cos γ ab sin γ ; cγ = cb = c c z y 1 1 fx = − 2 − ; fy = − 2 x z x y x 1 fz = + 2 y z
ca =
fx = e−xy (1 − xy) fy = −x2 e−xy
gx = (x +5t2t)2 ;
gt = −
5 (x + 2t)2
|x|y fx = − 2 x x2 − y 2 |x| fy = x x2 − y 2 3y fx = (3y − 2x)2 3x fy = − (3y − 2x)2 hx = cot (x − 2t) ht = −2 cot (x − 2t)
gx = 2 sin y cos (2x + y)
gy = 2 sin x cos (x + 2y)
fx = yxy−1 ; fy = xy ln x fx = − xy2 esin cos xy y x
1 sin y y e x cos x x xy 2 2x2 − 2y 2 fx = |y|(x4 − y 4 ) x2 y 2x2 − 2y 2 fy = − |y|(x4 − y 4 ) fy =
fx = yz(xy)z−1
fy = xz(xy)z−1 fz = (xy)z ln (xy)
fx = yz xy ln z; fz = xyz xy−1
fx = yz exyz ; fz = xy e
fy = xz xy ln z fy = xz exyz
xyz
fx = tan x;
fy = − tan y 2
+x ) zxx = − 2(y (y − x2 )2
2x (y − x2 )2 1 zyy = − (y − x2 )2 2u wuu = − ; wuv = 0 (1 + u2 )2 2v wvv = − (1 + v 2 )2 zxxx = 6; zxxy = 2; zxyy = 0 zyyy = 6 zxy =
αx =
t √ 2 x − x2 t2 x αt = 1 − xt2 fx = −a sin (ax − by) fy = b sin (ax − by)
zx = 2(x +1√xy) zy =
1 √ 2( xy + y)
√
zx = 2√1 x sin xy − yx2x cos xy
√ y x cos zy = x x 2
ux = t12 ex/t2 ; ut = − 2x ex/t t3 ux = ux ; uy = uy ; uz = uz 2 zxx = 2 (x −y y)3 xy zxy = −2 (x − y)3 x2 zyy = 2 (x − y)3 zx = y1 e xy ; zy = − yx2 e xy 1 x x x zxy = − 2 e y − 3 e y y y
!
y f + x · f − 2 ϕ y x xf 1 uy = − 2 + ϕ y x f + x · f 1 y uxy = − − 2 ϕ − 3 ϕ y2 x x 2xf 1 uyy = 3 + 2 ϕ y x ux =
zxy = zyx = abz 4z zxy = zyx = − xy 2 zxy = zyx = (x − 2y)2
grad z = (2x, 2y)Ì grad z|(3;4) = (6, 8)Ì (0; 0) r = 5 √ ∂z =x 3+y ∂a √ ∂z =3 3+4 ∂a (3;4)
x2 − y 2 = 3 −−−→ Ì a = M N = (3, 4) 1 3 ∂z · =1 = (−1, 2) 4 ∂a 5
(6, 3, 2)Ì 3 ! cos ϑ = √ ; ϑ ≈ 18, 43◦ 10 √ 9√ 2 − 3 2√ 2 68 3 # " − 13 3
6x1 + 6x2 , 6x1 + x4 ex2 , sin x3 , ex2
grad z = (2x, −2y)Ì
grad z|(2;1) = (4, −2)Ì
Δz = yΔx + xΔy + Δx · Δy $z = y $x + x$y Δz = −0, 62; $z = −0, 6 Δz = 2xyΔx + x2 Δy + y(Δx)2 +2xΔxΔy + (Δx)2 Δy $z = 2xy $x + x2 $y Δz = −0, 298602; $z = −0, 30 Δz = (2x − 3y)Δx + (2y − 3x)Δy +(Δx)2 − 3ΔxΔy + (Δy)2 $z = (2x − 3y)$x + (2y − 3x)$y Δz = −0, 79; $z = −0, 9 $f = $g = $f = $ $u = $f =
−y 2 $x + x2 $y (x − y)2 1 s/t s e $s − $t t t x$x + y $y x2 + y 2 x$x + y $y + z $z x2 + y 2 + z 2 2(x$y − y $x) x2 sin 2y x
grad z|(4;2)
Ì
√ 2 (1, 2)Ì = 4
1 x z−1 " y+ z $x $h = xy + y y 1 + 1 − 2 xz $y y x x 5 $z + xy + ln xy + y y $ϕ = (ex cos y + 3y cos 3x)$x +(sin 3x − ex sin y)$y (x2 + y 2 )$z − z(x$x + y $y) % $ψ = (x2 + y 2 )3
$f = (x2 − x3 )x1x2 −x3 −1 ln x4 $x1 +xx1 2 −x3 ln x1 ln x4 $x2 −xx1 2 −x3 ln x1 ln x4 $x3 $x4 +xx1 2 −x3 · x4 " 0, 075 ≈ −0, 738906 −0, 1 # Δz = ln 1, 044 ≈ 0, 04306 $z = 0, 04 2
$2 u = 4 3y 2 $x2 − 4xy $x$y x
+x2 $y 2
− xy) 2 u = − (yx xy 2
2 u = 2(xy + xz + yz) 2 u = −u(mx + ny)2
2
|Δz|Max
= f (2, 1; 3, 7) −f (2; 4) = 0, 06757
|ΔR| ≤ 0, 755Ω
ΔR R ≤ 0, 64 %
|ΔR| ≤ 0, 773Ω
|Δz| ≤ 0, 0625
ΔR R ≤ 0, 66 %
< |Δz|Max
|fx| ≤ 3,17 2, 1 3, 72 1 |Δz| ≤ = 3, 72 0, 07305 > |Δz|Max
2 |ΔA 0 |≤ 1, 363
|fy | ≤
ΔA0 A0 ≤ 0, 48 %
2 |ΔA 0 |≤ 1, 373
ΔA0 A0 ≤ 0, 49 %
≤ 4, 71 % ΔV V
ΔC ≤ 0, 63 % C ΔV ≤ 2 Δa V a Δh ≤ 10 % + h ΔV1 ≤ 0, 15 % V1
|ΔPa | ≤ 10 %
2x + 2y − z = 1 r = (1, 1, 3)Ì + λ(2, 2, −1)Ì x + 4y + 6z − 21 = 0 r = (1, 2, 2)Ì + λ(1, 4, 6)Ì
2x − z − 2 = 0 r = (1, 0, 0)Ì + λ(2, 0, −1)Ì x − y − 2z + 1 = 0; π π 1 Ì , , + λ(1, −1, −2)Ì r= 4 4 2 xy0 + yx0 = 2zz0 xy0 z0 + yx0 z0 + zx0 y0 = 3a3 xx1 yy1 zz1
2 + 2 − 2 = 1 a b c x y z + − =1 a b c
x + y − z = ±9
r = (3, 4, 5)Ì + λ(3, 4, −5)Ì
(0; 0; 0) 1 cos α = − cos β = cos γ = − √ 3 9 3 V = a = const 2 √ Sx a2/3 · 3 x0 ; 0; 0 √ Sy = 0; a2/3 · 3 y0 ; 0 √ Sz = 0; 0; a2/3 · 3 z0 r = (4, 3, 0)Ì + λ(4, 3, 5)Ì a ! √ 3 " P1 (0; 0; 4); z − 4 = 0 P2 (1; 1; 2); 2x + 2y + z = 6
z = 4t3 + 3t2 + 2t t z =0 t −2 cosh t (A − C) sin 2t + 2B cos 2t 2 e2t 4t e +1 u(w + 2vt2 ) − vw et tu2 (t2 + 1) tan t
2t · ln t · tan t + t 2 (t + 1) ln t + cos2 t z u v = vuv−1 + uv ln u x x x z y = ey +x ey x x n ∂f = k · tk−1 f xi ∂(tx ) i i=1
∂z z ex ex + ey ·x2 = x = ; ∂x e + ey x ex + ey ∂z = yxy−1 ∂x z y = xy ϕ (x) ln x + x x 2x x ∂z = 1− ∂u y y x ∂z x =− 4+ ∂v y y
∂z ∂x ∂z ∂y ∂z ∂x ∂z ∂y
∂z ∂z +p ∂u ∂v ∂z ∂z =n +q ∂u ∂v ∂z y ∂z =y − · ∂u x2 ∂v ∂z 1 ∂z =x + ∂u x ∂v =m
∂z ∂z 1 y ∂z = · + ∂x ∂u 2 x ∂v ∂z ∂z 1 y ∂z = · + ∂y ∂u 2 x ∂v
ur = ux cos ϕ + uy sin ϕ uϕ = −ux r sin ϕ + uy r cos ϕ zx = f · 2x; zy = 1 − 2f y 2−x 3+y y y = − 3 x 2x − e2y 2y e y = 2x e2y − e2x y y = − x x2 + xy + y 2
y = xy 1 y = 2 3−x y ; zy = − zx = z x y x ; zy = zx = 2z 2z a b zx = ; zy = c c y zx = 1; zy = x−z 3 3 y = ; y = − 4 4 y = −1 1 4 y = ; y = 5 5 (−1; −1) (−1; 3) (−3; 1) (1; 1)
y =
z z zx = − ; zy = − x y
zx = ϕ −
y ϕ ; zy = ϕ x
x20 + x0 y0 + y02 + (2x0 + y0 )h 2
+(x0 + 2y0 )k + h + hk + k
3
2
f (x, y) = mx + ny − (mx +3! ny)
f (x, y) = 9 + 11(x − 1) + 8(y − 2)
+3(x − 1)2 + 8(x − 1)(y − 1)+ 2(y − 2)2 + (x − 1)3 + 2(x − 1)(y − 2)2
f (x, y) = −1 − 2(x − 1) + (y + 1)
−(x − 1)2 + 2(x − 1)(y + 1) +(x − 1)2 (y + 1)
x2 y = [1 + (x − 1)]2 · [−1 + (y + 1)] 2
f (x, y) = x − (y + 1) − x2
1 +x(y + 1) − (y + 1)2 + R2 2 (x − y − 1)3 1 R2 = · 3 [ϑx + 1 − ϑ(y + 1)]3 00 zM in = −10 √ 3 xMin = 2; yMin = −3 xMin = yMin = 2V 1√ 3 zMin = 1; uMin = −14 zMin = 2V 2 1 xMax = yMax = zMax = VMax = 8 7 1 uMax = 7 7 √ √ 4 y = 2, 4 x − 0, 8 xMin = √ 2; yMin = 2√ 4 4 6, 59 zMin = 8; uMin = 4 · 2 y = −1, 21 + x 1 xMin = ; yMin = zMin = 1 3, 13 2 y = 2, 00x3 + uMin = 4 x 1 y = 7, 164 · x1,953 xMin = −3; yMin = 3 λ1 = λ2 = 3; λ3 = −3 2 44 zMin = ; uMin = 3 27 R = x21 + x22 + x23
2 1 +4x1 x2 + 4x1 x3 − 4x2 x3 xMin = − ; yMin = −
3 3 ÷ x21 + x22 + x23 4 zMin = 1; uMin = − ! " # R $ 3 R = −3; R = 3 z = −1 x = y = 1 x1 = 3; y1 = 0 √ x2 = 3; y2 = 3 √ x3 = 3; y3 = − 3 x4 = 2; y4 = 1 x5 = 4; y5 = −1
zxx 2>0 2>0 2>0 2>0 2>0
Min
zMax = 13 x = 2, y = −1 zMax = 1; zMin = −1 3√ zMax = 3; zMin = 0 2
%
&""
' ( ! )
! ") *+*
, xMin = yMin = 1; zMin = 2 xMax = yMax = −2 zMax = −4 xMin = yMin = 2; zMin = 4 xMax = yMax = ±1; zMax = 1 xMin = −yMin = ±1 zMin = −1 xMax = 1; yMax = −2
zMax = 2; uMax = 9 xMin = −1; yMin = 2 zMin = −2; uMin = −9 11 5
xMax = ; yMax = − 4 2 11 605 zMax = − ; uMax = 4 32 √ (± 5; 1)
r = 1 ;
A √ r = √ ; h = 2A π 3 π 3 √ a = b = c = r 3 xi = nc (i = 1, 2, . . . , n) ! a = b = c = A6
h = 2
AMax = 9 sin α ÷ sin β = v1 ÷ v2 AMax = 2ab
94
=1
" 6 + 4 ln 0, 5 " 20 56
" 15 − 8 ln 4 " 12 − 1e " 92 " a2
"
"
#
1 6
a
y=0
# √2a2 −y2 x y √
#
= #
x= ay x2 a # a
+ x=a
y x
y=0 x=0 √ √ a 2# 2a2 −x2
=
π 1 − 4 2 # 2−x2
√ 8 2 5 21 p # 1 y x " 0 x=0 y=x # 1 # y 88 " = x y 105 y=0 x=0 " 6 # 2 # √2−y + x y " 1 y=1 x=0 2
y=0 2
" 43 16 ln 2 − 9 38 ln 2 ! 16
y x Jx = ab3 ; 3
a (3π − 2) 12
a3 b ; 3 ab3 + a3 b Jp = 3 π π " 2 ; 8 " (3; 4, 8) Jy =
4
a 80
" 28 15 1 " 24
32 1 5 + − 16 2
a 12
1 110
"
ln 2
4
42 23 # a# 0
0
a−x# a−x−y 0
z z y x =
a4 24
"
a a a ; ; 4 4 4 a 0; 0; 3 5
" a4
√ 32 2a5 135
πa5
√ 2
2 π 3 3 πa4 2 # π/4 #
3 2 πa 8
8π ln 2 3 πa3 16 5πa3 16
1736 15 2/ cos ϕ f (r)r r ϕ π ϕ=0 r=0 16 √ aπ 8π 3 2 1
a3 4 3
3 πh2 R2 # 1 # u
4 f (x, y) π u=0 v=−u
1 10 × v u 2 8 # 2 # 2−u
a2 9 + f (x, y) u=1
6kπa2 k ! " # πa3 3 πa3
60
4
$ ! % VB = πa3 3 1 $ !% VC = πa3 3 VB − VC & =3 VC #
Q =
(x, y, z) b = # 2π #Bh # z zr r z ϕ
v=u−2
0 1 × v u 1 2
π · abc2 1 4 x = (u + v) 2 πabc 1
y = (u − v) 2 2 3 4πa
3a2 35 ln 2 2 18π 868 2 a
2πa3 15
0
0
0
=
y − xy = 0
ty˙ − 2y = 0 y − 2xy = 0
x + yy = 0 y˙ = y 2
2
3y − t = 2ty y˙
xyy xy 2 + 1 = 1
' y = xy ln
x y
x2 + y = xy (
2xy + y = 0
( y¨ − y˙ − 2y = 0
(
# y¨ + 4y = 0 ! y − 2¨ y + y˙ = 0 2
2
y − x = 2xyy
( (
()
π h4 4
O
( ) ' ! y = t O*
t" # O
# 3 + y = t $%"
&'
y = C e1/x x + y = ln C(x
+1)(y + 1)
r = C e1/φ +a 2 s2 = t − 1t + Ct 1 + y2
=0
y2 − 1
= 2 ln et +1 −2 ln(e +1)
r = C cos φ,
+ 1 + t2 = C
t2 + y2 = ln Ct2 y = a + 1 Ct + at y = C sin x y = C e−1/x 2 2y = (1Cx −1 + x)2 2
r = −2 cos φ √ y = x ln x−x+C, √ y = x ln x − x + 1 √ C 1 + x2 √ y= , x + 1 + x2 √ 1 + x2 √ y= x + 1 + x2
ln 16 kt = k · 10 ln 2 t = 40 '
, q = q(t) = CU
1 − e− RC t
I = 0, 316 * t ≈ 7 ' ( FH = H(−1 , 0)Ì
FT = FT (cos α, sin α)Ì FL = (0 , −qx)Ì
2
FH + FT + FL = o =⇒ H = FT cos α, qx = FT sin α 5 ' # FT : qx = tan α = y H
,- . & / t 0' " )$ T * 2 y = C x qx T = −k(T − 20 ◦1) , y(0) = a y = + x2 + a2 H t 2 k " qx2 +a 6$ y= C −x 3 " 2H y = 1 + Cx + * √ 7 vS = (vS , 0)Ì ln(T − 20 ◦ 1) = −kt y = C√e x vF = +C * $ y = e x−2 %Ì x2 t = 0 T = 2 0 , v0 1 − 2 100 ◦1 y = C sin 2x − 1 a C = ln 80 kt = 1 2 y = 2 sin2 x − 80 ◦ 1 v = vF + vS 2 ln * T − 20 ◦ 1 1 1 v ' 2 T1 =
x + y = C * y = −x 8# y = y(x) ◦ ◦ 25 1 T2 = 60 1 9 2 '' y = t
! " v2 =⇒ y = tan α = 2 t3 − y3
' 2 4" v1 k 2 +3 t2 − y 2 + 5
y =
v0 vS
x2 1− 2 , a y(−a) = 0
y=
p 1 + (p − 1) e−kt
*4
y PN = cos α 2 = y 1 + tan α
! " # v0 x3 2 Y − y = y (X − x) = y 1 + y2 y= x− 2 + a $ Y = 0% &' vS 3a 3 ! ,- 4v
, y(a) = 0 a
$ () A 3 vS x2 + y 2 = c2 4
# ./( x2 − y 2 = c2 y
0 x ' X = x − A p = −gp , p(0) = 0 y 2x p0 5 y = *
+ 1 −x y XA = 2x x = − p = p0 e−g x/p y k
T
! ' 6 e = − 4πe2 % y = Cx−λ C > 0
' k + c T = y(x) −→ ∞
,' 4πe 2 x −→ 0 - xy = −a % k c & ./( + k y(x) −→ Cμ−λ + c
20 ◦ 7 = 0 x2 + 2y 2 = c2 1' x −→ 0 4π2r ( k y(t + Δt) ≈ y(t) + c 100 ◦7 = 2 y − x2 = c ./(' 4πr p − y(t) +ky(t) · Δt · 160 ◦7 r p − 60 ◦ 7 T = 4 e y = cx e = 1, 6 r T = p−y
yx2 = c y, y = k 40 ◦ 7 p y(0) = 1 2 3 -)4 1 8 y = (cosh ax − 1) + b OP = x2 + y 2 % a
0
0
y = t eCt
y = t ln t2 + Ct
− 2t e y−t x
y − x = C e y−x x2 − y 2 = Cx
s2 = 2t2 ln
y + ln x = C x x y= C − ln x
sin
C t
2x % 1 − Cx2 2x y= 1 − 3x2 y = x eCx , y = x e−x/2
0 y =
Ct = x = y eCy+1
2 9 #' Y − y = y (X − x) X = 0 % &
V0 = −ON = y − xy % ON = xy − y = OM = x2 + y 2 x2 − c2 ! y = 2c
! $( 34 4( 44
5 x, y ',44 / y ' ' % x = 0 ) %
O
x=a
vF = (0 , vF )Ì vH = vH (−x, −y)Ì x2 + y 2 =⇒ v = vF + v H vH = −x, x2 + y 2 Ì vF 2 x + y2 − y vH v
y = y(x) v2 =⇒ y = v1
y
y =
x
vF − vH
1+
y 2
y = x sinh
, x y(a) = 0
vF a ln vH x
& arctan(x + y)
& x + 2y
=x+C
+3 ln |2x + 3y − 7| =C
& 5t + 10y + C
= x(vH −vF )/vH 2vF /vH 2vF /vH
−x a × 2avF /vH vF = vH : y = a x2 a − , 2 2a 2 vF < vH :
vF > vH : y = y(x) x = 0 !"
" # $ " % & 8t + 2y + 1 = 2 tan(4t + C)
& & & &
= 3 ln |10t − 5y + 6| 1 y =x− x+C t = −y y +2 tan C + 2 (y−2)2 −2(y−2)(t− 1)− (t− 1)2 + C = 0 y−1 arctan t+2 1 − ln (t + 2)2 2 +(y − 1)2 + C = 0
& y = C e−t − e−t
× ln |t − 1|
& y = − 12 e2t +C e4t
& y = (arctan et +C) & y = −1
& & &
× cosh t
π + lntan − 4 t + + C / sin t 2 y = (arcsin x + 1) × 1 − x2 ϕ = C ebt a − (b sin t 1 + b2 + cos t) t+C 1 y = sin t + 2 2 cos t
& y = t2 + Ct et ×(sin ωt − τ ω cos ωt) 3 y = ln x + Cx & z = ss2 ++ C1 '& y = t ln |t| + 1 + Ct * p˙ = r(t) " t t p p = pr(t) t & y = 1 −1 4+−t 3 +, 1 t & y = e t − 1 p = pr(t) t − a t - ., u0 p 4
3
ua (t) = 1 + (τ ω)2
×
t τ ω e− τ
+ sin ωt
( $ ) "$ −τ ω cos ωt
ua (t) =
u0 1 + (τ ω)2
= −a 1+t p = C − a ln(1 + t) ×(1 + t) $ C = p(0) p(t) = 0 =⇒ C − a ln(1 + t) = 0 =⇒ t = ep(0)/a −1 p˙ −
y =
1 t ln Ct
1 1 + C et2 2t y = 1 − Ct2 2t y= 1 − 3t2
y 2 =
2
y 2 =
et 2t + C
y 3 = t + C e−t y 3 = t − 2 e1−t 1 y = √ 3 1 − t2 − 1
4t2 + y 2 = Ct t3 ey −y = C
t e−y +y = C x3 ey −y = C y + x e−y = C x2 cos2 y + y 2 = C
m = e−2t ; y 2 = 2(C − t) e2t m = cos y; 2t2 sin y + cos 2y
=C m = 1/ sin y; t + t3 = C sin y m = 1/t; t sin y + y ln t = C m = (t2 y 2 +2ty)−1 ; t(2 + ty)5 =C y m = et+y ;
et+y y 2 + cos t
=C 1 y ; x+ = C x2 x 1 m = ; y xy − ln y = 0 1 m = 4 ; x y 2 = Cx3 + x2
m =
m = e−y ; e−y cos x = C + x
√
√
y = C1 +C2 x+C3 ex 2 +C4 e−x 2 y = (C1 + C2 x) cos 2x +(C3 + c4 x) sin 2x x
y = C1 e +C2 e3x y = (C1 + C2 x) e2x y = e2x (A cos 3x + B sin 3x) y = C1 e2x +C2 e−2x = A cosh 2x + B sinh 2x y = A cos 2x + B sin 2x = a sin(2x + φ) −4x y = C1 + C2 e x = C1 et +C2 e−4t φ φ = A cos + B sin 2 2 s = e−t (A cos t + B sin t); s = e−t (cos t + 2 sin t)
y = C1 ex +(C2 + C3 x) e2x y = C1 cosh 2x + C2 sin 2x +C3 cos 2x + C4 sin 2x 2x y = C1e + e−x √ √ × C2 cos x 3 + C3 sin x 3 y = (C1 + C2 x + C3 x2 ) e−ax y = A sin x sinh x +B sin x cosh x + C cos x sinh x +D cos x cosh x y = A cosh x + B sinh x x x +C cos + D sin 2 2 g ϕ¨ + ϕ = 0 l
g g + C2 sin t ϕ = C1 cos t l l
l g
F = −ky ey ! m¨ y + kx = 0; y(0) = −b, y(0) ˙ =0 mg k = b "# y = −b cos t gb T = 2π gb
$ % ! m¨ y + αy˙ + ky = 0 αt y = −b e− 2m cos ωt,
! ω =
T = 2π ω & ' z = C1 + (C2 + C3 t) e
'
'
( ' '
'
t
α2 g − b 4m2
+C4 e
−t
y = t(A0 + A1 t + A2 t2 ) + t2 (B0 +B1 t) et +(D0 + D1 t +D2 t2 + D3 t3 ) e−2t cos t +(E0 +E1 t+E2 t2 +E3 t3 ) e−2t sin t z = C1 e2t +C2 e−t +C3 cos t +C4 sin t
T = 2π
y = A + (B0 + B1 t + B2 t2 ) e0,5t +tD0 cos t + tE0 sin t z = C1 + C2 et +C3 e2t y = tA + B e−2t +(D0 + D1 t) e−t sin 3t +(E0 + E1 t) e−t cos 3t 1 5t e y = C1 cos 3t + C2 sin 3t + 34 1 3 1 2 1 t − t − t y = et 6 4 4 +C1 e−t +C2 et t y = C1 cos 2t + C2 + sin 2t 4
' y = 31 et + C1 − 21 t e2t +C2 e4t ' y = (C1 + C2 x) ex + e2x ' y = C1 e2x +C2 e−2x −2x3 − 3x ' y = C1 e−x +C2 e√−2x π − 2x 4 y = C1 cos x + C2 sin x + x + ex 3 y = C1 + C2 e−3x + x2 − x 2 y = e−2x (C1 cos x + C2 sin x) +x2 − 8x + 7 y = C1 e2x +(C2 − x) ex x = A sin k(t − t0 ) − t cos kt +0, 25 2 cos
' ' )'
' ' √ √ ' y = C1 ex 2 +C2 e−x 2
−(x − 2) e−x
3
' y = C1 + C2 e2x − x6
' y = 21 e−x +x e−2x +C1 e−2x ' *' ' '
' ' +' , '
+C2 e−3x x = e−kt (C1 cos kt + C2 sin kt) + sin kt − 2 cos kt y = C1 + C2 x + (C3 + x) e−x +x3 − 3x2 x e−3x y = C1 e3x + C2 − 4 +C3 cos 3x + C4 sin 3x x = C1 +C2 cos t+C3 sin t+t3 −6t x −2x e y = C1 + 12√ √ + C2 cos x 3 + C3 sin x 3 ex t2 x = C1 + C2 t + e−2t 2 t 1 t x = A cos + B sin + a a a t −t y = e$ + e− 2 √ √ % √ 3 3 3 t+ sin t × cos 2 3 2 +t − 2
t2 t 1 − − 2 3 18 √ √ e−t 61 + cos t 5 + √ sin t 5 18 5 1 1
y = cos 2t + sin 2t + sin t 3 3 1 t t2 t3 t + y = e2t + − − 8 2 4 4 6 x cos 2x y = C1 − 2 1 + C2 + ln sin 2x sin 2x 4 y = [(C1 + ln cos x) cos x + (C2 + x) sin x] e2x
y = C1 cos x + C2 sin x x π + − cos x ln tan 2 4 −x −x y = C1 + C2 e −(1 + e ) × ln(1 + ex ) + x 1
y = e−2x C1 + C2 x + 2x y = x2 ln x 3x2 − + C1 + C2 x e−2x 2 4 1 y = C1 sin x + C2 cos x + 2 cos x y = (C1− ln | sin x|) cos 2x 1 + C2 − x − cot x sin 2x 2 2 y = C1 + 4 − x x +x arcsin + C2 x ex 2 y
!
" F = yπr2 g ey # $ % & m¨ y + yπr2 g = 0, m = πr2 l & g g t + C2 sin t y = C1 cos l l
y =
' ( & T = 2π
l g
)* + # , & F y = (l − x) α - % & y(0) = y (0) = 0 F x3 2 & y = lx − 2α 3 x3 6 ) x = a (e−t +t − 1)
)) y =
). m¨ y = −mg − αy˙ (α > 0, '(/((( ) - % & y(0) = h, y(0) ˙ =0
g g & y = h − t + 2 1 − e−βt β β β = α/m ) - % & m¨ x + kx = 0; x(0) = x0 , x(0) ˙ =0 k & x = x0 cos t m )0 $ % & m¨ x + kx = 2ωm cos ωt k 1 (# ω = m & x = C1 cos ωt +C2 sin ωt ˆ sin (ωt + ϕ) + t sin ωt = x x ˆ = C12 + (C2 + t)2 lim x ˆ=∞ 2-/
t−→∞
$ % ∞
kl4 x − 2kl2 x3 + kx5 120a )4 & w(x) = C1 + C2 x +C3 cos λx + C4 sin λx w(0) = w (0) = 0 =⇒ C1 = C3 = 0 w(l) = lC2 + C4 sin λl = 0 w (l) = C2 + C4 λ cos λl = 0 "( 5# & λl cos λl − sin λl = 0 =⇒ tan λl = λl /(6 # & λl = 4, 4934 =⇒ 20, 1907α " & Fk = l2
)3 w(x) =
y = t(C1 cos ln t + C2 sin ln t) + t ln t y = − 23 t ln2 |t| + 14 t3 +
(C1 + C2 ln |t|) · |t| + C3 t2 C1 1 2 1 2
y = t + C2 t + 3 t − t ln |t| y = 2t ln |t| + C1 t−1 + C2 t y = − 21 t−1 ln |t| + C1 t−1 + C2 t y = 21 ln |t| sin ln |t| +C1 cos ln |t| + C2 sin ln |t| y = C1 x + C2 x−1 + C3 x3 y = Cx1 + C2 x2
y = C1 xn + C2 x−(n+1) y = x−2 (C1 + C2 ln x) y = C1 cos ln x + C2 sin ln x 2
y = 5x3
+ C1 x−1 + C2
y = C1 x3 + Cx22 − 2 ln x + 31 y = C1 x + C2 x2 − 4x ln x y = C1 + C2 lnx x + ln y =
3
x
x3 + C1 x + C2 x2 6
y = x2 + C1 cos ln x + C2 sin ln x
z1 = (C1 − C2 t) e−t −t
z2 = C2 e t 1 − + C1 et x1 = 2 4 1 t + − − − C2 e−t 2 4 t 1 x2 = + + C1 et 2 4 t 1 − + C2 e−t + 2 4
z1 = e−4t (A cos t + B sin t)
z2 = e−4t [(A − B) sin t −(A + B) cos t] t −2t z1 = C1 e +C2 e z2 = C1 et +C3 e−2t z3 = C1 et −C2 e−2t −C3 e−2t z1 = (C1 t + C2 ) e−4t z2 = (−C1 t − C2 − C1 ) e−4t z1 = C2 + 3C3 e2t z2 = C1 e−t −2C3 e2t z3 = −2C1 e−t +C2 + C3 e2t x1 = (C1 t2 + C2 t + C3 ) e−t
+t2 − 3t + 3 x2 = (−2C1 t − C2 ) e−t +t x3 = 2C1 e−t +t − 1 z1 = 4C1 + C2 e−3t +C3 t e−3t z2 = 4C1 − 2C2 e−3t +C3 (−2t + 1) e−3t −3t z3 = C1 + C2 e +C3 (t − 1) e−3t 1 y1 = C1 e5t +C2 e−t − et 2 1 y2 = C1 e5t − e−t 2 x = et +C1 + C2 e−2t y = et +C1 − C2 e−2t x = 2 e−t +C1 et +C2 e−2t y = 3 e−t +3C1 et +2C2 e−2t x = C1 et +C2 e−t +t cosh t x = et +C1 e3t +C2 e−3t +C3 cos(t + φ)
x1 = −3 − 3 e2t +8 e3t +2 e4t
x2 = −1 − 3 e2t −2 e4t x = e−2t (1 − 2t)
3 1 y ≈ 1 + (x − 1) + (x − 1)2 2 2 2 5 + (x − 1)3 + (x − 1)4 3 12 x4 x5 x6 x2 + + + y ≈ −x − 2 12 15 60
x2 x3 x4 x5 − + + 2 6 6 20 3 4 5x 4x + y ≈ 1 + x + x2 + 3 6
y ≈ 1 −
h = 0, 2 : tn
yn
0 0 0, 2 0, 0214 0, 4 0, 091818 h = 0, 1 : t4 = 0, 4 , y4 = 0, 0918242
h = 0, 1 : t2 = 1, 2 , y2 = 0, 941176 h = 0, 05 : t4 = 1, 2 , y4 = 0, 941176
h = 0, 2 : tn
y1n
y2n
0 0 1 0, 2 −0, 163733 0, 818733 0, 4 −0, 268108 0, 670324
h = 0, 1 : tn
y1n
y2n
0 0 1 0, 1 −0, 0904833 0, 904838 0, 2 −0, 163745 0, 818731 0, 3 −0, 222245 0, 740818 0, 4 −0, 268127 0, 67032 y1 := y, y2 := y˙ h = 0, 1 : t2 = 0, 2 , y12 = −0, 58 , y22 = −2, 8 h = 0, 05 : t4 = 0, 2 , y14 = −0, 58 , y24 = −2, 8 h = 0, 2 : t5 = 1 , y5 = 2, 99997 h = 0, 1 : t10 = 1 , y10 = 3
v(x0 , y0 ) x2 + y 2 = x20 +y02
x2 + y 2 = r2 v r v(x0 , y0 ) x2 + y 2 = x20 +y02
x2 + y 2 = r2 v r ! " # v(x0 , y0 ) v(x0 , y0 ) $% x2 − y 2 = a2 a2 = x20 − y02 " y 2 − x2 = a2 a2 = y02 − x20 & y = x " y = −x #
' x ( O ) O#
$% y · x = c * + c = 0 x " y , *-
9y 10; z = 10 − 4x Ì 9y 9 4, ,− z z √ Ì (4, 3, −2) ; 29
5;√(−1 , −4 , 1)Ì ;
3 2 0; (0 , 2)Ì ; 2
$% &%' % % f %%
− √36
11
!" "
# div v = 3f (|x|)
+|x| f (|x|) = 0 C =⇒ f (|x|) = |x|3 (C ∈ R) rot v ≡ 0 =⇒ v
ϕ = −xy − xz − yz
Δf = 0 ϕ = −x2 yz − xy 2 z −xyz 2 Δ f = div v = 2yz + 2xz + 2xy ϕ = −xyz Δf = 0
Ep = ϕ(x) = mgz
− 21
2 − √1 3 (x y + y x) = 8 −16 ( − 52 ∂P ∂Q 3 = ∂x ∂y
−12 2 ) 1, 5a2 # 3a2 a2 a2 2 8a2 11a2
6 πa2 ∂(x + y) πmab = = ∂y 4 ∂y ∂x # 1* " + % ( Py = 5π Qx %$ 6 #
13
√ 8 2 2 a 3 2πa2
2
√ 2−1
2 2πa 3 √ 2πp2 2 8a2
−πa2 1 ) 35 √ 2−1 2 c √ √ 2−1 2
√c
+ % (
rot |x|x 3
=o
( (
( ( 3a2 4a2 (π − 2) πa2 14 3
#
A= #
σ = F
β
ϕ=0
#
90◦ 2
R sin ϑ ϑ ϕ
ϑ=90◦ −α 2 ◦
= R β cos(90 − α) (R . . . )
α = 30◦ , β = 60◦ =⇒ πR2 A= 6 3 5 a 20 96π
4πc 16 πc 5 a3 2 π a4 4 + 3 5 16
! 3a4 9 − + 2 ln 2 2 "# $ % " ' (& # ) ' (& * v = u1 grad u2 − u2 grad u1 6 Z = v(x) · x k# (Qx − Py ) b = (14.16) B # 2 π = 1 b = + 3 8 B # Q=
b B# b div E =ε B # · σ = a2 E = (14.15) F # 2π # π × sin ϑ ϑ ϕ ϕ=0
6
k
ϑ=0
= 4a2 π ε
v · x = π #
= rot v(x) · σ F
k: x = cos ϕ, y = sin ϕ,
z = 0; ϕ ∈ [0 , 2π] F : x = cos ϕ sin ϑ, y = sin ϕ sin ϑ, z = cos ϑ; (ϕ , ϑ) ∈ B B: 0 ≤ ϕ ≤ 2π π 0≤ϑ≤ 2 # div v b = 84 π B # v(x) · σ = F
F = F1 ∪ F2 ∪ F3 F1 : x = 2 cos ϕ, y = 2 sin ϕ, z = z; (ϕ , z) ∈ B1 B1 : 0 ≤ ϕ ≤ 2π 0≤z≤3 F2 : x = r cos ϕ, y = r sin ϕ, z = 0; (r , ϕ) ∈ B2 B2 : 0≤r≤2 0 ≤ ϕ ≤ 2π F # 3 : # F2 + z = 3 ··· = 4π #F1 ··· = 0 F2
# · · · = 36 π F3
!, -' . / !, !! div v = 0 x = o 1(14.15) =⇒ v(x) · σ = 0+ F # F -0 1 ' )0 / U = 4 π !! 2 & '$ rot v = o x ) x2 + y 2 = 0.