Быстрое введение в систему Mathematica. Ч.1: Методические указания к спецкурсу ''Системы компьютерной математики''

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Jbk151: - Out [28 ] = Graphics3Db kh\f_klguc ihdZa g_kdhevdbo ]jZnbdh\/ ihkljh_gguo ih hl^_evghklb In [29 ]: =Show[%,%%]

Jbk151; Out [29 ] = Graphics3D

Kibkdb/ \_dlhju/ fZljbpu -------------- }, {x -> --------------}} 2 2 beb kjZam In[ ]:= Solve [x ∧ 3 + x - 2 == 0] Out[ ]= -1 - I Sqrt[7] -1 + I Sqrt[7] {{x -> 1}, {x -> -------------- }, {x ->-------------- }} 2 2 H[jZlbl_ \gbfZgb_ gZ fgbfmx _^bgbpm , \ aZibkb ^\mo dhjg_c1 Kbkl_fZ 0DWKHPDWLFD gZoh^bl k ihfhsvx dhfZg^u 6ROYH \k_ +\_s_kl\_ggu_ b dhfie_dkgu_, j_r_gby Ze]_[jZbq_kdbo mjZ\g_gbc kl_i_gb g_ \ur_ 7/ ijblhf b ^ey mjZ\g_gbc k iZjZf_ljhf = In[ ]:= Solve [x ∧ 4 + a∗ ∗ x - 2 == 0, x] Out[ ]= … hl\_l kebrdhf ]jhfha^hd/ \u [_a ljm^Z gZc^_l_ _]h kZfb1 J_r_gby Ze]_[jZbq_kdh]h mjZ\g_gby kl_i_gb 8 beb \ur_ fh]ml [ulv gZc^_gu/ \hh[s_ ]h\hjy/ qbke_ggh1 KdZ`_f/ mjZ\g_gb_ [∧8.[05 3 kbkl_fZ 0DWKHPDWLFD j_rbl lhqgh= In[ ]:= Solve [x ∧ 5 + x - 2 == 0] Out[ ] = … 3 0 hldZ`_lky= Z mjZ\g_gb_ [ ∧ 8 . [ 0 : In[ ]:= Solve [x ∧ 5 + x - 7 == 0] Out[ ]= {ToRules[Roots[x^5 + x == 7, x]]} Ijb^_lky j_rZlv mjZ\g_gb_ qbke_ggh= In[ ]:= NSolve [x^5 + x - 7

== 0]

Out[ ]= {{x -> -1.21388 - 0.924188 I}, {x -> -1.21388 + 0.924188 I}, {x->0.508469 -1.36862 I}, {x->0.508469 +1.36862 I}, {x ->1.41081}} LjZgkp_g^_glgu_ mjZ\g_gby j_rZxl/ aZ^Z\Zy gZqZevgh_ ijb[eb`_gb_ d dhjgx1 In[ ]:= FindRoot [Cos[x] == 2∗ x; {x,0}] Out[ ]= {x -> 0.450184} Fh`gh j_rZlv kbkl_fu mjZ\g_gbc= In[ ]:= Solve [{a*x +b*y == g, c*x + d*y == h} ,{x,y}] Out[ ]=

{{x ->

g a

+

b (c g - a h) c g - a h --------------, y -> -( ------------ ) } a (-(b c) + a d) -(b c) + a d

b In[ ]:= FindRoot [{ x + y == Sin[x], x - y == Cos[x], {x,0}, {y,0}] Out[ ]= {x -> 0.704812, y -> -0.0569214} Kbkl_fZ 0DWKHPDWLFD iha\hey_l j_rZlv ^bnn_j_gpbZevgu_ mjZ\g_gby b bo kbkl_fu= In[ ]:= DSolve [y''[x] - 2y'[x] + 1 == 0, y[x], x] x 2 x Out[ ] = {{y[x] -> - + C[1] + E C[2]}} 2 J_r_gb_ kh^_j`bl ijha\hevgu_ ihklhyggu_ K>4@ b &>5@1 Fh`gh gZclb qbke_ggh_ j_r_gb_ aZ^Zqb Dhrb ^ey ^bnn_j_gpbZevguo mjZ\g_gbc beb bo kbkl_f= In[ ]:= NDSolve[{x'[t]==-y[t]-x[t],y'[t]==2x[t]y[t],x[0]==1,y[0]==-1},{x[t],y[t]},{t,0,10}] Out[ ]={{x[t] -> InterpolatingFunction[{0., 10.},][t], y[t] -> InterpolatingFunction[{0., 10.}, ][t]}} In[ ]:= Plot[Evaluate[{x[t],y[t]}/.%],{t,0,10}, PlotRange->All]

4 318

5

7

9

;

43

0318 04

Out[ ]=-GraphicsFh`gh bkdZlv wdklj_fmfu nmgdpbc h^ghc b g_kdhevdbo i_j_f_gguo1

In[ ]:=FindMinimum[Exp[x]*Cos[x] ,{x,0}] Out[ ]= {-0.0670197, {x -> -2.35619}} In[ ]:=FindMinimum[Sin[x]*Cos[y] ,{x,0},{y,0}] Out[ ]= {-1., {x -> -1.5708, y -> 0.}} AZ^Zqb ebg_cgh]h ijh]jZffbjh\Zgby g_[hevrhc jZaf_jghklb fh`gh j_rZlv/ bkihevamy kbkl_fm 0DWKHPDWLFD In[ ]:= ConstrainedMax [19x - 47y + 28z, {x + y+ z > 0, x + y + z < 1, x > 0, y > 0, z > 0}, {x, y, z}] Out[ ]= {28, {x -> 0, y -> 0, z -> 1}}

:gZeba Ohly kbkl_fm ³0DWKHPDWLFD´ b _c ih^h[gu_ ^h\hevgh qZklh gZau\Zxl kbkl_fZfb dhfivxl_jghc Ze]_[ju/ \ gbo lZd beb bgZq_ ij_^klZ\e_gu \k_ nmg^Zf_glZevgu_ jZa^_eu fZl_fZlbdb1 ^Ze__ ± ^_jaZcl_$ >bnn_j_gpbjh\Zlv \ kbkl_f_ ³0DWKHPDWLFD´ g_ ijhklh/ Z hq_gv ijhklh$ < dZq_kl\_ Zj]mf_glh\ dhfZg^u ^bnn_j_gpbjh\Zgby '>1/1@ gm`gh mdZaZlv lm nmgdpbx/ dhlhjmx fu gZf_j_gu ijh^bnn_j_gpbjh\Zlv/ b lm i_j_f_ggmx +beb i_j_f_ggu_,/ ih dhlhjhc +dhlhjuf, ke_^m_l ^bnn_j_gpbjh\Zlv1 \@ b 'W>]@ 0 ^bnn_j_gpbZeu i_j_f_gguo [/ \ b ]1

DhfZg^Z 'W ijbf_gy_lky b ^ey \uqbke_gby iheguo ijhba\h^guo nmgdpbc fgh]bo i_j_f_gguo= In[ ]:=Dt[f[Sin[x y z]],x] Out[ ]= Cos[x y z] (y z + x z Dt[y, x] + x y Dt[z, x]) f'[Sin[x y z]] A^_kv 'W>\/[@ b 'W>]/[@ 0 ihegu_ ijhba\h^gu_ i_j_f_gguo \ b ] ih i_j_f_gghc o1 Gh fh`gh b kbkl_fm ³0DWKHPDWLFD´ ©ihkZ^blv \ dZehrmª1 HgZ ©agZ_lª/ qlh _[_ 0 g_^bnn_j_gpbjm_fZy nmgdpby/ b hldZau\Z_lky \uqbkeylv __ ijhba\h^gmx ^Z`_ \ l_o lhqdZo/ ]^_ nmgdpby ^bnn_j_gpbjm_fZ1 LZd kihdhcg__$ In[ ]:=D[Abs[x],x]/.x->1 Out[ ]= Abs'[1] Bgl_]jbjh\Zgb_ \ kbkl_f_ ³0DWKHPDWLFD´ +dZd b \ `bagb, keh`g__ ^bnn_j_gpbjh\Zgby1 NhjfZevgh \k_ ijhklh= g_hij_^_e_gguc bgl_]jZe \uqbkeyxl ihkj_^kl\hf dhfZg^u ,QWHJUDWH= ^ey ∫ [ Q G[ ihemqZ_f

In[ ]:=Integrate[x^n,x]

Out[ ]=

1 + n x -----1 + n

Gh In[ ]:=Integrate[Sin[Sin[x]],x] ^Zkl Out[ ]= Integrate[Sin[Sin[x]],x] LZdbf h[jZahf/ kbkl_fZ ³0DWKHPDWLFD´ hldZaZeZkv [jZlv g_[_jmsbcky bgl_]jZe1 Gh b g_ \k_ [_jmsb_ky bgl_]jZeu kbkl_fZ ³0DWKHPDWLFD´ ]hlh\Z [jZlv= \ hl\_l gZ aZ^Zgb_ In[ ]:= Integrate[1/(1+x^4+x^9+x^11),x] kbkl_fZ ³0DWKHPDWLFD´ ihdZ`_l \k_/ qlh hgZ fh`_l \ ^Zgghf kemqZ_ +kh\k_f dZd q_eh\_d,= Out[ ]= Integrate[(15/16 - (7*x)/8 + (13*x^2)/16 (3*x^3)/4 + (5*x^4)/8 - x^5/2 + (3*x^6)/8 - x^7/4 + x^8/8 - x^9/16)/ (1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + 2*x^8 - x^9 + x^10) , x] + Log[1 + x]/16 Dhg_qgh/ \ kemqZ_ hij_^_e_gguo bgl_]jZeh\ m^Z_lky k^_eZlv ih[hevr_ 0 dhfZg^Z ,QWHJUDWH iha\hey_l \uqbkeylv b hij_^_e_ggu_ bgl_]jZeu/ Z ^ey l_o/

dhlhju_ g_ [_jmlky k ihfhsvx dhfZg^u ,QWHJUDWH/ bf__lky dhfZg^Z 1,QWHJUDWH/ iha\heyxsZy \uqbkeylv hij_^_e_ggu_ bgl_]jZeu ijb[eb`_ggh1 E

BlZd1 \uqbkebf

∫ OQ[G[ D

In[ ]:=Integrate[Log[x],{x,a,b}] Integrate::gener: Unable to check convergence. Out[ ]= -(a (-1 + Log[a])) + b (-1 + Log[b]) Gh g_[_jmsb_ky bgl_]jZeu kbkl_fZ \havf_l lhevdh qbke_ggh= In[ ]:=Integrate[Sin[Sin[x]],{x,0,Pi}] General::intinit: Loading integration packages -- please wait. Out[ ]= Integrate[Sin[Sin[x]], {x,0,Pi }] In[ ]:=NIntegrate[Sin[Sin[x]], {x,0,Pi }] Out[ ]= 1.78649 Fh`gh \uqbkeylv ih\lhjgu_ bgl_]jZeu/ gZijbf_j/ 4

4− [ 5

∫ G[ ∫ +[ 3

5

+ \ 5 ,G\

− 4− [ 5

In[ ]:=Integrate[x^2+y^2,{x,0,1},{y,-Sqrt[1-x^2],Sqrt[1x^2]}] Pi Out[ ]= — 4 b 4

4− [ 5

∫ G[ ∫ 4 G\

−4

− 4− [ 5

In[ ]:=Integrate[1,{x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]}] Out[ ]= Pi Ijb bgl_]jbjh\Zgbb kbkl_fZ ³0DWKHPDWLFD´ fh`_l aZ[em`^Zlvky ± kh\k_f dZd q_eh\_d1 LZd i_j\hh[jZagZy nmgdpby ^ey [_a mdZaZgby ijhf_`mldh\/ gZ dhlhjuo wlh \_jgh1

4

[5

[m^_l gZc^_gZ dZd

: ^ey

4

∫ 7 + VLQ[G[

4 [

fu

ihemqbf ihkj_^kl\hf dhfZg^u ,QWHJUDWH>42+7.6LQ>[@,/[@ j_amevlZl/ g_ y\eyxsbcky i_j\hh[jZaghc ih^ugl_]jZevghc nmgdpbb1 LZdh\Zy i_j\hh[jZagZy h[yaZgZ [ulv g_ij_ju\ghc/ ihemq_ggZy `_ nmgdpby bf__l jZaju\u \ lhqdZo \b^Z 5+N + 4,Π/ \ q_f e_]dh m[_^blvky/ ihkljhb\ ]jZnbd ihkj_^kl\hf dhfZg^u 3ORW>(/^[/053L/53L`@ b ijhZgZebabjh\Z\ ih\_^_gb_ ihemq_gghc nmgdpbb \ mdZaZgguo lhqdZo1

>ey \uqbke_gby kmff \ kbkl_f_ ³0DWKHPDWLFD´ bf__lky dhfZg^Z 6XP= In[ ]:=Sum[x^i/i,{i,1,9}] 2

Out[ ]= x +

x - + 2

3 x - + 3

4 x - + 4

5 x - + 5

6 x - + 6

7 x - + 7

8 9 x x - + 8 9

In[ ]:=Sum[x^i y^j,{i,1,2},{j,1,3}] 2 2 2 2 3 2 3 Out[ ]=x y + x y + x y + x y + x y + x y Kbkl_fZ ³0DWKHPDWLFD´ mki_rgh jZaeZ]Z_l nmgdpbb \ jy^ L_cehjZ= In[ ]:= Series[Sin[x],{x,0,3}] 3 x 4 Out[ ]= x - - + O[x ] 6 Hl[jZku\Zlv hklZlhqguc qe_g iha\hey_l dhfZg^Z 1RUPDO= In[ ]:= Normal[%] 3 x Out[ ]= x - 6 In[ ]:= Normal[Series[Sin[x],{x,0,5}]]

3

5

x x Out[ ]= x - - + 6 120 In[ ]:= Normal[Series[Sin[x],{x,0,7}]] 3 5 x x Out[ ]=x - - + 6 120

-

7 x 5040

0] Out[ ]=1 GZ ihiuldm ih^klZ\blv 3 \ \ujZ`_gb_ kbkl_fZ ³aZjm]Z_lky´= In[ ]:= Sin[x]/x /.x->0 1 Power::infy: Infinite expression - encountered. 0 Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. Out[ ]=Indeterminate ohly ijb ^jm]bo agZq_gbyo [ agZq_gb_ \ujZ`_gby [m^_l \uqbke_gh= In[ ]:= Sin[x]/x /.x->1 Out[ ]=Sin[1] In[ ]:= Sin[x]/x /.x->2 Pi Out[ ]=0 Infinity] Out[ ]=E In[ ]:= Limit[(1+1/x)^x,x->-Infinity] Out[ ]=E : \hl ijbf_j ihkeh`g__=

H 42+[ − D, gZ ij_^eh`_gb_ \uqbkeblv OLP [→D kbkl_fZ ³0DWKHPDWLFD´ ^Zkl qZklbqgh \_jguc hl\_l= In[ ]:= Limit[E^(1/(x-a)),x->a] Out[ ]=Infinity HgZ mki_rgh \uqbkebl e_\uc b ijZ\uc ij_^_eu/ In[ ]:= Limit[E^(1/(x-a)),x->a,Direction->1]

Out[ ]=0 In[ ]:= Limit[E^(1/(x-a)),x->a,Direction->-1] Out[ ]=Infinity gh lZd b g_ kfh`_l ihgylv/ qlh ^\mklhjhgg_]h ij_^_eZ g_ kms_kl\m_l In[ ]:= Limit[E^(1/(x-a)),x->a] Out[ ]=Infinity Dhg_qgh/ \ dgb]_ K1