數(shù)學(xué)實(shí)驗(yàn)(第二版)課后習(xí)題答案

貴州師范學(xué)院

2012級(jí)數(shù)本一班李剛

數(shù)學(xué)實(shí)驗(yàn)課后練習(xí)答案

習(xí)題2.1

1.symsxy;

?x=-5:0.01:5;

?y=x.Al/2;

?Plot(x,y)

A

2?fplot('exp(-x.2)'z[-5z5])

AA

4.ezplot(*y.2-x.3/(l-x)'r[-5,5])

5?t=0:0.1:2*pi;

x=t-sin(t);

y=2*(1-cos(t));

plot(x,y)

6.t=0:0.1:2*pi;x=cos(t).A3;

?y=sin(t).A3;

?plot(t,y)

QF；gwr?10?'三?一回宮

7?t=0:0.1:2*pi;x=cos(t);y=2*sin(t);z=3*t;plot3(x,y,z)

8：x=0:0.1:2*pi;r=x;polar(xzr)

■

:x=0:0.1:2*pi;r=exp(x);polar(x,r)

10：x=0:0.1:2*pi;r=sqrt(cos(2*x));polar(x,r)

x=0:0.1:2*pi;r=sqrt(sin(2*x));polar(x,r)

12：x=0:0.1:2*pi;r=l+cos(x);polar(x,r)

OHgwr*1Q.國一53

練習(xí)2.2

i：(i)

?sy*sn;limit(*(n,3+3'n)'(l/n)#,n,inf)

ans=

(2)：symsn;limit('sqrt(n+2)-2*(sqrt(n+l))+sqrt(n)',n,inf)

Ans=

0

(3)：

CommandWindow________

ONewtoMATLAB?WatchthisVideo.seeDem。-orreadGetticaStarted.

SizeDateModified?5yMsx<iiaxtCx<cot<2?x*,x,0)

ans=

1/2

?

(4)：

CommandWindow

ONewtoMATLAB?WatchthisVideo.seeDemos,orreadGetti

Rd?syasx;liaitC^cos<m./x'?x',x,inf)

ans=

1

(5)：

CommandWindow

ONewtoMATLAB?WatchthisVideo.seeDemos.orreadGettingStarted.

?syihsx;liaitC1/x-l/(exp'x-l)*,x,1)

an$=

(exp-2)/(exp-1)

(6)：

CommandWindow

ONewtoMATLAB7WatchthisVideo,seeDemos.orreadGettingStarted.

?syasx;limitCsqrt(x*2+x)-x\x,inf)

axis=

1/2

2:

CommandWindow

NewtoMATLABWatchthisVideo.seeDemos.orreadGettii

?sywsxy;

y=845000*(l+x)-20/240

x=0.0504;

eval(y)

ans=

9.4133e+003

?

CcmmandWindow

QNewtoMATLABWatchthisVideo.seeDemos.orreadGetting

>>sy?sxy;

y=l00000*(1+x)*20/240；

x=0.0405;

eval(y)

ans=

921.7867

3:

fplot('x.A2*sin(x.A2-x-2)',[-2,2])

練習(xí)2.3

1：

D:\J$erData\$jxy\Document$\MATLAB▼一七一

CommandWindow,C

ONewtoMATLAB?NatchthisVideo一seeDemos-orreadGettingStarted.

ied?fplotCx*4-2*x*,[-2,2]);

[x,f,h]=fsolve(,x'4-2x*,-l)

Optimizationteramated:first-orderoptiMalityislessthanoptions.TolFun.

-0.8613

3.6580m2

x

(2)：

:oryD:\UserData\sjxy\Documents\MATLAB▼[…七

CommandWindow

ONewtoMATLABWatchthisVideo,seeDemos,orreadGettingStarted.

)dified?solveCx<log(sqrt(x,2-l)+x)-sqrt(x-2-1)-0.5<x*,*x*,[1,inf])

ans=

2.1155228843978670800804047839554

?

2：

CommandWindow

ONewtoMATLAB7WatchthisVideo.seeDemos,orreadGettingStarted.

?fun=?(t)[9*t(1)a2+36*t(2)*2+4*t(3)*2-36,;

t(l)-2-2*t(2)-2-20?t(3),;

16?t(Dr⑴?A2S(2)*2-16?t(3)-2];

t0=[0,0,0];

[t,f>h]=fsolve(fun,tO)

OptiMizationterainated:first-orderoptlaalityislessthanoptions.TolFun.

0.13420.9972-0.0985

1.0e-008,

0.7690

-0.0418

-0.1054

練習(xí)2.4

i：

(i)

CommandWindow

NewtoMATLAB?WatchthisVideo,seeDemos,orreadGettingStarted.

>>int(*1/(1+cos(x))*,*x*)

ans=

tan(l/2*x)

?diff(*tan(l/2*x)*,*x*)

ans=

l/2+l/2*tan(l/2*x)-2

?

(2)：

CommandWindow

ONewtoMATLAB?W^atchthisVideo,seeDemos.orreadGettingStarted.

?intC1/(l+exp(x))*,*x*)

ans=

loc(exp(x))-log(1+exp(x))

?diffClog(exp(x))-lot(l+exp(x))*,,x*)

ans=

1-1/(1+exp(x))*exp(x)

?

(3)

CcmmandWindow

QNewtoMATLAB?WatchthisVideo,seeDemos,orreadGettingStarted.

?intCx*sin(x)?//)

ans=

x*(-l/2*cos(x)*sm(x)+l/2*x)+l/4*sin(x)-2-l/4*x-2

?diffCx*(-l/2*cos(x)?sm(x)+l/2*x)-l/4*cos(x)"2-l/4*x*2,,*x*)

ans=

x*(l/2*sin(x)a2-l/2*cos(x)-2+l/2)

?

(4)：

CommandWindow

ONewtoMATLAB7WatchthisVideo.seeDemos,orreadGettingStarted.

?int(*sec(x)-3*,*x*)

ans=

1/2/cos(x)A2?sin(x)*l/2?loc(sec(x)+tan(x))

?diffC1/2/cos(x)*2*sin(xHl/2*log(sec(x)+tan(x))

ans=

l/cos(x)-3*sin(x)*2+l/2/cos(x)+l/2*(sec(x)*tan(x)+l+tan(x)*2)/(sec(x)+tan(x))

?

2：

(1)：

symsx;

int(x人(-x),x,0,1)

ans=

int(xA(-x),x=0..1)

vpa(ans,10)

ans=

1.291285997

(2)：

symsx；

int(exp(2*x)*cos(x)A3,x,0,2*pi)

ans=

-22/65+22/65*exp(4*pi)

(3)：

A

symsx;int(exp(x2/2)/sqrt(2*pi),x/0,l)

ans=

-1125899906842624/5644425081792261*i*erf(l/2*i*2A(l/2))*piA(l/2)*2A(l/2)

?vpa(ans,10)

ans=

.4767191345

(4)：

symsx;

AA

int(x*log(x4)*asin(l/x2)/x/l,3)

ans=

int(x*log(xA4)*asin(l/xA2),x=1..3)

?vpa(ans,10)

ans=

2.459772128

(5)：

symsx;

A

int(exp(x2/2)/sqrt(2*pi),x/-infzinf)

ans=

Inf

(6)：

symsx;

int(sin(x)/x,x,O,inf)

ans=

l/2*pi

(7)：

symsx;

int(tan(x)/sqrt(x)zx,O,l)

Warning:Explicitintegralcouldnotbefound.

>Iat58

ans=

int(tan(x)/xA(l/2),x=0..1)

?vpa(ans,10)

ans=

.7968288892

Lommana.vinaow

ONewtoMATLAB?WatchthisVideo,seeDemos,orreadGetting

?syasx;

int(tan(x)/sqrt(x),x,0,1)

Warning:Explicitintegralcouldnotbefound.

>Insy>.1ntat58

ans=

int(tan(x)/x*(1/2),x=0..1)

?vpa(ans,10)

ans=

?7968288892

(8)：

symsx;

AA

int(exp(-x2/2)/(l+x4)/x7-inf,inf)

ans=

l/4*piA(3/2)*2A(l/2)*(AngerJ(l/2,l/2)-2/piA(l/2)*sin(l/2)+2/piA(l/2)*cos(l/2)l-WeberE(l/2,l/2

))

?vpa(ans,10)

ans=

1.696392536

(9)：

symsx;

int(sin(x)/sqrt(l-x八2),x,0,l)

ans=

l/2*pi*StruveH(0zl)

?vpa(ans,10)

ans=

.8932437410

CommandWindow

ONewtoMATLAB?WatchthisVideo.seeDemos,orreadGettingStarted.

?syasx;

int(sm(x)/sqrt(l-x*2),x,0,1)

ans=

l/2*pitStruveH(0,1)

?vpaCans,10)

ans=

.8932437410

練習(xí)2.5

(1)：symsn;

AA

symsum(l/n2n/nzl,inf)

ans=

sum(l/((nA2)An),n=1..Inf)

(2)：Symsn;

symsum(sin(l/n)znzl,inf)

ans=

sum(sin(l/n),n=1..Inf)

(3)：

symsn;

A

symsum(log(n)/n3,nzl,inf)

ans=

-zeta(l,3)

(4)：

symsn;

A

symsum(l/(log(n))n/n,3/inf)

ans=

sum(l/(log(n)An),n=3..Inf)

(5)：

symsn;

symsum(l/(n*log(n))/n,2/inf)

ans=

sum(l/n/log(n)zn=2..Inf)

CommandWindow

?NewtoMATLAB?WatchthisVideo,seeDemos,orreadGetting

?sy>sn;

synsuad/(log(n))n,n,3,inf)

ans=

sun(1/(log(n)*n),n=3..Inf)

?syasn;

syjisua(1/(n*log(n)),n,2,inf)

ans=

suji(l/n/log(n),n=2..Inf)

(6)：

ymsn;

AA

symsum((-l)n*n/(n2+l)/n/l,inf)

ans=

-l/4*Psi(l-l/2*i)+l/4*Psi(l/2-l/2*i)-l/4*Psi(l+l/2*i)+l/4*Psi(l/24-l/2*i)

Conmand'.Vindow

NewtoMATLAB?'.VatchthisVideo,seeDemos,orreadGettingStarted.

?syasn;

syBsum(l/(log(n))*n,n,3,inf)

ans=

su?(l/(loj(n)*n),n=3..Inf)

?sy?sn;

syjhsua(1/(n*loj(n)),n,2,inf)

ans=

suB(l/n/log(n),n=2..Inf)

?sy?sn;

sy?suM((-1)*n*n/(n*2+l),n,I,inf)

ans=

-l/4?Psi(l-l/2*i)+l/4*Psi(l/2-l/2?i)-l/4*Psi(l+l/2?i)+l/4*Psi(l/2+l/2*i)

AW----"

第二早

練習(xí)3.1

1：

(i)：

a=-30:1:30;

b=-30:l:30;

[x,y]=meshgrid(a,b);

z=10*sin(sqrt(x?人2+y?人2))./(sqrt(1+x?人2+y?人2));

meshc(x,y,z)

(2)：

a=-30:l:30;

b=-30:l:30;

[x,y]=meshgrid(a,b);

z=4*x.A2/9+y.A2;

meshc(x,y,z)

(3)：

(4)：

a=-30:l:30;

"30:1:30;

[x,y]=meshgrid(a,b);

z=x.A2/3-y.A2/3;

meshc(x,y,z)

(5)：

a=-30:1:30;

>>b=-30:1:30;

[xzy]=meshgrid(a,b);

z=x*y;

meshc(x,y,z)

(6)：

(7)：a=-30:1:30;

>>b=-30:1:30;

>>[x,y]=meshgrid(afb);

>>z=sqrt(x.A2+y.^2);

>>meshc(x,y,z)

（8）：

(9)：

a=-30:1:30;

>>b=-30:1:30;

>>[xzy]=meshgrid(a,b);

>>z=atan(x./y);

>>meshc(x,y,z)

練習(xí)3.2

i;

a=-l:0.1:1;

>>b=0:0.1:2;

>>[xzy]=meshgrid(a,b);

>>z=x.*exp(-x.A2-y.A2);

>>[pxzpy]=gradient(z,0.1,0.1);

>>contour(a,b,z)

>>holdon

>>quiver(a,px,py)

JFigure1l。一f_絲一

ileEditViewInsertToolsDesktopWindowHelp、

5d'j)QR入⑦@&居?Q|口國?口

2

18

1.6

14

12

1

08

06

04

02

0

-148>06-0442002040608

2:

a=-2:0.1:1;

>>b=-7:0.1:1;

>>[x,y]=meshgrid(a,b);

AA

>>z=y.3/9+3*x.人2?*y+9*x?人2+y.2+x.*y+9;

>>plot3(x,y,z)

>>gridon

2JLgure1

3:

[x,y]=meshgrid(-2*pi:0.2:2*pi);

z=x,A2+2*y.A2;

plot3(x/y,z)

holdon

ezplot('xA2+yA2-l',[-2*pi,2*pi]);

gridon

■.____________

fileeditViewInsertToolsDesktopWindowHelp

UJH)Q?、-〈。⑨*4?③口用■□

4:

t=0:0.03:2*pi;

>>s=[0:0.03:2*pi]1;

>>x=(0*s+l)*cos(t);y=(0*s+l)*sin(t);z=s*(0*t+l);

>>mesh(x,y,z)

>>holdon

>>[x,y]=meshgrid(-1:0.1:1);

>>z=l-x+y;

>>mesh(x,y,z)

MFicure1,Io!-L

5:

symsxyzdxdy

z=75-xA2-yA2+x*y;

zx=diff(z,x)/zy=diff(z/y)

zy=

-2*y+x

練習(xí)3.3

1:

AA

ezplot(*x2+y2-2*x*,[-2r2]);

>>gridon

AA

s=int(int(x+y+l,y/-sqrt(l-(x-l)2),sqrt(l-(x-l)2)),x/0,2)

2*pi

2：

symsrt;

>>s=int(int(sqrt(1+rA2*sin(t)),t,0,2*pi)

int(l/2*((l+sin(t))A(l/2)*sin(t)A(l/2)+log(sin(t)A(l/2)+(l+sin(t))A(l/2)))/sin(t)A(l/2),t=0..2*pi)

3：

symsxyz;

>>s=int(int(int(1/(l+x+y+z)A3,z,0,1-x-y),y,0,l-x),x,0,l)

-5/16+l/2*log(2)

AA

4：s=vpa(int(int(x*exp(-x2-y2),y/0/2),x/-l,10))

0.16224980455070416645061789474030

練習(xí)3.4

11

y=dsolve('Dy=x+y',y(0)=1z*x')

得：y

-l-x+2*exp(x)

A,,

(2)：y=dsolve('Dy=2*x+y2/'y(0)=0)

tan(t*xA(l/2)*2A(l/2))*xA(l/2)*2A(l/2)

練習(xí)4.1

1：(1):p=[5000000000000000-68000-500];

>>x=roots(p)

0

0

0.9768

0.9388+0.26821

0.9388-0.2682i

0.8554+0.5363i

0.8554-0.5363i

0.6615+0.8064i

0.6615-0.8064i

0.3516+0.9878i

0.3516-0.9878i

-0.0345+1.0150i

-0.0345-1.0150i

-0.4609+0.9458i

-0.4609-0.9458i

-0.1150+0.8340i

-0.1150-0.8340i

-0.7821+0.7376i

-0.7821-0.7376i

-0.9859+0.4106i

-0.9859-0.4106i

-1.0416

-0.7927

(2):p=[8365423];

x=roots(p)

x=

-1.8969+0.6874i

-1.8969-0.6874i

-0.7063

2:

pl=[l0-3-2-l];p2=[l-25];

[q2,r2]=deconv(pl,p2)

q2二

12-4

r2=

000-2019

3:

symsx;

f=XA4+3*XA3.XA2-4*x-3;

g=3*xA3+10*xA2+2*x-3;

pl=factor(f)zp2=factor(g)

Pl=

(x+3)*(xA3-x-l)

(x+3)*(3*xA2+x-l)

4:

symsx;

f=xA12-l;

p=factor(f)

P=

(-l+x)*(l+xA2+x)*(l+x)*(l-x+xA2)*(l+xA2)*(xA4-xA2+l)

5：(1)：

P=[lo1];

q=[10001];

[a,b,r]=residue(p,q)

a=

-0.0000-0.3536i

-0.0000+0.3536i

0.0000-0.3536i

0.0000+0.3536i

b=

0.7071+0.7071i

0.7071-0.70711

-0.7071+0.70711

-0.7071-0.70711

[]

(2)：

p=[l];

q=[10001];

[a,b,r]=residue(p,q)

a=

-0.1768-0.1768i

-0.1768+0.1768i

0.1768-0.1768i

0.1768+0.1768i

b=

0.7071+0.7071i

0.7071-0.7071i

-0.7071+0.7071i

-0.7071-0.70711

r=

[]

(3)：

P=[101]；

q=[H-l-1]；

[a,b,r]=residue(p,q)

a=

0.5000

-1.0000

0.5000

b=

-1.0000

-1.0000

1.0000

r=

[]

(4)：p=[l1000-8];

q=[10-l0];

[a/b/r]=residue(p,q)

3=

-4

-3

8

b=

1

0

111

練習(xí)4.2

i：(1)：

D=[2131;3-121;1232;5062];

det(D)

ans=

6

(2)：

symsabed

D=[a100;-lbl0;0-1cl;00-1d];

det(D)

ans=

a*b*c*d+a*b+a*d+c*d+l

2：(1)：

symsabed

D=[l111;abcd;aA2bA2cA2dA2;aA3bA3cA3dA3];

det(D)

ans=

b*cA2*dA3-b*dA2*cA3-bA2*c*dA3+bA2*d*cA3+bA3*c*dA2-bA3*d*cA2-a*cA2*dA3+a*dA2*cA3+a

*bA2*dA3-a*bA2*cA3-a*bA3*dA2+a*bA3*cA2+aA2*c*dA3-aA2*d*cA3-aA2*b*dA3+aA2*b*cA3+aA

2*bA3*d-aA2*bA3*c-aA3*c*dA2+aA3*d*cA2+aA3*b*dA2-aA3*b*cA2-aA3*bA2*d+aA3*bA2*c

(2):Symsabxyz

D=[a*x+b*ya*y+b*za*z+b*x;a*y+b*za*z+b*xa*x+b*y;a*z+b*xa*x+b*ya*y+b*z];

det(D)

ans=

3*aA3*x*z*y+3*bA3*y*x*z-aA3*xA3-aA3*yA3-bA3*zA3-aA3*zA3-bA3*xA3-bA3*yA3

3:(1):D=[l111;12-14;2-3-1-5;31211];

Dl=[5111;-22-14;-2-3-1-5;01211];

D2=[l511;1-2-14;2-2-1-5;30211];

D3=[l151;12-24;2-3-2-5;31011];

D4=[l115;12-1-2;2-3-1-2;3120];

xl=det(Dl)/det(D);

x2=det(D2)/det(D);

x3=det(D3)/det(D);

x4=det(D4)/det(D);

xl,x2/x3/x4

xl=

1

x2=

2

x3=

3

x4=

-1

(2)：

D=[56000;l5600;01560;00156;00015];

Dl=[l6000;05600;01560;00156;10015];

D2=[51000;l0600;00560;00156;01015];

D3=[56100;l5000;01060;00056;00115];

D4=[56010;l5600;01500;00106;00015];

D5=[56001;15600;01560;00150;00011];

xl=det(Dl)/det(D);

x2=det(D2)/det(D);

x3=det(D3)/det(D);

x4=det(D4)/det(D);

x5=det(D5)/det(D);

xl,x2,x3,x4,x5

xl=

2.2662

x2=

-1.7218

x3=

1.0571

x4=

-0.5940

x5=

0.3188

練習(xí)4.3

1：A=[120;34-l;ll-l];

B=[123;-101;-24-3];

AA

A'/2+A,2*A-B,A*B,A2,A(-l)

ans=

131

241

0-1-1

ans=

342

561

331

ans=

12-3

78-3

4-21

ans=

-125

1216

2-27

ans=

710-2

1421-3

350

ans=

-3.00002.0000-2.0000

2.0000-1.00001.0000

-1.00001.0000-2.0000

2：(1)：

B=[243];

B'

ans=

2

4

3

(2)：

A=[l23];

B=[243];

A.*B,B.*A

ans=

289

ans=

289

3：(1)：

A=[010;100;001];

B=[100;001;010];

C=[l-43;20-l;l-20];

AA(-1),BA(-1),X=AA(-1)*C*BA(-1)

ans=

010

100

001

ans=

100

001

010

X=

2-10

13-4

10-2

(2)：

?A=[l23;223;351];

B=[l00;200;300];

AA(-l),x=AA(-l)*B

ans=

-1.00001.00000.0000

0.5385-0.61540.2308

0.30770.0769-0.1538

x=

100

000

000

練習(xí)4.4

1:(1):

A=[42-1;3-12;1130];

b=[2;10;8];

B=[A,b];

rank(A),rank(B)

ans=

2

ans=

3

(2)：A=[21-ll;3-2l-3;14-35];

b=[l;4;-2];

B=[A,b]；

rank(A),rank(B)

ans=

2

ans=

2

(3)：A=[1111;12-14;2-3-1-5;31211];

b，5;-2;-2;0];

B=[A,b];

rank(A),rank(B)

ans=

4

ans=

4

(4)：A=[112-l;211-l;2212];

b=[0;0;0];

B=[A,b];

rank(A),rank(B)

ans=

3

ans=

3

2：symsa;

A=[-211;1-21;11-2];

b=[-2;a;aA2];

B=[A,b];

rank(A),rank(B)

ans=

2

ans=

3

練習(xí)4.5

1：(1)：A=[01;-10];

[a,b]=eig(A)

a=

0.70710.7071

0+0.7071i0-0.7071i

b=

0+l.OOOOi0

00-l.OOOOi

(2)：

A=[001;010;100];

[a,b]=eig(A)

a=

0.70710.70710

00-1.0000

-0.70710.70710

b=

-100

010

001

(3)：

A=[41-1;32-6;1-53];

[a,b]=eig(A)

0.0185-0.9009-0.3066

-0.7693-0.1240-0.7248

-0.6386-0.41580.6170

b=

-3.052700

03.67600

008.3766

(4)：

A=[l111;11-1-1;1-11-1;11-11];

[a,b]=eig(A)

a=

0.56150.33660.2673-0.7683

-0.5615-0.33660.0000-0.0000

-0.5615-0.3366-0.5345-0.6236

-0.23260.81250.8018-0.1447

b=

-1.4142000

01.414200

002.00000

0002.0000

(5)：

A=[5765;71087;68109;57910];

[a,b]=eig(A)

0.83040.09330.39630.3803

-0.5016-0.30170.61490.5286

-0.20860.7603-0.27160.5520

0.1237-0.5676-0.62540.5209

b=

0.0102000

00.843100

003.85810

00030.2887

(6)：

A=[56000;l5600;01560;00156;00015];

[a,b]=eig(A)

a=

0.7843-0.7843-0.9860-0.9237-0.9237

0.55460.55460.00000.3771-0.3771

0.2614-0.26140.1643-0.00000.0000

0.09240.09240.0000-0.06280.0628

0.0218-0.0218-0.02740.02570.0257

b

9.24260000

00.7574000

005.000000

0002.55050

00007.4495

2：(1)：

A=[01;-10];

[a,b]=eig(A)

a=

0.70710.7071

0+0.707110-0.70711

b=

0+l.OOOOi0

00-l.OOOOi

?P=orth(a),B=P,*A*P,P*P,

P=

-0.7071-0.7071

0-0.707110+0.7071i

B=

0+l.OOOOi0-O.OOOOi

0-O.OOOOi0-l.OOOOi

ans=

1.00000+O.OOOOi

0-O.OOOOi1.0000

?inv(a)*A*a

ans=

0+l.OOOOi0

00-l.OOOOi

3：(1)：A=[2OO;O32;O23];

[a,b]=eig(A)

a=

01.00000

-0.707100.7071

0.707100.7071

b=

1.000000

0