matlab課后習習題答案 (附圖)

1、習題畫出下列常見曲線的圖形（1） 立方拋物線 命令：syms x y; ezplot('x.(1/3)')(2)高斯曲線y=e(-X2);命令：clearsyms x y;ezplot('exp(-x*x)')(3)笛卡爾曲線命令：>> clear>> syms x y;>> a=1;>> ezplot(x3+y3-3*a*x*y)(4) 蔓葉線命令：>> clear>> syms x y;>> a=1ezplot(y2-(x3)/(a-x)(5) 擺線:命令：>>

2、clear>> t=0:2*pi;>> x=t-sin(t);>>y=2*(1-cos(t);>> plot(x,y)7螺旋線命令：>> clear>> t=0:2*pi;>> x=cos(t);>> y=sin(t);>> z=t;>>plot3(x,y,z)(8) 阿基米德螺線命令：clear>> theta=0:2*pi;>> rho1=(theta);>> subplot(1,2,1),polar(theta,rho1)(9) 對數

3、螺線命令：cleartheta=0:2*pi;rho1=exp(theta);subplot(1,2,1),polar(theta,rho1)(12) 心形線命令：>> clear>> theta=0:2*pi;>> rho1=1+cos(theta);>> subplot(1,2,1),polar(theta,rho1)練習1. 求出下列極限值（1）命令：>>syms n>>limit(n3+3n)(1/n) ans =3（2）命令：>>syms n>>limit(n+2)(1/2)-2*(n+1

4、)(1/2)+n(1/2),n,inf) ans = 0（3）命令：syms x；>> limit(x*cot(2*x),x,0) ans =1/2（4）命令：syms x m；limit(cos(m/x)x,x,inf)ans =1（5）命令：syms x>> limit(1/x-1/(exp(x)-1),x,1)ans =(exp(1)-2)/(exp(1)-1)（6）命令：syms x>> limit(x2+x)(1/2)-x,x,inf)ans =1/2 練習1. 求下列不定積分，并用diff驗證：（1）>>Clear>> s

5、yms x y>> y=1/(1+cos(x);>> f=int(y,x) f = tan(1/2*x)>> y=tan(1/2*x);>> yx=diff(y,x);>> y1=simple(yx) y1 = 1/2+1/2*tan(1/2*x)2 (2)clearsyms x yy=1/(1+exp(x);f=int(y,x)f = -log(1+exp(x)+log(exp(x)syms x yy=-log(1+exp(x)+log(exp(x);yx=diff(y,x);y1=simple(yx)y1 = 1/(1+exp(x

6、)(3)syms x yy=x*sin(x)2;>> f=int(y,x) f =x*(-1/2*cos(x)*sin(x)+1/2*x)-1/4*cos(x)2-1/4*x2 clearsyms x y y=x*(-1/2*cos(x)*sin(x)+1/2*x)-1/4*cos(x)2-1/4*x2;yx=diff(y,x);>> y1=simple(yx) y1 =x*sin(x)2(4) syms x yy=sec(x)3;f=int(y,x)f =1/2/cos(x)2*sin(x)+1/2*log(sec(x)+tan(x) clearsyms x yy=1

7、/2/cos(x)2*sin(x)+1/2*log(sec(x)+tan(x);yx=diff(y,x);y1=simple(yx)y1 = 1/cos(x)32. 求下列積分的數值解1）clearsyms xy=int(x(-x),x,0,1)y =int(x(-x),x = 0 . 1)vpa(y,10)ans =2）clear syms xy=int(exp(2*x)*cos(x)3,x, clearsyms xy=int(1/(2*pi)(1/2)*exp(-x2/2),x,0,1)y = 713/0*erf(1/2*2(1/2)*2(1/2)*pi(1/0,2*pi)y = 22/6

8、5*exp(pi)4-22/65vpa(ans,10)(3) >> clear>> syms x>> y=int(1/(2*pi)(1/2)*exp(-x2/2),0,1);>> vpa(y,14)ans =.34 2（4）>> clear>> syms x>> y=int(x*log(x4)*asin(1/x2),1,3);Warning: Explicit integral could not be found.> In at 58>> vpa(y,14)ans = 2（5）>>

9、; clear>> syms x>> y=int(1/(2*pi)(1/2)*exp(-x2/2),-inf,inf);>> vpa(y,14)ans =.999999 練習1判斷下列級數的收斂性，若收斂，求出其收斂值。1）syms ns1=symsum(1/n(2n),n,1,inf)s1 = sum(1/(n(2n),n = 1 . Inf)vpa(s1,10)ans = 因此不收斂2）syms ns1=symsum(sin(1/n),n,1,inf) s1 =sum(sin(1/n),n = 1 . Inf)vpa(s1,10) ans =sum(si

10、n(1/n),n = 1 . Inf)不收斂(3) >> clear>> syms n>> s=symsum(log(n)/n3,n,1,inf)s =-zeta(1,3)收斂(4) syms n s1=symsum(1/(log10(n)n,n,3,inf)s1 =sum(1/(log(n)/log(10)n),n = 3 . inf)不收斂(5) syms ns1=symsum(1/n*log10(n),n,2,inf)s1 =sum(1/n*log(n)/log(10),n = 2 . Inf)不收斂(6) >> clear>>

11、 syms n>> s=symsum(-1)n*n/n2+1,n,1,inf)s =sum(-1)n/n+1,n = 1 . Inf)不收斂 習題1）clear;x,y=meshgrid(-30:30);z=10*sin(sqrt(x.2+y.2)./sqrt(1+x.2+y.2);>> meshc(x,y,z)clear>> x,y=meshgrid(-30:30);>> z=10*sin(x2+y2)(1/2)/(1+x2+y2)(1/2)mesh(x,y,z)1.2.取適當的參數繪制下列曲面的圖形。（1）clear>> a=-2

12、:2;>> b=-3:3;>> x,y=meshgrid(a,b);>> z=(1-(x.2)/4-(y.2)/9).(1/2);>> mesh(x,y,z)>> hold onmesh(x,y,-z)（2）clear>> a=-1:1;>> b=-2:2;x,y=meshgrid(a,b);>> z=(4/9)*(x.2)+(y.2);>> mesh(x,y,z)(4)clear>> x,y=meshgrid(-1:1);>> z=(1/3)*(x.2)-(1/

13、3)*(y.2);>> mesh(x,y,z) 習題P49/例命令:syms x y limit(limit(x2+y2)/(sin(x)+cos(y),0),pi),ans =-pi2 limit(limit(1-cos(x2+y2)/(x2+y2),0),0),ans =0P49/例命令：clear;syms x y z dx dy dz zxz zy zxx zxyz=atan(x2*y) z =atan(x2*y) zx=diff(z,x),zy=diff(z,y)zx 2*x*y/(1+x4*y2)zy =x2/(1+x4*y2) dz=zx*dx+zy*dy,dz =2

14、*x*y/(1+x4*y2)*dx+x2/(1+x4*y2)*dzxx=diff(zx,x),zxy=diff(zx,y)zxx =2*y/(1+x4*y2)-8*x4*y3/(1+x4*y2)2zxy =2*x/(1+x4*y2)-4*x5*y2/(1+x4*y2)2 作圖表示函數z=x*exp(-x2-y2) (-1<x<1,0<y<2)沿x軸方向梯度clear>> a=-1:1;>> b=0:2;>> x,y=meshgrid(a,b);>> z=x.*exp(-x.2-y.2);>> px,py=gra

15、dient(z,;contour(a,b,z),hold on,>> quiver(a,b,px,py),hold off 習題1. 解下列微分方程（1）y=dsolve('Dy=x+y','y(0)=1','x')y =-x-1+2*exp(x)x=1 2 3x = 1 2 3 -x-1+2*exp(x)ans = （2）x'=2*x+3*y,y'=2*x+y,x(0)=-2,y(0)=,0<t<10,做相平面圖新建M函數function dy=weifen1(t,y)dy=zeros(2,1);dy(1

16、)=2*y(1)+3*y(2);dy(2)=2*y(1)+y(2);輸入命令>> t=0:10;>> t,y=ode15s('weifen1',0,10,-2 );>> plot(t,y)（3）y''(y')2+2*y1=sin(t),y(0)=0,y'(0)=1,0<t<5,做y的圖新建M函數function dy=weifen2(t,y)dy=zeros(2,1);dy(1)=y(2);dy(2)=*y(2)2-2*y(1)+sin(t);輸入命令>> t,y=ode15s('

17、;weifen2',0,5,0 1);>> plot(t,y)1. 繪制飛船軌跡圖新建M函數function dy=weifen3(t,y)dy=zeros(4,1);dy(1)=y(3);dy(2)=y(4);dy(3)=2*y(4)+y(1)-(1-1/*(y(1)+ 1ans = 3 習題（3）clear>> A=1 1 1 1;1 2 -1 4;2 -3 -1 -5;3 1 2 11;>> B=1 1 1 1 5;1 2 -1 4 -2;2 -3 -1 -5 -2;3 1 2 11 0;>> rank(A),rank(B)ans

18、= 4 ans = 4 習題（3）>> clear>> A=4 1 -1;3 2 -6;1 -5 3;>> a,b=eig(A)a = 92/4963 -1237/1373 -424/1383 -627/815 -449/3622 -1301/1795 -1122/1757 -1097/2638 559/906 b = -4695/1538 0 0 0 1963/534 0 0 0 8318/993 （5）>> clear>> A=5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10;>> a,b=eig(

19、A)a = 431/519 308/3301 472/1191 551/1449 -641/1278 -2209/7323 1175/1911 2100/3973 -434/2081 1050/1381 -855/3148 494/895 368/2975 -1049/1848 -3157/5048 473/908 b = 23/2266 0 0 0 0 1639/1944 0 0 0 0 3615/937 0 0 0 0 2938/97 clear>> A=2 0 0;0 3 2;0 2 3;>> a,b=eig(A);>> a,b=eig(A)a = 0 1 0 -985/1393 0 985/1393 985/1393 0 985/1393 b = 1 0 0 0 2 0 0 0 5 >> p=orth(a)p = 0 -1 0 985/1393 0 985/1393 -985/1393 0 985/1393 >> B=p'*A*pB = 1 0 0 0 2 0 0 0 5 >> p*p'ans = 1 0 0