matlab课后习题答案.docx
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matlab课后习题答案
习题
画出下列常见曲线的图形
(1)立方抛物线
命令:
symsxy;
ezplot('x.^(1/3)')
(2)高斯曲线y=e^(-X^2);
命令:
clear
symsxy;
ezplot('exp(-x*x)')
(3)笛卡尔曲线
命令:
>>clear
>>symsxy;
>>a=1;
>>ezplot(x^3+y^3-3*a*x*y)
(4)蔓叶线
命令:
>>clear
>>symsxy;
>>a=1
ezplot(y^2-(x^3)/(a-x))
(5)摆线:
命令:
>>clear
>>t=0:
:
2*pi;
>>x=t-sin(t);
>>y=2*(1-cos(t));
>>plot(x,y)
7螺旋线
命令:
>>clear
>>t=0:
:
2*pi;
>>x=cos(t);
>>y=sin(t);
>>z=t;
>>plot3(x,y,z)
(8)阿基米德螺线
命令:
clear
>>theta=0:
:
2*pi;
>>rho1=(theta);
>>subplot(1,2,1),polar(theta,rho1)
(9)对数螺线
命令:
clear
theta=0:
:
2*pi;
rho1=exp(theta);
subplot(1,2,1),polar(theta,rho1)
(12)心形线
命令:
>>clear
>>theta=0:
:
2*pi;
>>rho1=1+cos(theta);
>>subplot(1,2,1),polar(theta,rho1)
练习
1.求出下列极限值
(1)
命令:
>>symsn
>>limit((n^3+3^n)^(1/n))
ans=
3
(2)
命令:
>>symsn
>>limit((n+2)^(1/2)-2*(n+1)^(1/2)+n^(1/2),n,inf)
ans=
0
(3)
命令:
symsx;
>>limit(x*cot(2*x),x,0)
ans=
1/2
(4)
命令:
symsxm;
limit((cos(m/x))^x,x,inf)
ans=
1
(5)
命令:
symsx
>>limit(1/x-1/(exp(x)-1),x,1)
ans=
(exp
(1)-2)/(exp
(1)-1)
(6)
命令:
symsx
>>limit((x^2+x)^(1/2)-x,x,inf)
ans=
1/2
练习
1.求下列不定积分,并用diff验证:
(1)
>>Clear
>>symsxy
>>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)
clear
symsxy
y=1/(1+exp(x));
f=int(y,x)
f=
-log(1+exp(x))+log(exp(x))
symsxy
y=-log(1+exp(x))+log(exp(x));
yx=diff(y,x);
y1=simple(yx)
y1=
1/(1+exp(x))
(3)
symsxy
y=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*x^2
clear
symsxyy=x*(-1/2*cos(x)*sin(x)+1/2*x)-1/4*cos(x)^2-1/4*x^2;
yx=diff(y,x);
>>y1=simple(yx)
y1=
x*sin(x)^2
(4)
symsxy
y=sec(x)^3;
f=int(y,x)
f=
1/2/cos(x)^2*sin(x)+1/2*log(sec(x)+tan(x))
clear
symsxy
y=1/2/cos(x)^2*sin(x)+1/2*log(sec(x)+tan(x));
yx=diff(y,x);
y1=simple(yx)
y1=
1/cos(x)^3
2.求下列积分的数值解
1)
clear
symsx
y=int(x^(-x),x,0,1)
y=
int(x^(-x),x=0..1)
vpa(y,10)
ans=
2)
clear
symsx
y=int(exp(2*x)*cos(x)^3,x,clear
symsx
y=int((1/(2*pi)^(1/2))*exp(-x^2/2),x,0,1)
y=
713/0*erf(1/2*2^(1/2))*2^(1/2)*pi^(1/0,2*pi)
y=
22/65*exp(pi)^4-22/65vpa(ans,10)
(3)
>>clear
>>symsx
>>y=int(1/(2*pi)^(1/2)*exp(-x^2/2),0,1);
>>vpa(y,14)
ans=
.34
2(4)
>>clear
>>symsx
>>y=int(x*log(x^4)*asin(1/x^2),1,3);
Warning:
Explicitintegralcouldnotbefound.
>Inat58
>>vpa(y,14)
ans=
2(5)
>>clear
>>symsx
>>y=int(1/(2*pi)^(1/2)*exp(-x^2/2),-inf,inf);
>>vpa(y,14)
ans=
.999999
练习
1判断下列级数的收敛性,若收敛,求出其收敛值。
1)symsn
s1=symsum(1/n^(2^n),n,1,inf)
s1=
sum(1/(n^(2^n)),n=1..Inf)
vpa(s1,10)
ans=
因此不收敛
2)symsn
s1=symsum(sin(1/n),n,1,inf)
s1=
sum(sin(1/n),n=1..Inf)
vpa(s1,10)
ans=
sum(sin(1/n),n=1..Inf)
不收敛
(3)
>>clear
>>symsn
>>s=symsum(log(n)/n^3,n,1,inf)
s=
-zeta(1,3)
收敛
(4)symsn
s1=symsum(1/(log10(n))^n,n,3,inf)
s1=
sum(1/((log(n)/log(10))^n),n=3..inf)
不收敛
(5)symsn
s1=symsum(1/n*log10(n),n,2,inf)
s1=
sum(1/n*log(n)/log(10),n=2..Inf)
不收敛
(6)
>>clear
>>symsn
>>s=symsum((-1)^n*n/n^2+1,n,1,inf)
s=
sum((-1)^n/n+1,n=1..Inf)
不收敛
习题
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((x^2+y^2)^(1/2))/(1+x^2+y^2)^(1/2)
mesh(x,y,z)
1.
2.取适当的参数绘制下列曲面的图形。
(1)
clear
>>a=-2:
:
2;
>>b=-3:
:
3;
>>[x,y]=meshgrid(a,b);
>>z=(1-(x.^2)/4-(y.^2)/9).^(1/2);
>>mesh(x,y,z)
>>holdon
mesh(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)-(1/3)*(y.^2);
>>mesh(x,y,z)
习题
P49/例命令:
symsxy
limit(limit((x^2+y^2)/(sin(x)+cos(y)),0),pi),
ans=
-pi^2
limit(limit((1-cos(x^2+y^2))/((x^2+y^2)),0),0),
ans=
0
P49/例命令:
clear;symsxyzdxdydzzxzzyzxxzxy
z=atan(x^2*y)
z=
atan(x^2*y)
zx=diff(z,x),zy=diff(z,y)
zx
2*x*y/(1+x^4*y^2)
zy=
x^2/(1+x^4*y^2)
dz=zx*dx+zy*dy,
dz=
2*x*y/(1+x^4*y^2)*dx+x^2/(1+x^4*y^2)*d
zxx=diff(zx,x),zxy=diff(zx,y)
zxx=
2*y/(1+x^4*y^2)-8*x^4*y^3/(1+x^4*y^2)^2
zxy=
2*x/(1+x^4*y^2)-4*x^5*y^2/(1+x^4*y^2)^2
作图表示函数z=x*exp(-x^2-y^2)(-1clear
>>a=-1:
:
1;
>>b=0:
:
2;
>>[x,y]=meshgrid(a,b);
>>z=x.*exp(-x.^2-y.^2);
>>[px,py]=gradient(z,,;
contour(a,b,z),holdon,
>>quiver(a,b,px,py),holdoff
习题
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=[123]
x=123
-x-1+2*exp(x)
ans=
(2)x'=2*x+3*y,y'=2*x+y,x(0)=-2,y(0)=,0新建M函数
functiondy=weifen1(t,y)
dy=zeros(2,1);
dy
(1)=2*y
(1)+3*y
(2);
dy
(2)=2*y
(1)+y
(2);
输入命令
>>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新建M函数
functiondy=weifen2(t,y)
dy=zeros(2,1);
dy
(1)=y
(2);
dy
(2)=*y
(2)^2-2*y
(1)+sin(t);
输入命令
>>[t,y]=ode15s('weifen2',[0,5],[01]);
>>plot(t,y)
1.绘制飞船轨迹图
新建M函数
functiondy=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
习题
(3)
clear
>>A=[1111;12-14;2-3-1-5;31211];
>>B=[11115;12-14-2;2-3-1-5-2;312110];
>>rank(A),rank(B)
ans=
4
ans=
4
习题
(3)
>>clear
>>A=[41-1;32-6;1-53];
>>[a,b]=eig(A)
a=
92/4963-1237/1373-424/1383
-627/815-449/3622-1301/1795
-1122/1757-1097/2638559/906
b=
-4695/153800
01963/5340
008318/993
(5)
>>clear
>>A=[5765;71087;68109;57910];
>>[a,b]=eig(A)
a=
431/519308/3301472/1191551/1449
-641/1278-2209/73231175/19112100/3973
-434/20811050/1381-855/3148494/895
368/2975-1049/1848-3157/5048473/908
b=
23/2266000
01639/194400
003615/9370
0002938/97
clear
>>A=[200;032;023];
>>[a,b]=eig(A);
>>[a,b]=eig(A)
a=
010
-985/13930985/1393
985/13930985/1393
b=
100
020
005
>>p=orth(a)
p=
0-10
985/13930985/1393
-985/13930985/1393
>>B=p'*A*p
B=
100
020
005
>>p*p'
ans=
100
010
001
习题
clear
>>x=0:
:
1;
>>y=exp(-x.^2/2);
>>plot(x,y);
>>symsx;
>>vpa(int(exp(-x.^2/2),x,0,1),6)
ans=
.855620
>>n=10000;
>>x=rand(n,1);
>>y=rand(n,1);
>>m=sum(ym=
8564
>>s=m/n
s=
>>
练习
求这两家煤场如何分配供煤能使总运输量最小
建立数学模型:
minz=10*x1+5*x2+6*x3+4*x4+8*x5+15*x6
.:
x1+x2+x3>=60
x4+x5+x6>=100
x1+x4=45
x2+x5=75
x3+x6=40
输入命令
>>c=[10;5;6;4;8;15];
>>A=[-1-1-1000;000-1-1-1];
>>b=[-60;-100];
>>Aeq=[100100;010010;001001];
>>beq=[45;75;40];
>>lb=zeros(6,1);
>>[x,fv]=linprog(c,A,b,Aeq,beq,lb)
Optimizationterminated.
x=
fv=