高等数学的实验报告册答案.docx
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高等数学的实验报告册答案.docx
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高等数学的实验报告册答案
《数学实验——高等数学分册》(郭科主编)
---《实验报告册》参考答案
------轩轩
第5章
1.
(1)
symsxy;
f=(1-cos(x^2+y^2))/((x^2+y^2)*exp(x^2*y^2));
limit(limit(f,x,0),y,0)
ans=
0
(2)
symsxy;
f=(log(x*exp(x)+exp(y)))/sqrt(x^2+y^2);
limit(limit(f,x,0),y,0)
ans=
NaN
另解
symsxy;
f=log(x*exp(x)+exp(y));
g=sqrt(x^2+y^2);
limit(limit(f/g,x,0),y,0)
ans=
NaN
注:
“()”多了以后,系统无法识别,但在matlab的语法上是合理的。
在有的一些matlab版本上可以识别。
在以下的题目答案中同理。
(3)
symsxy;
f=(2*x*sin(y))/(sqrt(x*y+1)-1);
limit(limit(f,x,0),y,0)
ans=
4
另解
symsxy;
f=2*x*sin(y);
g=sqrt(x*y+1)-1;
limit(limit(f/g,x,0),y,0)
ans=
4
2.
(1)
symsxy;
z=((x^2+y^2)/(x^2-y^2))*exp(x*y);
zx=diff(z,x)
zx=
(2*x*exp(x*y))/(x^2-y^2)-(2*x*exp(x*y)*(x^2+y^2))/(x^2-y^2)^2+(y*exp(x*y)*(x^2+y^2))/(x^2-y^2)
zy=diff(z,y)
zy=
(2*y*exp(x*y))/(x^2-y^2)+(x*exp(x*y)*(x^2+y^2))/(x^2-y^2)+(2*y*exp(x*y)*(x^2+y^2))/(x^2-y^2)^2
注:
所有的x在高的版本中都可以替换为x。
(即,不用单引号,结果任然正确。
前提为:
不与前面的函数冲突。
)
(2)symsxyz;
u=log(3*x-2*y+z);
ux=diff(u,x)
ux=
3/(3*x-2*y+z)
uy=diff(u,y)
uy=
-2/(3*x-2*y+z)
uz=diff(u,'z')
uz=
1/(3*x-2*y+z)
(3)symsxy;
z=sqrt(x)*sin(y/x);
zx=diff(z,x)
zx=
sin(y/x)/(2*x^(1/2))-(y*cos(y/x))/x^(3/2)
zy=diff(z,y)
zy=
cos(y/x)/x^(1/2)
(4)symsxy;
z=asin(y*sqrt(x));
zx=diff(z,x)
zx=
y/(2*x^(1/2)*(1-x*y^2)^(1/2))
zy=diff(z,y)
zy=
x^(1/2)/(1-x*y^2)^(1/2)
3.symsxy;
f=x+y-sqrt(x^2+y^2);
fx=diff(f,x);
fy=diff(f,y);
x=2;
y=4;
fx0=subs(fx)
fx0=
0.5528
fy0=subs(fy)
fy0=
0.1056
4.symsxyza;
f=x^2+y^2+z^2-2*a*x*y*z;
fx=diff(f,x);
fy=diff(f,y);
fz=diff(f,'z');
dzx=-fx/fz
dzx=
-(2*x-2*a*y*z)/(2*z-2*a*x*y)
dzy=-fy/fz
dzy=
-(2*y-2*a*x*z)/(2*z-2*a*x*y)
5.symsxyuvRduxdvxduydvy;
f=x^2+y^2+u^2+v^2-R^2;
g=x+y+u+v;
fx=diff(f,x);fy=diff(f,y);fu=diff(f,'u');fv=diff(f,'v');
gx=diff(g,x);gy=diff(g,y);gu=diff(g,'u');gv=diff(g,'v');
ffx=fu*dux+fv*dvx+fx;
ggx=gu*dux+gv*dvx+gx;
[dux,dvx]=solve(ffx,ggx,'dux','dvx')
dux=
(v-x)/(u-v)
dvx=
-(u-x)/(u-v)
ffy=fu*duy+fv*dvy+fy;
ggy=gu*duy+gv*dvy+gy;
[duy,dvy]=solve(ffy,ggy,'duy','dvy')
duy=
(v-y)/(u-v)
dvy=
-(u-y)/(u-v)
6.symsxyz;
f=x+y+z-exp(-(x+y+z));
fx=diff(f,x);fz=diff(f,'z');
dzx=-fx/fz;
g=dzx;
gy=diff(g,y);gz=diff(g,'z');
dzxy=gy+gz*dzx
dzxy=
0
另解
symsxyz
f=x+y+z-exp(-(x+y+z));
fx=diff(f,x);fz=diff(f,'z');
dzx=-fx/fz;
gy=diff(dzx,y);gz=diff(dzx,'z');
dzxy=gy+gz*dzx
dzxy=
0
7.symsxy;
z=x^2-x*y-2*y^2;
zx=diff(z,x);zy=diff(z,y);
fzx=inline(zx);fzy=inline(zy);
a=pi/3;b=pi/6;
f1=fzx(1,2)*cos(a)+fzy(1,2)*cos(b)
f1=
-7.7942
8.symst;
x=sin(t);
y=cos(t);
z=t;
dx=diff(x,'t');dy=diff(y,'t');dz=diff(z,'t');
x1=inline(dx);x2=inline(dy);x3=inline(dz);
t=pi/4;a=sin(t);b=cos(t);c=t;
x10=x1(a)
x10=
0.7602
x20=x2(b)
x20=
-0.6496
x30=x3(c)
x30=
1
9.symsxy;
f=x^2+y^2;
fx=diff(f,x);fy=diff(f,y);
x=1;y=2;
fx0=subs(fx)
fx0=
2
fy0=subs(fy)
fy0=
4
10.symsxy;
f=x^3-y^3+3*x^2+3*y^2-9*x;
fx=diff(f,x);fy=diff(f,y);
[x0,y0]=solve(fx,fy)
x0=
1
-3
1
-3
y0=
0
0
2
2
fxx=diff(diff(f,x),x)(此条可以写作:
fxx=diff(f,x,2),类似的同理)
fxx=
6*x+6
fxy=diff(diff(f,x),y);
fyy=diff(diff(f,y),y);
delta=inline(fxy^2-fxx*fyy);
delta(x0,y0)
ans=
-72
72
72
-72
x=1;y=0;
fmin=subs(f)
fmin=
-5
x=-3;y=2;
fmax=subs(f)
fmax=
31
11.symsxyzlamdaa;
L=x*y*z+lamda*((x^2+y^2)/4+z^2-a^2);
Lx=diff(L,x);
Ly=diff(L,y);
Lz=diff(L,'z');
Llamda=diff(L,'lamda');
[lamdaxyz]=solve(Lx,Ly,Lz,Llamda)
lamda=
0
0
0
0
0
0
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
x=
0
2*a
0
0
(-2)*a
0
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
y=
0
0
2*a
(-2)*a
0
0
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
z=
a
0
0
0
0
-a
(3^(1/2)*a)/3
(3^(1/2)*a)/3
(3^(1/2)*a)/3
-(3^(1/2)*a)/3
-(3^(1/2)*a)/3
-(3^(1/2)*a)/3
(3^(1/2)*a)/3
-(3^(1/2)*a)/3
V=x.*y.*z
V=
0
0
0
0
0
0
(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
第6章
1.symsxy;
f=x*y;
y1=2*x;
y2=x^2+1;
I=int(int(f,y,y1,y2),x,0,1)
I=
1/12
2.symsxyrt;
x=r*cos(t);y=r*sin(t);
f=exp(-(x^2+y^2));
I=int(int(f*r,r,0,1),t,0,2*pi)
I=
-pi*(1/exp
(1)-1)
3.symsxyzaAtrs;
x=r*sin(s)*cos(t);
y=r*sin(s)*sin(t);
z=r*cos(s);
f=x^2+y^2;
I1=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,a),t,0,2*pi);
I2=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,A),t,0,2*pi);
I=I2-I1
I=
(4*pi*A^5)/15-(4*pi*a^5)/15
4.symsxyztrs;
x=r*sin(s)*cos(t);
y=r*sin(s)*sin(t);
z=r*cos(s);
f=(z*log(x^2+y^2+z^2+1))/(x^2+y^2+z^2+1);
M=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,1),t,0,2*pi)
M=
-(pi*(log
(2)^2-log(16)+2))/4
5.x=1/2:
0.001:
1;
y=1./x;
plot(x,y,k);
xlabel(x);ylabel(y)
symsx;
f=1/x;
fx=diff(f,x);
I=int(x*sqrt(1+fx^2),x,1/2,1)
I=
int(x*(1/x^4+1)^(1/2),x=1/2..1)
6.symsxyrt;
x=r*cos(t);y=r*sin(t);
z=sqrt(x^2+y^2);
zx=diff(z,x);zy=diff(z,y);
f=(sqrt(x^2+y^2))*sqrt(1+zx^2+zy^2);
M=int(int(f*r,r,0,1),t,0,2*pi)
M=
(2*pi)/3
8、
symsxya;
p=exp(x)*sin(y)-y^2;
q=exp(x)*cos(y)-1;
f=diff(q,x)-diff(p,y);
x1=0;
x2=a;
y1=0;
y2=sqrt(a/4-(x-a/2)^2);
f1=int(f,y,y1,y2);
i=int(f1,x,x1,x2)
i=
-(a^2*(a-3))/12
9、
symsztr;
R=z^2;
Rz=diff(R,z);
t1=0;t2=2*pi;
z1=0;z2=1;
r1=0;r2=sqrt(z);
f1=int(r*Rz,r,r1,r2);
f2=int(f1,z,z1,z2);
i=int(f2,t,t1,t2)
i=
(2*pi)/3
10.symsxyz;
P=4*x;Q=-2*x*y;R=z^2;
px=diff(P,x);
qy=diff(Q,y);
rz=diff(R,'z');
x=1;y=1;z=3;
px0=subs(px)
px0=
4
qy0=subs(qy)
qy0=
-2
rz0=subs(rz)
rz0=
6
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