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

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第5章

1.（1）

syms x y;

f=(1-cos(x^2+y^2))/((x^2+y^2)*exp(x^2*y^2));

limit(limit(f,x,0),y,0)

ans =

（2）

syms x y;

f=(log(x*exp(x)+exp(y)))/sqrt(x^2+y^2);

limit(limit(f,x,0),y,0)

ans =

NaN

另解

syms x y;

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）

syms x y;

f=(2*x*sin(y))/(sqrt(x*y+1)-1);

limit(limit(f,x,0),y,0)

ans =

4

另解

syms x y;

f=2*x*sin(y);

g=sqrt(x*y+1)-1;

limit(limit(f/g,x,0),y,0)

ans =

4

2.（1）

syms x y;

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）syms x y z;

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）syms x y;

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）syms x y;

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.syms x y;

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. syms x y z a;

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. syms x y u v R dux dvx duy dvy;

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. syms x y z;

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 =

另解

syms x y z

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 =