《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）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 =42.（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 =0.5528fy0=subs(fy)fy0 =0.10564. 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/fzdzx =-(2*x - 2*a*y*z)/(2*z - 2*a*x*y)dzy=-fy/fzdzy =-(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*dzxdzxy =另解syms x y zf=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*dzxdzxy =7. syms x y;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 =-7.79428. syms t;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 =0.7602x20=x2(b)x20 =-0.6496x30=x3(c)x30 =19. syms x y;f=x^2+y^2;fx=diff(f,x);fy=diff(f,y);x=1;y=2;fx0=subs(fx)fx0 =2fy0=subs(fy)fy0 =410. syms x y;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 =1-31-3y0 =22fxx=diff(diff(f,x),x) (此条可以写作：fxx=diff(f,x,2),类似的同理) fxx =6*x + 6fxy=diff(diff(f,x),y);fyy=diff(diff(f,y),y);delta=inline(fxy^2-fxx*fyy);delta(x0,y0)ans =-727272-72x=1;y=0;fmin=subs(f)fmin =-5x=-3;y=2;fmax=subs(f)fmax =3111. syms x y z lamda a;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');[lamda x y z]=solve(Lx,Ly,Lz,Llamda)lamda =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)/3z =a-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)/3V=x.*y.*zV =(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. syms x y;f=x*y;y1=2*x;y2=x^2+1;I=int(int(f,y,y1,y2),x,0,1)I =1/122. syms x y r t;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. syms x y z a A t r s;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-I1I =(4*pi*A^5)/15 - (4*pi*a^5)/154. syms x y z t r s;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))/45．x=1/2:0.001:1;y=1./x;plot(x,y,k);xlabel(x);ylabel(y)syms x;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. syms x y r t;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)/38、syms x y a;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))/129、syms z t r;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)/310．syms x y z;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 =4qy0=subs(qy)qy0 =-2rz0=subs(rz)rz0 =6。