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=42.（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=0.5528fy0=subs(fy)fy0=0.10564.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/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.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*dzxdzxy=0另解symsxyzf=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=07.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=-7.79428.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=0.7602x20=x2(b)x20=-0.6496x30=x3(c)x30=19.symsxy;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.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=1-31-3y0=0022fxx=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.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=000000-(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)/3x=02*a00(-2)*a0(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)/3y=002*a(-2)*a00(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=a0000-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=000000(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=1/122.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-I1I=(4*pi*A^5)/15-(4*pi*a^5)/154.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))/45．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)/38、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))/129、s