成都理工大学数学实验高等数学分册习题(下)参考答案

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
zx = y/(2*x^(1/2)*(1 - x*y^2)^(1/2))
zy=diff(z,'y')
2
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)
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 =
1
(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
8. syms t;
x=sin(t); y=cos(t);
4
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)
p= 0 （3）syms n;
f='((n/(2*n+1))^n)^(1/n)'; p=limit(f,n,inf)
p= 1/2 （4）syms n;
f='(sqrt(n)/sqrt(n^4+1))/(1/sqrt(n^3))'; p=limit(f,n,inf)
rz0 = 6
第7章
1． （1）syms n;
symsum((n+1)^(1/3)-n^(1/3),n,1,inf)
10
ans = Inf
（2）syms n b c;
b='factorial(n+1)';c='factorial(n)'; f='((b/((n+1)^(n+1)))/(c/(n^n)))'; p=limit(f,n,inf)
fxy=diff(diff(f,'x'),'y'); fyy=diff(diff(f,'y'),'y'); delta=inlineຫໍສະໝຸດ Baidufxy^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)
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)
8
(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= 1/12 2. 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)
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
M= (2*pi)/3 10．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 = 4
qy0=subs(qy)
qy0 = -2
rz0=subs(rz)
dzxy = 0
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 = -7.7942
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
x10 = 0.7602
x20=x2(b)
x20 = -0.6496
x30=x3(c)
x30 = 1
9. syms x y;
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
fmax = 31
11. 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');
6
[lamda x y z]=solve(Lx,Ly,Lz,Llamda)
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
ans = NaN （3）
>>syms x y; >>f=(2*x*sin(y))/(sqrt(x*y+1)-1); >>limit(limit(f,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')
高等数学 II（下） 实验参考答案 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 = 0 （2）
>>syms x y; >>f=(log(x*exp(x)+exp(y)))/sqrt(x^2+y^2); >>limit(limit(f,x,0),y,0)
y = 0 0 2*a (-2)*a 0
7
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
3
[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')
10. 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 -3
5
1 -3 y0 = 0 0 2 2
fxx=diff(diff(f,'x'),'x')
fxx = 6*x + 6
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)
I= (4*pi*A^5)/15 - (4*pi*a^5)/15 4. 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)
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')
M= -(pi*(log(2)^2 - log(16) + 2))/4
9
5．x=1/2:0.01:1;
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= -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-I1
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;
（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;