高等数学的实验报告册答案

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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

8. 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 =

0.7602

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 =

4

fy0=subs(fy)

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 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 fmax =

31

x=-3;y=2; fmax=subs(f)

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');

[lamda x y z]=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. 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)

I =