MATLAB程序设计与应用 实验答案 第七章

第七章

1．

(1)

f=sym('x^9-1');

factor(f)

ans =

(x-1)*(x^2+x+1)*(x^6+x^3+1)

(2)

f=sym('x^4+x^3+2*x^2+x+1');

factor(f)

ans =

(x^2+1)*(x^2+x+1)

(3)

f=sym('125*x^6+75*x^4+15*x^2+1');

factor(f)

ans =

(5*x^2+1)^3

(4)

syms x y z;

f=x^2+y^2+z^2+2*(x*y+y*z+z*x);

factor(f)

ans =

(x+y+z)^2

2.

(1)

syms x y;

f=y/x+x/y;

simplify(f)

ans =

(x^2+y^2)/x/y

(2)

syms a b

f=sqrt((a+sqrt(a^2-b))/2)+sqrt((a-sqrt(a^2-b))/2);

simplify(f)

ans =

1/2*(2*a+2*(a^2-b)^(1/2))^(1/2)+1/2*(2*a-2*(a^2-b)^(1/2))^(1/2) (3)

sym x;

f=2*cos(x)^2-sin(x)^2;

simplify(f)

ans =

3*cos(x)^2-1

(4)

f=sqrt(3+2*sqrt(2));

simplify(f)

ans =

2.4142

3.

(1)

sym x;

f=(x^2+6*x+8)/(x^2-5*x+4);

limit(f,x,4)

ans =

NaN

(2)

sym x;

f=abs(x)/x;

limit(f,x,0)

ans =

NaN

(3)

sym x;

f=(sqrt(1+x^2)-1)/x;

limit(f,x,0)

ans =

(4)

sym x;

f=(x+1/x)^x;

limit(f,x,inf)

ans =

Inf

4.

(1)

sym x;

y=3*x^2-5*x+1;

diff(y,x)

diff(y,x,2)

ans =

6*x-5

ans =

6

(2)

sym x;

y=sqrt(x+sqrt(x+sqrt(x)));

diff(y,x)

diff(y,x,2)

ans =

1/2/(x+(x+x^(1/2))^(1/2))^(1/2)*(1+1/2/(x+x^(1/2))^(1/2)*(1+1/2/x^(1/2))) ans =

-

1/4/(x+(x+x^(1/2))^(1/2))^(3/2)*(1+1/2/(x+x^(1/2))^(1/2)*(1+1/2/x^(1/2))

)^2+1/2/(x+(x+x^(1/2))^(1/2))^(1/2)*(-

1/4/(x+x^(1/2))^(3/2)*(1+1/2/x^(1/2))^2-1/8/(x+x^(1/2))^(1/2)/x^(3/2)) (3)

sym x;

y=sin(x)-x^2/2;

diff(y,x)

diff(y,x,2)

ans =

cos(x)-x

ans =

-sin(x)-1

(4)

syms x y;

z=x+y-sqrt(x^2+y^2);

diff(z,x)*diff(z,y)

-diff(z,x)/diff(z,y)

ans =

(1-1/(x^2+y^2)^(1/2)*x)*(1-1/(x^2+y^2)^(1/2)*y)

ans =

(-1+1/(x^2+y^2)^(1/2)*x)/(1-1/(x^2+y^2)^(1/2)*y)

5.

(1)

x=sym('x');

y=1/(x+a);

int(y)

ans =

log(x+a)

(2)

x=sym('x');

y=(1-3*x)^(1/3);

int(y)

ans =

-1/4*(1-3*x)^(4/3)

(3)

x=sym('x');

y=1/(sin(x)^2*cos(x)^2);

int(y)

ans =

1/sin(x)/cos(x)-2/sin(x)*cos(x)

(4)

y=sym('x^2/(sqrt(a^2+x^2))');

int(y)

ans =

1/2*x*(a^2+x^2)^(1/2)-1/2*a^2*log(x+(a^2+x^2)^(1/2)) 6.

(1)

数值法：

y=inline('x.*(2-x.^2).^12');

quadl(y,0,1)

ans =

315.0385

符号法：

y=sym('x*(2-x^2)^12');

int(y,x,0,1)