实验十 符号计算基础与符号微积分

实验十 符号计算基础与符号微积分

一、实验目的

1、掌握定义符号对象的方法

2、掌握符号表达式的运算法则及符号矩阵运算

3、掌握求符号函数极限及导数的方法

4、掌握求符号函数定积分和不定积分的方法 二、实验内容

1、已知x=6,y=5，利用符号表达式求

z =

提示：定义符号常数()()'6','5'x

sym y sym ==。

x=sym('6') x = 6

>> y=sym('5');

>> z=(x+1)/[(sqrt(3+x))-sqrt(y)] z =

7/(3-5^(1/2))

>>

2、分解因式 （1）44x y -

syms x y;

>> A=x^4-y^4; >> factor(A) ans =

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

（2）5135

B=5135; >> factor(B)

ans =

5 13 79

3、化简表达式

（1）1212sin cos cos sin ββββ-

syms b1 b2;

>> s=sin(b1)*cos(b2)-cos(b1)*sin(b2) s =

sin(b1)*cos(b2)-cos(b1)*sin(b2)

>> simplify(s) ans =

sin(b1)*cos(b2)-cos(b1)*sin(b2)

（2）2483

21

x x x +++

syms x;

>> s=(4*x.^2+8*x+3)/(2*x+1) s =

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

>> simplify(s) ans =

2*x+3

4、已知

12010100100,010,001101a b c P P A d e f g

h

i ??????

??????===??????????????????

完成下列运算： （1）B=P 1? P 2?A

p1=sym('[0 1 0;1 0 0;0 0 1]')

p1 =

[ 0, 1, 0] [ 1, 0, 0] [ 0, 0, 1]

>> p2=sym('[1 0 0;0 1 0;1 0 1]'); >> A=sym('[a b c;d e f;g h l]') A =

[ a, b, c] [ d, e, f] [ g, h, l]

>> B=p1*p2*A B =

[ d, e, f] [ a, b, c] [ a+g, b+h, c+l]

（2）B 的逆矩阵并验证结果

inv(B) ans =

[ (b*l-c*h)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b), (-e*c-e*l+f*b+f*h)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b),

-(-e*c+f*b)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b)]

[ -(a*l-c*g)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b), -(-d*c-d*l+f*a+f*g)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b),

(-d*c+f*a)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b)]

[ (a*h-b*g)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b), (-d*b-d*h+e*a+e*g)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b),

-(-d*b+e*a)/(d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b)]

（3）包括B矩阵主对角线元素的下三角阵

tril(B)

ans =

[ d, 0, 0]

[ a, b, 0]

[ a+g, b+h, c+l]

（4）B的行列式值

determ(B)

ans =

d*b*l-d*c*h-a*e*l+a*f*h+g*e*c-g*f*b

5、用符号方法求下列极限或导数

()()()

(

)()()

()()()()22sin tan 31'''3222220,1

1211lim

sin 2lim

1cos 23,4cos ln 5,2,x x x x x y xy

x y x e e x x y y y x a t dA d A d A

A dx dt dxdt t x x y f f x y x x e

x x y +

→----==+---=

??=????

??=-???求已知，分别求

、、已知求、 1

sym x;

>> f=[x.*(exp(sin(x))+1)-2*(exp(tan(x))-1)]./sin(x)^3 f =

(x*(exp(sin(x))+1)-2*exp(tan(x))+2)/sin(x)^3

>> limit(f,x,0) ans = -1/2 2

syms x;

>> f=(sqrt(pi)-sqrt(acos(x)))/sqrt(x+1); >> limit(f,x,-1,'right') ans = -inf 3

syms x;

>> f=(1-cos(2*x))/x; diff(f,1) ans =

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

ans =

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

syms a t x;

>> f=sym('[a^x,t^3;t*cos(x),log(x)]')

f =

[ a^x, t^3]

[ t*cos(x), log(x)]

>> diff(f,x);

ans =

[ a^x*log(a), 0]

[ -t*sin(x), 1/x]

diff(f,t,2)

ans =

[ 0, 6*t]

[ 0, 0]

>> diff(f,x)/diff(f,t)

ans =

[ 0, 1/cos(x)*a^x*log(a)] [ 1/3/t^2/x, -1/cos(x)*t*sin(x)]

5

sym x y;

f=(x.^2-2.*x).*exp(-x.^2-y.^2-x.*y);

>> diff(f,x);

>> a=diff(f,x)/diff(f,t)

a =

Inf

>> a=diff(f,x)/diff(f,t);

>> eval(a)

ans =

Inf

6、用符号方法求下列积分

()()

()()()

48

20

4

2

ln 20

1121

3141x x dx

x x x dx x e e

dx

+∞+++++????

syms x;

>> f=1/(1+x.^4+x.^8); >> int(f) ans =

1/6*3^(1/2)*atan(1/3*(2*x-1)*3^(1/2))+1/6*3^(1/2)*atan(1/3*(1+2*x)*3^(1/2))-1/12*3^(1/2)*log (x^2-3^(1/2)*x+1)+1/12*3^(1/2)*log(x^2+3^(1/2)*x+1) 2

syms x;

>> f=1/a*sin(x).^2.*sqrt(1-x.^2); >> int(f)

Warning: Explicit integral could not be found.

> In C:\MATLAB6p5\toolbox\symbolic\@sym\int.m at line 58 ans =

int(1/inf*sin(x)^2*(1-x^2)^(1/2),x) 3

syms x;

>> f=(x.^2+1)/(x.^4+1);

ans =

1/2*pi*2^(1/2)

4

syms x;

>> f=exp(x).*(1+exp(x)).^2;

>> int(f,x,0,log(2))

ans =

-7/3+exp(6243314768165359/9007199254740992)+exp(6243314768165359/9007199254740992) ^2+1/3*exp(6243314768165359/9007199254740992)^3

>>