MATLAB符号运算练习题

第五章 符合运算 练习题

1.求符号函数f=ax+by+cx+d分别对x,y进行三次微分；对 y进行定积分和不定积分，对y的定积分区间为（0，1）；对y趋向于1求极限。

3

2

>> syms x y

>> f=sym('a*x^3+b*y^2+c*x+d'); >> diff(f,x,3) ans = 6*a

>> diff(f,y,3) ans = 0

>> int(f,y,0,1) ans =

a*x^3 + c*x + b/3 + d >> int(f,y) ans =

(b*y^3)/3 + (a*x^3 + c*x + d)*y

>> limit(f,y,1) ans =

a*x^3 + c*x + b + d

2. 已知f=1/(1+x^2),g=sin(y),求复合函数f(g(y)).

>> syms x y

>> f=sym('1/(1+x^2)'); >> g=sym('sin(y)'); >> compose(f,g,x,y) ans =

1/(sin(y)^2 + 1)

?x2?2x?1?0?x?3z?4?3.求三元非线性方程组

?y*z??1的解。 ?>> f1=sym('x^2+2*x+1'); >> f2=sym('x+3*z-4'); >> f3=sym('y*z+1'); >> [x,y,z]=solve(f1,f2,f3)

x =

-1

y =

-3/5

z =

5/3

?dy?dx?z?cosx?4.解方程组?dz当y(0)=1,z(0)=0时，求微分方程组的解。

?dx?y?1??>> [y,z]=dsolve('Dy-z=cos(x), Dz+y=1','y(0)=1','z(0)=0','x') y =

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

z =

-(x*sin(x))/2

5.求级数1?122?132???1k2??和1+x+x2+…+xk+…的和。

>> syms k

>> symsum(1/(k^2),k,1,inf)

ans =

pi^2/6

>> syms x k

>> symsum(x^k,k,0,inf) ans =

piecewise([1 <= x, Inf], [abs(x) < 1, -1/(x - 1)])

??xdx6计算积分 21（1+x）

?>> f=sym('x^(1/2)/(1+x)^2'); >> int(f,1,inf) ans =

pi/4 + 1/2

?? xdz7计算积分 1 2 1+z

?>> syms x z

>> f=sym('x/(1+z^2)'); >> int(f,z,1,inf) ans =

(pi*x)/4

8求函数 1 ? 2 x 3 ? 3 1 ? 3 x ? x 2 的5阶泰勒级数展开式 x ?

>> syms x a

>>f=sym('(1-2*x+x^3)^(1/2)-(1-3*x+x^2)^(1/3)'); >>taylor(f,x,a,'order',5) ans =

(a^3 - 2*a + 1)^(1/2) - (a - x)^4*(((5*(3*a^2 - 2)*((3*(a^3 - 2*a + 1)^(1/2))/2 - (3*(3*a^2 - 2)*(3*a*(a^3 - 2*a + 1)^(1/2) - (3*a^2 - 2)^2/(4*(a^3 - 2*a + 1)^(1/2))))/(4*(a^3 - 2*a + 1))))/(6*(a^3 - 2*a + 1)) - (3*a^2 - 2)/(4*(a^3 - 2*a + 1)^(1/2)) + (3*a*(3*a*(a^3 - 2*a + 1)^(1/2) - (3*a^2 - 2)^2/(4*(a^3 - 2*a + 1)^(1/2))))/(2*(a^3 - 2*a + 1)))/(4*(a^3 - 2*a + 1)) + ((2*((2*(2*a - 3)^2)/(9*(a^2 - 3*a + 1)^(2/3)) - (2*(a^2 - 3*a + 1)^(1/3))/3))/(3*(a^2 - 3*a + 1)) + (8*(2*a - 3)*((2*a - 3)/(9*(a^2 - 3*a + 1)^(2/3)) - (5*((2*(2*a - 3)^2)/(9*(a^2 - 3*a + 1)^(2/3)) - (2*(a^2 - 3*a + 1)^(1/3))/3)*(2*a - 3))/(6*(a^2 - 3*a + 1))))/(9*(a^2 - 3*a + 1)))/(4*(a^2 - 3*a + 1))) - ((3*a^2 - 2)/(2*(a^3 - 2*a + 1)^(1/2)) - (2*a - 3)/(3*(a^2 - 3*a + 1)^(2/3)))*(a - x) - (a^2 - 3*a + 1)^(1/3) - (((2*a - 3)/(9*(a^2 - 3*a + 1)^(2/3)) - (5*((2*(2*a - 3)^2)/(9*(a^2 - 3*a + 1)^(2/3)) - (2*(a^2 - 3*a + 1)^(1/3))/3)*(2*a - 3))/(6*(a^2 - 3*a + 1)))/(3*(a^2 - 3*a + 1)) + ((3*(a^3 - 2*a + 1)^(1/2))/2 - (3*(3*a^2 - 2)*(3*a*(a^3 - 2*a + 1)^(1/2) - (3*a^2 - 2)^2/(4*(a^3 - 2*a + 1)^(1/2))))/(4*(a^3 - 2*a + 1)))/(3*(a^3 - 2*a + 1)))*(a - x)^3 + (a - x)^2*((3*a*(a^3 - 2*a + 1)^(1/2) - (3*a^2 - 2)^2/(4*(a^3 - 2*a + 1)^(1/2)))/(2*(a^3 - 2*a + 1)) + ((2*(2*a - 3)^2)/(9*(a^2 - 3*a + 1)^(2/3)) - (2*(a^2 - 3*a + 1)^(1/3))/3)/(2*(a^2 - 3*a + 1)))

xey9.计算下列函数的倒数：z?2，求zx’, zy’。

y>> syms x y

>> z=sym('x*exp(y)/(y^2)'); >> diff(z,x)

ans =

exp(y)/y^2

>> diff(z,y)

ans =

(x*exp(y))/y^2 - (2*x*exp(y))/y^3

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