matlab习题 —— 符号运算相关练习

一、题目

1. 计算下列极限

• (1)

limlimits_{x to 0}frac{sqrt{1 x}-sqrt{1-x}}{sqrt[3]{1 x}-sqrt[3]{1-x}}

• (2)

limlimits_{x to 0}(frac{3x 2}{3x-1})^{2x-1}

• (3)

limlimits_{x to 0}(frac{1}{x^2}-frac{1}{sin^2x})

• (4)

limlimits_{x to 0}(frac{pi}{2}-arctan{x})^{frac{1}{ln{x}}}

2. 计算下列导数

• (1) 求

y^{'}

,

y=(sqrt{x} 1)arctan{x}

• (2) 求

y^{''}

，

y=frac{1 x^2}{sin{x} cos{x}}

• (3) 求

y^{'}

,

y=sqrt{x sqrt{x sqrt{x}}}

• (4)

left{ begin{aligned} x&=a(cos{t} tsin{t})\ y&=a(sin{t}-tcos{t}) end{aligned} right .

3. 计算下列定积分

• (1)

int_{0}^{frac{pi}{2}}frac{cos{x}}{1 sin^2x}dx

• (2)

int_{0}^{a}x^2sqrt{frac{a-x}{a x}}dx

• (3)

int_{0}^{1}frac{dx}{(x^2-x 1)^{frac{3}{2}}}

4. 计算下列级数的和

• (1)

sum_{n=1}^{infty}{frac{1}{n^2}}

• (2)

sum_{n=1}^{infty}{frac{(-1)^{n-1}}{n}}

• (3)

sum_{n=1}^{infty}{frac{x^{2n-1}}{2n-1}}

二、解答

题一，计算下列极限

①

代码语言：javascript复制

>> syms x
>> y = ((1 x)^(1/2)-(1-x)^(1/2))/((1 x)^(1/3)-(1-x)^(1/3))
 
y =
 
((x   1)^(1/2) - (1 - x)^(1/2))/((x   1)^(1/3) - (1 - x)^(1/3))
 
>> limit(y,x,0)
 
ans =
 
3/2

②

代码语言：javascript复制

>> syms x
>> y = ((3*x 2)/(3*x-1))^(2*x-1)
 
y =
 
((3*x   2)/(3*x - 1))^(2*x - 1)
 
>> limit(y,x,0)
 
ans =
 
-1/2

③

代码语言：javascript复制

>> syms x
>> y = ((1/x^2)-1/(sin(x)^2))
 
y =
 
1/x^2 - 1/sin(x)^2
 
>> limit(y,x,0)
 
ans =
 
-1/3

④

代码语言：javascript复制

>> syms x
>> y = (pi/2 - atan(x))^(1/(log(x)))
 
y =
 
(pi/2 - atan(x))^(1/log(x))
 
>> limit(y,x,0)
 
ans =
 
1

题二，计算下列导数

①

代码语言：javascript复制

>> y = (x^(1/2) 1)*atan(x)
 
y =
 
atan(x)*(x^(1/2)   1)
 
>> diff(y,x)
 
ans =
 
atan(x)/(2*x^(1/2))   (x^(1/2)   1)/(x^2   1)

②

代码语言：javascript复制

>> y = (1 x^2)/(sin(x) cos(x))
 
y =
 
(x^2   1)/(cos(x)   sin(x))
 
>> y1 = diff(y,x)
 
y1 =
 
(2*x)/(cos(x)   sin(x)) - ((x^2   1)*(cos(x) - sin(x)))/(cos(x)   sin(x))^2
 
>> y2 = diff(y1,x)
 
y2 =
 
(x^2   1)/(cos(x)   sin(x))   2/(cos(x)   sin(x))   (2*(x^2   1)*(cos(x) - sin(x))^2)/(cos(x)   sin(x))^3 - (4*x*(cos(x) - sin(x)))/(cos(x)   sin(x))^2

③

代码语言：javascript复制

>> y = (x (x (x)^(1/2))^(1/2))^(1/2)
 
y =
 
(x   (x   x^(1/2))^(1/2))^(1/2)
 
>> diff(y,x)
 
ans =
 
((1/(2*x^(1/2))   1)/(2*(x   x^(1/2))^(1/2))   1)/(2*(x   (x   x^(1/2))^(1/2))^(1/2))

④

代码语言：javascript复制

>> syms x y t a
>> x = a*(cos(t) t*sin(t))
 
x =
 
a*(cos(t)   t*sin(t))
 
>> y = a*(sin(t)-t*cos(t))
 
y =
 
a*(sin(t) - t*cos(t))

>> dx = diff(x,t)
 
dx =
 
a*t*cos(t)
 
>> dy = diff(y,t)
 
dy =
 
a*t*sin(t)
 
>> dy/dx
 
ans =
 
sin(t)/cos(t)

题三，计算下列定积分

①

代码语言：javascript复制

>> syms x y
>>  y = cos(x)/(1 (sin(x))^2)
 
y =
 
cos(x)/(sin(x)^2   1)
 
>> int(y,x,0,pi/2)
 
ans =
 
pi/4

②

代码语言：javascript复制

>> syms x y a
>> y = (x^2)*((a-x)/(a x))^(1/2)
 
y =
 
x^2*((a - x)/(a   x))^(1/2)
 
>> int(0,a)
 
ans =
 
0
 
>> int(y,x,0,a)
 
ans =
 
(a^3*(3*pi - 8))/12

③

代码语言：javascript复制

>> syms x y a
>> y = 1/((x^2-x 1)^(3/2))
 
y =
 
1/(x^2 - x   1)^(3/2)
 
>> int(y,x,0,1)
 
ans =
 
4/3

题四，计算下列级数的和

①

代码语言：javascript复制

>> syms x y n
>> y = 1/(n^2)
 
y =
 
1/n^2
 
>> symsum(y,n,1,Inf)
 
ans =
 
pi^2/6

②

代码语言：javascript复制

>> y = (-1)^(n-1)/n
 
y =
 
(-1)^(n - 1)/n
 
>> symsum(y,n,1,Inf)
 
ans =
 
log(2)

③

代码语言：javascript复制

>> y = (x^(2*n-1))/(2*n-1)
 
y =
 
x^(2*n - 1)/(2*n - 1)
 
>> symsum(y,n,1,Inf)
 
ans =
 
piecewise(abs(x) < 1, atanh(x))

matlab diff int limit sum

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