微积分基础(python)

1 函数与极限

# 导入sympy库
from sympy import *
# 将x符号化
x = Symbol("x")
x

$x$

# 利用sympy中solve函数求解方程
X = solve(x**2-10*x+21,x)
X
print("原方程的解为：",X)

原方程的解为： [3, 7]

# 定义集合
A = set("12345")
B = set("123")

print("集合AB的并为：",A | B)
print("集合AB的交为：",A & B)
print("集合AB的差为：",A - B)

集合AB的并为： {'4', '2', '1', '5', '3'}
集合AB的交为： {'1', '3', '2'}
集合AB的差为： {'4', '5'}

# 自变量趋近无穷
n = Symbol("n")
s = n**2/(n**2+1)
result = limit(s,n,oo)
print("数列极限为：",result)

数列极限为： 1

# 自变量趋近有限值
x = Symbol("x")
s = (2-5*x**2)/(2*x+1)
print("函数极限为:",limit(s,x,-1/2))

函数极限为: oo

# 自变量趋近无穷大
x = Symbol("x")
s = (x+x**3)/(6*x**3)
print("函数极限为：",limit(s,x,oo))

函数极限为： 1/6

2 求导与微分

# 导入sympy库
from sympy import *

# 常数导数
x = Symbol("x")
C = 2
y = C
diff(y,x)

$0$

# 幂函数导数
x = Symbol("x")
mu = Symbol("mu")
y = x**mu
diff(y,x)

$\frac{\mu x^{\mu }}{x}$

# 指数函数求导
a = Symbol("a")
x = Symbol("x")
y = a**x
diff(y,x)

$a^x\log \left(a\right)$

# 对数函数求导
a = Symbol("a")
x = Symbol("x")
y = log(x,a)
diff(y,x)

$\frac{1}{x\log \left(a\right)}$

# 正弦求导
x = Symbol("x")
y = sin(x)
diff(y,x)

$\cos \left(x\right)$

# 反正弦函数求导
x = Symbol("x")
y = asin(x)
diff(y,x)

$\frac{1}{\sqrt{1-x^2}}$

# 求导四则运算
x = Symbol("x")
u = log(x,a)
v = x**2+1
y = u+v
diff(y,x)

$2x+\frac{1}{x\log \left(a\right)}$

y = u - v
diff(y,x)

$-2x+\frac{1}{x\log \left(a\right)}$

y = u*v
diff(y,x)

$\frac{2x\log \left(x\right)}{\log \left(a\right)}+\frac{x^2+1}{x\log \left(a\right)}$

y = u/v
diff(y,x)

$-\frac{2x\log \left(x\right)}{{\left(x^2+1\right)}^2\log \left(a\right)}+\frac{1}{x\left(x^2+1\right)\log \left(a\right)}$

# 复合函数求导
x = Symbol("x")
u = Symbol("u")
u = x**2
y =sin(u)
diff(y,x)

$2x\cos \left(x^2\right)$

# 链式求导
x = Symbol("x")
u = Symbol("u")
v = Symbol("v")
v = sin(x)**2
u = tan(v)**2
y = log(u)**2
diff(y,x)

$\frac{8\left({\tan }^2\left({\sin }^2\left(x\right)\right)+1\right)\log \left({\tan }^2\left({\sin }^2\left(x\right)\right)\right)\sin \left(x\right)\cos \left(x\right)}{\tan \left({\sin }^2\left(x\right)\right)}$

# 二阶求导
diff(y,x,2)

$\left({\tan }^2\left(\sin \left(x\right)\right)+1\right)\left(-\frac{\left({\tan }^2\left(\sin \left(x\right)\right)+1\right){\cos }^2\left(x\right)}{{\tan }^2\left(\sin \left(x\right)\right)}-\frac{\sin \left(x\right)}{\tan \left(\sin \left(x\right)\right)}+2{\cos }^2\left(x\right)\right)$

# 计算函数拐点
from sympy import *
x = Symbol("x")
y = 2*x**3-12*x**2+18*x-2
# 一阶导数
df1 = diff(y,x)
df1

$6x^2-24x+18$

# 二阶导数
df2 = diff(y,x,2)
df2

$12\left(x-2\right)$

print("二阶导数取值为0的点为",solve(df2))
print("拐点值为",y.subs(x,2))

二阶导数取值为0的点为 [2]
拐点值为 2

# 第一充分条件求极值点
from sympy import *
x = Symbol("x")
y = (x+3)**2*(x-1)**3
df = diff(y,x)
print("函数驻点为：",solve(df,x))

函数驻点为： [-3, -7/5, 1]

print("函数极值为",y.subs(x,-3),y.subs(x,-7/5),y.subs(x,1))

函数极值为 0 -35.3894400000000 0

# 第二充分条件求极值点
from sympy import *
y = 2*x**3-6*x**2+7
df = diff(y,x)
print("函数极值点为",solve(df,x))

函数极值点为 [0, 2]

df2 = diff(y,x,2)
print("二阶导数驻点的值为：",df2.subs(x,0),df2.subs(x,2))

二阶导数驻点的值为： -12 12

print("函数的极值为",y.subs(x,0),y.subs(x,2))

函数的极值为 7 -1

3 不定积分

x = Symbol("x")
f = cos(x)
integrate(f,x)

$\sin \left(x\right)$

x = Symbol("x")
f = 1/(1+x**2)
integrate(f,x)

$\mathrm{atan}\left(x\right)$

x = Symbol("x")
f = exp(x)*sin(x)
integrate(f,x)

$\frac{e^x\sin \left(x\right)}{2}-\frac{e^x\cos \left(x\right)}{2}$

4 定积分

x = Symbol("x")
a = Symbol("a")
b = Symbol("b")
y = sin(a*x)*cos(b*x)
integrate(y,(x,a,b))

$\{\begin{array}{ll}0& \text{for}\left(a=0\wedge b=0\right)\vee \left(a=0\wedge a=b\wedge b=0\right)\vee \left(a=0\wedge a=-b\wedge b=0\right)\vee \left(a=0\wedge a=-b\wedge a=b\wedge b=0\right)\\ \frac{{\cos }^2\left(b^2\right)}{2b}-\frac{{\cos }^2\left(ab\right)}{2b}& \text{for}\left(a=0\wedge a=-b\right)\vee \left(a=-b\wedge a=b\right)\vee \left(a=-b\wedge b=0\right)\vee \left(a=0\wedge a=-b\wedge a=b\right)\vee \left(a=-b\wedge a=b\wedge b=0\right)\vee a=-b\\ -\frac{{\cos }^2\left(b^2\right)}{2b}+\frac{{\cos }^2\left(ab\right)}{2b}& \text{for}\left(a=0\wedge a=b\right)\vee \left(a=b\wedge b=0\right)\vee a=b\\ \frac{a\cos \left(a^2\right)\cos \left(ab\right)}{a^2-b^2}-\frac{a\cos \left(b^2\right)\cos \left(ab\right)}{a^2-b^2}+\frac{b\sin \left(a^2\right)\sin \left(ab\right)}{a^2-b^2}-\frac{b\sin \left(b^2\right)\sin \left(ab\right)}{a^2-b^2}& \text{otherwise}\end{array}$

x = Symbol("x")
f = sin(x)
integrate(f,(x,0,pi))

$2$

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