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有关网站建设文章,网站设计制作新报价图片,给手机开发网站吗,全屋设计培训哪里有?微积分基础(python) 文章目录 微积分基础(python)1 函数与极限2 求导与微分3 不定积分4 定积分 1 函数与极限

导入sympy库

from sympy import *

将x符号化

x Symbol(x) xx \displaystyle x x

利用sympy中solve函数求解方程

X solve(x**2-10*x21,x) X pri…微积分基础(python) 文章目录 微积分基础(python)1 函数与极限2 求导与微分3 不定积分4 定积分 1 函数与极限

导入sympy库

from sympy import *

将x符号化

x Symbol(x) xx \displaystyle x x

利用sympy中solve函数求解方程

X solve(x2-10*x21,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 n2/(n21) 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*x2)/(2*x1) 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 (xx3)/(6*x3) print(函数极限为,limit(s,x,oo))函数极限为 ¹⁄₆₂ 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 \displaystyle 0 0

幂函数导数

x Symbol(x) mu Symbol(mu) y x**mu diff(y,x)μ x μ x \displaystyle \frac{\mu x^{\mu}}{x} xμxμ​

指数函数求导

a Symbol(a) x Symbol(x) y a**x diff(y,x)a x log ⁡ ( a ) \displaystyle a^{x} \log{\left(a \right)} axlog(a)

对数函数求导

a Symbol(a) x Symbol(x) y log(x,a) diff(y,x)1 x log ⁡ ( a ) \displaystyle \frac{1}{x \log{\left(a \right)}} xlog(a)1​

正弦求导

x Symbol(x) y sin(x) diff(y,x)cos ⁡ ( x ) \displaystyle \cos{\left(x \right)} cos(x)

反正弦函数求导

x Symbol(x) y asin(x) diff(y,x)1 1 − x 2 \displaystyle \frac{1}{\sqrt{1 - x^{2}}} 1−x2 ​1​

求导四则运算

x Symbol(x) u log(x,a) v x**21 y uv diff(y,x)2 x 1 x log ⁡ ( a ) \displaystyle 2 x \frac{1}{x \log{\left(a \right)}} 2xxlog(a)1​ y u - v diff(y,x)− 2 x 1 x log ⁡ ( a ) \displaystyle - 2 x \frac{1}{x \log{\left(a \right)}} −2xxlog(a)1​ y u*v diff(y,x)2 x log ⁡ ( x ) log ⁡ ( a ) x 2 1 x log ⁡ ( a ) \displaystyle \frac{2 x \log{\left(x \right)}}{\log{\left(a \right)}} \frac{x^{2} 1}{x \log{\left(a \right)}} log(a)2xlog(x)​xlog(a)x21​ y u/v diff(y,x)− 2 x log ⁡ ( x ) ( x 2 1 ) 2 log ⁡ ( a ) 1 x ( x 2 1 ) log ⁡ ( a ) \displaystyle - \frac{2 x \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)}} −(x21)2log(a)2xlog(x)​x(x21)log(a)1​

复合函数求导

x Symbol(x) u Symbol(u) u x**2 y sin(u) diff(y,x)2 x cos ⁡ ( x 2 ) \displaystyle 2 x \cos{\left(x^{2} \right)} 2xcos(x2)

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)8 ( tan ⁡ 2 ( sin ⁡ 2 ( x ) ) 1 ) log ⁡ ( tan ⁡ 2 ( sin ⁡ 2 ( x ) ) ) sin ⁡ ( x ) cos ⁡ ( x ) tan ⁡ ( sin ⁡ 2 ( x ) ) \displaystyle \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)}} tan(sin2(x))8(tan2(sin2(x))1)log(tan2(sin2(x)))sin(x)cos(x)​

二阶求导

diff(y,x,2)( tan ⁡ 2 ( sin ⁡ ( x ) ) 1 ) ( − ( tan ⁡ 2 ( sin ⁡ ( x ) ) 1 ) cos ⁡ 2 ( x ) tan ⁡ 2 ( sin ⁡ ( x ) ) − sin ⁡ ( x ) tan ⁡ ( sin ⁡ ( x ) ) 2 cos ⁡ 2 ( x ) ) \displaystyle \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) (tan2(sin(x))1)(−tan2(sin(x))(tan2(sin(x))1)cos2(x)​−tan(sin(x))sin(x)​2cos2(x))

计算函数拐点

from sympy import * x Symbol(x) y 2*x3-12*x218*x-2

一阶导数

df1 diff(y,x) df16 x 2 − 24 x 18 \displaystyle 6 x^{2} - 24 x 18 6x2−24x18

二阶导数

df2 diff(y,x,2) df212 ( x − 2 ) \displaystyle 12 \left(x - 2\right) 12(x−2) 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 (x3)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*x3-6*x27 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 12print(函数的极值为,y.subs(x,0),y.subs(x,2))函数的极值为 7 -13 不定积分 x Symbol(x) f cos(x) integrate(f,x)sin ⁡ ( x ) \displaystyle \sin{\left(x \right)} sin(x) x Symbol(x) f 1/(1x**2) integrate(f,x)atan ⁡ ( x ) \displaystyle \operatorname{atan}{\left(x \right)} atan(x) x Symbol(x) f exp(x)*sin(x) integrate(f,x)e x sin ⁡ ( x ) 2 − e x cos ⁡ ( x ) 2 \displaystyle \frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2} 2exsin(x)​−2excos(x)​ 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)){ 0 for ( a 0 ∧ b 0 ) ∨ ( a 0 ∧ a b ∧ b 0 ) ∨ ( a 0 ∧ a − b ∧ b 0 ) ∨ ( a 0 ∧ a − b ∧ a b ∧ b 0 ) cos ⁡ 2 ( b 2 ) 2 b − cos ⁡ 2 ( a b ) 2 b for ( a 0 ∧ a − b ) ∨ ( a − b ∧ a b ) ∨ ( a − b ∧ b 0 ) ∨ ( a 0 ∧ a − b ∧ a b ) ∨ ( a − b ∧ a b ∧ b 0 ) ∨ a − b − cos ⁡ 2 ( b 2 ) 2 b cos ⁡ 2 ( a b ) 2 b for ( a 0 ∧ a b ) ∨ ( a b ∧ b 0 ) ∨ a b a cos ⁡ ( a 2 ) cos ⁡ ( a b ) a 2 − b 2 − a cos ⁡ ( b 2 ) cos ⁡ ( a b ) a 2 − b 2 b sin ⁡ ( a 2 ) sin ⁡ ( a b ) a 2 − b 2 − b sin ⁡ ( b 2 ) sin ⁡ ( a b ) a 2 − b 2 otherwise \displaystyle \begin{cases} 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)}}{2 b} - \frac{\cos^{2}{\left(a b \right)}}{2 b} \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)}}{2 b} \frac{\cos^{2}{\left(a b \right)}}{2 b} \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(a b \right)}}{a^{2} - b^{2}} - \frac{a \cos{\left(b^{2} \right)} \cos{\left(a b \right)}}{a^{2} - b^{2}} \frac{b \sin{\left(a^{2} \right)} \sin{\left(a b \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(b^{2} \right)} \sin{\left(a b \right)}}{a^{2} - b^{2}} \text{otherwise} \end{cases} ⎩ ⎨ ⎧​02bcos2(b2)​−2bcos2(ab)​−2bcos2(b2)​2bcos2(ab)​a2−b2acos(a2)cos(ab)​−a2−b2acos(b2)cos(ab)​a2−b2bsin(a2)sin(ab)​−a2−b2bsin(b2)sin(ab)​​for(a0∧b0)∨(a0∧ab∧b0)∨(a0∧a−b∧b0)∨(a0∧a−b∧ab∧b0)for(a0∧a−b)∨(a−b∧ab)∨(a−b∧b0)∨(a0∧a−b∧ab)∨(a−b∧ab∧b0)∨a−bfor(a0∧ab)∨(ab∧b0)∨abotherwise​ x Symbol(x) f sin(x) integrate(f,(x,0,pi))2 \displaystyle 2 2 -END-