%matplotlib inline

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

import sympy as sp

1/2 + 1/3

sp.Rational(1,2) + sp.Rational(1,3)

sp.sympify(1)/2 + sp.sympify(1)/3

from sympy import S

S(1)/2 + S(1)/3

sp.sqrt(2) * sp.sqrt(3)

π = sp.pi

π

sp.sin(π/3)

π.n(100)

'1215' in str(π.n(100000))

x = sp.Symbol('x')

x**2

x = sp.Symbol('🐶')

f = x**2

f

a = sp.Symbol('alpha')

sp.sqrt(a)

x, y, z = sp.symbols('x,y,z')

x**2 * y + 2*z

a, b, c = sp.symbols('alpha,beta,gamma')

a*b + c**2 + 3*a*b*c

from sympy.abc import x, y, z, t, a, b, c, alpha, beta, gamma

a*x + b*y + c*z + gamma*x**2

f = x**2

f

f.subs(x, 87)

g = x*y + 1

g.subs([(x, 94), (y, 87)])

sp.plot(f)

sp.plot(f, (x, -5, 5))

from sympy.plotting import PlotGrid

p1 = sp.plot(x**2,  show=False)
p2 = sp.plot(sp.exp(x), show=False)
p3 = sp.plot(sp.log(x), show=False)
p4 = sp.plot(sp.sin(x), show=False)

PlotGrid(2, 2, p1, p2, p3, p4)

sp.limit(x**2+1, x, 2)

f = (x**2+2*x-3)/(x-1)

f

sp.limit(f, x, 1)

f = sp.Piecewise((2*x+1, x<=2), (7-x, x<4), (x, True))

p1 = sp.plot(f, (x, 0, 3.99), show=False)
p2 = sp.plot(f, (x, 4, 6), show=False)
p1.extend(p2)
p1.show()

sp.limit(f, x, 5)

sp.limit(f, x, 4, '-')

sp.limit(f, x, 4, '+')

sp.limit(1/x, x, 0, '+')

sp.limit(1/x, x, 0, '-')

f = (4*x**3-5)/(x**3-2*x+9487)

f

sp.limit(f, x, sp.oo)

f = x**4 - 3*x**2 + 2*x - 5

f

sp.diff(f, x)

sp.diff(f, x, x)

sp.diff(f, x, 2)

sp.diff(sp.exp(x), x)

sp.diff(sp.log(x), x)

sp.diff(sp.sin(x), x)

sp.diff(sp.tan(x), x)

sp.diff(sp.sec(x), x)

sp.diff(sp.asin(x), x)

sp.diff(sp.acos(x), x)

sp.diff(sp.atan(x), x)

h = y**3 + x**3*y - 2*x*y**2 + 4*x**2 - 4

h

sp.idiff(h, y, x)

sp.idiff(h, y, x).subs([(x,1), (y,0)])

sp.plot_implicit(sp.Eq(h, 0))

f = x**4 - 8*x**2 + 1

f

f1 = sp.diff(f, x)

sp.solveset(f1)

f2 = sp.diff(f, x, x)

f2.subs(x, -2)

f2.subs(x,0)

f2.subs(x,2)

sp.is_decreasing(f, sp.Interval(-sp.oo, -2))

sp.is_increasing(f, sp.Interval(-2, 0))

sp.is_decreasing(f, sp.Interval(0, 2))

sp.is_increasing(f, sp.Interval(2, sp.oo))

f = x**3 + x

f

f1 = sp.diff(f, x)

f1

sp.solveset(f1, x)

sp.solveset(f1, x, domain=S.Reals)

from sympy.calculus import singularities

f = sp.Abs(x)

f

f1 = sp.diff(f, x)

singularities(f1, x)

f = 1/x + x

f1 = sp.diff(f, x)

singularities(f1, x)

f = sp.sqrt(31*x + 2)

f

sp.integrate(f, x)

f = x/(sp.sqrt(x**2+4))

f

sp.integrate(f, (x,0,2))

sp.integrate(f, (x,a,b))

sp.integrate(sp.sqrt(sp.log(x))/x, x)

sp.integrate(x*sp.exp(-x**2), (x, 0, 1))

f = t**3/(t**2+1)

f

sp.diff(sp.integrate(f, (t, a, x)),x)

sp.simplify(sp.diff(sp.integrate(f, (t, a, x)),x))

sp.integrate(x*sp.exp(x), x)

sp.integrate(sp.log(x), x)

sp.integrate(1/x**2, (x, 1, sp.oo))

sp.integrate(2*x*sp.exp(-x**2), (x, -sp.oo, sp.oo))

f = sp.exp(x)

sp.series(f, n=10)

f_taylor = sp.series(f, n=10).removeO()

f_taylor

e = f_taylor.subs(x,1)

e

e.n(100)

sp.exp(1).n(100)

from ipywidgets import interact

def taylor(n=10):
    f = sp.exp(x)
    egg = sp.series(f, n=n)
    f_taylor = egg.removeO()
    p1 = sp.plot(f, (x, -5, 5), line_color='blue', show=False)
    p2 = sp.plot(f_taylor, (x, -5, 5), line_color='red', show=False)
    p1.extend(p2)
    p1.show()

taylor(n=10)

interact(taylor, n=(1,50))

sp.tan(π/4)

f = sp.atan(x)

n = 100

f_taylor = sp.series(f, n=n).removeO()

pi = 4*f_taylor.subs(x, 1)

pi

pi.n(100)