summary.ipynb

Kernel: Python 2 (system-wide)

#MrG 2016.0624 Pythonic Calculus Executive Summary!
from __future__ import division
from math import sqrt,pi,sin,cos
#1) define a function, ie f(x)=x**2-2
def f(x):
    return x**2-2
print "f(sqrt(2)) = ", f(sqrt(2))

f(sqrt(2)) =  4.4408920985e-16

#2) find the numerical derivative as a limit
def diff(x,h):
    return (f(x+h)-f(x))/h

print 
print "f'(2) = "
for x in range(5):
    print diff(2,10**-x)

f'(2) = 
5.0
4.1
4.01
4.001
4.00010000001

#3) find roots using newton's method as a limit
def root(g):
    return g-f(g)/diff(g,10**-6)

print 
print "the closest root of f(x)==0 near x=1 is:"
g=1
for x in range(5):
    print g
    g=root(g)

the closest root of f(x)==0 near x=1 is:
1
1.49999975002
1.41666668056
1.41421568716
1.41421356238

#4) find the definite integral as a limit
print
print "the definite integral of f(x) from 0 to pi ="
a=0
b=pi
for x in range(5):
    n=10**x
    h=(b-a)/n
    l=sum([f(a+i*h)*h for i in range(n)])
    r=l-f(a)*h+f(b)*h
    print (l+r)/2

the definite integral of f(x) from 0 to pi =
9.21995303297
4.10391738072
4.0527570242
4.05224542063
4.0522403046

#5) plots using numpy and matplotlib
#5) f(x)=x**2-2 ==> f(1)==-1
#5) f'(x)=2*x ==> f'(1)==2
#5) tangent line to f(x) at x=1 is:
#5) y-y1==m*(x-x1) ==> y-f(x1)==f'(x1)*(x-x1)
#5) y+1==2*(x-1) ==> y+1==2*x-2 ==> y=2*x-3
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2, 8)
y1=x**2-2
y2=2*x-3
plt.plot(x, y1)
plt.plot(x, y2)
plt.xlim((0, 2))
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')

<matplotlib.text.Text at 0x7fb2b4687a10>

#6) what about parametric plots and polar plots?
from numpy import *
import matplotlib.pyplot as plt
t = arange(0,2*pi,0.1)
x = (2+3*sin(t))*cos(t) #x(theta)=r(theta)*cos(theta)
y = (2+3*sin(t))*sin(t) #y(theta)=r(theta)*sin(theta)
ll = plt.plot(x,y)
plt.show()