Content provided under a CC-BY license.  Copyright 2014 David I. Ketcheson.
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

x_values = (0,1,2,3,4)
y_values = (0,1,8,27,64)

plt.plot(x_values,y_values,'-o')

plt.plot(x_values,y_values,'-o')
plt.hold(True)
plt.plot(x_values[2:4],y_values[2:4],'-r',linewidth=5)

plt.plot(x_values,y_values,'-o')
plt.hold(True)
plt.plot(x_values[1:3],y_values[1:3],'-r',linewidth=5)

plt.plot(x_values,y_values,'-ok')
plt.hold(True)
plt.plot((x_values[1],x_values[3]),(y_values[1],y_values[3]),'-r',linewidth=5)

def f(x):
    return x**3

x = np.linspace(0,4)
tangent = 8 + 12*(x-2)
plt.plot(x,f(x),'k',linewidth=2)
plt.plot(x_values[2:4],y_values[2:4],'-g',linewidth=3)
plt.plot(x_values[1:3],y_values[1:3],'-b',linewidth=3)
plt.plot((x_values[1],x_values[3]),(y_values[1],y_values[3]),'-r',linewidth=3)
plt.plot(x,tangent,'--k',linewidth=3)
plt.legend(['f(x)','Forward','Backward','Centered','Tangent'],loc='best')
plt.axis((0.5,3.5,0,40))

x = 2.
df = 12.

for h in (0.1, 0.05, 0.025):
    forward = (f(x+h)-f(x))/h
    forward_error = forward - df
    print 'h: ', h, '  Error: ', forward_error