import numpy
import matplotlib.pyplot as plt

def f(x):
    return numpy.sin(x)

def taylor_f_10(x):
    return numpy.polyval([1./362880., 0, -1./5040., 0, 1./120., 0, -1./6., 0, 1., 0], x)

def taylor_f_5(x):
    return numpy.polyval([1./120., 0, -1./6., 0, 1., 0], x)

def taylor_f_3(x):
    return numpy.polyval([-1./6., 0, 1., 0], x)

x = numpy.linspace(0, 2.0*numpy.pi, 1000)
plt.figure()
plt.plot(x, f(x),
         x, taylor_f_3(x),
         x, taylor_f_5(x),
         x, taylor_f_10(x)
         )
plt.axis([0, 2.0*numpy.pi, -1, 1])
plt.legend(('f(x)', '3', '5', '10'))

def g(x):
    return 1/(1+x*x)

def taylor_g_10(x):
    return numpy.polyval([0, 1, 0, -1, 0, 1, 0, -1, 0, 1], x)

def taylor_g_5(x):
    return numpy.polyval([0, 1, 0, -1, 0, 1], x)

def taylor_g_3(x):
    return numpy.polyval([0, -1, 0, 1], x)

x = numpy.linspace(-5, 5, 1000)
plt.figure()
plt.plot(x, g(x), 
         x, taylor_g_3(x),
         x, taylor_g_5(x),
         x, taylor_g_10(x))
plt.axis([-5, 5, 0, 1.5])
plt.legend(('g(x)', '3', '5', '10'))

def h(x):
    return x**-12 - x**-6

def taylor_h_10(x):
    return numpy.polyval([349713, -165958, 74295, -31032, 11914, -4116, 1239, -308, 57, -6, 0], x-1)

def taylor_h_5(x):
    return numpy.polyval([-4116, 1239, -308, 57, -6, 0], x-1)

def taylor_h_3(x):
    return numpy.polyval([-308, 57, -6, 0], x-1)

x = numpy.linspace(1, 5, 1000)
plt.figure()
plt.plot(x, h(x),
         x, taylor_h_3(x),
         x, taylor_h_5(x),
         x, taylor_h_10(x))
plt.axis([1, 2, -0.3, 0.1])
plt.legend(('h(x)', '3', '5', '10'))

plt.show()