This is the code Below we need to manipulate to taylor formula

# Build and plot the Taylor Approximations of log(1+h) import numpy as np import matplotlib.pyplot as plt import math

h = np.linspace(0, 2, 10) # 100 h points plt.plot(h, np.log(1+h),'k') # plot sin(h) vs. h

Tlog = h # build and plot the 1st term in the Taylor Series plt.plot(h, Tlog, 'r--')

Tlog = Tlog - h**3/math.factorial(3) # add on the next term in the Taylor Series and plot plt.plot(h, Tlog, 'b--')

Tlog = Tlog + h**5/math.factorial(5) # add on the next term in the Taylor Series and plot plt.plot(h, Tlog, 'g--')

Tlog = Tlog - h**7/math.factorial(7) # add on the next term in the Taylor Series and plot plt.plot(h, Tlog, 'c--')

plt.grid('on') plt.legend(['$y=\sin(h)$','$y=h$','$y=h-h^3/3!$','$y=h-h^3/3!+h^5/5!$', '$y=h-h^3/3!+h^5/5!-h^7/7!$'],loc=(1.04,0)) plt.xlabel('h') plt.ylabel('y') plt.ylim([-2,2]) plt.title('Exploring the Tayor Series of sine');

Develop loglog in a Taylor series at =1

log(1+)log(1)+log(1)+log(1)2/2!+log(1)3/3!+log(1)4/4!

by explicitly computing, showing all of your work, each of these numbers, log(1)log(1), log(1)log(1), log(1)log(1), log(1)log(1),

and log(1)log(1). Here is the how you should start

log(1)=0,because integral from 1 to 1 is 0

log()=1/,so, by substitution log(1)=1

log()=???so, by substitutionlog(1)=???

Plot the true function in black and the approximating Taylor polynomials in dashed colors (using our old Taylor cell) and achieve the plot below for log(1+)log(1+) but with a useful legend that identifies each curve.

Develop exp in a Taylor series at =0

exp()exp(0)+exp(0)+exp(0)2/2!+exp(0)3/3!+exp(0)4/4!

by explicitly computing, showing all of your work, each of these numbers, exp(0)exp(0), exp(0)exp(0), exp(0)exp(0), exp(0)exp(0), and exp(0)exp(0). Plot the true function in black and the approximating Taylor polynomials in dashed colors (using our old Taylor cell) and achieve the plot below for exp()exp() but with a useful legend that identifies each curve.

Develop 2^ in a Taylor series at =0

2^20+(2^)(0)+(2^)(0)2/2!+(2^)(0)3/3!+(2^)(0)4/4!

!

by explicitly computing, showing all of your work, each of these numbers, 2^0, (2^)(0), (2^)(0), (2^)(0), and (2^)(0). Plot the true function in black and the approximating Taylor polynomials in dashed colors (using our old Taylor cell) and achieve the plot below for 2 but with a useful legend that identifies each curve.