import numpy as np

import matplotlib.pyplot as plt

$\%$matplotlib inline

# Factorial comes from 'math' package

from math import factorial

#

### Initialize section

#array of derivatives at $0$ compute by hand

deriv $=$ np$.$array$([1,0,-1,0,1,0,-1,0])$

# degree of Taylor polynomial

N$=$len$($deriv$)-1$

# Max abs$.$ value of N$+1$ deriv on interval $($Compute using calculus$)$

# $($used in error formula: error abs$($h$^($N$+1)/($N$+1))*$ max$($abs$($N$+1$th deriv.$))$

maxAbsNplus$1$deriv$=1$

#array with h values

h $=$ np$.$arange$(-3,3\mathrm{.}05,0\mathrm{.}01)$

# Give the expansion point for the Taylor series

expansion$=0$

# Create an array to contain Taylor series

taylor$=$np$.$zeros$($len$($h$))$

#

### Calculation section

#calculate Taylor series for all h at once using a loop

for n in range$($N$+1)$:

taylor $=$ taylor $+($h$**$n$)/$factorial$($n$)*$deriv$[$n$]$

#actual function values

actualFn $=$ np$.$cos$($expansion$+$h$)$

#error actual and taylor series

actualAbsError $=$ abs$($taylor$-$actualFn$)$

#theoretical error$=|$h$^($deg$+1)/($deg$+1)!*$max$|$n$+1$ deriv$||$

theoAbsError $=$ abs$($h$**($N$+1)/$factorial$($N$+1)*$maxAbsNplus$1$deriv$)$

#Compute actual x values $($displaced by expansion point$)$

xActual $=$ expansion$+$h

#

### Output section

#plot figures for cos and errors using

# Object$-$oriented style plotting

# Two subplots in a row

fig, axes $=$ plt$.$subplots$($nrows$=1,$ ncols$=2,$ figsize$=(8,4))$

# objects for the left subplot $($attached to axes #$0)$

axes$[0].$plot$($xActual$,$ taylor, $"$r$",$label$=$"Taylor series"$)$

axes$[0].$plot$($xActual$,$ actualFn, $"$b$--",$ label$=$"Cos$($x$)")$

axes$[0].$legend$($loc$=2)$ #legend in upper left

axes$[0].$set$\_$title$("$Taylor Series for cos$($x$)")$

# objects for the right subplot $($attached to axes #$1)$

axes$[1].$plot$($xActual$,$ actualAbsError, label$=$"Actual Absolute Error"$)$

axes$[1].$plot$($xActual$,$ theoAbsError, label$=$"Theoretical Absolute Error"$)$

axes$[1].$legend$($loc$=2)$ #legend in upper left

axes$[1].$set$\_$title$("$Actual and Theoretical Error"$)$

This code up here need to be change into codes that will match the exercises down below

Exercises

$1.$ Copy your code into another cell, and modify to plot the $10$ th order Taylor series approximation and theoretical error of $f(x)=cos(3x)$ with expansion point $x=0\mathrm{.}5,$

on the interval $-0\mathrm{.}5,2\mathrm{.}5$ with h$-$increment $0\mathrm{.}1.$ NOTE $($i$)\text{'}10$th order' means that you use $10$ derivatives. $($ii$)$ Since the expansion point is not zero, in your graph you will

need to plot $x+h$ on the $x-$axis versus the Taylor series value on the $y$ axis. $($iii$)$ Make sure to choose an appropriate value for when estimating the true error. will be

different than the warm$-$up$.$

$2.$ Copy your code into another cell, and modify it to do the $6$ th order Taylor series approximation and theoretical error for $f(x)=ln(x)$ with expansion point $x=1,$ on

the interval $0\mathrm{.}5,2$ with h$-$increment $0\mathrm{.}05.$ Note you must choose the point $c$ in the interval $0\mathrm{.}5,2$ which makes $f^{n+1}(c)$ as large as possible. You cannot assume that c

is equal to the expansion point.

$3.$ Copy your code into another cell, and compute the $12$ th order MacLaurin series approximation and theoretical error for $f(x)=e^{2x}-e^{-2x}$ on the interval $-1,1$ with

increment $0\mathrm{.}05.$ Use the expansion point $x=0.$

$4.$Create a new code cell. Consider the function $f(x)=e^{2x}.($a$)$ Write a python function that can output the nth term of a MacLaurin series with input $n$ and $h.($b$)$Write

code that estimates $e^2$ with minimum $6$ decimal points of accuracy by using the approximate relative error calcuation. $($c$)$Output your estimate, the true value $($using

pythons built in exponential function$),$ and the true error. Does your calculation have at least $6$ decimal points of accuracy?