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? This is the original code and it must be change to be able to be able to work with the exercises in different codes 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"$)$