Taylor Series Python Assignment

The $N$ th order Taylor series formula is:

$f(x+h)$~~$f(x)+\frac{h^1}{1!}{f}^{(1)}(x)+\frac{h^2}{2!}{f}^{(2)}(x)+$dots$+\frac{h^N}{N!}{f}^{(N)}(x),$

where $f^{(j)}(x)$ denotes the $j$ th derivative of the function $f$ at the point $x.$

The error in the formula will be less than the Taylor error bound, which is given by:

$R_N(x+h)=\frac{|h{|}^{N+1}}{(N+1)!}ma{x}_{yin|x,x+h}$

Warm$-$up exercise:

Complete the following code that computes and plots the $7$th order Taylor series approximation and Taylor error bound for $f(x)=cos(x)$ with expansion point

$x=0$ on the interval $-3,3$ with increment $0\mathrm{.}01.$

### Import section

# $3$ necessary statements for numerical plotting in Jupyter

import numpy as np

import matplotlib.pyplot as plt

$\%$matplotlib inline

# Factorial comes from 'math' package

from math import factorial

### Initialization 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 the remainder formula$)$

# @@@ The N$+1$ derivative will be proportional to cosine, and $|$cos$|$ is always

# @@@ less than $1$

maxAbsNplus$1$deriv$=1$

#array with h values

h $=$ np$.$arange$(-3,3\mathrm{.}005,0\mathrm{.}01)$

# 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 max 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

# 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, $\text{'}$r$\text{'},$label$=$'Taylor series'$)$

axes$[0].$plot$($xActual$,$ actualFn, $\text{'}$b$--\text{'},$ label$=$'cos$($x$)\text{'})$

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

axes$[0].$set$\_$title$(\text{'}$Taylor Series for cos$($x$)\text{'})$

# 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 Max Absolute Error'$)$

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

axes$[1].$set$\_$title$(\text{'}$Actual and Theoretical Max Errors'$)$Exercises

Copy your code into another cell, and modify to plot the $10$th order Taylor series approximation and theoretical max 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 $0\mathrm{.}1.$

NOTES

i$.\text{'}10$th order' means that you use up to the $10$th derivatives $($so the series has $11$ terms$).$

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. When calculating the theoretical max error, use the fact that the absolute value of the sine function is always less than $1($same is true for the cosine

function$).$ So everywhere that the sine function appears in your remainder expression, you may replace with $1.$

PLEASE HELP ME WITH EX $1$ AND USE PYTHON