Q$1)$ In this question, you will approximate the function $f(x)=e^{4x}$ using Taylor's approximation

in Python. Follow these steps:

Write the fourth order Taylor's approximation of $f(x)$ around $x_i=0.$ Do this by hand.

In Python, create an array named $x$ containing $50$ numbers evenly distributed between $0$

and $0\mathrm{.}5.$

For each value in $x,$ plot $f(x)$ and its Taylor approximations at the $0$ th$,1$ st$,2$nd$,3$rd$,$ and

$4$th orders, all around $x_i=0.$ See the hint at the end of the question. Include this plot in

your report.

Discuss how the accuracy of the approximation changes as you increase the order of the

Taylor approximations.

For each point in $x,$ plot the percentage of the absolute value of relative error for each

Taylor approximation order $(0$th to $4$th$)$ of $f(x)$ around $x_i=0.$ Add this plot to your

report.

Describe how the error changes as the value of $x$ increases.

Include the Python code used for these steps in your report.

Hint:

You can compute and plot $e^{4x}$ using the following commands:

from math import factorial

import matplotlib.pyplot as plt

import numpy as $np$

# Create an array X with $50$ numbers evenly distributed between $$ and $0\mathrm{.}5$

$x=np.$linspace$(0,0\mathrm{.}5,50)$

$Y=np*exp(4^{**}x)$ # Function $f(x)=e^{{?}^{{?}^{?}}}(4x)$

plt$.$figure$()$

plt$.$plot $($X$,$ Y$,$ label$="$f$($x$)=e^{{?}^{{?}^{?}}}(4x),$ color$=$"black"$)$

plt$.$xlabel$("$x$")$

plt$.$ylabel$("$f$($x$)")$

plt$.$ legend$()$

plt$.$grid$($True$)$

plt$.$show$()$Q$1)$ In this question, you will approximate the function $f(x)=e^{4x}$ using Taylor's approximation

in Python. Follow these steps:

Write the fourth order Taylor's approximation of $f(x)$ around $x_i=0.$ Do this by hand.

In Python, create an array named $x$ containing $50$ numbers evenly distributed between $0$

and $0\mathrm{.}5.$

For each value in $x,$ plot $f(x)$ and its Taylor approximations at the $0$ th$,1$ st$,2$nd$,3$rd$,$ and

$4$th orders, all around $x_i=0.$ See the hint at the end of the question. Include this plot in

your report.

Discuss how the accuracy of the approximation changes as you increase the order of the

Taylor approximations.

For each point in $x,$ plot the percentage of the absolute value of relative error for each

Taylor approximation order $(0$th to $4$th$)$ of $f(x)$ around $x_i=0.$ Add this plot to your

report.

Describe how the error changes as the value of $x$ increases.

Include the Python code used for these steps in your report.

Hint:

You can compute and plot $e^{4x}$ using the following commands:

from math import factorial

import matplotlib.pyplot as plt

import numpy as np

# Create an array X with $50$ numbers evenly distributed between $$ and $0\mathrm{.}5$

$x=np.$linspace$(0,0\mathrm{.}5,50)$

$Y=np*exp(4^{**}x)$ # Function $f(x)=e^{{?}^{{?}^{?}}}(4x)$

plt$.$figure$()$

plt$.$plot $($X$,$ Y$,$ label$="$f$($x$)=e^{{?}^{{?}^{?}}}(4x),$ color$=$"black"$)$

plt$.$xlabel$("$x$")$

plt$.$ylabel$("$f$($x$)")$

plt$.$legend$()$

plt$.$grid$($True$)$

plt$.$show $()$

You can compute $\frac{(4x{)}^2}{2!}$ using the following command:

$Y_{2}nd=Y_{1}st+(4**x)****\frac{2}{f}$actorial$(2)$

Calculate each term in Taylor's approximation using similar commands. Then, sum up the

terms to compute the Taylor's approximation.