Q$2$: In this question, you will use Python to perform numerical differentiation the function

$f(x)=e^{3x}.$ Follow these steps:

Create an array named $H$ containing $100$ evenly spaced values ranging from $0\mathrm{.}001$ to $0\mathrm{.}5.$

These values will serve as the step sizes for the numerical differentiation. See hint at the

end of the question.

For each step size in $H,$ calculate the forward difference approximation of $f(x)$ at $x_i=0.$

Then, determine the percentage of the absolute relative error for each step size. Plot this

percentage against the values in $H$ and include the plot in your report.

Repeat the process for the backward difference approximation of $f(x)$ at $x_i=0.$

Calculate the percentage of the absolute relative error for each step size in $H$ and plot

these percentages against $H.$ Include this plot in your report.

Do the same for the centred difference approximation of $f(x)$ at $x_i=0).$ Calculate the

percentage of the absolute relative error for each step size in $H$ and plot these

percentages against $H.$ This plot should also be included in your report.

Analyse and discuss what happens to the error as the step size decreases.

Compare the three methods and determine which approximation is the most accurate.

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

Hint:

import matplotlib.pyplot as plt

import numpy as $np$

def $f(x)$ :

return $np.$exp$(3**x)$

# Derivative of the function

def $df(x)$ :

return $3**np*exp(3**x)$

# Create an array H with $100$ numbers evenly distributed between $0\mathrm{.}001$ and $0\mathrm{.}5$

$H=np.$ inspace $(0\mathrm{.}001,0\mathrm{.}5,100)$

# Exact derivative at $x=0$

exact$\_$derivative $=df()$

# Forward difference approximation

forward$\_$diff $=\frac{f(H)-f()}{H}$

$...$

# Calculate the percentage of the absolute relative error

forward$\_$error $=$ np$.$abs$(($forward$\_$diff $-$ exact$\_$derivative