Task $3$: True and Approximate Relative Error $(15$ pts$)$

Solve problem $3\mathrm{.}7$ from the Chapra and Canale text $($below$)$ and follow the supplemental instructions

below. To clarify what the problem is asking, you are to compute the true percent relative error after

every iteration $($there should be $20$ of these error values$)$ and the approximate percent relative error

between successive approximations $($so there should be $19$ of these$).$

$3\mathrm{.}7$ Evaluate $e^{-5}$ using two approaches,

$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{3!}+$cdots

and

$e^{-x}=\frac{1}{e^x}=\frac{1}{1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\text{c}\text{d}\text{o}\text{t}\text{s}}$

and compare with the true value of $6\mathrm{.}737947{10}^{-3}.$ Use $20$ terms to evaluate each series and

compute true and approximate relative errors as terms are added.

You will:

Create a Python file called A$1\_$task$3.$py that:

a$.$ Defines a Python function $($or two$)$ to approximate $e^{-x}$ using both approaches. Your

function$($s$)$ should have at least two inputs: the value of $x($which will equal $5$ in this case$)$

and the number of terms to use, $n.$ In each of the approaches, $\text{'}1\text{'}$ counts as the first term.

b$.$ Uses the function$($s$)$ in part a$.$ to solve the textbook problem.

c$.$ Graphs the results $($approximation value, true percent relative error, and approximate

percent relative error $(y-$axis$)$ as functions of the number of terms $(x$ axis$)$ used in the

approximation$).$ Plots need to have sufficient resolution on the axes.