My code:

# $3$ key statements: numpy, pyplot, inline display

import numpy as np

import matplotlib.pyplot as plt

$\%$matplotlib inline

## Initialization

#Define the function that you're finding the root of

def fn$($x$)$:

return np$.$cos$($x$)$

# Desired error tolerance

tolerance $=1$E$-6$

# No$.$ iterations large enough to guarantee tolerance

maxIt $=100$

#guess for lower and upper bound on root

lower $=1$

upper $=2$

# Create array for error results

maxError $=$ np$.$zeros$($maxIt$+1)$

# Create array for actual error

actualError $=$ np$.$zeros$($maxIt$+1)$

#actual root for cos$($x$)$ between lower and upper $($theoretical$)$

actual $=$ np$.$pi$/2$

#first average to set up first iteration

avg$=($upper $+$ lower$)/2$

## Calculation

for i in range$($maxIt$+1)$:

maxError$[$i$]=$upper$-$avg

actualError$[$i$]=$abs$($avg$-$actual$)$

# If tolerance has been met, then exit loop

if maxError$[$itolerance:

break

## Narrow down location of root depending on signs

# First option: if avg is root, then exit loop $($done$)$

if fn$($avg$)==$actual:

break

# Second option: if fn changes sign on left, then move left

elif fn$($lower$)*$fn$($avg$)<mn>0</mn>$:

upper$=$avg

# Last resort: if fn changes sign on right, then move right

else:

lower$=$avg

avg$=($lower$+$upper$)/2$

# Done! Print result

print$("$Estimated root is$",$avg,"with tolerance",tolerance$)$

## Presentation in a figure

#plot max and actual error versus iteration number

plt$.$figure$(1)$

plt$.$plot$($np$.$arange$($i$+1),$ maxError$[$:i$+1],$ label$=$"Max Error"$)$

plt$.$plot$($np$.$arange$($i$+1),$ actualError$[$:i$+1],$ label$=$"Actual Error"$)$

# Change y scale to logarithmic

plt$.$yscale$(\text{'}$log$\text{'})$

# Add labels to axes

plt$.$xlabel$("$Iteration Number"$)$

plt$.$ylabel$("$Absolute Error"$)$

# Add legend and title.

plt$.$legend$($loc$=1)$

plt$.$title$("$Max and Actual Error for Bisection Method"$)$

plt$.$show$()$ Estimated root is $1\mathrm{.}570796012878418$ with tolerance $1$e$-86$

Modify your code to approximate the following

square root of $7$

the cube root of $37$

the fifth root of $300$

the tenth root of $39$

to within $0\mathrm{.}0000001.($You$\text{'}$ll probably need $3$ different loops$).$ Plot the theoretical and actual errors for all of these approximations on the same graph. What

does your graph show about the $($actual and theoretical$)$ rate of convergence for these three different cases? In particular, if the slopes for two different

methods are the same, what does that tell you about the relative accuracy of the two methods? $($Put your comments as a Markdown cell after your code.$)$

Consider the following system of equations:

$y-cos(x^2{e}^x)=2$

$2y+sin(x{e}^x)=4$

Use the bisection method to to find an approximate solution to the equaitons on the interval $(0,1).$ Set your level of tolerence to $0\mathrm{.}000001$

Questions $1$ and $2$ in python