Coding Exercises CAN YOU PLEASE JUST SOLVE PART C AND D$?$ THANKS

Exercise $1.($Component Skills $7\mathrm{.}1-7\mathrm{.}3)$

A former TA of this class studied caterpillar life cycles. She encountered the following equation in

her research:

$6-3x(1+e^{3(1-x)})=0.$

Here, $x$ represents the population of caterpillars in thousands. By guess and check, one can see

that $x=1$ is a solution of $(1).$ There are two other nontrivial solutions $x_1$ and $x_2.$ We take $x_2>x_1$

without loss of generality.

a$.$ Use the MATLAB built$-$in function fzero with initial guess $0\mathrm{.}1$ to determine the value of $x_1.$

Assign A$1$ to the value of $x_1.$

b$.$ Use the MATLAB built$-$in function fzero with initial guess $1\mathrm{.}9$ to determine the value of $x_2.$

Assign A$2$ to the value of $x_2.$

c$.$ Copy the MATLAB function bisectionMethod from the weekly lecture notes. Paste this

function at the very bottom of your hw$7.$m script. Use this function to perform the bisec$-$

tion method on the function $f(x)$ given by the LHS of $(1).$ Take the initial interval to be

$0x0\mathrm{.}5.$ Set the stop criterion to ${10}^{-15}.$ Assign A$3$ to the number of iterations necessary

to satisfy the stop criterion. d$.$ Copy the MATLAB function newtonMethod from the weekly lecture notes. Modify the func$-$

tion so that the Cauchy error is used as the stopping criterion. Have your function return the

approximation of the root according to Newton$$s method as well as the number of iterations

necessary to satisfy the stop criterion.

Once your function has been modified appropriately, paste this function at the very bottom

of your hw$7.$m script. Use this function to perform the Newton$$s method on the function f $($x$)$

given by the LHS of $(1).$ Take the initial guess to be x$0=0\mathrm{.}1.$ Set the stop criterion to $1015.$

Assign A$4$ to the number of iterations necessary to satisfy the stop criterion.

Note: You will need to compute f $($x$)$ by hand and write this as a function handle to run

Newton$$s method in MATLAB.