So$,$ you create the following function $f(T),$ for which the root of $f(T)=0$ is the desired Thrust $T$ :

For all the following problems,

don't change this function!

Just leave $f(T)$ exactly as shown.

$3\mathrm{.}1$ Applying Three Methods BY$-$HAND

$6$ pts Approximate the Optimal Trust $T($in kN $)$ by iterating on the root of $f(T)$ using the three methods

below. You picked the initial brackets $/$ guess$($es$)$ below based only on knowledge that these kits

tend to sell for somewhere around $20$ kN $.$

Do two iterations for each method, accurate to at least $4$ decimals $($yes$,$ I know that's a fraction of a

penny, but go with it$)."$By$-$hand" still means you should use your calculator for all the hard

arithmetic $($multiplying and dividing$),$ but I want to see all your steps: write out the formulae you

are using and key values as you iterate so I can confirm you're doing the process correctly.

$($i$)$ Bisection: start with the bracket $a=0\mathrm{.}50,b=40$ and calculate the midpoints $T_1$ and $T_2,$

clearly justifying $($based on the signs of $f(T))$ how the a$\text{'}$s and$/$or b$\text{'}$s are changing

after both iterations.

$($ii$)$ Secant: start with $T_1=0\mathrm{.}5$ and $T_2=40.$ Calculate $T_3$ and $T_4.$

$($iii$)$ N$-$R: start with $T_1=20.$ Calculate $T_2$ and $T_3.$