$3\mathrm{.}4$ PRETEND you are trying to computationally solve for the root of the Optimal Thrust equation $f(T)=$

$0$ using the secant method starting with the initial "bracket" $T_1=0\mathrm{.}5,T_2=40.$

$($a$)$ Calculate the values of tol ${?}_x$ and tol ${?}_f$ you would use for the two convergence criteria,

$T{T}_{12}=T_{13}=16\mathrm{.}407T{T}_{13}|f(x_k)|,if$ you were $to$ write your own secant method $in$ MATLAB.

Now $(pretend)$ you've completed $13$ iterations for $T,$ and the last two iterations have values $T_{12}=$

$16\mathrm{.}410$ and $T_{13}=16\mathrm{.}407.$ For the "exact solution" use the value you got for $Tin$ problem $3\mathrm{.}3$ above

$to$ answer the following $(and$ SHOW YOUR WORK!!$)$:

$(b)$ What $is$ the $(just$ plain old$)$ error $in$ the last iteration, $T_{13}?$

$(c)$ What $is$ the "proxy" for the error for the last iteration?

$(d)$ What $is$ the "residual" error for the last iteration?

$(e)$ Based $on$ all your values above, explain whether the process has "converged" yet $or$ not.$|x_k-x_{k-1}|$ and $|f(x_k)|,if$ you were $to$ write your own secant method $in$ MATLAB.

Now $(pretend)$ you've completed $13$ iterations for $T,$ and the last two iterations have values $T_{12}=$

$16\mathrm{.}410$ and $T_{13}=16\mathrm{.}407.$ For the "exact solution" use the value you got for $Tin$ problem $3\mathrm{.}3$ above

$to$ answer the following $(and$ SHOW YOUR WORK!!$)$:

$(b)$ What $is$ the $(just$ plain old$)$ error $in$ the last iteration, $T_{13}?$

$(c)$ What $is$ the "proxy" for the error for the last iteration?

$(d)$ What $is$ the "residual" error for the last iteration?

$(e)$ Based $on$ all your values above, explain whether the process has "converged" yet $or$ not.