$7$:$52$

CLASS TEST EXERCISE

A function is given by:

$f(t)=\frac{1}{t^3}$

Use zero$-$ through second$-$order Taylor series expansions to predict f $(2\mathrm{.}2)$by using the starting value at $t=2$ with step size $h=0\mathrm{.}2.$

Compute the remainder term $/$ truncation error for each approximation.

Show all working in $4$ decimal places.

A rectangular water tank is being filled, and the relationship between the height $h$ of water $($in m $)$ and the volume $($in $m^3)$ is modeled by:

$f(h)=h^3-6h^{2}5=0$

Engineers need to determine the height h at which the tank holds $5m^3$ of water.

$($a$)$

Using the bisection method, approximate the root of $f(h)$ between $h=0$ and $h=4$ up to three iterations, rounding answers to $4$ decimal places.

$($b$)$

Compute the percentage approximation error after the second and third iterations, to $4$ decimal places.

The displacement $s(t)$ of a moving object is modeled by:

$s(t)=5e^{-0\mathrm{.}4}tt-3$

The object crosses the reference point $($i$.$e$.,s(t)=0$ at some time t $).$

Tasks:

$($a$)$ Using the Newton$-$Raphson method, perform $3$ iterations to approximate the time t when the object is at the reference point.

Use an initial guess of $t=1\mathrm{.}0s$

Present your results to $4$ decimal places.

$($b$)$ After each iteration, compute the approximate error