$($This question should be completed by hand$)$ Consider an air bubble rising in a water column, where the

bubbles vertical position $($y$)$ varies with time $(t).$ Experimental measurements were made of the position of the

bubble at various instances in time. It is of interest to estimate the rising velocity of the bubble, $d\frac{y}{d}t$ at $t=t,$

based on the measured positions $y_3(t_t{t}_3),y_4(t=t_4),y_3(t=t_5)$ and $y_6(t=t_6),$ shown schematically on the sketch

below.

Given $y_3=1\mathrm{.}1[cm],y_4=1\mathrm{.}2[cm],y_5=1\mathrm{.}3[cm],y_6=1\mathrm{.}1[cm]$ estimate the following:

$($i$),\frac{dy}{dt}{|}_{\text{@}{t}_5}$ with the highest possible order of accuracy.

$($ii$)\frac{d^{2}y}{d{t}^2}{|}_{\text{@}{t}_5}$ with the highest possible order of accuracy.

$($iii$)\frac{d^{3}y}{d{t}^3}{|}_{\text{@}{t}_6}$ with the highest possible order of accuracy.

Based on your derivations, clearly identify, and explain the order of accuracy of your estimate..

This is a taylor serier expansion question with unequal spaced points.