Single Variable Calculus, Chapter 2, 2.1, Section 2.1, Problem 5

Suppose that a ball is thrown with a velocity of $40$ ft/s, its height in feet $t$ seconds later is given by $y= 40t-16t^2$

a. Find the average velocity for the time period beginning when $t=2$ and lasting




(i) 0.5 second
(ii) 0.1 second
(iii) 0.05 second
(iv) 0.01 second


$
\begin{equation}
\begin{aligned}


\begin{array}{|c|c|c|c|c|c|}
\hline\\
& t_{1_{(s)}} & t_{2_{(s)}} & d_{1_{(ft)}} = 40t_1 - 16t_1^2 & d_{2_{(ft)}} = 40t_2 - 16t_2^2 & V_{ave_{\left(\frac{ft}{s}\right)}} = \frac{d_2 - d_1}{t_2 - t_1} \\
\hline\\
(i) & 2 & 2.5 & 16 & 0 & -32 \\
\hline\\
(ii) & 2 & 2.10 & 16 & 13.44 & -25.6 \\
\hline\\
(iii) & 2 & 2.05 & 16 & 14.76 & -24.8 \\
\hline\\
(iv) & 2 & 2.01 & 16 & 15.7584 & -24.16\\
\hline
\end{array}

\end{aligned}
\end{equation}
$





b. Estimate the instantaneous velocity when $t=2$

Based on the table, the instantaneous velocity when $t = 2$ is approximately equal to $-24 \frac{ft}{s}$