Single Variable Calculus, Chapter 2, 2.1, Section 2.1, Problem 8

Given the equation of displacement $s= 2 \sin \pi t + 3 \cos \pi t$ in centimeters of a particle moving back and forth along a straight line. Where $t$ is measured in seconds.

a. Find the average velocity over the given time intervals:

(i) [1,2]
(ii) [1,1.1]
(iii) [1,1.01]
(iv) [1,1.001]


$
\begin{equation}
\begin{aligned}

\begin{array}{|c|c|c|c|c|c|}
\hline\\
& t_{1_{(s)}} & t_{2_{(s)}} & S_{1_{(cm)}} = 2\sin \pi t_1 + 3 \cos \pi t_1 & S_{2_{(cm)}} = 2 \sin \pi t_2 + 3 \cos \pi t_2 & V_{ave_{(\frac{m}{s})}} = \frac{S_2 - S_1}{t_2 - t_1} \\
\hline\\
\text{(i) } & 1 & 2 & -3 & 3 & 6 \\
\hline\\
\text{(ii) }& 1 & 1.1 & -3 & -3.4712 & -4.712 \\
\hline\\
\text{(iii) }& 1 & 1.01 & -3 & -3.0613 & -6.13 \\
\hline\\
\text{(iv) }& 1 & 1.001 & -3 & -3.0063 & -6.3\\
\hline
\end{array}


\end{aligned}
\end{equation}
$


b. Estimate the instantaneous velocity when $t=1$

Referring to the table, the instantaneous velocity at $t = 1$ is approximately equal to -6.