Chapter 2.4, Problem 24E

To determine

To find: The instantaneous rate of change of position function $y=3t^2+2t$ in feet at $t=3$ seconds.

Answer to Problem 24E

The instantaneous rate of change of position function $y=3t^2+2t$ in feet at $t=3\sec$ is equal to $20\text{ft}/\sec$.

Explanation of Solution

Given information:

The position function is $y=3t^2+2t$ and the time is $t=3$ second.

Calculation:

The formula for the instantaneous rate of change of position with respect to time at $t$ seconds is equal to $\underset{h\to 0}{\lim }\frac{f\left(t+h\right)-f\left(t\right)}{h}$.

The formula for the instantaneous rate of change of $y=3t^2+2t$ at $t=3$ is equal to $\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}$.

The function is given as $f\left(t\right)=3t^2+2t$.

Substitute $3+h$ for $x$ in the function.

  $\begin{array}{c}f\left(t\right)=3t^2+2t\\ f\left(3+h\right)=3{\left(3+h\right)}^2+2\left(3+h\right)\\ f\left(3+h\right)=3\left(9+h^2+6h\right)+\left(6+2h\right)\\ f\left(3+h\right)=3h^2+20h+33\end{array}$

Substitute $3$ for $x$ in the function.

  $\begin{array}{c}f\left(t\right)=3t^2+2t\\ f\left(3\right)=3{\left(3\right)}^2+2\left(3\right)\\ f\left(3\right)=27+6\\ f\left(3\right)=33\end{array}$

Substitute $3h^2+20h+33$ for $f\left(3+h\right)$ and $33$ for $f\left(3\right)$ in the formula for slope.

  $\begin{array}{c}\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}=\underset{h\to 0}{\lim }\frac{3h^2+20h+33-33}{h}\\ =\underset{h\to 0}{\lim }\frac{h\left(3h+20\right)}{h}\\ =\underset{h\to 0}{\lim }\left(3h+20\right)\\ =20\end{array}$

Therefore, the instantaneous rate of change of position function $y=3t^2+2t$ in feet at $t=3\sec$ is equal to $20\text{ft}/\sec$.