Chapter 2.4, Problem 23E

To determine

To find: The instantaneous rate of change of position function $y=3t-7$ in feet at $t=1$ seconds.

Answer to Problem 23E

The instantaneous rate of change of position function $y=3t-7$ in feet at $t=1\sec$ is equal to $3\text{ft}/\sec$.

Explanation of Solution

Given information:

The position function is $y=3t-7$ and the time is $t=1$ 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 $y=3t-7$ at $t=1$ is equal to $\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{h}$.

Substitute $1+h$ for $x$ in the function.

  $\begin{array}{c}f\left(t\right)=3t-7\\ f\left(1+h\right)=3\left(1+h\right)-7\\ f\left(1+h\right)=3+3h-7\\ f\left(1+h\right)=3h-4\end{array}$

Substitute $1$ for $x$ in the function.

  $\begin{array}{c}f\left(t\right)=3t-7\\ f\left(1\right)=3\left(1\right)-7\\ f\left(1\right)=3-7\\ f\left(1\right)=-4\end{array}$

Substitute $3h-4$ for $f\left(1+h\right)$ and $-4$ for $f\left(1\right)$ in the formula for slope.

  $\begin{array}{c}\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{h}=\underset{h\to 0}{\lim }\frac{3h-4-\left(-4\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{3h}{h}\\ =\underset{h\to 0}{\lim }\left(3\right)\\ =3\end{array}$

Therefore, the instantaneous rate of change of position function $y=3t-7$ in feet at $t=1\sec$ is equal to $3\text{ft}/\sec$.