Chapter 2.4, Problem 54E

To determine

To check: The reason for the same nature of rate of change of function $f$ over the interval $\left[3,3+h\right]$ and the difference quotient of $f$ at $a=3$.

Answer to Problem 54E

The function that represents the rate of change of function $f$ over the interval $\left[3,3+h\right]$ is same as the function of difference quotient of $f$ at $a=3$.

Explanation of Solution

Check the rate of change of function $f$ over the interval $\left[3,3+h\right]$.

  $\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(3+h\right)-f\left(3\right)}{3+h-3}\\ =\frac{f\left(3+h\right)-f\left(3\right)}{h}\end{array}$

The rate of change of function $f$ over the interval $\left[3,3+h\right]$ is equal to $\frac{f\left(3+h\right)-f\left(3\right)}{h}$.

Now check the difference quotient of $f$ at $a=3$.

Simplify the difference quotient.

  $\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{3+h-3}=\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}$

As the value of $h$ in both the cases is nearly equal to $0$ . So, the function for rate of change of function $f$ over the interval $\left[3,3+h\right]$ is same as the function of difference quotient of $f$ at $a=3$.

Therefore, the function that represents the rate of change of function $f$ over the interval $\left[3,3+h\right]$ is same as the function of difference quotient of $f$ at $a=3$.