Chapter 8, Problem 17CT

To determine

To Explain: The change in average rate of change when function is increasing.

Answer to Problem 17CT

When function is increasing the average rate of change is also too increasing.

Explanation of Solution

Given information:

The quadratic function $f(x)=x^2+4$ . The three intervals [0, 1], [1, 2] and [2, 3]

Formula used: Let’s use the formula for the average rate of change for a function f(x) over interval $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$ .

Calculation:

Let’s find the rate of change for the interval [0, 1]

  $\begin{array}{l}f(1)=1^2+4=5\\ f(0)=0^2+4=4\end{array}$

So, rate of change $\frac{5-4}{1-0}=1$

Now, let’s find the rate of change for the interval [1, 2]

  $f(1)=1^2+4=5$

  $f(2)=2^2+4=8$

Rate of change for this interval is $\frac{8-5}{2-1}=3$

Now, let’s find rate of change for the interval [2, 3]

  $f(2)=2^2+4=8$

  $f(3)=3^2+4=13$

Rate of change over this interval is $\frac{13-8}{3-2}=5$

Conclusion:

Notice the average rate of change over these intervals are increasing