Example 6 from Algebra and Trigonometry, 3.3 Rates of Change and Behavior of Graphs

Finding an Average Rate of Change as an Expression

Find the average rate of change of $g (t) = t^{2} + 3 t + 1$ on the interval $[0, a] .$ The answer will be an expression involving $a$ in simplest form.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We use the average rate of change formula.

$\begin{matrix} Average rate of change & = & \frac{g (a) - g (0)}{a - 0} & Evaluate . \\ = & \frac{(a^{2} + 3 a + 1) - (0^{2} + 3 (0) + 1)}{a - 0} & Simplify . \\ = & \frac{a^{2} + 3 a + 1 - 1}{a} & Simplify and factor . \\ = & \frac{a (a + 3)}{a} & Divide by the common factor a . \\ = & a + 3 \end{matrix}$
This result tells us the average rate of change in terms of $a$ between $t = 0$ and any other point $t = a .$ For example, on the interval $[0, 5],$ the average rate of change would be $5 + 3 = 8.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the average rate of change of $f (x) = x^{2} + 2 x - 8$ on the interval $[5, a]$ in simplest forms in terms
of $a .$

How did it go?

Solution

Your turn

Solution

Your turn