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Finding an Average Rate of Change as an Expression · Physics Playground

Exercises Functions

Example 6 from Precalculus, 1.3 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 $a .$

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.

$\begin{array}{l} \begin{array}{l} Average rate of change = \frac{g (a) - g (0)}{a - 0} & Evaluate . \\ = \frac{(a^{2} + 3 a + 1) - (0^{2} + 3 (0) + 1)}{a - 0} & Simplify . \\ = \frac{a^{2} + 3 a + 1 - 1}{a} & Simplify and factor . \\ = \frac{a (a + 3)}{a} & \begin{array}{l} Divide by the common factor a . \end{array} \end{array} \\ = a + 3 \end{array}$
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 $[5, a] .$

How did it go?

PreviousComputing Average Rate of Change from a Table Next Determine a Number within a Prescribed Distance

From Precalculus (1st edition) by OpenStax (Rice University), section 1.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

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