1.3 The Average Rate of Change of a Function

Introduction

Defining and interpreting the average rate of change of a function

How average rate of change indicates function trends

Let the height function for a ball tossed vertically be given by $s(t) = 64 - 16(t-1)^2$, where $t$ is measured in seconds and $s$ is measured in feet above the ground.

Compute the value of $AV_{[1.5,2.5]}$.

The table below gives the population of two counties in a state over time.

Let $K$ be the population of Kent County and $W$ the population of Ottawa County, each as a function of year.

Compute $AV_{[1990,2010]}$ for both $K$ and $W$.

What are the units on each of the quantities you computed in (a.)?

Write a careful sentence that explains the meaning of the average rate of change of the Ottawa county population on the time interval $[1990,2010]$. Your sentence should begin something like “In an average year between 1990 and 2010, the population of Ottawa County was $\ldots$”

Which county had a greater average rate of change during the time interval $[2000,2010]$? Were there any intervals in which one of the counties had a negative average rate of change?

Using the given data, what do you predict will be the population of Ottawa County in 2018? Why?

Consider the function $q(x) = 4-(x-2)^2$. Compute $AV_{[0,1]}$, $AV_{[1,2]}$, $AV_{[2,3]}$, and $AV_{[3,4]}$. What do your last two computations tell you about the behavior of the function $q$ on $[2,4]$?

Consider the function $h(t) = 3 - 2(0.5)^t$. Compute $AV_{[-1,1]}$, $AV_{[1,3]}$, and $AV_{[3,5]}$. What do your computations tell you about the behavior of the function $h$ on $[-1,5]$?

On the graphs that follow ($q$ at left, $h$ at right), plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b).

True or false: Since $AV_{[0,3]} = 1$, the function $q$ is increasing on the interval $(0,3)$. Justify your decision.

Give an example of a function that has the same average rate of change no matter what interval you choose. You can provide your example through a table, a graph, or a formula; regardless of your choice, write a sentence to explain.

Sketch a possible graph of a function with the stated properties, or explain why none exists.

$f$ is a function defined on $[-1,7]$ such that $f(1) = 4$ and $AV_{[1,3]} = -2$.

Sketch a possible graph of a function with the stated properties, or explain why none exists.

$g$ is a function defined on $[-1,7]$ such that $g(4) = 3$, $AV_{[0,4]} = 0.5$, and $g$ is not always increasing on $(0,4)$.

Sketch a possible graph of a function with the stated properties, or explain why none exists.

$h$ is a function defined on $[-1,7]$ such that $h(2) = 5$, $h(4) = 3$ and $AV_{[2,4]} = -2$.