Average rate of change; where a function increases or decreases

Warm-up

Three quick questions from earlier lessons.

Discover

A car drives 120 miles in 2 hours. What is its average speed?

60 miles per hour
$\tfrac{1}{60}$ hour per mile
120 miles per hour

Key idea

The average rate of change tells how fast the output changes, per unit of input.

The average rate of change of

f

from

x = a

to

x = b

is

\dfrac{f(b) - f(a)}{b - a}

, the slope of the line through the two points of the graph.

$f$ is increasing on an interval if its graph goes up from left to right there, and decreasing if it goes down.

Worked example

Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 4$ .

f(4) - f(1) = 16 - 1 = 15

Change in the output.

4 - 1 = 3

Change in the input.

Rate

= \dfrac{15}{3} = 5

Divide: output change per unit of input.

Your turn, guided

Your turn: $f(x) = 2x^2$ , from $x = 0$ to $x = 3$ .

f(3) - f(0) = 18

f(3) = 18, f(0) = 0.

Change in

x

:

3

3 − 0.

Rate

= 6

18 ÷ 3.

Practice ★

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 4$ .

Watch out!

Practice ★★

A tank holds 16 liters at time 0 and $61$ liters after 5 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 29 liters at time 0 and $49$ liters after 4 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 12 liters at time 0 and $36$ liters after 6 minutes. What is the average rate of change of the volume, in liters per minute?

$f(x) = -(x - 5)^2 + 8$ . On which interval is $f$ increasing?
1. $(-\infty, 5)$
2. $(5, \infty)$
3. $(-\infty, 8)$
$f(x) = -(x - 1)^2 + 7$ . On which interval is $f$ increasing?
1. $(-\infty, 1)$
2. $(1, \infty)$
3. $(-\infty, 7)$
$f(x) = -(x - 3)^2 + 2$ . On which interval is $f$ increasing?
1. $(-\infty, 3)$
2. $(3, \infty)$
3. $(-\infty, 2)$

Exit ticket

Three questions to check you have got it.

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .

Summary card

Average rate of change

From $a$ to $b$ : $\dfrac{f(b) - f(a)}{b - a}$ , output change over input change.
Increasing: the graph rises from left to right.
Give intervals of increase with $x$ -values.

Watch out: change in $f(x)$ on top, change in $x$ below.

About this lesson