Average Rates of Change and the Difference Quotient

Section 1.3 Average Rates of Change and the Difference Quotient

In this section, we’ll learn to identify the domain and range of functions given in various forms, as well as determine when a function exhibits important behaviors.

Subsection Textbook Reference

This relates to content in §3.3 of Algebra and Trigonometry 2e

http://tiny.cc/111Z-Textbook

Exercises Preparation Exercises

1.

Suppose you’re driving south on I-5 in Oregon and you pass mile marker 294 in Portland at 1:35 PM. Later, you pass mile marker 194 in Eugene at 3:05 PM.

(a)

What was your average speed (in miles per hour) of the trip from Portland to Eugene?

(b)

What was your speed at any particular moment, say as you drove past mile marker 256 in Salem?

2.

Let $f(x) = x^2 - 2x\text{.}$ Evaluate and simplify the following.

(a)

$f(4)$

(b)

$f(-6)$

(c)

$f(a)$

(d)

$f(a+b)$

Exercises Practice Exercises

1.

The function in Table 1.3.1 below shows the cost of movie tickets in the U.S. in the year $t\text{.}$

Table 1.3.1. Price of Movie Tickets in the U.S.

$t$ (year)	$m(t)$ (in dollars)
$1995$	$4.35$
$1999$	$5.06$
$2003$	$6.03$
$2009$	$7.50$
$2013$	$8.13$
$2017$	$8.97$
$2021$	$10.17$

(a)

What is the unit of the average rate of change in the price of a movie over any time period?

(b)

What is the average rate of change in the price of a movie ticket from 2003 to 2021?

2.

Given $f(n) = \frac{1}{3}n^2 - 1\text{,}$ find the average rate of change of $f$ on the interval $\left[3,9\right]\text{.}$

3.

Find and simplify the difference quotient for each of the following functions.

(a)

$f(x) = -6x + 8$

(b)

$g(x) = -2x^2-5x$

Subsection Definitions

Definition 1.3.2. Rate of Change.

A rate of change describes how the output values change in relation to a change in the input values. The unit for the rate of change is “output unit(s) per input unit.”

Definition 1.3.3. Average Rate of Change.

The average rate of change for a function $f$ between two input values $x_1$ and $x_2$ is the difference in their output values divided by the difference in the two input values. The average rate of change is calculated using the formula

\begin{equation*} \text{average rate of change } = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1}, x_1 \neq x_2 \end{equation*}

The average rate of change is the slope of the line between the two points $(x_1, f(x_1))$ and $(x_2, f(x_2))\text{.}$

Example 1.3.4.

The function $E(x)$ gives the cost of a dozen eggs (in dollars) $x$ years after 2010. If we know $E(19) = 1.362$ and $E(23) = 2.666\text{,}$ we can find the average rate of change as

\begin{align*} \dfrac{E(23)-E(19)}{23 \text{ years } - 19 \text{ years}} \amp = \dfrac{\$2.666 - \$1.362}{4 \text{ years}}\\ \amp = \dfrac{\$1.304}{4 \text{ years}}\\ \amp \approx \$0.33/\text{year} \end{align*}

This shows that between 2019 and 2023, the cost of a dozen eggs increased on average by about $0.33/year.

Definition 1.3.5. Difference Quotient.

The difference quotient for a function $f$ is given by the formula

\begin{equation*} \dfrac{f(x+h)-f(x)}{h}, h \neq 0 \end{equation*}

The difference quotient is the average rate of change between the two points $(x, f(x))$ and $(x+h, f(x+h))\text{.}$

Example 1.3.6.

Given the function $f(x) = 3x^2 - 4x\text{,}$ the difference quotient would be evaluated as

\begin{align*} \dfrac{f(x+h) - f(x)}{h} \amp = \dfrac{(3(x+h)^2 - 4(x+h)) - (3x^2 - 4x)}{h}\\ \amp = \dfrac{3x^2 + 6xh + 3h^2 - 4x - 4h - 3x^2 + 4x}{h}\\ \amp = \dfrac{6xh + 3h^2 - 4h}{h}\\ \amp = \dfrac{h(6x + 3h - 4)}{h}\\ \amp = 6x + 3h - 4, h \neq 0 \end{align*}

Exercises Exit Exercises

1.

(a)

What are two situations in your daily life that involve an average rate of change? What are the units for these rates of change?

(b)

What is the formula for the difference quotient for the function $k$ that has an input variable $p\text{?}$

(c)

If you have a function $m$ that gives the price of a gallon of milk in the year $t\text{,}$ what would be the unit for the average rate of change of $m\text{?}$

2.

Find and simplify the difference quotient for the function $f(t) = \dfrac{3}{t-6}\text{.}$

Reflection Reflection

1.

On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?

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\(t\) (year)	\(m(t)\) (in dollars)
\(1995\)	\(4.35\)
\(1999\)	\(5.06\)
\(2003\)	\(6.03\)
\(2009\)	\(7.50\)
\(2013\)	\(8.13\)
\(2017\)	\(8.97\)
\(2021\)	\(10.17\)