#Average Rate of Change

#Table of Contents

Introduction
Definition and Formula
Function Notation
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction

Understanding the average rate of change is crucial in analyzing how quantities change over intervals. This concept is foundational in calculus and provides insight into the behavior of functions.

#Definition and Formula

#What is the Average Rate of Change?

Average Rate of Change is the slope of the line segment connecting two points on a graph. It is calculated as:

$\text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Where $\left( x_1, y_1 \right)$ and $\left( x_2, y_2 \right)$ are the coordinates of the two points.

The average rate of change is undefined if the change in

x

is zero.

#Function Notation

#How Can I Write the Average Rate of Change Using Function Notation?

For a function $f$ :

A point with $y$ -coordinate $f(x)$
A point with $y$ -coordinate $f(a)$

The average rate of change can be written as:

$\frac{f(x) - f(a)}{x - a}$

If the second point is $h$ units to the right of $a$ :

$\frac{f(a + h) - f(a)}{h}$

Exam Tip

When using function notation, it's important to carefully substitute the values and simplify step by step.

#Worked Example

Let

f

be the function defined by

f(x) = 3x^3 + 2x - 8

. Find the average rate of change between the point with an

x

-coordinate of -1 and the point with an

x

-coordinate of 2. **Solution:**

Using the formula:

$\frac{f(2) - f(-1)}{2 - (-1)}$

Evaluate the function at $x = 2$ and $x = -1$ :

$f(2) = 3(2)^3 + 2(2) - 8 = 24 - 4 = 20$

$f(-1) = 3(-1)^3 + 2(-1) - 8 = -3 - 2 - 8 = -13$

Substitute these values back into the formula:

$\frac{20 - (-13)}{3} = \frac{33}{3} = 11$

The average rate of change is 11.

#Practice Questions

Practice Question

Question 1: Find the average rate of change of the function $g(t) = t^2 - 4t + 6$ between $t = 1$ and $t = 4$ .

Question 2: Determine the average rate of change for the function $h(x) = \sqrt{x}$ from $x = 9$ to $x = 16$ .

#Glossary

Average Rate of Change: The slope of the line segment connecting two points on a graph.
Function Notation: A way to represent functions using symbols like $f(x)$ .
Slope: A measure of how steep a line is, calculated as the ratio of the vertical change to the horizontal change.

#Summary and Key Takeaways

The average rate of change is a measure of how a function changes between two points.
It is calculated as the difference in $y$ -values divided by the difference in $x$ -values.
Using function notation simplifies calculations and helps in understanding the behavior of functions.