functions

Average Rate of Change

Algebra I25 minDifficulty

You should know: functions, slope, domain and range

Overview

The average rate of change of a function f over an interval [a, b] is (f(b) - f(a))/(b - a). This generalizes slope to non-linear functions: it is the slope of the secant line connecting two points on the graph. For linear functions, the average rate of change is constant (equals the slope). It precursors the derivative in calculus.

Intuition

Imagine driving from city A to city B. Your average speed is total distance / total time -- even if you varied your speed constantly. The average rate of change of position is average velocity. For a function on [a,b], draw the line between (a, f(a)) and (b, f(b)) -- its slope is the average rate of change. How steep is that 'shortcut line'?

Formal Definition

Definition

For a function f defined on [a, b]: Average rate of change = (f(b) - f(a)) / (b - a). This equals the slope of the secant line through (a, f(a)) and (b, f(b)). For linear f(x) = mx + c, the average rate of change equals m for any interval. For nonlinear functions, it depends on the interval chosen.

\text{AROC} = \frac{f(b) - f(a)}{b - a}

Average rate of change

= \frac{\Delta y}{\Delta x} = \text{slope of secant line}

Secant slope

Notation

Notation	Meaning
$\frac{\Delta y}{\Delta x}$	Change in output / change in input

Theorems

Theorem 1: Mean Value Theorem (preview)

For a differentiable function on [a, b], there exists c in (a, b) where f'(c) equals the average rate of change (f(b)-f(a))/(b-a). The instantaneous rate somewhere equals the average rate.

Worked Examples

1
f(3) = 9, f(1) = 1.
$\text{AROC} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$

✓ Answer

4 (the secant line from (1,1) to (3,9) has slope 4).

Practice Problems

Easyfill in blank

Find the average rate of change of f(x) = 2x^2 - 3x from x = 0 to x = 4.

Common Mistakes

Common Mistake

Confusing average rate of change (secant slope) with instantaneous rate of change (derivative/tangent slope).

Average rate uses two endpoints: (f(b)-f(a))/(b-a). Instantaneous rate is the limit as b->a. They differ for non-linear functions.

Quiz

For f(x) = 3x + 2, the average rate of change on any interval [a, b] is:

23a + bDepends on a and b

Historical Background

The notion of average rate of change is fundamental to kinematics. Galileo (c. 1600) studied average velocity (distance / time). Newton and Leibniz (1660s-1680s) developed calculus by considering what happens to the average rate of change as the interval shrinks -- the limit is the instantaneous rate of change (derivative).

1638
Galileo studies average and instantaneous velocity in projectile motion
Galileo Galilei
1665
Newton develops the method of fluxions (precursor to calculus)
Isaac Newton

Summary

AROC of f on [a,b] = (f(b) - f(a)) / (b - a) = secant line slope.
For linear f(x) = mx + c: AROC = m always.
For nonlinear f: AROC depends on the interval.
In calculus, letting b -> a gives the instantaneous rate = derivative.

References

BookHall, B. and Fabricant, M. Algebra 1. Prentice Hall, 2001.