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2.4 Average Rate of Change of a Function

Measure the change in output per unit input.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Distinguish net change, average rate, and unitsWorked steps · five questions02Calculate rates from polynomial formulasWorked steps · five questions03Simplify rational and radical difference quotientsWorked steps · five questions04Derive symbolic difference quotientsWorked steps · five questions05Read graphs and secant slopesWorked steps · five questions06Compare rates in tables and changing dataWorked steps · five questions07Relate signs and successive rates to behaviorWorked steps · five questions08Use constant rates and justify linear behaviorWorked steps · five questions09Interpret motion, laps, and race graphsWorked steps · five questions10Investigate shrinking intervals without assuming calculusWorked steps · five questions
Lesson brief and rules

2.4 Average Rate of Change of a Function

Average rate on [a,b][a,b] is [f(b)−f(a)]b−a\frac{[f(b) - f(a)]}{b - a}, where a≠ba \ne b.

Remember
Use the same order in the numerator and denominator.
Worked example 11
01 / A measured water volume changes from 3636 to 3232 liters during 22 minutes. Give the net change and average rate, including units.
02 / Net change is final minus initial: 32−36=−432-36=-4 liters.
03 / Divide by elapsed input 22: −42=−2-\dfrac{4}{2}=-2.
04 / The output/input units are liters per minute; the quotient is a secant rate.
Worked example 2 · Distinguish net change, average rate, and units

A measured altitude changes from 3939 to 3636 meters during 33 seconds. Give the net change and average rate, including units.

Net change is final minus initial: 36−39=−336-39=-3 meters.
Divide by elapsed input 33: −33=−1-\dfrac{3}{3}=-1.
The output/input units are meters per second; the quotient is a secant rate.

Read the graphs

Read graphs and secant slopes

Read the plotted polyline (( each marked vertex is exact )). Find the net change and average rate from x=0x=0 to x=4x=4. Draw its secant segment.
Rate graph with five exact plotted vertices-1012345-1012345xyf
Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Read
Given function values ⋅\cdot — means not provided
xxf(x)f(x)
0011
4411
from the graph.
Δf=1−1=0\Delta f=1-1=0; divide by 4−04-0 to get 00.
The secant is straight and joins the two endpoints, not all the intermediate vertices.
Written-work review:
  • Accurate graph endpoint readings.
  • Correct signed net change and quotient.
  • Draw straight secant through the two specified points.

Answer: Net change 00; rate 00; secant joins (0,1)(0,1), (4,1)(4,1).

Relate signs and successive rates to behavior

Using the three graphs, classify successive unit-interval rates as constant, increasing, or decreasing. All three functions increase.
Three increasing functions with different changes in rates-101234-10123456789101112131415xyFGHFGH

FF has unit changes 2,2,22,2,2.
GG has unit changes 1,3,51,3,5; HH has unit changes 4,2,14,2,1.
Increasing outputs do not imply an increasing rate.

Answer: FF: constant; GG: increasing; HH: decreasing.

Interpret motion, laps, and race graphs

The graph gives a cart’s position in meters at time in seconds. Find average speed and average velocity over the entire shown trip.
Cart goes out and returns; horizontal axis time in seconds-10123456789101112131415-1012345678910111213141516xyposition

The cart travels 15m15 m out and 15m15 m back, for total distance 30m30 m.
Divide by total time 14s14 s to get average speed 157\dfrac{15}{7} meters per second.
The final and initial positions agree, so average velocity is zero.

Answer: Speed 157\dfrac{15}{7} meters per second; velocity 00 meters per second.

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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