1.1: Average Rate of Change | AP Precalculus

How it works

Average rate of change is a secant slope: output change divided by input change. It has output units per input unit and depends on the entire interval. It differs from the output difference and from an instantaneous rate.

\frac{f(b)-f(a)}{b-a}\quad(a\ne b)

Worked examples

Example 1

Find the average rate of change of $\displaystyle f(x)=x^2$ on $\displaystyle [1,3]$ .

$\displaystyle [f(3)-f(1)]/(3-1)=(9-1)/2=4$ .

Example 2

A tank holds $\displaystyle 80$ L at $\displaystyle t=2$ min and $\displaystyle 50$ L at $\displaystyle t=8$ min. Find average rate.

The rate is $\displaystyle (50-80)/(8-2)=-5$ L/min. The volume decreases by an average of $\displaystyle 5$ L per minute.

Problems

Problem 1.1.1

For $\displaystyle f(x)=x^2$ , find the average rate of change on $\displaystyle [0,2]$ and $\displaystyle [2,4]$ .

Parabola y equals x squared with marked points 0,0; 2,4; 4,16. — Compare output change divided by input change on the two intervals.

Show worked solution

The rates are $\displaystyle (4-0)/2=2$ and $\displaystyle (16-4)/2=6$ . Later secants are steeper on these positive intervals.

Parabola y equals x squared with marked points 0,0; 2,4; 4,16. The solution adds labeled relationships. — Secant slopes on [0,2] and [2,4] are 2 and 6.

Problem 1.1.2

A car travels from mile marker $\displaystyle 12$ to mile marker $\displaystyle 72$ in $\displaystyle 1.5$ hours. Find average velocity.

Show worked solution

The rate is $\displaystyle (72-12)/1.5=40$ miles per hour in the increasing-marker direction.

Problem 1.1.3

For $\displaystyle f(x)=3x-7$ , show the average rate on any interval is $\displaystyle 3$ .

Show worked solution

$\displaystyle [3b-7-(3a-7)]/(b-a)=3(b-a)/(b-a)=3$ for distinct endpoints.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.1.1

Problem 1.1.2

Problem 1.1.3

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