Cogito
AP Precalculus · Chapter 1 · Lesson 1
Average Rate of Change
The slope between two points on any curve.
11 problems · about 22 minutes · AP Precalculus 1.1, 1.2
What this lesson teaches
The student computes the average rate of change of a function over an interval and interprets its sign.
- The average rate of change is the slope of the secant between the endpoints.
- It is (f(b) − f(a)) / (b − a), and it depends on the interval chosen.
- It compares endpoints only — it says nothing about what happened in between.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x². Average rate of change on [3, 6]?
Answer 9
Why 9.
f(x) = x². Average rate of change on [2, 5]?
Answer 7
Why (25 − 4) ÷ 3.
f(x) = 3x + 1. Average rate of change on [0, 10]?
Answer 3
Why A line has one slope everywhere.
f(x) = x². Average rate of change on [−3, 3]?
Answer 0
Why The endpoints have the same output.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = x³. Average rate of change on [0, 2]?
Answer 4
Why (8 − 0) ÷ 2.
Select every statement that must be true when the average rate of change on [a, b] is 0.
Answer f(a) and f(b) are equal.; The function need not be constant in between.
Why A zero rate says the endpoints match, nothing more.
Put these steps in the order you would carry them out.
Answer 1. Work out f(a) and f(b). 2. Subtract to get the change in output. 3. Subtract to get the change in input. 4. Divide output change by input change.
Why The division is last.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf(x) = −x². Average rate of change on [1, 3]?
Answer -4
Why (−9 − (−1)) ÷ 2.
Does a positive average rate of change guarantee the function increased throughout the interval?
Answer No. It compares only the endpoints.
Why A curve can dip and recover.
The Interval: For f(x) = x², what is the average rate of change on [1, 4]?
Answer 5
Why 15 ÷ 3 = 5.
Reading It Off the Graph: The secant shown joins x = −2 and x = 3 on y = x² − 4. What is its slope?
Answer 1
Why (5 − 0) ÷ (3 − (−2)) = 1.