AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/1/1-2)
Unit 1 · Topic 1.2
1.2 Rates of Change
A rate of change measures how fast the output changes compared with the input. You'll compute average rates of change over intervals, use small intervals to estimate the rate at a single point, and compare rates at different points.
Key terms
- average rate of change
- rate of change at a point
- positive and negative rates
- interval
Average rate of change
The average rate of change of f from x = a to x = b is the change in output divided by the change in input:
average rate of change = (f(b) − f(a)) / (b − a)
It is the constant rate that would produce the same total change over that interval. If a car's position changes by 120 miles in 2 hours, its average rate is 60 miles per hour, even if it sped up and slowed down along the way.
Units are always output units per input unit: dollars per year, meters per second, degrees per hour. On the exam, an answer in context needs these units.
The rate of change at a point
The rate of change at a single point says how fast the output is changing right at that input. You can't divide by a change of zero, so you estimate it instead.
Pick a small interval that contains the point and find the average rate of change over it. The smaller the interval, the better the estimate usually is. An interval centered on the point, like [0.9, 1.1] for x = 1, tends to give a good estimate.
To compare how fast a function changes at two points, estimate the rate at each one with small intervals and compare the estimates. In calculus this idea becomes the derivative, but in precalculus you only need the estimates.
What the sign tells you
A positive rate of change means the two quantities move in the same direction: as the input increases, the output increases (and as one decreases, so does the other).
A negative rate of change means they move in opposite directions: as the input increases, the output decreases.
A rate near zero means the output is barely changing at that point, which often happens near a peak or a valley of the graph.
Rates on graphs and tables
On a graph, the average rate of change from a to b is the slope of the line through (a, f(a)) and (b, f(b)). Steeper means a larger rate; a line falling to the right means a negative rate.
In a table, use the two rows for the endpoints of the interval. Subtract outputs and inputs in the same order (later minus earlier for both), and watch for negative values.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Average rate of change in context
A ball is thrown upward. Its height in meters t seconds after the throw is h(t) = −5t² + 30t + 2. Find the average rate of change of h from t = 1 to t = 3, and explain what it means.
Show the solutionHide the solution
- Step 1: Find the outputs: h(1) = −5 + 30 + 2 = 27 and h(3) = −45 + 90 + 2 = 47.
- Step 2: Divide the change in height by the change in time: (h(3) − h(1)) / (3 − 1) = (47 − 27) / 2 = 10.
- Step 3: Units: meters per second.
Answer: 10 meters per second. Between t = 1 and t = 3 seconds, the ball's height increased by an average of 10 meters each second.
- Example 2
Estimating and comparing rates at points
Using the same h(t) = −5t² + 30t + 2, estimate the rate of change of h at t = 1 and at t = 4 using intervals of width 0.2 centered at each point. At which time is the height changing faster?
Show the solutionHide the solution
- Step 1: For t = 1, use [0.9, 1.1]: h(0.9) = −4.05 + 27 + 2 = 24.95 and h(1.1) = −6.05 + 33 + 2 = 28.95. The estimate is (28.95 − 24.95) / 0.2 = 20.
- Step 2: For t = 4, use [3.9, 4.1]: h(3.9) = −76.05 + 117 + 2 = 42.95 and h(4.1) = −84.05 + 123 + 2 = 40.95. The estimate is (40.95 − 42.95) / 0.2 = −10.
- Step 3: At t = 1 the ball is rising at about 20 m/s. At t = 4 it is falling at about 10 m/s.
- Step 4: “Faster” compares how quickly the height changes, so compare sizes: 20 is larger than 10.
Answer: About 20 m/s at t = 1 and about −10 m/s at t = 4. The height is changing faster at t = 1.
- Example 3
Trap: signs and order in a table
A function g has g(−2) = 7 and g(3) = −8. Find the average rate of change of g on [−2, 3].
Show the solutionHide the solution
- Step 1: Subtract in the same order on top and bottom: (g(3) − g(−2)) / (3 − (−2)).
- Step 2: Top: −8 − 7 = −15. Bottom: 3 + 2 = 5. A common slip is to write 3 − 2 = 1 on the bottom.
- Step 3: Divide: −15 / 5 = −3.
Answer: −3. On average, g decreases by 3 for each 1-unit increase in the input.
Common mistakes
- Subtracting outputs in one order and inputs in the other, which flips the sign. Always do later minus earlier on both top and bottom.
- Dropping the parentheses around negative inputs, so 3 − (−2) becomes 3 − 2. Write the subtraction out in full.
- Leaving off units or describing the rate without “on average.” An average rate of change describes the whole interval, not every moment in it.
- Using a wide interval to estimate a rate at a point when a narrower one is available. The narrower interval gives the better estimate.
On the exam
- Expect to compute an average rate of change from a table, graph or formula, often in context. Show the quotient with numbers and give units.
- Questions that compare rates at two points expect estimates from small intervals around each point. Say which interval you used.
Connected topics
Videos
Check yourself
4 questions on 1.2 Rates of Change. Pick an answer to see if you got it, and why.
| t (hours) | 1.9 | 2.0 | 2.1 |
|---|---|---|---|
| g(t) (feet) | 5.62 | 6.00 | 6.42 |
Table of values
Using the average rate of change of g over the interval 1.9 ≤ t ≤ 2.1, which is the best estimate of the rate at which the depth of the water is changing at time t = 2 hours?
A ball is thrown upward. Its height, in feet, t seconds after it is thrown is h(t) = −16t² + 48t + 4. What is the average rate of change of h over the interval 1 ≤ t ≤ 2.5, and what does it tell you?
The average rate of change of the function g over the interval 0 ≤ x ≤ 4 is 3, and g(0) = −2. What is the value of g(4)?
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 3 | 4 | 7 | 12 | 19 |
Table of values
What is the average rate of change of f over the interval 1 ≤ x ≤ 4?
0 of 4 answered