Changing Rate Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Changing Rate.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A changing rate of change means the output grows by different amounts for equal increases in input — the hallmark of nonlinear functions like quadratics and exponentials.

Changing rate means accelerating or decelerating progress — like compound interest where each year's gain is larger than the last because the base keeps growing.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A changing rate means the amount the output moves per unit input is itself growing or shrinking — the signature of a curve.

Common stuck point: The procedure for changing rate is the easy part; the trap is reporting one average rate as 'the' rate of the function. Asking "Do equal steps in the input produce different changes in the output?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do equal steps in the input produce different changes in the output?

Worked Examples

Example 1

easy
For f(x)=x2f(x) = x^2, compute the average rate of change on [1,3][1, 3] and on [1,2][1, 2], and explain why these differ.

Answer

Average rate on [1,3][1,3] is 44; on [1,2][1,2] is 33

First step

1
On [1,3][1,3]: f(3)−f(1)3−1=9−12=82=4\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = \frac{8}{2} = 4.

Full solution

  1. 2
    On [1,2][1,2]: f(2)−f(1)2−1=4−11=3\frac{f(2)-f(1)}{2-1} = \frac{4-1}{1} = 3.
  2. 3
    These rates differ because f(x)=x2f(x)=x^2 is not linear — the slope of the secant line depends on which interval is chosen, reflecting the changing (non-constant) rate of change.
For non-linear functions, the average rate of change depends on the interval chosen. As the interval shrinks, the average rate approaches the instantaneous rate (derivative). This is the geometric meaning of the derivative as slope of the tangent line.

Example 2

hard
For g(x)=x3g(x) = x^3, find the average rate of change on [a,a+h][a, a+h] and simplify to see what happens as h→0h \to 0.

Example 3

medium
For f(x)=x2+1f(x)=x^2+1, compute the average rate of change on [1,4][1,4] and on [1,3][1,3]. What pattern do you notice?

Example 4

medium
For f(x)=x2−2xf(x)=x^2-2x, find the average rate on [a,a+1][a,a+1].

Example 5

hard
For f(x)=x2f(x)=x^2, average rates on [1,1+h][1,1+h] approach what value as h→0h\to0?

Example 6

hard
For f(x)=x2f(x)=x^2, find hh such that the average rate on [2,2+h][2,2+h] equals 55.

Example 7

hard
For f(x)=exf(x)=e^x, estimate the average rate of change on [0,1][0,1] using e≈2.718e\approx 2.718.

Example 8

challenge
Show that for f(x)=x2f(x)=x^2, the average rate on [a,b][a,b] equals the instantaneous rate at the midpoint a+b2\frac{a+b}{2}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A ball is thrown upward. Its height (m) is h(t)=−5t2+20th(t) = -5t^2 + 20t. Find the average rate of change from t=0t=0 to t=2t=2 seconds.

Example 2

medium
Explain why the average rate of change of f(x)=∣x∣f(x) = |x| from x=−2x=-2 to x=2x=2 is 00, even though ff is not constant on that interval.

Example 3

easy
Table (0,0),(1,1),(2,4),(3,9)(0,0),(1,1),(2,4),(3,9). Are the first differences equal?

Example 4

easy
Is the rate of change of y=x2y=x^2 constant?

Example 5

easy
Does a curved (non-straight) graph have a constant or changing rate?

Example 6

easy
From x=1x=1 to x=2x=2, y=x2y=x^2 changes by how much?

Example 7

easy
Does compound interest grow at a constant or changing rate?

Example 8

easy
Is the rate of y=2xy=2^x constant?

Example 9

easy
A ball's height each second is 0,5,8,90,5,8,9. Is its rate of rise increasing or decreasing?

Example 10

easy
Does a changing rate always mean the output is decreasing?

Example 11

medium
For y=x2y=x^2, find the average rate of change over [1,4][1,4].

Example 12

medium
Compare the change of y=x2y=x^2 over [0,1][0,1] and [3,4][3,4]. Which is larger?

Example 13

medium
A population is 100,120,150,200100,120,150,200 over four years. Is the growth rate increasing?

Example 14

medium
Average rate of f(x)=x3f(x)=x^3 over [0,2][0,2]?

Example 15

medium
Heights are 0,7,12,15,160,7,12,15,16 each second. Describe the rate of change.

Example 16

medium
Why can't you use a single slope to describe y=x2y=x^2 everywhere?

Example 17

medium
A function's average rate over [2,5][2,5] is 44 and f(2)=3f(2)=3. Find f(5)f(5).

Example 18

medium
Is a positive but decreasing rate the same as the output decreasing?

Example 19

challenge
For y=x2y=x^2, compute average rates over [1,2][1,2], [1,1.5][1,1.5], [1,1.1][1,1.1]. What value do they approach?

Example 20

challenge
Population doubles every 33 years from 10001000. Find the average annual rate of change over the first 33 years versus the next 33 years.

Example 21

challenge
A function has f(0)=0f(0)=0 and average rate 33 on [0,2][0,2] but average rate 77 on [2,4][2,4]. Find f(4)f(4) and decide if the rate is constant.

Example 22

medium
Outputs are 2,2,2,22,2,2,2 for equal input steps. Is the rate changing?

Example 23

easy
For f(x)=x2f(x)=x^2, find the average rate of change on [0,4][0,4].

Example 24

easy
Outputs of a function for x=0,1,2,3x=0,1,2,3 are 1,2,4,81,2,4,8. Are first differences constant?

Example 25

easy
For f(x)=x2f(x)=x^2, find the average rate of change on [2,5][2,5].

Example 26

easy
A car's distance is 0,10,25,450,10,25,45 km at 0,1,2,30,1,2,3 hours. Is the speed constant?

Example 27

medium
For f(x)=2xf(x)=2^x, find the average rate of change on [0,3][0,3].

Example 28

medium
A function has f(1)=4f(1)=4 and average rate 33 on [1,5][1,5]. Find f(5)f(5).

Example 29

medium
For f(x)=xf(x)=\sqrt{x}, find the average rate of change on [1,9][1,9].

Example 30

medium
A population grows from 200200 to 450450 in 55 years. Find the average rate of change per year.

Example 31

medium
A table shows f(2)=5f(2)=5, f(6)=21f(6)=21. Find the average rate of change.

Example 32

medium
For f(x)=3x+2f(x)=3x+2, find the average rate of change on [0,10][0,10] and on [2,7][2,7].

Example 33

hard
For f(x)=1xf(x)=\frac{1}{x}, find the average rate of change on [1,2][1,2].

Example 34

hard
A function satisfies f(0)=2f(0)=2, f(3)=14f(3)=14, f(6)=50f(6)=50. Compare the average rates on [0,3][0,3] and [3,6][3,6].

Example 35

hard
A car decelerates: position is 0,20,35,45,500,20,35,45,50 m at t=0,1,2,3,4t=0,1,2,3,4 s. Describe the rate of change.

Example 36

hard
A function has constant second differences of 22. Is the rate of change of ff constant?

Example 37

challenge
For f(x)=x3f(x)=x^3, find aa such that the average rate of change on [0,a][0,a] equals 99.

Example 38

challenge
Bacteria triple every hour starting at 100100. Find the average rate of change of the population over the first 22 hours.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

rate of change