Practice Changing Rate in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

medium
Compare the change of y=x2 over [0,1] and [3,4]. Which is larger?

Example 3

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

Example 4

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

Example 5

medium
Why can't you use a single slope to describe y=x2 everywhere?

Example 6

challenge
Population doubles every 3 years from 1000. Find the average annual rate of change over the first 3 years versus the next 3 years.

Example 7

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

Example 8

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

Example 9

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

Example 10

hard
A function has constant second differences of 2. Is the rate of change of f constant?

Example 11

challenge
For y=x2, compute average rates over [1,2], [1,1.5], [1,1.1]. What value do they approach?

Example 12

challenge
For f(x)=x3, find a such that the average rate of change on [0,a] equals 9.

Example 13

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

Example 14

easy
For f(x)=x2, find the average rate of change on [2,5].

Example 15

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

Example 16

easy
True or false: a linear function has a changing rate of change.

Example 17

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

Example 18

medium
A population grows from 200 to 450 in 5 years. Find the average rate of change per year.

Example 19

hard
For g(x)=x3, find the average rate of change on [a,a+h] and simplify to see what happens as h→0.

Example 20

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