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: In a nonlinear function, the slope of the graph changes as x changes. The derivative (rate of change) is itself a function, not a constant.

Common stuck point: Students often assume constant rate when none is stated — always ask whether the rate is the same at every input value before applying linear reasoning.

Sense of Study hint: Compute the average rate of change over several different intervals. If the rates differ, the rate is changing.

Worked Examples

Example 1

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

Solution

  1. 1
    On [1,3]: \frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = \frac{8}{2} = 4.
  2. 2
    On [1,2]: \frac{f(2)-f(1)}{2-1} = \frac{4-1}{1} = 3.
  3. 3
    These rates differ because f(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.

Answer

Average rate on [1,3] is 4; on [1,2] is 3
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) = x^3, find the average rate of change on [a, a+h] and simplify to see what happens as h \to 0.

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) = -5t^2 + 20t. Find the average rate of change from t=0 to t=2 seconds.

Example 2

medium
Explain why the average rate of change of f(x) = |x| from x=-2 to x=2 is 0, even though f is not constant on that interval.
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Background Knowledge

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

rate of change