Rate of Change Examples in Math

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

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 measure of how quickly one quantity changes with respect to another; the ratio of the change in output to the change in input.

How much does the output change for each unit increase in input? That ratio is the rate of change.

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: Average rate = slope of secant. Instantaneous rate = slope of tangent.

Common stuck point: Positive rate means the quantity is increasing; negative rate means it is decreasing.

Sense of Study hint: When you see a rate-of-change problem, first identify the two quantities and their units. Then compute the change in output (\Delta y = f(b) - f(a)) and the change in input (\Delta x = b - a), and divide: \frac{\Delta y}{\Delta x}. For instantaneous rate, take the limit as the interval shrinks to zero.

Worked Examples

Example 1

easy
The position of a particle at time t seconds is s(t) = 3t^2 - 2t + 1 metres. Find the average rate of change of position from t = 1 to t = 4, and the instantaneous rate of change at t = 2.

Solution

  1. 1
    Average rate of change: \frac{s(4) - s(1)}{4 - 1}.
  2. 2
    s(4) = 3(16) - 2(4) + 1 = 48 - 8 + 1 = 41. s(1) = 3 - 2 + 1 = 2.
  3. 3
    Average rate: \frac{41 - 2}{3} = \frac{39}{3} = 13 m/s.
  4. 4
    Instantaneous rate: s'(t) = 6t - 2. At t = 2: s'(2) = 12 - 2 = 10 m/s.

Answer

Average rate of change: 13 m/s; instantaneous rate at t=2: 10 m/s
The average rate of change is the slope of the secant line between two points. The instantaneous rate of change is the derivative evaluated at the specific time. They are equal only when the function is linear.

Example 2

medium
Water drains from a tank so that the volume remaining after t minutes is V(t) = 500 - 20t - t^2 litres (0 \leq t \leq 10). Find the rate at which water is draining at t = 3 minutes.

Example 3

medium
Find the average rate of change of f(x) = x^3 - 2x on [1, 3].

Practice Problems

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

Example 1

easy
Find the average rate of change of f(x) = x^2 + 3x from x = 0 to x = 5.

Example 2

medium
The temperature in a room is T(t) = 20 + 5\cos\left(\frac{\pi t}{12}\right) degrees Celsius at time t hours. Find the instantaneous rate of change of temperature at t = 6.
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Background Knowledge

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

slope