Changing Rate Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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.

Solution

  1. 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.
  2. 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.
  3. 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.

Answer

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

About Changing Rate

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.

Learn more about Changing Rate →

More Changing Rate Examples

Example 2 hard

For [formula], find the average rate of change on [formula] and simplify to see what happens as [for

Example 3 easy

A ball is thrown upward. Its height (m) is [formula]. Find the average rate of change from [formula]

Example 4 medium

Explain why the average rate of change of [formula] from [formula] to [formula] is [formula], even t

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