Changing Rate Math Example 4

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Example 4

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.

Solution

  1. 1
    Compute: f(2)−f(−2)2−(−2)=2−24=04=0\frac{f(2)-f(-2)}{2-(-2)} = \frac{2-2}{4} = \frac{0}{4} = 0.
  2. 2
    Yet ff is not constant: it decreases from 22 to 00 on [−2,0][-2,0] and increases from 00 to 22 on [0,2][0,2]. The average rate is zero because the net change in output is zero over the symmetric interval.

Answer

Average rate =0= 0; ff is not constant but changes cancel out over [−2,2][-2,2]
Average rate of change measures net change divided by interval length. Over a symmetric interval centered at the minimum of ∣x∣|x|, the function rises the same amount it fell, giving zero net change despite varying throughout.

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 1 easy

For [formula], compute the average rate of change on [formula] and on [formula], and explain why the

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]

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