Rate of Change Math Example 3

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Example 3

medium
Find the average rate of change of f(x)=x3−2xf(x) = x^3 - 2x on [1,3][1, 3].

Solution

  1. 1
    Evaluate f(1)=(1)3−2(1)=1−2=−1f(1) = (1)^3 - 2(1) = 1 - 2 = -1.
  2. 2
    Evaluate f(3)=(3)3−2(3)=27−6=21f(3) = (3)^3 - 2(3) = 27 - 6 = 21.
  3. 3
    Apply the average rate of change formula: f(3)−f(1)3−1=21−(−1)2=222=11\frac{f(3) - f(1)}{3 - 1} = \frac{21 - (-1)}{2} = \frac{22}{2} = 11.

Answer

1111
The average rate of change of a function on [a,b][a, b] is f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}, which is the slope of the secant line connecting (a,f(a))(a, f(a)) and (b,f(b))(b, f(b)). For f(x)=x3−2xf(x) = x^3 - 2x on [1,3][1, 3], this slope is 11.

About Rate of Change

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

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