Average rate of change

The average rate of change is the change in $y$ divided by the change in $x$. It is the same expression as the slope of the line through $(a,\,f(a))$ and $(b,\,f(b))$ on the graph.

For example, for $y=x^2$ from $x=2$ to $5$ it is $\dfrac{5^2-2^2}{5-2}=\dfrac{21}{3}=7$. For $y=x^2$, $\dfrac{b^2-a^2}{b-a}=a+b$, so the answer is the sum of the two endpoints.

Examples

1. 01What is the average rate of change of $y=x^2$ from $x=2$ to $x=5$?$\displaystyle \frac{5^2 - 2^2}{5 - 2}$AnswerHide

   Answer$\displaystyle 7$
2. 02What is the average rate of change of $y=x^2$ from $x=4$ to $x=5$?$\displaystyle \frac{5^2 - 4^2}{5 - 4}$AnswerHide

   Answer$\displaystyle 9$
3. 03What is the average rate of change of $y=x^2$ from $x=2$ to $x=4$?$\displaystyle \frac{4^2 - 2^2}{4 - 2}$AnswerHide

   Answer$\displaystyle 6$
4. 04What is the average rate of change of $y=x^2$ from $x=1$ to $x=2$?$\displaystyle \frac{2^2 - 1^2}{2 - 1}$AnswerHide

   Answer$\displaystyle 3$
5. 05What is the average rate of change of $y=x^2$ from $x=3$ to $x=4$?$\displaystyle \frac{4^2 - 3^2}{4 - 3}$AnswerHide

   Answer$\displaystyle 7$
6. 06What is the average rate of change of $y=x^2$ from $x=1$ to $x=4$?$\displaystyle \frac{4^2 - 1^2}{4 - 1}$AnswerHide

   Answer$\displaystyle 5$

FAQ

Q1In which order do I subtract?AnswerHide

A

Use "later value minus earlier value" in both the numerator and the denominator. $\dfrac{f(b)-f(a)}{b-a}$ and $\dfrac{f(a)-f(b)}{a-b}$ are equal, but mixing the order between top and bottom flips the sign.

Q2How is it related to the derivative?AnswerHide

A

Let $b$ approach $a$ in the average rate of change, and the value it approaches is $f'(a)$. The slope of the line through two points approaches the slope of the tangent at one point.