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Section 7.1 Computing Average Rate of Change

Lines are the only functions that actually have slope, but slope is such a useful measurement. Average rate of change gives us a way of computing something like slope but for non-lines.

Definition 7.1.

The average rate of change of a function \(f(x)\) on \([a,b]\) (sometimes also written "from \(x=a\) to \(x=b\)") is the slope of the line between the points \((a,f(a))\) and \((b,f(b))\)

Example 7.2.

Suppose \(f(x)=x^2+2x-1\) and we want to compute the average rate of change of \(f(x)\) on \([-1,0]\text{.}\) Both of these numbers are our \(x\)-values, so we need to plug them both into the function to get the \(y\)-values:

\begin{equation*} f(-1)=(-1)^2+2(-1)-1=-2 \end{equation*}
\begin{equation*} f(0)=(0)^2+2(0)-1=-1 \end{equation*}

Now, that means we just need to compute the slope between the points \((-1,f(-1))\) and \((0,f(0))\text{,}\) which we now know is \((-1,-2)\) and \((0,-1)\text{:}\)

\begin{equation*} \frac{-1-(-2)}{0-(-1)}=\frac{-1+2}{0+1}=\frac{1}{1}=1 \end{equation*}

Therefore, our answer is that the average rate of change of \(f(x)\) on \([-1,0]\) is \(1\text{.}\)

Checkpoint 7.3.

Suppose \(f(x)=x^2+3\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-1\) to \(x=3\text{.}\)
Answer.
\(2\)
Solution.

Both of the numbers given are our \(x\)-values, so we need to plug them both into the function to get the \(y\)-values:

\begin{equation*} f(-1)=(-1)^2+3=4 \end{equation*}
\begin{equation*} f(3)=(3)^2+3=12 \end{equation*}

Now, that means we just need to compute the slope between the points \((-1,f(-1))\) and \((3,f(3))\text{,}\) which we now know is \((-1,4)\) and \((3,12)\text{:}\)

\begin{equation*} \frac{4-12}{-1-3}=\frac{-8}{-4}=2 \end{equation*}

Therefore, our answer is that the average rate of change of \(f(x)\) on \([-1,3]\) is \(2\text{.}\)