Definition 7.1.1.
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 \((b,f(b))\)
Section 7.1 Computing Average Rate of Change
Example 7.1.2.
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 7.1.3.
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 \(2\text{.}\)
