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Section 8.1 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 8.1.

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 8.2.

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 8.3.

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 \(2\text{.}\)

Exercises Practice Problems

1.

Suppose \(f(x)=x^2-2\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=1\) to \(x=5\text{.}\)

Answer.
\(6\)

2.

Suppose \(f(x)\) is given in the graph below. Compute the average rate of change of \(f(x)\) from \(x=-5\) to \(x=2\text{.}\)

Answer.
\(1\)

3.

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

Answer.
\(2\)

4.

Suppose \(g(x)\) is given in the table below. Compute the average rate of change of \(g(x)\) from \(x=1\) to \(x=5\text{.}\)

Table 8.4.
\(x\) \(1\) \(2\) \(3\) \(4\) \(5\)
\(g(x)\) \(0\) \(2\) \(4\) \(1\) \(0\)
Answer.
\(0\)

5.

Suppose \(f(x)=\frac{x+3}{x-2}\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=0\) to \(x=4\text{.}\)

Answer.
\(\frac{5}{4}\)

6.

Suppose \(f(x)=x^3 + 5x\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-2\) to \(x=2\text{.}\)

Answer.
\(9\)

7.

Suppose \(f(x)\) is given in the table below. Compute the average rate of change of \(f(x)\) from \(x=2\) to \(x=5\text{.}\)

Table 8.5.
\(x\) \(2\) \(4\) \(5\) \(7\) \(9\)
\(f(x)\) \(1\) \(-2\) \(4\) \(10\) \(0\)
Answer.
\(1\)

8.

Suppose \(f(x)=x^2+16\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=1\) to \(x=10\text{.}\)

Answer.
\(11\)

9.

Suppose \(g(x)\) is given in the table below. Compute the average rate of change of \(g(x)\) from \(x=1\) to \(x=7\text{.}\)

Table 8.6.
\(x\) \(1\) \(3\) \(5\) \(7\) \(15\)
\(g(x)\) \(10\) \(2\) \(14\) \(21\) \(0\)
Answer.
\(\frac{11}{6}\)

10.

Suppose \(f(x)=3x-7\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-5\) to \(x=-2\text{.}\)

Answer.
\(3\)

11.

Suppose \(f(x)\) is given in the graph below. Compute the average rate of change of \(f(x)\) from \(x=0\) to \(x=4\text{.}\)

Answer.
\(1\)

12.

Suppose \(f(x)=x^2-21\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=3\) to \(x=5\text{.}\)

Answer.
\(8\)