Section 8.1 Computing Average Rate of Change
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:
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 \((0,-1)\text{:}\)
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 \(x=3\text{.}\)Both of the numbers given are our \(x\)-values, so we need to plug them both into the function to get the \(y\)-values:
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 \((3,12)\text{:}\)
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 \(x=5\text{.}\)
2.
Suppose \(f(x)\) is given in the graph below. Compute the average rate of change of \(f(x)\) from \(x=-5\) to \(x=2\text{.}\)
3.
Suppose \(f(x)=x^2 +2x +3\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-1\) to \(x=1\text{.}\)
4.
Suppose \(g(x)\) is given in the table below. Compute the average rate of change of \(g(x)\) from \(x=1\) to \(x=5\text{.}\)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
\(g(x)\) | \(0\) | \(2\) | \(4\) | \(1\) | \(0\) |
5.
Suppose \(f(x)=\frac{x+3}{x-2}\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=0\) to \(x=4\text{.}\)
6.
Suppose \(f(x)=x^3 + 5x\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-2\) to \(x=2\text{.}\)
7.
Suppose \(f(x)\) is given in the table below. Compute the average rate of change of \(f(x)\) from \(x=2\) to \(x=5\text{.}\)
\(x\) | \(2\) | \(4\) | \(5\) | \(7\) | \(9\) |
\(f(x)\) | \(1\) | \(-2\) | \(4\) | \(10\) | \(0\) |
8.
Suppose \(f(x)=x^2+16\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=1\) to \(x=10\text{.}\)
9.
Suppose \(g(x)\) is given in the table below. Compute the average rate of change of \(g(x)\) from \(x=1\) to \(x=7\text{.}\)
\(x\) | \(1\) | \(3\) | \(5\) | \(7\) | \(15\) |
\(g(x)\) | \(10\) | \(2\) | \(14\) | \(21\) | \(0\) |
10.
Suppose \(f(x)=3x-7\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=-5\) to \(x=-2\text{.}\)
11.
Suppose \(f(x)\) is given in the graph below. Compute the average rate of change of \(f(x)\) from \(x=0\) to \(x=4\text{.}\)
12.
Suppose \(f(x)=x^2-21\text{.}\) Compute the average rate of change of \(f(x)\) from \(x=3\) to \(x=5\text{.}\)