Find the average rate of change of f(x) = x^2 + 2 between x=1 and x=3.
Understand the Problem
The question is asking to find the average rate of change of the function f(x) = x^2 + 2 over the interval [1, 3]. This involves calculating the function's value at the endpoints of the interval, finding the difference in these values, and then dividing by the length of the interval (3 - 1).
Answer
The average rate of change is $4$.
Answer for screen readers
The average rate of change of $f(x) = x^2 + 2$ over the interval $[1, 3]$ is $4$.
Steps to Solve
- Calculate $f(1)$
Substitute $x = 1$ into the function $f(x) = x^2 + 2$:
$f(1) = (1)^2 + 2 = 1 + 2 = 3$
- Calculate $f(3)$
Substitute $x = 3$ into the function $f(x) = x^2 + 2$:
$f(3) = (3)^2 + 2 = 9 + 2 = 11$
- Calculate the change in $f(x)$
Find the difference between $f(3)$ and $f(1)$:
$f(3) - f(1) = 11 - 3 = 8$
- Calculate the change in $x$
Find the difference between the endpoints of the interval $[1, 3]$:
$3 - 1 = 2$
- Calculate the average rate of change
Divide the change in $f(x)$ by the change in $x$:
$\frac{f(3) - f(1)}{3 - 1} = \frac{8}{2} = 4$
The average rate of change of $f(x) = x^2 + 2$ over the interval $[1, 3]$ is $4$.
More Information
The average rate of change can be interpreted as the slope of the secant line connecting the points $(1, f(1))$ and $(3, f(3))$ on the graph of the function.
Tips
A common mistake is to incorrectly evaluate the function at the endpoints of the interval. Also, students may mix up the order of subtraction when calculating the change in $f(x)$ or the change in $x$, which would result in a sign error, but the absolute value would still be correct. Another mistake is not calculating the change in x at all, and only finding the change in f(x).
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