Question

find the average rate of change of f(x) = 2x^2 + 4 over each of the following intervals.
(a) from 3 to 5
(b) from 1 to 3
(c) from - 2 to 1
(a) the average rate of change from 3 to 5 is
(b) the average rate of change from 1 to 3 is
(c) the average rate of change from - 2 to 1 is

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate for interval [3,5]

First, find $f(3)$ and $f(5)$ for $f(x)=2x^{2}+4$.
$f(3)=2\times3^{2}+4=2\times9 + 4=18 + 4=22$.
$f(5)=2\times5^{2}+4=2\times25+4=50 + 4=54$.
Then, $\frac{f(5)-f(3)}{5 - 3}=\frac{54 - 22}{2}=\frac{32}{2}=16$.

Step3: Calculate for interval [1,3]

Find $f(1)$ and $f(3)$.
$f(1)=2\times1^{2}+4=2 + 4=6$.
We already know $f(3)=22$.
Then, $\frac{f(3)-f(1)}{3 - 1}=\frac{22 - 6}{2}=\frac{16}{2}=8$.

Step4: Calculate for interval [-2,1]

Find $f(-2)$ and $f(1)$.
$f(-2)=2\times(-2)^{2}+4=2\times4 + 4=8 + 4=12$.
We know $f(1)=6$.
Then, $\frac{f(1)-f(-2)}{1-(-2)}=\frac{6 - 12}{3}=\frac{-6}{3}=-2$.

Answer:

(a) 16
(b) 8
(c) -2