Given:
[tex]f(x)=2 x^{2}+6[/tex]
To find:
The average rate of change.
Solution:
Average rate of change formula:
[tex]$\frac{f(b)-f(a)}{b-a}[/tex]
(a) Average rate of change from 2 to 4:
Substitute x = 4 and x = 2 in f(x).
[tex]$\frac{f(4)-f(2)}{4-2}=\frac{(2(4)^2+6)-(2(2)^2+6)}{4-2}[/tex]
[tex]$=\frac{38-14}{2}[/tex]
= 12
Average rate of change from 2 to 4 is 12.
(b) Average rate of change from 1 to 3:
Substitute x = 3 and x = 1 in f(x).
[tex]$\frac{f(3)-f(1)}{3-1}=\frac{(2(3)^2+6)-(2(1)^2+6)}{3-1}[/tex]
[tex]$=\frac{24-8}{2}[/tex]
= 8
Average rate of change from 2 to 4 is 8.
(c) Average rate of change from -2 to 1:
Substitute x = 1 and x = -2 in f(x).
[tex]$\frac{f(1)-f(-2)}{1-(-2)}=\frac{(2(1)^2+6)-(2(-2)^2+6)}{1+2}[/tex]
[tex]$=\frac{8-14}{3}[/tex]
= -2
Average rate of change from 2 to 4 is -2.
