Answer:
[tex]Rate = 9(t+1)[/tex]
Step-by-step explanation:
Given
[tex]f(x) = 9x^2 - 2[/tex]
[tex]Interval = [1,t][/tex]
Required
Determine the average rate of change
This is calculated using:
[tex]Rate = \frac{f(b) - f(a)}{b - a}[/tex]
Where
[tex][a,b] = [1,t][/tex]
So, we have:
[tex]Rate = \frac{f(t) - f(1)}{t - 1}[/tex]
Solve for f(t) and f(1)
[tex]f(t) = 9t^2 - 2[/tex]
[tex]f(1) = 9*1^2 - 2[/tex]
[tex]f(1) = 9 - 2[/tex]
[tex]f(1) = 7[/tex]
So, we have:
[tex]Rate = \frac{9t^2 - 2 - 7}{t - 1}[/tex]
[tex]Rate = \frac{9t^2 - 9}{t - 1}[/tex]
Factorize:
[tex]Rate = \frac{9(t^2 - 1)}{t - 1}[/tex]
Apply difference of two squares
[tex]Rate = \frac{9(t- 1)(t+1)}{t - 1}[/tex]
[tex]Rate = 9(t+1)[/tex]
