Answer:
[tex]Rate = -2h-1[/tex]
Step-by-step explanation:
Given
[tex]f(x) = -2x^2 + 3x - 7[/tex]
[tex]Interval: [1,1+h][/tex]
Required
The average rate of change
This is calculated as:
[tex]Rate = \frac{f(b) - f(a)}{b - a}[/tex]
Where:
[tex][a,b] = [1,1+h][/tex]
So, we have:
[tex]Rate = \frac{f(1+h) - f(1)}{1+h - 1}[/tex]
[tex]Rate = \frac{f(1+h) - f(1)}{h}[/tex]
Calculate f(1+h) and f(1)
[tex]f(x) = -2x^2 + 3x - 7[/tex]
[tex]f(1) = -2 * 1^2 + 3 * 1 - 7 = -6[/tex]
[tex]f(1+h) = -2 * (1+h)^2 + 3 * (1+h) - 7[/tex]
Evaluate squares and open bracket
[tex]f(1+h) = -2 * (1+h+h+h^2) + 3+3h - 7[/tex]
[tex]f(1+h) = -2-2h-2h-2h^2 + 3+3h - 7[/tex]
[tex]f(1+h) = -2-4h-2h^2 + 3+3h - 7[/tex]
Collect like terms
[tex]f(1+h) = -2h^2-4h +3h-2+ 3 - 7[/tex]
[tex]f(1+h) = -2h^2-h-6[/tex]
So, we have:
[tex]Rate = \frac{f(1+h) - f(1)}{h}[/tex]
[tex]Rate = \frac{-2h^2-h-6 - -6}{h}[/tex]
[tex]Rate = \frac{-2h^2-h-6 +6}{h}[/tex]
[tex]Rate = \frac{-2h^2-h}{h}[/tex]
Factorize the numerator
[tex]Rate = \frac{h(-2h-1)}{h}[/tex]
Divide
[tex]Rate = -2h-1[/tex]
