Answer:
[tex]f(a)=5-8a+15a^2[/tex]
[tex]f(a+h)=5-8a-8h+15a^2+30ah+15h^2[/tex]
[tex]\frac{f(a+h)-f(a)}{h}=-8+30a+15h[/tex]
Step-by-step explanation:
We are given [tex]f(x)=5-8x+15x^2[/tex].
We want to find [tex]f(a)[/tex] so we just replace the x there with a giving us:
[tex]f(a)=5-8a+15a^2[/tex].
We want to find [tex]f(a+h)[/tex] so we just replace x with (a+h) now giving us:
[tex]f(a+h)=5-8(a+h)+15(a+h)^2[/tex].
We will need to distribute and multiply things out here for later use so let's go ahead and do that:
[tex]f(a+h)=5-8(a+h)+15(a+h)^2[/tex]
[tex]f(a+h)=5-8a-8h+15(a+h)(a+h)[/tex]
[tex]f(a+h)=5-8a-8h+15(a^2+2ah+h^2)[/tex]
[tex]f(a+h)=5-8a-8h+15a^2+30ah+15h^2[/tex]
We want to find [tex]\frac{f(a+h)-f(a)}{h}[/tex] where h is not 0.
This requires the parts we found above:
[tex]\frac{f(a+h)-f(a)}{h}[/tex]
[tex]\frac{(5-8a-8h+15a^2+30ah+15h^2)-(5-8a+15a^2)}{h}[/tex]
There are some thing that will zero out (cancel out) in the numerator.
You have 5-8a+15a^2 in both parenthesis and you are subtracting so that part zero's out so you have this now:
[tex]\frac{-8h+30ah+15h^2}{h}[/tex]
Now you can divide h from top and bottom giving you:
[tex]\frac{-8+30a+15h}{1}[/tex]
[tex]-8+30a+15h[/tex]
