Question

Let f(x)=5−8x+15x^2. Calculate the following values:

Answer 1

Answer:

$f(a)=5-8a+15a^2$

$f(a+h)=5-8a-8h+15a^2+30ah+15h^2$

$\frac{f(a+h)-f(a)}{h}=-8+30a+15h$

Step-by-step explanation:

We are given $f(x)=5-8x+15x^2$.

We want to find $f(a)$ so we just replace the x there with a giving us:

$f(a)=5-8a+15a^2$.

We want to find $f(a+h)$ so we just replace x with (a+h) now giving us:

$f(a+h)=5-8(a+h)+15(a+h)^2$.

We will need to distribute and multiply things out here for later use so let's go ahead and do that:

$f(a+h)=5-8(a+h)+15(a+h)^2$

$f(a+h)=5-8a-8h+15(a+h)(a+h)$

$f(a+h)=5-8a-8h+15(a^2+2ah+h^2)$

$f(a+h)=5-8a-8h+15a^2+30ah+15h^2$

We want to find $\frac{f(a+h)-f(a)}{h}$ where h is not 0.

This requires the parts we found above:

$\frac{f(a+h)-f(a)}{h}$

$\frac{(5-8a-8h+15a^2+30ah+15h^2)-(5-8a+15a^2)}{h}$

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:

$\frac{-8h+30ah+15h^2}{h}$

Now you can divide h from top and bottom giving you:

$\frac{-8+30a+15h}{1}$

$-8+30a+15h$