Question

Given the function f(x)=6x^2+8, calculate the following values:

f(a)=
f(a+h)=
f(a+h)-f(a)/h=

Answer 1

If
`f(x)=6x^2+8`, then

`f(a)=6a^2+8` since we just replace "x" with "a"

`f(a+h)=6(a+h)^2+8` since we just replace "x" with "a+h"

Now we can simplify
`f(a+h)` as follows:

`\begin{aligned}f(a+h)&=6(a+h)^2+8\\[0.5em] &=6(a^2+2ah+h^2)+8\\[0.5em] &=6a^2+12ah+6h^2+8\\[0.5em] \end{aligned}`

Now getting to the final piece, which puts the last two answers together:

`\begin{aligned} (f(a+h)-f(a))/(h) &= ((6a^2+12ah+6h^2+8) - (6a^2+8))/(h) \\[0.5em]&= (6a^2+12ah+6h^2+8 - 6a^2-8)/(h) \\[0.5em]&= (12ah+6h^2)/(h) \\[0.5em]&= 12a+6h ~~~~\text{for } h\\eq0 \\[0.5em]\end{aligned}`