Question

Given the function f(x)=1+5x^2, calculate the following values:

f(a)=

f(a+h)=

f(a+h)−f(a)/h=

Answer 1

`f(a)=1+5a^2`

`f(a+h)=1+5a^2+10ah+5h^2`

`(f(a+h)-f(a))/(h)=10a+5h`

Explanation:

We are given
`f(x)=1+5x^2`.

Find
`f(a)`. All this means is replace
`x` in
`f(x)=1+5x^2` with
`a`.

`f(x)=1+5x^2`

`f(a)=1+5a^2`

Find
`f(a+h)`. All this means is replace
`(a+h)` in
`f(x)=1+5x^2` with
`(a+h)`.

`f(x)=1+5x^2`

`f(a+h)=1+5(a+h)^2`

`f(a+h)=1+5(a+h)(a+h)`

`f(a+h)=1+5(a^2+2ah+h^2)`

`f(a+h)=1+5a^2+10ah+5h^2`

Find
`(f(a+h)-f(a))/(h)`. So we got to put some parts together; the parts above:

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

`((1+5a^2+10ah+5h^2)-(1+5a^2))/(h)`

Now in the first ( ) I see 1+5a^2 and in the second ( ) I see 1+5a^2, so this means you have (1+5a^2)-(1+5a^2) which equals 0.

`(10ah+5h^2)/(h)`

Now assuming h is not 0. we can divide top and bottom by h.

`(10a+5h)/(1)`

`10a+5h`