Question

Find the difference quotient f(x+h)-f(x)/h, where h doesn't equal 0 , for the function below:
f(x)=3x^2-9


Simplify your answer as much as possible.

I came up with 6x+3h, am I doing this right?

Answer 1

\[\frac{ f(x+h)-f(x) }{ h }, where (h) \neq 0\]

Answer 2

Okay, but you have not completed the exercise. Now, find the limit as h begins to vanish.

Answer 3

how do I do that?

Answer 4

You ponder it.

h = 10 ==> 6x + 30
h = 1 ==> 6x + 3
h = 0.1 ==> 6x + .3
h = 0.01 ==> 6x + .03
h = 0.001 ==> 6x + .003
h = 0.0001 ==> 6x + .0003
h = 0.00001 ==> 6x + .00003
h = 0.000001 ==> 6x + .000003

What's going on?

Answer 5

the answer is 3, right? thank you for helping me

Answer 6

@tkhunny

Answer 7

@tkhunny, it doesn't look like finding the limit is required; just the difference quotient.

@AwesomeAries, you're given \(f(x)=3x^2-9\), so
\[\begin{align*}\frac{f(x+h)-f(x)}{h}&=\frac{3(x+h)^2-9-\left(3x^2-9\right)}{h}\\
&=\frac{3\left(x^2+2xh+h^2\right)-9-3x^2+9}{h}\\
&=\frac{3x^2+6xh+3h^2-9-3x^2+9}{h}\\
&=\frac{6xh+3h^2}{h}\\
&=6x+3h
\end{align*}\]
So yes, your answer is correct.

Answer 8

Fair enough.

Answer 9

thank you both