Question

limit as h approaches 0 f(x+h)-f(x)/h for f(x)=3x2-5x

Answer 1

`Limit Definition of the Derivative`:
\[\Large f'(x)=\lim_{h\to0}\frac{f(x+h)-\color{royalblue}{f(x)}}{h}\]

We're given the function:
\[\Large \color{royalblue}{f(x)=3x^2-5x}\]Understand where this f(x) will be placed in our limit formula?

Answer 2

yes! i now understand that 3x^2-5x will be placed where the f(x) is, but what do i do with the f(x+h)?

Answer 3

Ok function notation can be a little tricky :)
Our function is essentially:\[\Large f(\quad)=3(\quad)^2-5(\quad)\]

When we plug x into the function we get,\[\Large f(x)=3(x)^2-5(x)\]

But we need to find f(x+h), so we'll need to plug x+h into our function.\[\Large f(x+h)=3(x+h)^2-5(x+h)\]Understand how that works.. kinda? :o

Answer 4

oh! so it's kinda like substitution? then would the modified equation be what ever the simplified answer is + 3x^2-5x?

Answer 5

Yes! You'll have to expand out that messy square involving the `x and h` and then you'll get a bunch of nice cancellations.

The goal is to get to a point where you can plug in h=0 without any problems.
Right now with the way the problem is written, plugging h=0 in would give us an indeterminate form because of the h in the denominator, right?

So do a little algebra, cancel some stuff out, then you'll be able to plug h=0 in at some point.

Answer 6

ahh! i see, but if i were to plug in 0 for the h in the denominator, wouldn't it be obsolete? or would the h eventually end up canceling out before i plug it in?

Answer 7

Yah the h in the denominator should cancel out at some point. :)
At THAT point, after you've done so, you should be able to plug in h=0.
\[\large f'(x)=\lim_{h\to0}\frac{3(x+h)^2-5(x+h)-\color{royalblue}{(3x^2-5x)}}{h}\]
Oh oh also, don't forget to DISTRIBUTE that negative sign to each blue value.
It helps to maybe put brackets around it when you plug it in.

Answer 8

gotcha! :) do i have to foil the (x+h)^2 and distribute the 5 to the x+h?

Answer 9

yes yes foil away :O yes distributeeee.
Oh and after you `foil`, don't forget to distribute that 3 to everything that you just foiled.

Answer 10

distribute the -5 to the x+h*

Answer 11

perfect! thank you so much! you're a life saver! :)

Answer 12

\c:/ np