Does anyone know the answer to this??

Complete the definition:

A real function on an open Interval I is differentiable at x ∈ I if...

Is it to do with Rolle's Theorem, although that uses the property but doesnt actually define it?

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- Aug 14th 2011, 11:30 AMrgjf1307Open Interval differention
Does anyone know the answer to this??

Complete the definition:

A real function on an open Interval I is differentiable at x ∈ I if...

Is it to do with Rolle's Theorem, although that uses the property but doesnt actually define it? - Aug 14th 2011, 01:58 PMPlatoRe: Open Interval differention
The answer to your question could be in many forms.

$\displaystyle \lim _{h \to 0} \frac{{f(x + h) - f(x)}}{h}=f'(x)$ if the limit exists.

But there are others. - Aug 14th 2011, 02:17 PMrgjf1307Re: Open Interval differention
Thank you for the response :)

What other forms could there be? - Aug 14th 2011, 02:20 PMPlatoRe: Open Interval differention
- Aug 14th 2011, 02:28 PMrgjf1307Re: Open Interval differention
The narrative used is: A function f: I → ℝ on an open interval I is differentiable at x ∈ I if ...

- Aug 14th 2011, 02:33 PMPlatoRe: Open Interval differention
- Aug 14th 2011, 02:44 PMrgjf1307Re: Open Interval differention
There is no definition in the text material so I guess I'll have to use the form you posted in your original reply!

- Aug 14th 2011, 02:57 PMPlatoRe: Open Interval differention
- Aug 14th 2011, 03:20 PMrgjf1307Re: Open Interval differention
OK, thanks for your help :)

On a similar note how would define a critical point of a differentiable function using the same notation? - Aug 14th 2011, 03:36 PMPlatoRe: Open Interval differention
- Aug 14th 2011, 03:47 PMrgjf1307Re: Open Interval differention
I am being asked to do these questions because my lecturer told me to? Sorry man, didn't mean to ignore you, no hard feelings!

- Aug 14th 2011, 04:06 PMPlatoRe: Open Interval differention
- Aug 14th 2011, 04:09 PMrgjf1307Re: Open Interval differention
Ok, I shall. But if possible could you first assist me on the last question? How would you define a critical point of a differentiable function using the same notation?

Thanks. - Aug 14th 2011, 04:53 PMHallsofIvyRe: Open Interval differention
Are you taking a course on Calculus or not? How could you possibly answer a question about "differentiable" if you have never been given a

**definition**of "differentiable"?

Okay, here's a similar question- is the function $\displaystyle f(x)= e^{x^2}- x$ "codsquafalic". I am not, of course, going to**define**"codsquafalic"! - Aug 15th 2011, 03:45 PMrgjf1307Re: Open Interval differention
Hello there! Yes I am taking a course on Calculus and whilst I am aware of the process of differentiation from previous courses I have not been given a formal definition during this particular course.

The format Plato used earlier is along the lines of what I was hoping to achieve, I was just wondering how, using the same notation, would you define a critical point of a differentiable function?