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April 16th, 2011, 08:52 AM   #1
 
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how do you know if a func is Differentiable?

The solution in my book starts out by stating that this func is Differentiable. How do you know it is Differentiable?

Let's say I'm asked to find all x values when this function is growing.

I know that if a function is Differentiable then, it's growing, when it's derivative function values are bigger than zero.

now solve for x & we get 2 x values: 1/3 & 3

Since 3x^2 is positive, we make the graph of this derivative function, starting from top right:


From the graph we see that the derivative function values are bigger than zero when x is less than 1/3 and bigger than 3:

so our answer is: this function is growing when x values are between [-infinity :1/3] & [3;infinity]

Second question is: Now let's suppose I do not know at first glance if a function is Differentiable or not, how can I find out when its growing?
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April 16th, 2011, 09:31 AM   #2
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Re: how do you know if a func is Differentiable?

Quote:
Originally Posted by peaceofmind
The solution in my book starts out by stating that this func is Differentiable. How do you know it is Differentiable?

Let's say I'm asked to find all x values when this function is growing.
...

Second question is: Now let's suppose I do not know at first glance if a function is Differentiable or not, how can I find out when it's growing?
Well, how I know it is differentiable is going to be different than how YOU know it is differentiable, since proving that something is differentiable on a metric space is ... a little advanced. But one result is:
Quote:
Polynomials (in any context that you will see them at this level) are differentiable.
You may need to review what exactly a polynomial IS ... the example you gave qualifies.


And generally, in this context, functions are going to be continuous EVERYWHERE, except at a point (or a few points). Take, for instance, the absolute value function y = |x|. This function is NOT differentiable at x = 0, since the limit that is the definition of the derivative does not exist.
But everywhere else (besides at x = 0), this function is differentiable.

Functions are not differentiable where they are discontinuous, so that might be another good starting point.

If you are asked to find where a function is increasing (i.e. "growing"), you would still take the derivative (whether or not it is differentiable everywhere!).
Then just keep in mind the x-values where the function is not differentiable, and set the derivative equal to zero to find the horizontal tangents.
etc.
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April 16th, 2011, 09:32 PM   #3
 
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Quote:
Originally Posted by peaceofmind
Let's say I'm asked to find all x values when this function is growing.
For x < ?, the function is increasing.
For x = ?, the function has relative extrema.
For ? < x < 3, the function is decreasing.
For x = 3, the function has relative extrema.
For x > 3, the function is increasing.
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April 16th, 2011, 10:43 PM   #4
 
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Re: how do you know if a func is Differentiable?

Simple, if a function is continuous i.e. it exists for a<x<b then it's differentiable.

Because differentiating means finding the gradient of a function at a specific point. All functions have gradients!
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April 17th, 2011, 02:57 AM   #5
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Re: how do you know if a func is Differentiable?

Quote:
Originally Posted by ikurwa
Simple, if a function is continuous i.e. it exists for a<x<b then it's differentiable...
http://en.wikipedia.org/wiki/Weierstrass_function
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April 17th, 2011, 05:22 PM   #6
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Quote:
Originally Posted by The Chaz
If you are asked to find where a function is increasing (i.e. "growing"), you would still take the derivative (whether or not it is differentiable everywhere!).
Hardly whether or not, as you can't take the derivative at all if the function is non-differentiable everywhere.

Quote:
Originally Posted by The Chaz
Then just keep in mind the x-values where the function is not differentiable, and set the derivative equal to zero to find the horizontal tangents.
We're not mind-readers, so "just keeping in mind" something may not suffice. Finding horizontal tangents won't tell you where the function is increasing.

Quote:
Originally Posted by ikurwa
if a function is continuous i.e. it exists for a<x<b then it's differentiable.
A function can exist and be discontinuous. It will be non-differentiable where it's discontinuous, and may also be non-differentiable at some points where it's continuous.
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April 17th, 2011, 05:59 PM   #7
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Re:

Quote:
Originally Posted by skipjack
Quote:
Originally Posted by The Chaz
If you are asked to find where a function is increasing (i.e. "growing"), you would still take the derivative (whether or not it is differentiable everywhere!).
Hardly whether or not, as you can't take the derivative at all if the function is non-differentiable everywhere.
Key word is "you" (=/= "skipjack"). This guy doesn't know of such functions.

Quote:
Quote:
Originally Posted by The Chaz
Then just keep in mind the x-values where the function is not differentiable, and set the derivative equal to zero to find the horizontal tangents.
We're not mind-readers, so "just keeping in mind" something may not suffice. Finding horizontal tangents won't tell you where the function is increasing.
That's where you shouldn't have truncated the quote... "etc." included (in my mind, at least!) using the zeros of the derivative to break the real number line into intervals... etc
Quote:
Quote:
Originally Posted by ikurwa
if a function is continuous i.e. it exists for a<x<b then it's differentiable.
A function can exist and be discontinuous. It will be non-differentiable where it's discontinuous, and may also be non-differentiable at some points where it's continuous.
[/quote]
I know.
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April 17th, 2011, 06:14 PM   #8
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Quote:
Originally Posted by The Chaz
This guy doesn't know of such functions.
If you know that, either you're a mind-reader or "this guy" was a reference to yourself (which seems unlikely). Feel free to demonstrate the efficacy of your approach in relation to the function y = x - arccos(cos(x)).
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April 17th, 2011, 06:32 PM   #9
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Re: how do you know if a func is Differentiable?

Case 1.
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April 17th, 2011, 07:16 PM   #10
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Which is what, and in reply to what?
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