## how do you know if a func is Differentiable?

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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?
$y=x^3-5x^2+3x+7$
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.
$y'=3x^2-10x+3$
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?
peaceofmind
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### Re: how do you know if a func is Differentiable?

peaceofmind wrote:The solution in my book starts out by stating that this func is Differentiable. How do you know it is Differentiable?
$y=x^3-5x^2+3x+7$
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:
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.

The Chaz
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peaceofmind wrote:Let's say I'm asked to find all x values when this function is growing.

For x < ⅓, the function is increasing.
For x = ⅓, the function has relative extrema.
For ⅓ < x < 3, the function is decreasing.
For x = 3, the function has relative extrema.
For x > 3, the function is increasing.
johnny
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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!
ikurwa
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### Re: how do you know if a func is Differentiable?

ikurwa wrote: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
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
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The Chaz wrote: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.

The Chaz wrote: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.

ikurwa wrote: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.
skipjack
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### Re:

skipjack wrote:
The Chaz wrote: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.

The Chaz wrote: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
ikurwa wrote: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.
Last edited by The Chaz on Sun Apr 17, 2011 9:02 pm, edited 2 times in total.
Reason: But I don't know how to use quotes!

The Chaz
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The Chaz wrote: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)).
skipjack
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### Re: how do you know if a func is Differentiable?

Case 1.

The Chaz
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Which is what, and in reply to what?
skipjack
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### Re: how do you know if a func is Differentiable?

I'm not a mind-reader, but I do make educated guesses. I'm not afraid to be wrong, but so far (in this thread - with certain assumptions about the context of the OP's questions) all I've seen is that some things that I didn't say are wrong.

Actually, looking back on the image uploaded in the first thread, I wouldn't be surprised if "differentiable" were roughly defined (in this context) as "no jumps, holes, sharp turns, or cusps". Seriously! Why aren't we trying to understand the question more?

I really don't know what the point of all this pedant-ery (and the now-deleted brown-nosing) is, but a similar practice of mine (not the concurring opinions...) cost me the most valuable relationship under the sun. Let's just say that whenever I got within delta of certain topics, my wife would run to at least a distance of epsilon from me...

The Chaz
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### Re: how do you know if a func is Differentiable?

Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
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The Chaz wrote:I wouldn't be surprised if "differentiable" were roughly defined (in this context) as "no jumps, holes, sharp turns, or cusps". Seriously!
There are very simple strictly increasing functions that have no jumps, holes, sharp turns, or cusps, and that are differentiable for all real x except for one value of x. Perhaps the phrase "in this context" is your get-out card.
skipjack
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