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Unread 10-16-2005, 10:49 PM
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Calculus Query

Is there a function f(x) such that the derivative of f(x) is equal to the inverse of f(x)?

Prove it.
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Unread 10-16-2005, 11:07 PM
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Re: Calculus Query

this is kind of cheating but:
f(x)=0 where x=0

I assume you are asking this for a reason. Do you have a proof that no such function exists or something?

Interesting question.

---edit----
I was thinking he meant inverse function not 1/f(x)
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Last edited by Rickertsen2 : 10-16-2005 at 11:45 PM.
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Unread 10-16-2005, 11:22 PM
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Re: Calculus Query

Quote:
Originally Posted by Rickertsen2
this is kind of cheating but:
f(x)=0 where x=0

Actually:

f(x)=0
f'(x)=0
f(x)^-1=1/0 which is a non-removable discontinuity. DNE

I dont think there is any, but then again I haven really taking time to look for a way.

Now you are gonna make me lose sleep on this.

Dave
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Unread 10-16-2005, 11:26 PM
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Re: Calculus Query

Quote:
Originally Posted by Rickertsen2
this is kind of cheating but:
f(x)=0 where x=0

I assume you are asking this for a reason. Do you have a proof that no such function exists or something?

Interesting question.
The function you proposed (a constant) is not invertible, unfortunately.
I'm still giving this some thought, but I don't think there's such a function.
It is, indeed an interesting question. Does look like an assignment, though

More later...

[EDIT: David Guzman beat me to it...]
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Unread 10-16-2005, 11:43 PM
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Re: Calculus Query

wait, when you say inverse, are you refering to 1/f(x) or the inverse function of f(x)

I was assuming he meant inverse function, but now that i think about it, he probably would have inserted the word "function" if that is what he meant. Please clarify this.
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Last edited by Rickertsen2 : 10-16-2005 at 11:46 PM.
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Unread 10-16-2005, 11:52 PM
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Re: Calculus Query

Quote:
Originally Posted by Rickertsen2
wait, when you say inverse, are you refering to 1/f(x) or the inverse function of f(x)

I was assuming he meant inverse function, but now that i think about it, he probably would have inserted the word "function" if that is what he meant. Please clarify this.
Judging by the question, phrontist seems to know his math. As such, if he meant 1/f(x), I believe he'd have used "the reciprocal" of f(x).
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Unread 10-16-2005, 11:59 PM
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Re: Calculus Query

I might be wrong but:

f(x)=1
f'(x)=0
f(x)^-1=0/1=0

EDIT: OK, I am wrong
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Last edited by David Guzman : 10-17-2005 at 06:32 AM.
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Unread 10-17-2005, 12:07 AM
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Re: Calculus Query

Quote:
Originally Posted by David Guzman
I might be wrong but:

f(x)=1
f'(x)=0
f(x)^-1=0/1=0
EDIT:
If you're considering the reciprocal, then 1/1 is not equal to 0.
If you're considering the inverse function, then I didn't understand what you tried to do.

f(x)^-1 is a poor notation because it usually refers to the inverse of a function but can also be mistaken for the reciprocal of the function.
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Last edited by Manoel : 10-17-2005 at 12:40 AM. Reason: Correction and clarification
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Unread 10-17-2005, 12:18 AM
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Re: Calculus Query

grr math things like this frusterate my simple mind. I am no math person and i don't entirely understand why my solution is wrong. Maybie i am just stubborn. Lets define the inverse of a function as a being a function where f(x)=y and f^-1(y)=x for all x. By this logic if f(0)=0 then shouldn't f^-1(0) be 0 as well and shouldn't my solution work?
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Unread 10-17-2005, 12:38 AM
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Re: Calculus Query

Quote:
Originally Posted by Rickertsen2
grr math things like this frusterate my simple mind. I am no math person and i don't entirely understand why my solution is wrong. Maybie i am just stubborn. Lets define the inverse of a function as a being a function where f(x)=y and f^-1(y)=x for all x. By this logic if f(0)=0 then shouldn't f^-1(0) be 0 as well and shouldn't my solution work?
Rickertsen2,

For a function to be invertible, it has to be bijective - "one on one" and "onto", that is, there's only one x value associated with a given y value, and that holds for every x. As an example, y = x^2 is not invertible, because for x = +- 2, y = 4. To think of it in another way, if you were to invert that function (rotate its graph about the line y = x), you'd get a function that, for one value of x, is defined for two y values. That is not a function.
For a quick test, draw an horizontal line in this graph and sweep it up and down. If it ever touches two points on the curve, then the function is not invertible.
You can see that your constant function does not meet the requisites for an invertible function.

A Google search will provide thousands of good sources for information, this site is a nice one, with a simple explanation and pictures (who doesn't like pictures? )
http://archives.math.utk.edu/visual....s/0/inverse.6/
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Unread 10-17-2005, 06:29 PM
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Re: Calculus Query

I meant (and said) inverse, not reciprocal. This isn't an assignment, just something that occured to me during my endless and tedious BC Caluclus (anyone else have Early Transcendentals?) homework.

I've been trying to approach the problem visually, the inverse of a function being that function "mirrored" about the line y=x. I think perhaps this is needlessly painful, I'm going to try some algerbraic manipulation of the definition of the derivative...

My guess is that no such function exists, but I wonder if there is an elegant proof of that.
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Unread 10-17-2005, 07:37 PM
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Re: Calculus Query

Quote:
Originally Posted by phrontist
(anyone else have Early Transcendentals?)
You mean the Stewart book? Yep, had that. I'd say it's not a bad book but there are some weird quirks every now and then where it won't explain some things very well. Mine came with CDs but I'd say they were pretty much useless.
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Unread 10-17-2005, 07:37 PM
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Re: Calculus Query

Quote:
Originally Posted by phrontist
I meant (and said) inverse, not reciprocal. This isn't an assignment, just something that occured to me during my endless and tedious BC Caluclus (anyone else have Early Transcendentals?) homework.

I've been trying to approach the problem visually, the inverse of a function being that function "mirrored" about the line y=x. I think perhaps this is needlessly painful, I'm going to try some algerbraic manipulation of the definition of the derivative...

My guess is that no such function exists, but I wonder if there is an elegant proof of that.
The following fact will lead you in the right direction.

Let f be a function that is differentialable on an interval I. Suppose that f has a defined inverse function, called g.

Using the defintion of an inverse, and the chain rule, it can be shown that

g'(x) = 1/f'(g(x)), where f'(g(x)) != 0

This should help with you algebraic manipulation.
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Unread 10-17-2005, 07:56 PM
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Re: Calculus Query

You know what? I was tempted to try this problem, but my hatred of calculus overcame me while I was working some stuff out. Calculus tends to have more letters than numbers. This is math, not grammar... These letters also tend to not be in the English alphabet. Again, this is math, not a foreign language.

Today's episode of Sesame Street was brought to you by... The number e ...And, the letter mu.

There also tends to be odd quirks. Take Gabriel's Horn. How can you possibly have something with an infinite surface area, infinite cross-section, but a finite volume!? And the volume isn't just finite, it happens to be a very specific number/non-English letter: pi.

Yes, I realize the power of calculus. I also respect it. Seeing it work in physics still amazes me. But, I just don't like it.
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Unread 10-17-2005, 09:23 PM
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Re: Calculus Query

This one is seriously making my head hurt. I recollect hearing a similar question proposed but my brain is too fuddled to remember the question in its entirety.

I say if 3rd semester Calc doesn't cover it, then it just isn't worth the headache.

...not yet, at least
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