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 June 23rd, 2018, 06:17 AM #1 Newbie   Joined: Jun 2018 From: Jordan Posts: 9 Thanks: 0 Using rate of change of a function to find an expression's value Hey. Simple question; Given that f(x) is function, if x changed from x to x + h, and the rate of that change is (x^2 * h) - (4*x*h^2), then f'(3) would be? how can this be solved?
 June 23rd, 2018, 07:32 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,244 Thanks: 887 If x changes from x to x+ h then f changes from f(x) to f(x+ h). That is a change of f(x+ h)- f(x) and so an average rate of change of $\displaystyle \frac{f(x+ h)- f(x)}{h}$. We get the derivative by taking the limit as h goes to 0. IF we were given that that average rate of change is $\displaystyle x^2h- 4xh^2$ then that limit, for all x, would be 0 and so f'(3) would be 0. However I suspect that you have mis-stated the problem. If $\displaystyle x^2h- 4xh^2$ is the change itself rather than the rate of change, then the average rate of change would be $\displaystyle \frac{x^2h- 4xh^2}{h}= x^2- 4xh$ and the limit of that, as h goes to 0, is $\displaystyle x^2$. In this case, $\displaystyle f'(3)= 3^2= 9$. Thanks from Integraluser
 June 24th, 2018, 03:02 AM #3 Global Moderator   Joined: Dec 2006 Posts: 19,186 Thanks: 1648 You haven't proved that x can equal 3. The question is flawed. Thanks from Integraluser
June 25th, 2018, 09:32 AM   #4
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Quote:
 Originally Posted by Country Boy If x changes from x to x+ h then f changes from f(x) to f(x+ h). That is a change of f(x+ h)- f(x) and so an average rate of change of $\displaystyle \frac{f(x+ h)- f(x)}{h}$. We get the derivative by taking the limit as h goes to 0. IF we were given that that average rate of change is $\displaystyle x^2h- 4xh^2$ then that limit, for all x, would be 0 and so f'(3) would be 0. However I suspect that you have mis-stated the problem. If $\displaystyle x^2h- 4xh^2$ is the change itself rather than the rate of change, then the average rate of change would be $\displaystyle \frac{x^2h- 4xh^2}{h}= x^2- 4xh$ and the limit of that, as h goes to 0, is $\displaystyle x^2$. In this case, $\displaystyle f'(3)= 3^2= 9$.

Exactly.

That's the problem, I've stated the problem here literally from the school math book, and it says "rate of change" rather than "change in the function" or "change in y", which made me think: that would absolutely be 0 since the result of that limit as a whole after we got rid of "h" in the denominator is an unimpeachable zero after substituting h with 0. So this is considered a mistake in the book.

@skipjack Yeah pardon, I should've mentioned that it's a multiple choice question which doesn't have a "f'(3) doesn't exist" choice and such. Neither it had "0" as a choice too.

 June 27th, 2018, 04:54 AM #5 Newbie   Joined: Jun 2018 From: Jordan Posts: 9 Thanks: 0 Edit: I've posted a reply days ago. Not sure why it isn't here. Anyways, yes. @Country Boy I've been thinking like that exactly, that it should of been "change of y" or "dy" rather than "rate of change", since rate of change is resulted of cancelling the h in the denominator so that we could process the limit. Otherwise rate of change would always be 0. @skipjack Yeah, pardon. I should've mentioned that it's a multi choice question with 4 choices and "f'(3) doesn't exist" or "0" ones don't exist. "9" was among the choices. Last edited by Integraluser; June 27th, 2018 at 05:01 AM.
 June 27th, 2018, 10:36 AM #6 Global Moderator   Joined: Dec 2006 Posts: 19,186 Thanks: 1648 What were the answer choices? Was the original question in English?
 June 28th, 2018, 04:22 AM #7 Newbie   Joined: Jun 2018 From: Jordan Posts: 9 Thanks: 0 Nope it's in Arabic, actually wait 0 happens to be there.. the choices are: 9, -9,0, -3
 June 28th, 2018, 05:57 AM #8 Global Moderator   Joined: Dec 2006 Posts: 19,186 Thanks: 1648 Does the latest edition of the book omit or modify this question?
 June 29th, 2018, 03:52 AM #9 Newbie   Joined: Jun 2018 From: Jordan Posts: 9 Thanks: 0 Nope. There's even another question that has the same subtext only with change of function.
 June 29th, 2018, 06:48 AM #10 Global Moderator   Joined: Dec 2006 Posts: 19,186 Thanks: 1648 Can you post that other question and its answer choices?

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