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It's been awhile since I've done anything with rates of change and I'm struggling with deriving a formula in terms of 'x.'

From what I can recall.. the average rate of change and instantaneous rate of change are basically the same? Instantaneous is just within a smaller interval?


EDIT:

I think I'm on the right track now. I use the difference quotient I think?

I'm working through them now, but if someone could possibly verify that I'm doing them correctly, that'd be awesome! Thanks.


For example, if I'm asked to find the instantaneous rate of change of the following functions for any value of x:

1.) $f(x) = a(x^2) + bx + c$

Solution: $ F(x) = 2ax + b $


2.) $ g(x) = \sqrt{x} $

Solution: $ G(x) = \cfrac{1}{2\sqrt{x}}$


3.) h(x) = 1/x

Solution: h(x) = 1


I don't even get how or where to start solving any of these to get a formula for the instantaneous rate of change for x. Would I substitute x into the formula for h?

Thanks

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    $\begingroup$ "The instantaneous rate of change" What do you mean by instantaneous, can you express this in terms of a limit? Also what is changing? And with respect to what? $\endgroup$
    – Ethan
    May 1, 2013 at 5:04
  • $\begingroup$ I didn't create a "new" account. That was my old one. And it's no longer allowed to ask questions on Stackoverflow because I asked a question that got downrated extremely fast by a lot of people. It's dumb but instead of using multiple accounts, I use this now. Sorry I realized it after I asked the question. $\endgroup$
    – Cozen
    May 1, 2013 at 5:06
  • $\begingroup$ @Ethan that's exactly my question. I have no Idea. That is quite literally what my review question is asking me. $\endgroup$
    – Cozen
    May 1, 2013 at 5:07
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    $\begingroup$ I think you should review the definition of the derivative. $\endgroup$
    – Jemmy
    May 1, 2013 at 5:10
  • $\begingroup$ Look up differential calculus on Wikipedia or in a high school textbook. The right answer for 1 is $f'(x)=2ax+b$ $\endgroup$
    – Dale M
    May 1, 2013 at 5:46

1 Answer 1

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The instantaneous rate of change formula is simply the derivative which can be obtained either by going through the difference quotient calculation or simply by following standard derivative rules. For example for the third question we have

$\lim_{h\to0}\frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\lim_{h\to0}\frac{\frac{x-(x+h)}{x(x+h)}}{h}=\lim_{h\to0}\frac{-1}{x(x+h)}=\frac{-1}{x^{2}}$.

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  • $\begingroup$ How is it x-(x+h) / x(x+h)? $\endgroup$
    – Cozen
    May 1, 2013 at 7:29
  • $\begingroup$ OK, I was getting ready to delete that lol I just realized that. $\endgroup$
    – Cozen
    May 1, 2013 at 7:34
  • $\begingroup$ You need a common denominator to add fractions so for the above case we have $\frac{1}{x+h}-\frac{1}{x}=\frac{x}{x(x+h)}-\frac{x+h}{x(x+h)}=\frac{x-(x+h)}{x(x+h)}$ $\endgroup$
    – user71352
    May 1, 2013 at 7:36
  • $\begingroup$ what happens to h in the denominator when you go from the 2nd to third step? $\endgroup$
    – Cozen
    May 1, 2013 at 7:36
  • $\begingroup$ Note that the numerator in my previous comment is $x-(x+h)=-h$. This will cancel with h in the denominator. $\endgroup$
    – user71352
    May 1, 2013 at 7:37

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