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MarcLeclair

  • 3 years ago

Really need help understanding this limit/derivative question... The limit Lim x approachs 5pi CosX +1 / x-5pi represents the derivative of some function f(x) at some number a. Find f and a . I don't even understand the wording of this question...

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  1. zepdrix
    • 3 years ago
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    Remember the limit definition of a derivative?\[\large f'(x)=\lim_{h \rightarrow 0}\frac{f(x-h)-f(x)}{h}\]Well there is also another definition that we see less often, of this form,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\]This is the one that we want to analyze.

  2. zepdrix
    • 3 years ago
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    If we compare this to the limit we were given, we can see that is looks like our A value will be 5pi yes? \[\large \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x-5\pi}\]

  3. zepdrix
    • 3 years ago
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    The top is still a little tricky though, we have to sort it out.

  4. MarcLeclair
    • 3 years ago
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    wait the second one just looks like the mean value theorem? it oddly looks like f(a)-b/a-b = f'(c)

  5. MarcLeclair
    • 3 years ago
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    oh nevermind me theres a limit sorry im a bit tired

  6. zepdrix
    • 3 years ago
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    heh, yah it looks similar :)

  7. zepdrix
    • 3 years ago
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    \[\large \cos(5\pi)=?\]

  8. MarcLeclair
    • 3 years ago
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    So yeah, but we're looking for the original function is that it or looking for the derivative? and cos5pi would be... .96 ? But I think i did it in degree so it might be wrong.

  9. zepdrix
    • 3 years ago
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    That's one of your special angles that you're going to want to remember. It will produce the same value as Pi. 5pi is Pi with an extra spin around the circle. -1 yes?

  10. zepdrix
    • 3 years ago
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    With 2 extra spins* my bad.

  11. MarcLeclair
    • 3 years ago
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    I'll ask you one thing before we continue, when taking pi in a derivative, do we ALWAYS use it in radians? I mean sometimes it works when I don't put it in my calculator as a radian

  12. zepdrix
    • 3 years ago
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    If you're dealing with Pi, then yes you need to be in radians :o You could convert to degrees if radians are confusing you though.

  13. MarcLeclair
    • 3 years ago
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    i mean pi i mean cos and sin... wow I' m sounding stupid

  14. zepdrix
    • 3 years ago
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    5pi is the same as 180 degrees.

  15. MarcLeclair
    • 3 years ago
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    so in the question am I using L'hospital rule or finding the limit? that's where I'm lost. the question is really confusing . And alright thanks for explaining the pi. :)

  16. zepdrix
    • 3 years ago
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    No L'Hop. We're relating a weird looking limit back to the Limit Definition of a Derivative. We need to match up the pieces so we can see what the original function was. So far we've established that our A value is 5pi. If you're unsure about that, compare the form of our limit with the Definition,\[\large \large f'(x)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\qquad\qquad \rightarrow \qquad \qquad \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x-5\pi}\]

  17. zepdrix
    • 3 years ago
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    See the a?

  18. MarcLeclair
    • 3 years ago
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    Yeah it's represented by 5pi?

  19. MarcLeclair
    • 3 years ago
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    so i have to equate both limit? and finding my h?

  20. zepdrix
    • 3 years ago
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    no h, we're using the second definition that i posted, not the one involving h.

  21. zepdrix
    • 3 years ago
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    It's another form of the limit definition. It comes up less often.

  22. MarcLeclair
    • 3 years ago
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    My bad I understood we had to manipulate it back to the other form

  23. MarcLeclair
    • 3 years ago
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    Then when its ask to find F, am I suppose to find cosx?

  24. MarcLeclair
    • 3 years ago
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    sorry, the wording is really throwing me off

  25. zepdrix
    • 3 years ago
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    If we can show that the limit matches the DEFINITION, then we can show what our F is. So we've established that 5pi matches the A we're looking for. We've also shown that cos(5pi)=-1. If we can somehow find a -1 in the top of that fraction, we can make it look like the Definition.

  26. MarcLeclair
    • 3 years ago
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    Oh wow. I understand. This was a very basic question but I have never seen that definition in my entire class. Thanks a lot! :)

  27. zepdrix
    • 3 years ago
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    \[\large \frac{\cos x+1}{x-5\pi} \qquad =\qquad \frac{\cos x-(-1)}{x-5\pi} \qquad = \qquad \frac{\cos x-(\cos 5\pi)}{x-5\pi}\]

  28. zepdrix
    • 3 years ago
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    Make sense? :) k cool!

  29. MarcLeclair
    • 3 years ago
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    Thanks a lot you're patient!

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