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How to find f'(0) from a left handed limit? (multiple questions)

  1. Sep 25, 2012 #1
    My homework is due really soon.
    Here are all the questions I have absolutely NO idea how to do.


    1.
    Let:
    f(X) = -6x^2+6x for x<0 and 7x^2-3 for x≥ 0

    According to the definition of derivative, to computer f'(0), we need to compute the left hand limit
    lim x-->0- =
    and the right hand limit
    lim x--> 0+
    We conclude that f'(0)=

    So...
    I've figured out that the right hand limit is
    [7x^2-3-(7(0)^2-3)]/(x-0)
    and that f'(0)=DNE

    My answer for the left hand limit is
    [-6x^2+6x-(-6(0)^2+6(0))]/(x-0)
    but the website won't accept my answer.

    2.
    Given the following table:

    x----- 0.0097 ------- 0.0098 -------- 0.0099 -------- 0.01---- 0.0101 ----- 0.0101
    f(x)-- 0.54783494--0.99814343 -- 0.46101272-- (-0.50636564)---- (-.9987636)

    Calculate the value of f'(0.0099) to two place of accuracy.

    3.
    Let f(x) = 2/(x-8)
    According to the definition of derivative, f'(x)= lim t-->x (2(x-8)-2(t-8))/((t-x)(t-8)(x-8))
    The expression inside the limit simplifies to: 2/[-(x-8)/(t-8)]
    Taking the limit of this fractional expression gives us
    f′(x)= ?

    Please, please, please help me. I am SO frustrated.

    Thanks!
     
  2. jcsd
  3. Sep 26, 2012 #2

    haruspex

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    Wrong. Fill in the missing steps.
     
  4. Sep 26, 2012 #3
    What do you mean 'missing steps'?
     
  5. Sep 26, 2012 #4

    haruspex

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    Pleae write out the steps between those last two statements so that I can see where you are going wrong.
     
  6. Sep 26, 2012 #5

    HallsofIvy

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    You are being far too "casual". What is "7x^2- 3- (7(0)^2- 3)" Actually write that out in detail.

    I don't see an answer! You haven't yet taken the limit.

    Okay, what have you done? Since you clearly know the formula, it's just a matter of arithmetic.

    Where did you get that?

    As t goes to x, what is [itex]\frac{x- 8}{t- 8}[/itex]?

     
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