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Homework Help: Slope of normal to a given function's inverse

  1. Apr 22, 2010 #1
    1. The problem statement, all variables and given/known data
    f(x) = 2x2 + 4e(5x)
    is invertible. Give the slope of the normal line to the graph of f-1 at x = 4.

    2. Relevant equations
    (Given in question)


    3. The attempt at a solution
    I don't know how to solve this question. But , I found the following:-

    f(4) = 32 + 4e20

    let y = f-1(x), then
    dy/dx = 1/f ' (y)
    f ' (x) = 4x + 20e(5x)
    Hence, dy / dx = 1/(4y + 20e(5y))
    I don't know if I am heading in the right direction.
    Please help!!
    Thank you.
     
  2. jcsd
  3. Apr 22, 2010 #2

    LCKurtz

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    Actually,the function is not invertible because it isn't 1-1. But you might observe that f(0) = 4 and it is invertible in a neighborhood of x = 0.

    But f(4) isn't relevant to the question. The question is about the inverse function's slope at x = 4. Remember that f(0) = 4 means (0,4) is on the graph of y = f(x) and it also means that f-1(4) = 0 and (4,0) is on the graph of y = f-1(x).

    The slope of that function at that point is what you are looking for. Does that help?
     
  4. Apr 23, 2010 #3
    So, (4,0) lies on the graph of f-1(x),
    y = f-1(x)
    dy / dx = 1/(4y + 20e(5y))
    Hence , slope of f-1(x) , dy / dx = 1/20
    which implies , slope of the normal to the graph = -1 /(1/20) = -20
    Ans = -20

    Am I correct?
    Thanks a lot!
     
  5. Apr 23, 2010 #4

    LCKurtz

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    If you calling y = f-1(x) then I would write

    y' = 1 / f'(x) = 1/(4x + 20e(5x))

    and yes, your answer looks correct.
     
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