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Tangent line that passes through origin

  1. Jan 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Find a > 0 such that the tangent line to the graph of

    f(x) = [itex]x^{2}[/itex]e[itex]^{-x}[/itex] at x = a passes through the origin.


    15dpf06.jpg

    2. Relevant equations



    3. The attempt at a solution

    First I found the derivative to be:

    [itex]-e^{-x}(x-2)x[/itex]

    , which is the slope of the function.

    I know the tangent line must pass through the origin (0,0), but I'm a bit stuck.

    I took a year off from math and am trying to get back in to math mode. Any help to lead me to the next step would be great.
     
  2. jcsd
  3. Jan 9, 2012 #2
    maybe this is the way
    http://en.wikipedia.org/wiki/Tangent
    find f '(x) and than substitute x=a
    use point-slopre formula and substitude k=f '(a) inside, than epress from there y
    than y=x=0 (through origin)
    and .... tell me your result
     
    Last edited: Jan 9, 2012
  4. Jan 9, 2012 #3
    Formula for a line through the origin is y = mx

    At a, y=f(a), m=f'(a), x=a
     
  5. Jan 9, 2012 #4
    you are right (my k is yours m,)
     
    Last edited: Jan 9, 2012
  6. Jan 10, 2012 #5
    continuing from Joffan and Elliptic: find where f'(x)x = f(x) = a. find f'(a). at that point you are basically done.
     
  7. Jan 10, 2012 #6
    Thanks guys for the responses.

    Here's what I have so far:

    [itex]f(a)=a^{2}-e^{-a}[/itex]

    [itex]f '(a)=-e^{-a}(a-2)a[/itex]


    [itex]y=f '(a)(x-a)+f(a)[/itex]

    which gave me

    [itex]a=0,3 [/itex](zero is not in the domain, so three is the only solution)

    There must be an error, just looking at it graphically I know that 'a' must be before the extrema (2).
     
  8. Jan 10, 2012 #7
    I found my error.
    I missed a negative sign :/
    My final answer is a=1.
    The problem doesn't require it, but I checked my answer by graphing the tangent line using y=f '(a)x, and it worked out.

    thanks again for all your help.
     
  9. Jan 10, 2012 #8
    Yes, a=1. Well done.
     
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