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Math Help - Equation of tangent lines, find intersection

  1. #1
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    Equation of tangent lines, find intersection

    I am having a bit of an issue with this problem.

    f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.

    "Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."

    Okay so for the tangent lines they want me to take the derivatives of f(x) and g(x) right?

    Well here they are:
    f'(x) = -ce^(-cx)
    g'(x) = ce^(-cx)

    f'(0) = -c
    g'(0) = c

    But these are constants! Just flat horizontal lines that won't intersect because they're parallel. So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?

    Any hints are much appreciated.
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Gryllis View Post
    I am having a bit of an issue with this problem.

    f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.

    "Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."

    Okay so for the tangent lines they want me to take the derivatives of f(x) and g(x) right?

    Well here they are:
    f'(x) = -ce^(-cx)
    g'(x) = ce^(-cx)

    f'(0) = -c
    g'(0) = c

    But these are constants! Just flat horizontal lines that won't intersect because they're parallel. they ain't parallel and also they are NOT horizontal. Horizontal lines have slope=0. Here the slopes of the two lines are the derivatives you have calculated. observe that the tangent lines considered have DIFFERENT slopes. one has slope c and other has -c. Of course slope of a line will be constant. Don't get confused over that
    So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?

    Any hints are much appreciated.
    did this help?
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  3. #3
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    Quote Originally Posted by Gryllis View Post
    f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.
    "Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."
    f'(x) = -ce^(-cx) & f'(0) = -c
    g'(x) = ce^(-cx) & g'(0) = c
    So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?
    Note that (0,1)\in f~\&~(0,-1)\in g.
    So the tangent lines are:
    y_f=f'(0)x+f(0)=-cx+1 and
    y_g=g'(0)x+g(0)=cx-1.
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  4. #4
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    Ooooh. How did I not see that...

    Thank-you!
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