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Thread: Applications of Differentiation involving e to power of x

  1. #1
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    Applications of Differentiation involving e to power of x

    Find the equation of the tangent to the curve y = e to the power x at the point where x = a. Hence find the equation to the tangent to the curve y = e to the power x which passes through the origin. The straight line y = mx intersects the curve y = e to the power x in two distinct points. Write down an inequality for m.
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  2. #2
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    Hi

    The equation of the tangent of the curve representing f at abscissa a is
    $\displaystyle y = f'(a)(x-a)+f(a)$

    When $\displaystyle f(x) = e^x$ you get $\displaystyle y = e^a x + e^a(1-a)$

    It passes through the origin when $\displaystyle e^a(1-a) = 0$ which gives a = 1
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  3. #3
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by puggie View Post
    Find the equation of the tangent to the curve y = e to the power x at the point where x = a. Hence find the equation to the tangent to the curve y = e to the power x which passes through the origin. The straight line y = mx intersects the curve y = e to the power x in two distinct points. Write down an inequality for m.

    the curve $\displaystyle y=e^x$ dose not pass through the origin how you can find the tangent line of it at (0,0)

    $\displaystyle e^x=0$ you can't find a value of x which make that equation true

    the tangent for $\displaystyle y=e^x$ at x=a first find
    y when x=a you will have

    $\displaystyle y=e^a$

    now derive y $\displaystyle y'=e^x$ sub a in it you will get the slope which equal $\displaystyle e^a$ finally the tangent line of y is

    $\displaystyle y-e^a=e^a(x-a)$

    then
    $\displaystyle y=e^x$ can't intersect $\displaystyle y=mx$ in tow points y=mx pass through the origin but $\displaystyle y=e^x$ dose not pass through it

    it is clear or not ........
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