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Thread: Need Help With a Tangent Line Problem

  1. #1
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    Need Help With a Tangent Line Problem

    Could someone please help me with the following problem I'm having difficulties with.

    1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

    2) Determine where this line intersects the y-axis
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by erimat89 View Post
    Could someone please help me with the following problem I'm having difficulties with.

    1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

    2) Determine where this line intersects the y-axis
    First find $\displaystyle y'$:

    $\displaystyle y'=2Ax$.

    Thus, the slope of the line at $\displaystyle x=c$ is $\displaystyle 2Ac$.

    Thus, the tangent has the equation $\displaystyle y-Ac^2=2Ac(x-c)\implies y=2Acx-Ac^2$

    Now find where it crosses the x axis:

    $\displaystyle 0=2Acx-Ac^2\implies x=\dots$.

    It crosses the y axis when x=0.

    So $\displaystyle y=\dots$

    Can you take it from here?

    --Chris
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  3. #3
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    Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by erimat89 View Post
    Could someone please help me with the following problem I'm having difficulties with.

    1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

    2) Determine where this line intersects the y-axis
    Quote Originally Posted by erimat89 View Post
    Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?
    It asks you to show that the line intersects the x-axis at the point (c/2,0).

    Since I got the equation of the line to be $\displaystyle y=2Acx-Ac^2$, at y=0, the line crosses the x-axis.

    Thus $\displaystyle 0=2Acx-Ac^2\implies 2Acx=Ac^2\implies x=\frac{Ac^2}{2Ac}\implies x=\frac{c}{2}$. So we see that it crosses the x-axis at $\displaystyle \left(\tfrac{1}{2}c,0\right)$

    $\displaystyle \mathbb{Q.E.D.}$

    Now, it intersects the y axis when x=0. Thus, we see that $\displaystyle y=2Ac(0)-Ac^2\implies y=-Ac^2$.

    Thus, the y intercept is $\displaystyle \left(0,-Ac^2\right)$.

    Does this make sense?

    --Chris
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  5. #5
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    Thanks

    Makes alot of sense you clarified things for me. I've never been good with any sort of math problem that involves words for some reason I get thrown off.
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