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Topic: [ap-calculus] Going crazy
Replies: 1   Last Post: Oct 10, 2012 9:08 PM

 Carl LaCombe Posts: 10 Registered: 10/6/11
Re: [ap-calculus] Going crazy
Posted: Oct 10, 2012 9:08 PM

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Rebecca,

The way I saw to do this, it is a challenging problem. The issue is that
the slopes are equal at the lines points of tangency with the parabolas,
but those don't happen at the same x-values.

I chose the first parabola to center on, since it has a simple equation,
and set the x-value of the point of tangency to the first parabola as x_1.
This means that the point of tangency is (x_1, x_1^2) and the slope is
2*x_1. I set the x-value of the point of tangency to the second parabola as
x_2. I found the equation of the tangent line using the slope and point of
tangency to the first parabola.

Then I get two equations in the two unknowns, x_1 and x_2. The first is
from knowing that the derivatives of the first parabola at x_1 and of the
second parabola at x_2 must be equal. The second one is from knowing that
the y-value of the tangent line at x_2 must equal the y-value of the second
parabola. Once you solve these two equations for x_2 and x_1 then you can
get the equations of the tangent lines.

Perhaps there is a more intuitive way, but this one works.

Carl

On Wed, Oct 10, 2012 at 4:41 PM, Rebecca Tackett <
rtackett@evansvilledayschool.org> wrote:

> NOTE:
> for the new AP Calculus
> Teacher Community Forum at
> https://apcommunity.collegeboard.org/getting-started
> and post messages there.
>
> ------------------------------------------------------------------------------------------------
> I need some help with this problem. What am I missing?
>
>
> Graph the two parabolas y=x^2 and y=-x^2+2x-5 in the same coordinate
> plane. Find the equations of the lines that are simultaneously tangent to
> both parabolas. I can visualize where the two equations are but now I can?t
> figure out how to write their equations?
>
> I keep setting the derivatives equal to each other as the slopes of the
> same linear equation and solved for x and got x=0 and x=.5, but I don't
> think that's correct. And I can't figure out how to write the equation at
> either of those points so that it is tangent simultaneously. Any thoughts?
> The problem is from Larson 4e, p.201 #2
>
> Thanks,
> Rebecca Tackett
> Upper School Mathematics Instructor
> Evansville Day School
>
> ---
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--
Carl LaCombe
Mathematics Teacher
The Cambridge School of Weston