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Topic: [ap-calculus] Going crazy
Replies: 1   Last Post: Oct 10, 2012 8:56 PM

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Stu Schwartz

Posts: 308
Registered: 8/23/06
Re: [ap-calculus] Going crazy
Posted: Oct 10, 2012 8:56 PM
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NOTE:
This ap-calculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus
Teacher Community Forum at https://apcommunity.collegeboard.org/getting-started
and post messages there.
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On Oct 10, 2012, at 4:41 PM, Rebecca Tackett <rtackett@evansvilledayschool.org> wrote:

> NOTE:
> This ap-calculus EDG will be closing in the next few weeks. Please sign up 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
>


A nice problem.

First of all, set the slopes equal to each other: 2x = -2x + 2 and get x = 1/2. Hold that aside for awhile,

Consider y = x^2. The slope = 2x. At some point (a, a^2), the equation of the tangent line will be y - a^2 = 2a (x - a) or y = 2ax - a^2.

Consider y = -x^2 + 2x - 5. The slope = 2 - 2x. At some point (a, -a^2 + 2a - 5), the equation of the tangent line will be y - (-a^2 + 2a - 5) = (2 - 2a)(x - a) or y = 2x - 2ax + a^2 - 5

Now use the x = 1/2 and set the equations equal to each other: a - a^2 = 1 - 2a + a^2 - 5 which simplifies to a^2 - a - 2 = 0. Solve and you get a = 2 or a = -1:

At a = 2: the equation line is y = 4x - 4

At a = -1, the equation line is y = -2x + 1

Stu Schwartz
www.mastermathmentor.com



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