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

 Doug Kuhlmann Posts: 3,630 Registered: 12/6/04
RE:[ap-calculus] Going crazy
Posted: Oct 10, 2012 10:10 PM

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Rebecca,

Let (a,a^2) and (b,-b^2+2b-5) be the points on the respective graphs that share a tangent line. Using the first function the tangent line is y=a^2+2a*(x?a)=2ax-a^2. Using the second function the tangent line is y=-b^2+2b-5+(-2b+2)*(x-b)=(-2b+2)x+b^2?5 with both tangent equations in slope intercept form. Since these are really the same line the slopes and intercepts must be equal, so 2a=?2b+2 (or a=1?b) and ?a^2=b^2?5. Substitute a=1?b into the other equation -(1?b)^2=b^2?5 which simplifies to 2b^2?2b?4=0 or b^2?b?2=0 from this we get b=2 or b=?1. When b=2, a=?1 and when b=?1, a=2.

So the line tangent to f(x)=x^2 at (2,4) is also tangent to g(x)=-x^2+2x-5 at (-1, -8) and the line tangent to f at (-1,1) is also tangent to g at (2,?5).

Doug
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Doug Kühlmann
Math Dept
Andover, MA 01810

________________________________________
From: Rebecca Tackett [rtackett@evansvilledayschool.org]
Sent: Wednesday, October 10, 2012 4:41 PM
To: AP Calculus
Subject: [ap-calculus] Going crazy

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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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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