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RE:[apcalculus] Going crazy
Posted:
Oct 10, 2012 10:10 PM


NOTE: This apcalculus 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/gettingstarted and post messages there.  Rebecca,
Let (a,a^2) and (b,b^2+2b5) 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)=2axa^2. Using the second function the tangent line is y=b^2+2b5+(2b+2)*(xb)=(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+2x5 at (1, 8) and the line tangent to f at (1,1) is also tangent to g at (2,?5).
Doug  Doug Kühlmann Math Dept Phillips Academy Andover, MA 01810
________________________________________ From: Rebecca Tackett [rtackett@evansvilledayschool.org] Sent: Wednesday, October 10, 2012 4:41 PM To: AP Calculus Subject: [apcalculus] Going crazy
NOTE: This apcalculus 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/gettingstarted 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+2x5 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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