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Topic: [ap-calculus] Going crazy
Replies: 1   Last Post: Oct 13, 2012 4:25 PM

 Ray Chaudhuri Posts: 42 Registered: 12/6/04
RE: [ap-calculus] Going crazy
Posted: Oct 13, 2012 4:25 PM

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
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Dear all,
This is a nice problem.
I did this way.. hope it is right.
Let y = mx + c be the common tangent to both parabolas y=x^2 and y=-x^2+2x-5
solving : mx + c = x^2 , but since it is a tangent , one root , hence Discriminant = 0 which gives , m^2 = -4c
For the second equation , mx + c = =-x^2+2x-5, using same logic , we get (m-2)^2 = 4( c+5)
solving this and using the info ..m^2 = -4c , we get a neat quadratic eqn m^2 -2m -8 =0 can be factorised into (m-2) ( m+2) = 0 which gives m= -2, 4 , so c = -1 , -4.
so the common tangents are y = -2x -1 and y = 4x -4
Thanks a bunch for this. I willchallenge my IBHL students with this.
All the best

> From: sharitzkevin@rockwood.k12.mo.us
> To: ap-calculus@lyris.collegeboard.com
> Date: Thu, 11 Oct 2012 15:18:04 -0500
> Subject: RE: [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.
> ------------------------------------------------------------------------------------------------
> Call the point of intersection between one tangent and the first curve (a,f(a)) and the intersection of that tangent and the other curve (b,g(b)). You can calculate the slope between those points and get the slope of the tangent line in terms of a and b(the unkown x values of the points of intersection). Next find f'(a) and g'(b) (the slope of that same tangent line)
> f'(a)=2a g'(b)=-2b+2 thus a=1-b. Substitute that into the equation you got for calculating the slope of the points and you should come up with two values for b(the two x values where the two tangent lines hit the second curve) then solve for a. So you get two pairs of points (a1,f(a1) and (b1, g(b1) and also (a2,f(a2)) and (b2,g(b2)). You can write the equations of those lines now.
>
> Sorry if this is rather roughly written, calculus was not meant to be typed in an email.
>
> -Kevin Sharitz
>
> ________________________________________
> From: Cope, Joellen M. [jmcope@stjames.edu]
> Sent: Wednesday, October 10, 2012 10:39 PM
> To: AP Calculus
> Cc: AP Calculus
> Subject: Re: [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.
> ------------------------------------------------------------------------------------------------
> Sorry! Read this way too quickly. Great problem!
>
> Joellen
>
> On Oct 10, 2012, at 8:10 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
> >
> > Thanks,
> > Rebecca Tackett
> > Upper School Mathematics Instructor
> > Evansville Day School
> >
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