I've used the same setting (from the same source!) Richard describes to investigate *errors* in linear approximations.
The error from using the tangent line to approximate a differentiable function at a point behaves like a quadratic with vertex at the point of tangency.
The error from using any other line behaves linearly!
Example: f(x) = e^x; g(x) = x+1; h(x) = 1.01x+1; errg(x)=g(x)-f(x); errh(x)=h(x)-f(x). Graph errg and errh in a standard decimal window, then set your vertical zoom factor to 16 and the horizontal zoom actor to 4 (this preserves the shape of parabolas). Zoom in repeatedly, and watch errh's shape get steeper and straighter, while errg's shape remains unchanged!
Try it. It's a marvelous foreshadowing of Taylor Polynomials, and really does nail down why the tangent line is so special.
> I have mentioned in a post some time ago that Donald > Kreider at Dartmouth created an > interesting investigation using a graphing > calculator of how the tangent line to a > particular funciton at a particular point compares > to a line with a slightly different > slope through the same point. Each year I take the > time with my AB students to run a > variation of his investigation. The students see > for themselves that the tangent line > dominates all competitors in a neat visual way. > This has led me to appreciate in a new > way the reasonableness of characterizing tangent > lines as "BEST" linear approximators. > > Sincerely, > > Richard Sisley > > > > > > > > > Doug > > > > Doug Kuhlmann > > Math Department > > Phillips Academy > > 180 Main Street > > Andover, MA 01810 > > email@example.com > > > > --- > > You are currently subscribed to ap-calculus as: > firstname.lastname@example.org > > To unsubscribe send a blank email to > leave-ap-calculus-5913831W@list.collegeboard.org > > > --- > You are currently subscribed to ap-calculus as: > email@example.com > To unsubscribe send a blank email to leave-ap-calculus-5913831W@list.collegeboard.org
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