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Topic: [ap-calculus] Re: Locally Linear
Replies: 0

 Mark Howell Posts: 1,221 Registered: 12/6/04
[ap-calculus] Re: Locally Linear
Posted: Nov 26, 2000 4:39 PM

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.

Mark Howell

> 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
> > 180 Main Street
> > Andover, MA 01810
> > dkuhlmann@andover.edu
> >
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