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Topic: [ap-calculus] Re: Locally Linear
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Richard Sisley

Posts: 4,189
Registered: 12/6/04
[ap-calculus] Re: Locally Linear
Posted: Nov 22, 2000 1:38 PM
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Doug Kuhlmann wrote:

> >Is the function y = x^(1/3) considered to be locally linear at x = 0?
> >
> >Mary Thomas

>
>
> I would come down on the side of saying that while y=x(1/3) has a vertical
> tangent at x=0, it is not locally linear there since there is no
> approximating linear function at x=0. Maybe we should just say that the
> function is locally vertical at the origin and locally affine elsewhere.


Thoughtful response to an interesting question. How about this--the function whose
graph is described by y = x^3 has the x-axis as its “best” local linear approximator at
the origin. I would propose that “best linear approximator” characterizes tangent
lines. Then, consider mapping the graph described by y = x^3 and its local linear
approximator at (0,0) with a reflection in the y = x line. Would this not lead to the
conclusion that the y-axis is the best local linear approximator (that is, the tangent
line) for the graph described by y = x^(1/3) at (0,0). In this case we would simple
ackowledge that the best linear approximator does not happen to be the graph of a
function.

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
> dkuhlmann@andover.edu
>
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