Re: Fundamental Theorem of Calculus

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 19:24:00 +0000 (UTC)

In article <aAVad.691576$Gx4.113373@bgtnsc04-news.ops.worldnet.att.net>,
Michael J. Strickland <qualityser@worldnet.att.net> wrote:
>Shouldn't the fundamental theorem of calculus define the derivative as:
>
> (1) f' = Limit as h->0 [ f(x+h) - f(x-h)] / 2h ]
>
>in order to get a chord balanced around f(x),

It gives you the exact same thing, though:

(f(x+h) - f(x-h))/2h = [f(x+h) - f(x)]/2h + [f(x)-f(x-h)]/2h

Taking the limit this gives

limit as h->0 [ f(x+h)-f(x-h) ] / 2h
   = (1/2)limit as h->0 [ f(x+h)-f(x)]/h
       + (1/2)limit as h->0 [f(x)-f(x-h)]h.

   = (1/2)limit as h->0 [f(x+h) - f(x)]/h
       + (1/2)limit as h'->0 [f(x+h')-f(x)]/h'

(by setting h' = -h)

   = limit as h->0 [f(x+h)-f(x)]/h

(since the name of the variable doesn't matter).

In any case, we are trying to find the slope of the tangent at the
point (x,f(x)) as an approximation of the slopes of secant lines: the
most natural way to do it is to take the secant lines that ->go
through<- (x,f(x)), not the ones that are "balanced around f(x)".

>instead of as:
>
>(2) f' = Limit as h->0 [ f(x+h) - f(x)] / h ]
>
>which provides a chord starting extending from f(x)?
>
>It seems to me that definition 2 above will only be accurate if f(x) is a
>straight line.

It seems wrong to you. Both "definitions" yield the exact same answer,
as you see above. 

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu

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