Web Components for Mathematics

From: js2479@columbia.edu
To: Tom Downey <tom@calculusapplets.com>
Subject: Re: calc website
References: <6.2.3.4.2.20080121164149.02ac57f0@pop.bizmail.yahoo.com>
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oh please leave it! this is a discrete version of f f' f'', which is
rarely discussed but i think could be quite useful.
for f'', as you said, the trick is to figure out delX. a visual way to
see it is to look at the rectangles in the f'' row; they are 1 unit
long, e.g. from .5 to 1.5, so delX = 1.
a symbolic way to see it is this. say you have x=0,1,2 then f'(0) or
f'(1) are approximately f(1)-f(0) / 1-0 so you can think of this as
f'(x) on [0,1] or f'(.5) Similarly you have f'(x) on [1,2], or f'(1.5)
So when you calculate f'', it is between [0,1] and [1,2] (or between
.5 and 1.5). the distance between them is 1.5-0.5=1. or, the distance
between those intervals (i.e. how far do you have to move one interval
to make it the other) is 1. you could think of "1" as the distance
between the left endpoints (1-0) or the distance between the right
endpoints (2-1).
in units, remember that the denominator of f'' is d(dx) or "the change
in the change in x". the first change in x is 0->1, the second change
in x is 1->2. the change between those is 1.
finally, if you need to associate this calculation with a single
x-value (i.e. f''(a)=result), it would be x=1 since that is the center
of [0,2] (or, the center of the first interval is .5, the center of
the second interval is 1.5, and the center of that pair is 1)
its probably easier just to use the midpoint (center) so you dont have
to keep track of intervals. i discussed intervals here to cover all
the angles.
i hope this explanation can be useful for you and the website and your
learners.
cheers,

Jason

Quoting Tom Downey <tom@calculusapplets.com>:

Hmm.... I need to think about this a bit more. The calculated first
derivatives give the average rate of change over intervals. The second
derivative will give the average rate of change of the derivative over
some interval. The question you raised is: what interval? I'm starting
to think that the way I built this example is confusing, and it would
be better to just show values for x, values for the first derivative at
specific x values, and average values for the second derivative,
leaving out the values of the function. I looked in the textbook we
use, and this is how they explain it.

Tom

At 04:52 PM 1/19/2008, you wrote:
Ive recently discovered your website. and i'm happy i found it. i
think you adapted the JCM applets nicely, both individually as well as
to present a coherent calculus sequence. i look forward to using them
with my students.
one note. perhaps i misunderstand the representation, but i think that
on {http://www.calculusapplets.com/secondderivtable.html} the f "
values are off by a factor of two. thats how it seemed to me after
some experimenting (e.g. try e^x). in your explanation below, you say
the denominator should be "2-0", but it should be "1-0" or "2-1" or
"1.5 - 0.5" depending on your interpretation.
thanks again for a great website.
cheers,

Jason Samuels