Elaborating a little more: TAN(a) is a useful function key
to introduce at this point, because the delta_y/delta_x
slope is going to become the "tangent to the curve or
graph of f(x)" as delta_x -> 0. This gives a geometric
basis for calling it TAN(a) i.e. 'tangent', but also bridges
over to SIN(a) and COS(a) as v/hyp and h/hyp respectively,
where v means 'vertical' or 'opposite to angle a' and h
means 'horizontal' or 'adjacent to angle a'.
/| cos(a) = h/hyp
hyp / | sin(a) = v/hyp
/ | v tan(a) = v/h
/a |
---------
h
Fig. 1
Trig
In sum: using the right triangle as our primary graphic, we have
a basis for talking about slope, tangent, change in vertical versus
change in horizontal, tangent to a curve, sine, cosine and, as h->0,
the derivative df(x)/dx.
Those of you tracking my web-based distance learning presentation
of these concepts know that I'm quick to segue 'change' (in the
sense of covariants, slopes or 'differences') to 'action-packed
frames' (as in a movie) and to a concept of energy as pdf where
pd=action (d=distance), p=mv (momentum) and f=1/t (frequency).
The minimum action is h (Planck's Constant) and time-increments
consisting of snapshots of h (actions) give us an idea of aggregating
energy quanta (energy increments).
Energy is time-independent (the potential to do work however long
it takes) whereas 'power' is a measure of energy/time (how quickly
energy is expended).
My school of thought holds that keeping the calculus embedded in
its original context (physics) keeps the subject more meaningful
to students and better prepares them to think scientifically,
whereas an overly abstract approach dwelling too one-sidedly on
theorems and axioms involving real numbers, is to put students
on some ivory tower trajectory probably dead ending in the
philosophy department.
Kirby
Major references:
http://www.inetarena.com/~pdx4d/ocn/chalkboard.html
http://www.inetarena.com/~pdx4d/ocn/calculus2.html
---------------------------------------------------------
Kirby T. Urner http://www.teleport.com/~pdx4d/kirby.html
4D Solutions http://www.teleport.com/~pdx4d/ [PGP OK]
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