Unformatted text preview: tants t 1 , t 2, t 3 o r t 4 ? We
may drop the subscript and let t be a general instant in [t0 , tn].
--- , y ’ (t), Dy(t), y (t)
So far we have only a NOTATION, some symbols that express the concept of
what we are trying to do: find the INSTANTANEOUS RATE OF CHANGE of the
particular function y(t) AT ANY INSTANT. 59 When we perform the calculation to find the INSTANTANEOUS RATE OF CHANGE of
y(t) at any instant we get (refer Preface iii) :
INSTANTANEOUS VERTICAL SPEED = --- = u . s i n θ -- g t
Compare this expression of INSTANTANEOUS VERTICAL SPEED to the
ear lier calculation of AVERAGE VERTICAL SPEED:
t = u . s i n θ -- -2
When we compare the AVERAGE SPEED with the INSTANTANEOUS SPEED we
make the profound distinction between an INTERVAL (no matter how small)
and an INSTANT. Only with this distinction and the calculation using INSTANT
are we able to get the INSTANTANEOUS SPEED.
AVERAGE VERTICAL SPEED = δ t is an INTERVAL.
δ x is an INTERVAL. L imit (a + δ t) is the INSTANT a. δt → 0 L imit (x + δ x) is any INSTANT x. δx → 0 Analysis
From the Analysis point of view : an inter val δ t or δ x, no matter how small, is
a continuous set of instants or points.
From the Algebra point of view : an inter val corresponds to a subset of the real
numbers R. This subset has UNCOUNTABLY many real numbers (both rational and
irrational numbers) in a contiguous manner, ie; with none missing in between.
Since we cannot even begin to COUNT them, there is no sense in talking about a
subscript to enumerate them.
We shall use the words DIFFERENTIATE or FIND THE DERIVATIVE to denote
the process of finding the expression for the INSTANTANEOUS RATE OF
CHANGE of a function. Let us now use this concept to define the INSTANTANEOUS
RATE OF CHANGE of a general function f(x). 60 INSTANT
1 3 . INSTANTANEOUS RATE OF CHANGE of f(x)
f (x) f(x) f(x)2
f(x)1 x1 (0 , 0 ) x2 x What is the instantaneous ra...
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- Fall '09
- Limit, Δx