Unformatted text preview: r t i e s S I N G L E VA LU E D, C O N T I N U O U S a n d
DIFFERENTIABLE must be satisfied. The first par t of the study of Calculus
deals with this.
VEDIC mathematicians understood the need for finding the INSTANTANEOUS
RATE OF CHANGE known in Sanskrit as Chal-na K al-na. This is illustrated in the
proof to find the INSTANTANEOUS RATE OF CHANGE OF xn at x = a.
The genius of this approach is in its simplicity. First DIVIDE to find the r ate of
c hange then SUBSTITUTE for the instant . This method works almost always,
especially if we note that almost all functions can be approximated by finite
polynomials or infinite polynomial series. olynomials are
Pol ynomials ar e to functions w ha t r a tionals ar e to r eal n umber s .
Without elaborating by analysis the concept, they developed formulas or
sutras : Gunaka-samuccaya-sutras f or differentiating various functions.
The methods were encoded in mnemonics (mental one-liners) known as
slokas o r stotras ( aphorisms). The more difficult functions could be
decomposed using the P ar a v ar ty a Sutr a .
ara tya Sutra
It was up to the genius of the western mind, the great Newton and Leibniz, to
develop the concepts of INFINITESIMAL and LIMIT. These concepts are illustrated
in the proof to find the INSTANTANEOUS RATE OF CHANGE of x n.
It is only with these concepts of INFINITESIMAL and LIMIT that we can
develop the concepts of CONTINUOUS and DIFFERENTIABLE. While the
concept of CONTINUOUS can be intuitively under stood the concept of
DIFFERENTIABLE is not so obvious. The reader should take hear t that
even the great Cauchy, master of rigorous analysis, failed to grasp the
difference. Put another way : f(x) is DIFFERENTIABLE if and only if f ’ (x) is CONTINUOUS.
P ar t 3 : ANALYTICAL GEOMETRY 95 Over view
Ov er vie w
We are familiar with the concept of the tangent to a cur ve: a straight line that
touches the cur ve at a point. As a r ule, for WELL-BEHAVED functions, to each point on
the graph or cur ve there is exactly one tangent . However, there are exceptio...
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- Fall '09