97 deriva angent 1 9 first derivative slope of tang

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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 Chal-na Kal-na 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 hange instant 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 wha ra hat are real number 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. G unaka-samuccaya-sutras The methods were encoded in mnemonics (mental one-liners) known as slokas o r stotras ( aphorisms). The more difficult functions could be s totras 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. 94 ANAL ALYTICAL GEOMETRY 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 tangent 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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