alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

In light we have the discr ete particle photon and

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Unformatted text preview: Light we have the discr ete particle photon and the continuous wave. Likewise, Mathematics is divided into two main branches: Algebra and Analysis. In Algebra we deal with discr ete operands, be they numbers like 1,2,3, . . . or symbols for numbers like x, y, z, . . . . The operands take on discr ete values. The 7 operations are + and its inverse -- , and its in its inverse , exponentiation y = x n and its two inverses ny = x and log x y = n. Usually the set of different values the operands may take on is finite, or at least the set of values can be counted or enumerated. In Analysis we deal with continuous expressions or operands or functions. Take for example a bouncing ball: each position it bounces is discrete and can be counted. The ball may bounce indefinitely. On the other hand a ball rolling in a straight line has infinitely many positions in a continuous manner. We cannot even begin to count all the different positions. So rather than discrete identification of position, we have a continuous expression or function to describe its position or motion over time. i The fundamental concepts in Calculus are INFINITESIMAL and LIMIT which are used to develop the concepts of INSTANT, INSTANTANEOUS, CONTINUITY and DIFFERENTIABLITY. Once these concepts are in place we can talk of functions that are WELL BEHAVED: SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE. We can then do two very beautiful calculations or operations: 1. Given a SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE function that expresses CHANGE we can DIFFERENTIATE the function and get the new function that expresses the INSTANTANEOUS RATE OF CHANGE. 2. Given the function that expresses the INSTANTANEOUS RATE OF CHANGE, we can INTEGRATE the function and get the new function that expresses CHANGE. Working with polynomials is easy since they are CONTINUOUS everywhere. The fundamental concepts and calculations (Differentiation and Integration) can be taught with ease and clarity. Getting more information about the function (increasing, decreasing, maximum, minimum, inflexion) from its Higher Order Derivatives can also be shown. I have...
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