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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
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
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.
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- Fall '09