alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

We shall then show that the time axis is a continuous

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Unformatted text preview: TIME AXIS is a CONTINUOUS set of INSTANTS and that each INSTANT corresponds to a Real number. We shall then define CONTINUOUS functions. 4 Par t 1 : CONTINUITY 5 Over view Ov er vie w continuous To show that the TIME AXIS is a continuous set of instants and that each instant contin instants instant corresponds to a Real number we shall proceed in three steps. Step continuous Step 1: We know from class 8 Geometry that a straight line is a continuous set of contin points. We are also familiar with the different kinds of sets of numbers in Algebra: N (natural numbers), Z (integers), Q (rational numbers), Irrational numbers and R (real numbers). We present the Analysis concept of the COMPLETENESS property Analysis Anal of the Real numbers. We thus show the connection between the set R of real numbers in Algebra and a straight line (a continuous set of points) in Geometry. Each point xi on the number continuous contin points line corresponds to a Real number and vice versa. We may call this number line the x-axis. And when we say x1 < x2 for x1, x2 ∈ R , the picture from the Geometry point of view is : ALGEBRA GEOMETRY ≡ set R x-axis x1 < x2 x1 x2 Step 2: In this step we learn the concepts: TENDS TO, LIMIT and INFINITESIMAL. Step We shall use these concepts to take an Analysis view of the x-axis as a continuous Analysis continuous Anal contin set of points. We shall then define an instant on the TIME AXIS. From this we shall points instant show the equivalence between the x-axis as a continuous set of points and the continuous points TIME AXIS as a continuous set of instants. continuous instants GEOMETRY x-axis x1 x2 ANALYSIS ≡ time axis t1 t2 Each point xi on the x-axis corresponds to an instant ti on the TIME AXIS. point instant 6 Step Step 3: Here we relate the TIME AXIS to the set of real numbers R. Now when we say instant t1 we mean some definite real number t1 ∈ R. And when we say t1 < t2 for t1, t2 in R , the picture from the Analysis point of view is : Analysis ALGEBRA set R ANALYSIS ≡ time axis t1 < t2 t1 t2 Since the TIME AXIS is a continuous set of instants, we may say t2 TENDS TO t1 in continuous instants a continuous manner. This is denoted by t...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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