Higher Concepts of Mathematics CALCULUSFigure 2 Motion Between Two PointsIf the time recorded by a clock, when the object is at position P1 is t1, and if the time when theobject is at position P2 is t2, then the average velocity of the object between points P1 and P2equals the distance traveled, divided by the elapsed time.(5-1)VavS2S1t2t1If positions P1 and P2 are close together, the distance traveled and the elapsed time are small.The symbol D, the Greek letter delta, is used to denote changes in quantities. Thus, the averagevelocity when positions P1 and P2 are close together is often written using deltas.(5-2)VavDSDtS2S1t2t1Although the average velocity isFigure 3 Displacement Versus Timeoften an important quantity, inmany cases it is necessary to knowthe velocity at a given instant oftime. This velocity, called theinstantaneous velocity, is not thesame as the average velocity,unless the velocity is not changingwith time.Using the graph of displacement,S, versus time, t, in Figure 3, wewill try to describe the concept ofthe derivative.Rev. 0 Page 31 MA-05
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