CALCULUSHigher Concepts of MathematicsUsing equation 5-1 we find the average velocity from S_{1}to S_{2} is . If we connect theS_{2 } S_{1}t_{2 } t_{1}points S_{1} and S_{2} by a straight line we see it does not accurately reflect the slope of the curvedline through all the points between S_{1} and S_{2}. Similarly, if we look at the average velocitybetween time t_{2} and t_{3} (a smaller period of time), we see the straight line connecting S_{2} and S_{3}more closely follows the curved line. Assuming the time between t_{3} and t_{4} is less than betweent_{2}and t_{3}, the straight line connecting S_{3}and S_{4}very closely approximates the curved line betweenS_{3}and S_{4}.As we further decrease the time interval between successive points, the expression moreD SD tclosely approximates the slope of the displacement curve. As approaches theD t 0,D SD tinstantaneous velocity. The expression for the derivative (in this case the slope of thedisplacement curve) can be written . In words, this expression would bedSdtlimD t oD SD t"the derivative of S with respect to time (t) is the limit of as Dt approaches 0."D SD t(5-3)V dsdtlimD t0D sD tThe symbols ds and dt are not products of d and s, or of d and t, as in algebra. Each representsa single quantity. They are pronounced "dee-ess" and "dee-tee," respectively. Theseexpressions and the quantities they represent are called differentials. Thus, ds is the differentialof s and dt is the differential of t. These expressions represent incremental changes, where dsrepresents an incremental change in distance s, and dt represents an incremental change in timet.The combined expression ds/dt is called a derivative; it is the derivative of s with respect tot. It is read as "dee-ess dee-tee." dz/dxis the derivative of zwith respect to x; it is read as"dee-zee dee-ex." In simplest terms, a derivative expresses the rate of change of one quantitywith respect to another. Thus, ds/dt is the rate of change of distance with respect to time.Referring to figure 3, the derivative ds/dt is the instantaneous velocity at any chosen pointalong the curve. This value of instantaneous velocity is numerically equal to the slope of thecurve at that chosen point.While the equation for instantaneous velocity, V = ds/dt, may seem like a complicatedexpression, it is a familiar relationship. Instantaneous velocity is precisely the value given bythe speedometer of a moving car. Thus, the speedometer gives the value of the rate of changeof distance with respect to time; it gives the derivative of s with respect to t; i.e. it gives thevalue of ds/dt.MA-05 Page 32Rev. 0

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