CALCULUSHigher Concepts of MathematicsThe instantaneous velocity again equals theFigure 5 Graph of Distance vs. Timevalue of the derivative ds/dt. This value ischanging with time. However, theinstantaneous velocity at any specified timecan be determined. First, small changes ins and t are considered.DsDt(sDs)s(tDt)tThe values of (s + Ds) and s in terms of(t + Dt) and t using Equation 5-5, can thenbe substituted into this expression. At timet, s = 10t2; at time t + Dt, s + Ds = 10(t +Dt)2. The value of (t + Dt)2equals t2+2t(Dt) + (Dt)2; however, for incrementalvalues of Dt, the term (Dt)2 is so small, itcan be neglected. Thus, (t + Dt)2 = t2 +2t(Dt).DsDt10[t22t(Dt)]10t2(tDt)tDsDt10t220t(Dt)]10t2tDttFigure 6 Slope of a CurveDsDt20tThe value of the derivative ds/dt in the caseplotted in Figure 5 equals 20t. Thus, at timet = 1 s, the instantaneous velocity equals 20ft/s; at time t = 2 s, the velocity equals 40ft/s, and so on.When the graph of a function is not a straightline, the slope of the plot is different atdifferent points. The slope of a curve at anypoint is defined as the slope of a line drawntangent to the curve at that point. Figure 6shows a line drawn tangent to a curve. Atangent line is a line that touches the curve atonly one point. The line AB is tangent to theMA-05 Page 36 Rev. 0
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