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1 A CD-ROM CDA.1 Illustration of Equal-Area Graphical Differentiation To illustrate this technique, consider the following data from which we wish to determine d f / dx as a function of x : First we calculate D f / D x (Table CDA-1) and then plot in the manner shown in Figure CDA-1. After drawing the smooth equal-area curve, we can complete our table to find d f / dx as a function of x (Table CDA-2). The func- tion used in this example was f ( x ) 5 1000(1 2 e 2 x ) (CDA-1) x 0 0.2 0.4 0.6 0.8 1.0 f ( x ) 0 182 330 451 551 631 T ABLE CDA-1 x f ( x ) 0.0 000 0.2 182 0.4 330 etc. D f Dx ------- 182 02 0.2 02------------------ 9105 330 1822 0.4 0.22------------------------ 7405 2 CD-ROM � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1100 1000 900 800 700 600 400 500 300 200 100 0 0.2 0.4 0.6 X 0.8 1.0 ∆f ∆x df dx , Fogler/PrenHall/CDA.1 S/S Equal area Differentiating Equation (CDA-1) with respect to x gives us The actual numerical values of the differential are given in the last column of Table CDA-2. T ABLE CDA-2 x f ( x ) Actual 0.0 000 — 1010 1000 0.2 182 910 0805 0818 0.4 332 740 0670 0670 0.6 451 605 0550 0548 0.8 551 500 0450 0449 1.0 631 400 0360 0368 Figure CDA-1 Equal-area differentiation. D f Dx ------- df dx----- df dx----- d f dx------ 1000 e x25
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