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Extensible Markup Language | 1995-08-15 | 7KB | 29 lines

<?xml version="1.0" encoding="utf-8" ?> <!DOCTYPE card PUBLIC "-//Apple, Inc.//DTD card V 2.0//EN" "" > <card> <id>47468</id> <filler1>0</filler1> <cantDelete> <false /> </cantDelete> <showPict> <true /> </showPict> <dontSearch> <false /> </dontSearch> <owner>5472</owner> <link rel="stylesheet" type="text/css" href="stylesheet_3106.css" /> <content> <layer>background</layer> <id>25</id> <text>unctions, Graphs and Change (4 of 6)Instantaneous rates of changeSimilarly if you want to know how fast the Orient Express was traveling as it flashed through the Simplon Tunnel, you obviously get a very poor answer if you divide the whole distance from London to Venice by the total time taken to cover it. You get a better approximation if you measure the distance and time between Paris and Milan, and a better approximation still if you time the train between the stations at either end of the tunnel. This suggests that if we had accurate enough clocks and measuring tapes we would be able to get closer and closer to a precise answer by timing the train over increasingly shorter distances. Although this still never tells us the speed at any one instant, it suggests that the instantaneous speed is the limit that this sequence of averages tends to as the length of the interval gets smaller.Let us now return to the example of the ball, and apply this reasoning formally: here the vertical height could be expressed in terms of elapsed time as h = 20t - 5t to the power 2; for example, the height after 0.5 second is 8.75 m, and after 1 second is 15 m. Now consider an arbitrary time t after the ball is thrown and take an interval of duration d on either side of it: We can see that the average velocity over this distance is the difference between the values of the distance function at the arguments t + d and at t - d, divided by the difference between these arguments. This is the slope of the chord PQ, that is [20(t+d)-5(t+d) to the power of 2]-[20(t-d)-5(t-d) to the power of 2] --------------- (t+d)-(t-d)which simplifies to: (40d - 20td)/2d = 20 - 10t. Since this value is independent of d, this remains the average velocity round t no matter how small the interval d becomes, so that we can infer that the instantaneous velocity at time t from the starting point is actually equal to 20 - 10t.This, of course, is another function of t, so that the rate of change of a function is another function of the same independent variable. Where the original function was y = f(x), the new function, known as its derivative, is written f'(x) or dy/dx (read as 'dy by dx'), where dy and dx represent small increments in y and x respectively; here, for example, the derivative of f(t) = 20t - 5t to the power of 2 is f'(t) = 20 - 10t. Similarly, in the expression y = x to the power of 2, f(x) = x to the power of 2, and the derivative of f(x) (x+d ) to the power of 2 - (x-d ) to the power of 2 4xd------------------------------------------------------------------- = --------- = 2x (x+d ) - (x-d ) 2d </text> </content> <content> <layer>background</layer> <id>23</id> <text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text> </content> <content> <layer>background</layer> <id>36</id> <text>20626668</text> </content> <name>p070-4</name> <script></script> </card>