<text><span class="style10">unctions, Graphs and Change (4 of 6)</span><span class="style7"></span><span class="style10">Instantaneous rates of change</span><span class="style7">Similarly 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 </span><span class="style26">h</span><span class="style7"> = 20</span><span class="style26">t</span><span class="style7"> - 5</span><span class="style26">t</span><span class="style7"> 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 </span><span class="style26">t</span><span class="style7"> after the ball is thrown and take an interval of duration </span><span class="style26">d</span><span class="style7"> 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 </span><span class="style26">t</span><span class="style7"> + </span><span class="style26">d</span><span class="style7"> and at </span><span class="style26">t</span><span class="style7"> - 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] --------------- (</span><span class="style26">t+d</span><span class="style7">)-(</span><span class="style26">t-d</span><span class="style7">)which simplifies to: (40</span><span class="style26">d</span><span class="style7"> - 20</span><span class="style26">td</span><span class="style7">)/2</span><span class="style26">d</span><span class="style7"> = 20 - 10</span><span class="style26">t</span><span class="style7">. Since this value is independent of </span><span class="style26">d</span><span class="style7">, this remains the average velocity round </span><span class="style26">t</span><span class="style7"> no matter how small the interval </span><span class="style26">d</span><span class="style7"> becomes, so that we can infer that the instantaneous velocity at time </span><span class="style26">t</span><span class="style7"> from the starting point is actually equal to 20 - 10</span><span class="style26">t</span><span class="style7">.This, of course, is another function of </span><span class="style26">t</span><span class="style7">, so that the rate of change of a function is another function of the same independent variable. Where the original function was </span><span class="style26">y = f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">), the new function, known as its </span><span class="style26">derivative</span><span class="style7">, is written </span><span class="style26">f'</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) or </span><span class="style26">dy/dx</span><span class="style7"> (read as '</span><span class="style26">dy</span><span class="style7"> by </span><span class="style26">dx</span><span class="style7">'), where </span><span class="style26">dy</span><span class="style7"> and </span><span class="style26">dx</span><span class="style7"> represent small </span><span class="style26">increments</span><span class="style7"> in </span><span class="style26">y</span><span class="style7"> and </span><span class="style26">x</span><span class="style7"> respectively; here, for example, the derivative of </span><span class="style26">f</span><span class="style7">(</span><span class="style26">t</span><span class="style7">) = 20</span><span class="style26">t</span><span class="style7"> - 5</span><span class="style26">t</span><span class="style7"> to the power of 2 is </span><span class="style26">f'</span><span class="style7">(</span><span class="style26">t</span><span class="style7">) = 20 - 10</span><span class="style26">t</span><span class="style7">. Similarly, in the expression </span><span class="style26">y</span><span class="style7"> = </span><span class="style26">x</span><span class="style7"> to the power of 2, </span><span class="style26">f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) = </span><span class="style26">x</span><span class="style7"> to the power of 2, and the derivative of </span><span class="style26">f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) (</span><span class="style26">x</span><span class="style7">+</span><span class="style26">d </span><span class="style7">) to the power of 2 - (</span><span class="style26">x</span><span class="style7">-</span><span class="style26">d </span><span class="style7">) to the power of 2 4</span><span class="style26">xd</span><span class="style7">------------------------------------------------------------------- = --------- = 2x (</span><span class="style26">x</span><span class="style7">+</span><span class="style26">d </span><span class="style7">) - (</span><span class="style26">x</span><span class="style7">-</span><span class="style26">d </span><span class="style7">) 2</span><span class="style26">d</span><span class="style7"> </span></text>
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<id>23</id>
<text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text>