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Extensible Markup Language  |  1995-08-15  |  6KB  |  29 lines
  1. <?xml version="1.0" encoding="utf-8" ?>
  2. <!DOCTYPE card PUBLIC "-//Apple, Inc.//DTD card V 2.0//EN" "" >
  3. <card>
  4.     <id>48015</id>
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  9.     <owner>5472</owner>
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  11.     <content>
  12.         <layer>background</layer>
  13.         <id>25</id>
  14.         <text><span class="style10">unctions, Graphs and Change (6 of 6)</span><span class="style7">The sum of the value of the function </span><span class="style26">y = f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) between arbitrary </span><span class="style26">x = a</span><span class="style7"> and </span><span class="style26">x = b</span><span class="style7"> (the shaded area on the figure) is written</span><span class="style26">ba</span><span class="style7"> </span><span class="style38">├Ü</span><span class="style26">f</span><span class="style7"> (</span><span class="style26">x</span><span class="style7">) </span><span class="style26">dx</span><span class="style7">, and is equal to </span><span class="style26">F</span><span class="style7"> (</span><span class="style26">b</span><span class="style7">) - </span><span class="style26">F</span><span class="style7"> (</span><span class="style26">a</span><span class="style7">), where </span><span class="style26">F</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) is the </span><span class="style26">indefinite integral</span><span class="style7"> of </span><span class="style26">f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">); this is another continuous function of </span><span class="style26">x</span><span class="style7">, written </span><span class="style38">├Ü</span><span class="style26">f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) </span><span class="style26">dx</span><span class="style7">. For example, in the ball example, the indefinite integral, </span><span class="style38">├Ü</span><span class="style7">(20 - 10</span><span class="style26">t</span><span class="style7">) </span><span class="style26">dt</span><span class="style7">, of </span><span class="style26">y</span><span class="style7"> = 20 - 10</span><span class="style26">t</span><span class="style7">, is 20</span><span class="style26">t</span><span class="style7"> - (10/2)</span><span class="style26">t  </span><span class="style7">to the power of 2 + </span><span class="style26">c</span><span class="style7"> (where </span><span class="style26">c</span><span class="style7"> can be any constant). If we denote this new function </span><span class="style26">Int</span><span class="style7">(</span><span class="style26">x</span><span class="style7">), then the definite integral between </span><span class="style26">a</span><span class="style7"> and </span><span class="style26">b</span><span class="style7"> is </span><span class="style26">Int</span><span class="style7">(</span><span class="style26">b</span><span class="style7">) - </span><span class="style26">Int</span><span class="style7">(</span><span class="style26">a</span><span class="style7">), that is (20</span><span class="style26">b</span><span class="style7"> - 5</span><span class="style26">b</span><span class="style7"> to the power of 2 + </span><span class="style26">c</span><span class="style7">) - (20</span><span class="style26">a</span><span class="style7"> - 5</span><span class="style26">a</span><span class="style7"> to the power of 2 + </span><span class="style26">c</span><span class="style7">) = 5(</span><span class="style26">a</span><span class="style7"> to the power of 2 -  </span><span class="style26">b</span><span class="style7"> to the power of 2) + 20(</span><span class="style26">b</span><span class="style7"> - </span><span class="style26">a</span><span class="style7">).In fact it turns out that this process of </span><span class="style26">integration</span><span class="style7"> is the inverse of differentiation and it is therefore sometimes called </span><span class="style26">antidifferentiation</span><span class="style7">. This means that the indefinite integral of the derivative of a given function, and the derivative of its indefinite integral, are both equal to the given function - a result so important that it is called the </span><span class="style26">fundamental theorem of calculus</span><span class="style7">.EJB</span><span class="style10">TABLE OF DERIVATIVES AND INTEGRALS Function            Derivative                               Integral</span><span class="style7">         x                             1                                          (x to the power of 2)/2 x to the power of 2        2x                                         (x to the power of 3)/3 x to the power of n        nx to the power of (n-1)        x to the power of (n+1)   /    (n+1) sin x                             cos x                                               -cos x cos x                            -sin x                                                sin x</span></text>
  15.     </content>
  16.     <content>
  17.         <layer>background</layer>
  18.         <id>23</id>
  19.         <text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text>
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  22.         <layer>background</layer>
  23.         <id>36</id>
  24.         <text>20626668</text>
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  28. </card>
  29.