Pre-Calc Exam Notes 136

Pre-Calc Exam Notes 136 - 4.1. That is, when taking...

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136 Chapter 6 Additional Topics §6.2 Below is the result of compiling and running the program using x 0 = 0 and x 1 = 1: javac secant.java java secant 0 1 x2 = 0.6850733573260451 x3 = 0.736298997613654 x4 = 0.7391193619116293 x5 = 0.7390851121274639 x6 = 0.7390851332150012 x7 = 0.7390851332151607 x8 = 0.7390851332151607 x = 0.73908513321516067229310920083662495017051696777344 Notice that the program only got up to x 8 , not x 10 . The reason is that the difference between x 8 and x 7 was small enough (less than ǫ error = 1.0 × 10 50 ) to stop at x 8 and call that our solution. The last line shows that solution to 50 decimal places. Does that number look familiar? It should, since it is the answer to Exercise 11 in Section
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Unformatted text preview: 4.1. That is, when taking repeated cosines starting with any number (in radians), you even-tually start getting the above number repeatedly after enough iterations. This turns out not to be a coincidence. Figure 6.2.2 gives an idea of why. 0.2 0.4 0.6 0.8 1 2-1 1 2 y x y = cos( x ) y = x Figure 6.2.2 Attractive xed point for cos x Since x = 0.73908513321516. .. is the solution of cos x = x , you would get cos (cos x ) = cos x = x , so cos (cos (cos x )) = cos x = x , and so on. This number x is called an attractive fxed...
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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