document.write( "Question 1012640: Determine the first 5 iterates of f(x)=1.2x (1-x) for x0=0.1 \n" ); document.write( "
Algebra.Com's Answer #634443 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Start by plugging in 0.1 for x in the equation\r\n" );
document.write( "for f(x).  Then take what you get, and plug that \r\n" );
document.write( "in for x in the equation for f(x).\r\n" );
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document.write( "Keep taking what you get each time and plugging \r\n" );
document.write( "that in for x in the equation for f(x) over and over. \r\n" );
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document.write( "f(x)=1.2x[1-x], x0 = 0.1\r\n" );
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document.write( "f(x0) = f(0.1)=1.2(0.1)[1-(0.1)] = 0.108 = x1 \r\n" );
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document.write( "f(x1) = f(0.108)=1.2(0.108)[1-(0.108)] = 0.1156032 = x2 \r\n" );
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document.write( "f(x2) = f(0.1156032)=1.2(0.1156032)[1-(0.1156032)] = 0.1226869202 = x3  \r\n" );
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document.write( "f(x3) = f(0.1226869202)=1.2(0.1226869202)[1-(0.1226869202)] = 0.1291618079 = x4 \r\n" );
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document.write( "f(x4) = f(0.1291618079)=1.2(0.1291618079)[1-(0.1291618079)] = 0.1349748422 = x5 \r\n" );
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document.write( "Interesting fact FYI:\r\n" );
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document.write( "If you keep getting iterates for this, they will get closer \r\n" );
document.write( "and closer to 0.1666666... which gets closer and closer to the \r\n" );
document.write( "fraction 1/6.\r\n" );
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document.write( "You can tell this by setting f(x) = x and solving\r\n" );
document.write( "                                 1.2x[1-x] = x\r\n" );
document.write( "                                1.2x-1.2x² = x\r\n" );
document.write( "                             -1.2x²+1.2x-x = 0  \r\n" );
document.write( "                              1.2x²-1.2x+x = 0\r\n" );
document.write( "                              12x²-12x+10x = 0\r\n" );
document.write( "                                   12x²-2x = 0\r\n" );
document.write( "                                     6x²-x = 0\r\n" );
document.write( "                                   x(6x-1) = 0\r\n" );
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document.write( "                                   x=0;  6x-1 = 0 \r\n" );
document.write( "                                           6x = 1\r\n" );
document.write( "                                            x = 1/6 \r\n" );
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document.write( "If we were beginning with x0 = 0 we would of\r\n" );
document.write( "course get 0, but if we started with any number x0 \r\n" );
document.write( "between 0 and 1,  0 < x0 < 1 the iterations would always \r\n" );
document.write( "converge to (get closer and closer to) 1/6.  \r\n" );
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document.write( "Edwin
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