document.write( "Question 388333: I have to use finite differences to write an equation for the following points: (0,-1) (1,-2) (2,5) (3.38) (4,115) (5,254) \n" ); document.write( "
Algebra.Com's Answer #275158 by Edwin McCravy(20086) $\"\"$ $\"About$
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document.write( "(0,-1) (1,-2) (2,5) (3.38) (4,115) (5,254)\r\n" );
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document.write( "List the second coordinates in a column\r\n" );
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document.write( " -1\r\n" );
document.write( " -2\r\n" );
document.write( "  5\r\n" );
document.write( " 38\r\n" );
document.write( "115\r\n" );
document.write( "254\r\n" );
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document.write( "Subtract each number from the number just below it and write the\r\n" );
document.write( "difference to the right of that number in a second column. These\r\n" );
document.write( "numbers are called the first differences.\r\n" );
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document.write( " -1    3\r\n" );
document.write( " -2    7\r\n" );
document.write( "  5   33\r\n" );
document.write( " 38   77\r\n" );
document.write( "115  139 \r\n" );
document.write( "254\r\n" );
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document.write( "Subtract each number in the second column from the number just below \r\n" );
document.write( "it and write the difference to the right of that number in a third \r\n" );
document.write( "column.  These numbers are called the second differences.\r\n" );
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document.write( " -1    3    4\r\n" );
document.write( " -2    7   26  \r\n" );
document.write( "  5   33   44\r\n" );
document.write( " 38   77   62\r\n" );
document.write( "115  139 \r\n" );
document.write( "254\r\n" );
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document.write( "Subtract each number in the third column from the number just below \r\n" );
document.write( "it and write the difference to the right of that number in a fourth \r\n" );
document.write( "column.  These numbers are called the third differences.\r\n" );
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document.write( " -1   -1    8   18\r\n" );
document.write( " -2    7   26   18\r\n" );
document.write( "  5   33   44   18\r\n" );
document.write( " 38   77   62\r\n" );
document.write( "115  139 \r\n" );
document.write( "254\r\n" );
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document.write( "We need go no further because we have reached a column in which all\r\n" );
document.write( "the numbers are the same.  Since this column was the column of 3rd\r\n" );
document.write( "differences, we assume the formula for the general term is a 3rd degree\r\n" );
document.write( "polynomial equation:\r\n" );
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document.write( "That has 4 unknown constants, so we will substitute the first\r\n" );
document.write( "four first coordinates for n and the second coordinate for \r\n" );
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document.write( "For n = 0\r\n" );
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document.write( "Since we know the value of D as -1, we can now just substitute in\r\n" );
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document.write( "For n = 1\r\n" );
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document.write( "For n = 2\r\n" );
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document.write( "For n = 3\r\n" );
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document.write( "So we have the system of 3 equations in 3 variables\r\n" );
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document.write( "Solve that system and get A=3, B=-5, C=1 and we already have D=-1,so\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "Edwin
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