Calculus Notes 12.11

Calculus Notes 12.11 - Calculus-Stewart Dr. Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 12.11 12.11 Applications of Taylor Polynomials Taylor Polynomial Estimates Example A Estimate e to three decimal places. For f ( x ) = e x P n ( x ) = 1 + x + x 2 2! +…+ x n n ! , and R n + 1 ( ) e n + 1 ( n + 1)! < 2 n + 1 ( n + = 1 2 n ( n + < 0.0005 only if 2 n ( n + > 1 0.0005 = 2000 . For n =3, 2 n ( n + = 2 3 (3 + = 8 24 = 192 . For n =4, 2 n ( n + = 2 4 (4 + = 16 120 = 1920 . For n =5, 2 n ( n + = 2 5 (5 + = 32 720 = 23040 > 2000 . Thus, e 1 + + ( ) 2 + ( ) 3 3! + ( ) 4 4! + ( ) 5 5! = 1 + 1 2 + 1 8 + 1 48 + 1 384 + 1 3840 1.648698 , so we use e 1.649 . Compare this to e 1.64872 . Example B Estimate e 2 to three decimal places. For f ( x ) = e x P n ( x ) = 1 + x + x 2 x n n ! , and R n + 1 (2) e 2 2 n + 1 ( n + < 8 2 n + 1 ( n + = 2 n + 4 ( n + < 0.0005 only if ( n + 2 n + 4 > 1 0.0005 = 2000 . For n =8, ( n + 2 n + 4 = (8 + 2 8 + 4 = 362880 4096 88.59 . For n =9, ( n + 2 n + 4 = (9 + 2 9 + 4 = 3628800 8192 442.97 . For n =10, ( n + 2 n + 4 = (10 + 2 10 + 4 = 39916800 16384 2436.33 > 2000 . Thus, e 2 1 + 2 + 2 2 + 2 3 + 2 4 + 2 5 + 2 6 6! + 2 7 7! + 2 8 8! + 2 9 9! + 2 10 10! = 1 + 2 + 4 2 + 8 6 + 16 24 + 32 120 + 64 720 + 128 5040 + 256 40320 + 512 362880 + 1024 3628800 7.388995 , so we use e 2 7.389 . Compare this to e 2 7.38906 .
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Calculus- Stewart Dr. Berg Summer ‘09 Page 2 12.11 Example C Were we to estimate e 8 to three decimal places we would need R n + 1 (8) e 8 8 n + 1 ( n + 1)!
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Calculus Notes 12.11 - Calculus-Stewart Dr. Berg Summer 09...

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