Pre-Calc Homework Solutions 380

Pre-Calc Homework Solutions 380 - 380 Section 9.3 e x. Then...

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25. Let f ( x ) 5 e x . Then P 3 ( x ) 5 1 1 x 1 } x 2 2 } 1 } x 6 3 } , so we may use the Remainder Estimation Theorem with n 5 3. Since ) f (4) ( x ) ) 5 e x , which is no more than e 0.1 when ) x ) # 0.1, we may use M 5 e 0.1 and r 5 1, giving ) R 3 ( x ) ) # } e 0. 4 1 ! ) x ) 4 } . Thus, for ) x ) # 0.1, the maximum possible absolute error is about } e 0.1 2 (0 4 .1) 4 } < 4.605 3 10 2 6 . Alternate method: Using graphing techniques, when ) x ) # 0.1, ) error ) 5 ) e x 2 1 1 1 x 1 } x 2 2 } 1 } x 6 3 } 2 ) # ) e 0.1 2 1 1 1 0.1 1 } 0. 2 01 } 1 } 0.0 6 01 } 2 ) < 4.251 3 10 2 6 . 26. Since the Maclaurin series is } 1 2 1 x } 5 1 1 x 1 x 2 1 1 x n 1 , P 3 ( x ) 5 1 1 x 1 x 2 1 x 3 . Since ) f (4) ( x ) ) 5 24(1 2 x ) 2 5 , which is no more than 24(0.9) 2 5 when ) x ) # 0.1, we may use M 5 24(0.9) 2 5 and r 5 1, giving ) R 3 ( x ) ) # } 24(0.9 4 ) ! 2 5 ) x ) 4 }5} 0 ) x .9 ) 4 5 } . Thus, for ) x ) # 0.1, an upper bound for the magnitude of the approximation error is } 0 0 . . 1
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