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Pre-Calc Homework Solutions 373

Pre-Calc Homework Solutions 373 - Section 9.2 8 Substitute...

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8. Substitute 2 2 x for x in the Maclaurin series for e x shown at the end of Section 9.2. e 2 2 x 5 1 1 ( 2 2 x ) 1 } ( 2 2 2 ! x ) 2 } 1 1 } ( 2 n 2 ! x ) n } 1 5 1 2 2 x 1 2 x 2 2 1 } ( 2 1 n ) n ! 2 n x n } 1 The series converges for all real x . 9. f (2) 5 } 1 x } ) x 5 2 5 } 1 2 } f 9 (2) 5 2 x 2 2 ) x 5 2 5 2 } 1 4 } f 0 (2) 5 2 x 2 3 ) x 5 2 5 } 1 4 } , so } f 0 2 ( ! 2) } 5 } 1 8 } f - (2) 5 2 6 x 2 4 ) x 5 2 5 2 } 3 8 } , so } f - 3 ( ! 2) } 5 2 } 1 1 6 } P 0 ( x ) 5 } 1 2 } P 1 ( x ) 5 } 1 2 } 2 } x 2 4 2 } P 2 ( x ) 5 } 1 2 } 2 } x 2 4 2 } 1 } ( x 2 8 2) 2 } P 3 ( x ) 5 } 1 2 } 2 } x 2 4 2 } 1 } ( x 2 8 2) 2 } 2 } ( x 2 16 2) 3 } 10. f 1 } p 4 } 2 5 sin x ) x 5 p /4 5 } ˇ 2 2 w } f 9 1 } p 4 } 2 5 cos x ) x 5 p /4 5 } ˇ 2 2 w } f 0 1 } p 4 } 2 5 2 sin x ) x 5 p /4 5 2 } ˇ 2 2 w } , so 5 2 } ˇ 4 2 w } f - 1 } p 4 } 2 5 2 cos x ) x 5 p /4 5 2 } ˇ 2 2 w } , so 5 2 } ˇ 12 2 w } P 0 ( x ) 5 } ˇ 2 2 w } P 1 ( x ) 5 } ˇ 2 2 w } 1 1 } ˇ 2 2 w } 21 x 2 } p 4 } 2 P 2 ( x ) 5 } ˇ 2 2 w } 1 1 } ˇ 2 2 w } 21 x 2 } p 4 } 2 2 1 } ˇ 4 2 w } 21 x 2 } p 4 } 2 2 P 3 ( x ) 5 } ˇ 2 2 w } 1 1 } ˇ 2 2 w } 21 x 2 } p 4 } 2 2 1 } ˇ 4 2 w } 21 x 2 } p 4 } 2 2 2 1 } ˇ 12 2 w } 21 x 2 } p 4 } 2 3 11. f 1 } p 4 } 2 5 cos x ) x 5 p /4 5 } ˇ 2 2 w } f 9 1 } p 4 } 2 5 2 sin x ) x 5 p /4 5 2 } ˇ 2 2 w } f 0 1 } p 4 } 2 5 2 cos x )
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