Calculus: Early Transcendentals, by Anton, 7th Edition,ch10

Calculus - Early Transcendentals

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397 CHAPTER 10 Infnite Series EXERCISE SET 10.1 1. (a) f ( k ) ( x )=( 1) k e x ,f ( k ) (0)=( 1) k ; e x 1 x + x 2 / 2 (quadratic), e x 1 x (linear) (b) f 0 ( x )= sin x,f 0 ( x cos (0)=1 0 (0)=0 0 (0) = 1 , cos x 1 x 2 / 2 (quadratic), cos x 1 (linear) (c) f 0 ( x ) = cos 0 ( x sin ( π/ 2)=1 0 ( 2)=0 0 ( 2) = 1 , sin x 1 ( x 2) 2 / 2 (quadratic), sin x 1 (linear) (d) f (1)=1 0 / 2 0 (1) = 1 / 4; x =1+ 1 2 ( x 1) 1 8 ( x 1) 2 (quadratic), x 1+ 1 2 ( x 1) (linear) 2. (a) p 2 ( x )=1+ x + x 2 / 2, p 1 ( x x (b) p 2 ( x )=3+ 1 6 ( x 9) 1 216 ( x 9) 2 , p 1 ( x 1 6 ( x 9) (c) p 2 ( x π 3 + 3 6 ( x 2) 7 72 3( x 2) 2 , p 1 ( x π 3 + 3 6 ( x 2) (d) p 2 ( x x , p 1 ( x x 3. (a) f 0 ( x 1 2 x 1 / 2 0 ( x 1 4 x 3 / 2 ; f 0 (1) = 1 2 0 (1) = 1 4 ; x 1 2 ( x 1) 1 8 ( x 1) 2 (b) x =1 . 1 ,x 0 , 1 . 1 1 2 (0 . 1) 1 8 (0 . 1) 2 . 04875, calculator value 1 . 0488088 4. (a) cos x 1 x 2 / 2 (b) 2 = 90 rad, cos 2 = cos( 90) 1 π 2 2 · 90 2 0 . 99939077, calculator value 0 . 99939083 5. f ( x ) = tan x, 61 = 3+ 180 rad; x 0 = 3 0 ( x ) = sec 2 x, f 0 ( x )=2sec 2 x tan x ; f ( 3) = 3 0 ( 3)=4 0 ( x )=8 3; tan x 3+4( x 3)+4 3( x 3) 2 , tan 61 = tan( 180) 3+4 180+4 3( 180) 2 1 . 80397443, calculator value 1 . 80404776 6. f ( x x, x 0 =36 0 ( x 1 2 x 1 / 2 0 ( x 1 4 x 3 / 2 ; f (36) = 6 0 (36) = 1 12 0 (36) = 1 864 ; x 6+ 1 12 ( x 36) 1 1728 ( x 36) 2 ; 36 . 03 0 . 03 12 (0 . 03) 2 1728 6 . 00249947917, calculator value 6 . 00249947938 7. f ( k ) ( x 1) k e x , f ( k ) 1) k ; p 0 ( x )=1 ,p 1 ( x x, p 2 ( x x + 1 2 x 2 , p 3 ( x x + 1 2 x 2 1 3! x 3 4 ( x x + 1 2 x 2 1 3! x 3 + 1 4! x 4 ; n X k =0 ( 1) k k ! x k
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398 Chapter 10 8. f ( k ) ( x )= a k e ax , f ( k ) (0) = a k ; p 0 ( x )=1 ,p 1 ( x )=1+ ax, p 2 ( x ax + a 2 2 x 2 , p 3 ( x ax + a 2 2 x 2 + a 3 3! x 3 4 ( x ax + a 2 2 x 2 + a 3 3! x 3 + a 4 4! x 4 ; n X k =0 a k k ! x k 9. f ( k ) (0) = 0 if k is odd, f ( k ) (0) is alternately π k and π k if k is even; p 0 ( x 1 ( x , p 2 ( x π 2 2! x 2 ; p 3 ( x π 2 2! x 2 4 ( x π 2 2! x 2 + π 4 4! x 4 ; [ n 2 ] X k =0 ( 1) k π 2 k (2 k )! x 2 k NB: The function [ x ] deFned for real x indicates the greatest integer which is x . 10. f ( k ) (0) = 0 if k is even, f ( k ) (0) is alternately π k and π k if k is odd; p 0 ( x )=0 1 ( x πx, p 2 ( x πx ; p 3 ( x π 3 3! x 3 4 ( x π 3 3! x 3 ; [ n 1 2 ] X k =0 ( 1) k π 2 k +1 (2 k + 1)! x 2 k +1 NB If n = 0 then [ n 1 2 ]= 1; by deFnition any sum which runs from k =0to k = 1 is called the ’empty sum’ and has value 0. 11. f (0) (0) = 0; for k 1, f ( k ) ( x ( 1) k +1 ( k 1)! (1 + x ) k , f ( k ) (0)=( 1) k +1 ( k 1)!; p 0 ( x , p 1 ( x x, p 2 ( x x 1 2 x 2 3 ( x x 1 2 x 2 + 1 3 x 3 4 ( x x 1 2 x 2 + 1 3 x 3 1 4 x 4 ; n X k =1 ( 1) k +1 k x k 12. f ( k ) ( x )=( 1) k k ! (1 + x ) k +1 ; f ( k ) 1) k k !; p 0 ( x 1 ( x x, p 2 ( x x + x 2 3 ( x x + x 2 x 3 4 ( x x + x 2 x 3 + x 4 ; n X k =0 ( 1) k x k 13. f ( k ) (0) = 0 if k is odd, f ( k ) (0) = 1 if k is even; p 0 ( x 1 ( x , p 2 ( x x 2 / 2 3 ( x x 2 / 2 4 ( x x 2 / 2+ x 4 / 4!; [ n 2 ] X k =0 1 (2 k )! x 2 k 14. f ( k ) (0) = 0 if k is even, f ( k ) (0) = 1 if k is odd; p 0 ( x 1 ( x x, p 2 ( x x, p 3 ( x x + x 3 / 3!
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Calculus: Early Transcendentals, by Anton, 7th Edition,ch10...

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