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

Calculus - Early Transcendentals

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397 CHAPTER 10 Infinite 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 ( x ) = sin x, f ( x ) = cos x, f (0) = 1 , f (0) = 0 , f (0) = 1 , cos x 1 x 2 / 2 (quadratic), cos x 1 (linear) (c) f ( x ) = cos x, f ( x ) = sin x, f ( π/ 2) = 1 , f ( π/ 2) = 0 , f ( π/ 2) = 1 , sin x 1 ( x π/ 2) 2 / 2 (quadratic), sin x 1 (linear) (d) f (1) = 1 , f (1) = 1 / 2 , f (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 ) = 1 + x (b) p 2 ( x ) = 3 + 1 6 ( x 9) 1 216 ( x 9) 2 , p 1 ( x ) = 3 + 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 ( x ) = 1 2 x 1 / 2 , f ( x ) = 1 4 x 3 / 2 ; f (1) = 1 , f (1) = 1 2 , f (1) = 1 4 ; x 1 + 1 2 ( x 1) 1 8 ( x 1) 2 (b) x = 1 . 1 , x 0 = 1 , 1 . 1 1 + 1 2 (0 . 1) 1 8 (0 . 1) 2 = 1 . 04875, calculator value 1 . 0488088 4. (a) cos x 1 x 2 / 2 (b) 2 = π/ 90 rad, cos 2 = cos( π/ 90) 1 π 2 2 · 90 2 0 . 99939077, calculator value 0 . 99939083 5. f ( x ) = tan x, 61 = π/ 3 + π/ 180 rad; x 0 = π/ 3 , f ( x ) = sec 2 x, f ( x ) = 2 sec 2 x tan x ; f ( π/ 3) = 3 , f ( π/ 3) = 4 , f ( x ) = 8 3; tan x 3 + 4( x π/ 3) + 4 3( x π/ 3) 2 , tan 61 = tan( π/ 3 + π/ 180) 3 + 4 π/ 180 + 4 3( π/ 180) 2 1 . 80397443, calculator value 1 . 80404776 6. f ( x ) = x, x 0 = 36 , f ( x ) = 1 2 x 1 / 2 , f ( x ) = 1 4 x 3 / 2 ; f (36) = 6 , f (36) = 1 12 , f (36) = 1 864 ; x 6 + 1 12 ( x 36) 1 1728 ( x 36) 2 ; 36 . 03 6 + 0 . 03 12 (0 . 03) 2 1728 6 . 00249947917, calculator value 6 . 00249947938 7. f ( k ) ( x ) = ( 1) k e x , f ( k ) (0) = ( 1) k ; p 0 ( x ) = 1 , p 1 ( x ) = 1 x, p 2 ( x ) = 1 x + 1 2 x 2 , p 3 ( x ) = 1 x + 1 2 x 2 1 3! x 3 , p 4 ( x ) = 1 x + 1 2 x 2 1 3! x 3 + 1 4! x 4 ; n 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 ) = 1 + ax + a 2 2 x 2 , p 3 ( x ) = 1 + ax + a 2 2 x 2 + a 3 3! x 3 , p 4 ( x ) = 1 + ax + a 2 2 x 2 + a 3 3! x 3 + a 4 4! x 4 ; n 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 , p 1 ( x ) = 1 , p 2 ( x ) = 1 π 2 2! x 2 ; p 3 ( x ) = 1 π 2 2! x 2 , p 4 ( x ) = 1 π 2 2! x 2 + π 4 4! x 4 ; [ n 2 ] k =0 ( 1) k π 2 k (2 k )! x 2 k NB: The function [ x ] defined 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 , p 1 ( x ) = πx, p 2 ( x ) = πx ; p 3 ( x ) = πx π 3 3! x 3 , p 4 ( x ) = πx π 3 3! x 3 ; [ n 1 2 ] 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 definition any sum which runs from k = 0 to 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 ) = 0 , p 1 ( x ) = x, p 2 ( x ) = x 1 2 x 2 , p 3 ( x ) = x 1 2 x 2 + 1 3 x 3 , p 4 ( x ) = x 1 2 x 2 + 1 3 x 3 1 4 x 4 ; n k =1 ( 1) k +1 k x k 12. f ( k ) ( x ) = ( 1) k k ! (1 + x ) k +1 ; f ( k ) (0) = ( 1) k k !; p 0 ( x ) = 1 , p 1 ( x ) = 1 x, p 2 ( x ) = 1 x + x 2 , p 3 ( x ) = 1 x + x 2 x 3 , p 4 ( x ) = 1 x + x 2 x 3 + x 4 ; n 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 , p 1 ( x ) = 1 , p 2 ( x ) = 1 + x 2 / 2 , p 3 ( x ) = 1 + x 2 / 2 , p 4 ( x ) = 1 + x 2 / 2 + x 4 / 4!; [ n 2 ] k =0 1 (2 k )!
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