MTH 201 Homework 6 Solution, Fall 2016

MATH 201. Homework 6 Solution Section 10.8: when � x � # # (c) (sec x)(tan x) 1 x œ � � � � á � � � � á Š ‹ Š ‹ x 5x 61x x 2x 17x 24 720 3 15 315 # x x x x x , œ � � � � � � � � � � á œ � � � � á ˆ ‰ ˆ ‰ ˆ ‰ " " " " # $ & ( 3 15 6 24 315 15 72 720 6 120 1008 2 5 17 5 61 5x 61x 277x x � � � 1 1 # # 55. (a) If f(x) a x , then f (x) n(n 1)(n 2) (n (k 1)) a x and f (0) k!a œ œ � � â � � œ ! ! _ _ œ œ n 0 n k n n k n k n k k Ð Ñ � Ð Ñ a ; likewise if f(x) b x , then b a b for every nonnegative integer k Ê œ œ œ Ê œ k n k k k f (0) f (0) k! k! n k k ! _ œ n 0 (b) If f(x) a x 0 for all x, then f (x) 0 for all x from part (a) that a 0 for every nonnegative integer k œ œ œ Ê œ ! _ œ n 0 n k n k Ð Ñ 10.8 TAYLOR AND MACLAURIN SERIES 1. f(x) e , f (x) 2e , f (x) 4e , f (x) 8e ; f(0) e , f (0) 2, f (0) 4, f (0) 8 P (x) 1, œ œ œ œ œ œ " œ œ œ Ê œ 2x 2x 2x 2x 2 0 w ww www w ww www ! a b P (x) 1 2x, P (x) 1 x 2x , P (x) 1 x 2x x " # $ # # œ � œ � � œ � � � 4 3 3 2. f(x) sin x, f (x) cos x , f (x) sin x , f (x) cos x; f(0) sin 0 0, f (0) 1, f (0) 0, f (0) 1 œ œ œ � œ � œ œ œ œ œ � w ww www w ww www P (x) 0, P (x) x, P (x) x, P (x) x x Ê œ œ œ œ � ! " # $ 1 6 3 3. f(x) ln x, f (x) , f (x) , f (x) ; f(1) ln 1 0, f (1) 1, f (1) 1, f (1) 2 P (x) 0, œ œ œ � œ œ œ œ œ � œ Ê œ w ww www w ww www " " ! x x x 2 P (x) (x 1), P (x) (x 1) (x 1) , P (x) (x 1) (x 1) (x 1) " # $ " " " # # # # $ œ � œ � � � œ � � � � � 3 4. f(x) ln (1 x), f (x) (1 x) , f (x) (1 x) , f (x) 2(1 x) ; f(0) ln 1 0, œ � œ œ � œ � � œ � œ œ w �" ww �# www �$ " 1 � x f (0) 1, f (0) (1) 1, f (0) 2(1) 2 P (x) 0, P (x) x, P (x) x , P (x) x w ww �# www �$ ! " # $ # # œ œ œ � œ � œ œ Ê œ œ œ � œ � � 1 x x x 1 3 5. f(x) x , f (x) x , f (x) 2x , f (x) 6x ; f(2) , f (2) , f (2) , f (x) œ œ œ � œ œ � œ œ � œ œ � " " " " �" w �# ww �$ www �% w ww www # x 4 4 8 3 P (x) , P (x) (x 2), P (x) (x 2) (x 2) , Ê œ œ � � œ � � � � ! " # " " " " " " # # # # 4 4 8 P (x) (x 2) (x 2) (x 2) $ " " " " # # $ œ � � � � � � 4 8 16 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 618 Chapter 10 Infinite Sequences and Series 6. f(x) (x 2) , f (x) (x 2) , f (x) 2(x 2) , f (x) 6(x 2) ; f(0) (2) , f (0) (2) œ � œ � � œ � œ � � œ œ œ � �" w �# ww �$ www �% �" w �# " # , f (0) 2(2) , f (0) 6(2) P (x) , P (x) , P (x) , œ � œ œ œ � œ � Ê œ œ � œ � � " " " " " ww �$ www �% ! " # # # # 4 4 8 4 4 8 3 x x x P (x) $ " œ # � � � x x x 4 8 16 7. f(x) sin x, f (x) cos x, f (x) sin x, f (x) cos x; f sin , f cos , œ œ œ � œ � œ œ œ œ w ww www w # # ˆ ‰ ˆ ‰ 1 1 1 1 4 4 4 4 2 2 È È f sin , f cos P , P (x) x , ww www # # # # # ! " ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 œ � œ � œ � œ � Ê œ œ � � È È È È È P (x) x x , P (x) x x x # $ # # # # # # # $ œ � � � � œ � � � � � � È È È È È È È 2 2 2 2 2 2 2 4 4 4 4 4 4 1 4 ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 8. f(x) tan x, f (x) sec x, f (x) 2sec x tan x, f (x) 2sec x 4sec x tan x; f tan 1 , œ œ œ œ � œ œ w ww www 2 2 4 2 2 4 4 ˆ ‰ 1 1 f sec 2 , f 2sec tan 4 , f 2sec 4sec tan 16 P (x) 1 , w ww www ! ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 2 2 4 2 2 œ œ œ œ œ � œ Ê œ P (x) 1 2 x , P (x) 1 2 x 2 x , P (x) 1 2 x 2 x x " # $ œ � � œ � � � � œ � � � � � � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 4 4 4 4 4 3 4 2 2 3 8 9. f(x) x x , f (x) x , f (x) x , f (x) x ; f(4) 4 2, œ œ œ œ � œ œ œ È ˆ ‰ ˆ ‰ ˆ ‰ È "Î# w �"Î# ww �$Î# www �&Î# " " # 4 8 3 f (4) 4 , f (4) 4 ,f (4) 4 P (x) 2, P (x) 2 (x 4), w �"Î# ww �$Î# www �&Î# " " " " " # ! " œ œ œ � œ � œ œ Ê œ œ � � ˆ ‰ ˆ ‰ ˆ ‰ 4 4 32 8 256 4 3 3 P (x) 2 (x 4) (x 4) , P (x) 2 (x 4) (x 4) (x 4) # $ " " " " " # # $ # œ � � � � œ � � � � � � 4 64 4 64 51 10. f(x) (1 x) , f (x) (1 x) , f (x) (1 x) , f (x) (1 x) ; f(0) (1) 1, œ � œ � � œ � � œ � � œ œ "Î# w �"Î# ww �$Î# www �&Î# "Î# " " # 4 8 3 f (0) (1) , f (0) (1) , f (0) (1) P (x) 1, w �"Î# ww �$Î# www �&Î# " " " " # # ! œ � œ � œ � œ � œ � œ � Ê œ 4 4 8 8 3 3 P (x) 1 x, P (x) 1 x x , P (x) 1 x x x " # $ " " " " " # # $ œ � œ � � œ � � � 2 2 8 2 8 16 1 11. f(x) e , f (x) e , f (x) e , f (x) e f (x) 1 e ; f(0) e , f (0) 1, œ œ � œ œ � Ê á œ � œ œ " œ � � w � ww � www � � � w x x x x x 0 k k a b a b f (0) 1, f (0) 1, f (0) ( 1) e 1 x x x x ww www � # � œ œ � á ß œ � Ê œ � � � � á œ k k x 3 n 1 1 2 6 n! ( 1) ! _ œ n 0 n 12. f(x) x e , f (x) x e e , f (x) x e 2e , f (x) x e 3e f (x) x e k e ; f(0) 0 e 0, œ œ � œ � œ � Ê á œ � œ œ x x x x x x x x x 0 w ww www k a b a b f (0) 1, f (0) 2, f (0) 3, f (0) k x x x x w ww www # � œ œ œ á ß œ Ê � � � á œ k 1 1 2 n 1 ! 3 n ! _ œ n 0 a b 13. f(x) (1 x) f (x) (1 x) , f (x) 2(1 x) , f (x) 3!(1 x) f (x) œ � Ê œ � � œ � œ � � Ê á �" w �# ww �$ www �% k ( 1) k!(1 x) ; f(0) 1, f (0) 1, f (0) 2, f (0) 3!, f (0) ( 1) k! œ � � œ œ � œ œ � á ß œ � k k 1 � � w ww www k k 1 x x x ( x) ( 1) x Ê � � � � á œ � œ � # $ ! ! _ _ œ œ n 0 n 0 n n n 14. f(x) f (x) , f (x) 6(1 x) , f (x) 18(1 x) f (x) 3 k! (1 x) ; f(0) 2, œ Ê œ œ � œ � Ê á œ � œ 2 x 3 1 x (1 x) � � � w ww �$ www �% k k 1 a b f (0) 3, f (0) 6, f (0) 18, f (0) 3 k! 2 3x 3x 3x 2 3x w ww www # $ œ œ œ á ß œ Ê � � � � á œ � k a b ! _ œ n 1 n 15 i i 3 3 Ê � ! ! ! _ _ _ ( ) x ( ) (3x) ( ) 3 x 3 x 3 x �" �" �" n 2n 1 n 2n 1 n 2n 1 2n 1 618 Chapter 10 Infinite Sequences and Series 6. f(x) (x 2) , f (x) (x 2) , f (x) 2(x 2) , f (x) 6(x 2) ; f(0) (2) , f (0) (2) œ � œ � � œ � œ � � œ œ œ � �" w �# ww �$ www �% �" w �# " # , f (0) 2(2) , f (0) 6(2) P (x) , P (x) , P (x) , œ � œ œ œ � œ � Ê œ œ � œ � � " " " " " ww �$ www �% ! " # # # # 4 4 8 4 4 8 3 x x x P (x) $ " œ # � � � x x x 4 8 16 7. f(x) sin x, f (x) cos x, f (x) sin x, f (x) cos x; f sin , f cos , œ œ œ � œ � œ œ œ œ w ww www w # # ˆ ‰ ˆ ‰ 1 1 1 1 4 4 4 4 2 2 È È f sin , f cos P , P (x) x , ww www # # # # # ! " ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 œ � œ � œ � œ � Ê œ œ � � È È È È È P (x) x x , P (x) x x x # $ # # # # # # # $ œ � � � � œ � � � � � � È È È È È È È 2 2 2 2 2 2 2 4 4 4 4 4 4 1 4 ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 8. f(x) tan x, f (x) sec x, f (x) 2sec x tan x, f (x) 2sec x 4sec x tan x; f tan 1 , œ œ œ œ � œ œ w ww www 2 2 4 2 2 4 4 ˆ ‰ 1 1 f sec 2 , f 2sec tan 4 , f 2sec 4sec tan 16 P (x) 1 , w ww www ! ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 2 2 4 2 2 œ œ œ œ œ � œ Ê œ P (x) 1 2 x , P (x) 1 2 x 2 x , P (x) 1 2 x 2 x x " # $ œ � � œ � � � � œ � � � � � � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 4 4 4 4 4 3 4 2 2 3 8 9. f(x) x x , f (x) x , f (x) x , f (x) x ; f(4) 4 2, œ œ œ œ � œ œ œ È ˆ ‰ ˆ ‰ ˆ ‰ È "Î# w �"Î# ww �$Î# www �&Î# " " # 4 8 3 f (4) 4 , f (4) 4 ,f (4) 4 P (x) 2, P (x) 2 (x 4), w �"Î# ww �$Î# www �&Î# " " " " " # ! " œ œ œ � œ � œ œ Ê œ œ � � ˆ ‰ ˆ ‰ ˆ ‰ 4 4 32 8 256 4 3 3 P (x) 2 (x 4) (x 4) , P (x) 2 (x 4) (x 4) (x 4) # $ " " " " " # # $ # œ � � � � œ � � � � � � 4 64 4 64 51 10. f(x) (1 x) , f (x) (1 x) , f (x) (1 x) , f (x) (1 x) ; f(0) (1) 1, œ � œ � � œ � � œ � � œ œ "Î# w �"Î# ww �$Î# www �&Î# "Î# " " # 4 8 3 f (0) (1) , f (0) (1) , f (0) (1) P (x) 1, w �"Î# ww �$Î# www �&Î# " " " " # # ! œ � œ � œ � œ � œ � œ � Ê œ 4 4 8 8 3 3 P (x) 1 x, P (x) 1 x x , P (x) 1 x x x " # $ " " " " " # # $ œ � œ � � œ � � � 2 2 8 2 8 16 1 11. f(x) e , f (x) e , f (x) e , f (x) e f (x) 1 e ; f(0) e , f (0) 1, œ œ � œ œ � Ê á œ � œ œ " œ � � w � ww � www � � � w x x x x x 0 k k a b a b f (0) 1, f (0) 1, f (0) ( 1) e 1 x x x x ww www � # � œ œ � á ß œ � Ê œ � � � � á œ k k x 3 n 1 1 2 6 n! ( 1) ! _ œ n 0 n 12. f(x) x e , f (x) x e e , f (x) x e 2e , f (x) x e 3e f (x) x e k e ; f(0) 0 e 0, œ œ � œ � œ � Ê á œ � œ œ x x x x x x x x x 0 w ww www k a b a b f (0) 1, f (0) 2, f (0) 3, f (0) k x x x x w ww www # � œ œ œ á ß œ Ê � � � á œ k 1 1 2 n 1 ! 3 n ! _ œ n 0 a b 13. f(x) (1 x) f (x) (1 x) , f (x) 2(1 x) , f (x) 3!(1 x) f (x) œ � Ê œ � � œ � œ � � Ê á �" w �# ww �$ www �% k ( 1) k!(1 x) ; f(0) 1, f (0) 1, f (0) 2, f (0) 3!, f (0) ( 1) k! œ � � œ œ � œ œ � á ß œ � k k 1 � � w ww www k k 1 x x x ( x) ( 1) x Ê � � � � á œ � œ � # $ ! ! _ _ œ œ n 0 n 0 n n n 14. f(x) f (x) , f (x) 6(1 x) , f (x) 18(1 x) f (x) 3 k! (1 x) ; f(0) 2, œ Ê œ œ � œ � Ê á œ � œ 2 x 3 1 x (1 x) � � � w ww �$ www �% k k 1 a b f (0) 3, f (0) 6, f (0) 18, f (0) 3 k! 2 3x 3x 3x 2 3x w ww www # $ œ œ œ á ß œ Ê � � � � á œ � k a b ! _ œ n 1 n 15. sin x sin 3x 3x œ Ê œ œ œ � � � á ! ! ! _ _ _ œ œ œ n 0 n 0 n 0 ( ) x ( ) (3x) ( ) 3 x ( n 1)! ( n 1)! ( n 1)! 3! 5! 3 x 3 x �" �" �" # � # � # � n 2n 1 n 2n 1 n 2n 1 2n 1 16. sin x sin œ Ê œ œ œ � � � á ! ! ! _ _ _ œ œ œ n 0 n 0 n 0 ( ) x ( ) x ( n 1)! ( n 1)! (2n 1)! 2 3! 2 5! x x x x ( ) �" �" # � # # � # � # �" n 2n 1 n 2n 1 n x 2n 1 2n 1 ˆ ‰ † † 17. 7 cos ( x) 7 cos x 7 7 , since the cosine is an even function � œ œ œ � � � � á ! _ œ n 0 ( ) x (2n)! ! 4! 6! 7x 7x 7x �" # n 2n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 7. f(x) sin x, f (x) cos x, f (x) sin x, f (x) cos x; f sin , f cos , œ œ œ � œ � œ œ œ œ w ww www w # # ˆ ‰ ˆ ‰ 1 1 1 1 4 4 4 4 2 2 È È f sin , f cos P , P (x) x , ww www # # # # # ! " ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 œ � œ � œ � œ � Ê œ œ � � È È È È È P (x) x x , P (x) x x x # $ # # # # # # # $ œ � � � � œ � � � � � � È È È È È È È 2 2 2 2 2 2 2 4 4 4 4 4 4 1 4 ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 8. f(x) tan x, f (x) sec x, f (x) 2sec x tan x, f (x) 2sec x 4sec x tan x; f tan 1 , œ œ œ œ � œ œ w ww www 2 2 4 2 2 4 4 ˆ ‰ 1 1 f sec 2 , f 2sec tan 4 , f 2sec 4sec tan 16 P (x) 1 , w ww www ! ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 2 2 4 2 2 œ œ œ œ œ � œ Ê œ P (x) 1 2 x , P (x) 1 2 x 2 x , P (x) 1 2 x 2 x x " # $ œ � � œ � � � � œ � � � � � � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 1 1 4 4 4 4 4 3 4 2 2 3 8 9. f(x) x x , f (x) x , f (x) x , f (x) x ; f(4) 4 2, œ œ œ œ � œ œ œ È ˆ ‰ ˆ ‰ ˆ ‰ È "Î# w �"Î# ww �$Î# www �&Î# " " # 4 8 3 f (4) 4 , f (4) 4 ,f (4) 4 P (x) 2, P (x) 2 (x 4), w �"Î# ww �$Î# www �&Î# " " " " " # ! " œ œ œ � œ � œ œ Ê œ œ � � ˆ ‰ ˆ ‰ ˆ ‰ 4 4 32 8 256 4 3 3 P (x) 2 (x 4) (x 4) , P (x) 2 (x 4) (x 4) (x 4) # $ " " " " " # # $ # œ � � � � œ � � � � � � 4 64 4 64 51 10. f(x) (1 x) , f (x) (1 x) , f (x) (1 x) , f (x) (1 x) ; f(0) (1) 1, œ � œ � � œ � � œ � � œ œ "Î# w �"Î# ww �$Î# www �&Î# "Î# " " # 4 8 3 f (0) (1) , f (0) (1) , f (0) (1) P (x) 1, w �"Î# ww �$Î# www �&Î# " " " " # # ! œ � œ � œ � œ � œ � œ � Ê œ 4 4 8 8 3 3 P (x) 1 x, P (x) 1 x x , P (x) 1 x x x " # $ " " " " " # # $ œ � œ � � œ � � � 2 2 8 2 8 16 1 11. f(x) e , f (x) e , f (x) e , f (x) e f (x) 1 e ; f(0) e , f (0) 1, œ œ � œ œ � Ê á œ � œ œ " œ � � w � ww � www � � � w x x x x x 0 k k a b a b f (0) 1, f (0) 1, f (0) ( 1) e 1 x x x x ww www � # � œ œ � á ß œ � Ê œ � � � � á œ k k x 3 n 1 1 2 6 n! ( 1) ! _ œ n 0 n 12. f(x) x e , f (x) x e e , f (x) x e 2e , f (x) x e 3e f (x) x e k e ; f(0) 0 e 0, œ œ � œ � œ � Ê á œ � œ œ x x x x x x x x x 0 w ww www k a b a b f (0) 1, f (0) 2, f (0) 3, f (0) k x x x x w ww www # � œ œ œ á ß œ Ê � � � á œ k 1 1 2 n 1 ! 3 n ! _ œ n 0 a b 13. f(x) (1 x) f (x) (1 x) , f (x) 2(1 x) , f (x) 3!(1 x) f (x) œ � Ê œ � � œ � œ � � Ê á �" w �# ww �$ www �% k ( 1) k!(1 x) ; f(0) 1, f (0) 1, f (0) 2, f (0) 3!, f (0) ( 1) k! œ � � œ œ � œ œ � á ß œ � k k 1 � � w ww www k k 1 x x x ( x) ( 1) x Ê � � � � á œ � œ � # $ ! ! _ _ œ œ n 0 n 0 n n n 14. f(x) f (x) , f (x) 6(1 x) , f (x) 18(1 x) f (x) 3 k! (1 x) ; f(0) 2, œ Ê œ œ � œ � Ê á œ � œ 2 x 3 1 x (1 x) � � � w ww �$ www �% k k 1 a b f (0) 3, f (0) 6, f (0) 18, f (0) 3 k! 2 3x 3x 3x 2 3x w ww www # $ œ œ œ á ß œ Ê � � � � á œ � k a b ! _ œ n 1 n 15. sin x sin 3x 3x œ Ê œ œ œ � � � á ! ! ! _ _ _ œ œ œ n 0 n 0 n 0 ( ) x ( ) (3x) ( ) 3 x ( n 1)! ( n 1)! ( n 1)! 3! 5! 3 x 3 x �" �" �" # � # � # � n 2n 1 n 2n 1 n 2n 1 2n 1 16. sin x sin œ Ê œ œ œ � � � á ! ! ! _ _ _ œ œ œ n 0 n 0 n 0 ( ) x ( ) x ( n 1)! ( n 1)! (2n 1)! 2 3! 2 5! x x x x ( ) �" �" # � # # � # � # �" n 2n 1 n 2n 1 n x 2n 1 2n 1 ˆ ‰ † † 17. 7 cos ( x) 7 cos x 7 7 , since the cosine is an even function � œ œ œ � � � � á ! _ œ n 0 ( ) x (2n)! ! 4! 6! 7x 7x 7x �" # n 2n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.8 Taylor and Maclaurin Series 619 18. cos x 5 cos x 5 5 œ Ê œ œ � � � � á ! ! _ _ œ œ n 0 n 0 ( 1) x ( 1) ( x) (2n)! ( n)! 2! 4! 6! 5 x 5 x 5 x � � # n 2n n 2n 1 1 1 1 1 19. cosh x 1 x 1 x 1 œ œ � � � � � á � � � � � � á œ � � � � á e e x x x x x x x x x ! 3! 4! ! 3! 4! ! 4! 6! x x � " # # # # # # ’ “ Š ‹ Š ‹ œ ! _ œ n 0 x (2n)! 2n 20. sinh x 1 x 1 x x œ œ � � � � � á � � � � � � á œ � � � � á e e x x x x x x x x x ! 3! 4! ! 3! 4! 3! 5! 6! x x � " # # # # ’ “ Š ‹ Š ‹ œ ! _ œ n 0 x (2n 1)! 2n 1 � 21. f(x) x 2x 5x 4 f (x) 4x 6x 5, f (x) 12x 12x, f (x) 24x 12, f (x) 24 œ � � � Ê œ � � œ � œ � œ % $ w $ # ww # www Ð Ñ 4 f (x) 0 if n 5; f(0) 4, f (0) 5, f (0) 0, f (0) 12, f (0) 24, f (0) 0 if n 5 Ê œ œ œ � œ œ � œ œ Ð Ñ w ww www Ð Ñ Ð Ñ n 4 n x 2x 5x 4 4 5x x x x 2x 5x 4 Ê � � � œ � � � œ � � � % $ $ % % $ 12 24 3! 4! 22. f(x) f (x) ; f (x) ; f (x) f (x) ; f(0) 0, f (0) 0, f (0) 2, œ Ê œ œ œ Ê œ œ œ œ x 2x x 2 6 x 1 x 1 x 1 x 1 x 1 n 1 n � w ww www Ð Ñ w ww � � � � � � � x a b a b a b a b a b 2 3 4 n 1 n f (0) 6, f (0) 1 n if n 2 x x x x 1 x www Ð Ñ # œ � œ � x Ê � � � � Þ Þ Þ œ � n 3 4 5 n n n a b a b ! _ œ n 2 23. f(x) x 2x 4 f (x) 3x 2, f (x) 6x, f (x) 6 f (x) 0 if n 4; f(2) 8, f (2) 10, œ � � Ê œ � œ œ Ê œ œ œ $ w # ww www Ð Ñ w n f (2) 12, f (2) 6, f (2) 0 if n 4 x 2x 4 8 10(x 2) (x 2) (x 2) ww www Ð Ñ $ # $ œ œ œ Ê � � œ � � � � � � n 12 6 2! 3! 8 10(x 2) 6(x 2) (x 2) œ � � � � � � # $ 24. f(x) 2x x 3x 8 f (x) 6x 2x 3, f (x) 12x 2, f (x) 12 f (x) 0 if n 4; f(1) 2, œ � � � Ê œ � � œ � œ Ê œ œ � $ # w # ww www Ð Ñ n f (1) 11, f (1) 14, f (1) 12, f (1) 0 if n 4 2x x 3x 8 w ww www Ð Ñ $ # œ œ œ œ Ê � � � n 2 11(x 1) (x 1) (x 1) 2 11(x 1) 7(x 1) 2(x 1) œ � � � � � � � œ � � � � � � � 14 12 2! 3! # $ # $ Section 10.8 Taylor and Maclaurin Series 619 18. cos x 5 cos x 5 5 œ Ê œ œ � � � � á ! ! _ _ œ œ n 0 n 0 ( 1) x ( 1) ( x) (2n)! ( n)! 2! 4! 6! 5 x 5 x 5 x � � # n 2n n 2n 1 1 1 1 1 19. cosh x 1 x 1 x 1 œ œ � � � � � á � � � � � � á œ � � � � á e e x x x x x x x x x ! 3! 4! ! 3! 4! ! 4! 6! x x � " # # # # # # ’ “ Š ‹ Š ‹ œ ! _ œ n 0 x (2n)! 2n 20. sinh x 1 x 1 x x œ œ � � � � � á � � � � � � á œ � � � � á e e x x x x x x x x x ! 3! 4! ! 3! 4! 3! 5! 6! x x � " # # # # ’ “ Š ‹ Š ‹ œ ! _ œ n 0 x (2n 1)! 2n 1 � 21. f(x) x 2x 5x 4 f (x) 4x 6x 5, f (x) 12x 12x, f (x) 24x 12, f (x) 24 œ � � � Ê œ � � œ � œ � œ % $ w $ # ww # www Ð Ñ 4 f (x) 0 if n 5; f(0) 4, f (0) 5, f (0) 0, f (0) 12, f (0) 24, f (0) 0 if n 5 Ê œ œ œ � œ œ � œ œ Ð Ñ w ww www Ð Ñ Ð Ñ n 4 n x 2x 5x 4 4 5x x x x 2x 5x 4 Ê � � � œ � � � œ � � � % $ $ % % $ 12 24 3! 4! 22. f(x) f (x) ; f (x) ; f (x) f (x) ; f(0) 0, f (0) 0, f (0) 2, œ Ê œ œ œ Ê œ œ œ œ x 2x x 2 6 x 1 x 1 x 1 x 1 x 1 n 1 n � w ww www Ð Ñ w ww � � � � � � � x a b a b a b a b a b 2 3 4 n 1 n f (0) 6, f (0) 1 n if n 2 x x x x 1 x www Ð Ñ # œ � œ � x Ê � � � � Þ Þ Þ œ � n 3 4 5 n n n a b a b ! _ œ n 2 23. f(x) x 2x 4 f (x) 3x 2, f (x) 6x, f (x) 6 f (x) 0 if n 4; f(2) 8, f (2) 10, œ � � Ê œ � œ œ Ê œ œ œ $ w # ww www Ð Ñ w n f (2) 12, f (2) 6, f (2) 0 if n 4 x 2x 4 8 10(x 2) (x 2) (x 2) ww www Ð Ñ $ # $ œ œ œ Ê � � œ � � � � � � n 12 6 2! 3! 8 10(x 2) 6(x 2) (x 2) œ � � � � � � # $ 24. f(x) 2x x 3x 8 f (x) 6x 2x 3, f (x) 12x 2, f (x) 12 f (x) 0 if n 4; f(1) 2, œ � � � Ê œ � � œ � œ Ê œ œ � $ # w # ww www Ð Ñ n f (1) 11, f (1) 14, f (1) 12, f (1) 0 if n 4 2x x 3x 8 w ww www Ð Ñ $ # œ œ œ œ Ê � � � n 2 11(x 1) (x 1) (x 1) 2 11(x 1) 7(x 1) 2(x 1) œ � � � � � � � œ � � � � � � � 14 12 2! 3! # $ # $ 25. f(x) x x 1 f (x) 4x 2x, f (x) 12x 2, f (x) 24x, f (x) 24, f (x) 0 if n 5; œ � � Ê œ � œ � œ œ œ % # w $ ww # www Ð Ñ Ð Ñ 4 n f( 2) 21, f ( 2) 36, f ( 2) 50, f ( 2) 48, f ( 2) 24, f ( 2) 0 if n 5 x x 1 � œ � œ � � œ � œ � � œ � œ Ê � � w ww www Ð Ñ Ð Ñ % # 4 n 21 36(x 2) (x 2) (x 2) (x 2) 21 36(x 2) 25(x 2) 8(x 2) (x 2) œ � � � � � � � � œ � � � � � � � � 50 48 24 2! 3! 4! # $ % # $ % 26. f(x) 3x x 2x x 2 f (x) 15x 4x 6x 2x, f (x) 60x 12x 12x 2, œ � � � � Ê œ � � � œ � � � & % $ # w % $ # ww $ # f (x) 180x 24x 12, f (x) 360x 24, f (x) 360, f (x) 0 if n 6; f( 1) 7, www # Ð Ñ Ð Ñ Ð Ñ œ � � œ � œ œ � œ � 4 5 n f ( 1) 23, f ( 1) 82, f ( 1) 216, f ( 1) 384, f ( 1) 360, f ( 1) 0 if n 6 w ww www Ð Ñ Ð Ñ Ð Ñ � œ � œ � � œ � œ � � œ � œ 4 5 n 3x x 2x x 2 7 23(x 1) (x 1) (x 1) (x 1) (x 1) Ê � � � � œ � � � � � � � � � � � & % $ # # $ % & 82 216 384 360 2! 3! 4! 5! 7 23(x 1) 41(x 1) 36(x 1) 16(x 1) 3(x 1) œ � � � � � � � � � � � # $ % & 26. f(x) 3x x 2x x 2 f (x) 15x 4x 6x 2x, f (x) 60x 12x 12x 2, œ � � � � Ê œ � � � œ � � � & % $ # w % $ # ww $ # f (x) 180x 24x 12, f (x) 360x 24, f (x) 360, f (x) 0 if n 6; f( 1) 7, www # Ð Ñ Ð Ñ Ð Ñ œ � � œ � œ œ � œ � 4 5 n f ( 1) 23, f ( 1) 82, f ( 1) 216, f ( 1) 384, f ( 1) 360, f ( 1) 0 if n 6 w ww www Ð Ñ Ð Ñ Ð Ñ � œ � œ � � œ � œ � � œ � œ 4 5 n 3x x 2x x 2 7 23(x 1) (x 1) (x 1) (x 1) (x 1) Ê � � � � œ � � � � � � � � � � � & % $ # # $ % & 82 216 384 360 2! 3! 4! 5! 7 23(x 1) 41(x 1) 36(x 1) 16(x 1) 3(x 1) œ � � � � � � � � � � � # $ % & 27. f(x) x f (x) 2x , f (x) 3! x , f (x) 4! x f (x) ( 1) (n 1)! x ; œ Ê œ � œ œ � Ê œ � � �# w �$ ww �% www �& Ð Ñ � � n n n 2 f(1) 1, f (1) 2, f (1) 3!, f (1) 4!, f (1) ( 1) (n 1)! œ œ � œ œ � œ � � Ê w ww www Ð Ñ " n n x 1 2(x 1) 3(x 1) 4(x 1) ( 1) (n 1)(x 1) œ � � � � � � � á œ � � � # $ ! _ œ n 0 n n 28. f(x) f (x) 3(1 x) , f (x) 12(1 x) , f (x) 60 (1 x) f (x) (1 x) ; œ Ê œ � œ � œ � Ê œ � 1 1 x 4 5 6 n n 3 n 2 ! 2 a b a b � w � ww � www � Ð Ñ � � � 3 f 0 1, f 0 3, f 0 12, f 0 60, , f 0 1 3x 6x 10x a b a b a b a b a b œ œ œ œ á œ Ê œ � � � � á w ww www Ð Ñ # � � n 3 n 2 ! 2 1 1 x a b a b3 x œ ! _ œ n 0 a ba b n 2 n 1 2 n � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 620 Chapter 10 Infinite Sequences and Series 29. f(x) e f (x) e , f (x) e f (x) e ; f(2) e , f (2) e , f (2) e œ Ê œ œ Ê œ œ œ á œ x x x n x n w ww Ð Ñ # w # Ð Ñ # e e e (x 2) (x 2) (x 2) (x 2) Ê œ � � � � � � � á œ � x n e e e 3! n! # # # $ # ! _ œ n 0 30. f(x) 2 f (x) 2 ln 2, f (x) 2 (ln 2) , f (x) 2 (ln 2) f (x) 2 (ln 2) ; f(1) 2, f (1) 2 ln 2, œ Ê œ œ œ Ê œ œ œ x x x x 3 n x n w ww # www Ð Ñ w f (1) 2(ln 2) , f (1) 2(ln 2) , , f (1) 2(ln 2) ww # www $ Ð Ñ œ œ á œ n n 2 2 (2 ln 2)(x 1) (x 1) (x 1) Ê œ � � � � � � � á œ x 3 2(ln 2) 2(ln 2) 2(ln 2) (x 1) 3! n! # # � 3 n n ! _ œ n 0 31. f(x) cos 2x , f (x) 2 sin 2x , f (x) 4 cos 2x , f (x) 8 sin 2x , œ � œ � � œ � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 2 2 2 2 w ww www f x 2 cos 2x f x 2 sin 2x . . ; f 1, f 0, f 4, f 0, f 2 , a b a b a b 4 4 5 5 4 4 2 2 4 4 4 4 4 a b a b ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ œ � ß œ � � ß œ � œ œ œ œ 1 1 1 1 1 1 1 w ww www f 0, . . ., f 1 2 cos 2x 1 2 x x . . . a b 5 2n 2n 4 4 2 4 3 4 n 2 4 2 ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ a b 1 1 1 1 1 œ œ � Ê � œ � � � � � � Ð Ñ x œ � ! ˆ ‰ _ œ n 0 a b a b � x 1 2 2n 4 2n n 2n 1 32. f(x) x 1, f (x) x 1 , f (x) x 1 , f (x) x 1 , f (x) x 1 , . . .; œ � œ � œ � � œ � œ � � È a b a b a b a b w ww www � Î � Î � Î � Î 1 1 3 15 2 4 8 16 1 2 3 2 5 2 7 2 4 a b f(0) 1, f (0) , f (0) , f (0) , f (0) , . . . x 1 1 x x x x œ œ œ � œ œ � Ê � œ � � � � � Þ Þ Þ w ww www 1 1 3 15 1 1 1 5 2 4 8 16 2 8 16 128 4 2 3 4 a b È 33. The Maclaurin series generated by cos x is x which converges on , and the Maclaurin series generated ! a b _ œ n 0 a b a b � x 1 2n 2n n �_ _ by is 2 x which converges on 1, 1 . Thus the Maclaurin series generated by f x cos x is given by 2 2 1 x 1 x n � � ! a b a b _ œ n 0 � œ � x 2 x 1 2x x which converges on the intersection of , and 1, 1 , so the ! ! a b a b _ _ œ œ n 0 n 0 a b a b � x 1 2n 2 2n n 2 5 n � œ � � � � Þ Þ Þ Þ �_ _ � interval of convergence is a 1, 1 . b � 34. The Maclaurin series generated by e is which converges on , . The Maclaurin series generated by x x ! n a b _ œ n 0 n x �_ _ f x 1 x x e is given by 1 x x 1 x x which converges on , a b a b a b a b ! œ � � � � œ � � Þ Þ Þ Þ �_ _ Þ 2 x 2 2 3 x 1 2 n 2 3 _ œ n 0 n x 35. The Maclaurin series generated by sin x is x which converges on , and the Maclaurin series ! a b _ œ n 0 a b a b � � x � 1 2n 1 2n 1 n �_ _ generated by ln 1 x is x which converges on 1, 1 . Thus the Maclaurin series genereated by a b a b ! � � _ œ n 1 a b �1 n n n 1 f x sin x ln 1 x is given by x x x x x which converges on a b a b Œ �Œ � ! ! œ † � œ � � � Þ Þ Þ Þ _ _ œ œ n 0 n 1 a b a b a b � � � x � 1 1 2n 1 n 2 6 2n 1 n 2 3 4 1 1 n n 1 the intersection of , and 1, 1 , so the interval of convergence is 1, 1 . a b a b a b �_ _ � � 36. The Maclaurin series generated by sin x is x which converges on , . The Maclaurin series ! a b _ œ n 0 a b a b � � x � 1 2n 1 2n 1 n �_ _ genereated by f x x sin x is given by x x x x x a b Œ � Œ �Œ � ! ! ! œ œ 2 2n 1 2n 1 2n 1 1 1 1 2n 1 2n 1 2n 1 2 _ _ _ œ œ œ n 0 n 0 n 0 a b a b a b a b a b a b � � � � x � x � x � � � n n n x x x . . . which converges on , œ � � � �_ _ Þ 3 5 7 1 2 3 45 a b 37. If e (x a) and f(x) e , we have f (a) e f or all n 0, 1, 2, 3, x œ � œ œ œ á ! _ œ n 0 f (a) n! n x n a n Ð Ñ e e e 1 (x a) at x a Ê œ � � � á œ � � � � á œ x a a (x a) (x a) (x a) (x a) 0! 1! 2! 2! ’ “ ’ “ � � � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 620 Chapter 10 Infinite Sequences and Series 29. f(x) e f (x) e , f (x) e f (x) e ; f(2) e , f (2) e , f (2) e œ Ê œ œ Ê œ œ œ á œ x x x n x n w ww Ð Ñ # w # Ð Ñ # e e e (x 2) (x 2) (x 2) (x 2) Ê œ � � � � � � � á œ � x n e e e 3! n! # # # $ # ! _ œ n 0 30. f(x) 2 f (x) 2 ln 2, f (x) 2 (ln 2) , f (x) 2 (ln 2) f (x) 2 (ln 2) ; f(1) 2, f (1) 2 ln 2, œ Ê œ œ œ Ê œ œ œ x x x x 3 n x n w ww # www Ð Ñ w f (1) 2(ln 2) , f (1) 2(ln 2) , , f (1) 2(ln 2) ww # www $ Ð Ñ œ œ á œ n n 2 2 (2 ln 2)(x 1) (x 1) (x 1) Ê œ � � � � � � � á œ x 3 2(ln 2) 2(ln 2) 2(ln 2) (x 1) 3! n! # # � 3 n n ! _ œ n 0 31. f(x) cos 2x , f (x) 2 sin 2x , f (x) 4 cos 2x , f (x) 8 sin 2x , œ � œ � � œ � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 1 2 2 2 2 w ww www f x 2 cos 2x f x 2 sin 2x . . ; f 1, f 0, f 4, f 0, f 2 , a b a b a b 4 4 5 5 4 4 2 2 4 4 4 4 4 a b a b ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ œ � ß œ � � ß œ � œ œ œ œ 1 1 1 1 1 1 1 w ww www f 0, . . ., f 1 2 cos 2x 1 2 x x . . . a b 5 2n 2n 4 4 2 4 3 4 n 2 4 2 ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ a b 1 1 1 1 1 œ œ � Ê � œ � � � � � � Ð Ñ x œ � ! ˆ ‰ _ œ n 0 a b a b � x 1 2 2n 4 2n n 2n 1 32. f(x) x 1, f (x) x 1 , f (x) x 1 , f (x) x 1 , f (x) x 1 , . . .; œ � œ � œ � � œ � œ � � È a b a b a b a b w ww www � Î � Î � Î � Î 1 1 3 15 2 4 8 16 1 2 3 2 5 2 7 2 4 a b f(0) 1, f (0) , f (0) , f (0) , f (0) , . . . x 1 1 x x x x œ œ œ � œ œ � Ê � œ � � � � � Þ Þ Þ w ww www 1 1 3 15 1 1 1 5 2 4 8 16 2 8 16 128 4 2 3 4 a b È 33. The Maclaurin series generated by cos x is x which converges on , and the Maclaurin series generated ! a b _ œ n 0 a b a b � x 1 2n 2n n �_ _ by is 2 x which converges on 1, 1 . Thus the Maclaurin series generated by f x cos x is given by 2 2 1 x 1 x n � � ! a b a b _ œ n 0 � œ � x 2 x 1 2x x which converges on the intersection of , and 1, 1 , so the ! ! a b a b _ _ œ œ n 0 n 0 a b a b � x 1 2n 2 2n n 2 5 n � œ � � � � Þ Þ Þ Þ �_ _ � interval of convergence is a 1, 1 . b � 34. The Maclaurin series generated by e is which converges on , . The Maclaurin series generated by x x ! n a b _ œ n 0 n x �_ _ f x 1 x x e is given by 1 x x 1 x x which converges on , a b a b a b a b ! œ � � � � œ � � Þ Þ Þ Þ �_ _ Þ 2 x 2 2 3 x 1 2 n 2 3 _ œ n 0 n x 35. The Maclaurin series generated by sin x is x which converges on , and the Maclaurin series ! a b _ œ n 0 a b a b � � x � 1 2n 1 2n 1 n �_ _ generated by ln 1 x is x which converges on 1, 1 . Thus the Maclaurin series genereated by a b a b ! � � _ œ n 1 a b �1 n n n 1 f x sin x ln 1 x is given by x x x x x which converges on a b a b Œ �Œ � ! ! œ † � œ � � � Þ Þ Þ Þ _ _ œ œ n 0 n 1 a b a b a b � � � x � 1 1 2n 1 n 2 6 2n 1 n 2 3 4 1 1 n n 1 the intersection of , and 1, 1 , so the interval of convergence is 1, 1 . a b a b a b �_ _ � � 36. The Maclaurin series generated by sin x is x which converges on , . The Maclaurin series ! a b _ œ n 0 a b a b � � x � 1 2n 1 2n 1 n �_ _ genereated by f x x sin x is given by x x x x x a b Œ � Œ �Œ � ! ! ! œ œ 2 2n 1 2n 1 2n 1 1 1 1 2n 1 2n 1 2n 1 2 _ _ _ œ œ œ n 0 n 0 n 0 a b a b a b a b a b a b � � � � x � x � x � � � n n n x x x . . . which converges on , œ � � � �_ _ Þ 3 5 7 1 2 3 45 a b 37. If e (x a) and f(x) e , we have f (a) e f or all n 0, 1, 2, 3, x œ � œ œ œ á ! _ œ n 0 f (a) n! n x n a n Ð Ñ e e e 1 (x a) at x a Ê œ � � � á œ � � � � á œ x a a (x a) (x a) (x a) (x a) 0! 1! 2! 2! ’ “ ’ “ � � � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.9 Convergence of Taylor Series 621 38. f(x) e f (x) e for all n f (1) e for all n 0, 1, 2, œ Ê œ Ê œ œ á x n x n Ð Ñ Ð Ñ e e e(x 1) (x 1) (x 1) e 1 (x 1) Ê œ � � � � � � � á œ � � � � � á x e e ! 3! 2! 3! (x 1) (x 1) # # $ � � ’ “ 39. f(x) f(a) f (a)(x a) (x a) (x a) f (x) œ � � � � � � � á Ê w # $ w # f (a) f (a) 3! f (a) f (a)(x a) 3(x a) f (x) f (a) f (a)(x a) 4 3(x a) œ � � � � � á Ê œ � � � � � á w ww # ww ww www # f (a) f (a) 3! 4! 4 † f (x) f (a) f (a)(x a) (x a) Ê œ � � � � � á Ð Ñ Ð Ñ Ð � Ñ # # n n n 1 f (a) n 2 f(a) f(a) 0, f (a) f (a) 0, , f (a) f (a) 0 Ê œ � œ � á œ � w w Ð Ñ Ð Ñ n n 40. E(x) f(x) b b (x a) b (x a) b (x a) b (x a) œ � � � � � � � � á � � ! " # $ # $ n n 0 E(a) f(a) b b f(a); from condition (b), Ê œ œ � Ê œ ! ! lim 0 x a Ä f(x) f(a) b (x a) b (x a) b (x a) b (x a) (x a) � � � � � � � � á � � � n n n œ lim 0 Ê œ x a Ä f (x) b 2b (x a) 3b (x a) nb (x a) n(x a) � � � � � � á � � � n n 1 n 1 b f (a) lim 0 Ê œ Ê œ " w � � � � á � � " � � � x a Ä f (x) 2b 3! b (x a) n(n )b (x a) n(n 1)(x a) n n 2 n 2 b f (a) lim 0 Ê œ Ê œ # " # � � # � ww � � á � � � � x a Ä f (x) 3! b n(n 1)(n 2)b (x a) n(n 1)(n )(x a) n n 3 n 3 b f (a) lim 0 b f (a); therefore, œ œ Ê œ Ê œ $ " " www Ð Ñ � 3! n! n! f (x) n! b n n x a Ä n n g(x) f(a) f (a)(x a) (x a) (x a) P (x) œ � � � � � á � � œ w # f (a) f (a) 2! n! n n n 41. f(x) ln (cos x) f (x) tan x and f (x) sec x; f(0) 0, f (0) 0, f (0) 1 L(x) 0 and Q(x) œ Ê œ � œ � œ œ œ � Ê œ œ � w ww # w ww x 2 42. f(x) e f (x) (cos x)e and f (x) ( sin x)e (cos x) e ; f(0) 1, f (0) 1, f (0) 1 œ Ê œ œ � � œ œ œ sin x sin x sin x sin x w ww # w ww L(x) 1 x and Q(x) 1 x Ê œ � œ � � x # 43. f(x) 1 x f (x) x 1 x and f (x) 1 x 3x 1 x ; f(0) 1, f (0) 0, œ � Ê œ � œ � � � œ œ a b a b a b a b # w # ww # # # w �"Î# �$Î# �$Î# �&Î# f (0) 1 L(x) 1 and Q(x) 1 ww # œ Ê œ œ � x 44. f(x) cosh x f (x) sinh x and f (x) cosh x; f(0) 1, f (0) 0, f (0) 1 L(x) 1 and Q(x) 1 œ Ê œ œ œ œ œ Ê œ œ � w ww w ww # x 45. f(x) sin x f (x) cos x and f (x) sin x; f(0) 0, f (0) 1, f (0) 0 L(x) x and Q(x) x œ Ê œ œ � œ œ œ Ê œ œ w ww w ww 46. f(x) tan x f (x) sec x and f (x) 2 sec x tan x; f(0) 0, f (0) 1, f 0 L(x) x and Q(x) x œ Ê œ œ œ œ œ Ê œ œ w # ww # w ww 10.9 CONVERGENCE OF TAYLOR SERIES 1. e 1 x e 1 ( 5x) 1 5x x 5x x x 5 x 5 x ! n! ! ! 3! n! ( 5x) ( 1) 5 x œ � � � á œ Ê œ � � � � á œ � � � � á œ # # # � � � ! ! _ _ œ œ n 0 n 0 n n n n Section 10.9 Convergence of Taylor Series 621 38. f(x) e f (x) e for all n f (1) e for all n 0, 1, 2, œ Ê œ Ê œ œ á x n x n Ð Ñ Ð Ñ e e e(x 1) (x 1) (x 1) e 1 (x 1) Ê œ � � � � � � � á œ � � � � � á x e e ! 3! 2! 3! (x 1) (x 1) # # $ � � ’ “ 39. f(x) f(a) f (a)(x a) (x a) (x a) f (x) œ � � � � � � � á Ê w # $ w # f (a) f (a) 3! f (a) f (a)(x a) 3(x a) f (x) f (a) f (a)(x a) 4 3(x a) œ � � � � � á Ê œ � � � � � á w ww # ww ww www # f (a) f (a) 3! 4! 4 † f (x) f (a) f (a)(x a) (x a) Ê œ � � � � � á Ð Ñ Ð Ñ Ð � Ñ # # n n n 1 f (a) n 2 f(a) f(a) 0, f (a) f (a) 0, , f (a) f (a) 0 Ê œ � œ � á œ � w w Ð Ñ Ð Ñ n n 40. E(x) f(x) b b (x a) b (x a) b (x a) b (x a) œ � � � � � � � � á � � ! " # $ # $ n n 0 E(a) f(a) b b f(a); from condition (b), Ê œ œ � Ê œ ! ! lim 0 x a Ä f(x) f(a) b (x a) b (x a) b (x a) b (x a) (x a) � � � � � � � � á � � � n n n œ lim 0 Ê œ x a Ä f (x) b 2b (x a) 3b (x a) nb (x a) n(x a) � � � � � � á � � � n n 1 n 1 b f (a) lim 0 Ê œ Ê œ " w � � � � á � � " � � � x a Ä f (x) 2b 3! b (x a) n(n )b (x a) n(n 1)(x a) n n 2 n 2 b f (a) lim 0 Ê œ Ê œ # " # � � # � ww � � á � � � � x a Ä f (x) 3! b n(n 1)(n 2)b (x a) n(n 1)(n )(x a) n n 3 n 3 b f (a) lim 0 b f (a); therefore, œ œ Ê œ Ê œ $ " " www Ð Ñ � 3! n! n! f (x) n! b n n x a Ä n n g(x) f(a) f (a)(x a) (x a) (x a) P (x) œ � � � � � á � � œ w # f (a) f (a) 2! n! n n n 41. f(x) ln (cos x) f (x) tan x and f (x) sec x; f(0) 0, f (0) 0, f (0) 1 L(x) 0 and Q(x) œ Ê œ � œ � œ œ œ � Ê œ œ � w ww # w ww x 2 42. f(x) e f (x) (cos x)e and f (x) ( sin x)e (cos x) e ; f(0) 1, f (0) 1, f (0) 1 œ Ê œ œ � � œ œ œ sin x sin x sin x sin x w ww # w ww L(x) 1 x and Q(x) 1 x Ê œ � œ � � x # 43. f(x) 1 x f (x) x 1 x and f (x) 1 x 3x 1 x ; f(0) 1, f (0) 0, œ � Ê œ � œ � � � œ œ a b a b a b a b # w # ww # # # w �"Î# �$Î# �$Î# �&Î# f (0) 1 L(x) 1 and Q(x) 1 ww # œ Ê œ œ � x 44. f(x) cosh x f (x) sinh x and f (x) cosh x; f(0) 1, f (0) 0, f (0) 1 L(x) 1 and Q(x) 1 œ Ê œ œ œ œ œ Ê œ œ � w ww w ww # x 45. f(x) sin x f (x) cos x and f (x) sin x; f(0) 0, f (0) 1, f (0) 0 L(x) x and Q(x) x œ Ê œ œ � œ œ œ Ê œ œ w ww w ww 46. f(x) tan x f (x) sec x and f (x) 2 sec x tan x; f(0) 0, f (0) 1, f 0 L(x) x and Q(x) x œ Ê œ œ œ œ œ Ê œ œ w # ww # w ww 10.9 CONVERGENCE OF TAYLOR SERIES 1 e 1 x e 1 ( 5x) 1 5x x 5x x x 5 x 5 x ( 5x) ( 1) 5 x œ � � � œ Ê œ � � � œ � � œ � � � ! ! _ _ n n n n Section 10.9: 43. f(x) 1 x f (x) x 1 x and f (x) 1 x 3x 1 x ; f(0) 1, f (0) 0, œ � Ê œ � œ � � � œ œ a b a b a b a b # w # ww # # # w �"Î# �$Î# �$Î# �&Î# f (0) 1 L(x) 1 and Q(x) 1 ww # œ Ê œ œ � x 44. f(x) cosh x f (x) sinh x and f (x) cosh x; f(0) 1, f (0) 0, f (0) 1 L(x) 1 and Q(x) 1 œ Ê œ œ œ œ œ Ê œ œ � w ww w ww # x 45. f(x) sin x f (x) cos x and f (x) sin x; f(0) 0, f (0) 1, f (0) 0 L(x) x and Q(x) x œ Ê œ œ � œ œ œ Ê œ œ w ww w ww 46. f(x) tan x f (x) sec x and f (x) 2 sec x tan x; f(0) 0, f (0) 1, f 0 L(