MA1014_HW21r_201012_v2

MA1014_HW21r_201012_v2 - ˆx c 4 ‰ c 4 ˆx c 4 ‰ , P$...

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Unformatted text preview: ˆx c 4 ‰ c 4 ˆx c 4 ‰ , P$ (x) œ # b # ˆx c 4 ‰ c 4 ˆx c 4 ‰ c 1# ˆx c 4 ‰ " 3 7. f(x) œ Èx œ x"Î# , f w (x) œ ˆ " ‰ xc"Î# , f ww (x) œ ˆc 4 ‰ xc$Î# , f www (x) œ ˆ 8 ‰ xc&Î# ; f(4) œ È4 œ 2, # 1 " " " " " 3 ww x, www ˆ . f(x) œ cos x, f w (x)f wœ cœ ˆ " ‰ 4ww (x) œ c ,cos(4) fœ (x) œ‰sin $Î# fœ 1c œ ,f www (4) œ È3 ‰ 4c&Î# œ 256 Ê P! (x) œ 2, P" (x) œ 2 b 4 (x c 4), x, (4) sin # f c"Î# c 4 4c x; ˆ 4 ‰ 32 cos 4 œ ˆ 8 , 4f 2 P# (x) œ # b # f w ˆ 1 ‰ œ c sin 4 P" (x) œ P$ (x) œ " È2 " È2 " " " " " " P" (x) œ 2 b 4 x, P# (x) œ 2 b 4 x c 64 x# , P$ (x) œ 2 b 4 x c 64 x# b 512 x$ " 3 . f(x) œ Èx œ x"Î#Pf w (x)II:ˆPower , f ww (x) œ ˆc 4 ‰ xc$Î# , f www (x) œ ˆ 8 ‰ xc&Î# ; f(4) œ È4 œ 2, , art œ " ‰ xc"Î# series. # _ww n textbook$Î# _ (Thomas), (4) œxˆ 3 ‰ x &Î# xœ 3 Ê 1, œ 4 P 44, and " (x 11. " " (c c " x f w (4) œ ˆ " ‰ 4c"Î# Fromf your Ê c cx‰ œc! œx)n œ ,f wwwchapterc 4c7, exercises:P4(x) 42 2, 3," (x) œ 2 b46. c 4), œ4 ! # 32 1 c x b 8 256 4 9. ex œ ,! (4) œ ˆ e 4 4 b cá " nœ nœ P# (x) œ 2 b 4 (x c 4) c 0" (x c 4)# , P$ (x) 0 2 b " (x c 4) c œ 64 4 # " " " c È ˆx c 1 ‰ , P# (x) œ wÈ c È" ˆx c 1 ‰ c"Î# È wwˆx c 1 ‰ , " c" 3 2 8. 2Part I: Intervals 2 œ ˆ # ‰ (x b 44) #, f2 (x) œ 4ˆc 4 ‰ (x b 4)c$Î# , f www (x) œ ˆ 8 ‰ (x b 4)c&Î# ; f(0) œ (4)"Î# œ 2, f(x) œ 4 b 4)"Î# , f (x) Convergence. (x of # $ "ˆ " 3 " c È2From 1your#Èc"Î# c 1 ‰(Thomas), chapter 11.c, " , f www (0) œ ˆ 3,‰9, 11, 15, and 25.! (x) œ 2, x c 4 ‰ c textbook ww (0)" 2 ˆc 1 (4) f w (0) œ ˆ " ‰ (4) 2 ˆx œ 4 , fb 6Èœ ˆx c" ‰ ‰ c$Î# œ 7 32 exercises: 78 (4)c&Î# œ 256 Ê P 4 44 # 1 4 " $ Mc œ ww " (x c–4) c 1 (x " #21:wwwPowerb " œ " Ê" (x c œ# b " ematics II Homework f ˆ 1 œ 2 Series 4) 1 " P ath" œ# (x)È2 ,2fbˆ 4 ‰ œ c cos 64 œ c 4)2 ,, P$ (x) ‰ œ sin 41(x cÈ2 c 64P! (x) 4) È2 , # (x c 4) 51 È 4 4 4 4 n! n! 10. œ e á ˆ b (x b b , . f(x) œ (x b 4) ,ef (x) œ ˆn! ‰Ê b 4) œ , f (x) œ 1c 4#‰ b 4†2!4) 2 †3!f b(x)†4! b 8 ‰ (x b 4) ; f(0) œ (4) n! 2 œˆ # (x nœ0 n œ0 3‰ 3 "‰ " "‰ " w c"Î# ww c$Î# www c&Î# f (0) œ ˆ # (4) Total, numberc 4 problems: 5+13+11=30. (4) œ 4 f (0) œ ˆ of (4) œ c 32 , f (0) œ ˆ 8 œ 256 Ê P! (x) œ 2, 51# Part III: Taylor and Maclaurin series. _ our textbook (Thomas), chapter 11.8, exercises: 9-16 (ALL), 19, 27, and 28. _ ˆ x ‰n From y x" x x x 2 c"Î# 3 "x "Î# xw c$Î# www x c&Î# "Î# !n ! ww 2 #! " 64 (x c 4)# b 3! 4! " (x c 4)$ œ 2, " 741 27. f(x) œ Ê Ê xf œ 1 œ exbf x(x) œ$eb áTaylorxœ ex ; f(2) œ e# , f wSeries # , á f ÐnÑ (2) œ e# ex c w (x) b xSection 11.8 œÐn! and Maclaurin (2) œ e , ww # b x x Ê f Ñ (x) n 1. f(x) œ (1 b x)c" Ê f w1(x) œ c(1 b x)c# , f ww (x) œ 2(1 bœx)c$ , f www (x) œ c3!(1 b x)c% Ê á f k (x) n0 _ _ 12. c )n x2n 7x 7x 7x œ 7 cos x œc")! ‰(x)"ckc1e_ b c (x1,2nf 2)4! œ c1, bwwá bsince www (0)cosine is œ ! ke(0) œ (2)n k k! 1 œ (c_ k(k!(1Ê 2nex œ œ 7 ec")n!xc w1(0) c (x c 2)# ,œe2,(x the2)œb á á even function1) 1) 7 n ˆb (2n)! ;#f(0)( # # b b e# 6! x (0) x3! f c $ c3!, an ß f n! (x c c œ f x x sin # œ ! (#nnœ01)! œ ! _2n 1 (2nnb2n 1 œ x c 2 †3! b_2_†5! b á 2n 1 _# _ nœ0n 2n 1 2n 1 b #$ 1)! n n )x x x x n" Ê 1 œ0x œ 1 c x1b x#œ0 ! b#á1)! ! (cx)n œœ !cc"#n(3x) 4. c 13. sin nœ x (c"nb œ Ê sin 3x ! ( ( 1)) xn1)! œ ! (c")(#3 b1)! œ 3x c 33! b 35! c á b ( (b n nœ0 nœ0 0 n0 nœ nœ0 _œw x www x $ Ðn Ñ x n w (c1)n x2n 28. f(x) œ 2x Ê f (x) n (1x)2n ln 2, f ww (x) xœ 2x51 x 2)# , 51 x Ê 5 cos 1x œ 5 ! (c1)#œ 2 œ 5 c 51 b (ln c f 6!(x) œ 2 (ln 2) Ê f (x) œ 2 (ln 2) ; f(1) œ 2, f (1) œ 2 ln 2, bá _ (2n)! n 2n ( n)! 2! 4! ( c" ) x ww 7x b 7x nc #7x www á , since the cosine Ñ an even function $ Ðn is n 0! œ0 œ 7 c œ 2(ln 2) , b œ 2(ln 2) 2. œ0 (2n)! (1 c f c"#!16. f4!(x) œ6!f (1)x)c# , f ww (x) , á , fc x)c$œf2(ln 2) 3!(1 c x)c% Ê á f k (x) f(x) œ x) (1) w (1 c œ 2(1 (1) , www (x) œ n _ n 2)n (x k œ k!(1 c x) Ê1 ; 2x œ œ b (2(0) 2)(x cww1) b 2(ln 2)www(x c 1)#xb 2(lnf2)k (0) c 1)$ b á x œ ! 2(ln x n! c1) f(0) 2 1, f w ln œ 1, f (0) œ 2, f (0)x œ 3!, á x3! (x œ k! ß # ex b e x " x x x x # n0 œ # ’Š1 b x nb8.2n b 3! b 4! b á _ b Š1 c x b #! c 3! b 4! c á ‹“ œ 1 b #! bœ4! b 6! b á ‹ 2 #! _ # " Ê c 5 ! c (1 # b x$ b á b 51 n b 5 cos 1x 1œ x œ 1 (b1)xn)!x)x œ 5 c 51 x œ ! xx c 51 x b á (# 2! 4! 6! n 0 nœ0 n 0.(ce)xn xœ 1! x " 2n n! Ê (#nb1)! nœ0 n c x " . ex œ ! x Ê ecx œ1! (œx) c x b x# b x!$ c x bœ ! (cx)n œ ! (c1)n xn Ê b x n! 1 œ 1 c x c # b 3! x á 4! c á n! nœ0 Answers fto EVEN problemsœ nARTx)c$ ,: f œ0(x) œ 3! (1 c x)c% Ê f ÐnÑ (x) œ n! (1 c x)cnc1 ; P œ0 III nwww x nœ0 w c# ww 26. f(x) œ 1 c x Ê (x) œ (1 c x) , f (x) 2(1 c Section 11.8 Taylor and Maclaurin Series 741 " " "w "# " " P" (x) œ 2 b 11. P# (x) œ 2 b x)c"cÊ x#(x)$œ c(12 b 4cx ,cww64 x# b 512b $ c$ , f www (x) œ c3!(1 b x)c% Ê á f k (x) f (x) œ 2(1 x x) 4 x, f(x) œ (1 b 4 x 64 f , P (x) œ b x) -œ (c1)k k!(1 b x)ckc1 ; f(0) œ 1, f w (0) œ c1, f ww (0) œ 2, f www (0) œ c3!, á ß f k (0) œ (c1)k k! _ 742 Chapter 11 n Infinite Sequences and Series _ _ _ _ x _ f(0) œ 20, f w! ˆœ‰n1, c1ww (0) _ f w2, f www (0) œ 3! c# ,xf ww (x) œ 2(1 c#x)c$ ,$ fb (x) œ 3!(1 xnb1 c% Ê á f k (x) (0) x f" œ Ê 1 c x œ x b x b x www á œ ! c x) 12. (1 2 _ Ê xexf(x) œ(c")ncx ‰2n 1 b x b (4x2!n x2n 2c3!x) x 2 †4! xb á x œ (1 n! x) Ê # (x)†"œb 1 x† b ˆœ nœ0 c) sin # œ !!(10(#nb1)!k 1 ;œ !œ#1, 1f(2nb1)! œ # ww (0) †3! 2, f2www†5! b á á ß f k (0) œ k! œ k nœ c x) f(0) 2n w (0) œ 1, f c 2 œ b (0) œ 3!, _ 10. 0 nœ _ ") ( 3. sin b œ ! x(c#nbx xb áa‹ bc a) œ ! c(c"#c 1)! c a)œ ! á ")(#3 ba x b œ b x b3!x(xb a)á c á b x# x x! b 3! (Ê x4! œ e ’ (x 3x c x(x a)!( ) n(3x) b x b ác ‹“n œ ’1 bx(x3x 4! bx6! c 35!b á “ at x œ a b 1)! Ê sin Š1 b b x b x 1 #! c c 3 b x (x 1)! # # b 3! 4! c x ex ce x " nœ0 x x nœ0 1! œ 0! b á ‹ c Š1 c x b nx 0 c“xœ e x c á ‹“ a) x b x b x b x b á 2! 2! œ # ’Š1 b x b #! b 3! b 4! œ # #! 3! b 4! 3! 5! 6! _ n 2n 1 )! 29. If ex œ ! nœ0 _ f n nœ0 nœ0 (a) n! (x c a)n and f(x) œ ex , we have f ÐnÑ (a) œ ea f or all n œ 0, 1, 2, 3, á _ n 2n 1 _ n 2n 1 2n 1 1 e e Ê ex œ e b e(x c 1) b #! (x c 1)# b 3! (x c 1)$ b á œ e ’1 b (x c 1) b (x c 1) b (x c 1) b á “ 2! 3! b x$b x! b x b x b á ‹ c Š1 c x#b x! cww x b x c # ‹“ œ xwwwb x b x b x b á á 4! 3! 5! 2x c#5x b3!4 Ê f w (x) œ 4x$ c 6x c # f (x) œ4! 5, 3! 12x c 12x, f (x) œ 24x c 6! f Ð4Ñ (x) œ 24 12, www (a) 4Ñ ) œ 0 if n 5; f(0) œ 4, f w (0)f œ c5,c ww (0) œ (a) (x c a)# b 12, f Ð(x(0)a)$ 24,áÐnÑÊ œ(x)if n 5 31. f(x) œ f(a) b w (a)(x f a) b f #0, f (0) œ c f 3! c œ b f (0) f w 0 12 $ 24 % % 2x$ c 5x b 4 œ 4 cw 5x c 3! x b 4! x œf x(a)c 2x$ c # b 4 itself ww 5x 4 œ f (a) b f ww (a)(x c a) b 3! 3(x c a) b á Ê f (x) œ f ww (a) b f www (a)(x c a) b f 4!(a) 4 † 3(x c a)# b á 2 (a) Ê f w (x) œ 4x$ c f Ðn# c 5,wwffwwÐ(x) œb f Ðn# cÑ12x, c a) œ n24x c 12, f Ð4Ñ (x) œ 24 f www Ê 6xÑ 1); f nÑ œ 12x b1Ñ (x) œ (x)n f 3; (x œ 1,# w (0) 1)# Ê f ww(x) œ 2(x b(x) œ (x) (a)2 Ê f Ðn(a)(xÐ4Ñ 0 ifb # Ðf(0) c a) fb ᜠ2, f ww (0) œ 2, f ÐnÑ (0) œ 0 if ww www f(0) œ 4, f (0) œ cf(a) œ f(a) 0, f f w (a) # cf12, fb 0, á ,24, Ñf(a) (0) fœ Ñ0 if b 5 5, f (0) œ (0) œ w (0) œ f Ðn nÑ œ Ðn (a) n 0 (x b 1)# œ 12 b$Ê b #!%x# œ%1b 0, $ b x œ (a) 1 2x 24 2 b 2x 4 c 5x c 3! x b 4! x œ x c 2x c 5x b 4 itself 1)! 30. f(x) œ ex Ê f ÐnÑ (x) œ ex for all n Ê f ÐnÑ (1) œ e for all n œ 0, 1, 2, á 32. E(x) œ f(x) c b c b (x c a) c b (x c a)# c b$ (x c a)$ c á c bn (x c a)n 2x b 4 Ê fwww (x) œ 3x# c 2,nÑ!ww (x) " 6x, f www (x) #œ 6 Ê fwÐnÑ (x) œ 0 ww n 4; f(2) œ 8, f w (2) œ 10, f œ if œ 2(x b 1); f (x) œ 2 Ê f Ðœ(x) œ 0bif n b! f(0) œ 1, f (0) œ 2, f (0) œ 2, f ÐnÑ (0) œ 0 if Ê 0 œ E(a) f(a) c Ê 3; œ f(a); from condition (b), 6 , f www (2) œ 6, f ÐnÑ (2) œ 0 if n 4 Ê x$! c 2x b 4 œ 8 b 10(x c 2)nb 12 (x c 2)# b 3! (x c 2)$ 2 b 2x b #! x# œ 1im 2x bc f(a) c b (x c a) c b (x c a) c b (x c a) c á c bn (x c a) œ2! b f(x) x# $ 0 (x c a)n (x c 2) b 6(x cxlÄ ab (x c 2) 2)# Ê xlim f (x) c b c 2b (x c a) c 3bc(x c a) c á c nbn (x c a) œ 0 n1 (x) # 3x c 2, f ww (x)wÄ a f www#(x) œ 6 Ê n(xÐnÑ (x) œ 0 if n www f(2) œ 8, f w (2)nœ 10, œ œ 6x, f a) 4; ww Ñ )b b x b 3x c 8 Ê b (x)$f w (a) Ê 2x b 3,f f(x) c 2b c12x (x122, c á c n(n c12 n (x c a)fn Ð 2$ (x)0œ 0 if n 4; f(1) œ c2, Ê f œ œ 6x b lim (x) œ 3! b bc a) f (x)#œ "6 Ê ÐnÑ " 1)(x (2) œ 0 if n 4 Ê x c Ñ2x b 4xœ 8 b 10(x c 2) $ n(n2! # c a)n 2 b 3! (x c 2) œ b c (x c 2) Äa , f ww (1) œ 14, f www (1) œ 12, f Ðn (1) œ 0 if n 4 (x) c 3! b c b c n(n c3x c 2)b (x c a)n 3 Ê 2x á x b 1)(n c 8 n 2)# b (x c 2)$ Ê b # œ " f ww (a) Ê xlim f œ0 # n(n c 1)(n c # c 1) n 3 1(x c 1) b 14 (x c 1)# b 12 (x c 1)$ œÄ a b 11(x c 1) b 7(x)(x c a)# b 2(x c 1)$ c2 2! 3! n c " " œ b$ œ 3! f www (a) Ê xlima f (x)n! n! bn œ 0 Ê bn œ n! f ÐnÑ (a); therefore, Ä Ê f w (x) œ 6x# b 2x b 3, f ww (x) œ 12x b 2, (a) (x)www 12 Ê f ÐnÐÑf4Ñn (a)œ 0 if n Ñ 4; f(1) œ c2, f www œ (x) n x# b 1 Ê f w (x) œ 4x$ b 2x, f ww (x) œ$ 12x#f b 2, fc(x)# œ 24x, f (x) œ 24, f Ðn (x) œ 0 if n 5; g(x) 0 if n w Ê 2x b b (x 8 www (1) œ 12, f ÐnÑ (1) œœ f(a) b f4 (a)(x c a)b x#2! 3x Ñc a) b á bÐnÑ n! (x c a) œ Pn (x) 1, f w (c2) œ 12 36, f ww (c2) œ 50, f www (c2) œ c48, f Ð4 (c2) œ 24, f $ (c2) œ 0 if n 5 Ê x% b x# b 1 c (x c 1)# b 50 (x c 1)$ œ 48 2 b 11(x c 24 b 7(x c 1)# b 2(x c 1) c 1) % % 3! w ww ww 6(x b 2) b 33.(x b 2)# ln (cos x) Ê$ b(x) (x b tan x and c (x) œb 2) b# 25(x b œ # c w8(x b 2)$fb (x œ 2)1 (0) b c 2! f(x) œ c 3! (x b 2) f 4! œ c 2) œ 21 f 36(x c sec x; f(0) 2) 0, f (0) œ 0, w # x fpm@itesm.mx 4 (x) œ 4x$ b 2x,Ê (x) œ œ 0 #and2, f www (x) c 24x, f Ð# Ñ (x) œ 24, f ÐnÑ (x) œ 0 if n 5; f ww L(x) 12x b Q(x) œ œ 2 c x% b 2x$ b x# c 2 Ê f w (x) œ 15x% c 4x$ b 6xnÑ b 2x, f ww (x) œ 60x$ c 12x# b # b 2, 12x ww www Ð 4Ñ Ð 36, f (c2) œ 50, f Ð4(Ñc2) œ c48, f (c2)Ñ œ 24, f (cÐ2) œ 0 if n 5 Ê x% b x b 1 # Ð5 nÑ 80x c#24x34. 12, f œ(x) 24 360x wc% 24, f(cos x)esin x and f ww (x)œ 0 c#sin x)esin cb (cos x)# esin x ; f(0) œ 1, f w (0) œ 1, b œ Ê (x) œ (x) œ 360, f (x) if n 6; f( x 1) œ c7, sin x x b ww c 48 (xf(x)2)www b 4! (x bf2) Ðœ 21 c 36(x b 2) Ð5Ñ 25(xœ (2) c 8(x b 2)$ b (x b 2)% 2) b $e b b 4Ñ ÐnÑ 3! n1 w ...
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