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hw-6-sec-12.11-solns

# hw-6-sec-12.11-solns - 1024 D CHAPTER 12 INFINITE SEQUENCES...

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Unformatted text preview: 1024 D CHAPTER 12 INFINITE SEQUENCES AND SERIES (n) 7 ‘ mm): i f (2)<m~2>" n=0 TL! 13 ‘5‘” 3 (")71' n 113(33): 2 U712 (“U—g) —11 n: n. 2 “(00” ’ Si“ 9”) I 2 f —-2e" cos :1: 26" (cos x + sin 9:) 3 (n) T3(a:) = 20 'f 711(0) :12" = a: — m2 + §x3 1026 El CHAPTER 12 INFINITE SEQUENCES AND SERIES 11. You may be able to simply ﬁnd the Taylor polynomials for 14 f (x) = cot :6 using your CAS. We will list the values of f (7‘) (7r / 4) forn: 0ton= 5. 5 (7') 7r 4 n 11(23):sz ('/)(\$_%) ”:0 n. _2 2 7r 3 1r 4 1r 5 =1'2(x—z)+2(\$—%) —§(\$‘z) + 130(95—2) —%(\$‘r) For n = 2 to n = 5, Tn(m) is the polynomial consisting of all the terms up to and including the (a: -— a" term. 12. You may be able to simply ﬁnd the Taylor polynomials for f (z) = \3/1 + 2:2 using your CAS. We will list the values of f (")(0) 15 forn: Oton =5. 5 (n) T5(\$) = 2 f 150) :13" = 1 + §x2 — \$254 n=0 77“ For n = 2 to n = 5, Tn (z) is the polynomial consisting of all the terms up to and including the 3:" term. Note thath = T3 and T4 = T5. 1 1 32 (a) ﬂea) = ﬁ~T2(z)=2+Z(m-4)— /— (m4)? ’ =2+§(m—4)~;(Ix—4)2 3. Q . M 3 III M (b) IR2(:I:)I_ <— !,I:c—4I where|f (z)I <M. Nowﬂ<x<4.2 => ,' (”5/ la: — 4I__ < 0. 2 => Ix — 4I3 < 0.008 Since f’”(a;) lS decreasing III .3 9.; oniﬁ-‘Hlflv ecantakeM: _256 6,50 M“ :I"?/3a8)\ 1| 3 < ——éO=968-)=——'— = ; 5625. - who» _ 37 6256 ”501°: 0&0“. m 2 2 «or (0) 0.00002 X y From the graph of |R2 (zII = If — T2 (1:)I, it seems that the error is less than 1.52 X 10’5 on [4, 4.2]. W Z % (3. . 3 acts 3220 : I 2 a 40 42 I IZzOOIé é, 87C\00\$v @IO 3 I IpszI x: 0. 000017763 ’/’/ SECTION 12.11 APPLICATIONS OF TAYLOR POLYNOMIALS U 1027 (a) f(2) = 90-2 % T2(x) = 1—2(:z:—1)+§—!(x—1)2 = 1—2(a:—1)+3(x—1)2 (b) |R2(a:)| <— M !|x — 1|3, where |f”’(:r )| g M. Now 0.9 S x S 1.1 => [m — 1| 3 0.1 2 [z — 1|3 3 0.001. Since f”’(a:) is decreasing on [0.9,“1.1], We can take M = |f’"(0.9)| = is, so (09) 24/(0.9)5 0.004 < —— . = z . . |R2(x)| _ 6 (0 001) 059049 000677404 From the graph oflR2(:c)| 2 [tr—2 —— T2 (a3)[, it seems that the error is less than 0.0046 on [0.97 1.1]. 2/9 (a)f(\$)=\$2/3~T3(\$)=1+§(\$—1) 2!(\$_1)2+8—'1/327 —(\$ — 1)3 :1+§(m—1)—5(m—1)2+ﬁ(x3—1)3 M 4 (4) (b) 1R3(z)| S —4—' |ac — 1| ,where [ f (x)[ S M. Now 0.8 g a; g 1.2 => [1‘ — 1| 3 0.2 => [cc — 1|4 _<_ 0.0016. Since [f(4)(r)[ is decreasing on [0.8, 1.2],we can take M = [f(4)(0.8)[ = %(0.8)_10/3, so 8—6(0' 8)— 10/3 1&0»: M (0.0016) m 0.000 096 97. From the graph of |R3(a:)| = [932/3 — T3(x)[, it seems that the error is less than 0.000 053 3 on [0.8, 1.2]. 1.2 (a) f(:r) 2 sins: z T4(x) =é+§0—%%&0-£Y-£0—%f+i0-ﬁ4 (b) um )I <— M —%3z—%:% 4 0—03% 4 h—lfsefsme [f(5)(z)[ is decreasing on [0, g], we can take M = [f(5)(0)[ = cosO = 1, 1 7r 5 so [R4(m)l 3 5(5) ~ 0.000328. 1020 D CHAPTER 12 INFINITE SEQUENCES AND SERIES £60. i 2 . I / (c) 0'00“ From the graph of |R4(a:)| = |sinx — T4 (m)|, it seems that the 19. error is less than 0. 000 297 on [0, ’3']. y = |R4(X)| al: 17. (a) f(a:) = secz z T2(a:) = 1 + §m2 secx seca: tans: secx (2 sec2 :1: —— 1) secx tan1:(6sec2 ac — 1) (b) |R2(z)| g % |mI3,where l f(3)(a:)1 g M. Now —0.2 g m g 0.2 :» |z| g 0.2 => |x|3 5 (0.2)3. 20. f (3) (a3) is an odd function and it is increasing on [0, 0.2] since sec :1: and tanx are increasing on [0, 0.2], (a) (a) . . ’ f‘3)(0-2) 3 so ' f (23' g f (0.2) m 1.085158 892. Thus, |R2(x)| g T (0.2) m 0.001447. (c) 0.0004 From the graph of|R2(a:)| = |secx — T2(.1:)|, it seems that the error is less than 0.000 339 on [—0.2, 0.2]. —0.2 0.2 0 (a) f(w) =1n<1 + 2m) w T30) 'n. f(n)(m) f(n)(1 i 16 27 I - =ln3+§(m—1) 4—/——g(m—1)2+ -/—, (z—1)3 ln(1 + 2x) 3 2/(1 + 21‘) (b) |R3(m)| g ‘24 la: — 0“, where lf(4)(:c)l g M. Now 0.5 g a: g 1.5 =>, —4 1 + 2:1: 2 ' - , /( ) 4531—1305 => |a:—1|§0.5 => Iz—1|4g+6,and t 21 16/0 + 2:12)3 6 1 1 : —96/(1 + 2x)4 letting 00 = 0.5 gives M = 6, so |R3(a:)| S a ' E = a = 0.015 625. (c) 0.005 From the graph of 112;; (03)] = |ln(1 + 29:) —— T3 (w)|, it seems that the I error is less than 0.005 on [0.5, 1.5]. 0.5 1.5 1030 D CHAPTER 12 INFINITE SEQUENCES AND SERIES (c) 0"” , 2 From the graph of |R4(a:)| = Ia: sin 1v — T4(x)I, it seems that the .' error is less than 0.0082 on [~—1, 1]. ' 1 0 1 22. (a) f(:c)=sinh2a:%T5(a2)=2x+ 8,333+ gz5=2m+4ar3+ﬁz5 ‘ f‘”)<°> ‘ sinh2z (b) |R5(m)| <— 6,6,Ia:I where I f(6)(x)I < M. Formin[— 1, 1], we have 2' 2 cosh 2w IxI S 1. Since f(6) (.71) is an increasing odd function on [—1, 1], we see 4 sinh 2a: 8cosh 236 that I f(6)(x)I g f<6>(1) = 64 sinh2 = 32(e2 — 3—2) m 232.119, 16sinh 2m so we can take M— _ 232.12 and get |R5(x)| < L733? 16~ ~ 0 3224. g 1 32 cosh 2m 64 sinh 2x =le(x)| From the graph of IR5(2:)| = |sinh 2a: — T5(\$)I, it seems that the 1 L error is less than 0.027 on [—1, 1]. ' . . 2! ,r M 7, 4 . I @From Exercise 5, cosm— — —(z — 3) + 6(116 —— —)3 + R3(:I:), where IR3(\$)|_ <— —'—4 Ia: — 3| w1th If(4)( m)I-— — |cosa:I < M— —- 1. Nowzv— — 80°: (90° — 10°): (— — ﬁ) = 4?" radians, so the error is IR3(9 (4—7' )I <§--14(1—8)4 m 0.000 039, which means our estimate would not be accurate to ﬁve decimal places. However, T3 = T4, so we can use IR4 (‘%’)I g 720 (ﬁ)5 ~ 0.000001. Therefore, to ﬁve decimal places, cos80° z — (1%) + %(—{—8)3 z 0.17365. 24. From Exercise 16, sinzz: = % + 3901: — g) — ﬂat: — 92 — 123 (an — %)3 + 4—18(z — —)4 + R4(x), where 29 IR4(x)I< _ 1;, Iz — %I5 with If(5)(a:)I = Icost S M = 1. Nowa: = 38° = (30° + 8°): (% + 2—75') radians, so the error is IR4 (”3—") I < + (2" )5 m 0.000 000 44, which means our estimate will be accurate to ﬁve decimal places. 180 45 Therefore, to ﬁve decimal places, sin 38°— — — + AC (3%) — l 4(i—§)2 — 31/75 (i—gf’ + 718 01—94 ~ 0.61566. ea: 25. All derivatives of ex are eI ,so IRn(a:)I< _ (11+ 1)! 1 0.1 I 0.1 S e 0.1 "+1 < 0.00001, and by trial and error we ﬁnd that n = 3 satisﬁes this inequality since (n + 1)! R3(0.1) < 0.0000046. Thus, by adding the four terms of the Maclaurin series for ex corresponding to n = 0, 1, 2, and 3, |mI”+1, where O < z < 0.1. Letting a: = 0.1, we can estimate «20'1 to within 0.00001. (In fact, this sum is 1.10516 and «30‘1 z 1.10517.) ...
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hw-6-sec-12.11-solns - 1024 D CHAPTER 12 INFINITE SEQUENCES...

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