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Chapter 3 36 - 186 U CHAPTER3 t’ ‘ 8 73(a Derive gives...

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Unformatted text preview: 186 U CHAPTER3 DIFFERENTIATIONRULES _/ t’ _ ‘ 8 73. (a) Derive gives g’(t) = 1),“ 2) —-—(2 + 1)”) without simplifying. With either Maple or Mathematica. we first get 9’“) :()M— 18 ”—2)“ -. ———.£ll‘ltlll1‘," 1 ‘ 1 .. v' .. ., 3..- (2t + 1)!) (2t + 1)“) L \lmpllllt. ttion comm ind results in the above expression. . . ‘ j 3 I .. _ I (b) Derive gives 1/ : 2(.L" — :1: + l) (2.1' + l)l(l 7.1" +6.1) — in + 3) without simplifying. With either Maple or Mathematica. we first get i I _ . , .1 ,3 , 4 . a _ . , . y — 1()(2.1 +1) (.1. — .1. +1) + 4(2.r + l) (.I" — .r + 1) (3:1’2 — I). If we use Mathematiea‘s Factor or Simplify. or Maples factor. we get the above expression. but Maple‘s simplify gives the polynomial expansion instead. For locating horizontal tangents. the factored form is the most helpful. 4 1/ :r ,1», 1 74. (a) fir) : (m) Mathematiea give f' (.1?) whereas either Maple or 1.) tb) f/(JIC) , 0 ,7 3:14 "0.7598. (c) f’(:1:) : 0 where f has horizontal tangents. f’ has two maxima and one minimum where f has inflection points. 75. (a) If f is exen. then f(:1‘) : f(i.r). Using the Chain Rule to differentiate this equation. we get (I f'(.l') : f'(‘.1') 1— (-1‘) : *f,(*.11). Thus. fl] 71'] : gf'(1‘). so f' is odd. ( .r (b) If f is odd. then f(.1') : ‘f(;.1'). Differentiating this equation. we get f'(.1:) : *f/(il')(#1) : f'(<2:). so f' is even. f(‘[), . -l ' ‘I, 7 .1771 ’72 /,‘ '. 76- (I) : {.fll') lg(-1‘)l } :f (I) M7 )i a (*1) lat-Ill 51 (Mill) 9 flirt) f(11‘)9’(v") f filial!) , filial-1') girl lgitll2 l!/l»I‘)l’ 77. (a) 3.] (sin' 1' cos 711') : n sitfl’l .1: cos .1‘ (:os'nt' 7L sin” .r (’71 sin 111') [Product Rule] (.1: : n Sin’"1 1' (cos 12.14 ('os .r 7* sin n.1- sin .L‘) [factor out 11 sin” 1 .17] : n sin”’1 I (:Os(n.r + 1*) [Addition Formula for cosine] : 71sin"’1 1' cos](n + l).ri [factor out I] (b) 11— (Cos” 1905171) r n (“o-3""l 1‘ (~ sin 1') cos 711‘ + cos’ .r (insin IIJ‘) [Product Rule] (11' . . \ "11,71 _ 1 ~11. cos;”’1 1‘ (cos Hi sin .r 1» sin [LI (051:) [lactor out in cos 1] : 711C0471 .1- sin(n,r + .r) [Addition Formulator sine] : Autos” ’1 .l‘ si11[(n + 1):] [factor out at] ...
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