236 Chapter 4 Applications of Derivatives

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1 26 Chapter Applications of Derivatives Î$ &Î$ Î$ 5 Î$ 0 "Î$ 5( 2) $È 26. (a) g() œ ( 5) œ 5 Ê g () œ œ Ê critical points at œ 2 and œ 0 Ê g œ ± )(, increasing on ( _ß 2) and (!ß _), decreasing on ( 2 ß!)! (b) local maimum is g( 2) œ $È.762 at œ 2, a local minimum is g(0) œ 0 at œ 0 no absolute etrema Š È7 2 Š È7 È$ "Î$ (Î$ "Î$ 7 %Î$ Î$ 27. (a) h() œ a b œ Ê h () œ œ Ê critical points at œ 0, È Ê h œ ± )( ±, increasing on Š _ß and ß _, decreasing on 7 È Š 7 È 7 Î È(! Î È( 2 2 È7 È7 Š ß! and Š!ß 2 2 È$ È$ 2 È7 7(Î' È7 È7 7(Î' (b) local maimum is h Š œ.2 at œ, the local minimum is h Š œ.2 no absolute etrema Î$ )Î$ Î$ &Î$ "Î$ 8( )( ) $È (a) k() œ a b œ Ê k () œ œ Ê critical points at œ 0, Ê k œ ± )( ±, increasing on ( "ß!) and ("ß _), decreasing on ( _ß ) "! " and (!ß ) (b) local maimum is k(0) œ 0 at œ 0, local minima are k a b œ at œ no absolute maimum; absolute minimum is at œ

2 Section. Monotonic Functions and the First Derivative Test " " " 29. (a) f() œ e e Ê f () œ 2e e œ 0 Ê e œ Ê a critical point at œ ln ˆ Ê f œ ±, increasing on ˆ " ln ˆ ", _, decreasing on ˆ " " _, ln ˆ " " ln ˆ (b) a local minimum is at ln ; no local maimum 22/ œ " ˆ " an absolute minimum at ln ˆ ; no absolute maimum 22/ œ " " 0. (a) f() œ È e Ê f () œ eè Ê no critical points Ê È f œ, increasing on 0, 0 ± a _ b 2 (b) A local minimum is at œ 0, no local maimum An absolute minimum is at œ 0, no absolute maimum. (a) f() œ ln Ê f () œ ln Ê a critical point at œ e Ê f œ [ ±, increasing on ae 0 e, _ b, decreasing on a0, e b (b) A local minimum is e at œ e, no local maimum An absolute minimum is e at œ e, no absolute maimum

3 28 Chapter Applications of Derivatives /2 2. (a) f() œ ln Ê f () œ 2 ln œ a 2 ln b Ê a critical point at œ e Ê f œ [ ±, 0 /2 e increasing on ˆ /2 e,, decreasing on ˆ /2 _ 0, e c e 2 /2 (b) A local minimum is at œ e, no local maimum c e 2 /2 An absolute minimum is at œ e, no absolute maimum. (a) f() œ 2 Ê f () œ 2 2 Ê a critical point at œ Ê f œ ± ] and fab œ and fa2b œ 0 2 a local maimum is at œ, a local minimum is 0 at œ 2. (b) There is an absolute maimum of at œ ; no absolute minimum.. (a) f() œ ( ) Ê f () œ 2( ) Ê a critical point at œ Ê f œ ± ] and f( ) œ 0, f(0) œ "! Ê a local maimum is at œ 0, a local minimum is 0 at œ (b) no absolute maimum; absolute minimum is 0 at œ 5. (a) g() œ Ê g () œ 2 œ 2( 2) Ê a critical point at œ 2 Ê g œ [ ± and " g() œ, g(2) œ 0 Ê a local maimum is at œ, a local minimum is g(2) œ 0 at œ 2 (b) no absolute maimum; absolute minimum is 0 at œ 2

4 Section. Monotonic Functions and the First Derivative Test (a) g() œ 6 9 Êg () œ 2 6 œ 2( ) Ê a critical point at œ Ê g œ[ ± and % $ g( ) œ, g( ) œ 0 Ê a local maimum is 0 at œ, a local minimum is at œ (b) absolute maimum is 0 at œ ; no absolute minimum $ 7. (a) f(t) œ 2t t Ê f (t) œ 2 t œ (2 t)(2 t) Ê critical points at t œ 2 Ê f œ [ ± ± $ and f( ) œ 9, f( 2) œ 6, f(2) œ 6 Ê local maima are 9 at t œ and 6 at t œ 2, a local minimum is 6 at t œ 2 (b) absolute maimum is 6 at t œ 2; no absolute minimum $ 8. (a) f(t) œ t t Ê f (t) œ t 6t œ t(t 2) Ê critical points at t œ 0 and t œ 2 Ê f œ ± ± ] and f(0) œ 0, f(2) œ, f() œ 0 Ê a local maimum is 0 at t œ 0 and t œ, a! $ local minimum is at t œ 2 (b) absolute maimum is 0 at t œ 0, ; no absolute minimum $ 9. (a) h() œ 2 Ê h () œ œ ( 2) Ê a critical point at œ 2 Ê h œ [ ± and! h(0) œ 0 Ê no local maimum, a local minimum is 0 at œ 0 (b) no absolute maimum; absolute minimum is 0 at œ 0

5 20 Chapter Applications of Derivatives $ 0. (a) k() œ Ê k () œ 6 œ ( ) Ê a critical point at œ Ê k œ ± ] and k( ) œ 0, k(0) œ Ê a local maimum is at œ 0, no local minimum "! (b) absolute maimum is at œ 0; no absolute minimum. (a) f() œ 2 sin ˆ " Ê f () œ cos ˆ, f () œ 0 Ê cos ˆ " 2 œ Ê a critical point at œ Ê f œ [ ± ] and f(0) œ 0, f ˆ 2 œ È, f(2 ) œ Ê local maima are 0 at œ 0 and! Î$ at œ 2, a local minimum is È 2 at œ (b) The graph of f rises hen f 0, falls hen f 0, and has a local minimum value at the point here f changes from negative to positive. 2. (a) f() œ 2 cos cos Ê f () œ2 sin 2 cos sin œ2(sin )( cos ) Ê critical points at œ, 0, Ê f œ[ ± ] and f( ) œ, f(0) œ, f( ) œ Ê a local maimum is at! œ, a local minimum is at œ 0 (b) The graph of f rises hen f 0, falls hen f 0, and has local etreme values here f œ 0. The function f has a local minimum value at œ 0, here the values of f change from negative to positive.

6 Section. Monotonic Functions and the First Derivative Test 2. (a) f() œ csc 2 cot Ê f () œ 2(csc )( csc )(cot ) 2 a csc b œ 2 acsc b(cot ) Ê a critical point at œ Ê f œ ( ± ) and f ˆ œ 0 Ê no local maimum, a local minimum is 0 at œ! Î% (b) The graph of f rises hen f 0, falls hen f 0, and has a local minimum value at the point here f œ 0 and the values of f change from negative to positive. The graph of f steepens as f () Ä _.. (a) f() œ sec 2 tan Ê f () œ 2(sec )(sec )(tan ) 2 sec œ a2 sec b(tan ) Ê a critical point at œ Ê f œ ( ± ) and f ˆ œ 0 Ê no local maimum, a local minimum is 0 at œ Î Î% Î (b) The graph of f rises hen f 0, falls hen f 0, and has a local minimum value here f œ 0 and the values of f change from negative to positive. 5. h( )) œ cos ˆ ) Ê h ()) œ sin ˆ ) Ê h œ [ ], (!ß $ ) and ( ß ) Ê a local maimum is at ) œ 0,! a local minimum is at ) œ 2 6. h( )) œ 5 sin ˆ ) 5 Ê h ()) œ cos ˆ ) Ê h œ [ ], (!ß 0) and ( ß 5) Ê a local maimum is 5 at ) œ, a local! minimum is 0 at ) œ 0 7. (a) (b) 8. (a) (b)

7 22 Chapter Applications of Derivatives 9. (a) (b) 50. (a) (b) sin 5. (a) f() œln (cos ) Ê f () œ cos œ tan œ0 Ê œ0; f () 0 for Ÿ 0 and f () 0 for 0 Ÿ Ê there is a relative maimum at œ 0 ith f(0) œ ln (cos 0) œ ln œ 0; f ˆ œ ln ˆ cos ˆ " " œ ln Š œ ln 2 and f ˆ œ ln ˆ cos ˆ " œ ln œ ln 2. Therefore, the absolute minimum occurs at È 2 œ ith f ˆ œ ln 2 and the absolute maimum occurs at œ 0 ith f(0) œ 0. sin (ln ) " (b) f() œ cos (ln ) Ê f () œ œ 0 Ê œ ; f () 0 for Ÿ and f () 0 for Ÿ 2 Ê there is a relative maimum at œ ith f() œ cos (ln ) œ cos 0 œ ; f ˆ " œ cos ˆ ln ˆ " œ cos ( ln 2) œ cos (ln 2) and f(2) œ cos (ln 2). Therefore, the absolute minimum occurs at œ " and œ 2 ith f ˆ " œ f(2) œ cos (ln 2), and the absolute maimum occurs at œ ith f() œ. " 52. (a) f() œ ln Ê f () œ ; if, then f () 0 hich means that f() is increasing (b) f() œ ln œ Ê f() œ ln 0, if by part (a) Ê ln if 5. f() œ e 2 Ê f () œ e 2; f () œ 0 Ê e œ 2 Ê œ ln 2; f(0) œ, the absolute maimum; f(ln 2) œ 2 2 ln , the absolute minimum; f() œ e , a relative or local maimum since f () e is alays positive. œ 5. The function f() œ 2e has a maimum henever sin œ and a minimum henever sin œ. Therefore the maimums occur at œ 2k(2) and the minimums occur at œ 2k(2), here k is any integer. The maimum is 2e and the minimum is sin ÐÎ2Ñ e " " " " " " "Î " " Èe Èe 55. f() œ ln Ê f () œ 2 ln Š a b œ 2 ln œ (2 ln ); f () œ 0 Ê œ 0 or ln œ. Since œ0 is not in the domain of f, œe œ. Also, f () 0 for 0 and f () 0 for " " " " "Î " ". Therefore, f Š œ ln Èe œ ln e œ ln e œ is the absolute maimum value of f assumed at œ ". Èe Èe Èe e e e e 56. f() œ a b c œ a ˆ b b b b c œ a c œ a ˆ b Š b ac, a parabola hose a a a a 2a a b verte is at œ. Thus hen a 0, f is increasing on ˆ bß _ and decreasing on ˆ b 2a 2a _ß a ; hen a 0, f is increasing on ˆ b and decreasing on ˆ b _ß ß _. Also note that f () œ 2a b œ 2a ˆ b Ê for a a a a 0, f œ ; for a 0, f œ ±. bî2a bî2a

8 $ Section. Concavity and Curve Sketching f() œ 2 Ê f () œ œ ( )( ) Ê f œ ± ± Ê rising for œ c œ since " " f () 0 for œ c œ (a) Let fab œ e Ê f ab œ e Ê a critical point at œ 0 Ê f œ ±, so f is increasing on 0 a0, _ b; since fa0b œ 0 it follos that fab œ e 0 for 0 Ê e for 0. " 2 (b) Let fab œ e Ê f ab œ e 0 for 0 by part (a), so f is increasing on a0, _ b; since " 2 " 2 fa0b œ 0 it follos that fab œ e 0 for 0 Ê e for Let " Á be to numbers in the domain of an increasing function f. Then, either " or " hich implies f( ") f( ) or f( ") f( ), since f() is increasing. In either case, f( ) Á f( ) and f is one-to-one. Similar arguments hold if f is decreasing. " " 5 " 5 df " df c" " " 6 " " 6 d d ˆ 60. f() is increasing since Ê ; œ Ê œ œ $ $ $ " "Î$ " " "Î$ " " c" df df " d œ 8 Ê d œ " " Î$ 8 œ Î$ œ f() is increasing since Ê ; y œ 27 Ê œ y Ê f () œ ; "Î$ $ $ $ " "Î$ " " "Î$ " " c" df df " d œ 2 Ê d œ " " Î$ 2 œ œ 6 ( ) "Î$ 6( " )Î$ 62. f() is decreasing since Ê 8 8 ; y œ 8 Ê œ ( y) Ê f () œ ( ) ; 2 Ð c Ñ $ $ $ "Î$ " "Î$ " " 6. f() is decreasing since Ê ( ) ( ) ; y œ ( ) Ê œ y Ê f () œ ; c" df df " " " Î$ d œ ( ) Ê d œ ( ) ¹ œ œ Î$ c "Î$ 6. f() is increasing since Ê ; y œ Ê œ y Ê f () œ ; " &Î$ &Î$ " df 5 Î$ df " d d 5 Î$ 5Î& 5 $Î& c" œ Ê œ ¹ œ œ. CONCAVITY AND CURVE SKETCHING Î& &Î$ $Î& " $Î& $ ". y œ 2 Ê y œ 2 œ ( 2)( ) Ê y œ 2 œ 2 ˆ ". The graph is rising on ( _ß ) and (ß _), falling on ( "ß ), concave up on ˆ " ß _ and concave don on ˆ " _ß. Consequently, " a local maimum is at œ, a local minimum is at œ2, and ˆ ß is a point of inflection. % $ 2. y œ 2 Ê y œ œ a b œ ( 2)( 2) Ê y œ œ Š È 2 Š È 2. The graph is rising on ( 2ß 0) and (ß _), falling on ( _ß ) and (!ß ), concave up on Š _ß and Š ß _ 2 2 È È 2 2 È È and concave don on Š ß. Consequently, a local maimum is at œ 0, local minima are 0 at È 9 È 9 œ 2, and Š ß and Š ß are points of inflection. Î$ "Î$ "Î$ 2. y œ a b Ê y œ ˆ ˆ a b (2) œ a b, y œ ) ( ± )( "! " Ê the graph is rising on ( "ß! ) and ("ß_), falling on ( _ß " ) and (!ß") Ê a local maimum is at œ0, local "Î$ %Î$ " minima are 0 at œ ; y œ a b () ˆ a b (2) œ, $ % Éa b

668 Chapter 11 Parametric Equatins and Polar Coordinates

668 Chapter 11 Parametric Equatins and Polar Coordinates 668 Chapter Parametric Equatins and Polar Coordinates 5. sin ( sin Á r and sin ( sin Á r Ê not symmetric about the x-axis; sin ( sin r Ê symmetric about the y-axis; therefore not symmetric about the origin