ma1012_hw19_201012

ma1012_hw19_201012 - Section 1: 1  ­9(ODD, 15,...

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Unformatted text preview: Section 1: 1  ­9 (ODD), 15, 23, 35, 43, 45, and 61 0.1 Ê 1.09999 Ÿ f(0.1) Ÿ 1.1. Section 3: 5, 7, 17, 23, and 27 Section 4: 1, 3, 9, 15, 21, 27, and 35 c" 4x 218 w% bc#œc4x1+5+7=2 f€ roblems c x% bc# Ê of c Ê% bf ww"(x) œfc(x) œ fTotal: 1bc#xœ 4xcÊb3 ww (x)for € Derivativesf w$ b Ê aincreasing when when0.1 Ê f w (x) is increasing when 1 c c a1 c x% c b a% b a1 4xx œ p 4x œ 0 0 for 0  x Ÿ 0.1 œ 1 c x b is€ 0 for 0 x Ÿ a a1 $ 0 ww a62. x (x)Chapter 4 Applications c a1 x Ÿ 0.1 ac4x (x) is w cx Ê f (x) increasing 4 Applications of Derivatives c $ b a1 c x b f(0.1) c 2 f(0.1) w and 1 Ÿ w 1.0001 in f w min f w max f w 0d) 1.0001. Now we havewe have0.1c 2 f(0.1) c 2 Ÿ 1.0001 .1 Ê œ 1 andœ 1 and(œŸ xfŸœ 1.0001. Nowœ11Ÿ max fŸœ 1.0001. Now we have 1 Ÿ 0.1 Ÿ 1.0001 max w 0.1 Ê min f 0.1 ven Problems 2 Ÿ c 2 Ÿ Ê 2.1Answers to EŸ22.10001. Ÿ 02.1Ÿ f(0.1) c f(0.1)0.100010.10001 Ê f(0.1) Ÿ 2.10001.Ÿ 0.10001 Ê 2.1 Ÿ f(0.1) Ÿ 2.10001. Ê .1 Ÿ f(0.1) Section 4 x 63. (a) Theorem  c then c Ê f(x) € f(1). Theorem f(x) c f(1)  by the 1, then by the Mean ValueSuppose .1f(x) 1,f(1) f(x)byf(1) Mean Value Suppose x € 1, then 0 Ê f(x) € f(1). Suppose x € 1, then by the  0 the 0.1) Ÿ 1.1. Ÿ 1.1. 99 Ÿ f(0.1) b) and 0 and lim (a), lim f(x) c f(1) Ÿ 0 f w (1) lim f(x) c f(1) 0. rt (a), lim f(x) c f(1) 3. 0 f(1) lim f(x) c f(1) f(x) 0. Since w (1) exists, these twoxone-sided Since w Ÿ x 0. c1 From part (a), lim1 (f(x) cYes.ŸFrom part c 1 x c f(1) c 1f Since and x exists, these two one-sidedf (1) exists, these two one-sided xc x Äc 1 Ä1 xÄ1 xÄ1 xc1 x1 xÄ1 xÄ1 w w w w al and have the value f5. limits wareÊ f w (1) Ÿhave f w (1) w0 (1) Ê f œ 0. 0 and f w (1) 0 Ê f w (1) œ 0. are equal and have the (1) Ê w (1) equaland f 0 and 0 value (1) œ f w (1) (1) Ÿ value f f (1) Ÿ 0 and (1) the Ê f f Ê 0. f(b) c f(a) 7. w w wfw Mean œ f Theorem lue Theorem we 64. From the f(b) c$f(a)Value(c)cwhere c is #have b.a aand fb. (c) whereœqis between a and b. But f (c) œ 2pc b q œ 0 have f(b) c f(a) œ f w (c) where is between a and b ean Value Theorem (a) b f(r) œ b c a b 16r w Ê f w (r) œwe between c Butœ (c) œ points c Ê f0 œ bbbbb, increasing on (c_ß _), never But 2pc b œ ca 15. # we have 3r 9r b 16 Ê no critical f (c) 2pc wb q œ 0 w w q b 16r c no as160 Êone critical is a c Ê f œ bbbbb ion c œÊ fqp(r) œ 9r. b Ádecreasingsince fpoints function.) p Á 0 , increasing on (c_ß _), never ne solution#c .œ(Note: h(Note:since fsolution c œ quadratic function.) since f is a quadratic function.) c #qp p onlyp Á 0is a quadratic #p . (Note: (b) no local extrema trema 4.3 AND THE DERIVATIVE TEST THE (c) no absolute FUNCTIONS AND UNCTIONS AND THE FIRST FIRST DERIVATIVE TEST FIRST DERIVATIVE TEST NIC FUNCTIONS MONOTONIC extrema e extrema (d) w 1. at andat 0 and 1 1) Ê critical points (a) 0 f (x)1œ x(x c 1) Ê critical points at 0 and 1 x(x c 1) Ê critical points b) w on (c_ß ( and ) and Ê increasing on on cc bbb bbb (Ê ifncreasing ±!) c_ß±"ß _), ("ß _), decreasing ) (!ß ") bb ± ccc ± Ê increasing œ bbbonccc (!bbb decreasing on (!ß "(c_ß !) and ("ß _), decreasing on (!ß ") ! " " ! " ( Local maximum maximum0 andœ 0 and minimum at x œ 1atat x œ 0 and a local minimum at x œ 1 at x œ at x a localc)a local minimum x œ 1 m xc1 se x  1, then by the Mean Value Theorem x c 1  0 Ê f(x) € f(1). Suppose x € 1, then by the xc1 f(x) c f(1) f(x) c f(1) 1. Mean Value Therefore f(x) c f(1) 1 for all xf(x) € f(1) œTherefore f(x) 1 for all x since f(1) œ 1. heorem x c 1 f(x) € f(x) Theorem f(x) 1 f(x) Ê since f(1). f(1) xc Value Theorem € 0 ʀ 0 Ê f(1). € f(1). Therefore € 0 1 for all x since1. œ 1. c# $ % s b Ê cww" Ê f ww fa1 b x% a1 b xb% cos x c"# a4x$f ww (x)xœ c sin x% œ w (x) c 1 b a4x$ c x x% c x bxx% cos xbf (x) 61.c(x) œœ acos xx% cos xbb cosÊ c cossin xb %a1 bxb cos xb a4x cos x c x sin xb c# c# % xŸ s b (4 xbc# c x sin œ cx0 ax)b Ÿ for xb (4 cosw (x) x fsin x)  0 for 0 0 Ÿ Ÿ Ÿ 0.1 x f w 0.1 x)  $ 1 bxx% cos cos x(4 cos x Thomas’ 0x0 cos 0 0.1x1th 0.1 cis decreasing when Ÿ when0.1ŸÊ Ÿ (x) is decreasing when 0 Ÿ x Ÿ 0.1 c x sin for  alculus 1 Ê Edition, Chapter 4. x x 0 Ÿ Ÿ f x Ê w (x) is decreasing C f(0.1) c " w w w 1 0.09999 Ÿ f(0.1) c Ÿ f(0.1) c " w 99 and max f œ 1. Now we have 0.9999 Ÿ 0.1 ¸ 0.9999 and max f w Ê 1. in f ¸ 0.9999 0.9999 Ÿ fŸœ 1." Now we have 0.9999f(0.1)0.1 1 Ÿ 0.1 Ê 0.09999 Ÿ f(0.1) c 1 Ÿ 0.1 œ m Now we have and max f(0.1) cÊ Ÿ 1 Ê 0.09999 Ÿ 1 Ÿ 0.1 Ÿ 1 c c" Section 4.3 Monotonic FunctionsSectionand the First Derivative Test the First Derivative Test and the First Derivative Test and Section 4.3 Monotonic Functions 4.3 Monotonic Functions215 215 Mathematics I: Homework 19. Extrema: Part I c# 215 xc1 2. (a) f w points c 1)(x b 2) )(x b 2) Ê critical points (x)c2 (xat c2 and 1 Ê critical points at c2 and 1 (x c 1)(x b 2) Ê criticalat œ and 1 b) f w on (c_ß (c and ("ß and ("ß _), decreasing _ß c ) ccc±± ccc ± Ê increasingœ bbb onccc ±c#) _),Ê increasing on (2ßon 3. 2ß ") ("ß _), decreasing on (c2ß " ) bb bbb bbb (15. increasing ± c#) _ß bbb decreasing on (c c"2(c #) and Ê c# " " c# " 35. local minimum at x œ 1 m at x œ at and a local a localmaximum at x œ 1 maximumc2x œ c2 (c) minimum at x œ 1at x œ c2 and a and Local minimum 16. (a) h(r) œ (r b 7)$ Ê hw (r) œ 3(r b 7)#w Ê a critical point at r œ c7 Ê hw œ bbb ± bbb, increasing on 7)$ Ê hw (r) œ 3(r b 7)#f wÊ œcritical point at r œ ccriticalhpoints at c± bbb, increasing on a 7Ê œ bbb 2 and 1 c( . points points at 1 # # ) (x c 1)# (x bcritical(a) (x) c2 and 1)2(x b 1 Ê (x b 2) Ê 2) 3Ê 43. at (x c c and 2) critical c( w _ß c7) (cœ cccr (bbb ±), never Ê increasing on (c2ß 1) and ("ß _), decreasing on (c_ß c2) c(ß _ bbb decreasing r (c±± _), never decreasing 1) bbb(ßbbb ± Ê increasing on (c2ßon (c2ß("ß _), ("ß _), decreasing on (2) _ß c2) cc bbb (b) f ncreasing ± and 1) and decreasing on (c_ß c c Êi c# (b) no local extrema " c" # " trema (c) No x œ at (c) no local maximum al maximum and a local minimumc2 extrema imum and a local minimum atabsolute x œ c2 and a local minimum at x œ c2 e extrema (45. d) c 1) 4 (a) f w (x) œ (x and 1# (x b 1# # c 1)# (x b 2)# . Ê critical points at c2 and2) Ê critical points at c2 and 1 # ) (x b 2) Ê critical points at c2 (x b) f w on (c_ß (c r 2) r Ê _), ("ß _ decreasing bb bbb (61. increasing ± c2) _ß bbb (c "ß ") r never on (c_ß c2) r (c Ê bbb±± bbb ± Ê increasingœ bbb onbbb (±c#ß ") r (#ß increasing ), never decreasing #ß ") r ("ß _), never decreasing c# " c" # " (c) No local extrema al maextrema Section 4.3 5c (a) f w critical c 1)(x (x c 1)(x b 2)(x . 3) Ê (x) œ (x points b 2)(x c 3) 3Ê critical points at c2, 1 and 3 )(x b 2)(x c 3) Ê critical points at4c2, 1at c2, 1 andof, and 3 216 5. aChapter 4 Applications of Derivatives wChapter 216 ) œ ccc ± Applications± Derivatives ± and ( Ê increasing _ß c c and (2) and cc ±± bbb ± ccc (b) fincreasing onbbbonccc$ß bbb$ß _), decreasing on (2) _ßand ($) _("ßdecreasing on (c_ß c2) and ("ß $) ±Ê bbb Ê increasingß 1) (c2ß(1) _), decreasing on (con (c 2ß 1) c"ß $ß ), $) bbb ccc bbb c# (c2 " and $ Chapter 4 Applications of Derivatives b) w , c" # " $ 7. $ f (x) œ xc"Î$ (x b 2) Ê critical points at c2 and 0 (a) f w (x) œ xc c) 7local minimac2x "Î$ (xat and x1, local minima atat c2c2 and x œ 3 . (c) Local maximum 2 x œ Ê critical points x œ and 0 (a) at x œ at and c œ 3 b 2) œ 3 maximum1, local 1, œx m at x œ at x œ minima w (b) Ê wcritical points at )(2 bbb Ê increasing on (c_ß c2) and (0ß _ decreasing on ( (c2ß 0) ) f w (x) œ xc"Î$ (x b 2) b) f f œ bbb ± ±ccc c)( and 0 Ê increasing on (c_ß c2) and (0ß _),), decreasing onc2ß 0) ( a) œ bbb ccc bbb , b) 7. , !! c# # c (a) f w critical points 7 % increasing1)(x(b $ ) (x w# 1)(x b 1)(x7. critical(x)œ Ê atc8x# cbcon c1 w5) c minimum ), x4x(x b 2)(xand2) Ê c 7)(x b ± 16. œ)( bbb œ (xcc7)(xat 2)(x c 2,andÊ critical points at x0œ 0c1 c œ 0) 7 critical points at c5, w b 5) $ points (c)4xLocal maximum b x œ 7 f (x) œ minimum œ Ê (a)c)f(x) 16x œ 4x(x b at 5,œ c2, local and (0ß _ œ x œ 0 and ( )(xfbœ bbb5) fcccc) ÊwLocalxmaximum116x Êc_ßÊ 2)4x c 16xatatdecreasing onx c2ß„ 2 critical points at x œ 0 and x œ „ 2 5, and c 2) local 8x b 16 Ê # (x) ! c ccc ± bbb ± ccc bbb Ê increasing ( c (c5 c and (7 _ œw cc ± bbb ± ccc(b) fÊ increasing on bbb!±and ±(7ß ±and decreasing on (on (_ß ß c 1)and5)"ßß7) ), decreasing on (c_ß c5) and (c"ß 7) bbb±± bbb ±±ccc ±±xbbb, fÊ iccc(c5c"1)x)5ß c1) _),), decreasing onon_ß c(5))_ß c(c!ß #) (decreasing on (c_ß c2) and (!ß #) Ê ic2, localcon ± (ßon ccc bbb increasing cc on c and ( _), cc & ccc " bbb 17. ancreasingminimum #ß(c œ ( (#ß _ (7ß,_), decreasing c#ß !2)and ( #ß and, c"ß 7) bbb œ ncreasing c at ! and # ( & c# c ) Local maximum at ( œ ) 0 c# c( c w c" ! 8. (c) Localœ xc"Î# (x c 3) œ ccritical points at at and 3 5 # w . (a) f minima xc"Î# (xat x Ê 1, criticalminima 0 œ c b) (x) maximum is maximumc1, œ 168minimawfat x maximum are3)7 „x2œlocalatœ 0, localxminimaand xaœ 7 b œ 0 at x œ „ 2 , local local at and c f a m at x is f(0)xlocal 1,(b) a0, (x) œ minima c5f(0)Ê b œat x points„ 2 and 3 are f „ 2 œ at œ cat x(a) local œ c5x œ x œand œ 167 0 x œ at 0 ximum œf œ ( ccc ± bbb Ê increasing on ($ß _), decreasing on (0ß 3) (b) Êf œ ( ccc ± bbb Ê ( ) f w (x) œ xc"Î# (x c 3) b)no wcritical points at 0 and 3increasing on ($ß _), decreasing on (0ß 3) c) c) absolute $ $œ „ 2 e maximum; absolute (minimum!is 0 atmaximum; absolute minimum is 0 at x œ „ 2 , x ! w ) f œ ( ccc ± bbb Ê increasing on ($ß _), decreasing on (0ß 3) x œ 3 (d) No local maximum and a local minimum at (c) No local maximum and a local minimum at x œ 3 (c) ! $ ) No local maximum and a local minimum at x œ 3 9.9. (a) g(t) œ ct#t# c 3t b 3Ê gwgw (t) œ c2t c 3Ê a a critical point at œ c 3 3 gwgœ bbb ± ± ccc, increasing on (a) g(t) œ c c 3t b 3 Ê (t) œ c2t c 3 Ê critical point at t t œ c # ; w œ bbb ccc, increasing on #; c$Î# c$Î# 3 # w w ) g(t) œ ct c 3t b 3 Ê g (t) œ 3 2t c 3 Ê a critical3point at t œ c # ; g œ bbb ± ccc, increasing on c ˆc_ß c # ‰ ‰ decreasing on ˆc # ß3_‰ ‰ ˆc_ß c 3 , , decreasing on ˆc # ß _ c$Î# # ([email protected] value of ˆ ˆc 3 œ 4 b) localˆmaximum ttœc ˆc_ß c 3 ‰ , decreasing localc 3 ß _‰ value of g g c 3 ‰ ‰ œ2121atat œ c 3 3 (b) on maximum ## ## # # 4 3 (c) absolute ‰ œ 21 at t is 21213 t t œ c ) local maximum value of absolute maximum isc atat œ c # 3 (c) g ˆc 3 maximum œ 4 (d) 21 (d) # 4 4# 3 # x (b) a local maximum is g(4) œ 16#Î$ x œ gw (x) localxminimumc"Î$ œ 5(xœ 2) Ê gcritical pointsccc )( bbb, increasing on (c_ß at Ê 4, a œ 5 #Î$ b 10 x is 0 at x b 0 and wxœ bbb ± at x œ c2 and œ5 x œÈx Ê 0 26. (a) g(x) œ x#Î$ (x b 5) œ x&Î$ b 5x 3 3 3 c) no absolute w 5(x b 2) Ê critical minimum œ at and ! c# 5x#Î$ Ê gw (x) œ 5 x(#Î$ b 10 xc"Î$ œmaximum; absolute points at xis 0c2 x œ 0, 5 3 3 3 bbb x œ 0 Ê g œ Èx ± ccc )( bbb, increasing on (c_ß c2) and (!ß _), decreasing on (c2ß !)$ (b) local maximum is g(c2) œ 3 È4 ¸ 4.762 at x œ c2, a loca (d) ! c# )( bbb, increasing on (c_ß c2) and (!ß _), decreasing on (c2ß !) x c3 c) 3b absolute œ#)c ") ! 2 g( (a) f(x) È x c # Ê f (x) c 2x(x c 2) c(a2) cno(1) is (x c extrema œ critical points at x œ 1, 3 (b) local maximum is3. c2) œ 3 $ œ4x¸ 4.762 at wx œ œ 2, a localcminimum œ g(0) 3)(x0 at xÊ 0 (x (x c $ È4 ¸ 4.762 at x œ cc) a local minimum is g(0) œ 0 at x œ 0 ( 2, no absolute3extrema 2x(x f œ bbb ± (x )( ccc (d) Ê cw2) c ax c 3b (1)cccc 3)(x c ") ±Ê critical points at x (c_ß 1) and ($ß _), decreasing on ("ß #) an bbb, increasing on œ 1, 3 xc w 23. (a) f(x) œ x c # Ê f (x) œ œ (x c #) (x c 2) # " $ (d) w 2) 2x(x c 2) c ax c 3b (1) œ#Î$ c 3)(x± ccc&Î$iscontinuous atwincreasing on (c_ßc"Î$ ± Ê , x œ 1, 3 w 26. (x c 2) g(x) fœœ bbb#)c ")œÊ )( ccc#Î$bbbat (x) œ 53x#Î$ b 10 x 1) and ($ßbx ), Ê critical points #) x œ (#ß $and (a) Ê x (x (xc 5) x d b 5x points g x œ 2 3 b œ 5(x _ decreasing on ("ß at and c2 ), # Ê f (x) œ (x " 3 # critical $ d) is f(1) œ 2 at x3œ 1, a local È (b) a local maximum minimum is f(3) œ 6 at x œ 3 bb ± ccc )( ccc ± bbb, increasingatbbb ± ccc )($ß _),,decreasing on ("ß _ßand (#ß $),, _), decreasing on (c2ß !) 2 xiscontinuous x c_ß 1) c #) c2) and (!ß 23.daœ 0 Ê gw œon (œ(c) no and ( bbb increasing ) # " c# absolute extrema ($ a local maximum is f(1) œ 2 at!x œ 1, a local minimum is f(3) œ 6 at x œ 3 , b) b) (d) œ 3 $ 4 ¸ 4.762 at x œ c2, a local minimum is g(0) œ 0 at x œ 0 ous at x œ 2 È (b) local maximum is g( (c) no absolute extrema c2) ximum is f(1) œ 2 at xc) 1, aabsolute extremais f(3) œ 6 at x œ 3 local minimum (œ (d) no x e extrema c3 23. (a) f(x) œ x c # Ê f w (x) œ 2x(x c 2) c a2) c 3b (1) œ (x c 3)(x)c ") Ê critical points at x œ 1, 3 (d) x (x c (x c # 3 17. (a)(c) no absolute maximum; absolute 4x$ c 16x is 04x(x b$2)(x c 2) Ê critical points at x œ 0 and x œ „ 2 f(x) œ x% c 8x# b 16 (d) f w (x) œ minimum œ at x œ 0, Ê (b) no wlocal maximum, a local minimum is f(c2) œ c6 È25¸ c7.56 at x œ c2 ximum is g(4) œ 16 at x Ê 4, a local minimum is ccc œbbb, increasing on (c#ß !) and (#ß _), decreasing on (c_ß c2) and (!ß #) œ f œ ccc ± bbb ± 0 at x ± 0 and x œ 5 (d) $ È c) no absolute c# œ ! # e maximum; absolute(a) g(x) œis 0Èmaximum; (5 c x)"Î# Ê gw (x) œ 2x(5 cat x œb x# ˆ " ‰ (5 c x)c"Î# (c1) œ 5x(4cx) xœ 22. (minimum x# at5 c x0, 5x# absolute minimum is c6 2 x)"Î# c2 # 2 È 5 cx (b)(d) local maximum is f(0)athematics I: Homework 1f9. Extrema: Pœ „I a x œ 0, local Mxœ 16 at and 5 Ê g minima are a „ 2b œ 0 at x art 2 Ê critical points at absolute minimum is w0œ ccc „bbb ± ccc Ñ , increasing on (0ß 4), decreasing on (c_ß !) œ 0, 4 ±2 (c) no absolute maximum; at x œ & ! % (d) and (%ß &) 26. (a) g(x) œ x#Î$ (x b 5) œ x&Î$ b 5x#Î$ Ê gw (x) œ 5 x#Î$ b 10 3 Ê f w œ bbb ± ccc )( ccc ± bbb, increasing on (c_ß 1) and ($ß _), decreasing on ("ß #) and (#ß $), # " $ ŠÈ7x x ‹ Š a 7x c 27. (a) h(x) œb 2"Î$ Èx#c #‹4b œ x(Î$ c 4x"Î$ Ê hw (x) œ 7 x%Î$ c 4 3 3 27. (a) discontinuous#at x b œ x(Î$ c 4x"Î$4.4 Concavity and Curve Sketching h(x) œ x"Î$ ax c 4 œ 2 Section Ê hw (x) œ 7 x%Î$ c 4 xc#Î$ œ Ê 229 critical points at 3 3 3 Èx (b) a local maximum7x bf(1) È7x2 at‹x œ 1, a local minimum is f(3) x œ at x 2 3 hw œ bbb ± is 2‹ Š œ c # œ 6 0, „œ Ê ŠÈ ccc )( ccc ± bbb, i „2 c 2 È7 4 "Î$ ccc critical points at , increasing on Šc_ß È2 ‹ and Š È7 ß _‹ , decreasing on 4x$ b 6x# hw (x) then(yw%Î$noœ 0,cb712x hw œ bbb ± Ê c 3, œ 7 x œxcabsoluteœ c 6x# È Ê x #Î$ extrema È Ê )( ccc ± bbb ! 7 3 c) 3 2x 3 d) È x c#ÎÈ( #ÎÈ( c) , c#Î ( ! #ÎÈ( ww (d) œ c12(x c 1). The 2) ccc ) ( ccc ±b 12 c, increasing on2 Šc_ß c2 ‹ and Š 2 ß _‹ , decreasing on Š c2 ß !‹ and Š!ß 2 ‹ and y œ c12x bbb 2 È7 È7 È È ! on((!ß #) and falls on (( _ßaÈ7and (#ß _). 2È7 (a) f(x) œ 3x xb 1 Ê f w (x) œ 3x a3x b 1b x (6x) œ 3x ax b 1b Ê ,a critical point at x œ 0 27. Š ) ß !‹ and Š!ß 4. ‹ Chapter 4 Applications ofa3x b 1c 7 a3x b 1b 7 #ÎÈ c 0) b 222 c2 Derivatives 24 È2 È2 c2 24 2 (b) h Š c2È2 ere is a local minimum b) local maximum is h Š È ‹ Ê 243xœ ¸b c x (6x)bbbÈaincreasing minimum !) h Š(È _œ and 7 ‹ œ c3.12¸ 3.12 at x œ È7 , the lo ( and at x œ 2 a local w 7 œ a w b 1 3.12 ± x œ c2x, b 1b local onlocal maximum‹is), c È never decreasing 3x f x bbb at œ 3x, 7 the Ê a critical point at7 x œ 0 7 ¸ 7 (c_ß is r !ß 7 24. (a) f(x) œ 3x b 1 Ê f (x) œ b) , a3x b 1b a3x b 1b The curve is concave up on (c_ß ") and ! (c) no absoluteb 2‹ ŠÈ7x c #‹ ŠÈ7x extrema (c) no absolute extrema (d) (Î$ "Î$ È2 24 È2 3x 7. (a) b c x (6x)w œ"Î$ aax# b 1bbb,xno local extremahw (x) )œ 7 !ß%Î$ ), and c#Î$ œdecreasing a3x c21 h(x) f x bbb cb4b(b) increasing 24 Ê _ß ! r ( x _ c 4 x never b 3x x ± Ê critical points at Ê f _œ 2 Ê a critical point at x c 0 wn on ("ßw3.12 at x œa3x b, 1the local minimum is œ Š È ‹ œ c on (c¸ œ 3.123 1 there œ a3x b h 2 c 4x 3 3 Èx b 1 Ê ¸ (x)). At x œ È7 b is a point of 1b 7 ! (c) no7 absolute 7extrema c 2 bb ± bbb, increasing on œ 0, È2) r (!ß _), bbb ± decreasing x local„extrema (b) no(c_ß ! Ê hw œ and never ccc )( ccc ± bbb, increasing on Šc_ß È2 ‹ and Š È7 ß _‹ , decreasing on 7 7 ! ! È( È( c#Î #Î (c) no absolute extrema trema c2 2 e c 9x c 6x# c x$ , then ywŠ È7 ß 9 ‹ a12xŠ!ß3x# ‹ extrema œ c ! c nd c È7 3)(B b 1) and yww 24. c12 c 6x œ 3x xb 1 b 2).h(x) œ‹3x a3x È2 b¸x3.12 œ x œxectionÊlocal minimum is Curve0‹ œ c 24 È2 ¸ c3.12 œ (a) f(x) œ c6(x Ê f w Š c2 œ 24 b 1 c (6x) at 3x a c21,b the aConcavity andathx œ Sketching Sb 4.4 critical point Š 2 123 (b) local maximum is a7 b 1b 3x a3x b 17 È7 Èb È7 7 ises on (c$ß c") and falls on (c_ß c3) and Ê f w œ bbb ± bbb ( CONCAVITY AND CURVE increasing on 4.4 c) no absolute extrema ,SKETCHING (c_ß !) r (!ß _), and never decreasing c) , d) t x œ c1 there is a local maximum and at! Local local extrema cal minimum. The 1.(b) no maximum up3 on x œ c1, a local minimum#is c3 at x œ 2, and ˆ "#Î$ 3 ‰ is a point of8 inflection. The"Î$ is rising c 1) curve is concave is at # 28. (a) œ #Î$ ax c 4 œ )Î$ 4x# ß c 4Ê kw (x) œ 3 &Î$ c 8 c graph 8(x b 1)(x œ Ê critical points 3 Èx no ). At x c2 c#ß _absoluteœ(#ß _), fallingk(x)c"ß x), concave upb on ˆx" ß _cand concave down on ˆcx " ‰ . 3 x nd concave down on ((c) n (c_ß c1) andextrema ‰ o on ( # # ection 4.4 Concavity and _ß # S Curve Sketching x œ 0, „ 1 Ê kw œ ccc ± bbb )( ccc ± bbb, increasing on (c"ß 123 ("ß _), decreasing !) and Section 4.4 int of inflection. ! c" " 2 2 4.4 CONCAVITY AND at x œ 0, local minima are 0 at x œ „ 2, and Šc È ß 16 ‹ and Š È ß 16 ‹ are points of 2. Local maximum is 4 CURVE SKETCHING and (!ß 1) 39 39 1. (b) (c2ß maximum is k(0) on ( at x œ and (!ß #minima up k „ 1b 2 c3 at inflection. The graph is rising onlocal 0) and (#ß _), falling œ 0c_ß c#)0, local ), concaveare onaŠc_ß œ ‹ and x œ „ 1 1. Local maximum is 3 at x œ c1, a local minimum is c3 at x œ 2, and ˆ " ß c 3 ‰ is a point of inflection. The È3 graph is rising # # 4 (c) no absolute maximum; absolute minimum is c3 at x œ „ 1 2 2 onÈcß _‹,1)nd concave),down ononcc"ß #),2concave up on ˆ " ß _‰ and concave down on ˆc_ß " ‰ . Š ( 3 _ß c a and (#ß _ falling Š ( È3 ß È3 ‹ . # # (d) 2 2 16 2.. Local maximum is 43 at x œ 0, local minima are 0 at x œ „ 2, and points3of 16 ‹ and Š È3Š „ ‹ are $È% ‹. The graph is 3 Local maximum is 4 at x œ 0, local minima are 0 at x œ „ 1, and Šc È ß inflection at ß 9 È3ß points of 9 % 2 irising on (The !) and is"ß _), falling2ß 0) c_ß(#ß") and (!ß "), concave c#)on Šc!ß #), concave up È3ß c_ß concave down nflection. c"ß graph ( rising on (c on ( and c _), falling on (c_ß up and ( _ß cÈ3‹ and Š on Š _‹, È3 ‹ and 3. Š È _Chapter 4 down on Šc È3 È3 Derivatives ‹ nd ‹. on 124 3ŠßcÈ3,ßaÈ3concaveApplicationsß of‹ . 2 2 2 x c 2)$ b 1, then yw œ 3(x c 2)# and È 4 .Local maximum is 7 a of 3 9Local maximum is 327at x œ 0, local minima are 0 at x œ27 1, œ points point of inflection atÈ3! $ % ‹ The graph is 2). The curve never..falls and there &are4noat x œ c1,w a local%minimum is c 7„at xand1and10. inflection at Š „ (!ß ß ).%The. graph is rising on % $ $ 21. When y œ x ("ß _ falling on (œ15x c 20x up 5x (x c 4) and œ (c c and c 5x , then y c ß on ( a. The curve is concave _ßon 1)on (cand#),"ß _), falling on"(), concave andon (!ß _), and concave down cÈc_ß !).ŠÈ3ß _‹, concave down rising ($ c_ß c") (!ß "), concave up on Šc_ß 3‹ and ww down c"ß !) _ß(#) # y œ 20x c 60x œ 20x (x c 3). The curve rises on e up on (#ß _). At x œon ŠcÈ3is a3point 21 È3 2 there È . È È È 1 21 5. (c_ß !)ß and‹(%ß _),b (a)fallsœ c (3 and x# Ê atwx œ œ, 2 c 2x œ 2(1 c x)cÊ 3 a critical1 point3 at x #3 at Ê f w œ bbb ± ccc Local maxima are c 29.and at f(x) œ 2!ß %). 1There is (x) 1 local minima are c 1 x on 2x c 3 b #3 f a local c œ1 3 # 3 3 # at x œ c 3 and . mœ 231 , and points of0, and a local minimum at ), and 4.1 ßTheThe graph is rising on ˆc 1 ß 1 ‰ , falling on ˆc #31 ß c 1 ‰ x aximum at x œ inflection at ˆc 1 ß c 1 ‰ , (!ß ! x œ ˆ # 1 ‰. " # # # 33 3 27 4. Local 1 ßis1concave7 down on1,fß !‰ andminimum local27 up x œ 1and a 1 c 2xof inflection !ß 1!ß. !). The graph is rising on (2) œ # ß 2 a is c 7 at on 1, a urve 23 ‰, concave up œ c 1 a local 0 1Ê ‰ , and concave down on ˆ at 3 c # a local ( ‰ cnd ˆ maximum is at xon ˆc (c_ß 3)ˆand31concavemaximum is point1 ßœ 1 ‰ and ˆ at # minimum is 0 at x œ 2 3 # ((3_ß c1) At x("ß _), falling on absolute inflection. is!ß _),x œ concave down on minimum c ß _). and œ 3 there(b) a point ), concave up on ( 1 at and 1; no absolute (c_ß !). is (c1ß " of maximum 1 6. 9. Local maximum is cÈ3 b c) at x œ 5. 1 , a local minimum is È3 c 43 at 21. 1 , and a point of inflection at (0ß !). The xœ 3 ( 431 1c 3 3 3 1 1 21 5. Local maxima are ˆc231 ßb 1 3 at x œ1c12‰ , and 1 b #ˆc 1x œ ,1concaveminimaˆare1c ,1and concaveœ c 1 on ˆc 1c0‰.3 at graph is rising on c 2 c 3 ‰ and ˆ 3 ß # 3 falling on at3 ß 1 ‰ 3 , local up on 0ß # ‰ 3 c # at x down and 3 2 ß # # 3 3 3 1 1‰ 1‰ % % $" ˆ1 ˆ1ß 1 ˆ 1 1‰ ˆ #1 x œ 23 y 2 œ When and #xandc 5‰ , then yw œ x c !ß b x(4) # x ‰ The graph c (x b 1)$ , then yw2. xc3(x, b œpoints of inflection at1 c # ߈c # ,5(‰ !), and ˆ # #c. 5‰ ˆ # ‰ is rising on c 3 ß 3 , falling on c 3 ß c 3 1) ˆ # # 1 21 1 1‰ 1 21 ‰ 13 ‰ 21 1‰ 7. aLocalxminima$are c1up x œ c # ß ! and 0 ˆ x œ‰!,, and concave down on ˆc 3„c #and 0 at !ß œ .„ #1, and points of local maxima are 1 at x œ ß # and ˆ x # nd ˆ ß 3 , concave at on ˆ „ # and at ß 3 b 1). The curve [email protected] 5‰ , and yww œ 3 ˆ#x c 5‰# ˆ " ‰ ˆ35x c 15‰ rises3 and 5‰ ˆ 5x c œ ˆ # c there #are 1 1 1 # inflection are (c1ß !) and (1ß !). The graph # rising on#ˆc # ß c # ‰ , ˆ!ß 1 ‰ and ˆ 3# ß #1‰ , falling on ˆc21ß c 3# ‰ , is # rema. The curve is concavex up on$(ˆ 5_ßÈ 1) x41 ‰# c 3‰ c ˆ 1 4). The curve is rising 41 1 ˆc ˆ !ca5‰ ˆ 1 is c 3 b cup œ cc , local minimum and 3 c at x œ , (c a 0) and ( inflection at (0ß !). The b 1 ‰ nd œ5 (x 6. Localßmaximum ß 1 ‰ , concaveat5x on (c#1ßac1) and (1ß #1),is Èconcave down onand1ß point of!ß 1). È È È È Section 4.4 Concavity and Curve Sketching 231 #3 3 3 3 # # e down on (c"ß _). At x œ# c1 there#ˆ# a is graphc_ß #) on c 1 ß ß _‰ and ˆfallingfalling on ˆc 1There is a up on ˆ0ß 1 ‰ , and concave down on ˆc 1 ß 0‰. on ( is rising and (10 c 1 ), and 1 ß 1 ‰ , on (#ß 10). 3 ß 1 ‰, concave 2 3 3# 3 # 2 lection. È È no extrema. oncave down ection. pter 4 Applications of Derivatives x c3 x c # , then x c 1) and 2) Mathematics I: Homework 19. Extrema: Part I yw œ 2x(x c 2) c ax c 3b (") (xc2) 4)(x c 2) c)Î& c 4x b 3b# (x c 2) c ax 6 c 25 x (x c 2) . œ 2 (x c 2) . e (crising).on (c_ß ") and ($ß _), and falling on is _ß ! At m. There is is a local maximum at x œ 1 and a (#ß $). There is concave imum at x œ 3. The curve is concave down on o points of up on (#ß _). There are no points and concave 27. ion because x œ 2 is not in the domain. c x4 x,ct'Î& . yw œ 3x a3x b 1b c x (6x) 3x 25 1 hen b a3x b 1b (c_ß !). At b 1b and 1b m. There is b 6xb a3x b is concave"b c 2 a3x b "b(6x)ˆ3x b 3x ‰ a3x b 1b o " b x) x)(points of . The curve is rising on (c_ß _) so b 1b 35. no local extrema. The curve is concave up on ) and (!ß "), and concave down on (c"ß !) and here are points of inflection at x œ c", x œ 0, 1. Î$ and ß !) and x# c 1, kxk 1 kx# c 1k œ , then al maximum œ 1 c x# , kxk  1 e kxk € 2x,curve is1 2, kxk € " and yww œ œ . The 2x, nok  " c#, kxk  " are kx s0. (c"ß !) and ("ß _) and falls on (c_ß c1) on . There is a local maximum at x œ 0 and local t x œ „ 1. The curve is concave up on (c_ß c1) ), and concave down on (c"ß "). There are no inflection $Î& c 1‰ y is not differentiable at x œ „ 1 2 œ 2 ˆxcbecause is 0ß 1) and line at those points). ( no tangent local minimum Ú x# c 2x, x  0 he curve is # kx c 2xk œ Û 2x c x# , 0 Ÿ x Ÿ 2 , then ere are no Ü x# c 2x, x € 2 0. Ú 2, x  0 x c 2, x  0 ww c 2x, 0  x  2 , and y œ Û c2, 0  x  2 . Ü 2, x € 2 x c 2, x € 2 e is rising on (!ß 1) and (#ß _), and falling on and ("ß #). There is a local maximum at x œ 1 and ima at x œ 0 and x œ 2. The curve is concave up !) and (#ß _), and concave down on (!ß #). no points of inflection because y is not able at x œ 0 and x œ 2 (so there is no tangent oints). [email protected] ...
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