SECTION. CURVE SKETCHING. CURVE SKETCHING A Click here for answers. S Click here for solutions. 9.. 8 Use the guidelines of this section to sketch the curve. ln ln. 5. 6 8 7. ln tan. e.. 9. ln. e 5. 6. 5. 6. e e 7. 8. 5 7. e 8. ln 9.. 9 9 Produce graphs of f that reveal all the important aspects. of the curve. In particular, ou should use graphs of f and f to. s s estimate the intervals of increase and decrease, etreme values, intervals of concavit, and inflection points.. s 5. s 9 9. f 7 6 5. s 6.. f 8 5 5 8 9 7. s s 8.. f sin, 7 7 s. f sin sin 9. cos sin. cot, Produce graphs of f that show all the important aspects of the curve. Estimate the local maimum and minimum values. cos sin. sin cos and then use calculus to find these values eactl. Use a graph of. sin s cos. sin sin f to estimate the inflection points.. f e 5. e 6. e. f e cos 7. ln cos 8. ln Copright, Cengage Learning. All rights reserved.
SECTION. CURVE SKETCHING. ANSWERS E Click here for eercises. S Click here for solutions.. A. R B. -int. C. None D. None E. Inc. on, ; dec. on,,, F. Loc. min. f =,loc. 7 ma. f = G. CU on, 5,CDon 5,,IP 5, 7 5. A. {, } B. -int. f = C. None D. HA =;VA =, = E. Inc. on,,, ;dec.on,,, F. Loc. ma. f = G. CD on, ; CUon,, 9,. A. R B. -int. 7 C. None D. None E. Inc. on,,, ; dec.on, F. Loc. ma. f = 7,loc.min.f = 7 G. CU on,, CD on,. IP, 5 6. A. {, } B. None C. None D. HA =; VA =, = E. Inc. on, ; dec.on,,,,, F. Loc. min. f = G. CU on,,, ; CDon,. A. R B. -int.; -int., C. None D. None E. Inc. on,,dec.on, F. Loc. ma. f = 7 G. CU on, ; CDon,,,. IP,,, 6 7. A. { ±} B.-int. C.About -ais D. HA =,VA = ± E. Inc. on,,, ; dec.on,,, F. Loc. min. f = G. CU on, ;CDon,,,. A.R B.-int., -int. C.None D.None E. Dec. on R F. None G. CU on,,cdon,. IP, 8. A. { 5} B. -int. C. None D. HA =,VA 5 =5 E. Inc. on, 5, dec. on 5, F. None G. CU on, 5, 5, Copright, Cengage Learning. All rights reserved. =5
SECTION. CURVE SKETCHING 9. A. { } B. -int., -int. C. None D. HA =, VA = E. Inc. on,,, F. None G. CU on,,cdon,. A., 5 5, B. -int. ±5 C. About the -ais D. None E. Inc. on 5,,dec.on, 5 F. None G. CD on, 5, 5, = =_. A. { }, ± B. None C. About the origin D. HA =,VA =, = ± E. Inc. on, ;dec.on,,,, F. Loc. min. f f =,,,,, =, loc. ma.,,, G. CU on,,, ;CDon. A., ] [, B. -int. are ± C. About the origin D. None E. Inc. on,,, F. None G. CU on,, ;CDon,,,,. IP ±, ± 9. A. { } B. -int. C. None D. VA = E. Inc. on,,, ; dec.on, F. Loc. min. f = G. CU on,,, ;CDon,. IP, _ 5. A. R B. -int., -int. C. None D. HA = ± E. Inc. on,, dec.on, F. Loc. ma. f = G. CU on, 7 +, 7, ;CD on 7, + 7 +. IP 7, 7+ 7, 7 7, 7 6 7 + 6 7 Copright, Cengage Learning. All rights reserved.. A. [, B. None C. None D. HA = E. Dec. on, F. None G. CU on, 6. A. R B.-int., 7; -int. C. None D. None E. Inc. on, 8,, ; dec.on 8, F. Loc. ma. f 8 =,loc.min.f = G. CD on,,,
SECTION. CURVE SKETCHING 7. A. [, B. -int. C. None D. None E. Inc. on, F. None G. CD on,. A.R B. -int. C.About the -ais, period D. None E. Inc. on n,n,dec.onn, n +, n an integer F. Loc. ma. f n =,loc.min. f n + =, n an integer G. CU on n +, n +,CDon n, n +.IP n ±, 8. A., B.-int., -int C.About -ais D. VA = ± E. Inc. on,,dec.on, F. Loc. min. f = G. CU on, =_ =. A. R Note: f is periodic with period,so in B G we consider onl [, ] B. -int., 7 ; -int. C. Period D. None E. Inc. on,, 5, ;dec.on, 5 F. Loc. ma. f =, loc. min. f 5 = G. CU on, 7 ;CDon,, 7,.IP,, 7, 9. A. R B. -int. n +, n an integer, -int. C. Period D. None E. Inc. on n +, n + 7,dec.on n, n +, n an integer F. Loc. ma. f n =,loc.min.f n + =, n an integer G. CU on n +, n + 5,CDon n, n +,IP n +,, n an integer ¹ _. A., B.None C.None D.VA =, = E. Inc. on, ;dec.on,,, F. Loc. min. f =+,loc.ma.f = G. CU on,,cdon,.ip,. A. R Note: f is periodic with period,so in B G we consider onl [, ] B. -int., 5 ; -int. C. Period D. None E. Inc. on, 6, 7, ;dec. 6 on, 7 6 6 F. Loc. ma. f 6 =,loc.min. f 7 6 = G. CU on, 5 ;CDon,, 5,.IP,, 5, ¹ _6, ¹ 5¹ Copright, Cengage Learning. All rights reserved., _ 7¹ 6
SECTION. CURVE SKETCHING 5. A. R Note: f is periodic with period,soinb Gwe consider onl [, ] B.-int.,, ; -int. C. Period D. None E. Inc. on,,, ;dec.on, F. Loc. ma. f =, loc. min. f = G. CU on, 6, 5, ;CDon, 5 6 6 6.IP, 5 6, 5, 5 6 5. A. { } B. -int. e C. None D. HA =, VA = E. Inc. on,,, F. None G. CU on,,, ;CDon,.IP, e G. CU on e /,,CDon,e /.IP e /, e / 9. A.,, B. -int. ± 5 C. None D. VA =, = E. Inc. on,,dec.on, F. None G. CD on,,,. A.R B.-int, -int C.About the -ais D. None E. Inc. on,,dec.on, F. Loc. min. f = G. CU on, ; CDon,,,. IP±, ln 6. A.R B.-int, -int C.About the origin D.None E. Inc. on R F. None G. CU on,,cdon,. IP,. A. { n/} Note: f is periodic with period,soin B-G we consider onl << B. -int., 7. A. { n <<n +, n =, ±, ±,...} Note: f is periodic with period, so in B G we consider onl [, ] B.-int., ; -int. C. About the -ais, period D. VA =, E. Inc. on,,dec. on, F. Loc. ma. f = f = G. CD on,,, 8. A., B. -int. C. None D. HA =,VA = E. Inc. on,e,dec.one, F. Loc. ma. f e =/e C. About the -ais, period D. VA =, = ± E. Inc. on,,dec.on, F. None G. CD on,,, ;CUon,,,.IP ±,. A. R B. -int, -int C. None D. HA = E. Inc. on, ; dec.on,,, F. Loc. ma. f = e, loc. min. f = G. CU on,, +, ;CDon, +. IP ±, 6 ± e ± inflection points Copright, Cengage Learning. All rights reserved. _ e@, -Ï +Ï
6 SECTION. CURVE SKETCHING. A., B. -int. C. None D. None E. Inc. on / e,, dec.on, / e F. Loc. min. f / e = /e G. CU on e /,,CDon, e /.IP e /, / e 6. A. { } B. None C. None D. HA =,VA = E. Inc. on,,, ; dec.on, F. Loc. min. f = e G. CU on,,, e@, _. A. { } B.None C.None D.VA = E. Inc. on,,, ; dec.on, F. Loc. min. f = e G. CU on,,cdon, 7. A. { } B. None C. None D. VA = E. Inc. on,,, ; dec.on, F. Loc. min. f = e G. CU on,,,. 5. A.R B.-int, -int. C.About the -ais D. HA = E. Inc. on,,, ; dec.on,,, F. Loc. ma. f ± = /e, loc.min. f = G. CU on, 5+ 7, 5 7, 5 7, CD on 5+ 7, 5 7, 5 7, 5+ 7, ; 5+ 7.IPat = ± 5+ 7, 8. A., B.-int., -int. C. None D. VA = E. Inc. on,,dec.on, F. Loc. min. f = G. CU on, =_ ± 5 7. Copright, Cengage Learning. All rights reserved. _Œ º º Œ
SECTION. CURVE SKETCHING 7 9. 999 Inc. on.,.,.7, ; dec. on,.,.,.7; loc.ma.f. 6.6; loc.min. f.., f.7 6.; CUon,.5,.5, ; CDon.5,.5; IP.5,.,.5, 6.5. Note: Due to periodicit, we consider the function onl on [, ]. Inc. on.,.6,.8,.8,.6,.; dec. on,.,.6,.8,.8,.6,.,; loc.ma. f.6.7, f.8.9, f..9;loc.min. f..9, f.8.9, f.6.7; CUon,.,.,,., ; CDon.,.,,.,.,; IP,,.,.8,.,.8,,,.,.8,.,.8,,. Inc. on,.5,., ; dec. on.5,.; loc. ma. f.5,loc.min.f 6;CUon.,,CDon,.; IP.,. Loc. ma. f = e /9.5, loc. min. f = e /9.7; IP.5,.5,.9,.8. Inc. on 7, 5.,.,., 5., 7 ; dec. on 5.,.,., 5.; loc. ma. f 5.., f..9; loc.min.f..9, f 5..; CUon 7, 6.8,.,.5,,.5,., 6.8; CDon 6.8,.,.5,,.5,., 6.8, 7; IP 6.8,.,.,.,.5,.,,,.5,.,.,., 6.8,.. Loc. ma. f = f =e.7, loc.min. f =/e.7;ip.9,.86, 5.8,.86 Copright, Cengage Learning. All rights reserved.
8 SECTION. CURVE SKETCHING. SOLUTIONS E Click here for eercises. Copright, Cengage Learning. All rights reserved.. = f = +5 A. D = R B. -intercept = f = C. No smmetr D. No asmptote E. f = + = > < <<. f < < or >. Sof is increasing on, and decreasing on, and,. F. The critical numbers occur when f = = =,. The local minimum is f = 7 and the local maimum is f =. G. f = 6 > < 5,sof is CU on, 5 and CD on 5,.IP 5, 7. = f = 6 8 +7 A. D = R B. -intercept = f = 7 C. No smmetr D. No asmptote E. f =6 8=6 + > + > < or >. f < <<. Sof is increasing on, and, and decreasing on,. F. The critical numbers are =,. The local maimum is f = 7 and the local minimum is f = 7. G. = > >,sof is CU on, and CD on,. IP, 5. = f = A. D = R B. -intercept = f =, -intercept = = =, C. No smmetr D. No asmptote E. = = > <,sof is increasing on, and decreasing on,. F. Local maimum is f = 7,nolocal minimum. G. = > <<,so f is CU on, and CD on, and,. IP, and, 6. = f = 9 = 8 + 7 + 6 + 5 + + + + + A. D = R B. -intercept: f = ; -intercept: f = =B part E below, f is decreasing on its domain, so it has onl one -intercept. C. No smmetr D. No asmptote E. f = 9 8 = 9 8 + < for all,sof is decreasing on R. F. No maimum or minimum G. f = 7 7 > <,sof is CU on, andcdon,. IPat, 5. = f = + = + A. D = {, } =,,, B. -intercept: f = ; no -intercept C. No smmetr D. ± + / ± +/ / = = so =is a HA. = and =are VA. E. f + + = + + = + > < ; f < >. So f is increasing on, and,,andf is
SECTION. CURVE SKETCHING 9 Copright, Cengage Learning. All rights reserved. decreasing on, and,. F. f = is a 9 local maimum. G. f + [ + ] + + = [ + ] = + [ + + + ] + = ++ + + + = + + + = 6 + + + The numerator is alwas positive, so the sign of f is determined b the denominator, which is negative onl for <<. Thus, f is CD on, andcuon, and,. NoIP. 6. = f = + A. D = {, } =,,, B. No intercept C. No smmetr D. =,so =is a HA. ± + + = and + + =, =,so =and = are VA. + E. f + = > <<; + f < < or >. So f is increasing on, and decreasing on,,,,and,. F. f = is a local minimum. G. f = + + [ + + + ] 6 + = 6 +8 +9 + Since +8 +9> for all, f > >, so f is CU on, and,,andcd on,. NoIP 7. = f = + = + A. D = { ±} B. No -intercept, -intercept = f = C. f =f,sof is even and the curve is smmetric about the -ais. + / D. ± + =,so = ± / is a HA. + + =, + =, + + =, + =. So = and = are VA. E. f = > >, so f increases on, and,, and decreases on, and,. F. f = is a local minimum. G. = = + > < <<, sof is CU on, and CD on, and,. No IP 8. = f =/ 5 A. D = { 5} =, 5 5, B. -intercept = f =,no-intercept C. No 5 smmetr D. =,so =is a ± 5 HA. 5 =, so =5is a VA. 5 E. f = 8/ 5 > <5 and f < > 5. Sof is increasing on, 5 and decreasing on 5,. F. No maimum or minimum G. f =/ 5 > for 5,sof is CU on, 5 and 5,.
SECTION. CURVE SKETCHING Copright, Cengage Learning. All rights reserved. =5 9. = f = / + A. D = { } =,, B. -intercept is, -intercept = f = C. No smmetr D. ± + / ± +/ =, so =is a HA. + = and =,so = is a VA. + + E. f = + + = 6 + f > sof is increasing on, and,. F. No maimum or minimum G. f = > <,sof is CU on +, and CD on,. NoIP =_ =. = f =/ [ 9 ] A. D = {, ± B. No intercept C. f = f, so the curve is smmetric about the origin. D. ± 9 =, so =is a HA. + 9 =, 9 =, / 9 =, / + 9 =, / + 9 =, and / 9 =,so =and = ± are VA. E. f = 9 9 > < < << and f < > or <,sof is increasing on, and,,, and decreasing on,,and,. F. f,,, = is a local } minimum, f G. f = is a local maimum. = 9 + 9 9 9 = 6 6 +7 9 Since 6 +7> for all, f > << or >,sof is CU on, and, and CD on, and,.. = f = = A. D = { } B. -intercept, no-intercept C. No smmetr D. =,sonoha. = ± + and =, so =is a VA. E. f = + = + > +> >, so f is increasing on, _,. and, and decreasing on F. f = is a local minimum. G. f = = f > > or <,sof is CU on, and, and CD on,. IPis,.. = f = A. D = { and } = { } =[, B. No intercept C. No smmetr D. = + + = + =,so = is a HA.
SECTION. CURVE SKETCHING Copright, Cengage Learning. All rights reserved. E. f = < for all >,since < <,sof is decreasing on,. F. No local maimum or minimum G. f = [ / / for >,sof is CU on,.. = f = 5 A. D = { 5 } =, 5] [5, ] f > B. -intercepts are ±5,no-intercept C. f =f, so the curve is smmetric about the -ais. D. 5 =, no asmptote ± E. f = 5 / = 5 > / if >5,sof is increasing on 5, and decreasing on, 5. F. No local maimum or minimum G. = 5 / 5 / 5 / = +5 5 7/ < so f is CD on, 5 and 5,. No IP. = f = 9 A. D = { 9 } =, ] [, B. -intercepts are ±, no-intercept. C. f = f, so the curve is smmetric about the origin. D. 9=, no asmptote E. f = 9+ 9=, 9 > for D, sof is increasing on, and,. F. No maimum or minimum G. f = 9 + 9 / 9 9 = 7 > and > 9 / or <<,sof is CU on, andcdon, and,,. IP ±, ± 9 5. = f = + + A. D = R B. -intercept, -intercept C. No smmetr D. + + =, + and =, so horizontal asmptotes are + = ±. + E. f + = + + = > <, / + so f is increasing on,, and decreasing on,. F. f = is a local maimum. G. f = + / + / + = + 5/ f = = = ± 9 = ± 7. f is CU on, 7 + and 7, andcdon 7, + 7 +. IP 7, 7+ 7 + 6, 7 7 7, 7 6 7 6. = f = + / A. D = R B. = + / = / / + =if =or 7 -intercepts, -intercept = f = C. No smmetr D. + / = + / =, =, / / + no asmptote E. f =+ / = / + / / > > or < 8,sof increases on, 8,, and
SECTION. CURVE SKETCHING decreases on 8,. F. Local maimum f 8 =, local minimum f = G. f = / < sof is CD on, and,. No IP G. f / / = = + > for all,sof is CU on,. 5/ 7. = f = A. D = { } = { } = { } =[, B. -intercept is. C. No smmetr D. =, no asmptote E. f = / / > for all >,sof is increasing on,. F. No local maimum or minimum. G. f = / / + / / = +6 6 < / since +6 < negative discriminant as a quadratic in. So f is CD on,. =_ = 9. = f =cos sin A. D = R B. = cos =sin = n +, n an integer -intercepts, -intercept = f =. C. Periodic with period D. No asmptote E. f = sin cos = cos = sin =n + or n + 7. f > cos < sin n + <<n + 7,sof is increasing on n +, n + 7 and decreasing on n, n +. F. Local maima f n =, local minima f n + =. G. f = cos +sin> sin >cos n +, n + 5,sof is CU on these intervals and CD on n, n +. IP n +, Copright, Cengage Learning. All rights reserved. 8. = f = / A. D = { < } =, B. -intercept ==-intercept C. f =f,sof is even. The curve is smmetric about the -ais. D. =,so = ± + are VA. E. f = / = / Since > and / >, f > if << and f < if <<,sof is increasing on, and decreasing on,. F. Local minimum f =. = f = +cot, << A. D =,. B. No -intercept C. No smmetr D. +cot =, +cot =,so + =and = are VA. E. f = csc > when csc < sin > <<,so f is increasing on, and decreasing on, and,. F. f =+ is a local minimum, f = is a local maimum. G. f = csc csc cot =csc cot > cot > <<,sof is CU on,,cdon,. IP,
SECTION. CURVE SKETCHING. = f =cos +sin A. D = R B. -intercept = f = C. f =f, sothe curve is smmetric about the -ais. Periodic with period D. No asmptote E. f = sin +sin cos =sin cos > sin < n <<n, so f is increasing on n, n and decreasing on n, n +. F. f n =is a local maimum. f n + = is a local minimum. G. f = cos +cos = cos cos =cos +cos > cos < n +, n +,sof is CU on these intervals and CD on n, n +. IP n ±,. = f =sin +cos A. D = R Note: f is periodic with period, so in B G we consider onl [, ]. B. -intercept = f =, -intercepts occur where sin = cos tan = =, 7. C. f + =f,sof is periodic with period. D. No asmptote E. f =cos sin > when cos >sin << or 5 <<, f < << 5,sof is increasing on, and 5, and decreasing on, 5. F. f = is a local maimum, f 5 = is a local minimum. G. f = sin cos > << 7,so f is CU on, 7 and CD on, and 7,.IP,, 7,.. = f =sin + cos A. D = R Note: f is periodic with period, so in B G we consider onl [, ]. B. -intercept =, -intercepts occur where sin = cos tan = =, 5. C. No smmetr other than periodicit. D. No asmptote E. f =cos sin =when cos = tan = = 6 or 7 6. f > << 6 or 7 6 <<, f < 6 << 7 6.Sof is increasing on, 6 and 7, and decreasing on, 7 6 6 6. F. f 6 =is a local maimum, f 7 6 = is a local minimum. G. f = sin cos =when tan = = or 5. f > << 5,sof is CU on, 5 and CD on, and 5,.IP,, 5,. ¹ _6, ¹ 5¹, _. = f =sin +sin A. D = R Note: f is periodic with period, soinb Gwe consider onl [, ]. B. -intercept =, -intercepts occur where sin +sin = sin = =,,. C. No smmetr other than periodicit D. No asmptote E. f =cos+ sin cos =cos + sin > cos > << or <<,sof is increasing on, and, and decreasing on,. F. f =is a local maimum, f = is a local minimum. G. f = sin +cos sin = sin + sin =+sin sin > sin> sin < < 6 or 5 <. Sof is CU on, 6 6, 5,,andCDon 6, 5 6 6.IP, 5 6 and 5, 5 6 7¹ 6 Copright, Cengage Learning. All rights reserved. ¹ _
SECTION. CURVE SKETCHING 5. = f =e /+ A. D = { } =,, B. No -intercept; -intercept = f = e C. No smmetr D. ± e /+ =since / +,so = is a HA. e /+ =since / +, + e /+ = since / +,so = is a VA. E. f =e /+ / + f > for all ecept,sof is increasing on, and,. F. No maimum or minimum G. f = e /+ + + e /+ + = e /+ + + f > +< <,sof is CU on, and,,andcdon,. f has an IP at, e. 6. = f =e A. D = R B. Both intercepts are. C. f = f, so the curve is smmetric about the origin. D. e =, e =, no asmptote E. f =e + e =e + >,sof is increasing on R. F. No maimum or minimum G. f =e + + e = e + > >,sof is CU on, and CD on,. f has an inflection point at,. about the -ais. f + =f. f has period, so in D G we consider onl <<. D. ln cos = and / ln cos =,so = and = are VA. / + No HA. E. f =/ cos sin = tan > <<,sof is increasing on, and decreasing on,. F. f = is a local maimum. G. f = sec < f is CD on,. No IP. 8. f =ln / A. D =, B. -intercept = ln H / C. No smmetr D. =,so ln =is a horizontal asmptote. Also + = since ln and +,so =is a vertical asmptote. E. f = ln =when ln = = e. f > ln > ln < <<e. f < >e. So f is increasing on,e and decreasing on e,. F. Thus, fe =/e is a local and absolute maimum. G. f = / ln = ln, so f > ln > ln > >e /. f < <<e /. So f is CU on e /, and CD on,e /. Inflection point: e /, e / Copright, Cengage Learning. All rights reserved. 7. = f = ln cos A. D = { cos >} =,, 5 = { n <<n +, n =, ±, ±,...} B. -intercepts occur when ln cos = cos = =n, -intercept = f =. C. f =f, so the curve is smmetric 9. = f =ln A. D = { > } = { < or >} =,, B. -intercepts occur when = = = ± 5. No -intercept C. No smmetr D. ln =,noha. ln =, ln =,so + =and =are VA. E. f = >
SECTION. CURVE SKETCHING 5 when > and f < when <,so f is increasing on, and decreasing on,. F. No maimum or minimum G. f = = + f < for all since + has a negative discriminant. So f is CD on, and,. No IP.. = f =ln + A. D = R B. Both intercepts are. C. f =f, so the curve is smmetric about the -ais. D. ln + =, no asmptote ± E. f = > >, sof is + increasing on, and decreasing on,. F. f = is a local and absolute minimum. G. f = + = + + > <,sof is CU on,,cdon, and,. IP, ln and, ln., and,.ipare ±,.. = f = e A. D = R B. Intercepts are C. No smmetr D. e H H e e e =, so =is a HA. Also e =. E. f =e e = e > when <<,sof is increasing on, and decreasing on, and,. F. f = is a local minimum, f = e is a local maimum. G. f = e e = + e =when += =±. f > < or >+,sof is CU on, and +, and CD on, +.IP ±, 6 ± e ± inflection points _ e@, Copright, Cengage Learning. All rights reserved.. =ln tan A. D = { n/} B. -intercepts n +,no-intercept. C. f =f, so the curve is smmetric about the -ais. Also f + =f,sof is periodic with period, and we consider D-G onl for <<. D. ln tan = and ln tan =, / ln tan =,so =, = ± are VA. / + E. f = tan sec = sec tan tan > tan > <<,sof is increasing on, and decreasing on,. F. No maimum or minimum G. f = sin cos = sin f = 8cos sin < cos > <<,sof is CD on, and, and CU on -Ï +Ï. = f = ln A. D =, B. -intercept when ln = =,no-intercept C. No smmetr D. ln =, ln + + + ln H / / + / =, no asmptote E. f = ln + = ln +> ln > >e /,sof is increasing on / e, and decreasing on, / e. F. f / e = /e is an absolute minimum. G. f =ln +> ln > >e /,sof is CU on e /, and CD on
6 SECTION. CURVE SKETCHING, e /.IPis e /, / e possibilities. Let α = 5+ 7 and β = 5 7. Then f > >αor <β,sof is CU on, α, β, β and α, andcdon α, β and β,α. IPat = ±α, ±β.. = f =e / A. D = { } B. No intercept C. No smmetr D. e/ =, e/ =,noha. e / e / + + / + e/ =, H + e / / / so =is a VA. Also e/ = since e/. E. f =e / + e / = e / > < < or >,sof is increasing on, and, and decreasing on,. F. f = e is a local minimum. G. f =e / / /+e / / = e / / > >, so f is CU on, and CD on,. NoIP _Œ º º Œ 6. = f =e / A. D = { } B. No intercept C. No smmetr e D. H e H e =, e so =is a HA. =,so =is a VA. e =, E. f = e e e = > < or >,sof is increasing on, and, and decreasing on,. F. f = e is a local minimum. G. f = e e 6 = e +6 > for all since +6has positive discriminant, so f is CU on, and,. e@, _ Copright, Cengage Learning. All rights reserved. 5. = f = e A. D = R B. Intercepts are C. f =f, so the graph is smmetric about the -ais. D. ± e = H ± e ± e =, so =is a HA. ± e E. f =e e = e > << or <,sof is increasing on, and, and decreasing on, and,. F. f = is a local minimum, f ± = /e are maima. G. f =e 5 + =where = 5 ± 7 = ± 5 ± 7 all four 7. = f = e / A. D = { } B. No intercept C. No smmetr D. ± e / =, + e / =, e / e / H e / / / / e / / H e / / / e / =,so =is a VA. E. f =e / + e / / = e / +> >,sof is increasing on, and, and
SECTION. CURVE SKETCHING 7 decreasing on,. F. f = e is a local minimum. G. f =e / / ++e / = e / + + / > for all since + +has positive discriminant, so f is CU on, and,. on.,. and.7, and decreasing on,. and.,.7,withalocalmaimumoff. 6.6 and minima of f.. and f.7 6.. Weestimate from the graph of f that f is CU on,.5 and.5, and CD on.5,.5,andthatf has inflection points at about.5,. and.5, 6.5..999 f =8 5 +5 +8 +9 + f = +8 + + 8 + f =6 + 5 +8 + 8 8. = f = ln + A. D = { > } =, B. Intercepts are. C. No smmetr D. [ ln + ] =,so = is a VA. + [ [ ln + ] ln + ] =, ln + H / + since =. E. f = + = > > since + +>. Sof is increasing on, and decreasing on,. F. f = is a local and absolute minimum. G. f =/ + >,sof is CU on,. After finding suitable viewing rectangles, we estimate from the graph of f that f is increasing on,.5 and., and decreasing on.5,.. Maimum: f.5. Minimum: f 6. We estimate from the graph of f that f is CU on., and CD on,., and has an IP at.,..999 f = sin f = sin + cos f =sin + cos sin =_ 999 9. 999 f = 7 + +6 f =6 + f =8 Copright, Cengage Learning. All rights reserved. After finding suitable viewing rectangles b ensuring that we have located all of the -values where either f =or f =weestimatefromthegraphoff that f is increasing We estimate from the graph of f that f is increasing on 7, 5.,.,.,and5., 7 and decreasing on 5.,., and., 5.. Local maima: f 5.., f..9. Local minima: f..9, f 5... From the graph of f, we estimate that f is CU on 7, 6.8,.,.5,,.5,and., 6.8,andCDon 6.8,.,.5,,.5,.,and6.8, 7. f has IP at 6.8,.,.,.,.5,.,,,.5,.,.,. and 6.8,..
8 SECTION. CURVE SKETCHING.999 f =sin + sin f =cos +cos f = sin sin From the graph, it appears that f changessignandthus f has inflection points at.5 and.9. From the graph of f, we see that these -values correspond to inflection points at about.5,.5 and.9,.8..999 Note that f is periodic with period,soweconsideriton the interval [, ].Fromthegraphoff,weestimatethatf is increasing on.,.6,.8,.8,and.6,. and decreasing on,.,.6,.8,.8,.6 and.,. Maima: f.6.7, f.8.9, f..9. Minima: f..9, f.8.9, f.6.7.weestimatefromthe graph of f that f is CD on.,.,,. and., and CU on,.,., and.,. f has IP at,,.,.8,.,.8,,,.,.8,.,.8,and,..999 The function f =e cos is periodic with period,sowe consider it onl on the interval [, ]. We see that it has local maima of about f.7 and f.7,and a local minimum of about f..7. To find the eact values, we calculate f = sin e cos.thisis when sin = =, or since we are onl considering [, ]. Also f > sin < <<. Sof = f =e both maima and f =e cos =/e minimum. To find the inflection points, we calculate and graph f = d d sin ecos = cos e cos sin e cos sin = e cos sin cos Copright, Cengage Learning. All rights reserved. f =e as,andf as. From the graph, it appears that f has a local minimum of about f.58 =.68, and a local maimum of about f.58 =.7. To find the eact values, we calculate f = e,whichis when = = ±.The negative root corresponds to the local maimum f = e / / = e /9, and the positive root corresponds to the local minimum f = e / / = e /9. To estimate the inflection points, we calculate and graph f = d [ ] e d = e + e 6 = e 9 6 +6 + From the graph of f,weseethatf has inflection points at.9 and at 5.8. These-coordinates correspond to inflection points.9,.86 and 5.8,.86.