3.4. Click here for solutions. Click here for answers. CURVE SKETCHING. y cos x sin x. x 1 x 2. x 2 x 3 4 y 1 x 2. x 5 2
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1 SECTION. CURVE SKETCHING. CURVE SKETCHING A Click here for answers. S Click here for solutions. 9. Use the guidelines of this section to sketch the curve. cos sin cot, cos sin. sin cos sin s cos. sin sin Produce graphs of f that reveal all the important aspects 5 of the curve. In particular, ou should use graphs of f and f to 9. estimate the intervals of increase and decrease, etreme values, 0. intervals of concavit, and inflection points. 9.. s s. s 5. s s 7. s s 8. s f sin,. f 7 6 f f sin sin 7 7 Copright 0, Cengage Learning. All rights reserved.
2 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 = 0 G. CU on, 5,CDon 5,,IP 5, 7 5. A. {, } B. -int. f 0 = C. None D. HA =0;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. { 0, } B. None C. None D. HA =0; VA =0, = E. Inc. on, 0; dec.on,,,, 0, F. Loc. min. f = G. CU on, 0, 0, ; CDon,. A. R B. -int.0; -int. 0, C. None D. None E. Inc. on,,dec.on, F. Loc. ma. f = 7 G. CU on 0, ; CDon, 0,,. IP0, 0,, 6 7. A. { ±} B.-int. C.About -ais D. HA =,VA = ± E. Inc. on 0,,, ; dec.on,,, 0 F. Loc. min. f 0 = G. CU on, ;CDon,,,. A.R B.-int., -int. C.None D.None E. Dec. on R F. None G. CU on, 0,CDon0,. IP0, 8. A. { 5} B. -int. C. None D. HA =0,VA 5 =5 E. Inc. on, 5, dec. on 5, F. None G. CU on, 5, 5, Copright 0, Cengage Learning. All rights reserved. 0 =5
3 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, = =_ 0 0. A. { } 0, ± B. None C. About the origin D. HA =0,VA =0, = ± E. Inc. on 0, ;dec.on,,,, F. Loc. min. f f =,, 0,,, =, loc. ma.,, 0, G. CU on, 0,, ;CDon. A., ] [, B. -int. are ± C. About the origin D. None E. Inc. on,,, F. None G. CU on,, ;CDon,,,,. IP ±, ± 9 0. A. { 0} B. -int. C. None D. VA =0 E. Inc. on, 0, 0, ; dec.on, F. Loc. min. f = G. CU on, 0,, ;CDon0,. IP, 0 _ 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, IP 7, 7+ 7, 7 7, Copright 0, Cengage Learning. All rights reserved.. A. [, B. None C. None D. HA =0 E. Dec. on, F. None G. CU on, 6. A. R B.-int. 0, 7; -int. 0 C. None D. None E. Inc. on, 8, 0, ; dec.on 8, 0 F. Loc. ma. f 8 =,loc.min.f 0 = 0 G. CD on, 0, 0,
4 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 ±, A., B.-int.0, -int 0 C.About -ais D. VA = ± E. Inc. on 0,,dec.on, 0 F. Loc. min. f 0 = 0 G. CU on, =_ 0 =. A. R Note: f is periodic with period,so in B G we consider onl [0, ] B. -int., 7 ; -int. C. Period D. None E. Inc. on 0,, 5, ;dec.on, 5 F. Loc. ma. f =, loc. min. f 5 = G. CU on, 7 ;CDon 0,, 7,.IP, 0, 7, 0 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 +, 0, n an integer 0 ¹ _ 0. A. 0, B.None C.None D.VA =0, = E. Inc. on, ;dec.on 0,,, F. Loc. min. f =+,loc.ma.f = G. CU on 0,,CDon,.IP,. A. R Note: f is periodic with period,so in B G we consider onl [0, ] B. -int., 5 ; -int. C. Period D. None E. Inc. on 0, 6, 7, ;dec. 6 on, F. Loc. ma. f 6 =,loc.min. f 7 6 = G. CU on, 5 ;CDon 0,, 5,.IP, 0, 5, 0 ¹ _6, 0 ¹ 5¹ Copright 0, Cengage Learning. All rights reserved., _ 7¹ 6
5 SECTION. CURVE SKETCHING 5. A. R Note: f is periodic with period,soinb Gwe consider onl [0, ] B.-int. 0,, ; -int.0 C. Period D. None E. Inc. on 0,,, ;dec.on, F. Loc. ma. f =, loc. min. f = G. CU on 0, 6, 5, ;CDon, IP, 5 6, 5, 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,.0,.5, 0,.5,.0, 6.8; CDon 6.8,.0,.5, 0,.5,.0, 6.8, 7; IP 6.8,.,.0,.0,.5,., 0, 0,.5,.,.0,.0, 6.8,. 5. Inc. on., 0., 0.7, ; dec. on,., 0., 0.7; loc.ma.f ; loc.min. f..0, f ; CUon, 0.5, 0.5, ; CDon 0.5, 0.5; IP 0.5,., 0.5, Inc. on,.5,.0, ; dec. on.5,.0; loc. ma. f.5,loc.min.f 6;CUon.,,CDon,.; IP., 8. Note: Due to periodicit, we consider the function onl on [, ]. Inc. on.,.6, 0.8, 0.8,.6,.; dec. on,.,.6, 0.8, 0.8,.6,.,; loc.ma. f.6 0.7, f , f. 0.9;loc.min. f. 0.9, f , f.6 0.7; CUon,.0,., 0,., ; CDon.0,., 0,.,.0,; IP, 0,.0, 0.8,., 0.8, 0, 0,., 0.8,.0, 0.8,, 0 Copright 0, Cengage Learning. All rights reserved.
6 6 SECTION. CURVE SKETCHING. SOLUTIONS E Click here for eercises. Copright 0, Cengage Learning. All rights reserved.. = f = +5 A. D = R B. -intercept = f 0 = C. No smmetr D. No asmptote E. f = +0 = > 0 < 0 <<. f < 0 < or >. Sof is increasing on, and decreasing on, and,. F. The critical numbers occur when f = = 0 =,. The local minimum is f = 7 and the local maimum is f = 0. G. f =0 6 >0 < 5,sof is CU on, 5 and CD on 5,.IP 5, 7. = f = A. D = R B. -intercept = f 0 = 7 C. No smmetr D. No asmptote E. f =6 8=6 + > 0 + > 0 < or >. f < 0 <<. 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. = > 0 >,sof is CU on, and CD on,. IP, 5. = f = A. D = R B. -intercept = f 0 = 0, -intercept =0 =0 =0, C. No smmetr D. No asmptote E. = = > 0 <,sof is increasing on, and decreasing on,. F. Local maimum is f = 7,nolocal minimum. G. = > 0 0 <<,so f is CU on 0, and CD on, 0 and,. IP0, 0 and, 6. = f = 9 = A. D = R B. -intercept: f 0 = ; -intercept: f =0 =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 = < 0 for all,sof is decreasing on R. F. No maimum or minimum G. f = 7 7 > 0 <0,sof is CU on, 0 andcdon0,. IPat0, 5. = f = + = + A. D = {, } =,,, B. -intercept: f 0 = ; no -intercept C. No smmetr D. ± + = / ± +/ / = 0 =0 so =0is a HA. = and =are VA. E. f = + + = + > 0 < ; f < 0 >. So f is increasing on, and,,andf is
7 SECTION. CURVE SKETCHING 7 Copright 0, Cengage Learning. All rights reserved. decreasing on, and,. F. f = is a 9 local maimum. G. f + [ + ] + + = [ + ] = + [ ] + = = = 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 = { 0, } =,, 0 0, B. No intercept C. No smmetr D. =0,so =0is a HA. ± = and + + =, =,so =0and = are VA. + E. f + = > 0 <<0; + f < 0 < or >0. So f is increasing on, 0 and decreasing on,,,,and 0,. F. f = is a local minimum. G. f = + + [ ] 6 + = Since +8 +9> 0 for all, f > 0 > 0, so f is CU on, 0 and 0,,andCD on,. NoIP 7. = f = + = + A. D = { ±} B. No -intercept, -intercept = f 0 = 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 = > 0 >0, so f increases on 0, and,, and decreases on, and, 0. F. f 0 = is a local minimum. G. = = + > 0 < <<, sof is CU on, and CD on, and,. No IP 8. = f =/ 5 A. D = { 5} =, 5 5, B. -intercept = f 0 =,no-intercept C. No 5 smmetr D. =0,so =0is a ± 5 HA. 5 =, so =5is a VA. 5 E. f = 8/ 5 > 0 <5 and f < 0 > 5. Sof is increasing on, 5 and decreasing on 5,. F. No maimum or minimum G. f =/ 5 > 0 for 5,sof is CU on, 5 and 5,.
8 8 SECTION. CURVE SKETCHING Copright 0, Cengage Learning. All rights reserved. 0 =5 9. = f = / + A. D = { } =,, B. -intercept is, -intercept = f 0 = C. No smmetr D. ± + = / ± +/ =, so =is a HA. + = and =,so = is a VA. + + E. f = + + = 6 + f > 0 sof is increasing on, and,. F. No maimum or minimum G. f = > 0 <,sof is CU on +, and CD on,. NoIP =_ 0 = 0. = f =/ [ 9 ] A. D = { 0, ± B. No intercept C. f = f, so the curve is smmetric about the origin. D. ± 9 =0, so =0is a HA =, 0 9 =, / 9 =, / + 9 =, / + 9 =, and / 9 =,so =0and = ± are VA. E. f = 9 9 > 0 < < << and f < 0 > or <,sof is increasing on, 0 and 0,,, and decreasing on,,and,. F. f,,, = is a local } minimum, f G. f = is a local maimum. = = Since 6 +7> 0 for all, f > 0 <<0 or >,sof is CU on, 0 and, and CD on, and 0,. 0. = f = = A. D = { 0} B. -intercept, no-intercept C. No smmetr D. =,sonoha. = ± 0 + and =, so =0is a VA. 0 E. f = + = + > 0 +> 0 > 0, so f is increasing on, 0 _,. and 0, and decreasing on F. f = is a local minimum. G. f = = f > 0 > or <0,sof is CU on, 0 and, and CD on 0,. IPis, 0.. = f = A. D = { 0 and } = { } =[, B. No intercept C. No smmetr D. = + + = + =0,so =0 is a HA.
9 SECTION. CURVE SKETCHING 9 Copright 0, Cengage Learning. All rights reserved. E. f = < 0 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 > 0 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 > 0 / if >5,sof is increasing on 5, and decreasing on, 5. F. No local maimum or minimum G. = 5 / 5 / 5 / = / < 0 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 > 0 for D, sof is increasing on, and,. F. No maimum or minimum G. f = / 9 9 = 7 > and > 0 9 / or <<0,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 + = + + = > 0 <, / + so f is increasing on,, and decreasing on,. F. f = is a local maimum. G. f = + / + / + = + 5/ f =0 =0 = ± 9 = ± 7. f is CU on, 7 + and 7, andcdon 7, IP 7, , 7 7 7, = f = + / A. D = R B. = + / = / / + =0if =0or 7 -intercepts, -intercept = f 0 = 0 C. No smmetr D. + / = + / =, =, / / + no asmptote E. f =+ / = / + / / > 0 >0 or < 8,sof increases on, 8, 0, and
10 0 SECTION. CURVE SKETCHING decreases on 8, 0. F. Local maimum f 8 =, local minimum f 0 = 0 G. f = / < 0 0sof is CD on, 0 and 0,. No IP G. f / / = = + > 0 for all,sof is CU on,. 5/ 7. = f = A. D = { } = { } = { } =[, B. -intercept is. C. No smmetr D. =, no asmptote E. f = / / > 0 for all >,sof is increasing on,. F. No local maimum or minimum. G. f = / / + / / = +6 6 < 0 / since +6 < 0 negative discriminant as a quadratic in. So f is CD on,. =_ 0 = 9. = f =cos sin A. D = R B. =0 cos =sin = n +, n an integer -intercepts, -intercept = f 0 =. C. Periodic with period D. No asmptote E. f = sin cos =0 cos = sin =n + or n + 7. f > 0 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>0 sin >cos n +, n + 5,sof is CU on these intervals and CD on n, n +. IP n +, 0 Copright 0, Cengage Learning. All rights reserved = f = / A. D = { < } =, B. -intercept =0=-intercept C. f =f,sof is even. The curve is smmetric about the -ais. D. = =,so = ± + are VA. E. f = / = / Since > 0 and / > 0, f > 0 if 0 << and f < 0 if <<0,sof is increasing on 0, and decreasing on, 0. F. Local minimum f 0 = 0 0. = f = +cot, 0 << A. D =0,. B. No -intercept C. No smmetr D. +cot =, +cot =,so 0 + =0and = are VA. E. f = csc >0 when csc < sin > <<,so f is increasing on, and decreasing on 0, and,. F. f =+ is a local minimum, f = is a local maimum. G. f = csc csc cot =csc cot >0 cot >0 0 <<,sof is CU on 0,,CDon,. IP,
11 SECTION. CURVE SKETCHING Copright 0, Cengage Learning. All rights reserved.. = f =cos +sin A. D = R B. -intercept = f 0 = C. f =f, sothe curve is smmetric about the -ais. Periodic with period D. No asmptote E. f = sin +sin cos =sin cos > 0 sin <0 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 > 0 cos < n +, n +,sof is CU on these intervals and CD on n, n +. IP n ±, 0. = f =sin +cos A. D = R Note: f is periodic with period, so in B G we consider onl [0, ]. B. -intercept = f 0 =, -intercepts occur where sin = cos tan = =, 7. C. f + =f,sof is periodic with period. D. No asmptote E. f =cos sin >0 when cos >sin 0 << or 5 <<, f < 0 << 5,sof is increasing on 0, and 5, and decreasing on, 5. F. f = is a local maimum, f 5 = is a local minimum. G. f = sin cos >0 << 7,so f is CU on, 7 and CD on 0, and 7,.IP, 0, 7, 0. 0 ¹ _. = f =sin + cos A. D = R Note: f is periodic with period, so in B G we consider onl [0, ]. B. -intercept =, -intercepts occur where sin = cos tan = =, 5. C. No smmetr other than periodicit. D. No asmptote E. f =cos sin =0when cos = tan = = 6 or 7 6. f > 0 0 << 6 or 7 6 <<, f < 0 6 << 7 6.Sof is increasing on 0, 6 and 7, and decreasing on, F. f 6 =is a local maimum, f 7 6 = is a local minimum. G. f = sin cos =0when tan = = or 5. f > 0 << 5,sof is CU on, 5 and CD on 0, and 5,.IP, 0, 5, 0. ¹ _6, 0 ¹ 5¹, _. = f =sin +sin A. D = R Note: f is periodic with period, soinb Gwe consider onl [0, ]. B. -intercept =0, -intercepts occur where sin +sin =0 sin =0 =0,,. C. No smmetr other than periodicit D. No asmptote E. f =cos+ sin cos =cos + sin > 0 cos >0 0 << or <<,sof is increasing on 0, and, and decreasing on,. F. f =is a local maimum, f = is a local minimum. G. f = sin +cos sin = sin + sin =+sin sin > 0 sin>0 sin < 0 < 6 or 5 <. Sof is CU on 0, 6 6, 5,,andCDon 6, IP, 5 6 and 5, 5 6 7¹ 6
12 SECTION. CURVE SKETCHING 5. f = f =6 + f =8 7. f = sin f = sin + cos f =sin + cos sin After finding suitable viewing rectangles b ensuring that we have located all of the -values where either f =0or f =0weestimatefromthegraphoff that f is increasing on., 0. and 0.7, and decreasing on,. and 0., 0.7,withalocalmaimumoff and minima of f..0 and f Weestimate from the graph of f that f is CU on, 0.5 and 0.5, and CD on 0.5, 0.5,andthatf has inflection points at about 0.5,. and 0.5, f = f = f = 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,.0,.5, 0,.5,and.0, 6.8,andCDon 6.8,.0,.5, 0,.5,.0,and6.8, 7. f has IP at 6.8,.,.0,.0,.5,., 0, 0,.5,.,.0,.0 and 6.8,.. 8. f =sin + sin f =cos +cos f = sin sin Copright 0, Cengage Learning. All rights reserved. After finding suitable viewing rectangles, we estimate from the graph of f that f is increasing on,.5 and.0, and decreasing on.5,.0. 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.,. Note that f is periodic with period,soweconsideriton the interval [, ].Fromthegraphoff,weestimatethatf is increasing on.,.6, 0.8, 0.8,and.6,. and decreasing on,.,.6, 0.8, 0.8,.6 and.,. Maima: f.6 0.7, f , f Minima: f. 0.9, f , f Weestimatefromthe graph of f that f is CD on.0,., 0,. and.0, and CU on,.0,., 0 and.,. f has IP at, 0,.0, 0.8,., 0.8, 0, 0,., 0.8,.0, 0.8,and, 0.
4.4. Click here for solutions. Click here for answers. CURVE SKETCHING. y ln x 2 x. y ln 1 x 2. y x 2 e x2. x 1 x 2. x 2 x 3. x 5 2. y x 3.
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