JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH 241) Final Review Fall 2016

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1 JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw Fall 06 Th Final Rviw is a starting point as you study for th final am. You should also study your ams and homwork. All topics listd in th Cours Objctivs or th Contnt Outlin listd blow ar possibl topics for Final Eam qustions. Th cours dscription is availabl at Cours Objctivs Upon succssful compltion of this cours th studnt should b abl to:. Evaluat th limits of functions.. Stat whthr a function is continuous or discontinuous basd on both th graph and th dfinition of continuity.. Us limits to dscrib instantanous rat of chang, th slop of th tangnt lin and th vlocity and acclration of a moving particl.. Diffrntiat algbraic, trigonomtric, and transcndntal functions plicitly and, whr appropriat, implicitly.. Us drivativs for curv sktching. 6. Us and intrprt th drivativs of functions to solv problms from a varity of filds, including physics and gomtry. 7. Intgrat algbraic, trigonomtric, and transcndntal functions. 8. Comput dfinit intgrals by th Fundamntal Thorm of Calculus, by numrical tchniqus, and by substitution. 9. Us intgration rsults to calculat aras and man valus. Contnt Outlin & Comptncis I. Using Limits A. Evaluation of limits. Evaluat th limit of a function at a point both algbraically and graphically.. Evaluat th limit of a function at infinity both algbraically and graphically.. Us th dfinition of a limit to vrify a valu of th limit of a function. B. Us of limits. Us th limit to dtrmin th continuity of a function.. Us th limit to dtrmin diffrntiability of a function.

2 C. Limiting procss.us th limiting procss to find th drivativ of a function. II. Finding Drivativs A. Find drivativs involving powrs, ponnts, and sums. B. Find drivativs involving products and quotints. C. Find drivativs involving th chain rul. D. Find drivativs involving ponntial and logarithmic functions. E. Find drivativs involving trigonomtric and invrs trigonomtric functions. F. Find drivativs involving implicit diffrntiation. G. Us th drivativ to find vlocity, acclration, and othr rats of chang. H. Us th drivativ to find th quation of a lin tangnt to a curv at a givn point. III. Using Drivativs A. Curv sktching. Us th first drivativ to find critical points.. Apply th Man-Valu Thorm for drivativs.. Dtrmin th bhavior of a function using th first drivativ.. Us th scond drivativ to find inflction points.. Dtrmin th concavity of a function using th scond drivativ. 6. Sktch th graph of th function using information gathrd from th first and scond drivativs. 7. Intrprt graphs of functions. B. Applications of th drivativ. Solv rlatd rats problms.. Us optimization tchniqus in conomics, th physical scincs, and gomtry.. Us diffrntials to stimat chang.. Us Nwton s Mthod. IV. Finding Intgrals A. Find ara using Rimann sums. B. Eprss th limit of a Rimann sum as a dfinit intgral. C. Evaluat th dfinit intgral using gomtry. D. Intgrat dfinit intgral using numrical approimation. E. Evaluat dfinit intgrals using th Fundamntal Thorm of Calculus. F. Intgrat algbraic, natural ponntial, natural logarithm, trigonomtric, and invrs trigonomtric functions. G. Intgrat indfinit intgrals. H. Intgration by substitution. V. Using th Intgral A. Utiliz th Man-Valu Thorm for Intgrals. B. Calculat ara.

3 JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw. Rfr to th graph of th function y f() to valuat or approimat ach of th following prssions or stat that th valu dos not ist. a. lim f + b. lim f c. lim f d. f. f f. lim f + g. lim f h. lim f i. f j. f k. lim f + l. lim f m. lim f n. f o. f p. f q. f. Evaluat th following limits or stat that th limit dos not ist. Justify your answrs. + cos tan a. h. lim o. lim tan lim i. b. lim + lim ln( ) + cos + p. lim + j. lim q. lim ( + ) c. lim + ( cos ) k. lim sin 0 r. lim d. lim l. lim s. lim sin. lim 0 + sin y m. lim y 0 y f. lim n. lim 9 g. lim +

4 cos. Suppos that th inqualitis < < hold for valus of clos to zro. What, if cos anything, dos this tll you about lim? Giv rasons for your answr. 0. Prov th following limit statmnts using th psilon-dlta dfinition of limit. a. lim ( ) b. lim. For ach function f ( ) blow, stat th intrvals on which th function is continuous. For ach discontinuity, stat which conditions rquird by th Continuity Tst ar not mt. Classify ach discontinuity as rmovabl or nonrmovabl. a. f + b. f + c. f +, < 6. For what valu of b is g continuous at vry? b, 7. Us th Intrmdiat Valu Thorm to show that thr is a root of th quation + 0 btwn and. ( π ) + cos, < +, < 0, 0 8. For th function dfind by f < 9, < 7, > a. Graph f ( ). Clarly indicat all important points. b. Discuss limits of f at all important points. c. Discuss th continuity of f at all important points. d. Discuss th diffrntiability of f at all important points. 9. Find th drivativ using th limit dfinition of th drivativ. a. f 7 + b. f c. f + d. f +

5 0. Find th drivativ of th function. a. f + b. f ln(tan ) c. f ln( 6 ) d.. f f ( 7 f. f + sin g. f h. f cos ) 6 i. f cos ( ) j. f sin k. f tan ( ) l. f 9 m. f sc n. f + o. f sin p. f dy. Find for ach of th following. d a. y ln b. + y c. y + + y d. y + y + y. y sin y y. Find th quation of th lin tangnt to f at th point (, ).. Find th quation of th tangnt lin to th graph of. For f tan a. Find th quation of th lin tangnt to f and paralll to th lin y 0 f at th point (, π ). f at th point π (, ) b. Find th quation of th lin tangnt to.. Givn that f 6 a. Show that f satisfis th hypothss of th Man Valu Thorm on th intrval [, ] b. Find all numbrs c that satisfy th conclusion of th Man Valu Thorm Find th coordinats of all rlativ/local trma and points of inflction, and any asymptots. Indicat any rlativ/local trma that ar also absolut trma. Giv th intrvals on which th function is incrasing, on which it is dcrasing, on which it is concav up, and on which it is concav down. Thn sktch th graph. ln a. f f. f + b. f sin + cos, on th intrval [0, π] g. f c. f sin, on th intrval (0, π) 9 h. f d. f 6 +. f ln f tan i.

6 7. For ach of th following functions, find th intrvals whr f is concav up and whr f is concav down. Stat any points of inflction. a. f + sin b. f, π π + cos 8. For ach of th following functions, find th intrvals whr th function is incrasing and whr th function is dcrasing. a. f + b. f cos sin, 0 π 9. Givn f, f ( ), f ( ), a. Find all critical numbrs of f. b. Find any local/rlativ trma using th Scond Drivativ Tst. c. Giv th intrvals on which f is incrasing, and thos on which f is dcrasing. d. Giv th intrvals on which f is concav up, and thos on which f is concav down.. Find all points of inflction of f. 0. Sktch th graph of a continuous function f that satisfis all of th statd conditions. f (0) ; f () ; f ( 0) f () 0 f () < 0 if > or < 0 f () > 0 if 0 < < f () > 0 if < ; f () < 0 if >. Sktch th graph of a continuous function f that satisfis all of th statd conditions. f (0) ; f () ; f () 6; f ( 0) f () 0 f () > 0 if > or < 0 f () < 0 if 0 < < f () < 0 if < or if < < f () > 0 if < < or if >. Th numbr of individuals (in thousands) in a population of animals is projctd to b givn by 6000 P( t) 0, whr t is masurd in yars sinc January, t a. Find dp and d P. b. Evaluat P( t), dp, and d P at t 0, t and t 0, and plain thir manings in trms of th population and its rat of growth. c. Whn is th population projctd to b growing most rapidly? 6

. A projctil is fird dirctly upward with a vlocity of ft/sc. Its hight in ft abov th ground aftr t sconds is givn by s( t) 00 + t 6t. Find th following. a. How long dos it tak for th projctil to hit th ground?

7 . A projctil is fird dirctly upward with a vlocity of ft/sc. Its hight in ft abov th ground aftr t sconds is givn by s( t) 00 + t 6t. Find th following. a. How long dos it tak for th projctil to hit th ground? b. What is th spd of th projctil at t 7 sconds? c. What is th vlocity of th projctil whn it hits th ground? d. What is th maimum hight of th projctil?. Th graph blow shows th position s f ( t) of an objct moving up and down, with displacmnt masurd from th origin. Us th graph to answr th following qustions about th objct s motion. a. During approimatly what tim intrvals is th objct moving toward th origin? b. During approimatly what tim intrvals is th objct moving away from th origin? c. At approimatly what tims dos th objct chang dirctions? d. During approimatly what tim intrvals is th objct stationary (not moving)?. At approimatly what tims (and, if applicabl, during what tim intrvals) is th vlocity of th objct qual to zro? f. At approimatly what tims (and, if applicabl, during what tim intrvals) is th acclration of th objct qual to zro? g. During approimatly what tim intrvals is th acclration of th objct positiv? h. During approimatly what tim intrvals is th acclration of th objct ngativ?. An opn-top rctangular bo is constructd from a 0- by 6- inch pic of cardboard by cutting squars of qual sid lngth from th cornrs and folding up th sids. Find th dimnsions of th bo of largst volum and find th maimum volum. 6. A rctangl is boundd by th -ais and th smicircl th rctangl hav so that its ara is a maimum? y. What lngth and wih should 7. Th sum of th primtrs of an quilatral triangl and a squar is 0. Find th dimnsions of th triangl and th squar that produc a minimum total ara. 8. Ptrolum is laking from an offshor wll and forms a circular oil slick on th surfac of th watr. If th ara of th slick is incrasing at th rat of 8 km pr day, at what rat is th radius incrasing at th instant whn th radius is km? 9. A sphrical balloon is bing inflatd. As th radius of th balloon incrass at th rat of ft./minut, find th rat of chang of th volum whn th radius is ft. nd bas 0. A basball diamond is a squar with ach sid 90 ft long. A battr hits th ball and runs from hom plat toward first bas at a spd of ft/sc. a. At what rat is his distanc from scond bas changing whn h is on third of th way to first bas? b. At what rat is his distanc from third bas changing at th sam instant? rd bas st bas 90 ft 7 hom plat

8 . Th hight of a rockt is givn by s ( t) 0t ft aftr t sconds. A ground lvl camra is locatd 00 ft from th launch pad. As sn by th camra, what is th rat of chang of th angl of lvation of th rockt 0 sconds aftr liftoff?. Find y and us th approimation d to find dy for th function y + 0 at with 0... Th radius of a circl incrasd from.00 to.0 mtrs. Us diffrntials to (a) stimat th rsulting chang in ara and (b) prss th stimat as a prcntag chang in ara.. If f cos and F() is an antidrivativ of f () with F(0), find F().. If f and F() is an antidrivativ of f () with F(0), find F(). 6. Find th indfinit intgrals and valuat th dfinit intgrals. π a. π 9 cos d b. d 6 c. ( + ) d 0 d. d. + 7 d f. (cos ) ( sin ) d g. sc( ) tan ( ) h. tan sc i. d j. d 9 7. Find F (). k. 0 + d l. d a. F() sin ( t ) d b. F() t + sin c. F() 0 t d 8 n. m. sin d d o. 9 d p. ln d q. ( + ) d r. d sin t s. t t. cot d u. d v. d w.. cos tan d d y. tan d

8. Us th trapzoidal rul to approimat. a. d, n 0 + π b. sin ( ) d, n 0 9. Us rctangls and (a) an uppr sum, and (b) a lowr sum to approimat th ara of th rgion btwn y and th -ais, on th intrval [0, ].

9 8. Us th trapzoidal rul to approimat. a. d, n 0 + π b. sin ( ) d, n 0 9. Us rctangls and (a) an uppr sum, and (b) a lowr sum to approimat th ara of th rgion btwn y and th -ais, on th intrval [0, ]. Us subintrvals of qual wih. 0. Find th avrag valu of on th intrval [,]. f. Th graph of y f is shown in th figur at right. Th shadd 6 rgion labld A has ara 8 squar units, and f d 6. Us this information to find th find th following valus: a. b. c f d f d f d d. 0 f d. [ f ] d 0 f. th avrag valu of f on [0,6]. Is th following statmnt Tru or Fals? You nd to justify you answr. a. If a function f is continuous at c, thn f must b diffrntiabl at c. b. If a function f is diffrntiabl at c, thn f must b continuous at c. c. If a function f is not diffrntiabl at c, thn f must b discontinuous at c. d. If c is a critical numbr of a function f, thn f must attain a local maimum or a local minimum at c.. Th function f, is continuous at., > f. Th function f,, > g. If f g, is diffrntiabl at. for all in [ a b ] and both f and g ar continuous on [, ] b b f d g d a. a a h. If f is an vn function that is continuous for all, thn 0 a f d. a i. If f is an odd function that is continuous for all, thn 0 a f d. a b, thn 9

10 . Stat ach of th following: a. Th dfinition of lim f ( ) b whr a and b ar ral numbrs. a b. Th dfinition of continuity for a function f ( ). c. Th Intrmdiat Valu Thorm. d. Th dfinition of th drivativ for a function f ( ).. Th Etrm Valu Thorm. f. Th Man Valu Thorm for drivativs. b g. Th dfinition of f ( ) d. a h. Th Man Valu Thorm for intgrals. i. Th Fundamntal Thorm of Calculus (both parts). 0

11 Answrs to th Math final rviw a. a. b. b. 0 c. DNE (Dos Not Eist) c. d. d.. DNE. f. f. DNE g. g. h. h. i. i. j. DNE j. k. k. 0 l. l. 6 m. m. n. DNE n. DNE o. DNE o. π p. p. 0 (S Thomas Calculus ET, th d, p. 07) q. q. DNE r. 0 s. 0. cos lim by Sandwich Thorm sinc 0 lim and lim 0 0 a. Lt ε > 0 b givn. ε Lt δ. Thn 0 < < δ ε < < ε ( ) < ε < ε < ε. Sinc this is tru for any ε > 0, lim ( ).

12 b. Lt ε > 0 b givn. Without loss of gnrality, w may assum ε<. Lt δ ε ε. Not that ε ε > 0 sinc 0< ε<. So 0 < < δ < ε ε ε ε < < ε ε Now not that for anyε > 0, ε ε < ε + ε so w hav ε ε < < ε ε < ε + ε ε ε + < < ε + ε + ε < < ε + W assumd ε<, so w will hav ε > 0, so w can writ ε < < ε + ε < < ε < ε Sinc this is tru for any ε > 0, lim. a. (For th Continuity Tst, s Thomas, p. 9.) Continuous on (, ) (, ) (, ) ; discontinuity at sinc f () is undfind, and discontinuity at sinc f is undfind and lim f DNE ; is a rmovabl discontinuity and is a nonrmovabl discontinuity. b. Continuous on (,) (, ) ; discontinuity at sinc f () is undfind and is a nonrmovabl discontinuity c. Continuous on (, ) (, ) ; discontinuity at sinc f ( ) is undfind and lim f DNE ; is a nonrmovabl discontinuity lim f DNE ; 6. b 7. f + is a polynomial, which is continuous for all valus of, and f () < 0, and f () > 0. Applying Intrmdiat Valu Thorm with a, b, and y 0 0, thr must b a valu c with c btwn and for which f ( c ) 0.

13 8a (0,) (, ) (,0) 8b. lim f 0 and lim f 0, so lim f 0 ; + lim f and lim f, so lim f ; lim f and lim f 0, so lim f DNE ; + lim f 7 and lim f 7, so lim f 7 ; + lim f and lim f, so lim f DNE ; + lim f DNE, and lim f (,0) (, 7) 8c. f is discontinuous at sinc f ( ) is undfind; f is continuous at 0 sinc f (0) lim f ; 0 f is discontinuous at sinc lim f DNE ; f is continuous at sinc f () 7 lim f ; f is discontinuous at sinc f () is undfind and f is continuous for all othr valus of. lim f DNE 8d. f cannot b diffrntiabl at,, or sinc f is not continuous at thos valus of. To chck diffrntiability at 0 and, amin th bhavior of th slops of th tangnt lins on ithr sid of ths valus. lim f lim d ( + ) d lim, and lim f ( ) lim d lim d so f is diffrntiabl at 0. lim f lim d ( 9 ) lim ( ) d 8, and 7 7 lim lim d f lim d. Th slops approach diffrnt valus on ithr + ( ) sid of so f is not diffrntiabl at Thrfor f is not diffrntiabl at,,, and, but is diffrntiabl for all othr valus of.

14 9a. Start: 9b. Start: 9c. Start: 9d. Start: ( + h) 7 ( 7) f ( ) lim h 0 h ( + h) + ( + h) ( f ( ) lim h 0 h f + h+ + lim Answr: h 0 h f ( ) lim h 0 ( + h) + h + ) f ( ) Answr: f Answr: f + ( + ) + Answr: f ( ) + 0a. 0b. 0c. 0d. 0 f ( ) + sc f ( ) tan f f + 0. f ( ) 6( ) ( 0) 0f. f ( ) ( + ) sin cos sin 0g. f ( ) 0h. f ( ) 8(cos )(sin ) 0i. f ( ) 0j. f ( ) sin 0k. f ( ) + 0l. f ( ) 9 (ln 9) + 9 0m. f ( ) sc tan + sc 0n. f ( ) 0o. f ( ) + 0p. f ( ln + ) a. ln dy ln d d. dy y + y d y + y dy b. d y c. dy y d +. dy y d y sin cos + y ( y ) ( y ). y. y and y + a. y π ( ) or y π + b. y ( π ) or 6 y + π

15 a. f is continuous for all cpt 0, so f is continuous on [,] ; is diffrntiabl on (, ). f ists for all cpt 0, so f b. c 6a. Rlativ/local and absolut min: (, ) rlativ/local and absolut ma: (, ); ; points of inflction:,, 0,0,, ; incrasing on (,) ; dcrasing on (, ) (, ) ; concav up on (,0) (, ) ; concav down on (, ) ( 0, ) ; horizontal asymptot y 0 6b. Rlativ/local and absolut min: (, ) rlativ/local and absolut ma: ( π, ) ; points of inflction: ( π,0) and ( 7π,0); incrasing on ) ( dcrasing on (, ) 0, π π, π ; π π ; concav up on( π, 7 π ) ; concav down on ) ( π 7 π π 0,, π ; c. Rlativ/local and absolut min: (π,-); rlativ/local and absolut ma: (π,); point of inflction: (π,0); incrasing on (0, π ) ( π, π ) ; dcrasing on ( π, π ) ; concav up on ( π, π ) ; concav up on ( π, π )

16 6d. No trma; point of inflction: (,8); incrasing on (, ) ; concav up on (, ) ; concav down on (, ) Rlativ/local and absolut min: (,); no maima; incrasing on (, ) ; dcrasing on (0,) ; no point of inflction; concav up on (0, ) ; vrtical asymptot f. No minima;, ; rlativ/local and absolut ma: ( ) point of inflction: incrasing on ( 0, ) ; dcrasing on (, ) ; concav up on 8, ; 8 8, ; concav down on 0, 8 ; horizontal asymptot y 0 ; vrtical asymptot g. No trma; point of inflction: ( 0, 0) ; dcrasing on (, ) (,) (, ) ; concav up on (,0) (, ) ; concav down on (, ) (0,) ; horizontal asymptot y 0 ; vrtical asymptots,

17 6h. No minima; rlativ/local and absolut ma: point of inflction:, ; incrasing on (,) ; dcrasing on (, ) ; concav up on (, ) ; concav down on (, ) ; horizontal asymptot y 0, ; i. Rlativ/local absolut min: (0,0) ; no maima; π points of inflction:, 6 and π, 6 ; incrasing on (0, ) ; dcrasing on (, 0) ; concav up on, ; concav down on,, ; horizontal asymptot y π 7a. Concav down on (, ); concav up on (, ) ; inflction point at (, ) 7b. Concav up on ( π, π ) ( 0,π ) ; concav down on ( π,0) ( π, π ); inflction point at ( 0, 0) 8 8a. Dcrasing on, 0, ; incrasing on 6 6, 0, 6 6 8b. Dcrasing on ( 0, π ) ( π, π ); incrasing on ( π ) 9a. f < 0, so local/rlativ min is at 9b. 9c. incrasing on (, ) ; dcrasing on 9d. concav up on (, ) ; concav down on (,) 9. point of inflction at (, ), 7, π

18 0.. Answrs may vary. Answrs may vary. a. dp,000 t ( + t ), ( t ) ( + t ) d P, 000 b. t P( t ) 0 0 On January, 00, th population was 0,000 animals 0 On January, 0, th population is projctd to b 0,000 animals 0 0 On January, 00, th population is projctd to b 0,000 animals dp 0 On January, 00, th population was not growing or dclining On January, 0, th population is projctd to b growing at a rat of,000 animals pr yar 7.68 On January, 00, th population is projctd to b growing at a rat of 7,680 animals pr yar d P 9. On January, 00, th rat of growth of th population was incrasing at a rat of 9,00 animals pr yar ach yar.8 On January, 0, th rat of growth of th population is projctd to b dcrasing at a rat of,800 animals pr yar ach yar.6896 On January, 00, th rat of growth of th population is projctd to b dcrasing at a rat of animals pr yar ach yar c. t.89 yars aftr January, 00. (Novmbr 9, 0) 8

19 9+ 06 a. sc 9.6 sc b. 80 ft/sc c ft/sc 6.7 ft/sc d. ft a. Estimats may vary slightly: [ 0,0. ), b. Estimats may vary slightly: ( 0.,. ), (,6), (7,9) sconds c. Estimats may vary slightly: t., t 6 sconds d. Estimats may vary slightly: (,) sconds. Estimats may vary slightly: t [ ] t f. Estimats may vary slightly: [ ].,, (,), (6,7) sconds.,,, 6 sconds t 0., t,,, t, t 7 sconds g. Estimats may vary slightly: [0,0.), (,), (,7) sconds h. Estimats may vary slightly: (0.,), (,), (7,9) sconds. Lngth '', wih 6'', hight ''; Volum in 6. Lngth, wih 7. Sid of th squar 0, sid of th triangl dr 8. km/day π dv 9. 6π ft /min 0a. ds 8. ft/sc 0b. dt 7.89 ft/sc 0 dθ 60. rad / sc.7 / sc 769. y 0.9, dy

20 a. 0.08π m b. %. F sin +. F a. 6 6b. 6c. 7 7,6,8 (7 ) 7 7 6d. ln 6. ( + 7) + C 6f. ( sin ) + C 6g. sc + C 6h. (tan ) + C 6i. ln + 6j. sin C 6k. π 6 6l. sc + C 6m. cos + sin + C 6n. 6o. + C 8 6p. ln + C + + C 6q. ( ) + C or r. cot + C 6s. t t t t t + + C 8 6t. cot + C 6u. 6v. + C ( ) + C 6w. tan + C 6. tan ln + + C 6y. ln cos + C or ln sc + C 7a. sin 7b. 6 + sin 7c. cos 0

21 a. T b. T π π π 9 π ( π ) 0 + sin + sin + sin + sin a. uppr sum /6 / 9/6 ( ) 9b. lowr sum /6 / 9/6 ( ) a. 8 b. 8 c. d. 6. f. a. F b. T c. F d. F. F f. F g. T h. F i. T Th pag rfrncs blow ar for Thomas Calculus, Early Transcndntals, th dition a. p. 77 b. p. 9 c. p. 99 d. p. 6. p. f. p. g. p. h. p. i. p. 7, 8

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl