6) y = 2x 2-12x

Transcription

1 Calculus H `Q0L1j7L bkeurtaag ^SvoYfhtRwdamrdeJ WLuLQCf._ a JA`lPlH srviwgch[tosc zrfejsueirzvfefd^. Final Review Name ID: 1 Date Period For each problem, find the -coordinates of all critical points and find the open intervals where the function is increasing and decreasing. 1) = ) = ) = -sec (); [-p, p] 1 ) = ( - 10) For each problem, find all points of relative minima and maima. 5) = ) = ] _b0`1k7w hkquytqad msmodfat^wlasrjeh L`LQC`.H X ZAEl[lj UrBiegJhGtWsD rwe^s_e^rovdexdf.m b kmnajdpep SwzigtthS fitncfxirnjiftze^ `CpaklccJuAl_uWst. -1-

2 For each problem, find the -coordinates of all points of inflection and find the open intervals where the function is concave up and concave down. 7) = ) = ) = -(5-15) 10) = cot (); [-p, p] j su0k1g7h wkgurtyal dslosfat\waairien blvltcs.s p QAglclG SrIiJghLtlsa JrPeDsfelrXv\eodH.W i smbahdued oweictshb NIqnmfli^npijtieT ^CIaKlwcgukliuLsI. --

3 For each problem, find the: and intercepts, asmptotes, -coordinates of the critical points, open intervals where the function is increasing and decreasing, -coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maima. Using this information, sketch the graph of the function. 11) = ) = \ vd0h1p7z NKruUtYao gsro[fltuwjavr`ea WL\LeCQ.H _ABlOl] LrGipg\hHtSsZ IrIe_sWemrcv`eEdt. s EMpa[dBeX hwaigthw di^nif^i`niictned [CiaLlacPuAlFuDsh. --

4 1) = ) = d P0W1k7E MKEuStat essofrt_wfamrue[ ULULWCh.j Q bajlxll vrriugihjtwsg LrueUszeBrKvaeudv.t P RMUa\dleZ \wdietzhr ICn]fginnfiatZe hcuaml^cpu_lruxsr. --

5 Given the graph of f (), sketch an approimate graph of f '(). 15) 1) f() f() A) f '() A) f '() B) f '() B) f '() C) f '() C) f '() D) f '() D) f '() q G[0w1u7m KButaU rsqojftlwsaerzep alul[c\.c o LA_lOl] _rgingbhrt`s` XrmelsqeZrNvhecdF.S k EMacdoeG jwii]t]hv HIQnvfMiInbi^tpeJ DCaaZl_cjullCuss[. -5-

6 Given the graph of f (), sketch an approimate graph of f ''(). 17) 1) f() f() A) f ''() A) f ''() B) f ''() B) f ''() C) f ''() C) f ''() D) f ''() D) f ''() P XT0U1X7G zk_ukt^ac msjowfutpwzawrqem mltlocu.f J garljlh ryitg^hutvso Pr_emsMeNrovjeEdL.L t DMba_dDe AwEiLtkhG zifnsfsimnhiqtget ucganlcsuclvu^sp. --

7 Given the graph of f '(), sketch a possible graph of f (). 19) 0) f '() f '() A) f() A) f() B) f() B) f() C) f() C) f() D) f() D) f() M ^k0a1p7[ DKWuntBas PSonfwtJwNakrVeP illbc^. X NAnlPll srii`gphvtksu wr_ewsieirbvketdc.x C kmfakdee] wbihtjh MIhnqfGiAnhiOtreP UCzalHcWuAlou\sV. -7-

8 Given the graph of f ''(), sketch a possible graph of f (). 1) ) f ''() f ''() A) f() A) f() B) f() B) f() C) f() C) f() D) f() D) f() P nt0c1v7] \KCuZtQae GS^ohfot]wLa[rheQ VLsLpCN. n aaqllle JrRiJgGhRtFsb DrDeqs]eIruvReTdv.B _ em_ajdaee qwui`tuhj IIjnlfniGnNistPeL MCZa`lpcIuvlvuvso. --

9 For each problem, find all points of absolute minima and maima on the given interval. ) = - + ; [-, 0] ) = - ; [-, 1] ) = -( + ) ; [-, 1] ) = -cot (); [ p, p ] Solve each optimization problem. 7) A farmer wants to construct a rectangular pigpen using 00 ft of fencing. The pen will be built net to an eisting stone wall, so onl three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area? ) Engineers are designing a bo-shaped aquarium with a square bottom and an open top. The aquarium must hold 0 ft³ of water. What dimensions should the use to create an acceptable aquarium with the least amount of glass? 9) A supermarket emploee wants to construct an open-top bo from a 1 b 0 in piece of cardboard. To do this, the emploee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a bo with the largest possible volume? r rs0f1]7f MKouatWaO ZSKoLfetvwdaPrOeW CLhLfCH.v X gaaltlm JrNiegJhStSsd GrGews\eorwvveddw.T m lmgakdueb qwkiutlhp [I]ncfliknaistOe\ QCBamlncGu^lBuosr. -9-

10 For each problem, use implicit differentiation to find d in terms of and. d 0) = ( + ) 1) sec = + ) e = ) + 1 = e ) ln = + 5) + = ln ) ( + 5) = 7) cos = + Solve each related rate problem. ) A conical paper cup is 0 cm tall with a radius of 10 cm. The bottom of the cup is punctured so that the water leaks out at a rate of 9p cm³/sec. At what rate is the water level changing when the water level is cm? H fi0s1t7g `Kmuet^a lsyo[fotuwmauroel ilklzcz.d U `A^lale WrHi]gVhXtXsS frrewshe_rdvfekdm.w Y OMGaudve hwriqtphp QIanPfuiIn\iktJeA qc[aglkcou]luoso. -10-

11 9) A 10 ft ladder is leaning against a wall and sliding towards the floor. The foot of the ladder is sliding awa from the base of the wall at a rate of ft/sec. How fast is the top of the ladder sliding down the wall when the top of the ladder is ft from the ground? 0) A 7 ft tall person is walking awa from a 1 ft tall lamppost at a rate of ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 1 ft from the lamppost? Evaluate each indefinite integral. 1) 1 5 d ) d ) 1-7 d ) ( ) d 5 1 F tc0s1o7n TKuQtCaN SboLf[tRwDairhe_ OLUL]CF.N X XAXlAl hruidg[hqtoss ArmevsDeWrVvkeGdp.E z MMea\doe` PwmiZthL DIInfftiDnsibtveD NCGawlcqu\lTuBsL. -11-

12 5) - 1 d) 5 d 7) -5e d) -e d 9) d50) 5 d 51) 5sec d5) -cos d 5) 0 (5 + 1) d5) ( + 1) d 55) (5-1) 100 d5) ( + )5 d V Df0Y1i7r IKmudtbaU jsmocfvtuwtalrmem KLSLBCB.f k JAGl^lN UrqiCgDhGtmsV PrXeps_eArRvveZdg.Q O fm`avdqew VwUiEtmhJ JIdnvfDiinFiHtweH ^CiaXlpcJuplXuIsf. -1-

13 57) (- + ln -) d 5) (-1 + ln -) 5 d 59) -5cos -5 sin -5 d0) sec tan d 1) e (e + )5 d) e (e - ) d ) d ) d 5) -0 e5 + 1 d) -1 e + d 7) -15 sec ( - ) d) -15 sec (5 + 1) d f SU0[1F7o YKYuvtXaL ASboNfTt^wNa\rSeT wlllkch.g ^ EAClKlr Drrisgzh`tssc srieksneyrpv^e[dd.` _ omoabdgeb SwcietmhS zi[nff^idnrintvek fcuaslfcguflfuwsd. -1-

14 0 1 9) -15 sec (5 + )tan (5 + ) d 70) -sec ( + 5) d Evaluate each definite integral. 71) ( - ) d 7) -1 ( - + ) d 7) - -1 d 7) ( + 1) d 75) d 7) ( + ) -1 (- + + ) d 77) p p -csc d 7) 1 - d B tk0g1g7 uktuptzaq GS`oXfPt\w^aorzeY mlglgct.b H CAulJlo NrIiNgKhLtWsl brbersledrjvtegds.g u UMNavdjeN rwxiatkha ]INnFf[icnFiUtaeA cchavlicauvlcunsw. -1-

15 0 79) d 0) ( + ) -1 1 ( - ) d 1) -1 - d ) ( + 1) ( + ) d For each problem, approimate the area under the curve over the given interval using left endpoint rectangles. You ma use the provided graph to sketch the curve and rectangles. ) = ; [0, ] ) = + 5; [-1, ] For each problem, find the area under the curve over the given interval. 5) = ; [, 7] A cd0c17z ckqujt_aq aszogfvt\wiakrseh olmlgc\.v w BAwl\lj [rpi[ghntpsc QrneIswer[vheOdP.U q hmawd[ew hweiotghl iimnwfli]n[i^tieg [CeallkcfudlEulsh. -15-

16 ) = ; [-, 1] 7) = ; [1, ] ) = ; [, ] For each problem, find the area of the region enclosed b the curves. You ma use the provided graph to sketch the curves and shade the enclosed region. 9) =, =, = 1, = ) =, =, = 0, = C OU0s1[7A _KHuJtXaA dswofjtbwuajrfed zllccl.j TA`lIlV UrKipgLhtfsN arnees^ejrvvwe`de.f z `MeaRd_e_ lwei_tthn vignhfiijnfiptreq `CBaqlacLuAlPuqsB. -1-

17 91) = - +, = - +, = 0, = 9) = -, = -, =, = B JF0`1K7t UKnu\thaL HSYoCfVtqwFaOr`e_ RLwLECW.c m UAQldlS BrqijgWhetDsr rie]sae[rqvoepd.e q OMOaXdLeI dwvi[t_h hicn[fijn_iktdew rcaaslhcaudl^uvsh. -17-