Conic Sections. Animation. Animation. 694 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

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1 9 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Section HYPATIA (7 5 AD) The Greeks discovered conic sections sometime between nd BC B the beginning of the Alendrin period, enough ws known bout conics for Apollonius ( 9 BC) to produce n eight-volume work on the subject Lter, towrd the end of the Alendrin period, Hpti wrote tetbook entitled On the Conics of Apollonius Her deth mrked the end of mjor mthemticl discoveries in Europe for severl hundred ers The erl Greeks were lrgel concerned with the geometric properties of conics It ws not until 9 ers lter, in the erl seventeenth centur, tht the broder pplicbilit of conics becme pprent Conics then pled prominent role in the development of clculus MthBio Conics nd Clculus Understnd the definition of conic section Anlze nd write equtions of prbols using properties of prbols Anlze nd write equtions of ellipses using properties of ellipses Anlze nd write equtions of hperbols using properties of hperbols Conic Sections Ech conic section (or simpl conic) cn be described s the intersection of plne nd double-npped cone Notice in Figure tht for the four bsic conics, the intersecting plne does not pss through the verte of the cone When the plne psses through the verte, the resulting figure is degenerte conic, s shown in Figure Circle Conic sections Figure Animtion Prbol Ellipse Hperbol Point Degenerte conics Figure Line Two intersecting lines Animtion There re severl ws to stud conics You could begin s the Greeks did b defining the conics in terms of the intersections of plnes nd cones, or ou could define them lgebricll in terms of the generl second-degree eqution FOR FURTHER INFORMATION To lern more bout the mthemticl ctivities of Hpti, see the rticle Hpti nd Her Mthemtics b Michel A B Dekin in The Americn Mthemticl Monthl A B C D E F Generl second-degree eqution However, third pproch, in which ech of the conics is defined s locus (collection) of points stisfing certin geometric propert, works best For emple, circle cn be defined s the collection of ll points (, tht re equidistnt from fied point h, k This locus definition esil produces the stndrd eqution of circle MthArticle h k r Stndrd eqution of circle

2 SECTION Conics nd Clculus 95 Prbol Directri Figure Focus p Verte Ais d (, ) d d d Prbols A prbol is the set of ll points, tht re equidistnt from fied line clled the directri nd fied point clled the focus not on the line The midpoint between the focus nd the directri is the verte, nd the line pssing through the focus nd the verte is the is of the prbol Note in Figure tht prbol is smmetric with respect to its is THEOREM Stndrd Eqution of Prbol The stndrd form of the eqution of prbol with verte h, k nd directri k p is h p k Verticl is For directri h p, the eqution is k p h Horizontl is The focus lies on the is p units (directed distnce) from the verte The coordintes of the focus re s follows h, k p h p, k Verticl is Horizontl is EXAMPLE Finding the Focus of Prbol Find the focus of the prbol given b Solution To find the focus, convert to stndrd form b completing the squre = + Verte p =, Focus ( ) Prbol with verticl is, Figure p < Rewrite originl eqution Fctor out Multipl ech side b Group terms Add nd subtrct on right side Write in stndrd form Compring this eqution with h p k, ou cn conclude tht h, k, nd Becuse p is negtive, the prbol opens downwrd, s shown in Figure So, the focus of the prbol is p units from the verte, or h, k p, p Focus Editble Grph Tr It Eplortion A Eplortion B A line segment tht psses through the focus of prbol nd hs endpoints on the prbol is clled focl chord The specific focl chord perpendiculr to the is of the prbol is the ltus rectum The net emple shows how to determine the length of the ltus rectum nd the length of the corresponding intercepted rc

3 9 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Focl Chord Length nd Arc Length = p Ltus rectum ( p, p) ( p, p) (, p) Length of ltus rectum: p Figure 5 Find the length of the ltus rectum of the prbol given b p Then find the length of the prbolic rc intercepted b the ltus rectum Solution Becuse the ltus rectum psses through the focus, p nd is perpendiculr to the -is, the coordintes of its endpoints re, p nd, p Substituting p for in the eqution of the prbol produces pp So, the endpoints of the ltus rectum re p, p nd p, p, nd ou cn conclude tht its length is p, s shown in Figure 5 In contrst, the length of the intercepted rc is p s d Use rc length formul p p p p p d d p p p p ln p p p p8p p lnp 8p p lnp p ln 59p ±p p Simplif Theorem 8 p Eplortion A Open Eplortion Light source t focus Ais One widel used propert of prbol is its reflective propert In phsics, surfce is clled reflective if the tngent line t n point on the surfce mkes equl ngles with n incoming r nd the resulting outgoing r The ngle corresponding to the incoming r is the ngle of incidence, nd the ngle corresponding to the outgoing r is the ngle of reflection One emple of reflective surfce is flt mirror Another tpe of reflective surfce is tht formed b revolving prbol bout its is A specil propert of prbolic reflectors is tht the llow us to direct ll incoming rs prllel to the is through the focus of the prbol this is the principle behind the design of the prbolic mirrors used in reflecting telescopes Conversel, ll light rs emnting from the focus of prbolic reflector used in flshlight re prllel, s shown in Figure Prbolic reflector: light is reflected in prllel rs Figure THEOREM Ref lective Propert of Prbol Let P be point on prbol The tngent line to the prbol t the point P mkes equl ngles with the following two lines The line pssing through P nd the focus The line pssing through P prllel to the is of the prbol

4 SECTION Conics nd Clculus 97 NICOLAUS COPERNICUS (7 5) Copernicus begn to stud plnetr motion when sked to revise the clendr At tht time, the ect length of the er could not be ccurtel predicted using the theor tht Erth ws the center of the universe MthBio Ellipses More thn thousnd ers fter the close of the Alendrin period of Greek mthemtics, Western civiliztion finll begn Renissnce of mthemticl nd scientific discover One of the principl figures in this rebirth ws the Polish stronomer Nicolus Copernicus In his work On the Revolutions of the Hevenl Spheres, Copernicus climed tht ll of the plnets,including Erth,revolved bout the sun in circulr orbits Although some of Copernicus s clims were invlid, the controvers set off b his heliocentric theor motivted stronomers to serch for mthemticl model to eplin the observed movements of the sun nd plnets The first to find n ccurte model ws the Germn stronomer Johnnes Kepler (57 ) Kepler discovered tht the plnets move bout the sun in ellipticl orbits, with the sun not s the center but s focl point of the orbit The use of ellipses to eplin the movements of the plnets is onl one of mn prcticl nd esthetic uses As with prbols, ou will begin our stud of this second tpe of conic b defining it s locus of points Now, however, two focl points re used rther thn one An ellipse is the set of ll points, the sum of whose distnces from two distinct fied points clled foci is constnt (See Figure 7) The line through the foci intersects the ellipse t two points, clled the vertices The chord joining the vertices is the mjor is, nd its midpoint is the center of the ellipse The chord perpendiculr to the mjor is t the center is the minor is of the ellipse (See Figure 8) (, ) FOR FURTHER INFORMATION To lern bout how n ellipse m be eploded into prbol, see the rticle Eploding the Ellipse b Arnold Good in Mthemtics Techer d d Verte Focus Focus Figure 7 Figure 8 Mjor is Focus ( h, k) Center Minor is Focus Verte MthArticle THEOREM Stndrd Eqution of n Ellipse The stndrd form of the eqution of n ellipse with center h, k nd mjor nd minor es of lengths nd b, where > b, is or h h b k b k Mjor is is horizontl Mjor is is verticl The foci lie on the mjor is, c units from the center, with c b Figure 9 Animtion NOTE You cn visulize the definition of n ellipse b imgining two thumbtcks plced t the foci, s shown in Figure 9 If the ends of fied length of string re fstened to the thumbtcks nd the string is drwn tut with pencil, the pth trced b the pencil will be n ellipse

5 98 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Completing the Squre ( ) ( + ) + = Ellipse with verticl mjor is Figure Verte Focus Center Focus Verte Find the center, vertices, nd foci of the ellipse given b Solution form 8 8 B completing the squre, ou cn write the originl eqution in stndrd Write originl eqution Write in stndrd form So, the mjor is is prllel to the -is, where h, k,, b, nd c So, ou obtin the following Center: Vertices: Foci: ,, nd,, nd, The grph of the ellipse is shown in Figure h, k h, k ± h, k ± c Editble Grph Tr It Eplortion A Eplortion B NOTE If the constnt term F 8 in the eqution in Emple hd been greter thn or equl to 8, ou would hve obtined one of the following degenerte cses F 8, single point,, : F > 8, no solution points: < EXAMPLE The Orbit of the Moon Moon The moon orbits Erth in n ellipticl pth with the center of Erth t one focus, s shown in Figure The mjor nd minor es of the orbit hve lengths of 78,8 kilometers nd 77, kilometers, respectivel Find the gretest nd lest distnces (the pogee nd perigee) from Erth s center to the moon s center Erth Perigee Figure Apogee Solution Begin b solving for nd b 78,8 Length of mjor is 8, Solve for b 77, Length of minor is b 8,8 Solve for b Now, using these vlues, ou cn solve for c s follows c b,8 The gretest distnce between the center of Erth nd the center of the moon is c 5,58 kilometers, nd the lest distnce is c,9 kilometers Tr It Eplortion A

6 SECTION Conics nd Clculus 99 FOR FURTHER INFORMATION For more informtion on some uses of the reflective properties of conics, see the rticle The Geometr of Microwve Antenns b Willim R Prznski in Mthemtics Techer MthArticle Theorem presented reflective propert of prbols Ellipses hve similr reflective propert You re sked to prove the following theorem in Eercise THEOREM Reflective Propert of n Ellipse Let P be point on n ellipse The tngent line to the ellipse t point P mkes equl ngles with the lines through P nd the foci One of the resons tht stronomers hd difficult in detecting tht the orbits of the plnets re ellipses is tht the foci of the plnetr orbits re reltivel close to the center of the sun, mking the orbits nerl circulr To mesure the ovlness of n ellipse, ou cn use the concept of eccentricit Definition of Eccentricit of n Ellipse The eccentricit e of n ellipse is given b the rtio e c c () is smll Foci c Foci c (b) is close to c Eccentricit is the rtio Figure c To see how this rtio is used to describe the shpe of n ellipse, note tht becuse the foci of n ellipse re locted long the mjor is between the vertices nd the center, it follows tht < c < For n ellipse tht is nerl circulr, the foci re close to the center nd the rtio c is smll, nd for n elongted ellipse, the foci re close to the vertices nd the rtio is close to, s shown in Figure Note tht < e < for ever ellipse The orbit of the moon hs n eccentricit of e 59, nd the eccentricities of the nine plnetr orbits re s follows Mercur: Venus: Erth: Mrs: Jupiter: Sturn: Urnus: Neptune: Pluto: You cn use integrtion to show tht the re of n ellipse is instnce, the re of the ellipse b is given b b A d b cos d e 5 e 8 e 7 e 9 e 8 e 5 e 7 e 8 e 88 Trigonometric substitution sin A b For However, it is not so simple to find the circumference of n ellipse The net emple shows how to use eccentricit to set up n elliptic integrl for the circumference of n ellipse

7 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE 5 Finding the Circumference of n Ellipse Show tht the circumference of the ellipse is b e sin d e c AREA AND CIRCUMFERENCE OF AN ELLIPSE In his work with elliptic orbits in the erl s, Johnnes Kepler successfull developed formul for the re of n ellipse, A b He ws less successful in developing formul for the circumference of n ellipse, however; the best he could do ws to give the pproimte formul C b Solution Becuse the given ellipse is smmetric with respect to both the -is nd the -is, ou know tht its circumference C is four times the rc length of b in the first qudrnt The function is differentible for ll in the intervl, ecept t So, the circumference is given b the improper integrl C lim d d Using the trigonometric substitution sin, ou obtin C b sin cos d cos d d cos b sin d sin b sin d b sin d Becuse e ou cn rewrite this integrl s C c b, e sin d Eplortion A Eplortion B Open Eplortion A gret del of time hs been devoted to the stud of elliptic integrls Such integrls generll do not hve elementr ntiderivtives To find the circumference of n ellipse, ou must usull resort to n pproimtion technique b d EXAMPLE Approimting the Vlue of n Elliptic Integrl + 5 = Use the elliptic integrl in Emple 5 to pproimte the circumference of the ellipse 5 Figure C 8 units Solution Becuse e ou hve C 5 c b 95, 9 sin d 5 Appling Simpson s Rule with n produces C So, the ellipse hs circumference of bout 8 units, s shown in Figure Tr It Eplortion A

8 SECTION Conics nd Clculus 7 Focus d (, ) d d is constnt d d = Verte c d Focus Hperbols The definition of hperbol is similr to tht of n ellipse For n ellipse, the sum of the distnces between the foci nd point on the ellipse is fied, wheres for hperbol, the bsolute vlue of the difference between these distnces is fied A hperbol is the set of ll points, for which the bsolute vlue of the difference between the distnces from two distinct fied points clled foci is constnt (See Figure ) The line through the two foci intersects hperbol t two points clled the vertices The line segment connecting the vertices is the trnsverse is, nd the midpoint of the trnsverse is is the center of the hperbol One distinguishing feture of hperbol is tht its grph hs two seprte brnches Figure Center Trnsverse is Verte THEOREM 5 Stndrd Eqution of Hperbol The stndrd form of the eqution of hperbol with center t h, k is or h k k b h b Trnsverse is is horizontl Trnsverse is is verticl The vertices re units from the center, nd the foci re c units from the center, where, c b NOTE The constnts, b, nd c do not hve the sme reltionship for hperbols s the do for ellipses For hperbols, c b, but for ellipses, c b An importnt id in sketching the grph of hperbol is the determintion of its smptotes, s shown in Figure 5 Ech hperbol hs two smptotes tht intersect t the center of the hperbol The smptotes pss through the vertices of rectngle of dimensions b b, with its center t h, k The line segment of length b joining h, k b nd h, k b is referred to s the conjugte is of the hperbol THEOREM Asmptotes of Hperbol For horizontl trnsverse is, the equtions of the smptotes re Conjugte is (h, k) ( h, k) (h, k + b) b Asmptote (h +, k) k b h nd For verticl trnsverse is, the equtions of the smptotes re k h b nd k b h k h b Figure 5 (h, k b) Asmptote In Figure 5 ou cn see tht the smptotes coincide with the digonls of the rectngle with dimensions nd b, centered t h, k This provides ou with quick mens of sketching the smptotes, which in turn ids in sketching the hperbol

9 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE 7 Using Asmptotes to Sketch Hperbol Sketch the grph of the hperbol whose eqution is TECHNOLOGY You cn use grphing utilit to verif the grph obtined in Emple 7 b solving the originl eqution for nd grphing the following equtions Solution Begin b rewriting the eqution in stndrd form The trnsverse is is horizontl nd the vertices occur t, nd, The ends of the conjugte is occur t, nd, Using these four points, ou cn sketch the rectngle shown in Figure () B drwing the smptotes through the corners of this rectngle, ou cn complete the sketch s shown in Figure (b) (, ) (, ) (, ) = (, ) () Figure Editble Grph (b) Tr It Eplortion A Eplortion B Eplortion C Eplortion D Eplortion E Open Eplortion Definition of Eccentricit of Hperbol The eccentricit e of hperbol is given b the rtio e c As with n ellipse, the eccentricit of hperbol is e c Becuse c > for hperbols, it follows tht e > for hperbols If the eccentricit is lrge, the brnches of the hperbol re nerl flt If the eccentricit is close to, the brnches of the hperbol re more pointed, s shown in Figure 7 Eccentricit is lrge Eccentricit is close to Verte Focus Verte Focus Focus Verte Verte Focus e = c c e = c c Figure 7

10 SECTION Conics nd Clculus 7 The following ppliction ws developed during World Wr II It shows how the properties of hperbols cn be used in rdr nd other detection sstems EXAMPLE 8 A Hperbolic Detection Sstem Two microphones, mile prt, record n eplosion Microphone sound seconds before microphone B Where ws the eplosion? A receives the B d d c 58 d d Figure 8 A Solution Assuming tht sound trvels t feet per second, ou know tht the eplosion took plce feet frther from B thn from A, s shown in Figure 8 The locus of ll points tht re feet closer to A thn to B is one brnch of the hperbol b, where nd c mile ft Becuse c b, it follows tht b c 5,759, 58 ft feet feet nd ou cn conclude tht the eplosion occurred somewhere on the right brnch of the hperbol given b,, 5,759, Tr It Eplortion A CAROLINE HERSCHEL (75 88) The first womn to be credited with detecting new comet ws the English stronomer Croline Herschel During her life, Croline Herschel discovered totl of eight new comets MthBio In Emple 8, ou were ble to determine onl the hperbol on which the eplosion occurred, but not the ect loction of the eplosion If, however, ou hd received the sound t third position C, then two other hperbols would be determined The ect loction of the eplosion would be the point t which these three hperbols intersect Another interesting ppliction of conics involves the orbits of comets in our solr sstem Of the comets identified prior to 97, 5 hve ellipticl orbits, 95 hve prbolic orbits, nd 7 hve hperbolic orbits The center of the sun is focus of ech orbit, nd ech orbit hs verte t the point t which the comet is closest to the sun Undoubtedl, mn comets with prbolic or hperbolic orbits hve not been identified such comets pss through our solr sstem onl once Onl comets with ellipticl orbits such s Hlle s comet remin in our solr sstem The tpe of orbit for comet cn be determined s follows Ellipse: Prbol: Hperbol: v < GMp v GMp v > GMp In these three formuls, p is the distnce between one verte nd one focus of the comet s orbit (in meters), v is the velocit of the comet t the verte (in meters per second), M 989 kilogrms is the mss of the sun, nd G 7 8 cubic meters per kilogrm-second squred is the grvittionl constnt View the video for more informtion bout comet with n ellipticl orbit Video

11 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises 8, mtch the eqution with its grph [The grphs re lbeled (), (b), (c), (d), (e), (f), (g), nd (h)] () (c) (e) (b) (d) (f) In Eercises 7, find the verte, focus, nd directri of the prbol Then use grphing utilit to grph the prbol In Eercises 8, find n eqution of the prbol Verte:, Verte:, Focus:, Focus:, Verte:, Focus:, Directri: 5 (, ) (, ) (, ) Directri: (, ) 7 Ais is prllel to -is; grph psses through,,, nd, 8 Directri: ; endpoints of ltus rectum re, nd 8, (, ) (, ) (g) (h) In Eercises 9, find the center, foci, vertices, nd eccentricit of the ellipse, nd sketch its grph In Eercises 5 8, find the center, foci, nd vertices of the ellipse Use grphing utilit to grph the ellipse In Eercises 9, find the verte, focus, nd directri of the prbol, nd sketch its grph In Eercises 9, find n eqution of the ellipse 9 Center:, Vertices:,,, Focus:, Verte:, Eccentricit: Vertices:,,, 9 Foci:, ±5 Minor is length: Mjor is length:

12 SECTION Conics nd Clculus 75 Center:, Center:, Mjor is: horizontl Points on the ellipse:,,, Mjor is: verticl Points on the ellipse:,,, In Eercises 5 5, find the center, foci, nd vertices of the hperbol, nd sketch its grph using smptotes s n id In Eercises 5 5, find the center, foci, nd vertices of the hperbol Use grphing utilit to grph the hperbol nd its smptotes In Eercises 57, find n eqution of the hperbol 57 Vertices: ±, 58 Vertices:, ± Asmptotes: ± Asmptotes: ± 59 Vertices:, ± Vertices:, ± Point on grph:, 5 Foci:, ±5 Center:, Center:, Verte:, Focus:, Verte:, Focus: 5, Vertices:,,, Focus:, Asmptotes: Asmptotes: ± In Eercises 5 nd, find equtions for () the tngent lines nd (b) the norml lines to the hperbol for the given vlue of 5 9, In Eercises 7 7, clssif the grph of the eqution s circle, prbol, n ellipse, or hperbol , 5 Writing About Concepts 77 () Give the definition of prbol (b) Give the stndrd forms of prbol with verte t h, k (c) In our own words, stte the reflective propert of prbol 78 () Give the definition of n ellipse (b) Give the stndrd forms of n ellipse with center t h, k 79 () Give the definition of hperbol (b) Give the stndrd forms of hperbol with center t h, k (c) Write equtions for the smptotes of hperbol 8 Define the eccentricit of n ellipse In our own words, describe how chnges in the eccentricit ffect the ellipse 8 Solr Collector A solr collector for heting wter is constructed with sheet of stinless steel tht is formed into the shpe of prbol (see figure) The wter will flow through pipe tht is locted t the focus of the prbol A wht distnce from the verte is the pipe? m m Figure for 8 Figure for 8 m cm Not drwn to scle 8 Bem Deflection A simpl supported bem tht is meters long hs lod concentrted t the center (see figure) The deflection of the bem t its center is centimeters Assume tht the shpe of the deflected bem is prbolic () Find n eqution of the prbol (Assume tht the origin is t the center of the bem) (b) How fr from the center of the bem is the deflection centimeter? 8 Find n eqution of the tngent line to the prbol Prove tht the -intercept of this tngent line is,

13 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes 8 () Prove tht n two distinct tngent lines to prbol intersect (b) Demonstrte the result of prt () b finding the point of intersection of the tngent lines to the prbol t the points, nd, 85 () Prove tht if n two tngent lines to prbol intersect t right ngles, their point of intersection must lie on the directri (b) Demonstrte the result of prt () b proving tht the tngent lines to the prbol 8 t the points, 5 nd, 5 intersect t right ngles, nd tht the point of intersection lies on the directri 8 Find the point on the grph of 8 tht is closest to the focus of the prbol 87 Rdio nd Television Reception In mountinous res, reception of rdio nd television is sometimes poor Consider n idelized cse where hill is represented b the grph of the prbol, trnsmitter is locted t the point,, nd receiver is locted on the other side of the hill t the point, Wht is the closest the receiver cn be to the hill so tht the reception is unobstructed? 88 Modeling Dt The tble shows the verge mounts of time A (in minutes) women spent wtching television ech d for the ers 99 to (Source: Nielsen Medi Reserch) Yer A () Use the regression cpbilities of grphing utilit to find model of the form A t bt c for the dt Let t represent the er, with t corresponding to 99 (b) Use grphing utilit to plot the dt nd grph the model (c) Find dadt nd sketch its grph for t Wht informtion bout the verge mount of time women spent wtching television is given b the grph of the derivtive? 89 Architecture A church window is bounded bove b prbol nd below b the rc of circle (see figure) Find the surfce re of the window 8 ft 8 ft Circle rdius ft Prbolic supporting cble (, ) 9 Bridge Design A cble of suspension bridge is suspended (in the shpe of prbol) between two towers tht re meters prt nd meters bove the rodw (see figure) The cbles touch the rodw midw between the towers () Find n eqution for the prbolic shpe of ech cble (b) Find the length of the prbolic supporting cble 9 Surfce Are A stellite-signl receiving dish is formed b revolving the prbol given b bout the -is The rdius of the dish is r feet Verif tht the surfce re of the dish is given b r d 9 Investigtion Sketch the grphs of p for p,, nd on the sme coordinte es Discuss the chnge in the grphs s p increses, 9 Are Find formul for the re of the shded region in the figure h = p Figure for 9 Figure for 9 95 Writing On pge 97, it ws noted tht n ellipse cn be drwn using two thumbtcks, string of fied length (greter thn the distnce between the tcks), nd pencil If the ends of the string re fstened t the tcks nd the string is drwn tu with pencil, the pth trced b the pencil will be n ellipse () Wht is the length of the string in terms of? (b) Eplin wh the pth is n ellipse 5 r 9 Construction of Semiellipticl Arch A fireplce rch is to be constructed in the shpe of semiellipse The opening is to hve height of feet t the center nd width of 5 feet long the bse (see figure) The contrctor drws the outline of the ellipse b the method shown in Eercise 95 Where should the tcks be plced nd wht should be the length of the piece of string? 97 Sketch the ellipse tht consists of ll points, such tht the sum of the distnces between, nd two fied points is units, nd the foci re locted t the centers of the two sets of concentric circles in the figure To print n enlrged cop of the grph, select the MthGrph button Figure for 89 Figure for 9 9 Arc Length Find the rc length of the prbol over the intervl

14 SECTION Conics nd Clculus Orbit of Erth Erth moves in n ellipticl orbit with the sun t one of the foci The length of hlf of the mjor is is 9,598, kilometers, nd the eccentricit is 7 Find the minimum distnce (perihelion) nd the mimum distnce (phelion) of Erth from the sun 99 Stellite Orbit The pogee (the point in orbit frthest from Erth) nd the perigee (the point in orbit closest to Erth) of n ellipticl orbit of n Erth stellite re given b A nd P Show tht the eccentricit of the orbit is Eplorer 8 On November 7, 9, the United Sttes lunched Eplorer 8 Its low nd high points bove the surfce of Erth were 9 miles nd, miles Find the eccentricit of its ellipticl orbit Hlle s Comet Probbl the most fmous of ll comets, Hlle s comet, hs n ellipticl orbit with the sun t the focus Its mimum distnce from the sun is pproimtel 59 AU (stronomicl unit 995 miles), nd its minimum distnce is pproimtel 59 AU Find the eccentricit of the orbit The eqution of n ellipse with its center t the origin cn be written s Show tht s e, with remining fied, the ellipse pproches circle Consider prticle trveling clockwise on the ellipticl pth The prticle leves the orbit t the point 8, nd trvels in stright line tngent to the ellipse At wht point will the prticle cross the -is? Volume The wter tnk on fire truck is feet long, nd its cross sections re ellipses Find the volume of wter in the prtill filled tnk s shown in the figure In Eercises 5 nd, determine the points t which d/d is zero or does not eist to locte the endpoints of the mjor nd minor es of the ellipse 5 e A P A P e 5 9 ft ft ft Are nd Volume In Eercises 7 nd 8, find () the re of the region bounded b the ellipse, (b) the volume nd surfce re of the solid generted b revolving the region bout its mjor is (prolte spheroid), nd (c) the volume nd surfce re of the solid generted b revolving the region bout its minor is (oblte spheroid) Arc Length Use the integrtion cpbilities of grphing utilit to pproimte to two-deciml-plce ccurc the ellipticl integrl representing the circumference of the ellipse 5 9 Prove tht the tngent line to n ellipse t point P mkes equl ngles with lines through P nd the foci (see figure) [Hint: () Find the slope of the tngent line t P, () find the slopes of the lines through P nd ech focus, nd () use the formul for the tngent of the ngle between two lines] + b = Figure for Figure for Geometr The re of the ellipse in the figure is twice the re of the circle Wht is the length of the mjor is? Conjecture Tngent line (c, ) (c, ) () Show tht the eqution of n ellipse cn be written s h k e (b) Use grphing utilit to grph the ellipse for e 95, P = (, ) e (, ) e 75, e 5, e 5, (, ) (, ) (, ) nd e (c) Use the results of prt (b) to mke conjecture bout the chnge in the shpe of the ellipse s e pproches Find n eqution of the hperbol such tht for n point on the hperbol, the difference between its distnces from the points, nd, is Find n eqution of the hperbol such tht for n point on the hperbol, the difference between its distnces from the points, nd, is

15 78 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes 5 Sketch the hperbol tht consists of ll points, such tht the difference of the distnces between, nd two fied points is units, nd the foci re locted t the centers of the two sets of concentric circles in the figure To print n enlrged cop of the grph, select the MthGrph button Consider hperbol centered t the origin with horizontl trnsverse is Use the definition of hperbol to derive its stndrd form: Sound Loction A rifle positioned t point c, is fired t trget positioned t point c, A person hers the sound of the rifle nd the sound of the bullet hitting the trget t the sme time Prove tht the person is positioned on one brnch of the hperbol given b where v m is the muzzle velocit of the rifle nd v s is the speed of sound, which is bout feet per second 8 Nvigtion LORAN (long distnce rdio nvigtion) for ircrft nd ships uses snchronized pulses trnsmitted b widel seprted trnsmitting sttions These pulses trvel t the speed of light (8, miles per second) The difference in the times of rrivl of these pulses t n ircrft or ship is constnt on hperbol hving the trnsmitting sttions s foci Assume tht two sttions, miles prt, re positioned on the rectngulr coordinte sstem t 5, nd 5, nd tht ship is trveling on pth with coordintes, 75 (see figure) Find the -coordinte of the position of the ship if the time difference between the pulses from the trnsmitting sttions is microseconds ( second) 5 b c vs vm c vm vs vm Figure for 8 Figure for 9 Mirror 8 9 Hperbolic Mirror A hperbolic mirror (used in some telescopes) hs the propert tht light r directed t the focus will be reflected to the other focus The mirror in the figure hs the eqution At which point on the mirror will light from the point, be reflected to the other focus? Show tht the eqution of the tngent line to b t the point is b, Show tht the grphs of the equtions intersect t right ngles b nd Prove tht the grph of the eqution A C D E F is one of the following (ecept in degenerte cses) Conic () Circle (b) Prbol A or C (but not both) (c) Ellipse (d) Hperbol True or Flse? In Eercises 8, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse It is possible for prbol to intersect its directri The point on prbol closest to its focus is its verte 5 If C is the circumference of the ellipse, b b < then b C b b Condition A C AC > AC < If D or E, then the grph of D E is hperbol 7 If the smptotes of the hperbol b intersect t right ngles, then b 8 Ever tngent line to hperbol intersects the hperbol onl t the point of tngenc Putnm Em Chllenge 9 For point P on n ellipse, let d be the distnce from the center of the ellipse to the line tngent to the ellipse t P Prove tht PF PF d is constnt s P vries on the ellipse where PF nd PF re the distnces from P to the foci F nd F of the ellipse Find the minimum vlue of for < u < nd v > u v u 9 v These problems were composed b the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved

16 SECTION Plne Curves nd Prmetric Equtions 79 Section Plne Curves nd Prmetric Equtions Sketch the grph of curve given b set of prmetric equtions Eliminte the prmeter in set of prmetric equtions Find set of prmetric equtions to represent curve Understnd two clssic clculus problems, the tutochrone nd brchistochrone problems 8 9 (, ) t = 9 Rectngulr eqution: = 7 +, 8 7 t = 5 5 Prmetric equtions: = t = t + t 7 Curviliner motion: two vribles for position, one vrible for time Figure 9 Plne Curves nd Prmetric Equtions Until now, ou hve been representing grph b single eqution involving two vribles In this section ou will stud situtions in which three vribles re used to represent curve in the plne Consider the pth followed b n object tht is propelled into the ir t n ngle of 5 If the initil velocit of the object is 8 feet per second, the object trvels the prbolic pth given b Rectngulr eqution s shown in Figure 9 However, this eqution does not tell the whole stor Although it does tell ou where the object hs been, it doesn t tell ou when the object ws t given point, To determine this time, ou cn introduce third vrible t, clled prmeter B writing both nd s functions of t, ou obtin the prmetric equtions nd 7 t t t Prmetric eqution for Prmetric eqution for From this set of equtions, ou cn determine tht t time t, the object is t the point (, ) Similrl, t time t, the object is t the point,, nd so on (You will lern method for determining this prticulr set of prmetric equtions the equtions of motion lter, in Section ) For this prticulr motion problem, nd re continuous functions of t, nd the resulting pth is clled plne curve Animtion Definition of Plne Curve If f nd g re continuous functions of t on n intervl I, then the equtions f t nd gt re clled prmetric equtions nd t is clled the prmeter The set of points, obtined s t vries over the intervl I is clled the grph of the prmetric equtions Tken together, the prmetric equtions nd the grph re clled plne curve, denoted b C NOTE At times it is importnt to distinguish between grph (the set of points) nd curve (the points together with their defining prmetric equtions) When it is importnt, we will mke the distinction eplicit When it is not importnt, we will use C to represent the grph or the curve

17 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes When sketching (b hnd) curve represented b set of prmetric equtions, ou cn plot points in the -plne Ech set of coordintes, is determined from vlue chosen for the prmeter t B plotting the resulting points in order of incresing vlues of t, the curve is trced out in specific direction This is clled the orienttion of the curve EXAMPLE Sketching Curve Sketch the curve described b the prmetric equtions t nd t, t Solution For vlues of t on the given intervl, the prmetric equtions ield the points, shown in the tble t = t = t = t = t = t = t 5 Prmetric equtions: = t nd = t, t Figure B plotting these points in order of incresing t nd using the continuit of f nd g, ou obtin the curve C shown in Figure Note tht the rrows on the curve indicte its orienttion s t increses from to Editble Grph Tr It Eplortion A Eplortion B t = t = t = t = t = t = NOTE From the Verticl Line Test, ou cn see tht the grph shown in Figure does not define s function of This points out one benefit of prmetric equtions the cn be used to represent grphs tht re more generl thn grphs of functions It often hppens tht two different sets of prmetric equtions hve the sme grph For emple, the set of prmetric equtions t nd t, t Figure Prmetric equtions: = t nd = t, t hs the sme grph s the set given in Emple However, compring the vlues of t in Figures nd, ou cn see tht the second grph is trced out more rpidl (considering t s time) thn the first grph So, in pplictions, different prmetric representtions cn be used to represent vrious speeds t which objects trvel long given pth TECHNOLOGY Most grphing utilities hve prmetric grphing mode If ou hve ccess to such utilit, use it to confirm the grphs shown in Figures nd Does the curve given b t 8t nd t, t represent the sme grph s tht shown in Figures nd? Wht do ou notice bout the orienttion of this curve?

18 SECTION Plne Curves nd Prmetric Equtions 7 Eliminting the Prmeter Finding rectngulr eqution tht represents the grph of set of prmetric equtions is clled eliminting the prmeter For instnce, ou cn eliminte the prmeter from the set of prmetric equtions in Emple s follows Prmetric equtions Solve for t in one eqution Substitute into second eqution Rectngulr eqution t t t Once ou hve eliminted the prmeter, ou cn recognize tht the eqution represents prbol with horizontl is nd verte t,, s shown in Figure The rnge of nd implied b the prmetric equtions m be ltered b the chnge to rectngulr form In such instnces the domin of the rectngulr eqution must be djusted so tht its grph mtches the grph of the prmetric equtions Such sitution is demonstrted in the net emple EXAMPLE Adjusting the Domin After Eliminting the Prmeter Sketch the curve represented b the equtions t = t = t nd t t, t > b eliminting the prmeter nd djusting the domin of the resulting rectngulr eqution =, > Figure t = 75 Prmetric equtions: =, = t, t > t + t + Rectngulr eqution: Solution Begin b solving one of the prmetric equtions for t For instnce, ou cn solve the first eqution for t s follows t Prmetric eqution for Squre ech side Solve for t Now, substituting into the prmetric eqution for produces t t t t t Prmetric eqution for Substitute for t Simplif The rectngulr eqution,, is defined for ll vlues of, but from the prmetric eqution for ou cn see tht the curve is defined onl when t > This implies tht ou should restrict the domin of to positive vlues, s shown in Figure Editble Grph Tr It Eplortion A

19 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes It is not necessr for the prmeter in set of prmetric equtions to represent time The net emple uses n ngle s the prmeter EXAMPLE Using Trigonometr to Eliminte Prmeter Sketch the curve represented b θ = cos nd sin, b eliminting the prmeter nd finding the corresponding rectngulr eqution Solution Begin b solving for cos nd sin in the given equtions θ = Prmetric equtions: = cos θ, = sin θ Rectngulr eqution: + = 9 Figure θ = θ = cos nd Solve for cos nd sin Net, mke use of the identit sin cos to form n eqution involving onl nd cos sin 9 sin Trigonometric identit Substitute Rectngulr eqution From this rectngulr eqution ou cn see tht the grph is n ellipse centered t,, with vertices t, nd, nd minor is of length b, s shown in Figure Note tht the ellipse is trced out counterclockwise s vries from to Editble Grph Tr It Eplortion A Open Eplortion Using the technique shown in Emple, ou cn conclude tht the grph of the prmetric equtions h cos nd is the ellipse (trced counterclockwise) given b h The grph of the prmetric equtions h sin nd is lso the ellipse (trced clockwise) given b h k b k b k b sin, k b cos, Use grphing utilit in prmetric mode to grph severl ellipses In Emples nd, it is importnt to relize tht eliminting the prmeter is primril n id to curve sketching If the prmetric equtions represent the pth of moving object, the grph lone is not sufficient to describe the object s motion You still need the prmetric equtions to tell ou the position, direction, nd speed t given time

20 SECTION Plne Curves nd Prmetric Equtions 7 Finding Prmetric Equtions The first three emples in this section illustrte techniques for sketching the grph represented b set of prmetric equtions You will now investigte the reverse problem How cn ou determine set of prmetric equtions for given grph or given phsicl description? From the discussion following Emple, ou know tht such representtion is not unique This is demonstrted further in the following emple, which finds two different prmetric representtions for given grph EXAMPLE Finding Prmetric Equtions for Given Grph Find set of prmetric equtions to represent the grph of, using ech of the following prmeters t b The slope m d t the point, d Solution Letting t produces the prmetric equtions t nd t b To write nd in terms of the prmeter m, ou cn proceed s follows m = m = m = m d d m Differentite Solve for This produces prmetric eqution for To obtin prmetric eqution for, substitute m for in the originl eqution Write originl rectngulr eqution m Substitute m for m = m = m So, the prmetric equtions re Simplif Rectngulr eqution: = Prmetric equtions: m m = =, Figure m nd m In Figure, note tht the resulting curve hs right-to-left orienttion s determined b the direction of incresing vlues of slope m For prt (), the curve would hve the opposite orienttion Editble Grph Tr It Eplortion A Eplortion B TECHNOLOGY To be efficient t using grphing utilit, it is importnt tht ou develop skill in representing grph b set of prmetric equtions The reson for this is tht mn grphing utilities hve onl three grphing modes () functions, () prmetric equtions, nd () polr equtions Most grphing utilities re not progrmmed to grph generl eqution For instnce, suppose ou wnt to grph the hperbol To grph the hperbol in function mode, ou need two equtions: nd In prmetric mode, ou cn represent the grph b sec t nd tn t

21 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes CYCLOIDS Glileo first clled ttention to the ccloid, once recommending tht it be used for the rches of bridges Pscl once spent 8 ds ttempting to solve mn of the problems of ccloids, such s finding the re under one rch, nd the volume of the solid of revolution formed b revolving the curve bout line The ccloid hs so mn interesting properties nd hs cused so mn qurrels mong mthemticins tht it hs been clled the Helen of geometr nd the pple of discord FOR FURTHER INFORMATION For more informtion on ccloids, see the rticle The Geometr of Rolling Curves b John Bloom nd Lee Whitt in The Americn Mthemticl Monthl MthArticle EXAMPLE 5 Prmetric Equtions for Ccloid Determine the curve trced b point P on the circumference of circle of rdius rolling long stright line in plne Such curve is clled ccloid View the nimtion to see how ccloid is drwn Animtion Solution Let the prmeter be the mesure of the circle s rottion, nd let the point P, begin t the origin When P is t the origin When P is t mimum point, When is bck on the -is t, From Figure 5, ou cn see tht APC 8 So, sin sin8 sinapc AC BD cos cos8 which implies tht AP cos nd Becuse the circle rolls long the -is, ou know tht OD PD Furthermore, becuse BA DC, ou hve OD BD sin BA AP cos So, the prmetric equtions re sin nd,, P cosapc AP BD sin cos, P = (, ) Ccloid: = ( θ sin θ) = ( cos θ) (, ) (, ) O A B Figure 5 θ D C (, ) (, ) Tr It Eplortion A TECHNOLOGY Some grphing utilities llow ou to simulte the motion of n object tht is moving in the plne or in spce If ou hve ccess to such utilit, use it to trce out the pth of the ccloid shown in Figure 5 The ccloid in Figure 5 hs shrp corners t the vlues n Notice tht the derivtives nd re both zero t the points for which sin cos cos sin n n Between these points, the ccloid is clled smooth n Definition of Smooth Curve A curve C represented b f t nd gt on n intervl I is clled smooth if nd re continuous on I nd not simultneousl, ecept possibl t the endpoints of I The curve C is clled piecewise smooth if it is smooth on ech subintervl of some prtition of I f g

22 SECTION Plne Curves nd Prmetric Equtions 75 A C The time required to complete full swing of the pendulum when strting from point C is onl pproimtel the sme s when strting from point A Figure B The Tutochrone nd Brchistochrone Problems The tpe of curve described in Emple 5 is relted to one of the most fmous pirs of problems in the histor of clculus The first problem (clled the tutochrone problem) begn with Glileo s discover tht the time required to complete full swing of given pendulum is pproimtel the sme whether it mkes lrge movement t high speed or smll movement t lower speed (see Figure ) Lte in his life, Glileo (5 ) relized tht he could use this principle to construct clock However, he ws not ble to conquer the mechnics of ctul construction Christin Hugens (9 95) ws the first to design nd construct working model In his work with pendulums, Hugens relized tht pendulum does not tke ectl the sme time to complete swings of vring lengths (This doesn t ffect pendulum clock, becuse the length of the circulr rc is kept constnt b giving the pendulum slight boost ech time it psses its lowest point) But, in studing the problem, Hugens discovered tht bll rolling bck nd forth on n inverted ccloid does complete ech ccle in ectl the sme time A JAMES BERNOULLI (5 75) Jmes Bernoulli, lso clled Jcques, ws the older brother of John He ws one of severl ccomplished mthemticins of the Swiss Bernoulli fmil Jmes s mthemticl ccomplishments hve given him prominent plce in the erl development of clculus MthBio B An inverted ccloid is the pth down which bll will roll in the shortest time Figure 7 The second problem, which ws posed b John Bernoulli in 9, is clled the brchistochrone problem in Greek, brchs mens short nd chronos mens time The problem ws to determine the pth down which prticle will slide from point A to point B in the shortest time Severl mthemticins took up the chllenge, nd the following er the problem ws solved b Newton, Leibniz, L Hôpitl, John Bernoulli, nd Jmes Bernoulli As it turns out, the solution is not stright line from A to B, but n inverted ccloid pssing through the points A nd B, s shown in Figure 7 The mzing prt of the solution is tht prticle strting t rest t n other point C of the ccloid between A nd B will tke ectl the sme time to rech B, s shown in Figure 8 A C B A bll strting t point C tkes the sme time to rech point B s one tht strts t point A Figure 8 FOR FURTHER INFORMATION To see proof of the fmous brchistochrone problem, see the rticle A New Minimiztion Proof for the Brchistochrone b Gr Lwlor in The Americn Mthemticl Monthl MthArticle

23 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph Consider the prmetric equtions t nd t () Complete the tble (b) Plot the points, generted in the tble, nd sketch grph of the prmetric equtions Indicte the orienttion of the grph (c) Use grphing utilit to confirm our grph in prt (b) (d) Find the rectngulr eqution b eliminting the prmeter, nd sketch its grph Compre the grph in prt (b) with the grph of the rectngulr eqution Consider the prmetric equtions cos nd sin () Complete the tble (b) Plot the points, generted in the tble, nd sketch grph of the prmetric equtions Indicte the orienttion of the grph (c) Use grphing utilit to confirm our grph in prt (b) (d) Find the rectngulr eqution b eliminting the prmeter, nd sketch its grph Compre the grph in prt (b) with the grph of the rectngulr eqution (e) If vlues of were selected from the intervl, for the tble in prt (), would the grph in prt (b) be different? Eplin In Eercises, sketch the curve represented b the prmetric equtions (indicte the orienttion of the curve), nd write the corresponding rectngulr eqution b eliminting the prmeter t, t t, t 5 t, t t, t t t, t t, 7 8 t sec, cos, 7 t, t 8 t t, 9 t, t t, t tn, sec 5 e t, e t e t, e t <, < t t t, t t t, t 9 cos, sin cos, sin In Eercises, use grphing utilit to grph the curve represented b the prmetric equtions (indicte the orienttion of the curve) Eliminte the prmeter nd write the corresponding rectngulr eqution sin, cos cos cos sin 5 cos sec sin 7 sec, tn 8 cos, sin 9 t, ln t ln t, t e t, e t e t, e t Compring Plne Curves In Eercises, determine n differences between the curves of the prmetric equtions Are the grphs the sme? Are the orienttions the sme? Are the curves smooth? () t (b) cos (c) t cos e t (d) e t () cos (b) (c) e t sin t t sin < (d) 5 () cos (b) cos () t, t (b) 7 Conjecture < () Use grphing utilit to grph the curves represented b the two sets of prmetric equtions cos t sin t e t < cost sint cos, sin sin tn e t t t t e t sin t, t (b) Describe the chnge in the grph when the sign of the prmeter is chnged (c) Mke conjecture bout the chnge in the grph of prmetric equtions when the sign of the prmeter is chnged (d) Test our conjecture with nother set of prmetric equtions 8 Writing Review Eercises nd write short prgrph describing how the grphs of curves represented b differen sets of prmetric equtions cn differ even though eliminting the prmeter from ech ields the sme rectngulr eqution <

24 SECTION Plne Curves nd Prmetric Equtions 77 In Eercises 9, eliminte the prmeter nd obtin the stndrd form of the rectngulr eqution 9 Line through, nd, : t, Circle: Ellipse: Hperbol: In Eercises 5, use the results of Eercises 9 to find set of prmetric equtions for the line or conic Line: psses through (, ) nd 5, Line: psses through (, ) nd 5, 5 Circle: center: (, ); rdius: Circle: center:, ; rdius: 7 Ellipse: vertices: ±5, ; foci: ±, 8 Ellipse: vertices: (, 7),, ; foci: (, 5),, 9 Hperbol: vertices: ±, ; foci: ±5, 5 Hperbol: vertices:, ±; foci:, ± In Eercises 5 5, find two different sets of prmetric equtions for the rectngulr eqution 5 5 In Eercises 55, use grphing utilit to grph the curve represented b the prmetric equtions Indicte the direction of the curve Identif n points t which the curve is not smooth 55 Ccloid: sin, cos 5 Ccloid: h cos, Prolte ccloid: h sec, t h r cos, k r sin sin, cos sin, cos 58 Prolte ccloid: sin, cos 59 Hpoccloid: cos, sin Curtte ccloid: sin, cos Witch of Agnesi: cot, sin Folium of Descrtes: t t t, t Writing About Concepts k b sin k b tn Stte the definition of plne curve given b prmetric equtions Eplin the process of sketching plne curve given b prmetric equtions Wht is ment b the orienttion of the curve? 5 Stte the definition of smooth curve 7 Curtte Ccloid A wheel of rdius rolls long line without slipping The curve trced b point P tht is b units from the center b < is clled curtte ccloid (see figure) Use the ngle to find set of prmetric equtions for this curve Writing About Concepts (continued) Mtch ech set of prmetric equtions with the correct grph [The grphs re lbeled (), (b), (c), (d), (e), nd (f)] Eplin our resoning P () (c) (e) (i) (ii) b (, b) t, sin, (, + b) t Figure for 7 Figure for 8 (b) (d) (f) (iii) Lissjous curve: cos, sin (iv) Evolute of ellipse: cos, sin (v) Involute of circle: cos sin cos sin (vi) Serpentine curve: cot, sin cos sin, (, )

25 78 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes 8 Epiccloid A circle of rdius rolls round the outside of circle of rdius without slipping The curve trced b point on the circumference of the smller circle is clled n epiccloid (see figure on previous pge) Use the ngle to find set of prmetric equtions for this curve True or Flse? In Eercises 9 nd 7, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse 9 The grph of the prmetric equtions t nd t is the line 7 If is function of t nd is function of t, then is function of Projectile Motion In Eercises 7 nd 7, consider projectile lunched t height h feet bove the ground nd t n ngle with the horizontl If the initil velocit is v feet per second, the pth of the projectile is modeled b the prmetric equtions v nd h v sin t t cos t 7 The center field fence in bllprk is feet high nd feet from home plte The bll is hit feet bove the ground It leves the bt t n ngle of degrees with the horizontl t speed of miles per hour (see figure) ft ft ft () Write set of prmetric equtions for the pth of the bll (b) Use grphing utilit to grph the pth of the bll when Is the hit home run? 5 (c) Use grphing utilit to grph the pth of the bll when Is the hit home run? (d) Find the minimum ngle t which the bll must leve the bt in order for the hit to be home run 7 A rectngulr eqution for the pth of projectile is 5 5 () Eliminte the prmeter t from the position function for the motion of projectile to show tht the rectngulr eqution is sec v tn h (b) Use the result of prt () to find h, v, nd Find the prmetric equtions of the pth (c) Use grphing utilit to grph the rectngulr eqution for the pth of the projectile Confirm our nswer in prt (b) b sketching the curve represented b the prmetric equtions (d) Use grphing utilit to pproimte the mimum height of the projectile nd its rnge

26 SECTION Prmetric Equtions nd Clculus 79 Section = t = t + t 5 At time t, the ngle of elevtion of the projectile is, the slope of the tngent line t tht point Figure 9 θ Prmetric Equtions nd Clculus Find the slope of tngent line to curve given b set of prmetric equtions Find the rc length of curve given b set of prmetric equtions Find the re of surfce of revolution (prmetric form) Slope nd Tngent Lines Now tht ou cn represent grph in the plne b set of prmetric equtions, it is nturl to sk how to use clculus to stud plne curves To begin, let s tke nother look t the projectile represented b the prmetric equtions t nd t t s shown in Figure 9 From Section, ou know tht these equtions enble ou to locte the position of the projectile t given time You lso know tht the object is initill projected t n ngle of 5 But how cn ou find the ngle representing the object s direction t some other time t? The following theorem nswers this question b giving formul for the slope of the tngent line s function of t THEOREM 7 Prmetric Form of the Derivtive If smooth curve C is given b the equtions f t nd gt, then the slope of C t, is d ddt d ddt, d dt ( f(t + t), g(t + t)) ( f(t), g(t)) The slope of the secnt line through the points ft, gt nd ft t, gt t is Figure Proof In Figure, consider t > nd let gt t gt nd Becuse s t, ou cn write d lim d gt t gt lim t f t t f t Dividing both the numertor nd denomintor b t, ou cn use the differentibilit of f nd g to conclude tht d gt t gtt lim d t f t t f tt gt t gt lim t t lim t gt ft ddt ddt f t t f t t f t t f t

27 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Differentition nd Prmetric Form Find dd for the curve given b sin t nd cos t STUDY TIP The curve trced out in Emple is circle Use the formul d d tn t to find the slopes t the points, nd, Solution d ddt sin t tn t d ddt cos t Becuse dd is function of t, ou cn use Theorem 7 repetedl to find higher-order derivtives For instnce, d d d d d d d d d d d d d dt d d ddt d dt d d ddt Second derivtive Third derivtive Tr It Eplortion A Eplortion B The editble grph feture below llows ou to edit the grph of function Editble Grph EXAMPLE Finding Slope nd Concvit For the curve given b t nd t t, find the slope nd concvit t the point, = t = (t ) (, ) t = m = 8 The grph is concve upwrd t,, when t Figure Solution Becuse d ddt d ddt t t t ou cn find the second derivtive to be d d d dt dd ddt At,,, it follows tht t, nd the slope is d d 8 Moreover, when t, the second derivtive is d > d d dt t ddt t t t nd ou cn conclude tht the grph is concve upwrd t,, Figure Prmetric form of first derivtive Prmetric form of second derivtive s shown in Editble Grph Tr It Eplortion A Eplortion B Becuse the prmetric equtions f t nd gt need not define s function of, it is possible for plne curve to loop round nd cross itself At such points the curve m hve more thn one tngent line, s shown in the net emple

28 SECTION Prmetric Equtions nd Clculus 7 EXAMPLE A Curve with Two Tngent Lines t Point = t sin t = cos t Tngent line (t = /) The prolte ccloid given b t sin t nd cos t crosses itself t the point,, s shown in Figure Find the equtions of both tngent lines t this point (, ) This prolte ccloid hs two tngent lines t the point, Figure Tngent line (t = /) Solution Becuse nd when t ±, nd d ddt d ddt ou hve dd when t nd dd when t So, the two tngent lines t, re sin t cos t Tngent line when t Tngent line when t Editble Grph Tr It Eplortion A Eplortion B Open Eplortion If ddt nd ddt when t t, the curve represented b f t nd gt hs horizontl tngent t f t, gt For instnce, in Emple, the given curve hs horizontl tngent t the point, when t Similrl, if ddt nd ddt when t t, the curve represented b f t nd gt hs verticl tngent t f t, gt Arc Length You hve seen how prmetric equtions cn be used to describe the pth of prticle moving in the plne You will now develop formul for determining the distnce trveled b the prticle long its pth Recll from Section 7 tht the formul for the rc length of curve C given b h over the intervl, is s h d d If C is represented b the prmetric equtions f t nd gt, t b, nd if ddt ft >, ou cn write s d d d ddt ddt d d d b b b d dt d dt dt ddt d ddt ddt dt dt ft gt dt

29 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes NOTE When ppling the rc length formul to curve, be sure tht the curve is trced out onl once on the intervl of integrtion For instnce, the circle given b cos t nd sin t is trced out once on the intervl t, but is trced out twice on the intervl t ARCH OF A CYCLOID The rc length of n rch of ccloid ws first clculted in 58 b British rchitect nd mthemticin Christopher Wren, fmous for rebuilding mn buildings nd churches in London, including St Pul s Cthedrl An epiccloid is trced b point on the smller circle s it rolls round the lrger circle Figure = 5 cos t cos 5t = 5 sin t sin 5t t in cr e ses In the preceding section ou sw tht if circle rolls long line, point on its circumference will trce pth clled ccloid If the circle rolls round the circumference of nother circle, the pth of the point is n epiccloid The net emple shows how to find the rc length of n epiccloid EXAMPLE Finding Arc Length A circle of rdius rolls round the circumference of lrger circle of rdius, s shown in Figure The epiccloid trced b point on the circumference of the smller circle is given b nd THEOREM 8 b s d dt d 5 cos t cos 5t 5 sin t sin 5t Find the distnce trveled b the point in one complete trip bout the lrger circle Solution Before ppling Theorem 8, note in Figure tht the curve hs shrp points when t nd t Between these two points, ddt nd ddt re not simultneousl So, the portion of the curve generted from t to t is smooth To find the totl distnce trveled b the point, ou cn find the rc length of tht portion ling in the first qudrnt nd multipl b s d Prmetric form for rc length dt d dt dt 5 sin t 5 sin 5t 5 cos t 5 cos 5t dt sin t sin 5t cos t cos 5t dt cos t dt sin t dt Trigonometric identit sin t dt cos t Arc Length in Prmetric Form If smooth curve C is given b f t nd gt such tht C does not intersect itself on the intervl t b (ecept possibl t the endpoints), then the rc length of C over the intervl is given b dt b dt ft gt dt For the epiccloid shown in Figure, n rc length of seems bout right becuse the circumference of circle of rdius is r 77 Editble Grph Tr It Eplortion A

30 SECTION Prmetric Equtions nd Clculus 7 5 in EXAMPLE 5 Length of Recording Tpe A recording tpe inch thick is wound round reel whose inner rdius is 5 inch nd whose outer rdius is inches, s shown in Figure How much tpe is required to fill the reel? Figure in in = r cos θ = r sin θ (, ) r θ Solution To crete model for this problem, ssume tht s the tpe is wound round the reel its distnce r from the center increses linerl t rte of inch per revolution, or where is mesured in rdins You cn determine the coordintes of the point, corresponding to given rdius to be nd r r cos r sin Substituting for r, ou obtin the prmetric equtions cos nd You cn use the rc length formul to determine the totl length of the tpe to be s d ln,78 inches 98 feet Tr It d d d d sin cos cos sin d d, Eplortion A sin Integrtion tbles (Appendi B), Formul NOTE The grph of r is clled the spirl of Archimedes The grph of r (in Emple 5) is of this form FOR FURTHER INFORMATION For more informtion on the mthemtics of recording tpe, see Tpe Counters b Richrd L Roth in The Americn Mthemticl Monthl MthArticle The length of the tpe in Emple 5 cn be pproimted b dding the circumferences of circulr pieces of tpe The smllest circle hs rdius of 5 nd the lrgest hs rdius of s i i 55 55,78 inches

31 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Are of Surfce of Revolution You cn use the formul for the re of surfce of revolution in rectngulr form to develop formul for surfce re in prmetric form THEOREM 9 Are of Surfce of Revolution If smooth curve C given b f t nd gt does not cross itself on n intervl t b, then the re S of the surfce of revolution formed b revolving C bout the coordinte es is given b the following b S S gt Revolution bout the -is: gt d dt d dt dt b f t Revolution bout the -is: f t d dt d dt dt These formuls re es to remember if ou think of the differentil of rc length s ds d dt d dt dt Then the formuls re written s follows b S gt ds b S f t ds EXAMPLE Finding the Are of Surfce of Revolution ( ), C (, ) This surfce of revolution hs surfce re of 9 Figure 5 Rottble Grph Let C be the rc of the circle 9 from, to,, s shown in Figure 5 Find the re of the surfce formed b revolving C bout the -is Solution You cn represent C prmetricll b the equtions cos t 9 nd Note tht ou cn determine the intervl for t b observing tht t when nd t when On this intervl, C is smooth nd is nonnegtive, nd ou cn ppl Theorem 9 to obtin surfce re of Formul for re of S sin t sin t cos t dt surfce of revolution sin t9sin t cos t dt sin t dt Trigonometric identit 8 cos t 8 Tr It sin t, Eplortion A t

32 SECTION Prmetric Equtions nd Clculus 75 Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, find d/d t, 5 t t, t sin, cos e, e In Eercises 5, find d/d nd d /d, nd find the slope nd concvit (if possible) t the given vlue of the prmeter In Eercises 5 nd, find n eqution of the tngent line t ech given point on the curve In Eercises 7, () use grphing utilit to grph the curve represented b the prmetric equtions, (b) use grphing utilit to find d/dt, d/dt, nd d/d t the given vlue of the prmeter, (c) find n eqution of the tngent line to the curve t the given vlue of the prmeter, nd (d) use grphing utilit to grph the curve nd the tngent line from prt (c) Prmetric Equtions t, t t, t t, t t t t, t cos, sin cos, sin sec, tn t, t cos, sin sin, cos sin (, ), ( ) t, t Prmetric Equtions t, t t t, t t cos, sin (, ) 5 Point t t t t t 5 cot cos (, 5) t t t sin (, ) Prmeter + (, ) 5 In Eercises, find the equtions of the tngent lines t the point where the curve crosses itself In Eercises 5 nd, find ll points (if n) of horizontl nd verticl tngenc to the portion of the curve shown 5 Involute of circle: In Eercises 7, find ll points (if n) of horizontl nd verticl tngenc to the curve Use grphing utilit to confirm our results In Eercises 7, determine the t intervls on which the curve is concve downwrd or concve upwrd sin t, cos t, t t, t t t t, cos sin t, t t, t, t t, cos, sin cos, sin cos, cos, sec, tn cos, cos t, t, t ln t, t, 8 sin t, sin cos t t t t t t ln t sin t t sin t t t ln t cos t, t t sin < t < cos t, sin t, < t < t sin t 8 cos 8 8

33 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Arc Length In Eercises, write n integrl tht represents the rc length of the curve on the given intervl Do not evlute the integrl 5 Arc Length In Eercises 7 5, find the rc length of the curve on the given intervl Prmetric Equtions t t, ln t, e t, t sin t, t, t e t cos t, Arc Length In Eercises 5 5, find the rc length of the curve on the intervl [, ] 5 Hpoccloid perimeter: cos, sin 5 Circle circumference: cos, sin 55 Ccloid rch: sin, cos 5 Involute of circle: cos 57 Pth of Projectile The pth of projectile is modeled b the prmetric equtions 9 cos t nd where nd re mesured in feet () Use grphing utilit to grph the pth of the projectile (b) Use grphing utilit to pproimte the rnge of the projectile (c) Use the integrtion cpbilities of grphing utilit to pproimte the rc length of the pth Compre this result with the rnge of the projectile 58 Pth of Projectile If the projectile in Eercise 57 is lunched t n ngle with the horizontl, its prmetric equtions re 9 cos t t t t Prmetric Equtions t, t t cos t t, t e t sin t rcsin t, ln t t, t 5 t nd Intervl t t t t Intervl t t t 9 sin t t Use grphing utilit to find the ngle tht mimizes the rnge of the projectile Wht ngle mimizes the rc length of the trjector? t t t sin, sin cos 9 sin t t 59 Folium of Descrtes Consider the prmetric equtions nd () Use grphing utilit to grph the curve represented b the prmetric equtions (b) Use grphing utilit to find the points of horizont tngenc to the curve (c) Use the integrtion cpbilities of grphing utilit to pproimte the rc length of the closed loop Hint: Use smmetr nd integrte over the intervl t Witch of Agnesi Consider the prmetric equtions cot nd () Use grphing utilit to grph the curve represented b the prmetric equtions (b) Use grphing utilit to find the points of horizont tngenc to the curve (c) Use the integrtion cpbilities of grphing utilit to pproimte the rc length over the interv Writing () Use grphing utilit to grph ech set of prmetric equtions (b) Compre the grphs of the two sets of prmetric equtions in prt () If the curve represents the motion of prticle nd t is time, wht cn ou infer bout the verge speeds of the prticle on the pths represented b the two sets of prmetric equtions? (c) Without grphing the curve, determine the time required for prticle to trverse the sme pth s in prts () nd (b) if the pth is modeled b nd cos t sint t Writing t t t sin t cos t t () Ech set of prmetric equtions represents the motion of prticle Use grphing utilit to grph ech set First Prticle cos t sin t t (b) Determine the number of points of intersection (c) Will the prticles ever be t the sme plce t the sme time? If so, identif the points (d) Eplin wht hppens if the motion of the second prticle is represented b sin t, t t sin, t sint cost t Second Prticle sin t cos t t cos t, t

34 SECTION Prmetric Equtions nd Clculus 77 Surfce Are In Eercises, write n integrl tht represents the re of the surfce generted b revolving the curve bout the -is Use grphing utilit to pproimte the integrl 5 Surfce Are In Eercises 7 7, find the re of the surfce generted b revolving the curve bout ech given is 7 t, t, t, () -is (b) -is 8 t, t, t, () -is (b) -is 9 cos, sin, -is, 7 t, t, t, -is 7 cos, sin,, -is 7 Prmetric Equtions t, t, cos, t sin, t cos cos, b sin, () -is (b) -is cos Writing About Concepts, 7 Give the prmetric form of the derivtive 7 Mentll determine dd () t, (b) t, Intervl t t t 75 Sketch grph of curve defined b the prmetric equtions gt nd f t such tht ddt > nd ddt < for ll rel numbers t 7 Sketch grph of curve defined b the prmetric equtions gt nd f t such tht ddt < nd ddt < for ll rel numbers t 77 Give the integrl formul for rc length in prmetric form 78 Give the integrl formuls for the res of the surfces of revolution formed when smooth curve C is revolved bout () the -is nd (b) the -is 8 Surfce Are A portion of sphere of rdius r is removed b cutting out circulr cone with its verte t the center of the sphere The verte of the cone forms n ngle of Find the surfce re removed from the sphere Are In Eercises 8 nd 8, find the re of the region (Use the result of Eercise 79) 8 sin 8 cot sin tn Ares of Simple Closed Curves In Eercises 8 88, use computer lgebr sstem nd the result of Eercise 79 to mtch the closed curve with its re (These eercises were dpted from the rticle The Surveor s Are Formul b Brt Brden in the September 98 issue of the College Mthemtics Journl, b permission of the uthor) 8 < () b (b) 8 (c) (d) b (e) b (f) 8 Ellipse: t 8 Astroid: t b cos t sin t MthArticle b sin < < cos t sin t 79 Use integrtion b substitution to show tht if is continuous function of on the intervl b, where ft nd gt, then b d t gt ft dt t where f t, f t b, nd both g nd re continuous on t, t f 85 Crdioid: t 8 Deltoid: t cos t cos t cos t cos t sin t sin t sin t sin t

35 78 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes 87 Hourglss: t 88 Terdrop: t sin t cos t sin t b sin t b sin t b b r r P Centroid In Eercises 89 nd 9, find the centroid of the region bounded b the grph of the prmetric equtions nd the coordinte es (Use the result of Eercise 79) Volume In Eercises 9 nd 9, find the volume of the solid formed b revolving the region bounded b the grphs of the given equtions bout the -is (Use the result of Eercise 79) 9 9 cos, sin 9 Ccloid Use the prmetric equtions nd to nswer the following () Find dd nd d d (b) Find the equtions of the tngent line t the point where (c) Find ll points (if n) of horizontl tngenc (d) Determine where the curve is concve upwrd or concve downwrd (e) Find the length of one rc of the curve 9 Use the prmetric equtions t nd to nswer the following () Use grphing utilit to grph the curve on the intervl t (b) Find dd nd d d (c) Find the eqution of the tngent line t the point, 8 (d) Find the length of the curve (e) Find the surfce re generted b revolving the curve bout the -is 95 Involute of Circle The involute of circle is described b the endpoint P of string tht is held tut s it is unwound from spool tht does not turn (see figure) Show tht prmetric representtion of the involute is rcos cos, sin, > sin sin 89 t, t 9 t, t t t nd cos, > rsin cos Figure for 95 9 Involute of Circle The figure shows piece of string tied to circle with rdius of one unit The string is just long enough to rech the opposite side of the circle Find the re tht is covered when the string is unwound counterclockwise 97 () Use grphing utilit to grph the curve given b t t, t t, (b) Describe the grph nd confirm our result nlticll (c) Discuss the speed t which the curve is trced s increses from to 98 Trctri A person moves from the origin long the positive -is pulling weight t the end of -meter rope Initill the weight is locted t the point, () In Eercise 8 of Section 87, it ws shown tht the pth of the weight is modeled b the rectngulr eqution ln where < Use grphing utilit to grph the rectngulr eqution (b) Use grphing utilit to grph the prmetric equtions sech t nd t t tnh t where t How does this grph compre with the grph in prt ()? Which grph (if either) do ou think is better representtion of the pth? (c) Use the prmetric equtions for the trctri to verif th the distnce from the -intercept of the tngent line to the point of tngenc is independent of the loction of the point of tngenc True or Flse? In Eercises 99 nd, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse 99 If f t nd gt, then d d gtft The curve given b t, t hs horizontl tngent the origin becuse ddt when t

36 SECTION Polr Coordintes nd Polr Grphs 79 O Polr coordintes Figure Section r = directed distnce θ = directed ngle P = (r, θ) Polr is Polr Coordintes nd Polr Grphs Understnd the polr coordinte sstem Rewrite rectngulr coordintes nd equtions in polr form nd vice vers Sketch the grph of n eqution given in polr form Find the slope of tngent line to polr grph Identif severl tpes of specil polr grphs Polr Coordintes So fr, ou hve been representing grphs s collections of points, on the rectngulr coordinte sstem The corresponding equtions for these grphs hve been in either rectngulr or prmetric form In this section ou will stud coordinte sstem clled the polr coordinte sstem To form the polr coordinte sstem in the plne, fi point O, clled the pole (or origin), nd construct from O n initil r clled the polr is, s shown in Figure Then ech point P in the plne cn be ssigned polr coordintes r,, s follows r directed distnce from O to P directed ngle, counterclockwise from polr is to segment OP Figure 7 shows three points on the polr coordinte sstem Notice tht in this sstem, it is convenient to locte points with respect to grid of concentric circles intersected b rdil lines through the pole θ = (, ) θ = (, ) θ = (, ) () Figure 7 (b) (c) POLAR COORDINATES The mthemticin credited with first using polr coordintes ws Jmes Bernoulli, who introduced them in 9 However, there is some evidence tht it m hve been Isc Newton who first used them With rectngulr coordintes, ech point, hs unique representtion This is not true with polr coordintes For instnce, the coordintes r, nd r, represent the sme point [see prts (b) nd (c) in Figure 7] Also, becuse r is directed distnce, the coordintes r, nd r, represent the sme point In generl, the point r, cn be written s r, r, n or r, r, n where n is n integer Moreover, the pole is represented b,, where is n ngle

37 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Pole θ (Origin) r (r, θ) (, ) Polr is (-is) Relting polr nd rectngulr coordintes Figure 8 Coordinte Conversion To estblish the reltionship between polr nd rectngulr coordintes, let the polr is coincide with the positive -is nd the pole with the origin, s shown in Figure 8 Becuse, lies on circle of rdius r, it follows tht r Moreover, for r >, the definition of the trigonometric functions implies tht tn cos nd sin, r, r If r <, ou cn show tht the sme reltionships hold THEOREM Coordinte Conversion The polr coordintes r, of point re relted to the rectngulr coordintes, of the point s follows r cos tn r sin r (r, θ) = (, ) (, ) = (, ) To convert from polr to rectngulr coordintes, let r cos nd r sin Figure 9 θ (, ) (r, ) = (, ) = (, ) EXAMPLE Polr-to-Rectngulr Conversion For the point r,,, r cos cos nd So, the rectngulr coordintes re,, b For the point r,,, cos nd So, the rectngulr coordintes re See Figure 9 Tr It Eplortion A sin,, r sin sin Eplortion B EXAMPLE Rectngulr-to-Polr Conversion θ (, ) (r, ) = (, ) = (, ) (r, θ ) = (, ) (, ) = (, ) To convert from rectngulr to polr coordintes, let tn nd r Figure For the second qudrnt point,,, tn Becuse ws chosen to be in the sme qudrnt s,, ou should use positive vlue of r r This implies tht one set of polr coordintes is r,, b Becuse the point,, lies on the positive -is, choose nd r, nd one set of polr coordintes is r,, See Figure Tr It Eplortion A

38 SECTION Polr Coordintes nd Polr Grphs 7 Polr Grphs One w to sketch the grph of polr eqution is to convert to rectngulr coordintes nd then sketch the grph of the rectngulr eqution EXAMPLE Grphing Polr Equtions () Circle: r (b) Rdil line: (c) Verticl line: r sec Figure Describe the grph of ech polr eqution Confirm ech description b converting to rectngulr eqution r b c r sec Solution The grph of the polr eqution r consists of ll points tht re two units from the pole In other words, this grph is circle centered t the origin with rdius of [See Figure ()] You cn confirm this b using the reltionship r to obtin the rectngulr eqution Rectngulr eqution b The grph of the polr eqution consists of ll points on the line tht mkes n ngle of with the positive -is [See Figure (b)] You cn confirm this b using the reltionship tn to obtin the rectngulr eqution Rectngulr eqution c The grph of the polr eqution r sec is not evident b simple inspection, so ou cn begin b converting to rectngulr form using the reltionship r cos r sec r cos Polr eqution Rectngulr eqution From the rectngulr eqution, ou cn see tht the grph is verticl line [See Figure (c)] Tr It Eplortion A 9 9 TECHNOLOGY Sketching the grphs of complicted polr equtions b hnd cn be tedious With technolog, however, the tsk is not difficult If our grphing utilit hs polr mode, use it to grph the equtions in the eercise set If our grphing utilit doesn t hve polr mode, but does hve prmetric mode, ou cn grph r f b writing the eqution s f cos f sin For instnce, the grph of r shown in Figure ws produced with grphing clcultor in prmetric mode This eqution ws grphed using the prmetric equtions cos Spirl of Archimedes Figure sin with the vlues of vring from to This curve is of the form r nd is clled spirl of Archimedes

39 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Sketching Polr Grph NOTE One w to sketch the grph of r cos b hnd is to mke tble of vlues B etending the tble nd plotting the points, ou will obtin the curve shown in Emple r Sketch the grph of r cos Solution Begin b writing the polr eqution in prmetric form cos cos nd After some eperimenttion, ou will find tht the entire curve, which is clled rose curve, cn be sketched b letting vr from to, s shown in Figure If ou tr duplicting this grph with grphing utilit, ou will find tht b letting vr from to, ou will ctull trce the entire curve twice cos sin Figure 5 Animtion Tr It Eplortion A Open Eplortion Use grphing utilit to eperiment with other rose curves the re of the form r cos n or r sin n For instnce, Figure shows the grphs of two other rose curves r = 5 cos θ r = sin 5θ Rose curves Figure Generted b Derive

40 SECTION Polr Coordintes nd Polr Grphs 7 r = f( θ) Tngent line (r, θ) Slope nd Tngent Lines To find the slope of tngent line to polr grph, consider differentible function given b r f To find the slope in polr form, use the prmetric equtions r cos f cos nd Using the prmetric form of dd given in Theorem 7, ou hve d dd d dd f cos f sin f sin f cos which estblishes the following theorem r sin f sin Tngent line to polr curve Figure 5 THEOREM Slope in Polr Form If f is differentible function of, then the slope of the tngent line to the grph of r f t the point r, is d d dd dd f cos f sin f sin f cos provided tht t r, See Figure 5 dd From Theorem, ou cn mke the following observtions d d Solutions to ield horizontl tngents, provided tht d d Solutions to ield verticl tngents, provided tht If nd re simultneousl, no conclusion cn be drwn bout tngent lines dd d d dd d d EXAMPLE 5 Finding Horizontl nd Verticl Tngent Lines Find the horizontl nd verticl tngent lines of r sin, ( ) (, ) (, ) ( ),, Horizontl nd verticl tngent lines of r sin Figure Solution nd Begin b writing the eqution in prmetric form r cos sin cos r sin sin sin sin Net, differentite nd with respect to nd set ech derivtive equl to d cos sin cos d d sin cos sin d,, So, the grph hs verticl tngent lines t, nd,, nd it hs horizontl tngent lines t, nd,, s shown in Figure Editble Grph Tr It Eplortion A

41 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Finding Horizontl nd Verticl Tngent Lines Horizontl nd verticl tngent lines of r cos Figure 7 (, ) (, ) (, ) Editble Grph (, ) (, 5 ) Find the horizontl nd verticl tngents to the grph of r cos Solution Using r sin, differentite nd set equl to d cos cos sin sin d So, cos nd cos, nd ou cn conclude tht when, nd Similrl, using r cos, ou hve, d sin cos sin sin cos d r sin cos sin cos cos r cos cos cos So, sin or cos, nd ou cn conclude tht when,, nd 5 From these results, nd from the grph shown in Figure 7, ou cn conclude tht the grph hs horizontl tngents t, nd,, nd hs verticl tngents t,,, 5, nd, This grph is clled crdioid Note tht both derivtives nd dd re when Using this informtion lone, ou don t know whether the grph hs horizontl or verticl tngent line t the pole From Figure 7, however, ou cn see tht the grph hs cusp t the pole dd dd dd dd, Tr It Eplortion A f( θ) = cos θ Theorem hs n importnt consequence Suppose the grph of r f psses through the pole when nd f Then the formul for dd simplifies s follows d d f sin f cos f sin f cos f sin f cos sin tn cos So, the line is tngent to the grph t the pole,, THEOREM Tngent Lines t the Pole If f nd f, then the line is tngent t the pole to the grph of r f This rose curve hs three tngent lines,, nd t the pole Figure 8 5 Theorem is useful becuse it sttes tht the zeros of r f cn be used to find the tngent lines t the pole Note tht becuse polr curve cn cross the pole more thn once, it cn hve more thn one tngent line t the pole For emple, the rose curve f cos hs three tngent lines t the pole, s shown in Figure 8 For this curve, f cos is when is,, nd 5 Moreover, the derivtive f sin is not for these vlues of

42 SECTION Polr Coordintes nd Polr Grphs 75 Specil Polr Grphs Severl importnt tpes of grphs hve equtions tht re simpler in polr form thn in rectngulr form For emple, the polr eqution of circle hving rdius of nd centered t the origin is simpl r Lter in the tet ou will come to pprecite this benefit For now, severl other tpes of grphs tht hve simpler equtions in polr form re shown below (Conics re considered in Section ) Limçons r ± b cos r ± b sin >, b > b < Limçon with inner loop b Crdioid (hert-shped) < b < Dimpled limçon b Conve limçon Rose Curves n petls if n is odd n petls if n is even n n = n = n = 5 n = r cos Rose curve n r cos Rose curve n r sin Rose curve n r sin Rose curve n Circles nd Lemnisctes r cos Circle r sin Circle r sin Lemniscte r cos Lemniscte TECHNOLOGY The rose curves described bove re of the form r cos n or r sin n, where n is positive integer tht is greter thn or equl to Use grphing utilit to grph r cos n or r sin n for some noninteger vlues of n Are these grphs lso rose curves? For emple, tr sketching the grph of r cos, FOR FURTHER INFORMATION For more informtion on rose curves nd relted curves, see the rticle A Rose is Rose b Peter M Murer in The Americn Mthemticl Monthl The computer-generted grph t the left is the result of n lgorithm tht Murer clls The Rose Generted b Mple MthArticle

43 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, plot the point in polr coordintes nd find the corresponding rectngulr coordintes for the point (c) (d),, 7,, 7 5,, 57 In Eercises 7, use the ngle feture of grphing utilit to find the rectngulr coordintes for the point given in polr coordintes Plot the point 7 5, 8, 9 5, 5 85, In Eercises, the rectngulr coordintes of point re given Plot the point nd find two sets of polr coordintes for the point for < In Eercises 7, use the ngle feture of grphing utilit to find one set of polr coordintes for the point given in rectngulr coordintes 7, 8 9 Plot the point, 5 if the point is given in () rectngulr coordintes nd (b) polr coordintes Grphicl Resoning () Set the window formt of grphing utilit to rectngulr coordintes nd locte the cursor t n position off the es Move the cursor horizontll nd verticll Describe n chnges in the displed coordintes of the points (b) Set the window formt of grphing utilit to polr coordintes nd locte the cursor t n position off the es Move the cursor horizontll nd verticll Describe n chnges in the displed coordintes of the points (c) Wh re the results in prts () nd (b) different? In Eercises, mtch the grph with its polr eqution [The grphs re lbeled (), (b), (c), nd (d)] () 5,,, 5,, 5, (b),,, 5 r sin r cos 5 r cos r sec In Eercises 7, convert the rectngulr eqution to polr form nd sketch its grph In Eercises 5, convert the polr eqution to rectngulr form nd sketch its grph 5 r r 7 r sin 8 r 5 cos 9 r r sec r csc In Eercises 5, use grphing utilit to grph the polr eqution Find n intervl for over which the grph is trced onl once r cos r 5 sin 5 r sin r cos 7 r cos 8 r 9 r cos 5 5 r 5 r sin 5 Convert the eqution r h cos k sin 5 sin r sin 5 to rectngulr form nd verif tht it is the eqution of circle Find the rdius nd the rectngulr coordintes of the center of the circle

44 SECTION Polr Coordintes nd Polr Grphs 77 5 Distnce Formul () Verif tht the Distnce Formul for the distnce between the two points r, nd r, in polr coordintes is (b) Describe the positions of the points reltive to ech other if Simplif the Distnce Formul for this cse Is the simplifiction wht ou epected? Eplin (c) Simplif the Distnce Formul if simplifiction wht ou epected? Eplin 9 Is the (d) Choose two points on the polr coordinte sstem nd find the distnce between them Then choose different polr representtions of the sme two points nd ppl the Distnce Formul gin Discuss the result In Eercises 55 58, use the result of Eercise 5 to pproimte the distnce between the two points in polr coordintes 55,, 5, In Eercises 59 nd, find d/d nd the slopes of the tngent lines shown on the grph of the polr eqution (, ) d r r r r cos 57, 5, 7, 58, 5, ( ) 5, ( ), In Eercises, use grphing utilit to () grph the polr eqution, (b) drw the tngent line t the given vlue of, nd (c) find d/d t the given vlue of Hint: Let the increment between the vlues of equl / r cos, r cos, r sin, r, In Eercises 5 nd, find the points of horizontl nd verticl tngenc (if n) to the polr curve 5 r sin r sin In Eercises 7 nd 8, find the points of horizontl tngenc (if n) to the polr curve 7 r csc 8 r sin cos 7,, ( ), 7,, 59 r sin r sin ( ), (, ) In Eercises 9 7, use grphing utilit to grph the polr eqution nd find ll points of horizontl tngenc 9 r sin cos 7 r cos sec 7 r csc 5 7 r cos In Eercises 7 8, sketch grph of the polr eqution nd find the tngents t the pole 7 r sin 7 r cos 75 r sin 7 r cos 77 r cos 78 r sin 5 79 r sin 8 r cos In Eercises 8 9, sketch grph of the polr eqution 8 r 5 8 r 8 r cos 8 r sin 85 r cos 8 r 5 sin 87 r csc 88 r sin cos 89 r 9 r 9 r cos 9 r sin In Eercises 9 9, use grphing utilit to grph the eqution nd show tht the given line is n smptote of the grph Nme of Grph 9 Conchoid 9 Conchoid 95 Hperbolic spirl 9 Strophoid Sketch the grph of r sin over ech intervl () (b) (c) Think About It Use grphing utilit to grph the polr eqution r cos for () (b) nd (c) Use the grphs to describe the effect of the ngle Write the eqution s function of sin for prt (c) Polr Eqution r sec r csc r r cos sec Writing About Concepts, Asmptote 97 Describe the differences between the rectngulr coordinte sstem nd the polr coordinte sstem 98 Give the equtions for the coordinte conversion from rectngulr to polr coordintes nd vice vers 99 For constnts nd b, describe the grphs of the equtions r nd in polr coordintes b How re the slopes of tngent lines determined in polr coordintes? Wht re tngent lines t the pole nd how re the determined?

45 78 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Verif tht if the curve whose polr eqution is r f is rotted bout the pole through n ngle, then n eqution for the rotted curve is r f The polr form of n eqution for curve is Show tht the form becomes () r f cos if the curve is rotted counterclockwise rdins bout the pole (b) r f sin if the curve is rotted counterclockwise rdins bout the pole (c) r f cos if the curve is rotted counterclockwise rdins bout the pole In Eercises 5 8, use the results of Eercises nd 5 Write n eqution for the limçon r sin fter it hs been rotted b the given mount Use grphing utilit to grph the rotted limçon () (b) (c) (d) Write n eqution for the rose curve r sin fter it hs been rotted b the given mount Verif the results b using grphing utilit to grph the rotted rose curve () (b) (c) (d) 7 Sketch the grph of ech eqution () r sin (b) r sin 8 Prove tht the tngent of the ngle between the rdil line nd the tngent line t the point r, on the grph of r f (see figure) is given b tn r f sin rdrd True or Flse? In Eercises 5 8, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse 5 If r, nd r, represent the sme point on the polr coordinte sstem, then If r, nd r, represent the sme point on the polr coordinte sstem, then n for some integer n 7 If >, then the point, on the rectngulr coordinte sstem cn be represented b r, on the polr coordinte sstem, where r nd rctn r r 8 The polr equtions r sin nd r sin hve the sme grph Polr curve: r = f( ) Tngent line P = (r, ) Rdil line O A Polr is In Eercises 9, use the result of Eercise 8 to find the ngle between the rdil nd tngent lines to the grph for the indicted vlue of Use grphing utilit to grph the polr eqution, the rdil line, nd the tngent line for the indicted vlue of Identif the ngle 9 Polr Eqution r cos r cos r cos r sin r r 5 cos Vlue of

46 SECTION 5 Are nd Arc Length in Polr Coordintes 79 θ Section 5 r The re of sector of circle is A r Figure 9 () β r = f( θ) α Are nd Arc Length in Polr Coordintes Find the re of region bounded b polr grph Find the points of intersection of two polr grphs Find the rc length of polr grph Find the re of surfce of revolution (polr form) Are of Polr Region The development of formul for the re of polr region prllels tht for the re of region on the rectngulr coordinte sstem, but uses sectors of circle insted of rectngles s the bsic element of re In Figure 9, note tht the re of circulr sector of rdius r is given b r, provided is mesured in rdins Consider the function given b r f, where f is continuous nd nonnegtive in the intervl given b The region bounded b the grph of f nd the rdil lines nd is shown in Figure 5() To find the re of this region, prtition the intervl, into n equl subintervls < < Then, pproimte the re of the region b the sum of the res of the n sectors, s shown in Figure 5(b) Rdius of ith sector fi Centrl ngle of ith sector Tking the limit s n produces A lim n n fi i f d < < A n which leds to the following theorem n < n n i fi β θ n r = f( θ) θ θ α THEOREM Are in Polr Coordintes If f is continuous nd nonnegtive on the intervl,, <, then the re of the region bounded b the grph of r f between the rdil lines nd is given b A f d r < d (b) Figure 5 NOTE You cn use the sme formul to find the re of region bounded b the grph of continuous nonpositive function However, the formul is not necessril vlid if f tkes on both positive nd negtive vlues in the intervl,

47 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes r = cos θ EXAMPLE Finding the Are of Polr Region Find the re of one petl of the rose curve given b r cos The re of one petl of the rose curve tht lies between the rdil lines nd is Figure 5 The re between the inner nd outer loops is pproimtel 8 Figure 5 Editble Grph NOTE To find the re of the region ling inside ll three petls of the rose curve in Emple, ou could not simpl integrte between nd In doing this ou would obtin 9, which is twice the re of the three petls The dupliction occurs becuse the rose curve is trced twice s increses from to θ = 5 Editble Grph θ = r = sin θ Solution In Figure 5, ou cn see tht the right petl is trced s increses from to So, the re is Formul for re in A r d cos d polr coordintes EXAMPLE Tr It Finding the Are Bounded b Single Curve Find the re of the region ling between the inner nd outer loops of the limçon r sin Solution In Figure 5, note tht the inner loop is trced s increses from to 5 So, the re inside the inner loop is A r d 9 9 sin 9 5 sin d 5 sin sin d 5 sin cos d 5 sin cos d cos sin 5 cos Eplortion A Simplif In similr w, ou cn integrte from 5 to to find tht the re of the region ling inside the outer loop is A The re of the region ling between the two loops is the difference of nd A A A A 8 d Open Eplortion A Trigonometric identit Formul for re in polr coordintes Trigonometric identit Tr It Eplortion A

48 SECTION 5 Are nd Arc Length in Polr Coordintes 7 FOR FURTHER INFORMATION For more informtion on using technolog to find points of intersection, see the rticle Finding Points of Intersection of Polr- Coordinte Grphs b Wrren W Est in Mthemtics Techer MthArticle Points of Intersection of Polr Grphs Becuse point m be represented in different ws in polr coordintes, cre must be tken in determining the points of intersection of two polr grphs For emple, consider the points of intersection of the grphs of r cos nd s shown in Figure 5 If, s with rectngulr equtions, ou ttempted to find the points of intersection b solving the two equtions simultneousl, ou would obtin cos First eqution Substitute r from nd eqution into st eqution Simplif Solve for The corresponding points of intersection re, nd, However, from Figure 5 ou cn see tht there is third point of intersection tht did not show up when the two polr equtions were solved simultneousl (This is one reson wh ou should sketch grph when finding the re of polr region) The reson the third point ws not found is tht it does not occur with the sme coordintes in the two grphs On the grph of r, the point occurs with coordintes,, but on the grph of r cos, the point occurs with coordintes, You cn compre the problem of finding points of intersection of two polr grphs with tht of finding collision points of two stellites in intersecting orbits bout Erth, s shown in Figure 5 The stellites will not collide s long s the rech the points of intersection t different times ( -vlues) Collisions will occur onl t the points of intersection tht re simultneous points those reched t the sme time ( -vlue) r cos cos, r NOTE Becuse the pole cn be represented b,, where is n ngle, ou should check seprtel for the pole when finding points of intersection Limçon: r = cos θ Circle: r = Three points of intersection:,, The pths of stellites cn cross without cusing,,, collision Figure 5 Figure 5 Animtion

49 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Finding the Are of Region Between Two Curves Find the re of the region common to the two regions bounded b the following curves r cos r cos Circle Crdioid Circle: r = cos θ Figure 55 Crdioid Editble Grph Circle Crdioid: r = cos θ Solution Becuse both curves re smmetric with respect to the -is, ou cn work with the upper hlf-plne, s shown in Figure 55 The gr shded region lies between the circle nd the rdil line Becuse the circle hs coordintes, t the pole, ou cn integrte between nd to obtin the re of this region The region tht is shded red is bounded b the rdil lines nd nd the crdioid So, ou cn find the re of this second region b integrting between nd The sum of these two integrls gives the re of the common region ling bove the rdil line Region between circle Region between crdioid nd nd rdil line rdil lines nd A cos d 8 cos d 9 cos d sin sin sin Finll, multipling b, ou cn conclude tht the totl re is 5 8 cos cos d cos d cos cos d NOTE To check the resonbleness of the result obtined in Emple, note tht the re of the circulr region is r 9 So, it seems resonble tht the re of the region ling inside the circle nd the crdioid is 5 To see the benefit of polr coordintes for finding the re in Emple, consider the following integrl, which gives the comprble re in rectngulr coordintes A d d Use the integrtion cpbilities of grphing utilit to show tht ou obtin the sme re s tht found in Emple Tr It Eplortion A Eplortion B

50 SECTION 5 Are nd Arc Length in Polr Coordintes 7 NOTE When ppling the rc length formul to polr curve, be sure tht the curve is trced out onl once on the intervl of integrtion For instnce, the rose curve given b r cos is trced out once on the intervl, but is trced out twice on the intervl Arc Length in Polr Form The formul for the length of polr rc cn be obtined from the rc length formul for curve described b prmetric equtions (See Eercise 77) THEOREM Arc Length of Polr Curve Let f be function whose derivtive is continuous on n intervl The length of the grph of r f from to is s f f d d d r dr EXAMPLE Finding the Length of Polr Curve Find the length of the rc from to for the crdioid r f cos s shown in Figure 5 r = cos θ Figure 5 Editble Grph Solution Becuse f sin, ou cn find the rc length s follows s cos sin d cos d sin d sin d 8 cos 8 f f d In the fifth step of the solution, it is legitimte to write Formul for rc length of polr curve Simplif Trigonometric identit sin for sin sin rther thn sin sin becuse sin for Tr It Eplortion A NOTE Using Figure 5, ou cn determine the resonbleness of this nswer b compring it with the circumference of circle For emple, circle of rdius hs circumference 5 of 5 57

51 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Are of Surfce of Revolution The polr coordinte versions of the formuls for the re of surfce of revolution cn be obtined from the prmetric versions given in Theorem 9, using the equtions r cos nd r sin NOTE When using Theorem 5, check to see tht the grph of r f is trced onl once on the intervl For emple, the circle given b r cos is trced onl once on the intervl THEOREM 5 Are of Surfce of Revolution Let f be function whose derivtive is continuous on n intervl The re of the surfce formed b revolving the grph of r f from to bout the indicted line is s follows S f sin f f d About the polr is S f cos f f d About the line EXAMPLE 5 Finding the Are of Surfce of Revolution Find the re of the surfce formed b revolving the circle r f cos bout the line s shown in Figure 57, r = cos θ Pinched torus () Figure 57 Rottble Grph (b) Solution You cn use the second formul given in Theorem 5 with f sin Becuse the circle is trced once s increses from to, ou hve Formul for re of surfce of S f cos f f d revolution cos cos cos sin d cos d Trigonometric identit cos d Trigonometric identit sin Tr It Eplortion A

52 SECTION 5 Are nd Arc Length in Polr Coordintes 75 Eercises for Section 5 The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, write n integrl tht represents the re of the shded region shown in the figure Do not evlute the integrl r sin r cos In Eercises 7, find the points of intersection of the grphs of the equtions 7 r cos 8 r sin r cos r sin 5 r sin 5 5 r cos 9 r cos r sin r cos r cos In Eercises 5 nd, find the re of the region bounded b the grph of the polr eqution using () geometric formul nd (b) integrtion 5 r 8 sin r cos In Eercises 7, find the re of the region 7 One petl of r cos 8 One petl of r sin 9 One petl of r cos One petl of r cos 5 Interior of r sin Interior of r sin (bove the polr is) In Eercises, use grphing utilit to grph the polr eqution nd find the re of the given region Inner loop of r cos Inner loop of r sin 5 Between the loops of r cos Between the loops of r sin r 5 sin r cos r sin r r 5 r sin r sin r In Eercises 7 nd 8, use grphing utilit to pproimte the points of intersection of the grphs of the polr equtions Confirm our results nlticll 7 r cos 8 r cos r sec Writing In Eercises 9 nd, use grphing utilit to find the points of intersection of the grphs of the polr equtions Wtch the grphs s the re trced in the viewing window Eplin wh the pole is not point of intersection obtined b solving the equtions simultneousl 9 r cos r sin r sin r cos r r csc r cos r sin

53 7 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes In Eercises, use grphing utilit to grph the polr equtions nd find the re of the given region Common interior of r sin nd r Common interior of r sin nd r sin Common interior of r sin nd r sin Common interior of r 5 sin nd r 5 cos 5 Common interior of r sin nd r Inside r sin nd outside r sin In Eercises 7, find the re of the region 7 Inside r cos nd outside r cos 8 Inside r cos nd outside r 9 Common interior of r cos nd r sin Common interior of r cos nd r sin where > Antenn Rdition The rdition from trnsmitting ntenn is not uniform in ll directions The intensit from prticulr ntenn is modeled b () Convert the polr eqution to rectngulr form (b) Use grphing utilit to grph the model for nd (c) Find the re of the geogrphicl region between the two curves in prt (b) Are The re inside one or more of the three interlocking circles nd is divided into seven regions Find the re of ech region Conjecture Find the re of the region enclosed b for n,,, Use the results to mke conjecture bout the re enclosed b the function if n is even nd if n is odd Are Sketch the strophoid Convert this eqution to rectngulr coordintes Find the re enclosed b the loop In Eercises 5 8, find the length of the curve over the given intervl r cos r cos, r cosn r sec cos, Polr Eqution r r cos r sin r 8 cos r sin, < < Intervl r In Eercises 9 5, use grphing utilit to grph the polr eqution over the given intervl Use the integrtion cpbilities of the grphing utilit to pproimte the length of the curve ccurte to two deciml plces 9 r, 5 r sec, 5 r, 5 r e, 5 5 In Eercises 55 58, find the re of the surfce formed b revolving the curve bout the given line 55 r cos Polr is 5 57 r sin cos, r sin cos, Polr Eqution r cos r e 58 r cos Polr is In Eercises 59 nd, use the integrtion cpbilities of grphing utilit to pproimte to two deciml plces the re of the surfce formed b revolving the curve bout the polr is Intervl 59 r cos, r, Writing About Concepts Ais of Revolution Give the integrl formuls for re nd rc length in polr coordintes Eplin wh finding points of intersection of polr grphs m require further nlsis beond solving two equtions simultneousl Which integrl ields the rc length of r cos? Stte wh the other integrls re incorrect () (b) (c) (d) cos sin d cos sin d cos sin d cos sin d Give the integrl formuls for the re of the surfce of revolution formed when the grph of r f is revolved bout () the -is nd (b) the -is

54 SECTION 5 Are nd Arc Length in Polr Coordintes 77 5 Surfce Are of Torus Find the surfce re of the torus generted b revolving the circle given b r bout the line r 5 sec Surfce Are of Torus Find the surfce re of the torus generted b revolving the circle given b r bout the line r b sec, where < < b 7 Approimting Are Consider the circle r 8 cos () Find the re of the circle (b) Complete the tble giving the res A of the sectors of the circle between nd the vlues of in the tble (c) Find the length of r (d) Find the re under the curve r over the intervl for 7 Logrithmic Spirl The curve represented b the eqution r e b, where nd b re constnts, is clled logrithmic spirl The figure below shows the grph of r e Find the re of the shded region A 8 (c) Use the tble in prt (b) to pproimte the vlues of for which the sector of the circle composes,, nd of the totl re of the circle (d) Use grphing utilit to pproimte, to two deciml plces, the ngles for which the sector of the circle composes,, nd of the totl re of the circle (e) Do the results of prt (d) depend on the rdius of the circle? Eplin 8 Approimte Are Consider the circle r sin () Find the re of the circle (b) Complete the tble giving the res A of the sectors of the circle between nd the vlues of in the tble A (c) Use the tble in prt (b) to pproimte the vlues of for which the sector of the circle composes 8,, nd of the totl re of the circle (d) Use grphing utilit to pproimte, to two deciml plces, the ngles for which the sector of the circle composes nd of the totl re of the circle 8,, 9 Wht conic section does the following polr eqution represent? r sin b cos 7 Are Find the re of the circle given b r sin cos Check our result b converting the polr eqution to rectngulr form, then using the formul for the re of circle 7 Spirl of Archimedes The curve represented b the eqution r, where is constnt, is clled the spirl of Archimedes () Use grphing utilit to grph r, where Wht hppens to the grph of r s increses? Wht hppens if? 8 (b) Determine the points on the spirl r >,, where the curve crosses the polr is 7 The lrger circle in the figure below is the grph of r Find the polr eqution of the smller circle such tht the shded regions re equl 7 Folium of Descrtes A curve clled the folium of Descrtes cn be represented b the prmetric equtions t t nd () Convert the prmetric equtions to polr form (b) Sketch the grph of the polr eqution from prt () (c) Use grphing utilit to pproimte the re enclosed b the loop of the curve True or Flse? In Eercises 75 nd 7, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse 75 If f > for ll nd g < for ll, then the grphs of r f nd r g do not intersect 7 If f g for nd, then the grphs of r f nd r g hve t lest four points of intersection,, t t 77 Use the formul for the rc length of curve in prmetric form to derive the formul for the rc length of polr curve

55 78 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Section EXPLORATION Grphing Conics Set grphing utilit to polr mode nd enter polr equtions of the form or r r ± b cos ± b sin As long s, the grph should be conic Describe the vlues of nd b tht produce prbols Wht vlues produce ellipses? Wht vlues produce hperbols? Polr Equtions of Conics nd Kepler s Lws Anlze nd write polr equtions of conics Understnd nd use Kepler s Lws of plnetr motion Polr Equtions of Conics In this chpter ou hve seen tht the rectngulr equtions of ellipses nd hperbols tke simple forms when the origin lies t their centers As it hppens, there re mn importnt pplictions of conics in which it is more convenient to use one of the foci s the reference point (the origin) for the coordinte sstem For emple, the sun lies t focus of Erth s orbit Similrl, the light source of prbolic reflector lies t its focus In this section ou will see tht polr equtions of conics tke simple forms if one of the foci lies t the pole The following theorem uses the concept of eccentricit, s defined in Section, to clssif the three bsic tpes of conics A proof of this theorem is given in Appendi A THEOREM Clssifiction of Conics b Eccentricit Let F be fied point (focus) nd D be fied line (directri) in the plne Let P be nother point in the plne nd let e (eccentricit) be the rtio of the distnce between P nd F to the distnce between P nd D The collection of ll points P with given eccentricit is conic The conic is n ellipse if < e < The conic is prbol if e The conic is hperbol if e > Directri Directri Directri Q P F = (, ) Q P F = (, ) P Q Q P F = (, ) Ellipse: < e < Prbol: e Hperbol: e > PF PF PQ PQ < PF PF > PQ Figure 58 PQ In Figure 58, note tht for ech tpe of conic the pole corresponds to the fied point (focus) given in the definition The benefit of this loction cn be seen in the proof of the following theorem

56 SECTION Polr Equtions of Conics nd Kepler s Lws 79 THEOREM 7 Polr Equtions of Conics The grph of polr eqution of the form r ed ± e cos or r ed ± e sin is conic, where e > is the eccentricit nd is the distnce between the focus t the pole nd its corresponding directri d d P = (r, θ) θ r F = (, ) Figure 59 Q Directri Proof The following is proof for r ed e cos with d > In Figure 59, consider verticl directri d units to the right of the focus F, If P r, is point on the grph of r ed e cos, the distnce between P nd the directri cn be shown to be PQ d d r cos r e cos e r cos r e Becuse the distnce between nd the pole is simpl the rtio of to is PFPQ r re P PF e r, PF PQ e nd, b Theorem, the grph of the eqution must be conic The proofs of the other cses re similr The four tpes of equtions indicted in Theorem 7 cn be clssified s follows, where d > Horizontl directri bove the pole: b Horizontl directri below the pole: c Verticl directri to the right of the pole: d Verticl directri to the left of the pole: r r r r ed e sin ed e sin ed e cos ed e cos Figure illustrtes these four possibilities for prbol Directri = d Directri = d Directri = d Directri = d r = ed + e sin θ r = ed e sin θ r = ed + e cosθ r = ed e cosθ () (b) (c) (d) The four tpes of polr equtions for prbol Figure

57 75 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Determining Conic from Its Eqution = 5 Directri 5 r = cosθ (, ) (5, ) 5 The grph of the conic is n ellipse with e Figure 5 Sketch the grph of the conic given b r cos Solution r To determine the tpe of conic, rewrite the eqution s 5 cos 5 cos Write originl eqution So, the grph is n ellipse with e You cn sketch the upper hlf of the ellipse b plotting points from to s shown in Figure Then, using smmetr with respect to the polr is, ou cn sketch the lower hlf, Divide numertor nd denomintor b Editble Grph Tr It Eplortion A For the ellipse in Figure, the mjor is is horizontl nd the vertices lie t (5, ) nd, So, the length of the mjor is is 8 To find the length of the minor is, ou cn use the equtions e c nd b c to conclude b c e e Ellipse Becuse e, ou hve b 9 5 which implies tht b 5 5 So, the length of the minor is is b 5 A similr nlsis for hperbols ields b c e e Hperbol EXAMPLE Sketching Conic from Its Polr Eqution Directri = 5 (, (, ) ) 8 r = + 5 sin θ = b = 8 The grph of the conic is hperbol with e 5 Figure Editble Grph Sketch the grph of the polr eqution r 5 sin Solution r Dividing the numertor nd denomintor b produces 5 sin Becuse e 5 the grph is hperbol Becuse d the directri is the line >, 5, 5 The trnsverse is of the hperbol lies on the line nd the vertices occur t r,, nd r,, Becuse the length of the trnsverse is is, ou cn see tht To find b, write b e 5, Therefore, b 8 Finll, ou cn use nd b to determine the smptotes of the hperbol nd obtin the sketch shown in Figure Tr It Eplortion A Open Eplortion

58 SECTION Polr Equtions of Conics nd Kepler s Lws 75 JOHANNES KEPLER (57 ) Kepler formulted his three lws from the etensive dt recorded b Dnish stronomer Tcho Brhe, nd from direct observtion of the orbit of Mrs MthBio Kepler s Lws Kepler s Lws, nmed fter the Germn stronomer Johnnes Kepler, cn be used to describe the orbits of the plnets bout the sun Ech plnet moves in n ellipticl orbit with the sun s focus A r from the sun to the plnet sweeps out equl res of the ellipse in equl times The squre of the period is proportionl to the cube of the men distnce between the plnet nd the sun* Although Kepler derived these lws empiricll, the were lter vlidted b Newton In fct, Newton ws ble to show tht ech lw cn be deduced from set of universl lws of motion nd grvittion tht govern the movement of ll hevenl bodies, including comets nd stellites This is shown in the net emple, involving the comet nmed fter the English mthemticin nd phsicist Edmund Hlle (5 7) EXAMPLE Hlle s Comet Sun Erth Hlle s comet hs n ellipticl orbit with the sun t one focus nd hs n eccentricit of e 97 The length of the mjor is of the orbit is pproimtel 588 stronomicl units (An stronomicl unit is defined to be the men distnce between Erth nd the sun, 9 million miles) Find polr eqution for the orbit How close does Hlle s comet come to the sun? Hlle's comet Solution r Using verticl is, ou cn choose n eqution of the form ed e sin, Becuse the vertices of the ellipse occur when nd ou cn determine the length of the mjor is to be the sum of the r-vlues of the vertices, s shown in Figure Tht is, 97d 97d d 588 So, d nd ed 97 Using this vlue in the eqution produces Figure r 97 sin where r is mesured in stronomicl units To find the closest point to the sun (the focus), ou cn write c e Becuse c is the distnce between the focus nd the center, the closest point is c AU 55,, miles Tr It Eplortion A View the video for more informtion bout Hlle s comet Video * If Erth is used s reference with period of er nd distnce of stronomicl unit, the proportionlit constnt is For emple, becuse Mrs hs men distnce to the sun of D 5 AU, its period P is given b D P So, the period for Mrs is P 88

59 75 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Kepler s Second Lw sttes tht s plnet moves bout the sun, r from the sun to the plnet sweeps out equl res in equl times This lw cn lso be pplied to comets or steroids with ellipticl orbits For emple, Figure shows the orbit of the steroid Apollo bout the sun Appling Kepler s Second Lw to this steroid, ou know tht the closer it is to the sun, the greter its velocit, becuse short r must be moving quickl to sweep out s much re s long r Sun Sun Sun A r from the sun to the steroid sweeps out equl res in equl times Figure EXAMPLE The Asteroid Apollo The steroid Apollo hs period of Erth ds, nd its orbit is pproimted b the ellipse r 9 59 cos 9 5 cos θ = where r is mesured in stronomicl units How long does it tke Apollo to move from the position given b to s shown in Figure 5?, Apollo Figure 5 Sun θ = Erth Solution Begin b finding the re swept out s increses from to A r Formul for re of polr grph d cos d Using the substitution u tn, s discussed in Section 8, ou obtin A 8 5 sin 8 5 tn rctn 9 5 cos 5 99 Becuse the mjor is of the ellipse hs length 88 nd the eccentricit is e 59, ou cn determine tht b e 95 So, the re of the ellipse is Are of ellipse b Becuse the time required to complete the orbit is ds, ou cn ppl Kepler s Second Lw to conclude tht the time t required to move from the position to is given b t re of ellipticl segment 99 re of ellipse 557 which implies tht t 9 ds Tr It Eplortion A Eplortion B

60 SECTION Polr Equtions of Conics nd Kepler s Lws 75 Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph Grphicl Resoning In Eercises, use grphing utilit to grph the polr eqution when () e, (b) e 5, nd (c) e 5 Identif the conic (e) (f) e e r r e cos e cos e e r r e sin e sin 5 Writing Consider the polr eqution () Use grphing utilit to grph the eqution for e, e 5, e 5, e 75, nd e 9 Identif the conic nd discuss the chnge in its shpe s e nd e (b) Use grphing utilit to grph the eqution for Identif the conic (c) Use grphing utilit to grph the eqution for e, e 5, nd e Identif the conic nd discuss the chnge in its shpe s e nd e Consider the polr eqution () Identif the conic without grphing the eqution (b) Without grphing the following polr equtions, describe how ech differs from the polr eqution bove (c) Verif the results of prt (b) grphicll In Eercises 7, mtch the polr eqution with the correct grph [The grphs re lbeled (), (b), (c), (d), (e), nd (f)] () (c) r r e sin cos r, cos r sin (b) (d) e 7 r 8 r cos cos 9 r r sin sin r r sin cos In Eercises, find the eccentricit nd the distnce from the pole to the directri of the conic Then sketch nd identif the grph Use grphing utilit to confirm our results In Eercises, use grphing utilit to grph the polr eqution Identif the grph 5 r r r r r r r r r r r r sin cos cos 5 5 sin r sin r cos 5 cos 7 sin sin cos sin sin cos sin

61 75 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes In Eercises 7, use grphing utilit to grph the conic Describe how the grph differs from tht in the indicted eercise 7 r (See Eercise ) sin 8 r (See Eercise ) cos 9 r (See Eercise 5) cos r (See Eercise ) 7 sin Write the eqution for the ellipse rotted rdin clockwise from the ellipse r Write the eqution for the prbol rotted rdin counterclockwise from the prbol r In Eercises, find polr eqution for the conic with its focus t the pole (For convenience, the eqution for the directri is given in rectngulr form) Conic Prbol Prbol 5 Ellipse Ellipse 7 Hperbol 8 Hperbol Conic 9 Prbol Prbol Ellipse Ellipse Hperbol Hperbol 5 5 cos sin Eccentricit e e e e e e Verte or Vertices, 5,,, 8,,,,,,,,, 9, Directri 9 Show tht the polr eqution for is b Ellipse 5 Show tht the polr eqution for is b Hperbol In Eercises 5 5, use the results of Eercises 9 nd 5 to write the polr form of the eqution of the conic 5 Ellipse: focus t (, ); vertices t (5, ), 5, 5 Hperbol: focus t (5, ); vertices t (, ),, 5 5 In Eercises 55 nd 5, use the integrtion cpbilities of grphing utilit to pproimte to two deciml plces the re of the region bounded b the grph of the polr eqution 55 Writing About Concepts 5 Clssif the conics b their eccentricities Eplin how the grph of ech conic differs from the grph 7 Identif ech conic r r r 5 r of r sin () r cos (c) r cos 5 () r cos 5 (c) r cos 8 Describe wht hppens to the distnce between the directri nd the center of n ellipse if the foci remin fied nd e pproches b e cos b e cos 9 cos sin (b) r sin (d) r sin 5 (b) r sin 5 (d) r sin

62 SECTION Polr Equtions of Conics nd Kepler s Lws Eplorer 8 On November 7, 9, the United Sttes lunched Eplorer 8 Its low nd high points bove the surfce of Erth were pproimtel 9 miles nd, miles (see figure) The center of Erth is the focus of the orbit Find the polr eqution for the orbit nd find the distnce between the surfce of Erth nd the stellite when (Assume tht the rdius of Erth is miles) 9 Eplorer 8 58 Plnetr Motion The plnets trvel in ellipticl orbits with the sun s focus, s shown in the figure r Erth r Sun Plnet () Show tht the polr eqution of the orbit is given b r e e cos where e is the eccentricit (b) Show tht the minimum distnce (perihelion) from the sun to the plnet is r e nd the mimum distnce (phelion) is r e In Eercises 59, use Eercise 58 to find the polr eqution of the ellipticl orbit of the plnet, nd the perihelion nd phelion distnces 59 Erth 9 8 kilometers e 7 Sturn 7 9 kilometers e 5 Pluto 59 9 kilometers e 88 Mercur kilometers e 5 Not drwn to scle Not drwn to scle Plnetr Motion In Eercise, the polr eqution for the ellipticl orbit of Pluto ws found Use the eqution nd computer lgebr sstem to perform ech of the following () Approimte the re swept out b r from the sun to the plnet s increses from to 9 Use this result to determine the number of ers for the plnet to move through this rc if the period of one revolution round the sun is 8 ers (b) B tril nd error, pproimte the ngle such tht the re swept out b r from the sun to the plnet s increses from to equls the re found in prt () (see figure) Does the r sweep through lrger or smller ngle thn in prt () to generte the sme re? Wh is this the cse? (c) Approimte the distnces the plnet trveled in prts () nd (b) Use these distnces to pproimte the verge number of kilometers per er the plnet trveled in the two cses Comet Hle-Bopp The comet Hle-Bopp hs n elliptic orbit with the sun t one focus nd hs n eccentricit of e 995 The length of the mjor is of the orbit is pproimtel 5 stronomicl units () Find the length of its minor is (b) Find polr eqution for the orbit (c) Find the perihelion nd phelion distnces In Eercises 5 nd, let r represent the distnce from the focus to the nerest verte, nd let r represent the distnce from the focus to the frthest verte 5 Show tht the eccentricit of n ellipse cn be written s e r r r Then show tht e r r e r Show tht the eccentricit of hperbol cn be written s e r r r Then show tht e r r e r In Eercises 7 nd 8, show tht the grphs of the given equtions intersect t right ngles ed 7 r nd sin = 9 r ed sin c d 8 r nd r cos cos