PARAMETRIC EQUATIONS AND POLAR COORDINATES

Transcription

1 PARAETRIC EQUATINS AND PLAR CRDINATES Prmetric equtions nd polr coordintes enble us to describe gret vriet of new curves some prcticl, some beutiful, some fnciful, some strnge. So fr we hve described plne curves b giving s function of f or s function of t or b giving reltion between nd tht defines implicitl s function of f,. In this chpter we discuss two new methods for describing curves. Some curves, such s the ccloid, re best hndled when both nd re given in terms of third vrible t clled prmeter f t, tt. ther curves, such s the crdioid, hve their most convenient description when we use new coordinte sstem, clled the polr coordinte sstem. 6

2 . CURVES DEFINED BY PARAETRIC EQUATINS FIGURE C (, )={ f(t), g(t)} Imgine tht prticle moves long the curve C shown in Figure. It is impossible to describe C b n eqution of the form f becuse C fils the Verticl Line Test. But the - nd -coordintes of the prticle re functions of time nd so we cn write f t nd tt. Such pir of equtions is often convenient w of describing curve nd gives rise to the following definition. Suppose tht nd re both given s functions of third vrible t (clled prmeter) b the equtions f t tt (clled prmetric equtions). Ech vlue of t determines point,, which we cn plot in coordinte plne. As t vries, the point, f t, tt vries nd trces out curve C, which we cll prmetric curve. The prmeter t does not necessril represent time nd, in fct, we could use letter other thn t for the prmeter. But in mn pplictions of prmetric curves, t does denote time nd therefore we cn interpret, f t, tt s the position of prticle t time t. EXAPLE Sketch nd identif the curve defined b the prmetric equtions t t t SLUTIN Ech vlue of t gives point on the curve, s shown in the tble. For instnce, if t, then, nd so the corresponding point is,. In Figure we plot the points, determined b severl vlues of the prmeter nd we join them to produce curve. t t= t= t= (, ) t=3 t=_ t=4 8 t=_ FIGURE N This eqution in nd describes where the prticle hs been, but it doesn t tell us when the prticle ws t prticulr point. The prmetric equtions hve n dvntge the tell us when the prticle ws t point. The lso indicte the direction of the motion. A prticle whose position is given b the prmetric equtions moves long the curve in the direction of the rrows s t increses. Notice tht the consecutive points mrked on the curve pper t equl time intervls but not t equl distnces. Tht is becuse the prticle slows down nd then speeds up s t increses. It ppers from Figure tht the curve trced out b the prticle m be prbol. This cn be confirmed b eliminting the prmeter t s follows. We obtin t from the second eqution nd substitute into the first eqution. This gives t t 4 3 nd so the curve represented b the given prmetric equtions is the prbol

3 6 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES (8, 5) No restriction ws plced on the prmeter t in Emple, so we ssumed tht t could be n rel number. But sometimes we restrict t to lie in finite intervl. For instnce, the prmetric curve t t t t 4 (, ) FIGURE 3 shown in Figure 3 is the prt of the prbol in Emple tht strts t the point, nd ends t the point 8, 5. The rrowhed indictes the direction in which the curve is trced s t increses from to 4. In generl, the curve with prmetric equtions f t tt t b hs initil point f, t nd terminl point f b, tb. V EXAPLE Wht curve is represented b the following prmetric equtions? cos t sin t t SLUTIN If we plot points, it ppers tht the curve is circle. We cn confirm this impression b eliminting t. bserve tht cos t sin t Thus the point, moves on the unit circle. Notice tht in this emple the prmeter t cn be interpreted s the ngle (in rdins) shown in Figure 4. As t increses from to, the point, cos t, sin t moves once round the circle in the counterclockwise direction strting from the point,. π t= (cos t, sin t) t=π t t= (, ) t=π FIGURE 4 3π t= t=, π, π EXAPLE 3 Wht curve is represented b the given prmetric equtions? sin t cos t t (, ) SLUTIN Agin we hve sin t cos t so the prmetric equtions gin represent the unit circle. But s t increses from to, the point, sin t, cos t strts t, nd moves twice round the circle in the clockwise direction s indicted in Figure 5. FIGURE 5 Emples nd 3 show tht different sets of prmetric equtions cn represent the sme curve. Thus we distinguish between curve, which is set of points, nd prmetric curve, in which the points re trced in prticulr w.

4 SECTIN. CURVES DEFINED BY PARAETRIC EQUATINS 63 EXAPLE 4 Find prmetric equtions for the circle with center h, k nd rdius r. SLUTIN If we tke the equtions of the unit circle in Emple nd multipl the epressions for nd b r, we get r cos t, r sin t. You cn verif tht these equtions represent circle with rdius r nd center the origin trced counterclockwise. We now shift h units in the -direction nd k units in the -direction nd obtin prmetric equtions of the circle (Figure 6) with center h, k nd rdius r: h r cos t k r sin t t r (h, k) FIGURE 6 =h+r cos t, =k+r sin t (_, ) (, ) V EXAPLE 5 Sketch the curve with prmetric equtions sin t, sin t. FIGURE 7 SLUTIN bserve tht sin t nd so the point, moves on the prbol. But note lso tht, since sin t, we hve, so the prmetric equtions represent onl the prt of the prbol for which. Since sin t is periodic, the point, sin t, sin t moves bck nd forth infinitel often long the prbol from, to,. (See Figure 7.) TEC odule.a gives n nimtion of the reltionship between motion long prmetric curve f t, tt nd motion long the grphs of f nd t s functions of t. Clicking on TRIG gives ou the fmil of prmetric curves =cos t cos bt c sin dt If ou choose b c d nd click on nimte, ou will see how the grphs of cos t nd sin t relte to the circle in Emple. If ou choose b c, d, ou will see grphs s in Figure 8. B clicking on nimte or moving the t-slider to the right, ou cn see from the color coding how motion long the grphs of cos t nd sin t corresponds to motion long the prmetric curve, which is clled Lissjous figure. t t FIGURE 8 =cos t =sin t =sin t

5 64 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES GRAPHING DEVICES ost grphing clcultors nd computer grphing progrms cn be used to grph curves defined b prmetric equtions. In fct, it s instructive to wtch prmetric curve being drwn b grphing clcultor becuse the points re plotted in order s the corresponding prmeter vlues increse. _3 3 3 _3 EXAPLE 6 Use grphing device to grph the curve 4 3. SLUTIN If we let the prmeter be t, then we hve the equtions t 4 3t t Using these prmetric equtions to grph the curve, we obtin Figure 9. It would be possible to solve the given eqution 4 3 for s four functions of nd grph them individull, but the prmetric equtions provide much esier method. FIGURE 9 In generl, if we need to grph n eqution of the form t, we cn use the prmetric equtions tt t 8 Notice lso tht curves with equtions f (the ones we re most fmilir with grphs of functions) cn lso be regrded s curves with prmetric equtions Grphing devices re prticulrl useful when sketching complicted curves. For instnce, the curves shown in Figures,, nd would be virtull impossible to produce b hnd..5 t f t _ _ _.5 _ FIGURE =t+ sin t =t+ cos 5t FIGURE =.5 cos t-cos 3t =.5 sin t-sin 3t FIGURE =sin(t+cos t) =cos(t+sin t) ne of the most importnt uses of prmetric curves is in computer-ided design (CAD). In the Lbortor Project fter Section. we will investigte specil prmetric curves, clled Bézier curves, tht re used etensivel in mnufcturing, especill in the utomotive industr. These curves re lso emploed in specifing the shpes of letters nd other smbols in lser printers. THE CYCLID TEC An nimtion in odule.b shows how the ccloid is formed s the circle moves. EXAPLE 7 The curve trced out b point P on the circumference of circle s the circle rolls long stright line is clled ccloid (see Figure 3). If the circle hs rdius r nd rolls long the -is nd if one position of P is the origin, find prmetric equtions for the ccloid.

6 SECTIN. CURVES DEFINED BY PARAETRIC EQUATINS 65 P P FIGURE 3 P r P r FIGURE 4 C(r, r) Q T SLUTIN We choose s prmeter the ngle of rottion of the circle when P is t the origin). Suppose the circle hs rotted through rdins. Becuse the circle hs been in contct with the line, we see from Figure 4 tht the distnce it hs rolled from the origin is rc PT r T Therefore the center of the circle is Cr, r. Let the coordintes of P be,. Then from Figure 4 we see tht T PQ r r sin r sin TC QC r r cos r cos Therefore prmetric equtions of the ccloid re r sin r cos A ne rch of the ccloid comes from one rottion of the circle nd so is described b. Although Equtions were derived from Figure 4, which illustrtes the cse where, it cn be seen tht these equtions re still vlid for other vlues of (see Eercise 39). Although it is possible to eliminte the prmeter from Equtions, the resulting Crtesin eqution in nd is ver complicted nd not s convenient to work with s the prmetric equtions. ccloid FIGURE 5 P P FIGURE 6 P P B P ne of the first people to stud the ccloid ws Glileo, who proposed tht bridges be built in the shpe of ccloids nd who tried to find the re under one rch of ccloid. Lter this curve rose in connection with the brchistochrone problem: Find the curve long which prticle will slide in the shortest time (under the influence of grvit) from point A to lower point B not directl beneth A. The Swiss mthemticin John Bernoulli, who posed this problem in 696, showed tht mong ll possible curves tht join A to B, s in Figure 5, the prticle will tke the lest time sliding from A to B if the curve is prt of n inverted rch of ccloid. The Dutch phsicist Hugens hd lred shown tht the ccloid is lso the solution to the tutochrone problem; tht is, no mtter where prticle P is plced on n inverted ccloid, it tkes the sme time to slide to the bottom (see Figure 6). Hugens proposed tht pendulum clocks (which he invented) swing in ccloidl rcs becuse then the pendulum tkes the sme time to mke complete oscilltion whether it swings through wide or smll rc. FAILIES F PARAETRIC CURVES V EXAPLE 8 Investigte the fmil of curves with prmetric equtions cos t tn t sin t Wht do these curves hve in common? How does the shpe chnge s increses?

7 66 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use grphing device to produce the grphs for the cses,,.5,.,,.5,, nd shown in Figure 7. Notice tht ll of these curves (ecept the cse ) hve two brnches, nd both brnches pproch the verticl smptote s pproches from the left or right. =_ =_ =_.5 =_. = =.5 = = FIGURE 7 embers of the fmil =+cos t, = tn t+sin t, ll grphed in the viewing rectngle _4, 4 b _4, 4 When, both brnches re smooth; but when reches, the right brnch cquires shrp point, clled cusp. For between nd the cusp turns into loop, which becomes lrger s pproches. When, both brnches come together nd form circle (see Emple ). For between nd, the left brnch hs loop, which shrinks to become cusp when. For, the brnches become smooth gin, nd s increses further, the become less curved. Notice tht the curves with positive re reflections bout the -is of the corresponding curves with negtive. These curves re clled conchoids of Nicomedes fter the ncient Greek scholr Nicomedes. He clled them conchoids becuse the shpe of their outer brnches resembles tht of conch shell or mussel shell.. EXERCISES 4 Sketch the curve b using the prmetric equtions to plot points. Indicte with n rrow the direction in which the curve is trced s t increses.. st, t 4t,. cos t, t cos t, 3. 5 sin t, t, 4. e t t, e t t, 5 () Sketch the curve b using the prmetric equtions to plot points. Indicte with n rrow the direction in which the curve is trced s t increses. (b) Eliminte the prmeter to find Crtesin eqution of the curve. 5. 3t 5, t 6. t, 5 t, t 5 t t t t 3 7. t, 5 t, 3 t t, 9. st,. t, 8 () Eliminte the prmeter to find Crtesin eqution of the curve. (b) Sketch the curve nd indicte with n rrow the direction in which the curve is trced s the prmeter increses.. sin, cos,. 4 cos, 5 sin, 3. sin t, csc t, 4. e t, 5. e t, t t 3 e t t 6. ln t, st, t 7. sinh t, cosh t t t

8 SECTIN. CURVES DEFINED BY PARAETRIC EQUATINS cosh t, 5 sinh t 9 Describe the motion of prticle with position, s t vries in the given intervl cos t, sin t, t 3. sin t, 4 cos t, t 3. 5 sin t, cos t,. sin t, cos t, t 5 t 5 7 Use the grphs of f t nd tt to sketch the prmetric curve f t, tt. Indicte with rrows the direction in which the curve is trced s t increses. 5. t t _ Suppose curve is given b the prmetric equtions f t, tt, where the rnge of f is, 4 nd the rnge of t is, 3. Wht cn ou s bout the curve? 4. tch the grphs of the prmetric equtions f t nd tt in () (d) with the prmetric curves lbeled I IV. Give resons for our choices. () I 7. t t t t (b) t t t t II 8. tch the prmetric equtions with the grphs lbeled I-VI. Give resons for our choices. (Do not use grphing device.) () t 4 t, t (b) t t, st (c) sin t, sint sin t (d) cos 5t, sin t (e) t sin 4t, t cos 3t sin t cos t (f), 4 t 4 t I II III (c) III t t IV V VI (d) IV t t ; 9. Grph the curve ; 3. Grph the curves 5 nd nd find their points of intersection correct to one deciml plce.

9 68 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES 3. () Show tht the prmetric equtions where t, describe the line segment tht joins the points P, nd P,. (b) Find prmetric equtions to represent the line segment from, 7 to 3,. ; 3. Use grphing device nd the result of Eercise 3() to drw the tringle with vertices A,, B4,, nd C, 5. ; 34. () Find prmetric equtions for the ellipse b. [Hint: odif the equtions of the circle in Emple.] (b) Use these prmetric equtions to grph the ellipse when 3 nd b,, 4, nd 8. (c) How does the shpe of the ellipse chnge s b vries? ; Use grphing clcultor or computer to reproduce the picture t t 33. Find prmetric equtions for the pth of prticle tht moves long the circle 4 in the mnner described. () nce round clockwise, strting t, (b) Three times round counterclockwise, strting t, (c) Hlfw round counterclockwise, strting t, If nd b re fied numbers, find prmetric equtions for the curve tht consists of ll possible positions of the point P in the figure, using the ngle s the prmeter. Then eliminte the prmeter nd identif the curve. b P 4. If nd b re fied numbers, find prmetric equtions for the curve tht consists of ll possible positions of the point P in the figure, using the ngle s the prmeter. The line segment AB is tngent to the lrger circle. A b P B Compre the curves represented b the prmetric equtions. How do the differ? 37. () t 3, t (b) t 6, t 4 (c) e 3t, e t 38. () t, t (b) cos t, (c) e t, e t 39. Derive Equtions for the cse. 4. Let P be point t distnce d from the center of circle of rdius r. The curve trced out b P s the circle rolls long stright line is clled trochoid. (Think of the motion of point on spoke of biccle wheel.) The ccloid is the specil cse of trochoid with d r. Using the sme prmeter s for the ccloid nd, ssuming the line is the -is nd when P is t one of its lowest points, show tht prmetric equtions of the trochoid re r d sin r d cos Sketch the trochoid for the cses d r nd d r. sec t 43. A curve, clled witch of ri Agnesi, consists of ll possible positions of the point P in the figure. Show tht prmetric equtions for this curve cn be written s cot Sketch the curve. = sin 44. () Find prmetric equtions for the set of ll points P s shown in the figure such tht P AB. (This curve is clled the cissoid of Diocles fter the Greek scholr Diocles, who introduced the cissoid s grphicl method for constructing the edge of cube whose volume is twice tht of given cube.) A C P

10 LABRATRY PRJECT RUNNING CIRCLES ARUND CIRCLES 69 (b) Use the geometric description of the curve to drw rough sketch of the curve b hnd. Check our work b using the prmetric equtions to grph the curve. ; 45. Suppose tht the position of one prticle t time t is given b 3 sin t nd the position of second prticle is given b 3 cos t () Grph the pths of both prticles. How mn points of intersection re there? (b) Are n of these points of intersection collision points? In other words, re the prticles ever t the sme plce t the sme time? If so, find the collision points. (c) Describe wht hppens if the pth of the second prticle is given b 3 cos t sin t t 46. If projectile is fired with n initil velocit of v meters per second t n ngle bove the horizontl nd ir resistnce is ssumed to be negligible, then its position fter t seconds is P A cos t B sin t = t t given b the prmetric equtions v cos t where t is the ccelertion due to grvit ( 9.8 ms ). () If gun is fired with nd v 5 ms, when will the bullet hit the ground? How fr from the gun will it hit the ground? Wht is the mimum height reched b the bullet? ; (b) Use grphing device to check our nswers to prt (). Then grph the pth of the projectile for severl other vlues of the ngle to see where it hits the ground. Summrize our findings. (c) Show tht the pth is prbolic b eliminting the prmeter. 3 v sin t tt ; 47. Investigte the fmil of curves defined b the prmetric equtions t, t 3 ct. How does the shpe chnge s c increses? Illustrte b grphing severl members of the fmil. ; 48. The swllowtil ctstrophe curves re defined b the prmetric equtions ct 4t 3, ct 3t 4. Grph severl of these curves. Wht fetures do the curves hve in common? How do the chnge when c increses? ; 49. The curves with equtions sin nt, b cos t re clled Lissjous figures. Investigte how these curves vr when, b, nd n vr. (Tke n to be positive integer.) ; 5. Investigte the fmil of curves defined b the prmetric equtions cos t, sin t sin ct, where c. Strt b letting c be positive integer nd see wht hppens to the shpe s c increses. Then eplore some of the possibilities tht occur when c is frction. LABRATRY PRJECT ; RUNNING CIRCLES ARUND CIRCLES In this project we investigte fmilies of curves, clled hpoccloids nd epiccloids, tht re generted b the motion of point on circle tht rolls inside or outside nother circle. C b P (, ) A. A hpoccloid is curve trced out b fied point P on circle C of rdius b s C rolls on the inside of circle with center nd rdius. Show tht if the initil position of P is, nd the prmeter is chosen s in the figure, then prmetric equtions of the hpoccloid re b cos b cos b b b sin b sin b b. Use grphing device (or the interctive grphic in TEC odule.b) to drw the grphs of hpoccloids with positive integer nd b. How does the vlue of ffect the grph? Show tht if we tke 4, then the prmetric equtions of the hpoccloid reduce to TEC Look t odule.b to see how hpoccloids nd epiccloids re formed b the motion of rolling circles. 4 cos 3 4 sin 3 This curve is clled hpoccloid of four cusps, or n stroid.

11 63 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES 3. Now tr b nd nd, frction where n nd d hve no common fctor. First let n nd tr to determine grphicll the effect of the denomintor d on the shpe of the grph. Then let n vr while keeping d constnt. Wht hppens when n d? 4. Wht hppens if b nd is irrtionl? Eperiment with n irrtionl number like s or e. Tke lrger nd lrger vlues for nd speculte on wht would hppen if we were to grph the hpoccloid for ll rel vlues of. 5. If the circle C rolls on the outside of the fied circle, the curve trced out b P is clled n epiccloid. Find prmetric equtions for the epiccloid. 6. Investigte the possible shpes for epiccloids. Use methods similr to Problems 4.. CALCULUS WITH PARAETRIC CURVES Hving seen how to represent curves b prmetric equtions, we now ppl the methods of clculus to these prmetric curves. In prticulr, we solve problems involving tngents, re, rc length, nd surfce re. TANGENTS In the preceding section we sw tht some curves defined b prmetric equtions f t nd tt cn lso be epressed, b eliminting the prmeter, in the form F. (See Eercise 67 for generl conditions under which this is possible.) If we substitute f t nd tt in the eqution F, we get tt F f t nd so, if t, F, nd f re differentible, the Chin Rule gives tt F f tf t F f t If f t, we cn solve for F: F tt f t Since the slope of the tngent to the curve F t, F is F, Eqution enbles us to find tngents to prmetric curves without hving to eliminte the prmeter. Using Leibniz nottion, we cn rewrite Eqution in n esil remembered form: N If we think of prmetric curve s being trced out b moving prticle, then ddt nd ddt re the verticl nd horizontl velocities of the prticle nd Formul ss tht the slope of the tngent is the rtio of these velocities. d d d dt d dt if d dt It cn be seen from Eqution tht the curve hs horizontl tngent when ddt (provided tht ddt ) nd it hs verticl tngent when ddt (provided tht ddt ). This informtion is useful for sketching prmetric curves.

12 SECTIN. CALCULUS WITH PARAETRIC CURVES 63 Note tht d d d dt d dt As we know from Chpter 4, it is lso useful to consider d d. This cn be found b replcing b dd in Eqution : d d dt d d d d d d d d dt EXAPLE A curve C is defined b the prmetric equtions t, t 3 3t. () Show tht C hs two tngents t the point (3, ) nd find their equtions. (b) Find the points on C where the tngent is horizontl or verticl. (c) Determine where the curve is concve upwrd or downwrd. (d) Sketch the curve. SLUTIN () Notice tht t 3 3t tt 3 when t or t s3. Therefore the point 3, on C rises from two vlues of the prmeter, t s3 nd t s3. This indictes tht C crosses itself t 3,. Since d ddt d ddt 3t 3 3 t t t the slope of the tngent when t s3 is dd 6(s3 ) s3,so the equtions of the tngents t 3, re s3 3 nd s3 3 t=_ (, ) t= (, _) FIGURE =œ 3(-3) (3, ) =_ œ 3(-3) (b) C hs horizontl tngent when dd, tht is, when ddt nd ddt. Since ddt 3t 3,this hppens when t,tht is, t. The corresponding points on C re, nd (, ). C hs verticl tngent when ddt t,tht is, t. (Note tht ddt there.) The corresponding point on C is (, ). (c) To determine concvit we clculte the second derivtive: d d d dt d d 3 t 3t d t 4t 3 dt Thus the curve is concve upwrd when t nd concve downwrd when t. (d) Using the informtion from prts (b) nd (c), we sketch C in Figure. V EXAPLE () Find the tngent to the ccloid r sin, r cos t the point where. (See Emple 7 in Section..) (b) At wht points is the tngent horizontl? When is it verticl? 3 SLUTIN () The slope of the tngent line is d d dd dd r sin r cos sin cos

13 63 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES When 3, we hve r 3 sin 3 r 3 s3 r cos 3 r nd d d sin3 cos3 s3 s3 Therefore the slope of the tngent is s3 nd its eqution is r s3 r 3 The tngent is sketched in Figure. rs3 or s3 r s3 (_πr, r) (πr, r) (3πr, r) (5πr, r) π = 3 FIGURE πr 4πr (b) The tngent is horizontl when dd, which occurs when sin nd cos,tht is,, n n integer. The corresponding point on the ccloid is n r, r. When, both nd re. It ppers from the grph tht there re verticl tngents t those points. We cn verif this b using l Hospitl s Rule s follows: n lim ln A similr computtion shows tht dd l s, so indeed there re verticl tngents when, tht is, when nr. n n dd d d lim ln dd sin lim cos ln l n cos sin AREAS N The limits of integrtion for t re found s usul with the Substitution Rule. When, t is either. When b, t is the remining vlue. or We know tht the re under curve F from to b is A b F d,where F. If the curve is trced out once b the prmetric equtions f t nd tt,,then we cn clculte n re formul b using the Substitution Rule for Definite Integrls s follows: A or tt f t dt b d tt f t dt t V EXAPLE 3 Find the re under one rch of the ccloid (See Figure 3.) r sin r cos

14 SECTIN. CALCULUS WITH PARAETRIC CURVES 633 FIGURE 3 πr N The result of Emple 3 ss tht the re under one rch of the ccloid is three times the re of the rolling circle tht genertes the ccloid (see Emple 7 in Section.). Glileo guessed this result but it ws first proved b the French mthemticin Robervl nd the Itlin mthemticin Torricelli. SLUTIN ne rch of the ccloid is given b. Using the Substitution Rule with r cos nd d r cos d, we hve ARC LENGTH We lred know how to find the length L of curve C given in the form F, b. Formul 8..3 ss tht if F is continuous, then 3 A r d r cos r cos d r cos d r cos cos d r [ cos cos ] d r [ 3 r ( 3 sin 4 sin ] ) 3r b L d d d Suppose tht C cn lso be described b the prmetric equtions f t nd tt,,where ddt f t. This mens tht C is trversed once,from left to right, s t increses from to nd f, f b. Putting Formul into Formul 3 nd using the Substitution Rule, we obtin t b L d d d ddt d ddt dt dt P P C P i_ P i Since ddt, we hve 4 L dt d dt d dt FIGURE 4 P P n Even if C cn t be epressed in the form F, Formul 4 is still vlid but we obtin it b polgonl pproimtions. We divide the prmeter intervl, into n subintervls of equl width t. If t, t, t,..., t n re the endpoints of these subintervls, then i f t i nd i tt i re the coordintes of points P i i, i tht lie on C nd the polgon with vertices P, P,..., P n pproimtes C. (See Figure 4.) As in Section 8., we define the length L of C to be the limit of the lengths of these pproimting polgons s n l : L lim nl n P ip i i The en Vlue Theorem, when pplied to f on the intervl t i, t i, gives number t i * in t i, t i such tht f t i f t i f t i *t i t i If we let i i i nd i i i,this eqution becomes i f t i * t

15 634 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES Similrl, when pplied to t, the en Vlue Theorem gives number t i ** in t i, t i such tht i tt i ** t Therefore P ip i s i i s f t i * tt i ** t nd so s f t i *t tt i **t 5 L lim n l n s f t i * tt i ** t i The sum in (5) resembles Riemnn sum for the function s f t tt but it is not ectl Riemnn sum becuse t i * t i ** in generl. Nevertheless, if f nd t re continuous, it cn be shown tht the limit in (5) is the sme s if t i * nd t i ** were equl, nmel, Thus, using Leibniz nottion, we hve the following result, which hs the sme form s Formul (4). 6 THERE If curve C is described b the prmetric equtions f t, tt,, where f nd t re continuous on, nd C is trversed ectl once s t increses from to, then the length of C is L dt dt d dt t L d s f t tt dt Notice tht the formul in Theorem 6 is consistent with the generl formuls L ds nd ds d d of Section 8.. EXAPLE 4 If we use the representtion of the unit circle given in Emple in Section., cos t sin t t then ddt sin t nd ddt cos t,so Theorem 6 gives L d dt dt dt d dt ssin t cos t dt s epected. If, on the other hnd, we use the representtion given in Emple 3 in Section., sin t cos t t then ddt cos t, ddt sin t,nd the integrl in Theorem 6 gives dt d dt d dt s4 cos t 4 sin t dt dt 4

16 SECTIN. CALCULUS WITH PARAETRIC CURVES 635 Notice tht the integrl gives twice the rc length of the circle becuse s t increses from to, the point sin t, cos t trverses the circle twice. In generl, when finding the length of curve C from prmetric representtion, we hve to be creful to ensure tht C is trversed onl once s t increses from to. V EXAPLE 5 Find the length of one rch of the ccloid r sin, r cos. SLUTIN From Emple 3 we see tht one rch is described b the prmeter intervl. Since d d r cos nd d d r sin N The result of Emple 5 ss tht the length of one rch of ccloid is eight times the rdius of the generting circle (see Figure 5). This ws first proved in 658 b Sir Christopher Wren, who lter becme the rchitect of St. Pul s Cthedrl in London. r L=8r we hve L sr cos cos sin d r s cos d d d d d d To evlute this integrl we use the identit sin cos with, which gives cos sin. Since, we hve nd so sin. Therefore s cos s4 sin sin sin sr cos r sin d πr nd so L r sin d r cos] FIGURE 5 r 8r SURFACE AREA In the sme w s for rc length, we cn dpt Formul 8..5 to obtin formul for surfce re. If the curve given b the prmetric equtions f t, tt,, is rotted bout the -is, where f, t re continuous nd tt, then the re of the resulting surfce is given b 7 S The generl smbolic formuls S ds nd S ds (Formuls 8..7 nd 8..8) re still vlid, but for prmetric curves we use ds dt d dt d dt EXAPLE 6 Show tht the surfce re of sphere of rdius r is 4r. SLUTIN The sphere is obtined b rotting the semicircle r cos t dt d dt d dt r sin t t t

17 636 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES bout the -is. Therefore, from Formul 7, we get S r sin t sr sin t r cos t dt r sin t sr sin t cos t dt r sin t r dt r sin t dt r cos t] 4r. EXERCISES Find dd.. t sin t, t t. t, st e t 9. cos,. cos 3, sin sin 3 6 Find n eqution of the tngent to the curve t the point corresponding to the given vlue of the prmeter. 3. t 4, t 3 t; 4. t t, t ; 5. e st, t ln t ; 6. cos sin, sin cos ; 7 8 Find n eqution of the tngent to the curve t the given point b two methods: () without eliminting the prmeter nd (b) b first eliminting the prmeter. ; 9 Find n eqution of the tngent(s) to the curve t the given point. Then grph the curve nd the tngent(s) sin t, t t;. cos t cos t, sin t sin t ; 6 Find dd nd d d. For which vlues of t is the curve concve upwrd?. 4 t, t t 3. t 3 t, t 3. t e t, t e t 4. t ln t, t ln t 5. sin t, 3 cos t, 6. cos t, cos t, 7 Find the points on the curve where the tngent is horizontl or verticl. If ou hve grphing device, grph the curve to check our work. 7. t, t 3 t t t t 7. ln t, t ;, 3 8. tn, sec ; (, s), t t 8. t 3 3t t, t 3 3t, ;. Use grph to estimte the coordintes of the rightmost point on the curve t t 6, e t. Then use clculus to find the ect coordintes. ;. Use grph to estimte the coordintes of the lowest point nd the leftmost point on the curve t 4 t, t t 4. Then find the ect coordintes. ; 3 4 Grph the curve in viewing rectngle tht displs ll the importnt spects of the curve. 3. t 4 t 3 t, 4. t 4 4t 3 8t, 5. Show tht the curve cos t, sin t cos t hs two tngents t, nd find their equtions. Sketch the curve. ; 6. Grph the curve cos t cos t, sin t sin t to discover where it crosses itself. Then find equtions of both tngents t tht point. 7. () Find the slope of the tngent line to the trochoid r d sin, r d cos in terms of. (See Eercise 4 in Section..) (b) Show tht if d r, then the trochoid does not hve verticl tngent. 8. () Find the slope of the tngent to the stroid, sin 3 cos 3 in terms of. (Astroids re eplored in the Lbortor Project on pge 69.) (b) At wht points is the tngent horizontl or verticl? (c) At wht points does the tngent hve slope or? 9. At wht points on the curve t 3, 4t t does the tngent line hve slope? 3. Find equtions of the tngents to the curve 3t, t 3 tht pss through the point 4, Use the prmetric equtions of n ellipse, cos, b sin,,to find the re tht it encloses. t 3 t t t

18 SECTIN. CALCULUS WITH PARAETRIC CURVES Find the re enclosed b the curve t t, st nd the -is. 33. Find the re enclosed b the -is nd the curve e t, t t. 34. Find the re of the region enclosed b the stroid cos 3, sin 3. (Astroids re eplored in the Lbortor Project on pge 69.) _ 49. Use Simpson s Rule with n 6 to estimte the length of the curve t e t, t e t, 6 t In Eercise 43 in Section. ou were sked to derive the prmetric equtions cot, sin for the curve clled the witch of ri Agnesi. Use Simpson s Rule with n 4 to estimte the length of the rc of this curve given b Find the distnce trveled b prticle with position, s t vries in the given time intervl. Compre with the length of the curve. 5. sin t, cos t, t 3 5. cos t, cos t, t Find the re under one rch of the trochoid of Eercise 4 in Section. for the cse d r. 36. Let be the region enclosed b the loop of the curve in Emple. () Find the re of. (b) If is rotted bout the -is, find the volume of the resulting solid. (c) Find the centroid of Set up n integrl tht represents the length of the curve. Then use our clcultor to find the length correct to four deciml plces. 37. t t, 4 3 t 3, t 38. e t, t, 3 t t cos t, t sin t, 4. ln t, st, 4 44 Find the ect length of the curve. 4. 3t, 4 t 3, t 4. e t e t, 5 t, t t, ln t, t cos t cos 3t, 3 sin t sin 3t, ; Grph the curve nd find its length. 45. e t cos t, e t sin t, t 46. cos t ln(tn t), sin t, 4 t e t t, 4e t, _ t 5 t 8 t 3 t t 48. Find the length of the loop of the curve 3t t 3, 3t. CAS CAS 53. Show tht the totl length of the ellipse sin, b cos, b, is L 4 where e is the eccentricit of the ellipse (e c, where c s b ). 54. Find the totl length of the stroid cos 3, sin 3, where. 55. () Grph the epitrochoid with equtions cos t 4cost sin t 4sint Wht prmeter intervl gives the complete curve? (b) Use our CAS to find the pproimte length of this curve. 56. A curve clled Cornu s spirl is defined b the prmetric equtions Ct t cosu du St t sinu du where C nd S re the Fresnel functions tht were introduced in Chpter 5. () Grph this curve. Wht hppens s t l nd s t l? (b) Find the length of Cornu s spirl from the origin to the point with prmeter vlue t Set up n integrl tht represents the re of the surfce obtined b rotting the given curve bout the -is. Then use our clcultor to find the surfce re correct to four deciml plces. 57. te t, t e t, t 58. sin t, sin 3t, t 3 s e sin d

19 638 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES 59 6 Find the ect re of the surfce obtined b rotting the given curve bout the -is. 59. t 3, t, t 6. 3t t 3, 3t, t 6. cos 3, sin 3, ; 6. Grph the curve If this curve is rotted bout the -is, find the re of the resulting surfce. (Use our grph to help find the correct prmeter intervl.) 63. If the curve cos cos is rotted bout the -is, use our clcultor to estimte the re of the resulting surfce to three deciml plces. 64. If the rc of the curve in Eercise 5 is rotted bout the -is, estimte the re of the resulting surfce using Simpson s Rule with n Find the surfce re generted b rotting the given curve bout the -is. 66. e t t, 4e t, 67. If f is continuous nd f t for t b, show tht the prmetric curve f t, tt, t b, cn be put in the form F. [Hint: Show tht f eists.] 68. Use Formul to derive Formul 7 from Formul 8..5 for the cse in which the curve cn be represented in the form F, b. 69. The curvture t point P of curve is defined s ds t t t, t 3, t t t 5 t d where is the ngle of inclintion of the tngent line t P, s shown in the figure. Thus the curvture is the bsolute vlue of the rte of chnge of with respect to rc length. It cn be regrded s mesure of the rte of chnge of direction of the curve t P nd will be studied in greter detil in Chpter 3. () For prmetric curve t, t, derive the formul 3 sin sin t (b) B regrding curve f s the prmetric curve, f, with prmeter, show tht the formul in prt () becomes 7. () Use the formul in Eercise 69(b) to find the curvture of the prbol t the point,. (b) At wht point does this prbol hve mimum curvture? 7. Use the formul in Eercise 69() to find the curvture of the ccloid, cos t the top of one of its rches. 7. () Show tht the curvture t ech point of stright line is. (b) Show tht the curvture t ech point of circle of rdius r is. sin 73. A string is wound round circle nd then unwound while being held tut. The curve trced b the point P t the end of the string is clled the involute of the circle. If the circle hs rdius r nd center nd the initil position of P is r,, nd if the prmeter is chosen s in the figure, show tht prmetric equtions of the involute re rcos d d r sin dd A cow is tied to silo with rdius r b rope just long enough to rech the opposite side of the silo. Find the re vilble for grzing b the cow. r T rsin P P cos where the dots indicte derivtives with respect to t, so ddt.[hint: Use nd Formul to find ddt. Then use the Chin Rule to find dds.] tn dd

20 SECTIN.3 PLAR CRDINATES 639 LABRATRY PRJECT ; BÉZIER CURVES The Bézier curves re used in computer-ided design nd re nmed fter the French mthemticin Pierre Bézier (9 999), who worked in the utomotive industr. A cubic Bézier curve is determined b four control points, P,, P,, P,, nd P 3 3, 3, nd is defined b the prmetric equtions t 3 3 t t 3 t t 3t 3 t 3 3 t t 3 t t 3t 3 where t. Notice tht when t we hve,, nd when t we hve, 3, 3,so the curve strts t nd ends t.. Grph the Bézier curve with control points P 4,, P 8, 48, P 5, 4, nd P 34, 5. Then, on the sme screen, grph the line segments P P, P P, nd P P 3. (Eercise 3 in Section. shows how to do this.) Notice tht the middle control points P nd P don t lie on the curve; the curve strts t P, heds towrd P nd P without reching them, nd ends t P 3.. From the grph in Problem, it ppers tht the tngent t P psses through P nd the tngent t P 3 psses through. Prove it. P P 3. Tr to produce Bézier curve with loop b chnging the second control point in Problem. 4. Some lser printers use Bézier curves to represent letters nd other smbols. Eperiment with control points until ou find Bézier curve tht gives resonble representtion of the letter C. 5. ore complicted shpes cn be represented b piecing together two or more Bézier curves. Suppose the first Bézier curve hs control points P, P, P, P 3 nd the second one hs control points P 3, P 4, P 5, P 6. If we wnt these two pieces to join together smoothl, then the tngents t P 3 should mtch nd so the points P, P 3, nd P 4 ll hve to lie on this common tngent line. Using this principle, find control points for pir of Bézier curves tht represent the letter S. P 3.3 PLAR CRDINATES FIGURE r polr is P(r, ) A coordinte sstem represents point in the plne b n ordered pir of numbers clled coordintes. Usull we use Crtesin coordintes, which re directed distnces from two perpendiculr es. Here we describe coordinte sstem introduced b Newton, clled the polr coordinte sstem, which is more convenient for mn purposes. We choose point in the plne tht is clled the pole (or origin) nd is lbeled. Then we drw r (hlf-line) strting t clled the polr is. This is is usull drwn horizontll to the right nd corresponds to the positive -is in Crtesin coordintes. If P is n other point in the plne, let r be the distnce from to P nd let be the ngle (usull mesured in rdins) between the polr is nd the line P s in Figure. Then the point P is represented b the ordered pir r, nd r, re clled polr coordintes of P. We use the convention tht n ngle is positive if mesured in the counterclockwise direction from the polr is nd negtive in the clockwise direction. If P, then r nd we gree tht, represents the pole for n vlue of.

21 64 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES +π (r, ) We etend the mening of polr coordintes r, to the cse in which r is negtive b greeing tht, s in Figure, the points r, nd r, lie on the sme line through nd t the sme distnce r from, but on opposite sides of. If r, the point r, lies in the sme qudrnt s ; if r, it lies in the qudrnt on the opposite side of the pole. Notice tht r, represents the sme point s r,. (_r, ) FIGURE 5π 4 5π, 4 (, 3π) EXAPLE Plot the points whose polr coordintes re given. (), 54 (b), 3 (c), 3 (d) 3, 34 SLUTIN The points re plotted in Figure 3. In prt (d) the point 3, 34 is locted three units from the pole in the fourth qudrnt becuse the ngle 34 is in the second qudrnt nd r 3 is negtive. 3π π _ 3 3π 4 FIGURE 3 π, _ 3 3π _3, 4 In the Crtesin coordinte sstem ever point hs onl one representtion, but in the polr coordinte sstem ech point hs mn representtions. For instnce, the point, 54 in Emple () could be written s, 34 or, 34 or, 4. (See Figure 4.) 5π 4 _ 3π 4 3π 4 π 4 5π, 4 3π, _ 4 3π, 4 π _, 4 FIGURE 4 In fct, since complete counterclockwise rottion is given b n ngle, the point represented b polr coordintes r, is lso represented b r, n nd r, n r P(r, )=P(, ) where n is n integer. The connection between polr nd Crtesin coordintes cn be seen from Figure 5, in which the pole corresponds to the origin nd the polr is coincides with the positive -is. If the point P hs Crtesin coordintes, nd polr coordintes r,, then, from the figure, we hve cos sin r r nd so FIGURE 5 r cos r sin Although Equtions were deduced from Figure 5, which illustrtes the cse where r nd, these equtions re vlid for ll vlues of r nd. (See the generl definition of sin nd cos in Appendi D.)

22 SECTIN.3 PLAR CRDINATES 64 Equtions llow us to find the Crtesin coordintes of point when the polr coordintes re known. To find r nd when nd re known, we use the equtions r tn which cn be deduced from Equtions or simpl red from Figure 5. EXAPLE Convert the point, 3 from polr to Crtesin coordintes. 3 SLUTIN Since r nd, Equtions give r cos cos 3 r sin sin 3 s3 s3 Therefore the point is (, s3 ) in Crtesin coordintes. EXAPLE 3 Represent the point with Crtesin coordintes, in terms of polr coordintes. SLUTIN If we choose r to be positive, then Equtions give r s s s tn Since the point, lies in the fourth qudrnt, we cn choose or. Thus one possible nswer is (s, 4) ; nother is s, NTE Equtions do not uniquel determine when nd re given becuse, s increses through the intervl, ech vlue of tn occurs twice. Therefore, in converting from Crtesin to polr coordintes, it s not good enough just to find r nd tht stisf Equtions. As in Emple 3, we must choose so tht the point r, lies in the correct qudrnt. r= r= r= r=4 PLAR CURVES The grph of polr eqution r f, or more generll Fr,, consists of ll points P tht hve t lest one polr representtion r, whose coordintes stisf the eqution. V EXAPLE 4 Wht curve is represented b the polr eqution r? FIGURE 6 SLUTIN The curve consists of ll points r, with r. Since r represents the distnce from the point to the pole, the curve r represents the circle with center nd rdius. In generl, the eqution r represents circle with center nd rdius. (See Figure 6.)

23 64 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES = (_, ) (_, ) (, ) (, ) (3, ) EXAPLE 5 Sketch the polr curve. SLUTIN This curve consists of ll points r, such tht the polr ngle is rdin. It is the stright line tht psses through nd mkes n ngle of rdin with the polr is (see Figure 7). Notice tht the points r, on the line with r re in the first qudrnt, wheres those with r re in the third qudrnt. EXAPLE 6 () Sketch the curve with polr eqution r cos. (b) Find Crtesin eqution for this curve. FIGURE 7 SLUTIN () In Figure 8 we find the vlues of r for some convenient vlues of nd plot the corresponding points r,. Then we join these points to sketch the curve, which ppers to be circle. We hve used onl vlues of between nd, since if we let increse beond, we obtin the sme points gin. FIGURE 8 Tble of vlues nd grph of r= cos 6 s3 4 s s 56 s3 r cos π, π, 3 π _, 3 π œ, 4 π œ, 3 6 (, ) 5π _ œ, 3 3π 6 _ œ, 4 (b) To convert the given eqution to Crtesin eqution we use Equtions nd. From r cos we hve cos r, so the eqution r cos becomes r r, which gives r or Completing the squre, we obtin which is n eqution of circle with center, nd rdius. N Figure 9 shows geometricl illustrtion tht the circle in Emple 6 hs the eqution r cos. The ngle PQ is right ngle (Wh?) nd so r cos. r P Q FIGURE 9

24 SECTIN.3 PLAR CRDINATES 643 r π π 3π π FIGURE r=+sin in Crtesin coordintes, π V EXAPLE 7 Sketch the curve r sin. SLUTIN Insted of plotting points s in Emple 6, we first sketch the grph of r sin in Crtesin coordintes in Figure b shifting the sine curve up one unit. This enbles us to red t glnce the vlues of r tht correspond to incresing vlues of. For instnce, we see tht s increses from to, r (the distnce from ) increses from to, so we sketch the corresponding prt of the polr curve in Figure (). As increses from to, Figure shows tht r decreses from to, so we sketch the net prt of the curve s in Figure (b). As increses from to 3, r decreses from to s shown in prt (c). Finll, s increses from 3 to, r increses from to s shown in prt (d). If we let increse beond or decrese beond, we would simpl retrce our pth. Putting together the prts of the curve from Figure () (d), we sketch the complete curve in prt (e). It is clled crdioid, becuse it s shped like hert. = π = π = =π =π =π = 3π = 3π () (b) (c) (d) (e) FIGURE Stges in sketching the crdioid r=+sin TEC odule.3 helps ou see how polr curves re trced out b showing nimtions similr to Figures 3. EXAPLE 8 Sketch the curve r cos. SLUTIN As in Emple 7, we first sketch r cos,, in Crtesin coordintes in Figure. As increses from to 4, Figure shows tht r decreses from to nd so we drw the corresponding portion of the polr curve in Figure 3 (indicted b!). As increses from 4 to, r goes from to. This mens tht the distnce from increses from to, but insted of being in the first qudrnt this portion of the polr curve (indicted lies on the opposite side of the pole in the third qudrnt. The reminder of the curve is drwn in similr fshion, with the rrows nd numbers indicting the order in which the portions re trced out. The resulting curve hs four loops nd is clled four-leved rose. r = π! $ % * = 3π 4 $ & ^! = π 4 π 4 π 3π 4 # ^ & 5π 4 3π 7π 4 π =π # 8 = FIGURE r=cos in Crtesin coordintes FIGURE 3 Four-leved rose r=cos

25 644 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES SYETRY When we sketch polr curves, it is sometimes helpful to tke dvntge of smmetr. The following three rules re eplined b Figure 4. () If polr eqution is unchnged when is replced b, the curve is smmetric bout the polr is. (b) If the eqution is unchnged when r is replced b r, or when is replced b, the curve is smmetric bout the pole. (This mens tht the curve remins unchnged if we rotte it through 8 bout the origin.) (c) If the eqution is unchnged when is replced b, the curve is smmetric bout the verticl line. (r, ) (r, π- ) (r, ) _ (_r, ) (r, ) π- (r, _ ) () (b) (c) FIGURE 4 The curves sketched in Emples 6 nd 8 re smmetric bout the polr is, since cos cos. The curves in Emples 7 nd 8 re smmetric bout becuse sin nd cos cos. The four-leved rose is lso smmetric bout the pole. These smmetr properties could hve been used in sketching the curves. For instnce, in Emple 6 we need onl hve plotted points for nd then reflected bout the polr is to obtin the complete circle. sin TANGENTS T PLAR CURVES To find tngent line to polr curve r f, we regrd s prmeter nd write its prmetric equtions s r cos f cos r sin f sin Then, using the method for finding slopes of prmetric curves (Eqution..) nd the Product Rule, we hve 3 We locte horizontl tngents b finding the points where (provided tht ). Likewise, we locte verticl tngents t the points where (provided tht ). Notice tht if we re looking for tngent lines t the pole, then r nd Eqution 3 simplifies to d dr if d tn dd dd d d d d d d dr d dr d sin r cos cos r sin d dd dd

26 SECTIN.3 PLAR CRDINATES 645 For instnce, in Emple 8 we found tht r cos when or 34. This mens tht the lines nd (or nd ) re tngent lines to r cos t the origin. EXAPLE 9 () For the crdioid r sin of Emple 7, find the slope of the tngent line when. (b) Find the points on the crdioid where the tngent line is horizontl or verticl. 3 SLUTIN Using Eqution 3 with r sin, we hve () The slope of the tngent t the point where is d cos3 sin3 d sin3 sin3 (b) bserve tht d d 3 4 dr d dr d 34 sin r cos cos r sin cos sin sin cos sin sin sin sin s3 ( s3 )( s3 ) cos sin sin cos cos cos sin sin 3 s3 s3 4 ( s3 ) ( s3 )( s3 ) d d cos sin when, 3, 7 6, 6 d d sin sin when 3, 6, 5 6 π, m=_ œ 3 π +, 3 Therefore there re horizontl tngents t the points,, (, 76), (, 6) nd verticl tngents t ( 3, 6) nd ( 3, 56). When, both nd re, so we must be creful. Using l Hospitl s Rule, we hve d lim l3 d sin cos lim lim l3 sin l3 sin 3 dd dd 3, 5π 6 (, ) 3 π, 6 3 cos lim l3 sin 3 sin lim l3 cos 7π π, 6, 6 FIGURE 5 Tngent lines for r=+sin B smmetr, d lim l3 d Thus there is verticl tngent line t the pole (see Figure 5).

27 646 CHAPTER PARAETRIC EQUATINS AND PLAR CRDINATES NTE Insted of hving to remember Eqution 3, we could emplo the method used to derive it. For instnce, in Emple 9 we could hve written Then we would hve d d r cos sin cos cos sin r sin sin sin sin sin dd dd which is equivlent to our previous epression. cos sin cos cos sin sin cos sin cos GRAPHING PLAR CURVES WITH GRAPHING DEVICES Although it s useful to be ble to sketch simple polr curves b hnd, we need to use grphing clcultor or computer when we re fced with curve s complicted s the ones shown in Figures 6 nd _ FIGURE 6 r=sin@(.4 )+cos$(.4 ) _.7 FIGURE 7 r=sin@(. )+cos#(6 ) Some grphing devices hve commnds tht enble us to grph polr curves directl. With other mchines we need to convert to prmetric equtions first. In this cse we tke the polr eqution r f nd write its prmetric equtions s r cos f cos Some mchines require tht the prmeter be clled t rther thn. EXAPLE Grph the curve r sin85. r sin f sin SLUTIN Let s ssume tht our grphing device doesn t hve built-in polr grphing commnd. In this cse we need to work with the corresponding prmetric equtions, which re r cos sin85 cos r sin sin85 sin In n cse, we need to determine the domin for. So we sk ourselves: How mn complete rottions re required until the curve strts to repet itself? If the nswer is n, then sin 8 n 5 sin 8 5 6n sin 8 5 5

28 SECTIN.3 PLAR CRDINATES 647 _ nd so we require tht 6n5 be n even multiple of. This will first occur when n 5. Therefore we will grph the entire curve if we specif tht. Switching from to t, we hve the equtions sin8t5 cos t sin8t5 sin t t nd Figure 8 shows the resulting curve. Notice tht this rose hs 6 loops. FIGURE 8 r=sin(8 /5) _ N In Eercise 55 ou re sked to prove nlticll wht we hve discovered from the grphs in Figure 9. V EXAPLE Investigte the fmil of polr curves given b r c sin. How does the shpe chnge s c chnges? (These curves re clled limçons, fter French word for snil, becuse of the shpe of the curves for certin vlues of c.) SLUTIN Figure 9 shows computer-drwn grphs for vrious vlues of c. For c there is loop tht decreses in size s c decreses. When c the loop disppers nd the curve becomes the crdioid tht we sketched in Emple 7. For c between nd the crdioid s cusp is smoothed out nd becomes dimple. When c decreses from to, the limçon is shped like n ovl. This ovl becomes more circulr s c l, nd when c the curve is just the circle r. c=.7 c= c=.7 c=.5 c=. c=.5 c=_ c= c=_. c=_.5 c=_.8 c=_ FIGURE 9 embers of the fmil of limçons r=+c sin The remining prts of Figure 9 show tht s c becomes negtive, the shpes chnge in reverse order. In fct, these curves re reflections bout the horizontl is of the corresponding curves with positive c..3 EXERCISES Plot the point whose polr coordintes re given. Then find two other pirs of polr coordintes of this point, one with r nd one with r.. (), 3 (b), 34 (c),. (), 74 (b) 3, 6 (c), 3 4 Plot the point whose polr coordintes re given. Then find the Crtesin coordintes of the point. 3. (), (b) (, 3) (c), () (s, 54) (b), 5 (c), The Crtesin coordintes of point re given. (i) Find polr coordintes r, of the point, where r nd. (ii) Find polr coordintes r, of the point, where r nd. 5. (), (b) (, s3 ) 6. () (3s3, 3) (b),