Polar Functions Polar coordinates

Transcription

1 548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar equations enable us to define some interesting and important curves that would be difficult or impossible to define in the form f ). Polar Functions Polar Coordinates If ou graph the two functions sin and cos 5 on the same pair of aes, ou will get two sinusoids. But if ou graph the curve defined parametricall b sin t and cos 5t, ou will get the figure shown. Parametric graphing opens up a whole new world of curves that can be defined using our familiar basic functions. Another wa to enter that world is to use a different coordinate sstem. In polar coordinates we identif the origin as the pole and the positive -ais as the initial ra of angles measured in the usual trigonometric wa. We can then identif each point P in the plane b polar coordinates r, ), where r gives the directed distance from to P and gives the directed angle from the initial ra to the ra P. In Figure 1.19 we see that the point P with rectangular Cartesian) coordinates, ) has polar coordinates, 4). P, ) P, p/4) p/4 Rectangular coordinates Polar coordinates Figure 1.19 Point P has rectangular coordinates, ) and polar coordinates, 4). As ou would epect, we can also coordinatize point P with the polar coordinates, 9 4) or, 7 4), since those angles determine the same ra P. Less obviousl, we can also coordinatize P with polar coordinates, 4), since the directed distance in the 4 direction is the same as the directed distance in the 4 direction Figure 1.). So, although each pair r, ) determines a unique point in the plane, each point in the plane can be coordinatized b an infinite number of polar ordered pairs. p/4 p/4 Figure 1. The directed negative distance in the 4 direction is the same as the directed positive distance in the 4 direction. Thus the polar coordinates, 4) and, 4) determine the same point.

2 Section 1. Polar Functions 549 EXAMPLE 1 Rectangular and Polar Coordinates a) Find rectangular coordinates for the points with given polar coordinates. i) 4, ) ii), ) iii) 16, 5 6) iv), 4) b) Find two different sets of polar coordinates for the points with given rectangular coordinates. i) 1, ) ii), ) iii), 4) iv) 1, ) a) i), 4) ii), ) iii) 8,8) iv) 1, 1) b) A point has infinitel man sets of polar coordinates, so here we list just two tpical eamples for each given point. i) 1, ), 1, ) ii), 4),, 4) iii) 4, ), 4, ) iv), ),, 4 ) Now tr Eercises 1 and. EXAMPLE Graphing with Polar Coordinates Graph all points in the plane that satisf the given polar equation a) r b) r c) 6 First, note that we do not label our aes r and. We are graphing polar equations in the usual -plane, not renaming our rectangular variables! a) The set of all points with directed distance units from the pole is a circle of radius centered at the origin Figure 1.1a). b) The set of all points with directed distance units from the pole is also a circle of radius centered at the origin Figure 1.1b). c) The set of all points of positive or negative directed distance from the pole in the 6 direction is a line through the origin with slope tan 6) Figure 1.1c). Now tr Eercise 7. p/6 a) b) c) Figure 1.1 Polar graphs of a) r, b) r, and c) 6. Eample ) Polar Curves The curves in Eample are a start, but we would not introduce a new coordinate sstem just to graph circles and lines; there are far more interesting polar curves to stud. In the past it was hard work to produce reasonable polar graphs b hand, but toda, thanks to graphing technolog, it is

3 55 Chapter 1 Parametric, Vector, and Polar Functions just a matter of finding the right window and pushing the right buttons. ur intent in this section is to use the technolog to produce the graphs and then concentrate on how calculus can be used to give us further information. EXAMPLE Polar Graphing with Technolog Find an appropriate graphing window and produce a graph of the polar curve. a) r sin 6 b) r 1 cos c) r 4 sin For all these graphs, set our calculator to PLAR mode. a) First we find the window. Notice that r sin 6 1 for all, so points on the graph are all within 1 unit from the pole. We want a window at least as large as [ 1, 1] b [ 1, 1], but we choose the window [ 1.5, 1.5] b [ 1, 1] in order to keep the aspect ratio close to the screen dimensions, which have a ratio of :. We choose a -range of to get a full rotation around the graph, after which we know that sin 6 will repeat the same graph periodicall. Choose step.5. The result is shown in Figure 1.a. b) In this graph we notice that r 1 cos, so we choose [, ] for our -range and, to get the right aspect ratio, [ 4.5, 4.5] for our -range. Due to the cosine s period, again suffices for our -range. The graph is shown in Figure 1.b. c) Since r 4 sin 4, we choose [ 4, 4] for our -range and [ 6, 6] for our -range. Due to the sine s period, again suffices for our -range. The graph is shown in Figure 1.c. Now tr Eercise 1. [ 1.5, 1.5] b [ 1, 1] u p a) [ 4.5, 4.5] b [, ] u p b) [ 6, 6] b [ 4, 4] u p c) Figure 1. The graphs of the three polar curves in Eample. The curves are a) a 1-petaled rose, b) a limaçon, and c) a circle. A Rose is a Rose The graph in Figure 1.a is called a 1-petaled rose, because it looks like a flower and some flowers are roses. The graph in Figure 1.b is called a limaçon LEE-ma-sohn) from an old French word for snail. We will have more names for ou at the end of the section. With a little eperimentation, it is possible to improve on the safe windows we chose in Eample at least in parts b) and c)), but it is alwas a good idea to keep a : ratio of the -range to the -range so that shapes do not become distorted. Also, an astute observer ma have noticed that the graph in part c) was traversed twice as went from to, so a range of would have sufficed to produce the entire graph. From to, the circle is swept out b positive r values; then from to, the same circle is swept out b negative r values. Although the graph in Figure 1.c certainl looks like a circle, how can we tell for sure that it reall is? ne wa is to convert the polar equation to a Cartesian equation and verif that it is the equation of a circle. Trigonometr gives us a simple wa to convert polar equations to rectangular equations and vice versa.

4 Section 1. Polar Functions 551 Polar Rectangular Conversion Formulas r cos r r sin tan EXAMPLE 4 Converting Polar to Rectangular Use the polar rectangular conversion formulas to show that the polar graph of r 4 sin is a circle. To facilitate the substitutions, multipl both sides of the original equation b r. This could introduce etraneous solutions with r, but the pole is the onl such point, and we notice that it is alread on the graph.) r 4 sin r 4r sin Multipl b r. 4 Polar rectangular conversion ) Completing the square Circle in standard form Sure enough, the graph is a circle centered at, ) with radius. Now tr Eercise 5. The polar rectangular conversion formulas also reveal the calculator s secret to polar graphing: It is reall just parametric graphing with as the parameter. Parametric Equations of Polar Curves The polar graph of r f ) is the curve defined parametricall b: r cos f ) cos r sin f ) sin EXPLRATIN 1 Graphing Polar Curves Parametricall Switch our grapher to parametric mode and enter the equations sin 6t) cos t sin 6t) sin t. 1. Set an appropriate window and see if ou can reproduce the polar graph in Figure 1.a.. Then produce the graphs in Figures 1.b and 1.c in the same wa.

5 55 Chapter 1 Parametric, Vector, and Polar Functions Slopes of Polar Curves Since polar curves are drawn in the -plane, the slope of a polar curve is still the slope of the tangent line, which is d d. The polar rectangular conversion formulas enable us to write and as functions of, so we can find d d as we did with parametricall defined functions: d d d. d d d EXAMPLE 5 Finding Slope of a Polar Curve Find the slope of the rose curve r sin at the point where 6 and use it to find the equation of the tangent line Figure 1.). The slope is [, ] b [, ] u p Figure 1. The -petaled rose curve r sin. Eample 5 shows how to find the tangent line to the curve at 6. d sin sin ) d d d d. d d 6 d d 6 sin cos ) d This epression can be computed b hand, but it is an ecellent candidate for our calculator s numerical derivative functionalit Section.). NDERIV quickl gives an answer of , which ou might recognize as. When 6, 6 sin ) cos 6) and sin ) sin 6) 1. So the tangent line has equation 1 ). Now tr Eercise 9. Areas Enclosed b Polar Curves We would like to be able to use numerical integration to find areas enclosed b polar curves just as we did with curves defined b their rectangular coordinates. Converting the equations to rectangular coordinates is not a reasonable option for most polar curves, so we would like to have a formula involving small changes in rather than small changes in. While a small change produces a thin rectangular strip of area, a small change produces a thin circular sector of area Figure 1.4). u Figure 1.4 A small change in produces a rectangular strip of area, while a small change in produces a thin sector of area.

6 Section 1. Polar Functions 55 Recall from geometr that the area of a sector of a circle is 1 r, where r is the radius and is the central angle measured in radians. If we replace b the differential d, we get the area differential da 1 r d Figure 1.5), which is eactl the quantit that we need to integrate to get an area in polar coordinates. 1 da r d Pr, ) d r Figure 1.5 The area differential da. Area in Polar Coordinates The area of the region between the origin and the curve r f for is A 1 r d 1 f d. r r 1 cos ) Pr, ) Figure 1.6 The cardioid in Eample 6. 4, EXAMPLE 6 Finding Area Find the area of the region in the plane enclosed b the cardioid r 1 cos. We graph the cardioid Figure 1.6) and determine that the radius P sweeps out the region eactl once as runs from to. Solve Analticall The area is therefore 1 r d p cos d p p p 1 cos cos d 4 cos 1 c os ) d 4 cos cos d ] [ 4 sin sin 6 6. Support Numericall NINT 1 cos,,, , which agrees with 6 to eight decimal places. Now tr Eercise 4.

7 554 Chapter 1 Parametric, Vector, and Polar Functions 4 Figure 1.7 The limaçon in Eample 7. r cos 1 EXAMPLE 7 Finding Area Find the area inside the smaller loop of the limaçon r cos 1. After watching the grapher generate the curve over the interval Figure 1.7), we see that the smaller loop is traced b the point r, as increases from to 4 the values for which r cos 1 ). The area we seek is 4 A 1 r d 1 4 cos 1 d. Solve Numericall 1 NINT cos 1,,, Now tr Eercise 47. r To find the area of a region like the one in Figure 1.8, which lies between two polar curves r 1 r 1 ) and r r ) from to, we subtract the integral of 1 )r 1 from the integral of 1 )r. This leads to the following formula. r 1 Area Between Polar Curves The area of the region between r 1 and r for is A 1 r d 1 r 1 d 1 r r 1 d. Figure 1.8 The area of the shaded region is calculated b subtracting the area of the region between r 1 and the origin from the area of the region between r and the origin. r 1 1 cos Upper limit / r 1 Lower limit / Figure 1.9 The region and limits of integration in Eample 8. EXAMPLE 8 Finding Area Between Curves Find the area of the region that lies inside the circle r 1 and outside the cardioid r 1 cos. The region is shown in Figure 1.9. The outer curve is r 1, the inner curve is r 1 1 cos, and runs from to. Using the formula for the area between polar curves, the area is A 1 r r 1 d 1 r r 1 d 1 1 cos cos d cos cos d In case ou are interested, the eact value is 4. Smmetr Using NINT Now tr Eercise 5.

8 Section 1. Polar Functions 555 A SMALL PLAR GALLERY Here are a few of the more common polar graphs and the -intervals that can be used to produce them. CIRCLES r constant r a sin r a cos RSE CURVES r a sin n, n odd n petals -ais smmetr r a sin n, n even n petals -ais smmetr and -ais smmetr r a cos n, n odd n petals -ais smmetr r a cos n, n even n petals -ais smmetr and -ais smmetr

9 556 Chapter 1 Parametric, Vector, and Polar Functions LIMAÇN CURVES r a b sin or r a b cos with a and b r a b sin has -ais smmetr; r a b cos has -ais smmetr.) a 1 b Limaçon with loop a 1 b Cardioid 1 a b Dimpled limaçon a b Conve limaçon LEMNISCATE CURVES r a sin r a cos

10 Section 1. Polar Functions 557 SPIRAL F ARCHIMEDES r Quick Review 1. For help, go to Sections 1.1 and 1..) 1. Find the component form of a vector with magnitude 4 and direction angle º.,. Find the area of a º sector of a circle of radius 6.. Find the area of a sector of a circle of radius 8 that has a central angle of 8 radians Find the rectangular equation of a circle of radius 5 centered at the origin Eplain how to use our calculator in function mode to graph the curve 4. Graph 4 1/ and 4 1/ Eercises 6 1 refer to the parametrized curve cos t, 5 sin t, t. 6. Find d d. 5 cot t 7. Find the slope of the curve at t. 5 cot Find the points on the curve where the slope is zero., 5) and, 5) 9. Find the points on the curve where the slope is undefined., ) and, ) 1. Find the length of the curve from t to t Section 1. Eercises In Eercises 1 and, plot each point with the given polar coordinates and find the corresponding rectangular coordinates. 1. a), 4 b) 1, c), d), 4. a), 5 6 b) 5, tan 1 4 c) 1, 7 d), In Eercises and 4, plot each point with the given rectangular coordinates and find two sets of corresponding polar coordinates.. a) 1, 1 b) 1, c), d) 1, 4. a), 1 b), 4 c), d), In Eercises 5 1, graph the set of points whose polar coordinates satisf the given equation. 5. r 6. r 7. r r 8 6r. 1, a line slope 1, -intercept 1) 4. 1, a circle center, ), radius 1) 5. 5, a line slope, -intercept 5) In Eercises 11, find an appropriate window and use a graphing calculator to produce the polar curve. Then sketch the complete curve and identif the tpe of curve b name. 11. r 1 cos 1. r cos 1. r cos 14. r sin 15. r 1 sin 16. r cos 17. r 4 cos 18. r sin 19. r 4 sin. r cos In Eercises 1, replace the polar equation b an equivalent Cartesian rectangular) equation. Then identif or describe the graph. 1. r 4 csc. r sec 4, a horizontal line, a vertical line. r cos r sin 1 4. r r 6. r sin cos sin 1, a hperbola 7. cos sin 8. r 4r cos 9. r 8 sin. r cos sin 7., the union of two lines: 8. ) 4, a circle center, ), radius ) 9. 4) 16, a circle center, 4), radius 4). 1) 1), a circle center 1, 1), radius )

11 558 Chapter 1 Parametric, Vector, and Polar Functions In Eercises 1 8, find an appropriate window and use a graphing calculator to produce the polar curve. Then sketch the complete curve and identif the tpe of curve b name. Note:You won t find these in the Polar Galler.) 1. r sec tan. r csc cot 1. r 4. r 1 cos 1 sin r 6. r 5 9 cos 8 6 cos r 8. r 1. 8 cos 1 1. cos In Eercises 9 4, find the slope of the curve at each indicated point. 9. r 1 sin,, At : 1; At :1 4. r cos,,, At : undefined; At : At : ; At : undefined 41. r sin At, ): / At 1: ):, p), ) At, ): / At 5, ): 1, p 4. r 1 cos At 1.5, ): undefined At 4.5, ): At 6, ): undefined At, ): 1 6, p) 4.5 5, p p In Eercises 4 56, find the area of the region described. 4. inside the conve limaçon r 4 cos inside the cardioid r sin inside one petal of the four-petaled rose r cos inside the eight-petaled rose r sin inside one loop of the lemniscate r 4 cos 48. inside the si-petaled rose r sin inside the dimpled limaçon r cos inside the inner loop of the limaçon r sin shared b the circles r cos and r sin ) 1 5. shared b the circles r 1 and r sin ) ) 5. shared b the circle r and the cardioid r 1 cos ) shared b the cardioids r 1 cos ) and r 1 cos ) inside the circle r and outside the cardioid r 1 sin ) inside the four-petaled rose r 4 cos and outside the circle r 4 8 ) 1.5 p p 6. False. Integrating from to traverses the curve twice, giving twice the area. The correct upper limit of integration is. 57. Sketch the polar curves r cos and r 1 cos and find the area that lies inside the circle and outside the cardioid. 58. Sketch the polar curves r and r 1 sin ) and find the area that lies inside the circle and outside the cardioid. 59. Sketch the polar curve r sin. Find the area enclosed b the curve and find the slope of the curve at the point where The accompaning figure shows the parts of the graphs of the line 5 and the curve 1 that lie in the first quadrant. Region R is enclosed b the line, the curve, and the -ais. R 1 /4 1 5 d.47 a) Set up and evaluate an integral epression with respect to that gives the area of R. b) Show that the curve 1 can be described in polar coordinates b r 1 Let r cos and cos. sin r sin and solve for r. c) Use the polar equation in part b) to set up an integral epression with respect to that gives the area of R. Let tan 1 /5). Then the area is 1 1 cos sin d. Standardized Test Questions You ma use a graphing calculator to solve the following problems. 61. True or False There is eactl one point in the plane with polar coordinates, ). Justif our answer. True. Polar coordinates determine a unique point. 6. True or False The total area enclosed b the -petaled rose r sin is 1 sin d. Justif our answer. 6. Multiple Choice The area of the region enclosed b the polar graph of r s co is given b which integral? D A) s co d B) s co d C) cos ) d D) cos ) d E) s co d 64. Multiple Choice The area enclosed b one petal of the -petaled rose r 4 cos ) is given b which integral? E A) 16 cos ) d C) 8 cos ) d E) cos ) d B) cos ) d D) cos ) d