Polar Curves

10 Key excerpts on "Polar Curves"

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In Section 1.9 we learned how to graph points in rectangular coordinates. In this chapter we study a different way of locating points in the plane, called polar coordinates. Using rectangular coordinates is like describing a location in a city by saying that it’s at the corner of 2nd Street and 4th Avenue; these directions would help a taxi driver find the location. But we may also describe this same location “as the crow flies”; we can say that it’s 1.5 miles northeast of City Hall. These directions would help an airplane or hot air balloon pilot find the location. So instead of specifying the location with respect to a grid of streets and avenues, we specify it by giving its distance and direction from a fixed reference point. That’s what we do in the polar coordinate system. In polar coordinates the location of a point is given by an ordered pair of numbers: the distance of the point from the origin (or pole) and the angle from the positive x-axis. Why do we study different coordinate systems? It’s because certain curves are more naturally described in one coordinate system rather than another. For example, in rectangular coordinates lines and parabolas have simple equations, but equations of circles are rather complicated. We’ll see that in polar coordinates circles have very simple equations. 587 Polar Coordinates and Parametric Equations 8 8.1 Polar Coordinates 8.2 Graphs of Polar Equations 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 8.4 Plane Curves and Parametric Equations FOCUS ON MODELING The Path of a Projectile © gary718/Shutterstock.com Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Calculus

  Late Transcendental

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Spirals generally have “left-hand” and “right-hand” versions that coil in opposite directions, depending on the restrictions on the polar angle and the signs of constants that appear in their equations. Some of the more common types of spirals are shown in Figure 10.2.22 for nonnegative values of θ, a, and b. Figure 10.2.22 626 Chapter 10 / Parametric and Polar Curves; Conic Sections SPIRALS IN NATURE Spirals of many kinds occur in nature. For example, the shell of the chambered nautilus forms a logarithmic spiral, and a coiled sailor’s rope forms an Archimedean spiral. Spirals also occur in flowers, the tusks of certain animals, and in the shapes of galaxies. GENERATING Polar Curves WITH GRAPHING UTILITIES For Polar Curves that are too complicated for hand computation, graphing utilities can be used. Although many graphing utilities are capable of graphing Polar Curves directly, some are not. However, if a graphing utility is capable of graphing parametric equations, then it can be used to graph a polar curve r = f (θ) by converting this equation to parametric form. This can be done by substituting f (θ) for r in (1). This yields x = f (θ) cos θ, y = f (θ) sin θ (10) which is a pair of parametric equations for the polar curve in terms of the parameter θ. Example 11 Express the polar equation r = 2 + cos 5θ 2 parametrically, and generate the polar graph from the parametric equations using a graph- ing utility. Solution. Substituting the given expression for r in x = r cos θ and y = r sin θ yields the parametric equations x =  2 + cos 5θ 2  cos θ, y =  2 + cos 5θ 2  sin θ Next, we need to find an interval over which to vary θ to produce the entire graph. To find such an interval, we will look for the smallest number of complete revolutions that must occur until the value of r begins to repeat. Algebraically, this amounts to finding the TECHNOLOGY MASTERY Use a graphing utility to duplicate the curve in Figure 10.2.23.

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  Calculus

  Early Transcendental Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  However, it is often useful to be able to make quick sketches of these graphs that show their orientations and give some sense of their dimensions. The orientation of a conic relative to the polar axis can be deduced by matching its equation with one of the four forms in Theorem 10.6.2. The key dimensions of a parabola are determined by the constant p (Figure 10.4.5) and those of ellipses and hyperbolas by the constants a, b, and c (Figures 10.4.11 and 10.4.20). Thus, we need to show how these constants can be obtained from the polar equations. 664 Chapter 10 / Parametric and Polar Curves; Conic Sections Example 1 Sketch the graph of r = 2 1 − cos θ in polar coordinates. Solution. The equation is an exact match to (4) with d = 2 and e = 1. Thus, the graph is a parabola with the focus at the pole and the directrix 2 units to the left of the pole. This tells us that the parabola opens to the right along the polar axis and p = 1. Thus, the parabola looks roughly like that sketched in Figure 10.6.4. Figure 10.6.4 All of the important geometric information about an ellipse can be obtained from the values of a, b, and c in Figure 10.6.5. One way to find these values from the polar equation Figure 10.6.5 of an ellipse is based on finding the distances from the focus to the vertices. As shown in the figure, let r 0 be the distance from the focus to the closest vertex and r 1 the distance to the farthest vertex. Thus, r 0 = a − c and r 1 = a + c (7) from which it follows that a = 1 2 (r 1 + r 0 ) c = 1 2 (r 1 − r 0 ) (8–9) Moreover, it also follows from (7) that r 0 r 1 = a 2 − c 2 = b 2 Thus, b = √ r 0 r 1 (10) In words, Formula (8) states that a is the arithmetic average (also called the arithmetic mean) of r 0 and r 1 , and Formula (10) states that b is the geometric mean of r 0 and r 1 . Example 2 Find the constants a, b, and c for the ellipse r = 6 2 + cos θ .

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  Multivariable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  For instance, in Example 6 we need only have plotted points for 0 <  < y2 and then reflected about the polar axis to obtain the complete circle. Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 690 CHAPTER 10 Parametric Equations and Polar Coordinates ■ Graphing Polar Curves with Technology Although it’s useful to be able to sketch simple Polar Curves by hand, we need to use a graphing calculator or computer when we are faced with a curve as complicated as the ones shown in Figures 15 and 16. FIGURE 16 r - sin 2 s3y2d 1 cos 2 s2y3d 2 _2 _2 2 1 _1 _1 1 FIGURE 15 r - sin 3 s2.5d 1 cos 3 s2.5d EXAMPLE 9 Graph the curve r - sins8y5d. SOLUTION First we need to determine the domain for . So we ask ourselves: how many complete rotations are required until the curve starts to repeat itself ? If the answer is n, then sin 8s 1 2nd 5 - sin S 8  5 1 16n 5 D - sin 8  5 and so we require that 16ny5 be an even multiple of . This will first occur when n - 5. Therefore we will graph the entire curve if we specify that 0 <  < 10. Figure 17 shows the resulting curve. Notice that this curve has 16 loops. ■ EXAMPLE 10 Investigate the family of Polar Curves given by r - 1 1 c sin . How does the shape change as c changes? (These curves are called limaçons, after a French word for snail, because of the shape of the curves for certain values of c.) SOLUTION Figure 18 shows computer-drawn graphs for various values of c. (Note that we obtain the complete graph for 0 <  < 2.) For c .

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  FOR FURTHER INFORMATION For more information on rose curves and related curves, see the article “A Rose is a Rose . . .” by Peter M. Maurer in The American Mathematical Monthly. The computer-generated graph at the left is the result of an algorithm that Maurer calls “The Rose.” To view this article, go to MathArticles.com. Generated by Maple 0 π 2 π 3 π 2 a b ≥ 2 Convex limaçon 0 π 2 π 3 π 2 1 < a b < 2 Dimpled limaçon 0 π 2 π 3 π 2 a b = 1 Cardioid (heart-shaped) 0 π 2 π 3 π 2 a b < 1 Limaçon with inner loop Limaçons r = a ± b cos θ r = a ± b sin θ (a > 0, b > 0) n = 2 a 0 π 2 π 3 π 2 r = a sin nθ Rose curve n = 5 a 0 π 2 π 3 π 2 r = a sin nθ Rose curve 0 π n = 4 a 2 π 3 π 2 r = a cos nθ Rose curve 0 π n = 3 a 2 π 3 π 2 r = a cos nθ Rose curve Rose Curves n petals when n is odd 2n petals when n is even (n ≥ 2) a 0 π 2 π 3 π 2 r 2 = a 2 cos 2θ Lemniscate a 0 π 2 π 3 π 2 r 2 = a 2 sin 2θ Lemniscate a 0 π 2 π 3 π 2 r = a sin θ Circle a 0 π 2 π 3 π 2 r = a cos θ Circle Circles and Lemniscates 726 Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.4 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CONCEPT CHECK 1. Polar Coordinates Consider the polar coordinates (r, θ ). What does r represent? What does θ represent? 2. Plotting Points Plot the points below on the same set of coordinate axes. (r, θ ) = parenleft.alt4 2, π 2 parenright.alt4 and (x, y) = parenleft.alt4 2, π 2 parenright.alt4 3. Comparing Coordinate Systems Describe the differences between the rectangular coordinate system and the polar coordinate system. 4. Parametric Form of a Polar Equation Explain how to write a polar equation in parametric form. Polar-to-Rectangular Conversion In Exercises 5–14, the polar coordinates of a point are given. Plot the point and find the corresponding rectangular coordinates for the point. 5. parenleft.alt4 8, π 2 parenright.alt4 6. parenleft.alt4 -2, 5π 3 parenright.alt4 7.

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  Calculus

  Single Variable

  • Carl V. Lutzer, H. T. Goodwill(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  . . . . . . . . 718 9.3 Polar Coordinates . . . . . . . . . . 730 9.4 Calculus in Polar Coordinates . . . . . . . . . . 745 9.5 Complex Numbers . . . . . . . . . . 752 Chapter Review . . . . . . . . . . . . . . . . 764 Projects & Applications . . . . . . . . 768 In this section we extend our ability to calculate slopes, areas, and lengths into the realm of polar coordinates. z Slopes of Tangent Lines to Polar Curves The cardioid seen in Figure 4.1 implicitly defines y as a function of x, so let’s think of it as y = F (x) in a small patch. Figure 4.1: y as an implicit function of x. Though we don’t have an explicit formula for F , we can still determine the slope of the tangent line to its graph by using the Chain Rule, just as we did with parametric curves: We could also calculate the slope of the tangent line by us- ing what we know of differen- tials (i.e., linear approximation of change). Since y = r sin(θ) and r is a function of θ, we can think of y as a function of θ, so dy = y 0 (θ) dθ. Similarly, dx = x 0 (θ) dθ so the rise-to-run ratio is dy dx = y 0 (θ) x 0 (θ) . dy dθ = dF dx dx dθ ⇒ dF dx = y 0 (θ) x 0 (θ) when x 0 (θ) 6= 0. Since y = r sin(θ) and x = r cos(θ), and r is a function of θ as we move along the curve, we have to use the Product Rule to determine y 0 (θ) and x 0 (θ). So doing leads us to the following fact: Tangent lines to Polar Curves: Suppose C is the curve r = f (θ), where f is differentiable at θ 0 , and y is a differentiable function of x at the point P 0 = (r(θ 0 ), θ 0 ). Then the slope of the tangent line to C at P 0 is dy dx     P0 = r 0 (θ 0 ) sin(θ 0 ) + r(θ 0 ) cos(θ 0 ) r 0 (θ 0 ) cos(θ 0 ) - r(θ 0 ) sin(θ 0 ) (4.1) provided that the denominator is not zero. Tangent to a cardioid Example 4.1. Determine the slope of the tangent line to the cardioid r = 1 -sin(θ) when θ = 7π/4.

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  These curves, which are called Cassini ovals, have one of the three basic shapes shown in the accompanying figure. a. Show that if a = b, then the polar equation of the Cassini oval is r 2 = 2a 2 cos 2θ, which is a lemniscate. b. Use the formula in Exercise 71 to show that the lemniscate in part (a) is the curve traced by a point that moves in such a way that the product of its distances from the polar points (a, 0) and (a, π ) is a 2 . a < b a > b x y a = b FIGURE Ex-75 76–78 Vertical and horizontal asymptotes of Polar Curves can some- times be detected by investigating the behavior of x = r cos θ and y = r sin θ as θ varies. This idea is used in these exercises. 76. Show that the hyperbolic spiral r = 1/θ (θ > 0) has a horizon- tal asymptote at y = 1 by showing that y → 1 and x → +∞ as θ → 0 + . Confirm this result by generating the spiral with a graph- ing utility. 77. Show that the spiral r = 1/θ 2 does not have any horizontal asymptotes. 78. Does the spiral r = 1/ √ θ have a horizontal asymptote? 79. Prove that a rose with an even number of petals is traced out exactly once as θ varies over the interval 0 ≤ θ < 2π and a rose with an odd number of petals is traced out exactly once as θ varies over the interval 0 ≤ θ < π. 80. a. Use a graphing utility to confirm that the graph of r = 2 − sin (θ/2) (0 ≤ θ ≤ 4π ) is symmetric about the x-axis. b. Show that replacing θ by −θ in the polar equation r = 2 − sin (θ/2) does not produce an equivalent equation. Why does this not contradict the symmetry demonstrated in part (a)? 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 613 81. Writing Use a graphing utility to investigate how the family of Polar Curves r = 1 + a cos nθ is affected by changing the values of a and n, where a is a positive real number and n is a positive integer. Write a brief paragraph to explain your conclusions.

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  Calculus Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  600 Chapter 9 / Parametric and Polar Curves; Conic Sections Intersections of Polar Graphs In the last example we found the intersections of the cardioid and circle by equating their expressions for r and solving for . However, because a point can be represented in different ways in polar coordinates, this procedure will not always produce all of the intersections. For example, the cardioids r = 1 − cos  and r = 1 + cos  (7) intersect at three points: the pole, the point (1, ∕2), and the point (1, 3∕2) (Figure 9.3.13). Equating the right-hand sides of the equations in (7) yields 1 − cos  = 1 + cos  or cos  = 0, so  =  2 + k, k = 0, ±1, ±2, … Substituting any of these values in (7) yields r = 1 so that we have found only two distinct points of intersection, (1, ∕2) and (1, 3∕2) ; the pole has been missed. This problem occurs because the two cardioids pass through the pole at different values of —the cardioid r = 1 − cos  passes through the pole at  = 0, and the cardioid r = 1 + cos  passes through the pole at  = . r = 1 - cos θ r = 1 + cos θ π/2 0 ▴ Figure 9.3.13 The situation with the cardioids is analogous to two satellites circling the Earth in intersecting orbits (Figure 9.3.14).The satellites will not collide unless they reach the same point at the same time. In general, when looking for intersections of Polar Curves, it is a good idea to graph the curves to determine how many intersections there should be. The orbits intersect, but the satellites do not collide. ▴ Figure 9.3.14 ✓Quick Check Exercises 9.3 (See page 603 for answers.) 1. a. To obtain dy∕dx directly from the polar equation r = f (), we can use the formula dy dx = dy∕d dx∕d = b. Use the formula in part (a) to find dy∕dx directly from the polar equation r = csc . 2. a. What conditions on f (  0 ) and f ′ ( 0 ) guarantee that the line  =  0 is tangent to the polar curve r = f () at the origin? b.

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  Single Variable Calculus

  Early Transcendentals, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  ■ A Unified Description of Conics If we place the focus at the origin, then a conic section has a simple polar equation, which provides a convenient description of the motion of planets, satellites, and comets. 1 Theorem Let F be a fixed point (called the focus) and l be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). The set of all points P in the plane such that | PF | | Pl | - e (that is, the ratio of the distance from F to the distance from l is the constant e) is a conic section. The conic is (a) an ellipse if e , 1 (b) a parabola if e - 1 (c) a hyperbola if e . 1 10.6 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 712 CHAPTER 10 Parametric Equations and Polar Coordinates PROOF Notice that if the eccentricity is e - 1, then | PF | - | Pl | and so the given condition simply becomes the definition of a parabola as given in Section 10.5. Let us place the focus F at the origin and the directrix parallel to the y-axis and d units to the right. Thus the directrix has equation x - d and is perpendicular to the polar axis.

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  Single Variable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  ■ A Unified Description of Conics If we place the focus at the origin, then a conic section has a simple polar equation, which provides a convenient description of the motion of planets, satellites, and comets. 1 Theorem Let F be a fixed point (called the focus) and l be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). The set of all points P in the plane such that | PF | | Pl | - e (that is, the ratio of the distance from F to the distance from l is the constant e) is a conic section. The conic is (a) an ellipse if e , 1 (b) a parabola if e - 1 (c) a hyperbola if e . 1 10.6 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 750 CHAPTER 10 Parametric Equations and Polar Coordinates PROOF Notice that if the eccentricity is e - 1, then | PF | - | Pl | and so the given condition simply becomes the definition of a parabola as given in Section 10.5. Let us place the focus F at the origin and the directrix parallel to the y-axis and d units to the right. Thus the directrix has equation x - d and is perpendicular to the polar axis.

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