Solution: Polar CoordinatesExercise gives the eccentricities

Problem 1AAE Finding Conic Sections Find an equation for the parabola with focus (4, 0) and directrix x = 3. Sketch the parabola together with its vertex, focus, and directrix.

Problem 1PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 1QGY What is a parametrization of a curve in the xy-plane? Does a function y = f(x) always have a parametrization? Are parametrizations of a curve unique? Give examples.

Problem 2AAE Finding Conic Sections Find the vertex, focus, and directrix of the parabola

Problem 2PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 2QGY Give some typical parametrizations for lines, circles, parabolas, ellipses, and hyperbolas. How might the parametrized curve differ from the graph of its Cartesian equation?

Problem 3AAE Finding Conic Sections Find an equation for the curve traced by the point P(x, y) if the distance from P to the vertex of the parabola x2 = 4y is twice the distance from P to the focus. Identify the curve.

Problem 3PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 3QGY What is a cycloid? What are typical parametric equations for cycloids? What physical properties account for the importance of cycloids?

Problem 4AAE Finding Conic Sections A line segment of length a + b runs from the x-axis to the y-axis. The point P on the segment lies a units from one end and b units from the other end. Show that P traces an ellipse as the ends of the segment slide along the axes.

Problem 4PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 4QGY What is the formula for the slope dy/dx of a parametrized curve x = ƒ(t), y = g(t) ? When does the formula apply? When can you expect to be able to find d2y/dx2as well? Give examples.

Problem 5AAE Finding Conic Sections The vertices of an ellipse of eccentricity 0.5 lie at the points Where do the foci lie?

Problem 5PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 5QGY How can you sometimes find the area bounded by a parametrized curve and one of the coordinate axes?

Problem 6AAE Finding Conic Sections Find an equation for the ellipse of eccentricity 2/3 that has the line x = 2 as a directrix and the point (4, 0) as the corresponding focus.

Problem 6PE Identifying Parametric Equations in the Plane Exercise gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

Problem 6QGY How do you find the length of a smooth parametrized curve x = ƒ(t), y = g(t), a ? t ? b ? What does smoothness have to do with length? What else do you need to know about the parametrization in order to find the curve’s length? Give examples.

Problem 7AAE Finding Conic Sections One focus of a hyperbola lies at the point (0, -7) and the corresponding directrix is the line y = -1. Find an equation for the hyperbola if its eccentricity is (a) 2, (b) 5.

Problem 7PE Finding Parametric Equations and Tangent Lines Find parametric equations and a parameter interval for the motion of a particle in the xy-plane that traces the ellipse 16x2 + 9y2 = 144 once counterclockwise. (There are many ways to do this.)

Problem 7QGY What is the arc length function for a smooth parametrized curve? What is its arc length differential?

Problem 8AAE Finding Conic Sections Find an equation for the hyperbola with foci (0, – 2) and (0, 2) that passes through the point (12, 7).

Problem 8PE Finding Parametric Equations and Tangent Lines Find parametric equations and a parameter interval for the motion of a particle that starts at the point (-2, 0) in the xy-plane and traces the circle x2 + y2 = 4 three times clockwise. (There are many ways to do this.)

Problem 8QGY Under what conditions can you find the area of the surface generated by revolving a curve x = ƒ(t), y = g(t), a ? t ? b, about the x-axis? the y-axis? Give examples.

Problem 9AAE Finding Conic Sections Show that the line is tangent to the ellipse at the point (x1, y1 ) on the ellipse.

Problem 9PE Finding Parametric Equations and Tangent Lines In Exercise, find an equation for the line in the xy-plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d2y/dx2 at this point.

Problem 9QGY What are polar coordinates? What equations relate polar coordinates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?

Problem 10AAE Finding Conic Sections Show that the line is tangent to the hyperbola at the point (x1, y1 ) on the ellipse.

Problem 10PE Finding Parametric Equations and Tangent Lines In Exercise, find an equation for the line in the xy-plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d2y/dx2 at this point.

Problem 10QGY What consequence does the lack of uniqueness of polar coordinates have for graphing? Give an example.

Problem 11AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 11PE Finding Parametric Equations and Tangent Lines Eliminate the parameter to express the curve in the form y = f(x)

Problem 11QGY How do you graph equations in polar coordinates? Include in your discussion symmetry, slope, behavior at the origin, and the use of Cartesian graphs. Give examples.

Problem 12AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 12PE Finding Parametric Equations and Tangent Lines Find parametric equations for the given curve. a. Line through (1, -2) with slope 3

Problem 12QGY How do you find the area of a region in the polar coordinate plane? Give examples.

Problem 13AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 13PE Lengths of Curves Find the lengths of the curves.

Problem 13QGY Under what conditions can you find the length of a curve in the polar coordinate plane? Give an example of a typical calculation.

Problem 14AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 14PE Lengths of Curves Find the lengths of the curves.

Problem 14QGY What is a parabola? What are the Cartesian equations for parabolas whose vertices lie at the origin and whose foci lie on the coordinate axes? How can you find the focus and directrix of such a parabola from its equation?

Problem 15AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 15PE Lengths of Curves Find the lengths of the curves in Exercise

Problem 15QGY What is an ellipse? What are the Cartesian equations for ellipses centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?

Problem 16AAE Equations and Inequalities What points in the xy-plane satisfy the equations and inequalities in Exercise? Draw a figure for each exercise.

Problem 16PE Lengths of Curves Find the lengths of the curves.

Problem 16QGY What is a hyperbola? What are the Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?

Problem 17AAE Polar Coordinates a. Find an equation in polar coordinates for the curve b. Find the length of the curve from

Problem 17PE Lengths of Curves Find the lengths of the curves in Exercise

Problem 17QGY What is the eccentricity of a conic section? How can you classify conic sections by eccentricity? How are an ellipse’s shape and eccentricity related?

Problem 18AAE Polar Coordinates Find the length of the curve in the polar coordinate plane.

Problem 18PE Lengths of Curves Find the lengths of the curves in Exercise

Problem 18QGY Explain the equation

Problem 19AAE Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 2, r cos ? = 2

Problem 19PE Lengths of Curves Find the lengths of the curves in Exercise

Problem 19QGY What are the standard equations for lines and conic sections in polar coordinates? Give examples.

Problem 20AAE Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1, r cos ? = -4

Problem 20PE Lengths of Curves Find the length of the enclosed loop shown here. The loop starts at

Problem 21AAE Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1/2, r sin ? = 2

Problem 21PE Surface Areas Find the areas of the surfaces generated by revolving the curves in Exercise about the indicated axes.

Problem 22AAE Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1/3, r sin ? = -6

Problem 22PE Surface Areas Find the areas of the surfaces generated by revolving the curves in Exercise about the indicated axes.

Problem 23AAE Theory and Examples Epicycloids When a circle rolls externally along the circumference of a second, fixed circle, any point P on the circumference of the rolling circle describes an epicycloid, as shown here. Let the fixed circle have its center at the origin O and have radius a. Let the radius of the rolling circle be b and let the initial position of the tracing point P be A(a, 0). Find parametric equations for the epicycloid, using as the parameter the angle ? from the positive x-axis to the line through the circles’ centers.

Problem 23PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 24AAE Theory and Examples Find the centroid of the region enclosed by the x-axis and the cycloid arch

Problem 24PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 25AAE Theory and Examples The Angle Between the Radius Vector and the Tangent Line to a Polar Coordinate Curve In Cartesian coordinates, when we want to discuss the direction of a curve at a point, we use the angle measured counterclockwise from the positive x-axis to the tangent line. In polar coordinates, it is more convenient to calculate the angle from the radius vector to the tangent line (see the accompanying figure). The angle can then be calculated from the relation which comes from applying the Exterior Angle Theorem to the triangle in the accompanying figure. Suppose the equation of the curve is given in the form r = ƒ(?) , where ƒ(?) is a differentiable function of ?. Then (2) are differentiable functions of ? with Furthermore, because is the slope of the curve at P. Also, Hence The numerator in the last expression in Equation (4) is found from Equations (2) and (3) to be Similarly, the denominator is When we substitute these into Equation (4), we obtain This is the equation we use for finding as a function of ? Show, by reference to a figure, that the angle between the tangents to two curves at a point of intersection may be found from the formula When will the two curves intersect at right angles?

Problem 25PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 26AAE Theory and Examples Find the value of for the curve

Problem 26PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 27AAE Theory and Examples Find the angle between the radius vector to the curve r = 2a sin 3? and its tangent when

Problem 27PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 28AAE Theory and Examples a. Graph the hyperbolic spiral What appears to happen to as the spiral winds in around the origin? b. Confirm your finding in part (a) analytically.

Problem 28PE Polar to Cartesian Equations Sketch the lines in Exercise. Also, find a Cartesian equation for each line.

Problem 29AAE Theory and Examples The circles intersect at the point Show that their tangents are perpendicular there.

Problem 29PE Polar to Cartesian Equations Find Cartesian equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

Problem 30AAE Theory and Examples Find the angle at which the cardioid r = a(1 - cos ?) crosses the ray

Problem 30PE Polar to Cartesian Equations Find Cartesian equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

Problem 31PE Polar to Cartesian Equations Find Cartesian equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

Problem 32PE Polar to Cartesian Equations Find Cartesian equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

Problem 33PE Cartesian to Polar Equations Find polar equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. x 2 + y2 + 5y = 0

Problem 34PE Cartesian to Polar Equations Find polar equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. x 2 + y2 - 2y = 0

Problem 35PE Cartesian to Polar Equations Find polar equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. x 2 + y2 - 3x = 0

Problem 36PE Cartesian to Polar Equations Find polar equations for the circles in Exercise. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. x 2 + y2 + 4x = 0

Problem 37PE Graphs in Polar Coordinates Sketch the regions defined by the polar coordinate inequalities in Exercise.

Problem 38PE Graphs in Polar Coordinates Sketch the regions defined by the polar coordinate inequalities in Exercise.

Problem 39PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Four-leaved rose

Problem 40PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Spiral

Problem 41PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Limaçon

Problem 42PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Lemniscate

Problem 43PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Circle

Problem 44PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Cardioid

Problem 45PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Parabola

Problem 46PE Graphs in Polar Coordinates Match each graph in Exercise with the appropriate equation (a)–(1). There are more equations than graphs, so some equations will not be matched. Lemniscate

Problem 47PE Area in Polar Coordinates Find the areas of the regions in the polar coordinate plane described in Exercise. Enclosed by the limaçon r = 2 - cos ?

Problem 48PE Area in Polar Coordinates Find the areas of the regions in the polar coordinate plane described in Exercise. Enclosed by one leaf of the three-leaved rose r = sin 3?

Problem 49PE Area in Polar Coordinates Find the areas of the regions in the polar coordinate plane described in Exercise. Inside the “figure eight” r = 1 + cos 2? and outside the circle r = 1

Problem 50PE Area in Polar Coordinates Find the areas of the regions in the polar coordinate plane described in Exercise. Inside the cardioid r = 2(1 + sin ?) and outside the circle r = 2 sin ?

Problem 51PE Length in Polar Coordinates Find the lengths of the curves given by the polar coordinate equations in Exercise

Problem 52PE Length in Polar Coordinates Find the lengths of the curves given by the polar coordinate equations in Exercise

Problem 53PE Length in Polar Coordinates Find the lengths of the curves given by the polar coordinate equations in Exercise

Problem 54PE Length in Polar Coordinates Find the lengths of the curves given by the polar coordinate equations in Exercise

Problem 55PE Graphing Conic Sections Sketch the parabolas in Exercise. Include the focus and directrix in each sketch. x 2 = -4y

Problem 56PE Graphing Conic Sections Sketch the parabolas in Exercise. Include the focus and directrix in each sketch. x 2 = 2y

Problem 57PE Graphing Conic Sections Sketch the parabolas in Exercise. Include the focus and directrix in each sketch. y 2 = 3x

Problem 58PE Graphing Conic Sections Sketch the parabolas in Exercise. Include the focus and directrix in each sketch. y 2 = -(8/3)x

Problem 59PE Graphing Conic Sections Find the eccentricities of the ellipses and hyperbolas in Exercise. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch. 16x2 + 7y2 = 112

Problem 60PE Graphing Conic Sections Find the eccentricities of the ellipses and hyperbolas in Exercise. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch. x 2 + 2y2 = 4

Problem 61PE Graphing Conic Sections Find the eccentricities of the ellipses and hyperbolas in Exercise. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch. 3x2 - y2 = 3

Problem 62PE Graphing Conic Sections Find the eccentricities of the ellipses and hyperbolas in Exercise. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch. 5y2 - 4x2 = 20

Problem 63PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 64PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 65PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 66PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 67PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 68PE Graphing Conic Sections Exercises give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

Problem 69PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 70PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 71PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 72PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 73PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 74PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 75PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 76PE Identifying Conic Sections Complete the squares to identify the conic sections in Exercise. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

Problem 77PE Conics in Polar Coordinates Sketch the conic sections whose polar coordinate equations are given in Exercise. Give polar coordinates for the vertices and, in the case of ellipses, for the centers as well.

Problem 78PE Conics in Polar Coordinates Sketch the conic sections whose polar coordinate equations are given in Exercise. Give polar coordinates for the vertices and, in the case of ellipses, for the centers as well.

Problem 79PE Conics in Polar Coordinates Sketch the conic sections whose polar coordinate equations are given in Exercise. Give polar coordinates for the vertices and, in the case of ellipses, for the centers as well.

Problem 80PE Conics in Polar Coordinates Sketch the conic sections whose polar coordinate equations are given in Exercise. Give polar coordinates for the vertices and, in the case of ellipses, for the centers as well.

Problem 81PE Conics in Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 2, r cos ? = 2

Problem 82PE Conics in Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1, r cos ? = -4

Problem 83PE Conics in Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1/2, r sin ? = 2

Problem 84PE Conics in Polar Coordinates Exercise gives the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. e = 1/3, r sin ? = -6

Problem 85PE Theory and Examples Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x2 + 4y2 = 36 about (a) the x-axis, (b) the y-axis.

Problem 86PE Theory and Examples The “triangular” region in the first quadrant bounded by the x-axis, the line x = 4, and the hyperbola 9x2 – 4y2 = 36 is revolved about the x-axis to generate a solid. Find the volume of the solid.

Problem 87PE Theory and Examples Show that the equations transform the polar equation into the Cartesian equation

Problem 88PE Theory and Examples Archimedes spirals The graph of an equation of the form r = a?, where a is a nonzero constant, is called an Archimedes spiral. Is there anything special about the widths between the successive turns of such a spiral?