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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
More parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 52. Involute of a circle x = cos t + t sin t, y = sin t t cos tMore parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 53. Evolute of an ellipse x=a2b2acos3t,y=a2b2bsin3t; a = 4 and b = 3More parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 54. Cissoid of Diocles x = 2 sin 2t,y=2sin3tcostImplicit function graph Explain and carry out a method for graphing the curve x = 1 + cos2 y sin2 y using parametric equations and a graphing utility.Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by x=80t,y=4.9t2+3000,fort0, where the origin is the point on the ground directly beneath the plane at the moment of the release. Graph the trajectory of the packet and find the coordinates of the point where the packet lands.Air dropinverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by x=100t,y=4.9t2+4000,fort0, where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?66E67EDerivatives Consider the following parametric curves. a. Determine dy/dx in terms of t and evaluate it at the given value of t. b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t. 60. x = 3 sin t, y = 3 cos t; t = /2Derivatives Consider the following parametric curves. a. Determine dy/dx in terms of t and evaluate it at the given value of t. b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t. 61. x = cos t, y = 8 sin t; t = /270EDerivatives Consider the following parametric curves. a. Determine dy/dx in terms of t and evaluate it at the given value of t. b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t. 63. x = t + 1/t, y = t 1/t; t = 172ETangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t. 67. x = t2 1, y = t3 + t; t = 2Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t. 66. x = sin t, y = cos t; t = /4Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t. 69. x = cos t + t sin t, y = sin t cos t; t = /4Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t. 68. x = et, y = ln (t + 1); t = 0Slopes of tangent lines Find all the points at which the following curves have the given slope. 89. x = 4 cos t, y = 4 sin t; slope=12Slopes of tangent lines Find all the points at which the following curves have the given slope. 90. x = 2 cos t, y = 8 sin t, slope = 1Slopes of tangent lines Find all the points at which the following curves have the given slope. 91. x = t + 1/t, y = t 1/t; slope = 1Slopes of tangent lines Find all the points at which the following curves have the given slope. 92. x=2+t, y = 2 4t; slope = 8Arc length Find the arc length of the following curves on the given interval. 81. x = 3t + 1, y = 4t + 2; 0 t 2Arc length Find the arc length of the following curves on the given interval. 82. x = 3 cos t, y = 3 sin t + 1; 0 t 2Arc length Find the arc length of the following curves on the given interval. 83. x = cos t sin t, y = cos t + sin t; 0 tArc length Find the arc length of the following curves on the given interval. 84. x = et sin t, y = et cos t; 0 t 2Arc length Find the arc length of the following curves on the given interval. 85. x = t4, y=t63; 0 t 1Arc length Find the arc length of the following curves on the given interval. 86. x = 2t sin t t2 cos t, y = 2t cos t + t2 sin t; 0 tArc length Find the arc length of the following curves on the given interval. 87. x = cos3 2t, y = sin3 2t; 0 t /4Arc length Find the arc length of the following curves on the given interval. 88. x = sin t, y = t cos t; 0 t /2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The equations x = cos t, y = sin t, for 0 t 2t, generate a circle in the clockwise direction. b. An object following the parametric curve x = 2 cos 2t, y = 2 sin 2t circles the origin once every 1 time unit. c. The parametric equations x = cos t, y = t2, for t 0, describe the complete parabola y = x2. d. The parametric equations x = cos t, y = sin t, for /2 t /2, describe a semicircle. e. There are two points on the curve x = 4 cos t, y = sin t, for 0 t 2 , at which there is a vertical tangent line.90E91E92EParametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 9192). Graph the ellipse and find a description in terms of x and y. 93. An ellipse centered at the origin with major axis of length 6 on the x-axis and minor axis of length 3 on the y-axis, generated counterclockwise94E95E96E97EBeautiful curves Consider the family of curves x=(2+12sinat)cos(t+sinbtc),y=(2+12sinat)sin(t+sinbtc). Plot the curve for the given values of a, b, and c with 0 t 2. (Source: Mathematica in Action, Stan Wagon, Springer, 2010; created by Norton Starr, Amherst College) 56. a = 6, b = 12, c = 399E100E101ELissajous curves Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) a horizontal tangent line and (b) a vertical tangent line. (See the Guided Project Parametric art for more on Lissajous curves.) 96. x = sin 4t, y = sin 3t; 0 t 2Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [, ], which implies that the area bounded by the graph of h and x-axis on [, ] equals h(x)dx or ydx. If the graph of y = h(x) on [, ] is traced exactly once by the parametric equations x = f(t), y = g(t), for a t b, then it follows by substitution that the area bounded by h is h(x)dx=ydx=abg(t)f(t)dtif=f(a)and=f(b) (orh(x)dx=abg(t)f(t)dtif=f(b)and=f(a)). 103. Find the area under one arch of the cycloid x = 3(t sin t), y = 3(1 cos t) (see Example 5).Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [, ], which implies that the area bounded by the graph of h and x-axis on [, ] equals h(x)dx or ydx. If the graph of y = h(x) on [, ] is traced exactly once by the parametric equations x = f(t), y = g(t), for a t b, then it follows by substitution that the area bounded by h is h(x)dx=ydx=abg(t)f(t)dtif=f(a)and=f(b) (orh(x)dx=abg(t)f(t)dtif=f(b)and=f(a)). 104. Show that the area of the region bounded by the ellipse x = 3 cos t, y = 4 sin t, for 0 t 2, equals 4/20sint(3sint)dt. Then evaluate the integral.Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [, ], which implies that the area bounded by the graph of h and x-axis on [, ] equals h(x)dx or ydx. If the graph of y = h(x) on [, ] is traced exactly once by the parametric equations x = f(t), y = g(t), for a t b, then it follows by substitution that the area bounded by h is h(x)dx=ydx=abg(t)f(t)dtif=f(a)and=f(b) (orh(x)dx=abg(t)f(t)dtif=f(b)and=f(a)). 105. Find the area of the region bounded by the asteroid x = cos3 t, y = sin3 t, for 0 t 2 (see Example 8, Figure 12.17).106E107E108ESurfaces of revolution Let C be the curve x = f(t), y = g(t), for a t b, where f and g are continuous on [a, b] and C does not intersect itself, except possibly at its endpoints. If g is nonnegative on [a, b], then the area of the surface obtained by revolving C about the x-axis is S=ab2g(t)f(t)2+g(t)2dt. Likewise, if f is nonnegative on [a, b], then the area of the surface obtained by revolving C about the y-axis is S=ab2f(t)f(t)2+g(t)2dt. (These results can be derived in a manner similar to the derivations given in Section 6.6 for surfaces of revolution generated by the curve y = f(x).) 109. Find the area of the surface obtained by revolving one arch of the cycloid x = t sin t, y = 1 cos t, for 0 t 2110ESurfaces of revolution Let C be the curve x = f(t), y = g(t), for a t b, where f and g are continuous on [a, b] and C does not intersect itself, except possibly at its endpoints. If g is nonnegative on [a, b], then the area of the surface obtained by revolving C about the x-axis is S=ab2g(t)f(t)2+g(t)2dt. Likewise, if f is nonnegative on [a, b], then the area of the surface obtained by revolving C about the y-axis is S=ab2f(t)f(t)2+g(t)2dt. (These results can be derived in a manner similar to the derivations given in Section 6.6 for surfaces of revolution generated by the curve y = f(x).) 111. A surface is obtained by revolving the curve x = e3t + 1, y = e2t, for 0 t 1, about the y-axis. Find an integral that gives the area of the surface and approximate the value of the integral.112E113E114EWhich of the following coordinates represent the same point: (3, /2), (3, 3/2), (3, 5/2), (3, /2), and (3, 3/2)?Draw versions of Figure 12.21 with P in the second, third, and fourth quadrants. Verify that the same conversion formulas hold in all cases.Give two polar coordinate descriptions of the point with Cartes an coordinates (1, 0). What are the Cartesian coordinates of the point with polar coordinates(2,2)?Describe the polar curves r = 12, r = 6, and r sin = 10.5QC6QCPlot the points with polar coordinates (2,6) and (3,2). Give two alternative sets of coordinate pairs for both points.2E3E4EWhat is the polar equation of the vertical line x = 5?What is the polar equation of the horizontal line y = 5?7E8EGraph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. 9. (2,4)Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. 10. (3,23)11E12E13EPoints in polar coordinates Give two sets of polar coordinates for each of the points AF in the figure.15E16E17E18E19E20E21E22ERader Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane typically measured in degrees clockwise from north. 23. A plane is 100 miles from a radar station at an angle of 135 clockwise from north. Find polar coordinates for the location of the plane.24EConverting coordinates Express the following polar coordinates in Cartesian coordinates. 15. (3,4)Converting coordinates Express the following polar coordinates in Cartesian coordinates. 16. (1,23)Converting coordinates Express the following polar coordinates in Cartesian coordinates. 17. (1,3)Converting coordinates Express the following polar coordinates in Cartesian coordinates. 18. (2,74)Converting coordinates Express the following polar coordinates in Cartesian coordinates. 19. (4,34)Converting coordinates Express the following polar coordinates in Cartesian coordinates. 20. (4, 5)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 21. (2, 2)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 22. (1.0)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 23. (1,3)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 34. (0, 9)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 25. (4,43)Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways. 36. (83, 8)37E38E39E40E41E42E43E44E45E46E47E48ECartesian-to-polar coordinates Convert the following equations to polar coordinates. 63. y = x2Cartesian-to-polar coordinates Convert the following equations to polar coordinates. 62. y = 3Cartesian-to-polar coordinates Convert the following equations to polar coordinates. 65. y = 1/xCartesian-to-polar coordinates Convert the following equations to polar coordinates. 64. (x l)2 + y2 = l53E54E55E56EGraphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 57. r = 2 sin 1Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 42. r = 2 2 sin59E60EGraphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 45. r2 = 16 cosGraphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 46. r2 = 16 sin 2Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 47. r = sin 3Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph. 48. r = 2 sin 565E66E67E68EUsing a graphing utility Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve. 53. r=sin4Using a graphing utility Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve. 54. r = 2 4 cos 571EUsing a graphing utility Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve. 56. r=2sin23Using a graphing utility Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve. 57. r=cos35Using a graphing utility Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve. 58. r=sin3775E76E77E78ECircles in general Show that the polar equation r22rr0cos(0)=R2r02 describes a circle of radius R whose center has polar coordinates (r0, 0).80E81E82E83EEquations of circles Find equations of the circles in the figure. Determine whether the combined area of the circles is greater than or less than the area of the region inside the square but outside the circles.Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period. a. Find the distance traveled during this 30-minute period. b. Determine the average velocity of the plane (relative to the ground) during this 30-minute period.86E87E88E89E90E91ELimiting limaon Consider the family of limaons r = 1 + b cos . Describe how the curves change as b .93E94E95EThe lemniscate family Equations of the form r2 = a sin 2 and r2 = a cos 2 describe lemniscates (see Example 7). Graph the following lemniscates. 94. r2 = 8 cos 2The rose family Equations of the form r = a sin m or r = a cos m. where a is a real number and m is a positive integer, have graphs known as roses (see Example 6). Graph the following roses. 95. r = sin 298E99EThe rose family Equations of the form r = a sin m or r = a cos m. where a is a real number and m is a positive integer, have graphs known as roses (see Example 6). Graph the following roses. 98. r = 6 sin 5101E102E103ESpirals Graph the following spirals. Indicate the direction in which the spiral is generated as increases, where 0. Let a = 1 and a = 1. 102. Hyperbolic spiral: r = a/Enhanced butterfly curve The butterfly curve of Example 8 is enhanced by adding a term: r=esin2cos4+sin5(/12),for024. a. Graph the curve. b. Explain why the new term produces the observed effect. (Source: S. Wagon and E. Packel, Animating Calculus. Freeman, 1994)106E107E108E109E110ECartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r2 = a2 cos 2, where a is a real number.Verify that if y = f() sin , then y'() =f'() sin + f() cos (which was used earlier to find dy/dx).2QC3QC4QC1EExplain why the slope of the line = /2 is undefined.Explain why the slope of the line tangent to the polar graph of r = f() is not drd.What integral must be evaluated to find the area of the region bounded by the polar graphs of r = f() and r = g() on the interval , where f() g() 0?What is the slope of the line = /3?6EFind the area of the shaded region.8EExplain why the point with polar coordinates (0, 0) is an intersection point of the curves r = cos and r = sin even though cos sin when = 0.10ESlopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 5. r = 1 sin ; (12,6)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 6. r = 4 cos ; (2,3)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 7. r = 8 sin ; (4,56)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 8. r = 4 + sin ; (4, 0) and (3,32)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 9. r = 6 + 3 cos ; (3, ) and (9, 0)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 10. r = 2 sin 3; at the tips of the leavesSlopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 11. r = 4 cos 2; at the tips of the leavesSlopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 12. r = 1 + 2 sin 2; (3,4)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 13. r2 = 4 cos 2; (0,4)Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. 14. r = 2; (2,4)Tangent line at the origin Find the polar equation of the line tangent to the polar curve r = cos + sin at the origin, and then find the slope of this tangent line.22EMultiple tangent lines at a point a. Give the smallest interval [0, P] that generates the entire polar curve and use a graphing utility to graph the curve. b. Find a polar equation and the slope of each line tangent to the curve at the origin. 23. r = cos 3 + sin 3Multiple tangent lines at a point a. Give the smallest interval [0, P] that generates the entire polar curve and use a graphing utility to graph the curve. b. Find a polar equation and the slope of each line tangent to the curve at the origin. 24. r = 1 + 2 cos 2Horizontal and vertical tangents Find the points at which the following polar curves have a horizontal or a vertical tangent line. 15. r = 4 cosHorizontal and vertical tangents Find the points at which the following polar curves have a horizontal or a vertical tangent line. 16. r = 2 + 2 sinHorizontal and vertical tangents Find the points at which the following polar curves have a horizontal or a vertical tangent line. 17. r = sin 228E29E30E31E32EAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 21. The region inside the curve r=cosAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 22. The region inside the right lobe of r=cos2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 23. The region inside the circle r = 8 sinAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 24. The region inside the cardioid r = 4 + 4 sinAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 25. The region inside the limaon r = 2 + cosAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 26. The region inside all the leaves of the rose r = 3 sin 2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 27. The region inside one leaf of r = cos 3Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 28. The region inside the inner loop of r=cos12Intersection points and area a. Find all the intersection points of the following curves. b. Find the area of the entire region that lies within both curves. 41. r = 3 sin and r = 3 cosIntersection points and area a. Find all the intersection points of the following curves. b. Find the area of the entire region that lies with in both curves. 42. r = 2 + 2 sin and r = 2 2 sinIntersection points and area a. Find all the intersection points of the following curves. b. Find the area of the entire region that lies within both curves. 43. r = 1 + sin and r = 1 + cosIntersection points and area a. Find all the intersection points of the following curves. b. Find the area of the entire region that lies within both curves. 44. r = 1 and r=2cos2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 29. The region outside the circle r=12 and inside the circle r = cosAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 30. The region inside the curve r=cos and outside the circle r=1/2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 31. The region inside the curve r=cos and inside the circle r=1/2 in the first quadrantAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 32. The region inside the right lobe of r=cos2 and inside the circle r=1/2 in the first quadrantAreas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 33. The region inside one leaf of the rose r = cos 5Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 34. The region inside the rose r = 4 cos 2 and outside the circle r = 2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 35. The region inside the rose r = 4 sin 2 and inside the circle r = 2Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. 36. The region inside the lemniscate r2 = 2 sin 2 and outside the circle r = 1Area of plane regions Find the areas of the following regions. 47. The region common to the circles r = 2 sin and r = 1Area of plane regions Find the areas of the following regions. 48. The region inside the inner loop of the limaon r = 2 + 4 cosArea of plane regions Find the areas of the following regions. 49. The region inside the outer loop but outside the inner loop of the limaon r = 3 6 sinArea of plane regions Find the areas of the following regions. 50. The region common to the circle r = 3 cos and the cardioid r = 1 + cosArea of polar regions Find the area of the regions bounded by the following curves. 55. The lemniscate r2 = 6 sin 2Area of polar regions Find the area of the regions bounded by the following curves. 56. The limaon r = 2 4 sin59EArea of polar regions Find the area of the regions bounded by the following curves. 54. The complete three-leaf rose r = 2 cos 3Two curves, three regions Determine the intersection points of the polar curves r = 4 2 cos and r = 2 + 2 cos , and then find areas of regions A, B, and C (see figure).62EArc length of polar curves Find the length of the following polar curves. 63. The Complete circle r a sin . where a 064E65E66E67EArc length of polar curves Find the length of the following polar curves. 68. The spiral r 42, for 0 6Arc length of polar curves Find the length of the following polar curves. 69. The spiral r = 2c20, for 0 ln 8Arc length of polar curves Find the length of the following polar curves. 70. The parabola r 21+cos, for 00271E72E73E74E75E76E77E78E79E80ERegions bounded by a spiral Let Rn be the region bounded by the nth turn and the (n + 1)st turn of the spiral r = e in the first and second quadrants, for 0 (see figure). a. Find the area An of Rn. b. Evaluate limnAn. c. Evaluate limnAn+1/An.Tangents and normals Let a polar curve be described by r = f() and let be the line tangent to the curve at the point P(x, y) = P(r, ) (see figure). a. Explain why =dydx. b. Explain why tan = y/x. c. Let be the angle between and the line through O and P. Prove that tan = f()/f(). d. Prove that the values of for which is parallel to the x-axis satisfy tan = f()/f(). e. Prove that the values of for which is parallel to the y-axis satisfy tan = f()/f().83E84EGrazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.) 59. A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0 a 2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a = 0 and a = 2.Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.) 60. A circular concrete slab of unit radius is surrounded by grass. A goat is tied to the edge of the slab with a rope of length 0 a 2 (see figure). What is the area of the grassy region that the goat can graze? Note that the rope can extend over the concrete slab. Check your answer with the special cases a = 0 and a = 2.87EVerify that x2+(yp)2=y+p is equivalent to x2 = 4py.2QCIn the case that the vertices and foci are on the x-axis, show that the length of the minor axis of an ellipse is 2b.4QC5QC6QCGive the property that defines all parabolas.2EGive the property that defines all hyperbolas.4E5EWhat is the equation of the standard parabola with its vertex at the origin that opens downward?7E8EGiven vertices (a, 0) and eccentricity e, what are the coordinates of the foci of an ellipse and a hyperbola?10EWhat are the equations of the asymptotes of a standard hyperbola with vertices on the x-axis?12EGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 13. x2 = 12yGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 14. y2 = 20xGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 15. x24+y2=116EGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 17. x=y21618E19E20EGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 21. 4x2 y2 = 16Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 22. 25y2 4x2 = 10023E24EGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 25. x24+y216=1Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 26. x2+y29=127EGraphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 28. 12x2 + 5y2 = 60Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 29. x23y25=1Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. 30. 10x2 7y2 = 14031E