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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
50RE51RE52REFind an equation for the ellipse that shares a vertex and a focus with the parabola x2 + y = 100 and that has its other focus at the origin.54RE55RE56RE57REA curve called the folium of Descartes is defined by the parametric equations x=3t1+t3y=3t21+t3 (a) Show that if (a, b) lies on the curve, then so does (b, a); that is, the curve is symmetric with respect to the line y = x. Where does the curve intersect this line? (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Show that the line y = x 1 is a slant asymptote. (d) Sketch the curve. (e) Show that a Cartesian equation of this curve is x3 + y3 = 3xy. (f) Show that the polar equation can be written in the form r=3sectan1+tan3 (g) Find the area enclosed by the loop of this curve. (h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.)1P(a) Find the highest and lowest points on the curve x4 + y4 = x2 + y2. (b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines y = x, so it suffices to consider y x 0 initially.) (c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve.3PFour bugs are placed at the four comers of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find the polar equation of a bugs path assuming the pole is at the center of the square. (Use the fact that the line joining one bug to the next is tangent to the bugs path.) (b) Find the distance traveled by a bug by the time it meets the other bugs at the center. FIGURE FOR PROBLEM 45PA circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0 b r. [See parts (i) and (ii) of the figure.] Let L be the line from the center of C to the center of the rolling circle and let be the angle that L makes with the positive x-axis. (a) Using as a parameter, show that parametric equations of the path traced out by P are x=bcos3+3rcosy=bsin3+3rsin Note: If b = 0, the path is a circle of radius 3r; if b = r, the path is an epicycloid. The path traced out by P for 0 b r is called an epitrochoid. (b) Graph the curve for various values of b between 0 and r. (c) Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin. Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve. (d) In most rotary engines the sides of the equilateral triangles are replaced by arcs of circles centered at the opposite vertices as in part (iii) of the figure. (Then the diameter of the rotor is constant.) Show that the rotor will fit in the epitrochoid if b32(23)r.Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. 1. x = 1 t2, y = 2t t2, 1 t 22E3E4E5E6E7E8E9E10E11E12E13E14E(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 15. x = t2, y = ln t16E17E18E19E20E21E22E23E24E25E26E27EMatch the parametric equations with the graphs labeled IVI. Give reasons for your choices. (Do not use a graphing device.) (a) x = t4 t + 1, y = t2 (b) x = t2 2t, y=t (c) x = sin 2t, y = sin(t + sin 2t) (d) x = cos 5t, y = sin 2t (e) x = t + sin 4t, y = t2 + cos 3t (f) x=sin2t4+t2, y=cos2t4+t229E30E31E32EFind parametric equations for the path of a particle that moves along the circle x2 + (y 1)2 = 4 in the manner described. (a) Once around clockwise, starting at (2, 1) (b) Three times around counterclockwise, starting at (2, 1) (c) Halfway around counterclockwise, starting at (0, 3)34E35E36E37E38E39E40E41E42E43E44ESuppose that the position of one particle at time t is given by x1=3sinty1=2cost0t2 and the position of a second particle is given by x2=3+costy2=1+sint0t2 (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, arc the particles ever at the same place at the same time? If so, find the collision points, (c) Describe what happens if the path of the second particle is given by x2=3+costy2=1+sint0t246E47E48E49E50E51E52EFind dy/dx. 1. x=t1+t,y=1+t2E3E4EFind an equation of the tangent to the curve at the point corresponding to the given value of the parameter. 5. x = t cos t, y = t sin t; t =Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. 6. x et sin t, y = e2t; t = 07E8E9EFind an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. 10. x = sin t, y = t2 + t; (0, 2)11E12E13E14EFind dy/dx and d2y/dx2. For which values of t is the curve concave upward? 15. x = t ln t, y = t + ln t16E17E18E19EFind the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. 20. x = esin , y = ecos21E22E23E24EShow that the curve x = cos t, y = sin t cos t has two tangents at (0, 0) and find their equations. Sketch the curve.26E(a) Find the slope of the tangent line to the trochoid x = r d sin , y = r d cos in terms of . (See Exercise 10.1.40.) (b) Show that if d r, then the trochoid does not have a vertical tangent.28E29E30EUse the parametric equations of an ellipse, x = a cos , y = b sin , 0 2, to find the area that it encloses.32EFind the area enclosed by the x-axis and the curve x = t3 + 1, y = 2t t2.34E35ELet R be the region enclosed by the loop of the curve in Example 1. (a) Find the area of R. (b) If R is rotated about the x-axis, find the volume of the resulting solid. (c) Find the centroid of R.37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E(a) Graph the epitrochoid with equations x=11cost4cos(11t/2)y=11sint4sin(11t/2) What parameter interval gives the complete curve? (b) Use your CAS to find the approximate length of this curve.56E57E58E59E60E61EFind the exact area of the surface obtained by rotating the given curve about the x-axis. 62. x = 2t2 + 1/t, y=8t, 1 t 363E64E65E66E67E68E69E70E71E72E73E74E1E2EPlot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. 3. (a) (2, 3/2) (a) (2,/4) (b) (1, /6)4E5E6E7E8E9E10E11E12E13E14E15E16EIdentify the curve by finding a Cartesian equation for the curve. 17. r =5 cos18E19E20E21E22E23E24E25E26E27E28E29ESketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 30. r = 1 cos31E32E33E34E35E36E37E38E39E40ESketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 41. r2 = 9 sin 242E43ESketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 44. r2 = 145E46E47E48E49E50E51E52E(a) In Example 11 the graphs suggest that the limaon r = 1 + c sin has an inner loop when |c| 1. Prove that this is true, and find the values of that correspond to the inner loop. (b) From Figure 19 it appears that the limaon loses its dimple when c=12. Prove this.54E55EFind the slope of the tangent line to the given polar curve at the point specified by the value of . 56. r = 2 + sin 3, = /457E58E59E60E61EFind the points on the given curve where the tangent line is horizontal or vertical. 62. r = 1 sin63E64E65E66EUse a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. 67. r = 1 + 2 sin(/2) (nephroid of Freeth)68E69EUse a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. 70. r = |tan ||cot | (valentine curve)Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. 71. r = 1 + cos999 (Pac-Man curve)72E73E74E75E76E77E78EFind the area of the region that is bounded by the given curve and lies in the specified sector. 1. r = e/4, /2Find the area of the region that is bounded by the given curve and lies in the specified sector. 2. r = cos , 0 /6Find the area of the region that is bounded by the given curve and lies in the specified sector. 3. r = sin + cos , 04EFind the area of the shaded region. 5.6E7E8E9E10E11E12E13E14E15EGraph the curve and find the area that it encloses. 16. r = 1 + 5 sin 6Find the area of the region enclosed by one loop of the curve. 17. r = 4 cos 318E19E20E21EFind the area enclosed by the loop of the strophoid r = 2 cos sec .Find the area of the region that lies inside the first curve and outside the second curve. 23. r = 4 sin , r = 224E25E26E27E28EFind the area of the region that lies inside both curves. 29. r = 3 sin , r = 3 cos30E31E32EFind the area of the region that lies inside both curves. 33. r2 = 2 sin 2, r = 134EFind the area inside the larger loop and outside the smaller loop of the limaon r=12+cos.Find the area between a large loop and the enclosed small loop of the curve r = 1 + 2 cos 3.Find all points of intersection of the given curves. 37. r = sin, r = 1 sin38E39EFind all points of intersection of the given curves. 40. r = cos 3, r = sin 341E42EThe points of intersection of the cardioid r = 1 + sin and the spiral loop r = 2, /2 /2, cant be found exactly. Use a graphing device to find the approximate values of at which they intersect. Then use these values to estimate the area that lies inside both curves.When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the cardioid r = 8 + 8 sin , where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.45EFind the exact length of the polar curve. 46. r = 5, 0 247E48E49E50E51E52E53E54E55E(a) Find a formula for the area of the surface generated by rotating the polar curve r = f(), a b (where f is continuous and 0 a b ), about the line = /2. (b) Find the surface area generated by rotating the lemniscate r2 = cos 2 about the line = /2.1E2E3E4E5E6EFind the vertex, focus, and directrix of the parabola and sketch its graph. 7. y2 + 6y + 2x + 1 = 0Find the vertex, focus, and directrix of the parabola and sketch its graph. 8. 2x2 16x 3y + 38 = 09E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E