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All Textbook Solutions for Multivariable Calculus

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 2 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. 2. x = t3 + t, y = t2 + 2, 2 t 2 3E 4E 5E 6E 7E 8E 9E 10E (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. 11. x=sin12, y=cos12,(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. 12. x=12cos, y = 2 sin , 0 (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. 13. x = sin t, y = csc t, 0 t /2 14E (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 t 16E 17E (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. 18. x = tan2 , y = sec , /2 /2 Describe the motion of a particle with position (x, y) as t varies in the given interval. 19. x = 5 + 2 cos t, y = 3 + 2 sin t, 1 t 2 Describe the motion of a particle with position (x, y) as t varies in the given interval. 20. x = 2 + sin t, y = 1 + 3 cos t, /2 t 2 Describe the motion of a particle with position (x, y) as t varies in the given interval. 21. x = 5 sin t, y = 2 cos t, t 5 Describe the motion of a particle with position (x, y) as t varies in the given interval. 22. x = sin t, y = cos2 t, 2 t 2 Suppose a curve is given by the parametric equations x = f(t), y = g(t), where the range of f is [1, 4] and the range of g is [2, 3]. What can you say about the curve?24E 25E 26E 27E Match 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+t2 Graph the curve x = y 2 sin y.Graph the curves y = x3 4x and x = y3 4y and find their points of intersection correct to one decimal place.(a) Show that the parametric equations x=x1+(x2x1)ty=y1+(y2y1)t where 0 t 1, describe the line segment that joins the points P1(x1, y1) and P2(x2, y2). (b) Find parametric equations to represent the line segment from (2, 7) to (3, 1).Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices A(1, 1), B(4, 2), and C(1, 5).Find 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)(a) Find parametric equations for the ellipse x2/a2 + y2/b2 = 1. [Hint: Modify the equations of the circle in Example 2.] (b) Use these parametric equations to graph the ellipse when a = 3 and b = 1, 2, 4, and 8. (c) How does the shape of the ellipse change as b varies?Use a graphing calculator or computer to reproduce the picture. 35.Use a graphing calculator or computer to reproduce the picture. 36.Compare the curves represented by the parametric equations. How do they differ? 37. (a) x = t3, y = t2 (b) x = t6, y = t4 (c) x = e3t, y = e2t 38E Derive Equations 1 for the case /2 .Let P be a point at a distance d from the center of a circle of radius r. The curve traced out by P as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with d = r. Using the same parameter as for the cycloid, and assuming the line is the x-axis and = 0 when P is at one of its lowest points, show that parametric equations of the trochoid are x=rdsiny=rdcos Sketch the trochoid for the cases d r and d r.If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as the parameter. Then eliminate the parameter and identify the curve.If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as the parameter. The line segment AB is tangent to the larger circle.43E (a) Find parametric equations for the set of all points P as shown in the figure such that |OP| = |AB|. (This curve is called the cissoid of Diodes after the Greek scholar Diodes, who introduced the cissoid as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube.) (b) Use the geometric description of the curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.Suppose 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+sint0t2 46E Investigate the family of curves defined by the parametric equations x = t2, y = t3 ct. How does the shape change as c increases? Illustrate by graphing several members of the family.48E 49E Graph several members of the family of curves x = sin t + sin nt, y = cos t + cos nt, where n is a positive integer. What features do the curves have in common? What happens as n increases?The curves with equations x = a sin nt, y = b cos t are called Lissajous figures. Investigate how these curves vary when a, b, and n vary. (Take n to be a positive integer.)52E Find dy/dx. 1. x=t1+t,y=1+t Find dy/dx. 2. x = tet, y = t + sin t Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. 3. x = t3 + l, y = t 4 + t; t = 1 Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. 4. x=t, y = t2 2t; t = 4 Find 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 = 0 Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. 7. x = 1 + ln t, y = t2 + 2; (1,3)8E 9E 10E 11E Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 12. x = t3 + 1, y = t2 t Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 13. x = et, y = tet 14E Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 15. x = t ln t, y = t + ln t Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 16. x = cos t, y = sin 2t, 0 t Find 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. 17. x = t3 3t, y = t2 3 18E Find 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. 19. x = cos , y = cos 3 20E 21E Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x = t4 2t, y = t + t4. Then find the exact coordinates.23E Graph the curve in a viewing rectangle that displays all the important aspects of the curve. 24. x = r4 + 4t3 8t2, y = 2t2 t 25E Graph the curve x = 2 cos t, y = sin t + sin 2t to discover where it crosses itself. Then find equations of both tangents at that point.(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.(a) Find the slope of the tangent to the astroid x = a cos3, y = a sin3 in terms of . (Astroids are explored in the Laboratory Project on page 689.) (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope 1 or 1?At what point(s) on the curve x = 3t2 + 1, y = t3 1 does the tangent line have slope 12?Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).Use the parametric equations of an ellipse, x = a cos , y = b sin , 0 2, to find the area that it encloses.Find the area enclosed by the curve x = t2 2t, y=t and the y-axis.Find the area enclosed by the x-axis and the curve x = t3 + 1, y = 2t t2.Find the area of the region enclosed by the astroid x = a cos3, y = a sin3. (Astroids are explored in the Laboratory Project on page 689.)Find the area under one arch of the trochoid of Exercise 10.1.40 for the case d r.Let 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.37E Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. 38. x = t2 t, y = t4, 1 t 4 39E 40E Find the exact length of the curve. 41. x = 1 + 3t2, y = 4 + 2t3, 0 t 1 Find the exact length of the curve. 42. x = et t, y = 4et/2, 0 t 2 43E 44E 45E 46E 47E Find the length of the loop of the curve x = 3t t3, y = 3t2.Use Simpsons Rule with n = 6 to estimate the length of the curve x = t et, y = t + et, 6 t 6.In Exercise 10.1.43 you were asked to derive the parametric equations x = 2a cot , y = 2a sin2 for the curve called the witch of Maria Agnesi. Use Simpsons Rule with n = 4 to estimate the length of the arc of this curve given by /4 /2.Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve. 51. x = sin2t, y = cos2t, 0 t 3 Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve. 52. x = cos2t, y = cos t, 0 t 4 53E Find the total length of the astroid x = a cos3, y = a sin3, where a 0.(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.57E 58E 59E 60E 61E 62E Find the exact area of the surface obtained by rotating the given curve about the x-axis. 63. x = a cos3, y = a sin3, 0 /2 64E Find the surface area generated by rotating the given curve about the y-axis. 65. x = 3t2, y = 2t3, 0 t 5 Find the surface area generated by rotating the given curve about the y-axis. 66. x = et t, y = 4et/2, 0 t 1 If f is continuous and f(t) 0 for a t b, show that the parametric curve x = f(t), y = g(t), a t b, can be put in the form y = F(x). [Hint: Show that f 1 exists.]68E The curvature at a point P of a curve is defined as =|dds| where is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 13. (a) For a parametric curve x = x(t), y = y(t), derive the formula =|xyxy|[x2+y2]3/2 (b) where the dots indicate derivatives with respect to t, so x=dx/dt. [Hint: Use = tan1(dy/dx) and Formula 2 to find d/dt. Then use the Chain Rule to find d/ds.] (c) By regarding a curve y = f(x) as the parametric curve x = x, y = f(x), with parameter x, show that the formula in part (a) becomes =|d2y/dx2|[1+(dy/dx)2]3/2 (a) Use the formula in Exercise 69(b) to find the curvature of the parabola y = x2 at the point (1, 1). (b) At what point does this parabola have maximum curvature?Use the formula in Exercise 69(a) to find the curvature of the cycloid x = sin , y = 1 cos at the top of one of its arches.(a) Show that the curvature at each point of a straight line is = 0. (b) Show that the curvature at each point of a circle of radius r is = 1/r.A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the siring is called the involute of the circle. If the circle has radius r and center O and the initial position of P is (r, 0), and if the parameter is chosen as in the figure, show that parametric equations of the involute are x=r(cos+sin)y=r(sincos)A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow.1E 2E Plot 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)Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. 4. (a) (4, 4/3) (b) (2, 3/4) (c) (3, /3)5E 6E Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 7. r 1 8E Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 9. r 0, /4 3/4 10E 11E 12E 13E 14E Identify the curve by finding a Cartesian equation for the curve. 15. r2 = 5 16E Identify the curve by finding a Cartesian equation for the curve. 17. r =5 cos 18E Identify the curve by finding a Cartesian equation for the curve. 19. r2 cos 2 = 1 20E Find a polar equation for the curve represented by the given Cartesian equation. 21. y = 2 22E 23E 24E 25E Find a polar equation for the curve represented by the given Cartesian equation. 26. x2 y2 = 4 27E 28E 29E 30E 31E 32E Sketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 33. r = , 0 34E 35E 36E Sketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 37. r = 2 cos 4 Sketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 38. r = 2 sin 6 Sketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates. 39. r = 1 + 3 cos 40E Sketch 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 2 42E 43E 44E 45E 46E 47E 48E 49E 50E Show that the curve r = sin tan (called a cissoid of Diodes) has the line x = 1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0 x 1. Use these facts to help sketch the cissoid.52E (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.54E Find the slope of the tangent line to the given polar curve at the point specified by the value of . 55. r = 2 cos , = /3 Find the slope of the tangent line to the given polar curve at the point specified by the value of . 56. r = 2 + sin 3, = /4 Find the slope of the tangent line to the given polar curve at the point specified by the value of . 57. r = 1/, =Find the slope of the tangent line to the given polar curve at the point specified by the value of . 58. r = cos(/3), =Find the slope of the tangent line to the given polar curve at the point specified by the value of . 59. r = cos 2, = /4 60E Find the points on the given curve where the tangent line is horizontal or vertical. 61. r = 3cos Find the points on the given curve where the tangent line is horizontal or vertical. 62. r = 1 sin 63E 64E 65E 66E Use 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)68E 69E 70E 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)72E 73E Use a graph to estimate the y-coordinate of the highest points on the curve r = sin 2. Then use calculus to find the exact value.75E 76E Let P be any point (except the origin) on the curve r = f(). If is the angle between the tangent line at P and the radial line OP, show that tan=rdr/d [Hint: Observe that = in the figure.]78E Find the area of the region that is bounded by the given curve and lies in the specified sector. 1. r = e/4, /2 Find the area of the region that is bounded by the given curve and lies in the specified sector. 2. r = cos , 0 /6 Find the area of the region that is bounded by the given curve and lies in the specified sector. 3. r = sin + cos , 0 4E Find the area of the shaded region. 5.Find the area of the shaded region. 6.7E 8E 9E Sketch the curve and find the area that it encloses. 10. r = 1 sin 11E 12E 13E 14E Graph the curve and find the area that it encloses. 15. r=1+cos2(5)16E Find the area of the region enclosed by one loop of the curve. 17. r = 4 cos 3 18E 19E Find the area of the region enclosed by one loop of the curve. 20. r = 2 sin 5 21E 22E 23E 24E Find the area of the region that lies inside the first curve and outside the second curve. 25. r2 = 8 cos 2, r = 2 Find the area of the region that lies inside the first curve and outside the second curve. 26. r = 1 + cos , r = 2 cos Find the area of the region that lies inside the first curve and outside the second curve. 27. r = 3 cos , r = 1 + cos 28E Find the area of the region that lies inside both curves. 29. r = 3 sin , r = 3 cos 30E Find the area of the region that lies inside both curves. 31. r = sin 2, r = cos 2 32E Find the area of the region that lies inside both curves. 33. r2 = 2 sin 2, r = 1 34E Find the area inside the larger loop and outside the smaller loop of the limaon r=12+cos.36E Find all points of intersection of the given curves. 37. r = sin, r = 1 sin 38E Find all points of intersection of the given curves. 39. r = 2 sin 2, r = 1 40E 41E 42E The 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.45E 46E Find the exact length of the polar curve. 47. r = 2, 0 2 48E Find the exact length of the curve. Use a graph to determine the parameter interval. 49. r = cos4(/4)50E Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. 51. One loop of the curve r = cos 2 52E 53E 54E 55E 56E Find the vertex, focus, and directrix of the parabola and sketch its graph. 1. x2 = 6y 2E 3E Find the vertex, focus, and directrix of the parabola and sketch its graph. 4. 3x2 + 8y = 0 5E Find the vertex, focus, and directrix of the parabola and sketch its graph. 6. (y 2)2 = 2x + 1 Find the vertex, focus, and directrix of the parabola and sketch its graph. 7. y2 + 6y + 2x + 1 = 0 8E 9E 10E 11E 12E 13E Find the vertices and foci of the ellipse and sketch its graph. 14. 100x2 + 36y2 = 225 15E Find the vertices and foci of the ellipse and sketch its graph. 16. x2 + 3y2 + 2x 12y + 10 = 0 17E 18E 19E 20E Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 21. x2 y2 = 100 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E

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