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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)

The elliptic paraboloid x=y23+z27 is a bowl-shaped surface. Along which axis does the bowl open?Which coordinate axis is the axis of the hyperboloid y2a2+z2b2+x2c2=1?6QC To which coordinate axes are the following cylinders in 3 parallel: x2: + 2y2: = 8, z2 + 2y2 = 8, and x2 + 2z2 = 8?Describe the graph of x = z2 in 3.What is a trace of a surface?What is the name of the surface defined by the equation y=x24+z28?What is the name of the surface defined by the equation x2+y23+2z2=1?What is the name of the surface defined by the equation y2z22+x2=1?Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 39.z = y2 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 40.x2 + 4y2 = 4 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 41.x2 + z2 = 4 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 42.x = z2 4 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 43.y x3 = 0 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 44.x 2z2 = 0 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 45.z ln y = 0 Cylinders in 3 Consider the following cylinders in 3. a.Identify the coordinate axis to which the cylinder is parallel. b.Sketch the cylinder. 46.x 1/y = 0 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 15. 25x2 + 25y2 + z2 = 25 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 16. 25x2 + 25y2 z2 = 25 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 17. 25x2 + 25y2 z = 0 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 18. 25x2 25y2 z = 0 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 19. 25x2 25y2 + z2 = 25 Identifying quadric surfaces Identify the following quadric surfaces by name. Find and describe the xy-, xz-, and yz-traces, when they exist. 20. 25x2 25y2 + z2 = 0 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 81.y = 4z2 x2 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 82.y2 9z2 + x2/4 = 1 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 83.y = x2/6 + z2/16 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 80.z2 + 4y2 x2 = 1 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 87.y2 z2 = 2 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 86.x2 + 4y2 = 1 Identifying surfaces Identify the following surfaces by name. 27. 9x2 + 4z2 36y = 0 Identifying surfaces Identify the following surfaces by name. 28. 9y2 + 4z2 36x2 = 0 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Ellipsoids 47.x2+y24+z29=1 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Ellipsoids 48.4x2+y2+z22=1 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Elliptic paraboloids 51.x = y2 + z2 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Elliptic paraboloids 52.z=x24+y29 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of one sheet 55.x225+y29z2=1 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of one sheet 56.y24+z29x216=1 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperbolic paraboloids 59.z=x29y2 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperbolic paraboloids 60.y=x2164z2 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Elliptic cones 63.x2+y24=z2 38E Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Ellipsoids 49.x23+3y2+z212=3 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Ellipsoids 50.x26+24y2+z2246=0 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Elliptic paraboloids 53.9x81y2z24=0 42E Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of one sheet 57.y216+36z2x249=0 44E Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperbolic paraboloids 61.5xy25+z220=0 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperbolic paraboloids 62.6y+x26z224=0 47E 48E Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of two sheets 67.x2+y24z29=1 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of two sheets 70.x2624y2+z2246=0 Quadric surfaces Consider the following equations of quadric surfaces. a.Find the intercepts with the three coordinate axes, when they exist. b.Find the equations of the xy-, xz-, and yz-traces, when they exist. c.Sketch a graph of the surface. Hyperboloids of two sheets 69.x23+3y2z212=1 52E 53E Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 84.x2 + y2 + 4z2 + 2x = 0 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 85.9x2 + y2 4z2 + 2y = 0 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 88.x2 y2 + z2/9 + 6x 8y = 26 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 89.x2/4 + y2 2x 10y z2 + 41 = 0 Identifying surfaces Identify and briefly describe the surfaces defined by the following equations. 58. z = x2 y2 59E Matching graphs with equations Match equations af with surfaces AF. a.yz2=0 b.2x+3yz=5 c.4x2+y29+z2=1 d.x2+y29z2=1 e.x2+y29=z2 f.y=|x|Explorations and Challenges 61. Solids of revolution Which of the quadric surfaces in Table 13.1 can be generated by revolving a curve in are of the coordinate planes about a coordinate axis, assuming a = b = c 0? Table 13.1 62E 63E Light cones The idea of a light cone appears in the Special Theory of Relativity. The xy-plane (see figure) represents all of three-dimensional space, and the z-axis is the time axis (t-axis). If an event E occurs at the origin, the interior of the future light cone (t 0) represents all events in the future that could be affected by E, assuming that no signal travels faster than the speed of light. The interior of the past light cone (t 0) represents all events in the past that could have affected E, again assuming that no signal travels faster than the speed of light. a.If time is measured in seconds and distance (x and y) is measured in light-seconds (the distance light travels in 1 s), the light cone makes a 45 angle with the .xy-plane. Write the equation of the light cone in this case. b.Suppose distance is measured in meters and time is measured in seconds. Write the equation of the light cone in this case given that the speed of light is 3 108 m/s.65E Hand tracking Researchers are developing hand tracking Software that will allow computers to track and recognize detailed hand movements for batter human-computer interaction. One three-dimensional hand model under investigation is constructed from a set of truncated quadrics (see figure) For example the calm of the hand consists of a truncated elliptic cylinder, capped off by the upper half of an ellipsoid. Suppose the palm of the hand is modeled by the truncated elliptic cylinder, capped off by the upper half of an ellipsoid. Suppose the palm of the hand is modeled by the truncated cylinder 4x2/9 + 4y2 = 1, for 0 z 3. find an equator of the upper pair of an ellipsoid corresponds with the top of the cylinder. if the distance from the top of the located cylinder to the top of the ellipsoid is 1/2. 3-D model of hand Designing a snow cone A surface, having the shape of an oblong snow cone, consists of a truncated cone, x22+y2=z28for 0 z 3, capped off by the upper half of an ellipsoid. Find an equation for the upper half of the ellipsoid so that the bottom edge of the truncated ellipsoid and the top edge of the cone coincide, and the distance from the top of the cone to the top of the ellipsoid is 3/2.Designing a glass The outer, lateral side of a 6-inch-tall glass has the shape of the truncated hyperboloid of one sheet x2a2+y2b2z2c2=1,for 0 z 6. If the base of the glass has a radius of 1 inch and the top of the glass has a radius of 2 inches, find the values of a2, b2, and c2 that satisfy these conditions. Assume horizontal traces of the glass are circular.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Given two vectors u and v, it is always true that 2u + v = v + 2u. b. The vector in the direction of u with the length of v equals the vector in the direction of v with the length of u. c. If u 0 and u + v = 0, then u and v are parallel. d. The lines x = 3 + t, y = 4 + 2t, z = 2 t and x = 2t, y = 4t, z = t are parallel. e. The lines x = 3 + t, y = 4 + 2t, z = 2 t and the plane x + 2y + 5z = 3 are parallel. f. There is always a plane orthogonal to both of two distinct intersecting planes.2RE 3RE 4RE 5RE Working with vectors Let u = 2, 4, 5 and v = 6, 10, 2. 6. Compute u 3v.Working with vectors Let u = 2, 4, 5 and v = 6, 10, 2. 7. Compute |u + v|.8RE working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 9. Write the vector w as a product of its magnitude and a unit vector in the direction of w.working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 10. Find a vector in the direction of w that is 10 times as long as w.working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 11. Find a vector in the direction of w with a length of 10.working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 12. Compute u v working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 13. Compute u v working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 14. For what value of a is the vector v orthogonal to y = a,1,3?working with vectors Let u = 2,4,5 , v = 6,10,2 and w = 4,8,8. 15. For what value of a is the vector w parallel toy = a,6,6?Scalar multiples Find scalars a, b, and c such that 2,2,2=a1,1,0+b0,1,1+c1,0,1.Velocity vectors Assume the positive x-axis points east and the positive y-axis points north. a. An airliner flies northwest at a constant altitude at 550 mi/hr in calm air. Find a and b such that its velocity may be expressed in the form v = ai + bj. b. An airliner flies northwest at a constant altitude at 550 mi/hr relative to the air in a southerly crosswind w = 0, 40. Find the velocity of the airliner relative to the ground.18RE Spheres and balls Use set notation to describe the following sets. 15. The sphere of radius 4 centered at (1, 0, 1)Spheres and balls Use set notation to describe the following sets. 16. The points inside the sphere of radius 10 centered at (2, 4, 3)Spheres and balls Use set notation to describe the following sets. 17. The points outside the sphere of radius 2 centered at (0, 1, 0)Identifying sets. Give a geometric description of the following sets of points. 18. x2 6x + y2 + 8y + z2 2z 23 = 0 Identifying sets. Give a geometric description of the following sets of points. 19. x2 x + y2 + 4y + z2 6z + 11 0 Identifying sets. Give a geometric description of the following sets of points. 20. x2 + y2 10y + z2 6z = 34 Identifying sets. Give a geometric description of the following sets of points. 21. x2 6x + y2 + z2 20z + 9 0 26RE 27RE Cross winds A small plane is flying north in calm air at 250 mi/hr when it is hit by a horizontal crosswind blowing northeast at 40 mi/hr and a 25 mi/hr downdraft. Find the resulting velocity and speed of the plane.29RE Canoe in a current A woman in a canoe paddles cue west at 4 mi/hr relative to the water in a current that flows northwest at 2mi/hr. Find the Speed and direction of the canoe relative to the shore.Sets of points Describe the set of points satisfying both the equation x2 + z2 = 1 and y = 2.Angles and projections a. Find the angle between u and v. b. Compute projvu and scalvu. c. Compute projuv and scaluv. 26. u = 3j + 4k, v = 4i + j + 5k 33RE 34RE Computing work Calculate the work done in the following situations. 35. A suitcase is pulled 25 ft along a horizontal sidewalk. with a constant force of 20 lb at an angle of 15 above the horizontal.Computing work Calculate the work done in the following situations. 36. A constant force F = 2,3,4 (in nektons)moves an object from (0,0,0) to (2,1,6). (Distance is measured in meters.)37RE Inclined plane A 1804b map stands on a hillside that makes an angle of 30. wish the horizontal producing a force of w = 0,180 38. How much work is done when the man moves 10ft up the hillside?Area of a parallelogram Find the area of the parallelogram with vertices (1, 2, 3), (1, 0, 6), and (4, 2, 4).Area of a triangle Find the area of the triangle with vertices (1, 0, 3), (5, 0, 1), and (0, 2, 2).Vectors normal to a plane Find a unit vector normal to the vectors 2, 6, 9 and 1, 0, 6.Angle in two ways Find the angle between 2, 0, 2 and 2, 2, 0 using (a) the dot product and (b) the cross product.43RE Suppose you apply a force of |F| = 50 N near the end of a wrench attached to a bolt (see figure). Determine the magnitude of the torque when the force is applied at an angle of 60 to the wrench. Assume the distance along the wrench from the center of the bolt to the point where the force is applied is |r| = 0.25 m.45RE Lines in space Find an equation of the following lines or line segments. 32. The line that passes through the points (2, 6, 1) and (6, 4, 0)Lines in space Find an equation of the following lines or line segments. 33. The line segment that joins the points (0, 3, 9) and (2, 8, 1)Lines in space Find an equation of the following lines or line segments. 34. The line through the point (0, 1, 1) and parallel to the line R(t) = 1 + 2t, 3 5t, 7 + 6t Lines in space Find an equation of the following lines or line segments. 35. The line through the point (0, 1, 1) that is orthogonal to both 0, 1, 3 and 2, 1, 2 Lines in space Find an equation of the following lines or line segments. 36. The line through the point (0, 1, 4) and orthogonal to the vector 2, 1, 7 and the y-axis Equations of planes Consider the plane passing through the points (0, 0, 3), (1, 0, 6), and (1, 2, 3). a.Find an equation of the plane. b.Find the intercepts of the plane with the three coordinate axes. c.Make a sketch of the plane.Intersecting planes Find an equation of the line of intersection of the planes Q and R 4.Q: 2x + y z = 0, R: x + y + z = 1 Intersecting planes Find an equation of the line of intersection of the planes Q and R 5.Q: 3x + y + 2z = 0, R: 3x + 3y + 4z 12 = 0 Equations of planes Find an equation of the following planes. 54. The plane passing through (5, 0, 2) that is parallel to the plane 2x + y z = 0 55RE 56RE Equations of planes Find an equation of the following planes. 7.The plane passing through (2, 3, 1), (1, 1, 0), and (1, 0, 1)Distance from a point to a line Find the distance from the point (1, 2, 3) to the line x = 2 + t, y = 3, z = 1 3t.Distance from a point to a plane Find the distance from the point (2, 2, 2) to the plane x + 2y + 2z = 1.Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 8.zx=0 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 9.3z=x212y248 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 10.x2100+4y2+z216=1 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 11.y2 = 4x2 + z2/25 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 12.4x29+9z24=y2 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 13.4z=x24+y29 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 14.x216+z236y2100=1 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 15.y2 + 4z2 2x2 = 1 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 16.x216+z236y225=4 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 17.x24+y216z2=4 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 18.x=y264z29 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 19.x24+y216+z2=4 Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 20.y ex = 0 73RE Identifying surfaces Consider the surfaces defined by the following equations. a.Identify and briefly describe the surface. b.Find the xy-, xz-, and yz-traces, when they exist. c.Find the intercepts with the three coordinate axes, when they exist. d.Make a sketch of the surface. 22.y=4x2+z29 75RE Designing a water bottle The lateral surface of a water bottle consists of a circular cylinder of radius 2 and height 6, topped off by a truncated hyperboloid of one sheet of height 2 (see figure). Assume the top of the truncated hyperboloid has a radius of 1/2. Find two equations that, when graphed together, form the lateral surface of the bottle. Answers may vary.Restrict the domain o f the vector function in Example 1 to produce a line segment that goes from P(2, −1, 4) to R(5, 1, 8). Example 1 Lines as vector-valued functions Find a vector function for the line that passes through the points P(2, −1, 4) and Q(3, 0,6). Explain why the curve in Example 5 lies on the cylinder x2+y2=1,as shown in Figure 14.7 Example 5 Limits and continuity consider the function r(t)=costi+sintj+etk for t0. Figure 14.7 How many independent variables does the function r(t) = f(t), g(t), h(t) have?How many dependent scalar variables does the function r(t) = f(t), g(t), h(t) have?3E 4E How do you evaluate limtar(t), where r(t) = f(t), g(t), h(t)?How do you determine whether r(t) = f(t) i + g(t) j + h(t) k is continuous at t = a?Find a function r(t) for the line passing through the points P(0,0,0) and Q(1,2,3). Express your answer in terms of i, j and k.Find a function r(t) whose graph is a circle of radius 1 parallel to the xyplane and centered at (0,0,10).9E 10E Lines and line segments Find a function r(t) that describes the given line or line segment. 11. The line through P(3, 4, 5) that is orthogonal to the plane 2x z = 4 914. Lines and line segments Find a function r(t) that describes the given line or line segment. The line of intersection of the planes 2x + 3y + 4z = 7 and 2x + 3y + 5z = 8 13E 14E Graphing curves Graph the curves described by the following functions, indicating the positive orientation. 15. r(t)=2cost,2sint, for 0t2 Graphing curves Graph the curves described by the following functions, indicating the positive orientation. 16. r(t)=2cost,2sint,for0t2 Graphing curves Graph the curves described by the following functions, indicating the positive orientation. 17. r(t)=1+cost,2+sint,for0t2 Graphing curves Graph the curves described by the following functions, indicating the positive orientation. 18. r(t)=3cost,2sint,for0t2 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 29. r(t) = cos t, 0, sin t for 0 t 2 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 30. r(t) = 0, 4 cos t, 16 sin t for 0 t 2 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 31. r(t) = cos t i + j + sin t k, for 0 t 2 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 32. r(t) = 2 cos t i + 2 sin t j + 2 k, for 0 t 2 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 33. r(t) = t cos t i + t sin t j + t k, for 0 t 6 Curves in space Graph the curves described by the following functions, indicating the positive orientation. 34. r(t) = 4 sin t i +4 cos t j + et/10 k, for 0 t =Curves in space Graph the curves described by the following functions, indicating the positive orientation. 35. r(t) = et/20 sin t i + et/20 cos t j + t k, for 0 t Curves in space Graph the curves described by the following functions, indicating the positive orientation. 36. r(t) = et/10i + 3 cos t j + 3 sin t k, for 0 t Exotic curves Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility. 27. r(t) = cos 15ti + (4 +sin 15t) cos tj + (4 + sin 15t) sin tk, for 0 t 2 Exotic curves Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility. 38. r(t) = 2 cos t i + 4 sin t j + cos 10t k, for 0 t 2 Exotic curves Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility. 39. r(t) = sin t i + sin2 t j + t/(5) k, for 0 t 10 Exotic curves Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility. 40. r(t)=costsin3ti+sintsin3tj+tk, for 0 t 9 Limits Evaluate the following limits. 41. limt/2(cos2ti4sintj+2tk)Limits Evaluate the following limits. 42. limtln/2(2eti+6etj4e2tk)Limits Evaluate the following limits. 43. limt(eti2tt+1j+tan1tk)Limits Evaluate the following limits. 44. limt2(tt2+1i4etsintj+14t+1k)Limits Evaluate the following limits. 45. limt0(sinttiett1tj+cost+t2/21t2k)Limits Evaluate the following limits. 46. limt0(tantti3tsintj+t+1k)37E Domains Find the domain of the following vector-valued functions. 56. r(t)=2t1i+3t+2j Domains Find the domain of the following vector-valued functions. 57. r(t)=t+2i+2tj Domains Find the domain of the following vector-valued functions. 58. r(t)=cos2ti+etj+12tk 41E Curve-plane intersections Find the points (if they exist) at which the following planes and curves intersect. 64. y = 1; r(t) = 10 cos t, 2 sin t, 1, for 0 t 2 Curve-plane intersections Find the points (if they exist) at which the following planes and curves intersect. 65. z = 16; r(t) = t, 2t, 4 + 3t, for t Curve-plane intersections Find the points (if they exist) at which the following planes and curves intersect. 66. y + x = 0; r(t) = cos t, sin t, t, for 0 t 4 Matching functions with graphs Match functions af with the appropriate graphs AF. a. r(t) = t, t, t b. r(t) = t2, t, t c. r(t) = 4 cos t, 4 sin t, 2 d. r(t) = 2t, sin t, cos t e. r(t) = sin t, cos t, sin 2t) f. r(t) = sin t, 2t, cos t)46E 4750. Curve of intersection Find a function r(t) that describes the curve where the following surfaces intersect. Answers are not unique. z = 4; z = x2 + y2 4750. Curve of intersection Find a function r(t) that describes the curve where the following surfaces intersect. Answers are not unique. z = 3x2 + y2 + 1; z = 5 x2 3y2 4750. Curve of intersection Find a function r(t) that describes the curve where the following surfaces intersect. Answers are not unique. x2 + y2 = 25; z = 2x + 2y Curve of intersection Find a function r(t) that describes the curve where the following surfaces intersect. Answers are not unique. 50. z = y + 1; z = x2 + 1 Golf slice A golfer launches a tee shot down a horizontal fairway; it follows a path given by r(t) = , where t ≥ 0 measures time in seconds and r has units of feet. The y-axis points straight down the fairway and the z-axis points vertically upward. The parameter a is the slice factor that determines how much the shot deviates from a straight path down the fairway. With no slice (a = 0), describe the shot. How far does the ball travel horizontally (the distance between the point where the ball leaves the ground and the point where it first strikes the ground)? With a slice (a = 0.2), how far does the ball travel horizontally? How far does the ball travel horizontally with a = 2.5? Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. 52. r(t) = (t cos t) i + (t sin t) j + t k; x2 + y2 = z2 5256. Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. r(t)=(t2+1cost)i+(t2+1sint)j+tk; x2 + y2 z2 = 1 Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. 54. r(t)=tcost,tsint,t;z=x2+y2 Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. 55. r(t)=0,2cost,3sint;x2+y24+z29=1 5256. Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. r(t)=(3+cos15t)cost,(3+cos15t)sint,sin15t; (3x2+y2)2+z2=1 (Hint: See Example 4.)5758. Closest point on a curve Find the point P on the curve r(t) that lies closest to P0 and state the distance between P0 and P. r(t) = t2 i + t j + t k; P0 (1, 1, 15)5758. Closest point on a curve Find the point P on the curve r(t) that lies closest to P0 and state the distance between P0 and P. r(t) =cos t i + sin t j + t k; P0(1, 1, 3)Curves on spheres 75. Graph the curve r(t)=12sin2t,12(1cos2t),cost and prove that it lies on the surface of a sphere centered at the origin.60E 61E Closed plane curves Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane. 70. Assuming the curve lies in a plane, show that it is a circle centered at the origin with radius R provided a2 + c2 + e2 = b2 + d2 + f2 = R2 and ab + cd + ef = 0.Closed plane curves Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane. 71. Graph the following curve and describe it. r(t)=(12cost+13sint)i+(12cost+13sint)j+(13sint)k Closed plane curves Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane. 72. Graph the following curve and describe it. r(t)=(2cost+2sint)i+(cost+2sint)j+(cost2sint)k Closed plane curves Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane. 73. Find a general expression for a nonzero vector orthogonal to the plane containing the curve. r(t)=(acost+bsint)i+(ccost+dsint)j+(ecost+fsint)k, where a, c, e b, d, f 0.Limits of vector functions Let r(t) = (f(t), g(t), h(t). a. Assume that limtar(t)=L=L1,L2,L3, which means that limtar(t)L=0. Prove that limtaf(t)=L1, limtag(t)=L2, and limtah(t)=L3. b. Assume that limtaf(t)=L1, limtag(t)=L2, and limtah(t)=L3. Prove that limtar(t)=L=L1,L2,L3, which means that limtar(t)L=0.1QC Suppose r(t) has units of m/s. Explain why T(t) = r'(t)/|r(t)| is dimensionless (has no units) and carries information only about direction.Let u(t)=t,t,t and v(t)=1,1,1 compute ddt(u(t)v(t)) using Derivative Rule 5, and show that it agrees with the result obtained by first computing the dot product and differentiating directly.Let r(t)=1,2t,3t2. Compute r(t)dt.1E Explain the geometric meaning of r(t).3E Compute r(t) when r(t) = t10, 8t, cos t.How do you find the indefinite integral of r(t) = f(t), g(t), h(t)?How do you evaluate abr(t)dt?Find C if r(t)=et,3cost,t+10+C and r(0)=0,0,0.Find the unit tangent vector at t = 0 for the parameterized curve r(t) if r(t)=et+5,sint+2,cost+2.Derivatives of vector-valued functions Differentiate the following functions. 7. r(t) = cos t, t2, sin t 10E 11E Derivatives of vector-valued functions Differentiate the following functions. 10. r(t) = 4, 3 cos 2t, 2 sin 3t 13E Derivatives of vector-valued functions Differentiate the following functions. 12. r(t) = tan t i + sec t j + cos2 t k 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 31. (t12 + 3t)u(t)Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 32. (4t8 6t3)v(t)Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 33. u(t4 2t)Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 34. v(t)Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 35. u(t) v(t)Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following functions. 36. u(t) v(t)39E 40E 41E Derivative rules Suppose u and v are differentiable functions at t = 0 withu(0)=0,1,1,u(0)=0,7,1,v(0)=0,1,1, andv(0)=1,1,2. Evaluate the following expressions. 42. ddt(u(sint))|t=0 Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the following functions. 73. v(et)44E Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the following functions. 75. v(g(t))46E Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the following functions. 77. u(t) v(t)Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the following functions. 76. u(t) v(t)Derivative rules Compute the following derivatives. 49. ddt(t2(i+2i2tk)(eti+2etj-3etk))Derivative rules Compute the following derivatives. 38. ddt((t3i2tj2k)(tit2jt3k))Derivative rules Compute the following derivatives. 39. ddt((3t2i+tj2t1k)(costi+sin2tj3tk))Derivative rules Compute the following derivatives. 40. ddt((t3i+6j2tk)(3ti12t2j6t2k))Higher-order derivatives Compute r(t) and r(t) for the following functions. 41. r(t) = t2 + 1, t + 1, 1 54E Higher-order derivatives Compute r(t) and r(t) for the following functions. 43. r(t) = cos 3t, sin 4t, cos 6t Higher-order derivatives Compute r(t) and r(t) for the following functions. 44. r(t) = e4t, 2e4t + 1, 2et Higher-order derivatives Compute r(t) and r(t) for the following functions. 45. r(t)=t+4i+tt+1jet2k Higher-order derivatives Compute r(t) and r(t) for the following functions. 46. r(t)=tanti+(t+1t)jln(t+1)k Indefinite integrals Compute the indefinite integral of the following functions. 47. r(t) = t4 3t, 2t 1, 10 60E Indefinite integrals Compute the indefinite integral of the following functions. 49. r(t) = 2 cos t, 2 sin 3t, 4 cos 8t Indefinite integrals Compute the indefinite integral of the following functions. 50. r(t)=teti+tsint2j2tt2+4k Indefinite integrals Compute the indefinite integral of the following functions. 51. r(t)=e3ti+11+t2j12tk Indefinite integrals Compute the indefinite integral of the following functions. 52. r(t)=2ti+11+2tj+lntk Finding r from r Find the function r that satisfies the given conditions. 53. r(t) = et, sin t, sec2 t; r(0) = 2, 2, 2 66E 67E Finding r from r Find the function r that satisfies the given conditions. 56. r(t)=t,cost,4/t; r(1) = 2, 3, 4 Finding r from r Find the function r that satisfies the given conditions. 57. r(t) = e2t, 1 2et, 1 2et; r(0) = 1, 1, 1 Finding r from r Find the function r that satisfies the given conditions. 58. r(t)=tt2+1i+tet2j2tt2+4k;r(0)=i+32j3k Definite integrals Evaluate the following definite integrals. 59. 11(i+tj+3t2k)dt Definite integrals Evaluate the following definite integrals. 60. 14(6t2i+8t3j+9t2k)dt Definite integrals Evaluate the following definite integrals. 61. 0ln2(eti+etcos(et)j)dt Definite integrals Evaluate the following definite integrals. 62. 1/21(31+2ticsc2(2t)k)dt Definite integrals Evaluate the following definite integrals. 63. (sinti+costj+2tk)dt Definite integrals Evaluate the following definite integrals. 64. 0ln2(eti+2e2tj4etk)dt Definite integrals Evaluate the following definite integrals. 65. 02tet(i+2jk)dt Definite integrals Evaluate the following definite integrals. 66. 0/4(sec2ti2costjk)dt 79E 80E 81E

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