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All Textbook Solutions for Calculus

16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26. Using an Ellipse Consider the ellipse . (a) Find the area of the region bounded by the ellipse. (b) Find the volume of the solid generated by revolving the region about its major axis. 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 57095-10-47RE-Question-Digital.docx Finding an Equation of a Tangent Line In Exercises 47 and 48, (a) use a graphing utility to graph the curve representedby the parametric equations, (b) use a graphing utility to find dx/d, dy/d, and dy/dx at the given value of the parameter, (c)find an equation of the tangent line to the curve at the givenvalue of the parameter, and (d) use a graphing utility to graphthe curve and the tangent line from part (c). Parametric Equations Parameter x=cot,y=sin2 =6 48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE 64RE 65RE 66RE 67RE 68RE 69RE 70RE 71RE 72RE 73RE 74RE 75RE 76RE 77RE 78RE 79RE 80RE 81RE 82RE 83RE 84RE 85RE 86RE 87RE 88RE 89RE 90RE 91RE 92RE 93RE 94RE 95RE 96RE 97RE 98RE 99RE 100RE 101RE 102RE 107RE 108RE 103RE 104RE 105RE 106RE 109RE 110RE 111RE 112RE 113RE 114RE 115RE 116RE 117RE 118RE 119RE 120RE 121RE 122RE 123RE 124RE 1PS 2PS 3PS Flight Paths An air traffic controller spots two planes at the same altitude flying toward each other (see figure). Their flight paths are 20 and 315. One plane is 150 miles from point P with a speed of 375 miles per hour. The other is 190 miles from point P with a speed of 450 miles per hour. (a) Find parametric equations for the path of each plane where t is the time in hours, with t=0 corresponding to the time at which the air traffic controller spots the planes. (b) Use the result of part (a) to write the distance between the planes as a function of t. (c) Use a graphing utility to graph the function in part (b). When will the distance between the planes be minimum? If the planes must keep a separation of at least 3 miles, is the requirement met?5PS 6PS Cornu Spiral Consider the cornu spiral given by x(t)=0tcosu22du and y(t)=0tsinu22du. (a) Use a graphing utility to graph the spiral over the interval t. (b) Show that the cornu spiral is symmetric with respect to the origin. (c) Find the length of the comu spiral from t=0 to t=a. What is the length of the spiral from t= to t=?8PS 9PS 10PS 11PS 12PS 13PS 14PS 15PS 16PS 17PS CONCEPT CHECK Scalar and Vector Describe the difference between a scalar and a vector. Give example of each.Sketching a Vector In Exercises 3 and 4, (a) find the component form of the vector v and (b) sketch the vector with its initial point at the origin.3E 2E 4E 5E 6E 7E 8E 9E Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form,(c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Terminal Initial Point (4,6) (3,6)11E 12E 13E 14E 15E 16E Sketching Scalar MultipliesIn Exercises 25 and 26, sketch each scalar multiple of v. v=3,5 (a) 2v (b) 3v (c) 72v (d) 23v 18E 19E 20E 27E 28E 21E 22E 23E 24E 25E 26E 29E 30E 31E 32E 33E 34E 35E Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has length 1. v=5,15 37E 38E Finding MagnitudesIn Exercises 3942, find the following. (a) u (b) v (c) u+v (d) uu (e) vv (f) u+vu+v u=1,1,v=1,2 Finding Magnitudes In Exercises 39-42, find the following. (a) u (b) v (c) u+v (d) uu (e) vv (f) u+vu+v u=0,1,v=3,3 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E Finding a Vector In Exercises 53-56, find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. u=1,u=0v=3,v=45 54E 55E 56E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E Finding Unit Vectors In Exercises 67-72, find a unit vector (a) parallel to and (b) perpendicular to the graph of fat the given point. Then sketch the graph of f and sketch the vectors at the given point. f(x)=25x2,(3,4)72E Finding a Vector In Exercises 73 and 74, find the component form of v given the magnitudes of u and u+v and the angles that u and u+v make with the positive x-axis. u =1,=45 u+v =2,=90 74E 75E Numerical and Graphical Analysis Forces with magnitudes of 180 newtons and 275 newtons act on a hook (see figure). The angle between the two forces is degrees. (a) When =30, find the direction and magnitude of the resultant force. (b) Write the magnitude M and direction of the resultant force as functions of , where 0180. (c) Use a graphing utility to complete the table. 0 30 60 90 120 150 180 M (d) Use a graphing utility to graph the two functions M and . (e) Explain why one of the functions decreases for increasing values of , whereas the other does not.77E 78E 79E Cable TensionIn Exercises 79 and 80, determine the tension in the cable supporting the given load.Projectile Motion A gun with a muzzle velocity of 1200 feet per second is fired at an angle of 6 above thehorizontal. Find the vertical and horizontal components of the velocity.82E Navigation A plane is flying with a bearing of 302. Its speed with respect to the air is 900 kilometers per hour. The wind at the planes altitude is from the southwest at100 kilometers per hour (see figure). What is the true direction of die plane, and what is its speed with respect to the ground?NavigationA plane flies at a constant groundspeed of 400 miles per hour due east and encounters a 50-mile-per-hour wind from the northwest. Find the airspeed and compass direction that will allow the plane to maintain its groundspeed and eastward direction.85E 86E 87E 88E 89E 90E 91E 92E 93E 94E 95E 96E CONCEPT CHECK Describing Coordinates A point in the three- dimensional coordinate system has coordinates (x0, y0, z0,) Describe what each coordinate measures.1E 2E 3E 4E 5E 6E Finding Coordinates of a PointIn Exercises 912, find the coordinates of the point. The point is located on the x-axis, 12 units in front of the yz-plane.8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E Classifying a TriangleIn Exercises 2932, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (0,0,4),(2,6,7),(6,4,8)28E 29E 30E 31E 32E Finding the Midpoint In Exercises 33-36, find the coordinates of the midpoint of the line segment joining the points. (4,0,6),(8,8,20)34E 33E 35E 37E 38E 39E 40E Finding the Equation of a SphereIn Exercises 4346, complete the square to write the equation of the sphere in standard form. Find the center and radius. x2+y2+z22x+6y+8z+1=0 Finding the Equation of a SphereIn Exercises 4346, complete the square to write the equation of the sphere in standard form. Find the center and radius. x2+y2+z2+9x2y+10z+19=0 Finding the Equation of a SphereIn Exercises 4346, complete the square to write the equation of the sphere in standard form. Find the center and radius. 9x2+9y2+9z26x+18y+1=0 Finding the Equation of a Sphere In Exercises 4346, complete the square to write the equation of the sphere in standard form. Find the center and radius. 4x2+4y2+4z224x4y+8z23=0 45E 46E 47E 48E 49E 51E 52E Finding the Component Form of a Vector in SpaceIn Exercises 5154, find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial point: (3,2,0) Terminal point: (4,1,6)53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E

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