tapia jat4858 HW11 clark 52990 This print out should have 22 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 x x x x x 1 x 10 0 points Find the domain of the function f x y ln 8 x2 5y 2 x x o 1 2 1 2 x y x y 1 5 8 o n 5 1 x y x2 y 2 1 correct 8 8 o n 5 1 x y x2 y 2 1 8 8 n o 1 1 x y x2 y 2 1 5 8 n o 5 1 x y x2 y 2 1 8 8 o n 1 1 x y x2 y 2 1 5 8 Determine the volume V of the box as a function of x and y Explanation Since ln x is defined only for x 0 the function Explanation The rectangular sheet of cardboard will have dimensions y 4y Thus the length of the shorter side of the base of the box is y 2x while the length of the longer side is 4y 2x Since the height of the box is x the volume will thus be 1 2 3 4 5 6 n f x y ln 8 x2 5y 2 is defined only for n o 1 2 5 2 x y x y 1 8 8 keywords function several variables ln function domain 002 10 0 points A rectangular piece of cardboard is 4 times as long as it is wide If the length of the shorter side is y inches and an open box is constructed by cutting equal squares of sidelength x inches from the corners of the piece of cardboard and turning up the sides as shown in the figure 1 V x y 4xy 2 10x2 y 2x3 cu ins 2 V x y 4xy 2 10x2 y 4x3 cu ins correct 3 V x y 10xy 2 4x2 y 2x3 cu ins 4 V x y 10xy 2 4x2 y 2x3 cu ins 5 V x y 4xy 2 10x2 y 4x3 cu ins 6 V x y 10xy 2 4x2 y 4x3 cu ins V x length width height x 4y 2x y 2x Consequently V x 4xy 2 10x2 y 4x3 cu ins 003 10 0 points Which one of the following surfaces is the graph of f x y 2x2 tapia jat4858 HW11 clark 52990 2 y y 1 5 2 1 1 0 0 2 1 1 2 4 2 8 2 6 z 0 z 4 2 2 4 2 0 2 1 1 0 0 1 1 2 x x 2 y 2 2 1 Explanation Note that the function z does not depend on the variable y Hence the cross sections of the graph by any plane parallel to x z plane would give a parabola with branches directed upwards This is exactly the case with 0 1 2 4 2 z 0 2 y 4 2 2 1 1 0 1 0 2 8 1 2 x y 3 2 6 1 0 1 2 z 4 4 2 3 0 2 2 1 z 0 1 1 x 2 0 2 1 004 0 10 0 points 1 x 2 y 4 Which of the following surfaces could have contour map 2 1 0 1 2 8 z 6 6 5 4 4 5 3 2 4 2 0 2 1 0 0 1 x correct 3 1 2 1 2 3 4 5 6 2 1 0 tapia jat4858 HW11 clark 52990 3 z z 5 1 y x x y z 2 Explanation The graphs in the contour map show that the horizontal cross sections of the surface are all circles as in cones and paraboloids On the other hand the values of the contours and the grid show that these cross sections increase at a constant rate This occurs for a cone but not for a paraboloid Consequently of the surfaces shown the only one having the given contour map is a y x z 3 z x y y z 4 x correct keywords 005 10 0 points y x Which one of the following functions has tapia jat4858 HW11 clark 52990 4 y 2 1 5 1516 0 1 2 4 Explanation 2 z 0 007 2 Find 4 2 1 lim x y 6 2 10 0 points xy cos x 3y 1 6 0 1 x 2 2 12 correct as its graph 1 f x y y 2 x2 correct 3 2 2 f x y x2 y 2 4 0 3 f x y 2x2 5 2 1 4 f x y x2 y 2 2 Explanation 1 5 f x y 8 x2 y 2 2 008 Explanation The cross section of the graph by the plane y 0 gives a parabola opening downwards with vertex at the point O 0 0 0 Similarly the cross section of the graph by the plane x 0 gives a parabola opening upwards with vertex at the origin O Hence 4xy 2 if it exists x y 0 0 x2 y 2 lim 1 4 2 8 3 2 f x y y 2 x2 4 The limit does not exist keywords 006 Find Find 10 0 points lim x y 4 3 5 0 correct 10 0 points x5 2x3 y 3xy 2 Explanation 1 424 2 748 009 Find 3 532 correct 1 14 4 1300 2 7 10 0 points 7xy p if it exists x y 0 0 x2 y 2 lim tapia jat4858 HW11 clark 52990 3 0 correct 4 The limit does not exist 5 is continuous n o 1 x y x 0 y 0 n o p 2 x y y 6 5x 6 y 6 1 5 3 5 Explanation n o 3 x y x 0 010 10 0 points 4xy 4 Find lim if it exists x y 0 0 x2 y 8 n o 4 x y y 0 n o 5 x y y 6 correct 1 8 2 The limit does not exist correct 3 0 Explanation 013 Determine the set of points at which the function 4 2 5 4 f x y z Explanation 011 Find 10 0 points 10 0 points xy 6yz 2 5xz 2 if it x2 y 2 z 4 x y z 0 0 0 lim exists 3x2 xyz 2y 2 z is continuous n o 1 x y z z 6 3x2 2y 2 xyz 0 1 The limit does not exist correct o n 2 x y z z 6 3x2 2y 2 correct 2 0 n o 3 x y z xyz 0 3 5 o n 4 x y z z 6 3x2 2y 2 4 11 n o 2 2 5 x y z z 6 3x 2y xyz 0 5 6 Explanation Explanation 012 10 0 points Determine the set of points at which the function p F x y arctan 5x 6 y 6 014 10 0 points From the contour …