abdallah haa2348 Homework 10 Due by 11 59pm on Sun 5 3 reyes 52535 This print out should have 31 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 Consequently V x 3xy 2 8x2 y 4x3 cu ins 10 0 points A rectangular piece of cardboard is 3 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 002 10 0 points Which of the following surfaces could have contour map 3 x x x 2 2 x 1 1 0 1 0 2 x x 1 x 3 x 1 4 Determine the volume V of the box as a function of x and y 1 V x y 3xy 2 8x2 y 2x3 cu ins z 1 correct 2 V x y 3xy 2 8x2 y 4x3 cu ins 3 V x y 8xy 2 3x2 y 2x3 cu ins 2 2 y 3 4 V x y 8xy 3x y 2x cu ins 5 V x y 3xy 2 8x2 y 4x3 cu ins correct x 6 V x y 8xy 2 3x2 y 4x3 cu ins Explanation The rectangular sheet of cardboard will have dimensions y 3y Thus the length of the shorter side of the base of the box is y 2x while the length of the longer side is 3y 2x Since the height of the box is x the volume will thus be V x length width height x 3y 2x y 2x z 2 x y abdallah haa2348 Homework 10 Due by 11 59pm on Sun 5 3 reyes 52535 2 only one having the given contour map is a z 3 z y x y x z 4 keywords 003 y 10 0 points Which one of the following surfaces is the graph of x 1 f x y 8 x2 y 2 2 z y 5 1 2 1 0 1 2 4 3 z 2 1 0 2 y 1 0 x 1 2 x Explanation The graphs in the contour map are parallel lines so the horizontal cross sections of the surface are all lines as in planes and parabolic cylinders On the other hand the values of the contours and the grid show that these crosssections grow at a constant rate Now for a plane the contours are equally spaced lines whereas for a parabolic cylinder the contours will be increasingly close together Consequently of the surfaces shown the correct 2 y 2 1 0 1 2 4 2 z 0 2 4 2 1 0 1 x 2 abdallah haa2348 Homework 10 Due by 11 59pm on Sun 5 3 reyes 52535 3 y y 3 2 2 1 1 0 0 1 1 2 4 2 4 3 3 z z 2 2 1 1 0 2 0 2 1 1 0 0 1 1 2 x x 2 y 4 2 1 0 004 1 2 8 Use a table of numerical values of f x y for x y near the origin to make a conjecture about the value of the limit of f x y as x y 0 0 6 z 10 0 points 4 2 0 2 f x y 1 0 1 x 2 y 5 2 1 0 x2 y 3 x3 y 2 9 3 xy 1 Cannot determine the value from the given information 1 2 4 2 3 correct 3 2 z 3 9 1 0 4 3 2 1 0 1 x 2 5 9 Explanation 005 Explanation When y 0 the function becomes z 1 2 8 x2 Its graph is a parabola in the x z plane opening downwards i e a parabola opening downwards in the plane y 0 Thus the cross section of the graph of f by the plane y 0 should give a parabola opening downwards Similarly the cross section of the graph by the plane x 0 gives a parabola opening downwards This is exactly the case with 10 0 points Determine fx fy when f x y 3x2 2xy 3y 2 x 3y 1 fx fy 8x 4y 4 2 fx fy 8x 8y 4 3 fx fy 4x 8y 2 correct 4 fx fy 4x 8y 4 abdallah haa2348 Homework 10 Due by 11 59pm on Sun 5 3 reyes 52535 4 z 5 fx fy 8x 4y 2 6 fx fy 4x 4y 2 Explanation After differentiation we see that fx 6x 2y 1 fy 2x 6y 3 Consequently P y x fx fy 4x 8y 2 006 10 0 points Determine fx fy when f x y x2 3xy 3y 2 3x y 1 fx 0 fy 0 2 fx 0 fy 0 3 fx 0 fy 0 4 fx 0 fy 0 correct 1 fx fy x 3y 2 correct 2 fx fy 5x 9y 4 3 fx fy x 9y 2 4 fx fy x 3y 4 5 fx fy 5x 3y 4 6 fx fy 5x 9y 2 Explanation After differentiation we see that fx 2x 3y 3 fy 3x 6y 1 Consequently fx fy x 3y 2 007 10 0 points Determine whether the partial derivatives fx fy of f are positive negative or zero at the point P on the graph of f shown in 5 fx 0 fy 0 6 fx 0 fy 0 7 fx 0 fy 0 8 fx 0 fy 0 Explanation The value of fx at P is the slope of the tangent line to graph of f at P in the xdirection while fy is the slope of the tangent line in the y direction Thus the sign of fx indicates whether f is increasing or decreasing in the x direction or whether the tangent line in that direction at P is horizontal Similarly the value of fy at P is the slope of the tangent line at P in the y direction and so the sign of fy indicates whether f is increasing or decreasing in the y direction or whether the tangent line in that direction at P is horizontal From the graph it thus follows that at P fx 0 fy 0 abdallah haa2348 Homework 10 Due by 11 59pm on Sun 5 3 reyes 52535 keywords surface partial derivative first order partial derivative graphical interpretation 5 fx 6 fx 008 10 0 points 4x x 2y 2 4y x 2y 2 Explanation From the Quotient Rule we see that Determine fx when f x y x2 y 2y 2 x 1 fx 2y 2xy 2 3x2 fx 2 3 fx 2xy 2y 3x 2 x 2y 2x y x 2y 2 Consequently 2 fx 4xy 2 y 3x2 fx 2 4 fx 4xy 2 y 3x2 correct 010 3y x 2y 2 10 0 points 5 fx y 4xy 2 3x2 Determine fx when 6 fx 2y 2xy 2 3x2 f x y x sin 2y x cos 2y …