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MVC Quiz #2 NAME: Show all appropriate work and thinking to receive full credit. Calculator allowed. Explain briefly how you are using your calculator. #1(5 pt's) Write the parametric equations of the line that is perpendicular to the plane with equation 3x 2 y 1 and goes through the point (1,1,1) . #2(5 pt’s) Write the equation of the plane containing that contains the point (3,5,0) and the line x 5 2t y 1 t z 4t #3(5 pt's) Prove that for a , b 3 , a b a b 2a b #4(6 pt's) Find the parametric equations of a line that is perpendicular to the line given by x 1 3t y 1 2t and is in the plane given by 2 x y 3 . z 3 2t #5(6 pt's) Determine whether there is a plane that contains the two lines x 3t y 1 t . If so, find the equation of this plane. If not, explain. z 2 5t x 3 3s y 1 2s and z 4s Spherical/Cylindrical r sin z cos 2 r2 z2 r tan z Spherical/Cartesian x sin cos y sin sin z cos 2 2 2 2 x y z x2 y 2 tan z y tan x #6(5 pts) Convert the equation sin( ) csc( ) into an equivalent equation in Cartesian coordinates and then sketch its graph. [If you want to clarify your picture you may describe the graph in words] #7(6 pts) Describe the region inside the solid cylinder of radius 4 and height 4 pictured below. Assume that the cylinder is centered on the positive z-axis and “sits on the xy-plane. [For example, a description of a solid sphere of radius 1 centered at the origin would be 1 in spherical and 0 r 1, 0 2 , 1 r 2 z 1 r 2 in cylindrical]. a. Using cylindrical coordinates. 4 4 b.Using spherical coordinates. Do one of the following – your choice. Indicate clearly which one you want graded. 2 #8(3 pt's) Determine whether the function f : linear function. Explain clearly and completely. 2 defined f ( x, y ) (3x 2 y 1,3x y ) is a Remember: A function f : 2 2 is a linear function if and only if, for all u , v 1. f (ku ) kf (u ) , and 2. f (u v ) f (u ) f (v ) 2 , and k #9(3 pts) Show that the area of a triangle, two of whose sides are determined by the vectors 2 1 a b (a b ) 2 . a and b is given by the formula: A 2 a b