TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS LAST NAME FIRST NAME ID MATH 251 EXAM 2 Section 504 FALL 2008 O.Shatalov Show all work. 1. [17%] For f (x, y, z) = x3 + sin(xyz) (a) find the gradient; (b) find the directional derivative at the point (1, π2 , 1) in the direction of the vector h2, 1, 2i; (c) find the maximum rate of change of f at the point (1, π2 , 1). 2. [13%] Find the equation of the plane which is tangent to the surface zexyz = 1 at the point (5, 0, 1). 3. [15%] Find all critical points of f (x, y) = x3 + y 3 − 3xy and classify them as local maxima, local minima, or saddle points. 4. [15%] Let f (x, y) = xy − 2x + 5. Find the absolute maximum and minimum values of f on the set D which is the closed triangular region with vertices A(0,0), B(1,1), C(0,1). 5. [15%] For Z 3Z 9 0 y2 f (x, y) dxdy (a) sketch the region of integration; (b) change the order of integration. 6. [15%]Find the volume of the solid that lies under the paraboloid z = x2 + y 2 , above the xy-plane, and inside the cylinder x2 + y 2 = 4. 7. [15%]Convert the integral √ 2Z √ Z Z 3 0 9−y 0 0 9−x2 −y 2 (x2 + y 2 + z 2 ) dz dx dy to an integral in spherical coordinates, but don’t evaluate it. spherical coordinates: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.