MATH 201 Final, Fall 2009

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2009-2010 Final Exam Exercise 1 a. (3 points) Find the directional derivative of f(x, y) = x2e−2y at P(1, 0) in the direction of the vector v = 3i + 4j b. (3 points) The equation 1 − x − y2 − sin(xy) defines y as a differentiable function of x. Find dy/dx at the point P(0, 1). c. (5 points) Find the points on the surface xy + yz + zx − x − z2 = 0 , where the tangent plane is parallel to the xy-plane. Exercise 2 (12 points) Find the absolute minimum and maximum values of f(x, y) = x2+xy+ y2 − 3x + 3y on the triangular region R cut from the first quadrant by the line x + y = 4. Exercise 3 (6 points) Use Lagrange Multipliers to find the maximum and the minimum values of f(x, y) = xy subject to the constraint x2 + y2 = 10. Exercise 4 (10 points) Reverse the order of integration, then evaluate the integral ∫ 2 0 ∫ 4−x2 0 xe2y 4 − y dydx Exercise 5 (10 points) Let V be the volume of the region D that is bounded below by the xy-plane, above by the paraboloid z = 9 − x2 − y2, and lying outside the cylinder x2 + y2 = 1. a. express V as an iterated triple integral in cylindrical coordinates, then evaluate the resulting integral. b. express, but do not evaluate , V as an iterated triple integral in cylindrical coordinates in the order drdzdθ. Exercise 6 (10 points) Let V be the volume of the region that is bounded form below by the sphere x2 + y2 + (z − 1)2 = 1 and from above by the cone z = √ x2 + y2. Express V as an iterated triple integral in spherical coordinates, then evaluate the resulting integral (sketch the region of integration). Exercise 7 (12 points) Write six different iterated triple integrals for the volume of the tetra- hedron cut from the first octant by the plane 6x + 3y + 2z = 6. (do not evaluate any of the integrals) −→ Exercise 8 a. (6 points) Find the line integral of f(x, y) = (x + y2)/ √ 1 + x2 along the curve C : y = x2/2 from (1, 1/2) to (0, 0). b. (8 points) Show that the differential form 2 cos ydx+( 1 y −2x sin y)dy +(1/z)dz is exact, then evaluate ∫ (1,π/2,2) (0,2,1) 2 cos ydx + (1 y − 2x sin y ) dy + (1/z)dz c. Find the counterclockwise circulation of the field F = 2xy3i + 4x2y2j across the curve C in the first quadrant, bounded by the lines y = 0, x = 1 and the curve y = x3. i. (8 points) by direct calculation ii. (8 points) by Green’s theorem good luck