MATH 201 Exam 2, Fall 2012

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2012 quiz # 2 Exercise 1 (answer on page 1 of the booklet) a) Find the domain and the range of the function f(x, y) = ln ( 1 e − x2 − y2 ) ; determine if the domain open or closed, also determine if the domain is bounded or unbounded; find the equation of the level curve of f that passes through the point (1, 0) b) Find dy dx at P(0, 1) if the equation 1−x−y2−sin(xy) = 0 define y as differentiable function of x Exercise 2 (answer on page 2 of the booklet) a) Can the function f(x, y) = y2 sin x 2x2 + 3y2 be extended by continuity at (0, 0) ? b) Use the two path test to show that f(x, y) = xy − x x2 + 2(y − 1)2 has no limit at (0, 1) Exercise 3 (answer on pages 3 and 4 of the booklet) Suppose that the derivative of the function f(x, y, z) at P(1, 1, 1) is greatest in the direction of ⃗u = 6⃗i − 3⃗j + 3⃗k, and that in this direction the value of the derivative is √ 6. Also suppose that f(3, 0, 0) = 1 , ⃗∇f(3, 0, 0) = 3⃗i −⃗j + 5⃗k , f(3, 2, 1) = 3 , and ⃗∇f(3, 2, 1) = 6⃗i − 2⃗j +⃗k a) Find the derivative of f at the point (3, 2, 1) in the direction of ⃗v =⃗i +⃗j + √ 2⃗k b) Find ⃗∇f(1, 1, 1) c) Give an approximate value of f(2.99, 2.02, 1) d) Let x = r2 + s , y = rs , z = 2r s , and w = f(x, y, z) Find ∂w ∂s at the point (r, s) = (1, 2) e) Let w = w(r, s) be defined as in part d). Find a plane tangent to the surface 2 1 − t + 3 = w(r, s) in the rst-space (hint: start by finding a point (r0, s0, t0) on the surface) Exercise 4 (answer on page 5 of the booklet) Find the points on the surface xy+yz+zx−x−z2 = 0 where the tangent plane is perpendicular to the xz-plane Exercise 5 (answer on page 6 of the booklet, and last page if needed) Find the extreme values of f(x, y) = x2 + y2 − 3x − xy on the region R in plane defined by: R = {x2 + y2 ≤ 9 , x ≥ 0 , y ≥ 0}