MATH 201 Problems Quiz 2, Fall 2012

Exercise 1 Let f(x, y) = √ 9 − x2 − y2 a) give the domain and the range of f; b) sketch some level curves of f; c) sketch the graph of f; d) find fx(0, 0) and fy(0, 0) (by using the definition) Exercise 2 The derivative of of f(x, y) at P(1, 2) in the direction of i+j is 2 √ 2 and in direction of −2j is −3. Find the derivative of f in the direction of −i − 2j Exercise 3 The derivative of of f(x, y, z) at P is greatest in the direction of the vector −→v = −→i + −→j + −→k . In this direction, the value of the derivative is 2 √ 3. Find ∇f at P. Find the derivative of f at P in the direction of −→i + −→j Exercise 4 Find the points on the surface (y + z)2 + (z − x)2 = 16 where the normal line is parallel to the yz-plane Exercise 5 Locate all relative extremum and saddle points of f(x, y) = 4xy − x4 − y4 Exercise 6 A flat circular plate has the shape of the region x2 + y2 ≤ 1. The plate, including the boundary where x2 + y2 = 1, is heated so that the temperature at the point (x, y) is T(x, y) = x2 + 2y2 − x Find the temperature at the hottest and the coldest point on the plate Exercise 7 The function w = f(u, v) is twice differentiable in u and v, and satisfying fuu+fvv = 0. Suppose that u = x2−y2 2 and v = xy. Use the chain rule to show that wxx + wyy = 0 Exercise 8 Suppose that the derivative of the function f(x, y, z) at the point (1, 1, 1) is greatest in the direction of ⃗p = 6⃗i − 3⃗j + 3⃗k, and that in this direction the value of the derivative is √ 6. Also suppose that f(3, 0, −1) = 1 , ⃗∇f(3, 0, −1) = 3⃗i−⃗j+5⃗k , ⃗∇f(3, 2, 1) = 6⃗i−2⃗j+⃗k , ⃗∇f(0, −1, 1) =⃗i+⃗j+⃗k a) Find the directional derivative of f at the point (3, 2, 1) in the direction of the vector⃗i+⃗j+ √ 2⃗k b) Find ⃗∇f(1, 1, 1) c) Is there a unit vector ⃗q such that (D⃗qf)(3, 0, −1) = 6 ? justify your answer d) Find the normal line to the surface f(x, y, z) at the point (3, 0, −1) e) Let x = u , y = v − 2 , and z = v − u , and g = f(x, y, z). Find ∂g ∂u and ∂g ∂v at the point (u, v) = (3, 2) f) Let g = g(u, v) be as in part e). Find a plane tangent to the surface g(u, v) = 2w2 − 1 in the uvw-space (hint: start by finding a point (u, v, w) = (?, ?, ?) on the surface)