MATH 201 Problems Sample Exam 2, Fall 2011

Exercise 1 Let f(x, y) = � (x2 + y2) sin( 1 x2+y2 ) (x, y) ̸= (0, 0) 0 (x, y) = (0, 0) a) Show that f is continuous at (0, 0) b) Find ∂f ∂x(0, 0), and ∂f ∂y (0, 0) c) Show that f is differentiable at (0, 0) Exercise 2 Consider a function f(x, y) such that f(1, 2) = 10 , ⃗∇f(1, 2) = 3⃗i + 4⃗j , ⃗∇f(3, 4) = 5⃗i + 6⃗j , ⃗∇f(5, 6) =⃗i + 2⃗j a) Find the directional derivative of f at the point P0(5, 6) in the direction of the vector ⃗v = (3, 4) b) Find an approximate value for f(1.02, 1.99) (you can use either methods: 1) ∆f = fx ∆x + fy ∆y + ε � x2 + y2 or 2) df = (Duf) × ds) c) Find the partial derivative ∂ ∂s � f(s2 + t2, 3st) � (s,t)=(1,2) Exercise 3 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 Duf(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)