Quiz 2B for MATH 105 SECTION 205
January 23, 2015
Given Name
Family Name
Student Number
1. (a) (1 point) Find the value of a such that the gradient of f (x, y, z) = aex + xy + sin(z) at point (0, 0, 0) is
orthogonal to a normal vector of the plane 2x − z = 1.
(a)
(b) (1 point) Can you find a function h(x, y) such that ∇h(x, y) = hcos(y), sin(x)i? (Do not need to find
h(x, y), just put ‘Yes’ or ‘No’)
(b)
(c) (2 points) Solve the system
x3 = xy
.
y−x=2
(c)
2. (4 points) Use the Lagrange multipliers to find the maximum and minimum values of f (x, y) = xy subject to
x2 + y 2 − xy = 9.
3. Let f (x, y) = 2x2 + 2y 2 − 6x and R := {(x, y) : x2 + y 2 ≤ 9}, then
(a) (1 point) Use polar coordinates to find the maximum and minimum values of f (x, y) on the boundary of
R.
(b) (1 point) Use Lagrange multipliers to find the maximum and minimum values of f (x, y) on the boundary
of R.
(c) (1 point) Find the absolute maximum and minimum values of f (x, y) on R.
Your Score:
/11