Find a normal vector to the level curve found in part a at

Question

1. Use polar coordinates to find the limit if it exists.

lim_{(x, y)→(0, 0)} (x² + y²)In (x² + y²)

2. Given w = f(x, y), x = r cos θ and y = r sin θ

a. Compute ∂w/∂x and ∂w/∂y.

b. Prove that (∂w/∂x)² + (∂w/∂y)² = (∂w/∂r)² + (1/r2) (∂w/∂θ)².

3. f(x, y) = x / (x² + y²)

a. Find an equation of the level curve which contains point P(1, 1)

b. Find a normal vector to the level curve found in part a, at point P.

4. a. Find all critical points of f(x, y) = (4y + x² y² + 8x) / xy

b. Find all extrema of f.

5. Find the max and min values of f(x, y) = x² - 3xy - y² + 2y- 6x constrained to region R. R = {(x, y) | |x| ≤ 3, |y| = ≤ 2}.

In problems 6 and 7 use lagrange multipliers to find the extrema of the objective functions subject to the given constraints.

6. Find the point on the intersection line of planes x + 3y - 2z = 11, 2x - y + z= 3 which is closest to the origin.

7.Find the extrema of f(x, y, z) = x + 2y - 3z subject to z= 4x²" + y², then use the discriminant to determine whether it is a max or a min.