Find all relative extrema and saddle points for the

Question

1. Determine and sketch domains of the given functions

(a) f(x, y) = √(x² + y² -4)               

(b) f(x, y) = (√(4-x²)/y²+3)

(c) f(x, y, z) = 1/In(x² + y² + z² -9)

2. Sketch the graph of f

(a) f(x, y) = √(x²+y²)

(b) f(x, y) = √(x²+y²-1)

3. Let z = sin(y² - 4x)

(a) Find the rate of change of z with respect to x at the point (2, 1) with y held fixed

(b) Find the rate of change of z with respect to y at the point (2, 1) with x held fixed

4. Find ∂z/∂x and ∂z/∂y  for each of the following:

(a) z = 4e^x2y3        

(b) z = xy/x² +y²

5. Find f_x, f_y, f_z for the function

f(x, y, z) = zln(x²y cosz)

6. Use Implicit differentiation to compute ∂z/∂x, ∂z/∂y, ∂z/∂z

X²+ zsin(xyz) = 0

7. Use the chain rule to find ∂z/∂u and ∂z/∂v

(a) z = x² - y tan(x); x = u/v, y = u²v²

(b) z = cosx siny; x = u - v, y = u² + v²

8. Find a unit vector in the direction in which f increases most rapidly at P, and find the rate of change of f in that direction

(a) f(x, y) = 3x -lny; P(2, 4)

(b) f(x, y, z) = x/z + z/y²; P(1, 2, -2)

9. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere x² + y²+ z² - 6x - 8y - 8z + 24 =0 common tangent point at P(1, 1, 2).

10. (a) Find all relative extrema and saddle points for the function f(x, y) = xy + 2/x + 4/y

(b) Find absolute max and min values of f(x, y) = 4x² - 3y² + 2xy on the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.