Review V- (Sec 10.4- Sec 10.8) 1. (10 pts) Find the equations of the tangent plane to the surface

Review V- (Sec 10.4- Sec 10.8)

1. (10 pts) Find the equations of the tangent plane to the surface x²e^xy− z² = 0 at the point(1, 0, 1).

2. (5 pts) Suppose w = x² − 2xy + yz, x = 2e^r2 − s, y = ln (r + 2s), z = r + s². Find

∂w

∂s with r = 0, s = 1.

3. (5 pts) Assume that the equation xz + y ln x− y²+ 4 = 0 defines x as a differentiable function of two independent variables y and z. Find ^∂x_∂z at (1, 1,−3).

4. (5 pts) Let f (x, y) = x² + y². Find ∇f(x, y) and the directional derivative of f at (2, 1) toward the point (1, 0).

5. (10 pts) Find all critical points of the function f (x, y) = x⁴+y⁴−4xy, then determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

6. (10 pts) Find all extreme values of f (x, y) = x² + y² in the region 2x² + y² ≤ 3.