Find first and second partial derivatives of f x y x3 e12y

Question

1. Find the first and second partial derivatives of f (x, y) = x³ _e^1+2y + sin(x² + y²)

2. Find ∂z/∂y if x³+ y³ + 4z² + 6xyz = 1

3. Find the equation of the tangent plane and equation of the normal line to the surface x² + 2y² + 3z² = 3 at the point (2, 1, 1).

4. Use the chain rule to find ∂z/∂s and ∂z/∂t for z = 4x² + x³y if x = s + 2t and y = st² at the point s = 2, t = 1.

5. Filld the linear approximation of the function f (x, y) = √(y + cos (x)) at the point (0.0) and use it to appoxiznate f (0.1, 0.2).

6. Find the directional derivative of the function f (x, y) = x²e^-y it at the point (-2, 0) in the direction of <2, 3>.

7. Find the maximum rate of change of f(x, y) = x²y + √y at the point (2, 1). In what direction does it occur?

8. Find the local maximum and minimum values and saddle points of the function

f(x, y) = xy (1 x y).