Find the volume under the paraboloid z x2 y2 and above

Question

1. If w = f(x, y, z), where x = x(t, u), y = y (t, u), and z = z (t, u), use the chain rule to write out expression for f_t and f_u.  

2. Find the maximum directional derivates and the direction in which it occurs for f(x, y) = 3x² + siny at the point (1, 0).

3. Find the classify the local extrema of f(x, y) = x⁴ + y⁴ - 4xy.

4. Find the points on the surface z² = xy + 1 that are closest to the origin.

5. Calculate the exact value of _-1∫¹_x2∫¹(x+2y)dA.

6. Sketch the region of integration and change the order of integration for ₀∫⁴₀∫^√xf(x, y) dy dx.

7. Find the volume under the paraboloid z = x² + y² and above the disk x² + y² ≤ 1.

8. Set up, but do not evaluate the integrals required to compute the centre of mass for the region bounded by x = 0, x = 1, y = 0 and y = x² if density is proportional to distance from the x-axis.

9. Find the area of the portion of the surface z = x + y² that lies above a triangle with vertices (0, 0), (1, 1), and (0, 1).