2. True. If you find the partial derivatives at the point (0, 0), you find they are both 0, so the tangent plane approximation gives z = 0 (you'll want to check this on your own if you're not sure of it). Thus, f(0.01, 0.01) is approximately the z-coordinate of the tangent plane at the point (0.01, 0.01), which is 0.

3. True. The equation xy = 0 is satisfied when either x = 0 or y = 0. x = 0 is satisfied for any point on the yz-plane, and similarly y = 0 is satisfied for any point on the xz-plane, so the points where xy = 0 are these two intersecting planes. Note that if the equation was xy = c, with c non-zero, or if it were x = 0 or xyz = 0, this would be false.

4. False. If the three points happen to lie on a line, then you can get any plane that is just rotated around that line. One way to think about it is if the points were on the edge of a door next to a hinge - if you open or close the door, those points don't move, but the plane - the door - does change. If the three points weren't colinear - that is, they don't lie on the same line - then this is true.

5. False. You can get a surface with the tangent plane x = 1, for example by looking at the surface that is a sphere of radius one centered at the origin - x = 1 is the tangent plane to that surface through the point (0, 0, 1). To make this true, the surface would have to be given by the formula z = f(x, y), where f is some function, or the plane would have to be of the most general form Ax + By + Cz + D = 0, with A, B, C, D all constants.