Thanks for your kind help. Hope you can accommodate more follow-up.
1. Real Analysis: I don't see this listed among the remainder of the Multivariate sessions nor among MIT's Open Courseware math listings.
2. Based on your most recent answer, in the case of Example 1 it seems you are telling me we want an answer that functions as both a range and literally a square. Yes?
3. #2 and the Christine Breiner Tangent Plane Approximation recitation video leads me to believe there are additional levels of understanding to the Tangent Plane Approximation that are eluding me. In trying to visually grasp the things taught in Session 27, I was trying to get my mind around the idea of finding planes and lines that are tangent to a two-variable function. Just when I thought I understood the concept (a plane tanget or roughly tangent to a point on a function), Christine introduces using Tangent Plane Approximation to find the area of a shape, in her example a square!! Now I'm having trouble wrapping my mind around the idea of using Tangent Plane Approx. to find area. How can a plane be tangent to a square? But maybe that leads back to your comments about Example 1, in which ultimately we find a square (from which we can compute area if we wanted to). So does this mean Tangent Plane Approx. can be used to find the area of functions that describe squares or rectangles (x times y functions), but not shapes that are curvy (e.g., hyperbolic)?
4. In her video Christine says we can simply calculate the area using the numbers 2 & 3 given, but she wants us to use Tangent Plane Approx. to get the answer. OK, but why are we doing this? What deep-level thing are we supposed to understand by taking a Tangent Plane Approx. rather than doing a straightforward calculation, as Christine says we can also do?
Sorry for the length. I've been OK with the Multivariate course until now, but the point of this Approx. stuff is confusing me.