Observation 2- Starting 12.6-

Introduction

Showed the handin problems (PDF)
Also asks them for the fun fact stuff.
They need to know how to do 12.6 problems, but they really start turning in 14.1

Goes into 14.1.44 [[James Stewart - Calculus-Brooks Cole (2015).pdf#page=973]] they were able to understand this problem pretty well. They got it was a pyramid quickly.
Then they did another example (sketching contour diagrams) for $f (x, y) = \ln (x^{2} + 4 y^{2})$ , which is problem 14.1.48 above
- It might be good to have them try to realize that if they have $c = f (x, y)$ and solve for $y$ in terms of $x$ and $c$ , they don't have to solve for $y$ like 4 or however many times.
- They realized that the curve was an ellipse (it sounds like they suprisingly really know their 12.6 material)
- They intersected $x = 0$ to graph the sideview. Brussel really led them through this tho himself.
- I really like when Brussel questions himself, as it makes a scenario where the students don't want to be stuck in the Wait Time Zone, so they try to get the right answer.
For 14.1 they need to focus on:
- Level Curves
- Contour Diagrams
- 12.6 shapes (just come up in this chapter)
- Level Surfaces (3D abstraction).
Does the example of drawing level surfaces for $g (x, y, z) = x^{2} + y^{2} - z^{2}$ through the point $(1, 1, 2)$
- They need to plug in $g (p) = - 2$ , and that's the level of our surface: $x^{2} + y^{2} - z^{2} = - 2$
- He really wants them to memorize the 12.6 table at the end of the chapter.
- Then they find the level curves of this surface to be able to draw this itself.