Class Summary 20120221

[Section 1]
9.2.23 About three iterations.
9.2.22 About three iterations.
Emphasized being able to do calculations by hand.
Drew a picture, but didn't have time to explain the equation in full. Thought perhaps it was too conceptual to be repeated in class.
Was trying to just help with problem 9.3.42 but basically did it only without determining the constants.
pg 499 15-25 odd method recognition.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
25 Allowed to discuss the degree and needing to simplify by using partial fractions. While there, I was able to sneak in my blurb about simplifying common factors and Heaviside cover-up method.

[Section 2]
I started with
pg 499 15-25 odd method recognition.
I gave each one it's own column making the work a little more neat.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
A student for 19 made an important simplification before the u-substitution that I overlooked.
$e^{x+e^x}=e^xe^{e^x}$
I couldn't figure out on the spot if the u-substitution $x+e^x$ really works or not, but I didn't linger, as the purpose of the exercise is to test recognition.
I was able to be more thorough about Heaviside cover-up method, throwing in an example of a repeated factor and using an easy to compute $x$ to compute any lingering variables.
After that I went to discus 9.2.23 up to $y_3$ but without computing $y_3$. I spend a moment to explain the meaning of the formula and drew a little picture.
A student asked about multiple solutions and what that means. I first refer to Figure 8. Then I explain there are infinitely many solutions. If you give an initial condition, then you get a particular solution.
Then I talked about level sets but with #5 instead of #3. (I did #3 in Section 1)
After class a student asked about finishing problem 7.5.25, that is after we have divided to make sure the numerator is less than the denominator. I wanted to hand back homework to those who were waiting for it, while he had to leave.
Another student asked about $y(5)$. I suppose the confusion was why we stop at $y_5$. After some discussion and drawing a picture, it was made apparent that it depends on the step size $h$.

Comments:
There's a lot to cover and there's only so much I can make sure students pay attention to. Ultimately it's up to them to ask the questions and do whatever they can to secure their understanding. For example, by reading the book, going to math help room, going to the professor's or TA's office hours, and comparing solutions with a friend.