Day 4 in Mat329

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Today:

Announcements
- Reminder: stow those rectangles....
- Your quizzes are returned.
  1. A Haiku, by A. Long.
    
    A Function
    rule -- associates
    with each x in domain D
    just one y in E.
  2. Actually, the domain could include complex numbers, but no one went there.... The domain is the entire real plane: $(x,y) \in R^2$.
  3. parabola, paraboloid; both focus (e.g. satellite dishes, headlamps in cars)
  4. There are a few modellers in our midst! I have to mention two gold stars....
- 14.2 problems are now assigned: pp. 923: 1-3, 5, 10, 14, 15, 17, 24, 29, 37 (due Friday)
- Please read 14.3 for Thursday.
- Weird side note: I'm using a new method for putting mathematics into web pages (MathJax), and it's causing me fits. There may be hiccups....
I want to continue talking about limits, section 14.2:
- Let's go back to the univariate case (found in section 1.7 of your book, p. 73). Let's take a look at that, modifying my picture slightly from last time.
- We then ask "how must we modify this for multivariate functions?"
In the univariate case, how can a limit fail to exist?
- Single holes are no problem;
- Jumps are bad (including infinite ones);
- Functions may fail to "settle down": $\sin\left(\frac{1}{x}\right)$ around x=0.
The same sorts of problems will occur for multivariate functions. But, ironically, limits may exist along particular lines of approach but not agree. This is something new and different. Let's take a look at p. 917, Tables 1 and 2.
Example 2, p. 919 is also interesting.
The generalization of the univariate limit of section 1.7 is on p. 917, and well illustrated on p. 918.
- The difference is in the domain, as usual. Instead of a segment surrounding a, we require a circle surrounding (a,b); in general, a ball surrounding the point in the domain at which we desire the limit, if it exists.
- Issued a challenge about getting within a tiny value $\epsilon$ (epsilon) of the limiting value L, we must be able to find a ball small enough (as indicated by radius $\delta$ ) that function values within that "delta-neighborhood" are within $\epsilon$ of L.

Now let's take a look at some problems:

Examples 1-3 illustrate ways in which functions fail to have a limit at a point.

p. 923, #4:

1.0000	0.8824	0.3846	-0.3846	-0.8824	-1.0000
0.8824	1.0000	0.6000	-0.6000	-1.0000	-0.8824
0.3846	0.6000	1.0000	-1.0000	-0.6000	-0.3846
-0.3846	-0.6000	-1.0000	1.0000	0.6000	0.3846
-0.8824	-1.0000	-0.6000	0.6000	1.0000	0.8824
-1.0000	-0.8824	-0.3846	0.3846	0.8824	1.0000

#6 -- use theorems to solve this one.
#9
#14 -- uses a trick, and theorems.
Let's take a peek at Example 4, and see it working in #16. These arguments are a little tricky!

The next step is to generalize continuity. That's pretty easy: A function $f$ of $x$ is called ${\bf{continuous}}$ at $(a,b)$ if \[ \lim_{(x,y) \to (a,b)} f(x,y) = f(a,b) \] We say $f$ is ${\bf{continuous}}$ on $D$ if $f$ is continuous at every point $(a,b)$ in $D$.
Problems involving continuity: again, invoke theorems wherever you can:
1. #30, p. 924
2. #32
3. #38

Next time: 14.3: partial derivatives

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