Day 5 in Mat329

• Announcements
  • Reminder: stow those rectangles....
  • Your homeworks are returned: graded #4, 28, 46.
    • 4: in each case, you need to do this in general. Some of you just gave one example for part a, which isn't sufficient. You might have just gotten lucky.... Patrick?

      Amit noticed that to do part b is sufficient.

    • 28: talked about how to do this one in class; label your axes!
    • 46: ditto; label your axes! A lot of you threw out the lower half of the plane. What were you thinking?

      Colton asked a great question -- can contour lines cross?

• Let's wrap up limits, section 14.2:
  • Let's go back to some problems:
    1. #6 -- use theorems to solve this one. Univariate limit laws are in section 1.6. Review! There are a few new ones in your section 14.2 summary (perhaps relating to continuity -- getting to that in a minute!).
    2. #9
    3. #14 -- uses a trick, and theorems.
    4. 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

• Today I hope to start section 14.3, on partial derivatives.

  Here is what I call "the most important definition in calculus" -- the limit definition of the derivative function, $f'(x)$: \[ f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} \]

  This is what we want to generalize.

  • Take a look at Table 1, p. 925. How can we use the univariate idea?
  • What about other directions? Think about a topo map -- you can set off in any direction from a point (often): what does the partial derivative mean in any direction?
  • #2, p. 935
  So here's the partial with respect to $x$: \[ f_x(x,y) = \lim_{h \to 0}\frac{f(x+h,y)-f(x,y)}{h} \]

  and the partial with respect to $y$: \[ f_y(x,y) = \lim_{h \to 0}\frac{f(x,y+h)-f(x,y)}{h} \]

  They're easy to compute, actually: e.g., for $f_x$, just imagine that $y$ is a parameter, and differentiate in the univariate way with respect to $x$.

  Problems:

  1. #15, p. 936
  2. #24
  Higher partial derivatives are easy to calculate, as well, since each partial is again a multivariate function in its own right. So do it again!
  • #53, p. 937 (note the notation, p. 930)

Next time: 14.3: more partial derivatives