Day 6 in Mat329

• Announcements
  • Reminder: stow those rectangles....

  • I'm going to give you more time on this assignment for 14.2, since we're still talking about it today -- you can turn that in next Tuesday.

  • Quiz today, at the end of the hour.

• We're wrapping up limits and continuity section 14.2:

  We ended the hour talking about continuity: Let's take a look at a few problems involving continuity: again, invoke theorems wherever you can:

  1. #30, p. 924
  2. #32
  3. #38

• Today we 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 $x$.

  Figures 4 and 5 on p. 928 give us this idea very nicely.

  Problems:

  1. #10, p. 936
  2. #15
  3. #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!

  We think of second derivatives in the univariate world as saying something about curvature, and the same is true in the bivariate case.

  • #53, p. 937 (note the notation, p. 930)

Next time: 14.3: more partial derivatives