Math 205
HWK #11
Due Mon Mar 8

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Here's a guide for 14.8, what's in this section, what to concentrate on for now, and so forth. But first, a few comments before we start.

  1. Although the definition of differentiability is technical, hence a little difficult to absorb and begin to work with, the underlying idea is pretty simple: f is differentiable at a point if it has a good linear approximation at that point. For functions of two variables this can be thought of as having a tangent plane. The technical details are need to make "good linear approximation" precise.
  2. Although the definition is not very useful as a test for differentiability, the definition is very useful for establishing properties of or facts about differentiable functions. For instance, we can use it to rigorously establish the formula we've been using for finding directional derivatives; and we can use it to establish that differentiability implies continuity; and more.
  3. Some of you will probably like this somewhat technical, somewhat abstract stuff, and those of you who major in mathematics definitely need to be exposed to it (as do, I suspect, those of you who major in economics and want to be well grounded in mathematics for economics).
  4. Some of you will probably hate this somewhat technical, somewhat abstract stuff, and perhaps you could get by quite nicely without most of it. That's ok. I might wax enthusiastic about it, but there's some personal taste involved here, and my taste (or some other mathematician's) doesn't have to be your taste. I hope there'll be other things in the topics that you do like, as well as other topics that you find more immediately useful.
  5. You do have to learn this stuff. No, it won't (at least for the most part) be on Test 1. Yes, it will be on some other quiz or test or final.

Now here's the promised outline or reading guide.


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