Math 205 HWK #11 Due Mon Mar 8

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Reading

• Begin reading/studying Section 14.8 Differentiability. A study guide for this section appears below.

Problems

• 14.4 p663: 69
• 14.5 p671: 31, 33
• 14.7 p686: 19, 27

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.

• Page 689. What's the definition of differentiability?
  • Introductory paragraphs. We will decree f(x,y) to be differentiable at (a,b) if it has a "good" linear approximation near (a,b). But what's "good"?
  • Definition in the blue box. Gives definition for differentiability of f(x,y) at (a,b). The notation is slightly different than what I used in class but not too much.
  • One way of summarizing this definition: f(x,y) is differentiable at (a,b) iff there exists a linear function
    • L(x,y) = f(a,b) + m(x-a)+n(y-b)

    for which the error expression

    • E(x,y) = f(x,y)-L(x,y)

    not only tends to zero as (x,y) --> (a,b). What more is required is that we can divide this error term by the (tiny!) distance between (x,y) and (a,b) and the result will still tend to zero as (x,y) --> (a,b) !
    
    It is in this (explicit) sense that L(x,y) is a good (really good!) approximation for f(x,y) when (x,y) is close to (a,b). See text or class notes for precise version of definition.

• Pages 690-691. So how are partial derivatives and differentiability related?
  • Example 1 gives one answer. Suppose that f(x,y) is differentiable at (a,b).
    • Example 1 then shows that f(x,y) will also have partial derivatives at the point (a,b). In other words, differentiability at (a,b) implies existence of partial derivatives at (a,b).
    • Example 1 also shows that the numbers m and n that appear in the good linear approximation L(x,y) are exactly what we would expect them to be: m is f_x(a,b) and n is f_y(a,b).
    • Bottom line: if f is differentiable at (a,b) then there's one and only one good linear approximation (think: one and only one tangent plane) and the good linear approximation is exactly what we would want it to be, namely
      • L(x,y) = f(a,b) +f_x(a,b)(x-a) + f_y(a,b)(y-b).
  • Example 2. This example shows that the particular function
    • f(x,y) = square root of (x² + y² )

    is not differentiable at (0,0) because it fails to have partial derivatives there. This function is cone-shaped. See the sketch in Example 2.
    So Example 2 illustrates: nonexistence of partial derivatives implies nondifferentiability.

  • Example 3. This example shows that the particular function
    • f(x,y) = x^1/3y^1/3

    does have partial derivatives at (0,0) but still fails to be differentiable at (0,0). So Example 3 illustrates: existence of partial derivatives is not enough to guarantee differentiability.

• Page 692. Summary at the top. Repeats:
  • differentiability implies partial derivatives exist
  • partial derivatives exist does not imply differentiability
• Rest of page 692. So how are differentiability and continuity related? How are partial derivatives and continuity related?
  • Opening paragraph states:
    • differentiability (at a point) implies continuity (at that point).

    No proof is given. Problem 14 asks the student to supply a proof. Problem 14 will be on either HWK 13 or HWK 14. After we've worked with the definitions a little longer, this problem will be pretty easy.

  • Example 4 shows: the particular function given by
    • f(x,y) = 0 if (x,y) = (0,0) while

      f(x,y) = xy/(x² +y²) otherwise

    fails to be continuous at (0,0) [ because it has no limit there] and yet it does have partial derivatives at (0,0). So this example illustrates:

    • existence of partial derivatives is not enough to guarantee continuity.
  • Second summary box repeats
    • differentiability does imply continuity
    • existence of partial derivatives does not imply continuity (hence can't imply differentiability either)
  • One bottom line: existence of partial derivatives is a pretty weak condition since it doesn't even guarantee continuity! Differentiability (existence of good linear approximation) is a much stronger condition.
    
• Bottom of page 692 and top of page 693. Well, then, how do we decide, in practice, whether a function is differentiable or not? Do we have to actually use that annoyingly technical definition?
  • Theorem 14.2. Tells us: one condition that is strong enough to guarantee differentiability of function f(x,y) at point (a,b) is that f has continuous partial derivatives at every point of some (potentially small) disk centered at (a,b). Briefly (and loosely): continuous partial derivatives imply differentiability.
  • Example 5. This example uses Theorem 14.2 to show(quite easily) that the particular function
    • f(x,y) = ln(x² +y²)

    is differentiable at every point of its domain. How does it do this? Easy: just find the partial derivatives for f. Observe that they are defined at every point where f is defined. Observe that they are also known to be continuous at all these points. Invoke Thm 14.2. Done.

  • Concluding sentences on p693 reassure us: most of the time we are working with functions that can easily be seen to possess continuous partial derivatives. So most of the time Thm 14.2 assures us our functions are differentiable.
    
  • Second line of p693: introduces the notation C¹ as a name for the collection of functions that possess continuous partial derivatives. In other words, to say that a function is in C¹ (on some domain) is to say that the function possesses continuous partial derivatives at all points (of that domain).

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 18, 2000
• Last Modified: March 3, 2004
• Expires: June 30, 2004