- Begin reading/studying Section 14.8 Differentiability. A study guide for
this section appears below.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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
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
- 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
- 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 fx(a,b) and
n is fy(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) +fx(a,b)(x-a)
- Example 2. This example shows that the particular
- 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
- Example 3. This example shows that the particular
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
- 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
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
- 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?
Go back to the top
- Alexia Sontag, Mathematics
- Wellesley College
- Date Created: January 18, 2000
- Last Modified: March 3, 2004
- Expires: June 30, 2004