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Let me start by basically
listing the main things we have
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00:00:25 --> 00:00:28
learned over the past three
weeks or so.
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00:00:28 --> 00:00:31
And I will add a few
complements of information about
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00:00:31 --> 00:00:34
that because there are a few
small details that I didn't
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00:00:34 --> 00:00:38
quite clarify and that I should
probably make a bit clearer,
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00:00:38 --> 00:00:48
especially what happened at the
very end of yesterday's class.
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Here is a list of things that
should be on your review sheet
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00:00:56 --> 00:01:01
for the exam.
The first thing we learned
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about, the main topic of this
unit is about functions of
16
00:01:08 --> 00:01:12
several variables.
We have learned how to think of
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00:01:12 --> 00:01:16
functions of two or three
variables in terms of plotting
18
00:01:16 --> 00:01:17
them.
In particular,
19
00:01:17 --> 00:01:19
well, not only the graph but
also the contour plot and how to
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00:01:19 --> 00:01:27
read a contour plot.
And we have learned how to
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study variations of these
functions using partial
22
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derivatives.
Remember, we have defined the
23
00:01:44 --> 00:01:47
partial of f with respect to
some variable,
24
00:01:47 --> 00:01:52
say, x to be the rate of change
with respect to x when we hold
25
00:01:52 --> 00:01:55
all the other variables
constant.
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00:01:55 --> 00:02:01
If you have a function of x and
y, this symbol means you
27
00:02:01 --> 00:02:07
differentiate with respect to x
treating y as a constant.
28
00:02:07 --> 00:02:15
And we have learned how to
package partial derivatives into
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00:02:15 --> 00:02:20
a vector,the gradient vector.
For example,
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00:02:20 --> 00:02:24
if we have a function of three
variables, the vector whose
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components are the partial
derivatives.
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00:02:26 --> 00:02:33
And we have seen how to use the
gradient vector or the partial
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00:02:33 --> 00:02:39
derivatives to derive various
things such as approximation
34
00:02:39 --> 00:02:43
formulas.
The change in f,
35
00:02:43 --> 00:02:48
when we change x,
y, z slightly,
36
00:02:48 --> 00:02:57
is approximately equal to,
well, there are several terms.
37
00:02:57 --> 00:03:03
And I can rewrite this in
vector form as the gradient dot
38
00:03:03 --> 00:03:08
product the amount by which the
position vector has changed.
39
00:03:08 --> 00:03:11
Basically, what causes f to
change is that I am changing x,
40
00:03:11 --> 00:03:16
y and z by small amounts and
how sensitive f is to each
41
00:03:16 --> 00:03:22
variable is precisely what the
partial derivatives measure.
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00:03:22 --> 00:03:26
And, in particular,
this approximation is called
43
00:03:26 --> 00:03:30
the tangent plane approximation
because it tells us,
44
00:03:30 --> 00:03:35
in fact,
it amounts to identifying the
45
00:03:35 --> 00:03:38
graph of the function with its
tangent plane.
46
00:03:38 --> 00:03:43
It means that we assume that
the function depends more or
47
00:03:43 --> 00:03:45
less linearly on x,
y and z.
48
00:03:45 --> 00:03:48
And, if we set these things
equal, what we get is actually,
49
00:03:48 --> 00:03:52
we are replacing the function
by its linear approximation.
50
00:03:52 --> 00:03:56
We are replacing the graph by
its tangent plane.
51
00:03:56 --> 00:03:58
Except, of course,
we haven't see the graph of a
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00:03:58 --> 00:04:00
function of three variables
because that would live in
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00:04:00 --> 00:04:04
4-dimensional space.
So, when we think of a graph,
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00:04:04 --> 00:04:08
really, it is a function of two
variables.
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00:04:08 --> 00:04:12
That also tells us how to find
tangent planes to level
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00:04:12 --> 00:04:12
surfaces.
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00:04:12 --> 00:04:22
58
00:04:22 --> 00:04:30
Recall that the tangent plane
to a surface,
59
00:04:30 --> 00:04:37
given by the equation f of x,
y, z equals z,
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00:04:37 --> 00:04:43
at a given point can be found
by looking first for its normal
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00:04:43 --> 00:04:47
vector.
And we know that the normal
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00:04:47 --> 00:04:49
vector is actually,
well,
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00:04:49 --> 00:04:53
one normal vector is given by
the gradient of a function
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because we know that the
gradient is actually pointing
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perpendicularly to the level
sets towards higher values of a
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function.
And it gives us the direction
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of fastest increase of a
function.
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OK.
Any questions about these
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topics?
No.
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OK.
Let me add, actually,
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a cultural note to what we have
seen so far about partial
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derivatives and how to use them,
which is maybe something I
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should have mentioned a couple
of weeks ago.
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Why do we like partial
derivatives?
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Well, one obvious reason is we
can do all these things.
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But another reason is that,
really,
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you need partial derivatives to
do physics and to understand
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much of the world that is around
you because a lot of things
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00:05:46 --> 00:05:50
actually are governed by what is
called partial differentiation
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00:05:50 --> 00:05:51
equations.
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00:05:51 --> 00:05:59
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00:05:59 --> 00:06:07
So if you want a cultural
remark about what this is good
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00:06:07 --> 00:06:09
for.
A partial differential equation
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00:06:09 --> 00:06:13
is an equation that involves the
partial derivatives of a
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function.
So you have some function that
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00:06:15 --> 00:06:18
is unknown that depends on a
bunch of variables.
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00:06:18 --> 00:06:23
And a partial differential
equation is some relation
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00:06:23 --> 00:06:28
between its partial derivatives.
Let me see.
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These are equations involving
the partial derivatives -- -- of
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00:06:45 --> 00:06:54
an unknown function.
Let me give you an example to
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see how that works.
For example,
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00:06:57 --> 00:07:02
the heat equation is one
example of a partial
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00:07:02 --> 00:07:09
differential equation.
It is the equation -- Well,
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00:07:09 --> 00:07:15
let me write for you the space
version of it.
95
00:07:15 --> 00:07:21
It is the equation partial f
over partial t equals some
96
00:07:21 --> 00:07:27
constant times the sum of the
second partials with respect to
97
00:07:27 --> 00:07:32
x, y and z.
So this is an equation where we
98
00:07:32 --> 00:07:38
are trying to solve for a
function f that depends,
99
00:07:38 --> 00:07:42
actually, on four variables,
x, y, z, t.
100
00:07:42 --> 00:07:47
And what should you have in
mind?
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00:07:47 --> 00:07:50
Well, this equation governs
temperature.
102
00:07:50 --> 00:07:55
If you think that f of x, y, z,
t will be the temperature at a
103
00:07:55 --> 00:07:59
point in space at position x,
y, z and at time t,
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00:07:59 --> 00:08:04
then this tells you how
temperature changes over time.
105
00:08:04 --> 00:08:07
It tells you that at any given
point,
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00:08:07 --> 00:08:10
the rate of change of
temperature over time is given
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00:08:10 --> 00:08:15
by this complicated expression
in the partial derivatives in
108
00:08:15 --> 00:08:18
terms of the space coordinates
x, y, z.
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00:08:18 --> 00:08:21
If you know, for example,
the initial distribution of
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00:08:21 --> 00:08:24
temperature in this room,
and if you assume that nothing
111
00:08:24 --> 00:08:26
is generating heat or taking
heat away,
112
00:08:26 --> 00:08:29
so if you don't have any air
conditioning or heating going
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00:08:29 --> 00:08:31
on,
then it will tell you how the
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00:08:31 --> 00:08:35
temperature will change over
time and eventually stabilize to
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00:08:35 --> 00:08:41
some final value.
Yes?
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00:08:41 --> 00:08:43
Why do we take the partial
derivative twice?
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00:08:43 --> 00:08:45
Well, that is a question,
I would say,
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00:08:45 --> 00:08:48
for a physics person.
But in a few weeks we will
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00:08:48 --> 00:08:52
actually see a derivation of
where this equation comes from
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00:08:52 --> 00:08:55
and try to justify it.
But, really,
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00:08:55 --> 00:08:57
that is something you will see
in a physics class.
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00:08:57 --> 00:09:02
The reason for that is
basically physics of how heat is
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00:09:02 --> 00:09:09
transported between particles in
fluid, or actually any medium.
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00:09:09 --> 00:09:12
This constant k actually is
called the heat conductivity.
125
00:09:12 --> 00:09:17
It tells you how well the heat
flows through the material that
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00:09:17 --> 00:09:20
you are looking at.
Anyway, I am giving it to you
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00:09:20 --> 00:09:23
just to show you an example of a
real life problem where,
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00:09:23 --> 00:09:26
in fact, you have to solve one
of these things.
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00:09:26 --> 00:09:29
Now, how to solve partial
differential equations is not a
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00:09:29 --> 00:09:32
topic for this class.
It is not even a topic for
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00:09:32 --> 00:09:34
18.03 which is called
Differential Equations,
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00:09:34 --> 00:09:38
without partial,
which means there actually you
133
00:09:38 --> 00:09:41
will learn tools to study and
solve these equations but when
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00:09:41 --> 00:09:43
there is only one variable
involved.
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00:09:43 --> 00:09:47
And you will see it is already
quite hard.
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00:09:47 --> 00:09:50
And, if you want more on that
one, we have many fine classes
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00:09:50 --> 00:09:52
about partial differential
equations.
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00:09:52 --> 00:09:58
But one thing at a time.
I wanted to point out to you
139
00:09:58 --> 00:10:03
that very often functions that
you see in real life satisfy
140
00:10:03 --> 00:10:08
many nice relations between the
partial derivatives.
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00:10:08 --> 00:10:10
That was in case you were
wondering why on the syllabus
142
00:10:10 --> 00:10:13
for today it said partial
differential equations.
143
00:10:13 --> 00:10:15
Now we have officially covered
the topic.
144
00:10:15 --> 00:10:20
That is basically all we need
to know about it.
145
00:10:20 --> 00:10:22
But we will come back to that a
bit later.
146
00:10:22 --> 00:10:27
You will see.
OK.
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00:10:27 --> 00:10:30
If there are no further
questions, let me continue and
148
00:10:30 --> 00:10:33
go back to my list of topics.
Oh, sorry.
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00:10:33 --> 00:10:42
I should have written down that
this equation is solved by
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00:10:42 --> 00:10:48
temperature for point x,
y, z at time t.
151
00:10:48 --> 00:10:52
OK.
And there are, actually,
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00:10:52 --> 00:10:56
many other interesting partial
differential equations you will
153
00:10:56 --> 00:10:59
maybe sometimes learn about the
wave equation that governs how
154
00:10:59 --> 00:11:02
waves propagate in space,
about the diffusion equation,
155
00:11:02 --> 00:11:07
when you have maybe a mixture
of two fluids,
156
00:11:07 --> 00:11:11
how they somehow mix over time
and so on.
157
00:11:11 --> 00:11:16
Basically, to every problem you
might want to consider there is
158
00:11:16 --> 00:11:19
a partial differential equation
to solve.
159
00:11:19 --> 00:11:23
OK. Anyway. Sorry.
Back to my list of topics.
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00:11:23 --> 00:11:27
One important application we
have seen of partial derivatives
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00:11:27 --> 00:11:30
is to try to optimize things,
try to solve minimum/maximum
162
00:11:30 --> 00:11:31
problems.
163
00:11:31 --> 00:11:42
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00:11:42 --> 00:11:47
Remember that we have
introduced the notion of
165
00:11:47 --> 00:11:56
critical points of a function.
A critical point is when all
166
00:11:56 --> 00:12:03
the partial derivatives are
zero.
167
00:12:03 --> 00:12:05
And then there are various
kinds of critical points.
168
00:12:05 --> 00:12:09
There is maxima and there is
minimum, but there is also
169
00:12:09 --> 00:12:15
saddle points.
And we have seen a method using
170
00:12:15 --> 00:12:24
second derivatives -- -- to
decide which kind of critical
171
00:12:24 --> 00:12:29
point we have.
I should say that is for a
172
00:12:29 --> 00:12:35
function of two variables to try
to decide whether a given
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00:12:35 --> 00:12:41
critical point is a minimum,
a maximum or a saddle point.
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00:12:41 --> 00:12:44
And we have also seen that
actually that is not enough to
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00:12:44 --> 00:12:48
find the minimum of a maximum of
a function because the minimum
176
00:12:48 --> 00:12:50
of a maximum could occur on the
boundary.
177
00:12:50 --> 00:12:53
Just to give you a small
reminder,
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00:12:53 --> 00:12:55
when you have a function of one
variables,
179
00:12:55 --> 00:13:00
if you are trying to find the
minimum and the maximum of a
180
00:13:00 --> 00:13:03
function whose graph looks like
this,
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00:13:03 --> 00:13:05
well, you are going to tell me,
quite obviously,
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00:13:05 --> 00:13:07
that the maximum is this point
up here.
183
00:13:07 --> 00:13:11
And that is a point where the
first derivative is zero.
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00:13:11 --> 00:13:14
That is a critical point.
And we used the second
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00:13:14 --> 00:13:18
derivative to see that this
critical point is a local
186
00:13:18 --> 00:13:20
maximum.
But then, when we are looking
187
00:13:20 --> 00:13:23
for the minimum of a function,
well, it is not at a critical
188
00:13:23 --> 00:13:26
point.
It is actually here at the
189
00:13:26 --> 00:13:30
boundary of the domain,
you know, the range of values
190
00:13:30 --> 00:13:38
that we are going to consider.
Here the minimum is at the
191
00:13:38 --> 00:13:44
boundary.
And the maximum is at a
192
00:13:44 --> 00:13:50
critical point.
Similarly, when you have a
193
00:13:50 --> 00:13:53
function of several variables,
say of two variables,
194
00:13:53 --> 00:13:55
for example,
then the minimum and the
195
00:13:55 --> 00:13:58
maximum will be achieved either
at a critical point.
196
00:13:58 --> 00:14:01
And then we can use these
methods to find where they are.
197
00:14:01 --> 00:14:06
Or, somewhere on the boundary
of a set of values that are
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00:14:06 --> 00:14:09
allowed.
It could be that we actually
199
00:14:09 --> 00:14:13
achieve a minimum by making x
and y as small as possible.
200
00:14:13 --> 00:14:16
Maybe letting them go to zero
if they had to be positive or
201
00:14:16 --> 00:14:19
maybe by making them go to
infinity.
202
00:14:19 --> 00:14:23
So, we have to keep our minds
open and look at various
203
00:14:23 --> 00:14:26
possibilities.
We are going to do a problem
204
00:14:26 --> 00:14:29
like that.
We are going to go over a
205
00:14:29 --> 00:14:34
practice problem from the
practice test to clarify this.
206
00:14:34 --> 00:14:38
Another important cultural
application of minimum/maximum
207
00:14:38 --> 00:14:42
problems in two variables that
we have seen in class is the
208
00:14:42 --> 00:14:45
least squared method to find the
best fit line,
209
00:14:45 --> 00:14:49
or the best fit anything,
really,
210
00:14:49 --> 00:14:56
to find when you have a set of
data points what is the best
211
00:14:56 --> 00:15:01
linear approximately for these
data points.
212
00:15:01 --> 00:15:03
And here I have some good news
for you.
213
00:15:03 --> 00:15:07
While you should definitely
know what this is about,
214
00:15:07 --> 00:15:09
it will not be on the test.
215
00:15:09 --> 00:15:30
216
00:15:30 --> 00:15:35
[APPLAUSE]
That doesn't mean that you
217
00:15:35 --> 00:15:41
should forget everything we have
seen about it,
218
00:15:41 --> 00:15:51
OK?
Now what is next on my list of
219
00:15:51 --> 00:15:58
topics?
We have seen differentials.
220
00:15:58 --> 00:16:03
Remember the differential of f,
by definition,
221
00:16:03 --> 00:16:09
would be this kind of quantity.
At first it looks just like a
222
00:16:09 --> 00:16:12
new way to package partial
derivatives together into some
223
00:16:12 --> 00:16:15
new kind of object.
Now, what is this good for?
224
00:16:15 --> 00:16:18
Well, it is a good way to
remember approximation formulas.
225
00:16:18 --> 00:16:22
It is a good way to also study
how variations in x,
226
00:16:22 --> 00:16:26
y, z relate to variations in f.
In particular,
227
00:16:26 --> 00:16:30
we can divide this by
variations,
228
00:16:30 --> 00:16:34
actually, by dx or by dy or by
dz in any situation that we
229
00:16:34 --> 00:16:40
want,
or by d of some other variable
230
00:16:40 --> 00:16:46
to get chain rules.
The chain rule says,
231
00:16:46 --> 00:16:50
for example,
there are many situations.
232
00:16:50 --> 00:16:56
But, for example,
if x, y and z depend on some
233
00:16:56 --> 00:17:04
other variable,
say of variables maybe even u
234
00:17:04 --> 00:17:08
and v,
then that means that f becomes
235
00:17:08 --> 00:17:13
a function of u and v.
And then we can ask ourselves,
236
00:17:13 --> 00:17:18
how sensitive is f to a value
of u?
237
00:17:18 --> 00:17:25
Well, we can answer that.
The chain rule is something
238
00:17:25 --> 00:17:33
like this.
And let me explain to you again
239
00:17:33 --> 00:17:41
where this comes from.
Basically, what this quantity
240
00:17:41 --> 00:17:46
means is if we change u and keep
v constant, what happens to the
241
00:17:46 --> 00:17:48
value of f?
Well, why would the value of f
242
00:17:48 --> 00:17:51
change in the first place when f
is just a function of x,
243
00:17:51 --> 00:17:55
y, z and not directly of you?
Well, it changes because x,
244
00:17:55 --> 00:17:59
y and z depend on u.
First we have to figure out how
245
00:17:59 --> 00:18:02
quickly x, y and z change when
we change u.
246
00:18:02 --> 00:18:05
Well, how quickly they do that
is precisely partial x over
247
00:18:05 --> 00:18:08
partial u, partial y over
partial u, partial z over
248
00:18:08 --> 00:18:10
partial u.
These are the rates of change
249
00:18:10 --> 00:18:14
of x, y, z when we change u.
And now, when we change x,
250
00:18:14 --> 00:18:17
y and z, that causes f to
change.
251
00:18:17 --> 00:18:21
How much does f change?
Well, partial f over partial x
252
00:18:21 --> 00:18:25
tells us how quickly f changes
if I just change x.
253
00:18:25 --> 00:18:29
I get this.
That is the change in f caused
254
00:18:29 --> 00:18:33
just by the fact that x changes
when u changes.
255
00:18:33 --> 00:18:36
But then y also changes.
y changes at this rate.
256
00:18:36 --> 00:18:39
And that causes f to change at
that rate.
257
00:18:39 --> 00:18:42
And z changes as well,
and that causes f to change at
258
00:18:42 --> 00:18:45
that rate.
And the effects add up together.
259
00:18:45 --> 00:18:57
Does that make sense?
OK.
260
00:18:57 --> 00:19:00
And so, in particular,
we can use the chain rule to do
261
00:19:00 --> 00:19:03
changes of variables.
If we have, say,
262
00:19:03 --> 00:19:08
a function in terms of polar
coordinates on theta and we like
263
00:19:08 --> 00:19:14
to switch it to rectangular
coordinates x and y then we can
264
00:19:14 --> 00:19:19
use chain rules to relate the
partial derivatives.
265
00:19:19 --> 00:19:23
And finally,
last but not least,
266
00:19:23 --> 00:19:31
we have seen how to deal with
non-independent variables.
267
00:19:31 --> 00:19:37
When our variables say x,
y, z related by some equation.
268
00:19:37 --> 00:19:41
One way we can deal with this
is to solve for one of the
269
00:19:41 --> 00:19:44
variables and go back to two
independent variables,
270
00:19:44 --> 00:19:47
but we cannot always do that.
Of course, on the exam,
271
00:19:47 --> 00:19:50
you can be sure that I will
make sure that you cannot solve
272
00:19:50 --> 00:19:53
for a variable you want to
remove because that would be too
273
00:19:53 --> 00:19:56
easy.
Then when we have to look at
274
00:19:56 --> 00:20:02
all of them, we will have to
take into account this relation,
275
00:20:02 --> 00:20:05
we have seen two useful
methods.
276
00:20:05 --> 00:20:09
One of them is to find the
minimum of a maximum of a
277
00:20:09 --> 00:20:13
function when the variables are
not independent,
278
00:20:13 --> 00:20:17
and that is the method of
Lagrange multipliers.
279
00:20:17 --> 00:20:33
280
00:20:33 --> 00:20:39
Remember, to find the minimum
or the maximum of the function
281
00:20:39 --> 00:20:45
f,
subject to the constraint g
282
00:20:45 --> 00:20:52
equals constant,
well, we write down equations
283
00:20:52 --> 00:20:59
that say that the gradient of f
is actually proportional to the
284
00:20:59 --> 00:21:04
gradient of g.
There is a new variable here,
285
00:21:04 --> 00:21:08
lambda, the multiplier.
And so, for example,
286
00:21:08 --> 00:21:12
well, I guess here I had
functions of three variables,
287
00:21:12 --> 00:21:14
so this becomes three
equations.
288
00:21:14 --> 00:21:21
f sub x equals lambda g sub x,
f sub y equals lambda g sub y,
289
00:21:21 --> 00:21:25
and f sub z equals lambda g sub
z.
290
00:21:25 --> 00:21:27
And, when we plug in the
formulas for f and g,
291
00:21:27 --> 00:21:31
well, we are left with three
equations involving the four
292
00:21:31 --> 00:21:33
variables, x,
y, z and lambda.
293
00:21:33 --> 00:21:36
What is wrong?
Well, we don't have actually
294
00:21:36 --> 00:21:41
four independent variables.
We also have this relation,
295
00:21:41 --> 00:21:48
whatever the constraint was
relating x, y and z together.
296
00:21:48 --> 00:21:51
Then we can try to solve this.
And, depending on the
297
00:21:51 --> 00:21:56
situation, it is sometimes easy.
And it sometimes it is very
298
00:21:56 --> 00:22:01
hard or even impossible.
But on the test,
299
00:22:01 --> 00:22:03
I haven't decided yet,
but it could well be that the
300
00:22:03 --> 00:22:06
problem about Lagrange
multipliers just asks you to
301
00:22:06 --> 00:22:08
write the equations and not to
solve them.
302
00:22:08 --> 00:22:14
[APPLAUSE]
Well, I don't know yet.
303
00:22:14 --> 00:22:18
I am not promising anything.
But, before you start solving,
304
00:22:18 --> 00:22:23
check whether the problem asks
you to solve them or not.
305
00:22:23 --> 00:22:26
If it doesn't then probably you
shouldn't.
306
00:22:26 --> 00:23:02
307
00:23:02 --> 00:23:09
Another topic that we solved
just yesterday is constrained
308
00:23:09 --> 00:23:13
partial derivatives.
And I guess I have to
309
00:23:13 --> 00:23:19
re-explain a little bit because
my guess is that things were not
310
00:23:19 --> 00:23:23
extremely clear at the end of
class yesterday.
311
00:23:23 --> 00:23:25
Now we are in the same
situation.
312
00:23:25 --> 00:23:29
We have a function,
let's say, f of x,
313
00:23:29 --> 00:23:34
y, z where variables x,
y and z are not independent but
314
00:23:34 --> 00:23:39
are constrained by some relation
of this form.
315
00:23:39 --> 00:23:43
Some quantity involving x,
y and z is equal to maybe zero
316
00:23:43 --> 00:23:47
or some other constant.
And then, what we want to know,
317
00:23:47 --> 00:23:51
is what is the rate of change
of f with respect to one of the
318
00:23:51 --> 00:23:57
variables,
say, x, y or z when I keep the
319
00:23:57 --> 00:24:02
others constant?
Well, I cannot keep all the
320
00:24:02 --> 00:24:07
other constant because that
would not be compatible with
321
00:24:07 --> 00:24:11
this condition.
I mean that would be the usual
322
00:24:11 --> 00:24:15
or so-called formal partial
derivative of f ignoring the
323
00:24:15 --> 00:24:18
constraint.
To take this into account means
324
00:24:18 --> 00:24:23
that if we vary one variable
while keeping another one fixed
325
00:24:23 --> 00:24:26
then the third one,
since it depends on them,
326
00:24:26 --> 00:24:31
must also change somehow.
And we must take that into
327
00:24:31 --> 00:24:34
account.
Let's say, for example,
328
00:24:34 --> 00:24:39
we want to find -- I am going
to do a different example from
329
00:24:39 --> 00:24:42
yesterday.
So, if you really didn't like
330
00:24:42 --> 00:24:46
that one, you don't have to see
it again.
331
00:24:46 --> 00:24:51
Let's say that we want to find
the partial derivative of f with
332
00:24:51 --> 00:24:56
respect to z keeping y constant.
What does that mean?
333
00:24:56 --> 00:25:03
That means y is constant,
z varies and x somehow is
334
00:25:03 --> 00:25:11
mysteriously a function of y and
z for this equation.
335
00:25:11 --> 00:25:14
And then, of course because it
depends on y,
336
00:25:14 --> 00:25:19
that means x will vary.
Sorry, depends on y and z and z
337
00:25:19 --> 00:25:21
varies.
Now we are asking ourselves
338
00:25:21 --> 00:25:25
what is the rate of change of f
with respect to z in this
339
00:25:25 --> 00:25:26
situation?
340
00:25:26 --> 00:25:42
341
00:25:42 --> 00:25:47
And so we have two methods to
do that.
342
00:25:47 --> 00:25:55
Let me start with the one with
differentials that hopefully you
343
00:25:55 --> 00:26:02
kind of understood yesterday,
but if not here is a second
344
00:26:02 --> 00:26:06
chance.
Using differentials means that
345
00:26:06 --> 00:26:10
we will try to express df in
terms of dz in this particular
346
00:26:10 --> 00:26:14
situation.
What do we know about df in
347
00:26:14 --> 00:26:19
general?
Well, we know that df is f sub
348
00:26:19 --> 00:26:25
x dx plus f sub y dy plus f sub
z dz.
349
00:26:25 --> 00:26:28
That is the general statement.
But, of course,
350
00:26:28 --> 00:26:31
we are in a special case.
We are in a special case where
351
00:26:31 --> 00:26:41
first y is constant.
y is constant means that we can
352
00:26:41 --> 00:26:50
set dy to be zero.
This goes away and becomes zero.
353
00:26:50 --> 00:26:53
The second thing is actually we
don't care about x.
354
00:26:53 --> 00:26:57
We would like to get rid of x
because it is this dependent
355
00:26:57 --> 00:27:00
variable.
What we really want to do is
356
00:27:00 --> 00:27:12
express df only in terms of dz.
What we need is to relate dx
357
00:27:12 --> 00:27:16
with dz.
Well, to do that,
358
00:27:16 --> 00:27:20
we need to look at how the
variables are related so we need
359
00:27:20 --> 00:27:24
to look at the constraint g.
Well, how do we do that?
360
00:27:24 --> 00:27:31
We look at the differential g.
So dg is g sub x dx plus g sub
361
00:27:31 --> 00:27:37
y dy plus g sub z dz.
And that is zero because we are
362
00:27:37 --> 00:27:40
setting g to always stay
constant.
363
00:27:40 --> 00:27:44
So, g doesn't change.
If g doesn't change then we
364
00:27:44 --> 00:27:48
have a relation between dx,
dy and dz.
365
00:27:48 --> 00:27:50
Well, in fact,
we say we are going to look
366
00:27:50 --> 00:27:52
only at the case where y is
constant.
367
00:27:52 --> 00:27:56
y doesn't change and this
becomes zero.
368
00:27:56 --> 00:27:59
Well, now we have a relation
between dx and dz.
369
00:27:59 --> 00:28:05
We know how x depends on z.
And when we know how x depends
370
00:28:05 --> 00:28:10
on z, we can plug that into here
and get how f depends on z.
371
00:28:10 --> 00:28:11
Let's do that.
372
00:28:11 --> 00:28:28
373
00:28:28 --> 00:28:33
Again, saying that g cannot
change and keeping y constant
374
00:28:33 --> 00:28:39
tells us g sub x dx plus g sub z
dz is zero and we would like to
375
00:28:39 --> 00:28:46
solve for dx in terms of dz.
That tells us dx should be
376
00:28:46 --> 00:28:53
minus g sub z dz divided by g
sub x.
377
00:28:53 --> 00:28:57
If you want,
this is the rate of change of x
378
00:28:57 --> 00:29:00
with respect to z when we keep y
constant.
379
00:29:00 --> 00:29:13
In our new terminology this is
partial x over partial z with y
380
00:29:13 --> 00:29:18
held constant.
This is the rate of change of x
381
00:29:18 --> 00:29:23
with respect to z.
Now, when we know that,
382
00:29:23 --> 00:29:30
we are going to plug that into
this equation.
383
00:29:30 --> 00:29:37
And that will tell us that df
is f sub x times dx.
384
00:29:37 --> 00:29:43
Well, what is dx?
dx is now minus g sub z over g
385
00:29:43 --> 00:29:51
sub x dz plus f sub z dz.
So that will be minus fx g sub
386
00:29:51 --> 00:29:56
z over g sub x plus f sub z
times dz.
387
00:29:56 --> 00:30:02
And so this coefficient here is
the rate of change of f with
388
00:30:02 --> 00:30:06
respect to z in the situation we
are considering.
389
00:30:06 --> 00:30:13
This quantity is what we call
partial f over partial z with y
390
00:30:13 --> 00:30:21
held constant.
That is what we wanted to find.
391
00:30:21 --> 00:30:25
Now, let's see another way to
do the same calculation and then
392
00:30:25 --> 00:30:28
you can choose which one you
prefer.
393
00:30:28 --> 00:30:57
394
00:30:57 --> 00:31:09
The other method is using the
chain rule.
395
00:31:09 --> 00:31:14
We use the chain rule to
understand how f depends on z
396
00:31:14 --> 00:31:19
when y is held constant.
Let me first try the chain rule
397
00:31:19 --> 00:31:24
brutally and then we will try to
analyze what is going on.
398
00:31:24 --> 00:31:29
You can just use the version
that I have up there as a
399
00:31:29 --> 00:31:35
template to see what is going
on, but I am going to explain it
400
00:31:35 --> 00:31:37
more carefully again.
401
00:31:37 --> 00:31:50
402
00:31:50 --> 00:31:57
That is the most mechanical and
mindless way of writing down the
403
00:31:57 --> 00:32:01
chain rule.
I am just saying here that I am
404
00:32:01 --> 00:32:04
varying z, keeping y constant,
and I want to know how f
405
00:32:04 --> 00:32:07
changes.
Well, f might change because x
406
00:32:07 --> 00:32:10
might change,
y might change and z might
407
00:32:10 --> 00:32:14
change.
Now, how quickly does x change?
408
00:32:14 --> 00:32:18
Well, the rate of change of x
in this situation is partial x,
409
00:32:18 --> 00:32:24
partial z with y held constant.
If I change x at this rate then
410
00:32:24 --> 00:32:29
f will change at that rate.
Now, y might change,
411
00:32:29 --> 00:32:32
so the rate of change of y
would be the rate of change of y
412
00:32:32 --> 00:32:35
with respect to z holding y
constant.
413
00:32:35 --> 00:32:38
Wait a second.
If y is held constant then y
414
00:32:38 --> 00:32:40
doesn't change.
So, actually,
415
00:32:40 --> 00:32:43
this guy is zero and you didn't
really have to write that term.
416
00:32:43 --> 00:32:47
But I wrote it just to be
systematic.
417
00:32:47 --> 00:32:51
If y had been somehow able to
change at a certain rate then
418
00:32:51 --> 00:32:54
that would have caused f to
change at that rate.
419
00:32:54 --> 00:32:57
And, of course,
if y is held constant then
420
00:32:57 --> 00:33:01
nothing happens here.
Finally, while z is changing at
421
00:33:01 --> 00:33:05
a certain rate,
this rate is this one and that
422
00:33:05 --> 00:33:10
causes f to change at that rate.
And then we add the effects
423
00:33:10 --> 00:33:12
together.
See, it is nothing but the
424
00:33:12 --> 00:33:16
good-old chain rule.
Just I have put these extra
425
00:33:16 --> 00:33:22
subscripts to tell us what is
held constant and what isn't.
426
00:33:22 --> 00:33:23
Now, of course we can simplify
it a little bit more.
427
00:33:23 --> 00:33:27
Because, here,
how quickly does z change if I
428
00:33:27 --> 00:33:32
am changing z?
Well, the rate of change of z,
429
00:33:32 --> 00:33:37
with respect to itself,
is just one.
430
00:33:37 --> 00:33:41
In fact, the really mysterious
part of this is the one here,
431
00:33:41 --> 00:33:45
which is the rate of change of
x with respect to z.
432
00:33:45 --> 00:33:49
And, to find that,
we have to understand the
433
00:33:49 --> 00:33:52
constraint.
How can we find the rate of
434
00:33:52 --> 00:33:54
change of x with respect to z?
Well, we could use
435
00:33:54 --> 00:33:56
differentials,
like we did here,
436
00:33:56 --> 00:33:58
but we can also keep using the
chain rule.
437
00:33:58 --> 00:34:17
438
00:34:17 --> 00:34:20
How can I do that?
Well, I can just look at how g
439
00:34:20 --> 00:34:24
would change with respect to z
when y is held constant.
440
00:34:24 --> 00:34:33
I just do the same calculation
with g instead of f.
441
00:34:33 --> 00:34:37
But, before I do it,
let's ask ourselves first what
442
00:34:37 --> 00:34:40
is this equal to.
Well, if g is held constant
443
00:34:40 --> 00:34:44
then, when we vary z keeping y
constant and changing x,
444
00:34:44 --> 00:34:53
well, g still doesn't change.
It is held constant.
445
00:34:53 --> 00:34:58
In fact, that should be zero.
But, if we just say that,
446
00:34:58 --> 00:35:01
we are not going to get to
that.
447
00:35:01 --> 00:35:04
Let's see how we can compute
that using the chain rule.
448
00:35:04 --> 00:35:09
Well, the chain rule tells us g
changes because x,
449
00:35:09 --> 00:35:12
y and z change.
How does it change because of x?
450
00:35:12 --> 00:35:18
Well, partial g over partial x
times the rate of change of x.
451
00:35:18 --> 00:35:21
How does it change because of y?
Well, partial g over partial y
452
00:35:21 --> 00:35:24
times the rate of change of y.
But, of course,
453
00:35:24 --> 00:35:28
if you are smarter than me then
you don't need to actually write
454
00:35:28 --> 00:35:31
this one because y is held
constant.
455
00:35:31 --> 00:35:38
And then there is the rate of
change because z changes.
456
00:35:38 --> 00:35:45
And how quickly z changes here,
of course, is one.
457
00:35:45 --> 00:35:50
Out of this you get,
well, I am tired of writing
458
00:35:50 --> 00:35:58
partial g over partial x.
We can just write g sub x times
459
00:35:58 --> 00:36:05
partial x over partial z y
constant plus g sub z.
460
00:36:05 --> 00:36:11
And now we found how x depends
on z.
461
00:36:11 --> 00:36:17
Partial x over partial z with y
held constant is negative g sub
462
00:36:17 --> 00:36:24
z over g sub x.
Now we plug that into that and
463
00:36:24 --> 00:36:32
we get our answer.
It goes all the way up here.
464
00:36:32 --> 00:36:34
And then we get the answer.
I am not going to,
465
00:36:34 --> 00:36:35
well, I guess I can write it
again.
466
00:36:35 --> 00:36:47
467
00:36:47 --> 00:36:52
There was partial f over
partial x times this guy,
468
00:36:52 --> 00:36:59
minus g sub z over g sub x,
plus partial f over partial z.
469
00:36:59 --> 00:37:03
And you can observe that this
is exactly the same formula that
470
00:37:03 --> 00:37:07
we had over here.
In fact, let's compare this to
471
00:37:07 --> 00:37:10
make it side by side.
I claim we did exactly the same
472
00:37:10 --> 00:37:13
thing, just with different
notations.
473
00:37:13 --> 00:37:17
If you take the differential of
f and you divide it by dz in
474
00:37:17 --> 00:37:20
this situation where y is held
constant and so on,
475
00:37:20 --> 00:37:23
you get exactly this chain rule
up there.
476
00:37:23 --> 00:37:28
That chain rule up there is
this guy, df,
477
00:37:28 --> 00:37:33
divided by dz with y held
constant.
478
00:37:33 --> 00:37:38
And the term involving dy was
replaced by zero on both sides
479
00:37:38 --> 00:37:41
because we knew,
actually, that y is held
480
00:37:41 --> 00:37:44
constant.
Now, the real difficulty in
481
00:37:44 --> 00:37:48
both cases comes from dx.
And what we do about dx is we
482
00:37:48 --> 00:37:52
use the constant.
Here we use it by writing dg
483
00:37:52 --> 00:37:55
equals zero.
Here we write the chain rule
484
00:37:55 --> 00:38:00
for g, which is the same thing,
just divided by dz with y held
485
00:38:00 --> 00:38:03
constant.
This formula or that formula
486
00:38:03 --> 00:38:07
are the same,
just divided by dz with y held
487
00:38:07 --> 00:38:11
constant.
And then, in both cases,
488
00:38:11 --> 00:38:16
we used that to solve for dx.
And then we plugged into the
489
00:38:16 --> 00:38:21
formula of df to express df over
dz, or partial f,
490
00:38:21 --> 00:38:26
partial z with y held constant.
So, the two methods are pretty
491
00:38:26 --> 00:38:27
much the same.
Quick poll.
492
00:38:27 --> 00:38:33
Who prefers this one?
Who prefers that one?
493
00:38:33 --> 00:38:34
OK.
Majority vote seems to be for
494
00:38:34 --> 00:38:36
differentials,
but it doesn't mean that it is
495
00:38:36 --> 00:38:39
better.
Both are fine.
496
00:38:39 --> 00:38:42
You can use whichever one you
want.
497
00:38:42 --> 00:38:50
But you should give both a try.
OK. Any questions?
498
00:38:50 --> 00:38:58
Yes?
Yes. Thank you.
499
00:38:58 --> 00:39:02
I forgot to mention it.
Where did that go?
500
00:39:02 --> 00:39:11
I think I erased that part.
We need to know -- --
501
00:39:11 --> 00:39:20
directional derivatives.
Pretty much the only thing to
502
00:39:20 --> 00:39:23
remember about them is that df
over ds,
503
00:39:23 --> 00:39:25
in the direction of some unit
vector u,
504
00:39:25 --> 00:39:30
is just the gradient f dot
product with u.
505
00:39:30 --> 00:39:35
That is pretty much all we know
about them.
506
00:39:35 --> 00:39:39
Any other topics that I forgot
to list?
507
00:39:39 --> 00:39:45
No.
Yes?
508
00:39:45 --> 00:39:46
Can I erase three boards at a
time?
509
00:39:46 --> 00:39:47
No, I would need three hands to
do that.
510
00:39:47 --> 00:40:03
511
00:40:03 --> 00:40:07
I think what we should do now
is look quickly at the practice
512
00:40:07 --> 00:40:10
test.
I mean, given the time,
513
00:40:10 --> 00:40:15
you will mostly have to think
about it yourselves.
514
00:40:15 --> 00:40:23
Hopefully you have a copy of
the practice exam.
515
00:40:23 --> 00:40:26
The first problem is a simple
problem.
516
00:40:26 --> 00:40:28
Find the gradient.
Find an approximation formula.
517
00:40:28 --> 00:40:30
Hopefully you know how to do
that.
518
00:40:30 --> 00:40:33
The second problem is one about
writing a contour plot.
519
00:40:33 --> 00:40:41
And so, before I let you go for
the weekend, I want to make sure
520
00:40:41 --> 00:40:47
that you actually know how to
read a contour plot.
521
00:40:47 --> 00:40:51
One thing I should mention is
this problem asks you to
522
00:40:51 --> 00:40:55
estimate partial derivatives by
writing a contour plot.
523
00:40:55 --> 00:40:57
We have not done that,
so that will not actually be on
524
00:40:57 --> 00:40:59
the test.
We will be doing qualitative
525
00:40:59 --> 00:41:01
questions like what is the sine
of a partial derivative.
526
00:41:01 --> 00:41:04
Is it zero, less than zero or
more than zero?
527
00:41:04 --> 00:41:07
You don't need to bring a ruler
to estimate partial derivatives
528
00:41:07 --> 00:41:09
the way that this problem asks
you to.
529
00:41:09 --> 00:41:35
530
00:41:35 --> 00:41:38
[APPLAUSE]
Let's look at problem 2B.
531
00:41:38 --> 00:41:43
Problem 2B is asking you to
find the point at which h equals
532
00:41:43 --> 00:41:46
2200,
partial h over partial x equals
533
00:41:46 --> 00:41:49
zero and partial h over partial
y is less than zero.
534
00:41:49 --> 00:41:53
Let's try and see what is going
on here.
535
00:41:53 --> 00:41:57
A point where f equals 2200,
well, that should be probably
536
00:41:57 --> 00:41:59
on the level curve that says
2200.
537
00:41:59 --> 00:42:09
We can actually zoom in.
Here is the level 2200.
538
00:42:09 --> 00:42:12
Now I want partial h over
partial x to be zero.
539
00:42:12 --> 00:42:17
That means if I change x,
keeping y constant,
540
00:42:17 --> 00:42:24
the value of h doesn't change.
Which points on the level curve
541
00:42:24 --> 00:42:30
satisfy that property?
It is the top and the bottom.
542
00:42:30 --> 00:42:34
If you are here, for example,
and you move in the x
543
00:42:34 --> 00:42:36
direction,
well, you see,
544
00:42:36 --> 00:42:38
as you get to there from the
left,
545
00:42:38 --> 00:42:41
the height first increases and
then decreases.
546
00:42:41 --> 00:42:44
It goes for a maximum at that
point.
547
00:42:44 --> 00:42:47
So, at that point,
the partial derivative is zero
548
00:42:47 --> 00:42:53
with respect to x.
And the same here.
549
00:42:53 --> 00:42:59
Now, let's find partial h over
partial y less than zero.
550
00:42:59 --> 00:43:03
That means if we go north we
should go down.
551
00:43:03 --> 00:43:07
Well, which one is it,
top or bottom?
552
00:43:07 --> 00:43:11
Top. Yes.
Here, if you go north,
553
00:43:11 --> 00:43:16
then you go from 2200 down to
2100.
554
00:43:16 --> 00:43:23
This is where the point is.
Now, the problem here was also
555
00:43:23 --> 00:43:25
asking you to estimate partial h
over partial y.
556
00:43:25 --> 00:43:28
And if you were curious how you
would do that,
557
00:43:28 --> 00:43:33
well, you would try to figure
out how long it takes before you
558
00:43:33 --> 00:43:42
reach the next level curve.
To go from here to here,
559
00:43:42 --> 00:43:47
to go from Q to this new point,
say Q prime,
560
00:43:47 --> 00:43:49
the change in y,
well, you would have to read
561
00:43:49 --> 00:43:56
the scale,
which was down here,
562
00:43:56 --> 00:44:00
would be about something like
300.
563
00:44:00 --> 00:44:04
What is the change in height
when you go from Q to Q prime?
564
00:44:04 --> 00:44:07
Well, you go down from 2200 to
2100.
565
00:44:07 --> 00:44:14
That is actually minus 100
exactly.
566
00:44:14 --> 00:44:19
OK?
And so delta h over delta y is
567
00:44:19 --> 00:44:27
about minus one-third,
well, minus 100 over 300 which
568
00:44:27 --> 00:44:35
is minus one-third.
And that is an approximation
569
00:44:35 --> 00:44:43
for partial derivative.
So, that is how you would do it.
570
00:44:43 --> 00:44:48
Now, let me go back to other
things.
571
00:44:48 --> 00:44:52
If you look at this practice
exam, basically there is a bit
572
00:44:52 --> 00:44:56
of everything and it is kind of
fairly representative of what
573
00:44:56 --> 00:45:00
might happen on Tuesday.
There will be a mix of easy
574
00:45:00 --> 00:45:03
problems and of harder problems.
Expect something about
575
00:45:03 --> 00:45:05
computing gradients,
approximations,
576
00:45:05 --> 00:45:08
rate of change.
Expect a problem about reading
577
00:45:08 --> 00:45:13
a contour plot.
Expect one about a min/max
578
00:45:13 --> 00:45:15
problem,
something about Lagrange
579
00:45:15 --> 00:45:17
multipliers,
something about the chain rule
580
00:45:17 --> 00:45:20
and something about constrained
partial derivatives.
581
00:45:20 --> 00:45:22
I mean pretty much all the
topics are going to be there.
582
00:45:22 --> 00:45:23