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PROFESSOR: Today we're moving
on from theoretical things from
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the mean value theorem to the
introduction to what's going to
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occupy us for the whole rest of
the course, which
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is integration.
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00:00:34 --> 00:00:39
So, in order to introduce that
subject, I need to introduce
13
00:00:39 --> 00:00:52
for you a new notation, which
is called differentials.
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I'm going to tell you what a
differential is, and we'll get
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00:00:56 --> 00:01:02
used to using it over time.
16
00:01:02 --> 00:01:11
If you have a function which
is y = f ( x), then the
17
00:01:11 --> 00:01:28
differential of y is going to
be denoted dy, and it's by
18
00:01:28 --> 00:01:34
definition f' ( x ) dx.
19
00:01:34 --> 00:01:41
So here's the notation.
20
00:01:41 --> 00:01:45
And because y is really equal
to f, sometimes we also call
21
00:01:45 --> 00:01:50
it the differential of f.
22
00:01:50 --> 00:01:58
It's also called the
differential of f.
23
00:01:58 --> 00:02:07
That's the notation, and it's
the same thing as what happens
24
00:02:07 --> 00:02:12
if you formally just take this
dx, act like it's a number
25
00:02:12 --> 00:02:14
and divide it into dy.
26
00:02:14 --> 00:02:22
So it means the same thing
as this statement here.
27
00:02:22 --> 00:02:27
And this is more or less
the Leibniz, not Leibniz,
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00:02:27 --> 00:02:38
interpretation of derivatives.
29
00:02:38 --> 00:02:50
Of a derivative as a ratio of
these so called differentials.
30
00:02:50 --> 00:03:04
It's a ratio of what are
known as infinitesimals.
31
00:03:04 --> 00:03:10
Now, this is kind of a vague
notion, this little bit here
32
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being an infinitesimal.
33
00:03:12 --> 00:03:16
It's sort of like an
infinitely small quantity.
34
00:03:16 --> 00:03:21
And Leibniz perfected the
idea of dealing with
35
00:03:21 --> 00:03:23
these intuitively.
36
00:03:23 --> 00:03:25
And subsequently,
mathematicians use
37
00:03:25 --> 00:03:26
them all the time.
38
00:03:26 --> 00:03:33
They're way more effective than
the notation that Newton used.
39
00:03:33 --> 00:03:38
You might think that notations
are a small matter, but they
40
00:03:38 --> 00:03:40
allow you to think much
faster, sometimes.
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00:03:40 --> 00:03:42
When you have the right
names and the right
42
00:03:42 --> 00:03:44
symbols for everything.
43
00:03:44 --> 00:03:47
And in this case it made
it very big difference.
44
00:03:47 --> 00:03:52
Leibniz's notation was adopted
on the Continent and Newton
45
00:03:52 --> 00:03:58
dominated in Britain and, as a
result, the British fell behind
46
00:03:58 --> 00:04:01
by one or two hundred years in
the development of calculus.
47
00:04:01 --> 00:04:03
It was really a serious matter.
48
00:04:03 --> 00:04:06
So it's really well worth
your while to get used
49
00:04:06 --> 00:04:08
to this idea of ratios.
50
00:04:08 --> 00:04:12
And it comes up all over the
place, both in this class and
51
00:04:12 --> 00:04:14
also in multivariable calculus.
52
00:04:14 --> 00:04:17
It's used in many contexts.
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So first of all, just to
go a little bit easy.
54
00:04:20 --> 00:04:25
We'll illustrate it by its use
in linear approximations,
55
00:04:25 --> 00:04:36
which we've already done.
56
00:04:36 --> 00:04:39
The picture here, which we've
drawn a number of times, is
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00:04:39 --> 00:04:41
that you have some function.
58
00:04:41 --> 00:04:44
And here's a value
of the function.
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And it's coming up like that.
60
00:04:47 --> 00:04:48
So here's our function.
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00:04:48 --> 00:04:53
And we go forward a little
increment to a place which
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00:04:53 --> 00:04:56
is dx further along.
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00:04:56 --> 00:05:03
The idea of this notation is
that dx is going to replace
64
00:05:03 --> 00:05:07
the symbol delta x, which
is the change in x.
65
00:05:07 --> 00:05:11
And we won't think too hard
about - well, this is a small
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quantity, this is a small
quantity, we're not going to
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think too hard about
what that means.
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00:05:16 --> 00:05:20
Now, similarly, if you see how
much we've gone up - well,
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00:05:20 --> 00:05:26
this is kind of low, so
it's a small bit here.
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So this distance here
is, previously we
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called it delta y.
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00:05:36 --> 00:05:41
But now we're just
going to call it dy.
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So dy replaces delta y.
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00:05:51 --> 00:05:57
So this is the change in
level of the function.
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00:05:57 --> 00:05:59
And we'll represent it
symbolically this way.
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Very frequently, this just
saves a little bit of notation.
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00:06:04 --> 00:06:07
For the purposes of this, we'll
be doing the same things we did
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with delta x and delta y,
but this is the way that
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Leibniz thought of it.
80
00:06:12 --> 00:06:14
And he would just have
drawn it with this.
81
00:06:14 --> 00:06:24
So this distance here is dx
and this distance here is dy.
82
00:06:24 --> 00:06:32
So for an example of linear
approximation, we'll say what's
83
00:06:32 --> 00:06:39
64.1, say, to the 1/3 power,
approximately equal to?
84
00:06:39 --> 00:06:43
Now, I'm going to carry this
out in this new notation here.
85
00:06:43 --> 00:06:47
The function involved
is x ^ 1/3.
86
00:06:47 --> 00:06:50
And then it's a
differential, dy.
87
00:06:50 --> 00:06:53
Now, I want to use this
rule to get used to it.
88
00:06:53 --> 00:06:56
Because this is what we're
going to be doing all of today
89
00:06:56 --> 00:07:00
is, we're differentiatating, or
taking the differential of y.
90
00:07:00 --> 00:07:02
So that is going to be
just the derivative.
91
00:07:02 --> 00:07:11
That's 1/3 x ^ - 2/3 dx.
92
00:07:11 --> 00:07:18
And now I'm just going to fill
in exactly what this is.
93
00:07:18 --> 00:07:25
At x = 64, which is the natural
place close by where it's easy
94
00:07:25 --> 00:07:36
to do the evaluations, we have
y = 64 ^ 1/3, which is just 4.
95
00:07:36 --> 00:07:39
And how about dy?
96
00:07:39 --> 00:07:42
Well, so this is a little
bit more complicated.
97
00:07:42 --> 00:07:43
Put it over here.
98
00:07:43 --> 00:07:55
So dy = 1/3 ( 64) ^ - 2/3 dx.
99
00:07:55 --> 00:08:16
And that is (1/3 ) 1/16
dx, which is 1/48 dx.
100
00:08:16 --> 00:08:22
And now I'm going to work out
what 64 to the, whatever it is
101
00:08:22 --> 00:08:26
here, this strange fraction.
102
00:08:26 --> 00:08:30
I just want to be very
careful to explain to
103
00:08:30 --> 00:08:33
you one more thing.
104
00:08:33 --> 00:08:39
Which is that we're using x =
64, and so we're thinking of
105
00:08:39 --> 00:08:45
x + dx is going to be 64.1.
106
00:08:45 --> 00:08:53
So that means that dx
is going to be 1/10.
107
00:08:53 --> 00:08:59
So that's the increment
that we're interested in.
108
00:08:59 --> 00:09:03
And now I can carry out
the approximation.
109
00:09:03 --> 00:09:11
The approximation says that
64.1 ^ 1/3 is, well, it's
110
00:09:11 --> 00:09:14
approximately what I'm
going to call y + dy.
111
00:09:14 --> 00:09:18
Because really, the dy that I'm
determining here is determined
112
00:09:18 --> 00:09:26
by this linear relation.
dy = 1/48 dx.
113
00:09:26 --> 00:09:29
And so this is only
approximately true.
114
00:09:29 --> 00:09:37
Because what's really true
is that this = y + delta y.
115
00:09:37 --> 00:09:39
In our previous notation.
116
00:09:39 --> 00:09:41
So this is in disguise.
117
00:09:41 --> 00:09:42
What this is equal to.
118
00:09:42 --> 00:09:45
And that's the only
approximately equal to what
119
00:09:45 --> 00:09:47
the linear approximation
would give you.
120
00:09:47 --> 00:09:51
So, really, even though I wrote
dy is this increment here, what
121
00:09:51 --> 00:09:56
it really is if dx is exactly
that, is it's the amount it
122
00:09:56 --> 00:10:00
would go up if you went
straight up the tangent line.
123
00:10:00 --> 00:10:02
So I'm not going to do
that because that's
124
00:10:02 --> 00:10:03
not what people write.
125
00:10:03 --> 00:10:06
And that's not even
what they think.
126
00:10:06 --> 00:10:08
They're really thinking of
both dx and dy as being
127
00:10:08 --> 00:10:10
infinitesimally small.
128
00:10:10 --> 00:10:15
And here we're going to the
finite level and doing it.
129
00:10:15 --> 00:10:21
So this is just something you
have to live with, is a little
130
00:10:21 --> 00:10:28
ambiguity in this notation.
131
00:10:28 --> 00:10:29
This is the approximation.
132
00:10:29 --> 00:10:33
And now I can just calculate
these numbers here. y
133
00:10:33 --> 00:10:36
at this value is 4.
134
00:10:36 --> 00:10:43
And dy, as I said, is 1/48 dx.
135
00:10:43 --> 00:10:50
And that turns out to be 4 +
1/480, because dx is 1/10.
136
00:10:50 --> 00:10:54
So that's approximately 4.002.
137
00:10:54 --> 00:11:04
And that's our approximation.
138
00:11:04 --> 00:11:20
Now, let's just compare it
to our previous notation.
139
00:11:20 --> 00:11:23
This will serve as a review
of, if you like, of
140
00:11:23 --> 00:11:35
linear approximation.
141
00:11:35 --> 00:11:39
But what I want to emphasize
is that these things are
142
00:11:39 --> 00:11:43
supposed to be the same.
143
00:11:43 --> 00:11:45
Just that it's really
the same thing.
144
00:11:45 --> 00:11:52
It's just a different
notation for the same thing.
145
00:11:52 --> 00:11:56
I remind you the basic formula
for linear approximation is
146
00:11:56 --> 00:12:05
that f ( x ) is approximately
f ( a) + f' ( a )( x - a).
147
00:12:05 --> 00:12:07
And we're applying it in
the situation that a =
148
00:12:07 --> 00:12:17
64 and f(x) = x ^ 1/3.
149
00:12:17 --> 00:12:27
And so f ( a ), which is
f ( 64 ) is of course 4.
150
00:12:27 --> 00:12:38
And f' ( a ), which is
a, (1/3)a ^ - 2/3,
151
00:12:38 --> 00:12:43
is in our case 1/16.
152
00:12:43 --> 00:12:49
No, 1/48.
153
00:12:49 --> 00:12:52
OK, that's the same
calculation as before.
154
00:12:52 --> 00:12:59
And then our relationship
becomes x ^ 1/3 is
155
00:12:59 --> 00:13:12
approximately equal to 4 +
1/48 ( x - a ), which is 64.
156
00:13:12 --> 00:13:14
So look, every single number
that I've written over here
157
00:13:14 --> 00:13:20
has a corresponding number
for this other method.
158
00:13:20 --> 00:13:26
And now if I plug in the value
we happen to want, which is the
159
00:13:26 --> 00:13:33
64.1, this would be 4 + 1/48 (
1/10 ), which is just the
160
00:13:33 --> 00:13:38
same thing we had before.
161
00:13:38 --> 00:13:45
So again, same answer.
162
00:13:45 --> 00:13:55
Same method, new notation.
163
00:13:55 --> 00:14:02
Well, now I get to use this
notation in a novel way.
164
00:14:02 --> 00:14:04
So again, here's the notation.
165
00:14:04 --> 00:14:16
This notation of differential.
166
00:14:16 --> 00:14:21
The way I'm going to use it is
in discussing something called
167
00:14:21 --> 00:14:32
antiderivative Again, this
is a new notation now.
168
00:14:32 --> 00:14:33
But it's also a new idea.
169
00:14:33 --> 00:14:37
It's one that we
haven't discussed yet.
170
00:14:37 --> 00:14:42
Namely, the notation that
I want to describe here
171
00:14:42 --> 00:14:48
is what's called the
integral of g ( x ) dx.
172
00:14:48 --> 00:14:51
And I'll denote that by a
function capital G (x).
173
00:14:51 --> 00:14:54
So it's, you start with a
function g ( x ) and you
174
00:14:54 --> 00:14:58
produce a function capital
G ( x ), which is called
175
00:14:58 --> 00:15:12
the antiderivative of G.
176
00:15:12 --> 00:15:15
Notice there's a differential
sitting in here.
177
00:15:15 --> 00:15:31
This symbol, this guy here,
is called an integral sign.
178
00:15:31 --> 00:15:34
Or an integral, or this whole
thing is called an integral.
179
00:15:34 --> 00:15:40
And another name for the
antiderivative of g is the
180
00:15:40 --> 00:15:50
indefinite integral of g.
181
00:15:50 --> 00:15:58
And I'll explain to you why
it's indefinite in just,
182
00:15:58 --> 00:16:04
very shortly here.
183
00:16:04 --> 00:16:13
Well, so let's carry
out some examples.
184
00:16:13 --> 00:16:17
Basically what I'd like to do
is as many examples along the
185
00:16:17 --> 00:16:20
lines of all the derivatives
that we derived at the
186
00:16:20 --> 00:16:21
beginning of the course.
187
00:16:21 --> 00:16:23
In other words, in principle
you want to be able
188
00:16:23 --> 00:16:26
to integrate as many
things as possible.
189
00:16:26 --> 00:16:34
We're going to start out with
the integral of sine x dx.
190
00:16:34 --> 00:16:40
That's a function whose
derivative is sine x.
191
00:16:40 --> 00:16:44
So what function would that be?
192
00:16:44 --> 00:16:48
Cosine x, minus, right.
193
00:16:48 --> 00:16:52
It's - cos x.
194
00:16:52 --> 00:16:56
So - cos x differentiated
gives you sine x.
195
00:16:56 --> 00:17:00
So that is an
antiderivative of sine.
196
00:17:00 --> 00:17:02
And it satisfies this property.
197
00:17:02 --> 00:17:10
So this function, capital G ( x
) = - cos x, has the property
198
00:17:10 --> 00:17:15
that its derivative is sine x.
199
00:17:15 --> 00:17:20
On the other hand, if
you differentiate a
200
00:17:20 --> 00:17:22
constant, you get 0.
201
00:17:22 --> 00:17:25
So this answer is what's
called indefinite.
202
00:17:25 --> 00:17:28
Because you can also
add any constant here.
203
00:17:28 --> 00:17:33
And the same thing
will be true.
204
00:17:33 --> 00:17:38
So, c is constant.
205
00:17:38 --> 00:17:41
And as I said, the integral
is called indefinite.
206
00:17:41 --> 00:17:45
So that's an explanation
for this modifier in
207
00:17:45 --> 00:17:46
front of the integral.
208
00:17:46 --> 00:17:49
It's indefinite because we
actually didn't specify
209
00:17:49 --> 00:17:50
a single function.
210
00:17:50 --> 00:17:52
We don't get a single answer.
211
00:17:52 --> 00:17:54
Whenever you take the
antiderivative of something
212
00:17:54 --> 00:18:08
it's ambiguous up
to a constant.
213
00:18:08 --> 00:18:12
Next, let's do some other
standard functions
214
00:18:12 --> 00:18:13
from our repertoire.
215
00:18:13 --> 00:18:17
We have an integral
of (x ^ a)dx.
216
00:18:17 --> 00:18:20
Some power, the
integral of a power.
217
00:18:20 --> 00:18:24
And if you think about it, what
you should be differentiating
218
00:18:24 --> 00:18:27
is one power larger than that.
219
00:18:27 --> 00:18:33
But then you have to divide by
1 / a + 1, in order that the
220
00:18:33 --> 00:18:36
differentiatiation be correct.
221
00:18:36 --> 00:18:45
So this just is the fact that d
/ dx of x ^ a + 1, or maybe I
222
00:18:45 --> 00:18:46
should even say it this way.
223
00:18:46 --> 00:18:49
Maybe I'll do it in
differential notation. d ( x ^
224
00:18:49 --> 00:18:57
a + 1) = (a + 1) (x ^ a) dx.
225
00:18:57 --> 00:19:03
So if I divide that through
by a + 1, then I get
226
00:19:03 --> 00:19:06
the relation above.
227
00:19:06 --> 00:19:12
And because this is ambiguous
up to a constant, it could be
228
00:19:12 --> 00:19:20
any additional constant
added to that function.
229
00:19:20 --> 00:19:26
Now, the identity that I
wrote down below is correct.
230
00:19:26 --> 00:19:35
But this one is not always
correct What's the exception?
231
00:19:35 --> 00:19:38
Yeah. a equals
232
00:19:38 --> 00:19:38
STUDENT: 0.
233
00:19:38 --> 00:19:42
PROFESSOR: Negative 1.
234
00:19:42 --> 00:19:47
So this one is OK for all a.
235
00:19:47 --> 00:19:49
But this one fails
because we've divided
236
00:19:49 --> 00:19:52
by 0 when a = - 1.
237
00:19:52 --> 00:20:04
So this is only true when
a is not equal to - 1.
238
00:20:04 --> 00:20:07
And in fact, of course,
what's happening when a = 0,
239
00:20:07 --> 00:20:11
you're getting 0 when you
differentiate the constant.
240
00:20:11 --> 00:20:15
So there's a third case
that we have to carry out.
241
00:20:15 --> 00:20:25
Which is the exceptional case,
namely the integral of dx/x.
242
00:20:25 --> 00:20:32
And this time, if we just think
back to what are - so what
243
00:20:32 --> 00:20:34
we're doing is thinking
backwards here, which a
244
00:20:34 --> 00:20:38
very important thing to do
in math at all stages.
245
00:20:38 --> 00:20:41
We got all of our formulas, now
we're reading them backwards.
246
00:20:41 --> 00:20:49
And so this one, you
may remember, is ln x.
247
00:20:49 --> 00:20:53
The reason why I want to do
this carefully and slowly now,
248
00:20:53 --> 00:20:56
is right now I also want
to write the more standard
249
00:20:56 --> 00:20:58
form which is presented.
250
00:20:58 --> 00:21:01
So first of all, first we
have to add a constant.
251
00:21:01 --> 00:21:04
And please don't put
the parentheses here.
252
00:21:04 --> 00:21:10
The parentheses go there.
253
00:21:10 --> 00:21:14
But there's another formula
hiding in the woodwork
254
00:21:14 --> 00:21:16
here behind this one.
255
00:21:16 --> 00:21:19
Which is that you can also
get the correct formula
256
00:21:19 --> 00:21:20
when x is negative.
257
00:21:20 --> 00:21:27
And that turns out to
be this one here.
258
00:21:27 --> 00:21:31
So I'm treating the case,
x positive, as being
259
00:21:31 --> 00:21:34
something that you know.
260
00:21:34 --> 00:21:43
But let's check the
case, x negative.
261
00:21:43 --> 00:21:46
In order to check the case
x negative, I have to
262
00:21:46 --> 00:21:51
differentiate the logarithm
of the absolute value
263
00:21:51 --> 00:21:55
of x in that case.
264
00:21:55 --> 00:21:58
And that's the same thing,
again, for x negative as
265
00:21:58 --> 00:22:02
the derivative of the
lograrithm of negative x.
266
00:22:02 --> 00:22:08
That's the formula,
when x is negative.
267
00:22:08 --> 00:22:12
And if you carry that out, what
you get, maybe I'll put this
268
00:22:12 --> 00:22:20
over here, is, well,
it's the chain rule.
269
00:22:20 --> 00:22:23
It's (1 / -x) d/dx(-x).
270
00:22:23 --> 00:22:27
271
00:22:27 --> 00:22:30
So see that there are
two minus signs.
272
00:22:30 --> 00:22:32
There's a - x in the
denominator and then there's
273
00:22:32 --> 00:22:35
the derivative of -
x in the numerator.
274
00:22:35 --> 00:22:38
That's just - 1.
275
00:22:38 --> 00:22:39
This part is - 1.
276
00:22:39 --> 00:22:43
So this negative 1 over
negative x, which is 1 / x.
277
00:22:43 --> 00:22:53
So the negative signs cancel.
278
00:22:53 --> 00:22:59
If you just keep track of this
in terms of ln negative x and
279
00:22:59 --> 00:23:05
its graph, that's a function
that looks like this.
280
00:23:05 --> 00:23:08
For x negative.
281
00:23:08 --> 00:23:14
And its derivative
is 1 / x, I claim.
282
00:23:14 --> 00:23:18
And if you just look at it a
little bit carefully, you see
283
00:23:18 --> 00:23:23
that the slope is
always negative.
284
00:23:23 --> 00:23:23
Right?
285
00:23:23 --> 00:23:26
So here the slope is negative.
286
00:23:26 --> 00:23:30
So it's going to be
below the axis.
287
00:23:30 --> 00:23:32
And, in fact, it's getting
steeper and steeper
288
00:23:32 --> 00:23:34
negative as we go down.
289
00:23:34 --> 00:23:37
And it's getting less and less
negative as we go horizontally.
290
00:23:37 --> 00:23:41
So it's going like this, which
is indeed the graph of this
291
00:23:41 --> 00:23:43
function, for x negative.
292
00:23:43 --> 00:23:53
Again, x negative.
293
00:23:53 --> 00:23:56
So that's one other
standard formula.
294
00:23:56 --> 00:24:00
And very quickly, very
often, we won't put the
295
00:24:00 --> 00:24:01
absolute value signs.
296
00:24:01 --> 00:24:03
We'll only consider the
case x positive here.
297
00:24:03 --> 00:24:07
But I just want you to have the
tools to do it in case we want
298
00:24:07 --> 00:24:14
to use, we want to handle,
both positive and negative x.
299
00:24:14 --> 00:24:28
Now, let's do two
more examples.
300
00:24:28 --> 00:24:35
The integral of sec^2 x dx.
301
00:24:35 --> 00:24:38
These are supposed to get you
to remember all of your
302
00:24:38 --> 00:24:41
differentiatation formulas,
the standard ones.
303
00:24:41 --> 00:24:45
So this one, integral
of sec^2 dx is what?
304
00:24:45 --> 00:24:48
Tan x.
305
00:24:48 --> 00:24:50
And here we have + c, alright?
306
00:24:50 --> 00:24:54
And then the last one of,
a couple of, this type
307
00:24:54 --> 00:24:56
would be, let's see.
308
00:24:56 --> 00:25:04
I should do at least this one
here, square root of 1 - x ^2.
309
00:25:04 --> 00:25:05
This is another notation,
by the way, which is
310
00:25:05 --> 00:25:07
perfectly acceptable.
311
00:25:07 --> 00:25:10
Notice I've put the dx in the
numerator and the function
312
00:25:10 --> 00:25:13
in the denominator here.
313
00:25:13 --> 00:25:18
So this one turns
out to be sin-1 x.
314
00:25:18 --> 00:25:23
And, finally, let's see.
315
00:25:23 --> 00:25:28
About the integral
of dx / 1 + x ^2.
316
00:25:28 --> 00:25:41
That one is tan -1 x.
317
00:25:41 --> 00:25:43
For a little while, because
you're reading these things
318
00:25:43 --> 00:25:45
backwards and forwards,
you'll find this happens
319
00:25:45 --> 00:25:46
to you on exams.
320
00:25:46 --> 00:25:49
It gets slightly worse
for a little while.
321
00:25:49 --> 00:25:53
You will antidifferentiate when
you meant to differentiate.
322
00:25:53 --> 00:25:54
And you'll differentiate
when you're meant to
323
00:25:54 --> 00:25:57
antidifferentiate.
324
00:25:57 --> 00:26:00
Don't get too
frustrated by that.
325
00:26:00 --> 00:26:05
But eventually, you'll
get them squared away.
326
00:26:05 --> 00:26:11
And it actually helps to do
a lot of practice with
327
00:26:11 --> 00:26:15
antidifferentiations, or
integrations, as they're
328
00:26:15 --> 00:26:17
sometimes called.
329
00:26:17 --> 00:26:20
Because that will solidify
your remembering all of the
330
00:26:20 --> 00:26:25
differentiation formulas.
331
00:26:25 --> 00:26:31
So, last bit of information
that I want to emphasize before
332
00:26:31 --> 00:26:45
we go on some more complicated
examples is this It's obvious
333
00:26:45 --> 00:26:49
because the derivative
of a constant is 0.
334
00:26:49 --> 00:26:55
That the antiderivative is
ambiguous up to a constant.
335
00:26:55 --> 00:26:59
But it's very important to
realize that this is the only
336
00:26:59 --> 00:27:01
ambiguity that there is.
337
00:27:01 --> 00:27:04
So the last thing that I want
to tell you about is
338
00:27:04 --> 00:27:24
uniqueness of antiderivatives
up to a constant.
339
00:27:24 --> 00:27:30
The theorem is the following.
340
00:27:30 --> 00:27:41
The theorem is if F'
= F', then F = G.
341
00:27:41 --> 00:27:44
So F ( x ) = G ( x) + c.
342
00:27:44 --> 00:27:48
343
00:27:48 --> 00:27:54
But that means, not only that
these are antiderivatives, all
344
00:27:54 --> 00:27:56
these things with these +
c's are antiderivatives.
345
00:27:56 --> 00:28:02
But these are the only ones.
346
00:28:02 --> 00:28:03
Which is very reassuring.
347
00:28:03 --> 00:28:05
And that's a kind of
uniqueness, although its
348
00:28:05 --> 00:28:09
uniqueness up to a constant,
it's acceptable to us.
349
00:28:09 --> 00:28:12
Now, the proof of
this is very quick.
350
00:28:12 --> 00:28:18
But this is a fundamental fact.
351
00:28:18 --> 00:28:19
The proof is the following.
352
00:28:19 --> 00:28:29
If F' = G', then if you take
the difference between the two
353
00:28:29 --> 00:28:40
functions, its derivative,
which of course is F' - G' = 0.
354
00:28:40 --> 00:28:55
Hence, F( x) - G
(x) is a constant.
355
00:28:55 --> 00:28:58
Now, this is a key fact.
356
00:28:58 --> 00:28:59
Very important fact.
357
00:28:59 --> 00:29:03
We deduced it last time from
the mean value theorem.
358
00:29:03 --> 00:29:05
It's not a small matter.
359
00:29:05 --> 00:29:06
It's a very, very
important thing.
360
00:29:06 --> 00:29:08
It's the basis for calculus.
361
00:29:08 --> 00:29:11
It's the reason why
calculus make sense.
362
00:29:11 --> 00:29:14
If we didn't have the fact that
the derivative is 0 implied
363
00:29:14 --> 00:29:18
that the function was
constant, we would be done.
364
00:29:18 --> 00:29:23
We would have, calculus would
be just useless for us.
365
00:29:23 --> 00:29:24
The point is, the rate of
change is supposed to
366
00:29:24 --> 00:29:29
determine the function up
to this starting value.
367
00:29:29 --> 00:29:32
So this conclusion
is very important.
368
00:29:32 --> 00:29:35
And we already checked it
last time, this conclusion.
369
00:29:35 --> 00:29:39
And now just by algebra, I
can rearrange this to say
370
00:29:39 --> 00:30:03
that f ( x) = G ( x) + c.
371
00:30:03 --> 00:30:07
Now, maybe I should leave
differentials up here.
372
00:30:07 --> 00:30:20
Because I want to illustrate.
373
00:30:20 --> 00:30:22
So let's go on to some
trickier, slightly
374
00:30:22 --> 00:30:29
trickier, integrals.
375
00:30:29 --> 00:30:35
Here's an example.
376
00:30:35 --> 00:30:51
The integral of, say, x^3
( x ^ 4 + 2) ^ 5 dx.
377
00:30:51 --> 00:30:54
This is a function which you
actually do know how to
378
00:30:54 --> 00:30:59
integrate, because we already
have a formula for all powers.
379
00:30:59 --> 00:31:03
Namely, the integral of
x ^ a is equal to this.
380
00:31:03 --> 00:31:06
And even if it were a negative
power, we could do it.
381
00:31:06 --> 00:31:08
So it's OK.
382
00:31:08 --> 00:31:11
On the other hand, to
expand the 5th power
383
00:31:11 --> 00:31:14
here is quite a mess.
384
00:31:14 --> 00:31:18
And this is just a
very, very bad idea.
385
00:31:18 --> 00:31:21
There's another trick for doing
this that evaluates this
386
00:31:21 --> 00:31:23
much more efficiently.
387
00:31:23 --> 00:31:27
And it's the only device
that we're going to learn
388
00:31:27 --> 00:31:31
now for integrating.
389
00:31:31 --> 00:31:36
Integration actually is much
harder than differentiation.
390
00:31:36 --> 00:31:37
Symbolically.
391
00:31:37 --> 00:31:39
It's quite difficult.
392
00:31:39 --> 00:31:42
And occasionally impossible.
393
00:31:42 --> 00:31:45
And so we have to go
about it gently.
394
00:31:45 --> 00:31:47
But for the purposes of this
unit, we're only going
395
00:31:47 --> 00:31:50
to use one method.
396
00:31:50 --> 00:31:50
Which is very good.
397
00:31:50 --> 00:31:53
That means whenever you see an
integral, either you'll be able
398
00:31:53 --> 00:31:56
to divine immediately what the
answer is, or you'll
399
00:31:56 --> 00:31:57
use this method.
400
00:31:57 --> 00:31:59
So this is it.
401
00:31:59 --> 00:32:09
The trick is called the
method of substitution.
402
00:32:09 --> 00:32:17
And it is tailor-made for
notion of differentials.
403
00:32:17 --> 00:32:36
So tailor-made. for
differential notation.
404
00:32:36 --> 00:32:37
The idea is the following.
405
00:32:37 --> 00:32:40
I'm going to to define
a new function.
406
00:32:40 --> 00:32:43
And it's the messiest
function that I see here.
407
00:32:43 --> 00:32:50
It's u = x ^ 4 + 2.
408
00:32:50 --> 00:32:56
And then, I'm going to take
its differential and what I
409
00:32:56 --> 00:32:59
discover, if I look at its
formula, is and the rule for
410
00:32:59 --> 00:33:02
differentials, which
is right here.
411
00:33:02 --> 00:33:06
Its formula is what?
412
00:33:06 --> 00:33:10
4x^3 dx.
413
00:33:10 --> 00:33:14
Now, lo and behold with these
two quantities, I can
414
00:33:14 --> 00:33:17
substitute, I can plug
in to this integral.
415
00:33:17 --> 00:33:21
And I will simplify
it considerably.
416
00:33:21 --> 00:33:23
So how does that happen?
417
00:33:23 --> 00:33:35
Well, this integral is the same
thing as, well, really I should
418
00:33:35 --> 00:33:36
combine it the other way.
419
00:33:36 --> 00:33:41
So let me move this over.
420
00:33:41 --> 00:33:43
So there are two pieces here.
421
00:33:43 --> 00:33:46
And this one is u ^ 5.
422
00:33:46 --> 00:33:54
And this one is 1/4 du.
423
00:33:54 --> 00:34:01
Now, that makes it the
integral of (u ^ 5 du) / 4.
424
00:34:01 --> 00:34:04
And that's relatively
easy to integrate.
425
00:34:04 --> 00:34:05
That is just a power.
426
00:34:05 --> 00:34:06
So let's see.
427
00:34:06 --> 00:34:11
It's just 1/20 u to
the - not 1/20.
428
00:34:11 --> 00:34:15
The antiderivative
of u ^ 5 is u ^ 6.
429
00:34:15 --> 00:34:25
With the 1/6, so it's
1/24 u ^ 6 + c.
430
00:34:25 --> 00:34:29
Now, that's not the
answer to the question.
431
00:34:29 --> 00:34:32
It's almost the answer
to the question.
432
00:34:32 --> 00:34:33
Why isn't it the answer?
433
00:34:33 --> 00:34:35
It isn't the answer
because now the answer's
434
00:34:35 --> 00:34:37
expressed in terms of u.
435
00:34:37 --> 00:34:41
Whereas the problem was posed
in terms of this variable x.
436
00:34:41 --> 00:34:45
So we must change back
to our variable here.
437
00:34:45 --> 00:34:47
And we do that just
by writing it in.
438
00:34:47 --> 00:34:56
So it's 1/24 (x ^
4 + 2) ^ 6 + c.
439
00:34:56 --> 00:35:02
And this is the end
of the problem.
440
00:35:02 --> 00:35:02
Yeah, question.
441
00:35:02 --> 00:35:16
STUDENT: [INAUDIBLE]
442
00:35:16 --> 00:35:19
PROFESSOR: The question is,
can you see it directly?
443
00:35:19 --> 00:35:20
Yeah.
444
00:35:20 --> 00:35:23
And we're going to talk about
that in just one second.
445
00:35:23 --> 00:35:30
OK.
446
00:35:30 --> 00:35:35
Now, I'm going to do
one more example and
447
00:35:35 --> 00:35:44
illustrate this method.
448
00:35:44 --> 00:35:45
Here's another example.
449
00:35:45 --> 00:35:51
The integral of x dx /
squre root of 1 + x ^2.
450
00:35:51 --> 00:35:56
Now, here's another example.
451
00:35:56 --> 00:36:03
Now, the method of substitution
leads us to the idea u = 1
452
00:36:03 --> 00:36:11
+ x ^2. du = 2x dx, etc.
453
00:36:11 --> 00:36:14
It takes about as long as
this other problem did.
454
00:36:14 --> 00:36:15
To figure out what's going on.
455
00:36:15 --> 00:36:17
It's a very similar
sort of thing.
456
00:36:17 --> 00:36:20
You end up integrating
u ^ - 1/2.
457
00:36:20 --> 00:36:28
It needs to the integral
of u ^ - 1/2 du.
458
00:36:28 --> 00:36:31
Is everybody seeing
where this..?
459
00:36:31 --> 00:36:37
However, there is a
slightly better method.
460
00:36:37 --> 00:36:46
So recommended method.
461
00:36:46 --> 00:36:59
And I call this method
advanced guessing.
462
00:36:59 --> 00:37:01
What advanced guessing means
is that you've done enough
463
00:37:01 --> 00:37:04
of these problems that you
can see two steps ahead.
464
00:37:04 --> 00:37:08
And you know what's
going to happen.
465
00:37:08 --> 00:37:11
So the advanced guessing leads
you to believe that here you
466
00:37:11 --> 00:37:14
had a power - 1/2, here you
have the differential
467
00:37:14 --> 00:37:14
of the thing.
468
00:37:14 --> 00:37:16
So it's going to
work out somehow.
469
00:37:16 --> 00:37:19
And the advanced guessing
allows you to guess that the
470
00:37:19 --> 00:37:26
answer should be something
like this. (1 + x ^2) ^ 1/2.
471
00:37:26 --> 00:37:27
So this is your advanced guess.
472
00:37:27 --> 00:37:31
And now you just differentiate
it, and see whether it works.
473
00:37:31 --> 00:37:32
Well, here it is.
474
00:37:32 --> 00:37:38
It's 1/2 (1 + x ^2) ^ -
1/2( 2x), that's the
475
00:37:38 --> 00:37:39
chain rule here.
476
00:37:39 --> 00:37:44
Which, sure enough, gives you
x / square root of 1 + x ^2.
477
00:37:44 --> 00:37:45
So we're done.
478
00:37:45 --> 00:37:56
And so the answer is square
root of (1 + x^2) + c.
479
00:37:56 --> 00:38:02
Let me illustrate this further
with another example.
480
00:38:02 --> 00:38:06
I strongly recommend that
you do this, but you
481
00:38:06 --> 00:38:09
have to get used to it.
482
00:38:09 --> 00:38:18
So here's another
example. e ^ 6x dx.
483
00:38:18 --> 00:38:26
My advanced guess is e ^ 6x.
484
00:38:26 --> 00:38:29
And if I check, when
I differentiate
485
00:38:29 --> 00:38:33
it, I get 6e ^ 6x.
486
00:38:33 --> 00:38:35
That's the derivative.
487
00:38:35 --> 00:38:38
And so I know that the
answer, so now I know
488
00:38:38 --> 00:38:39
what the answer is.
489
00:38:39 --> 00:38:46
It's 1/6 e ^ 6x + c.
490
00:38:46 --> 00:38:57
Now, OK, you could, it's also
OK, but slow, to use a
491
00:38:57 --> 00:39:02
substitution, to use u = 6x.
492
00:39:02 --> 00:39:07
Then you're going to
get du = 6dx ...
493
00:39:07 --> 00:39:23
It's going to work, it's
just a waste of time.
494
00:39:23 --> 00:39:26
Well, I'm going to give you
a couple more examples.
495
00:39:26 --> 00:39:41
So how about this one.
x ( e^ - x^2) dx.
496
00:39:41 --> 00:39:45
What's the guess?
497
00:39:45 --> 00:39:51
Anybody have a guess?
498
00:39:51 --> 00:39:52
Well, you could also correct.
499
00:39:52 --> 00:39:54
So I don't want you to
bother - yeah, go ahead.
500
00:39:54 --> 00:39:57
STUDENT: [INAUDIBLE]
501
00:39:57 --> 00:39:59
PROFESSOR: Yeah, so you're
already one step ahead of me.
502
00:39:59 --> 00:40:02
Because this is too easy.
503
00:40:02 --> 00:40:04
When they get more complicated,
you just want to make
504
00:40:04 --> 00:40:05
this guess here.
505
00:40:05 --> 00:40:09
So various people have said
1/2, and they understand that
506
00:40:09 --> 00:40:10
there's 1/2 going here.
507
00:40:10 --> 00:40:13
But let me just show
you what happens, OK?
508
00:40:13 --> 00:40:19
If you make this guess and you
differentiate it, what you get
509
00:40:19 --> 00:40:25
here is e^ - x ^2 times the
derivative of negative 2x, so
510
00:40:25 --> 00:40:30
that's - 2x. - x^2,
so it's - 2x.
511
00:40:30 --> 00:40:37
So now you see that you're off
by a factor of not 2, but - 2.
512
00:40:37 --> 00:40:39
So a number of you
were saying that.
513
00:40:39 --> 00:40:46
So the answer is -
1/2 e^ - x ^2 + c.
514
00:40:46 --> 00:40:50
And I can guarantee you, having
watched this on various
515
00:40:50 --> 00:40:55
problems, that people who
don't write this out make
516
00:40:55 --> 00:40:57
arithmetic mistakes.
517
00:40:57 --> 00:41:00
In other words, there is a
limit to how much people
518
00:41:00 --> 00:41:02
can think ahead and
guess correctly.
519
00:41:02 --> 00:41:05
Another way of doing it, by the
way, is simply to write this
520
00:41:05 --> 00:41:08
thing in and then fix the
coefficient by doing the
521
00:41:08 --> 00:41:10
differentiation here.
522
00:41:10 --> 00:41:14
That's perfectly OK as well.
523
00:41:14 --> 00:41:18
Alright, one more example.
524
00:41:18 --> 00:41:30
We're going to integrate
sin x cos x dx.
525
00:41:30 --> 00:41:33
So what's a good
guess for this one?
526
00:41:33 --> 00:41:36
STUDENT: [INAUDIBLE]
527
00:41:36 --> 00:41:38
PROFESSOR: Someone
suggesting sine ^2 x.
528
00:41:38 --> 00:41:41
So let's try that.
529
00:41:41 --> 00:41:45
Over 2 - well, we'll get the
coefficient in just a second.
530
00:41:45 --> 00:41:47
So sine ^2 x, if I
differentiate I get
531
00:41:47 --> 00:41:50
2 sine x cosine x.
532
00:41:50 --> 00:41:53
So that's off by a factor of 2.
533
00:41:53 --> 00:42:04
So the answer is 1/2 sine ^2 x.
534
00:42:04 --> 00:42:14
But now I want to point out
to you that there's another
535
00:42:14 --> 00:42:17
way of doing this problem.
536
00:42:17 --> 00:42:31
It's also true that if you
differentiate cosine ^2 x,
537
00:42:31 --> 00:42:38
you get 2 cos x ( - sine x).
538
00:42:38 --> 00:42:49
So another answer is that
the integral of sin x cos
539
00:42:49 --> 00:43:01
x dx = - 1/2 cos^2 x + c.
540
00:43:01 --> 00:43:03
So what is going on here?
541
00:43:03 --> 00:43:06
What's the problem with this?
542
00:43:06 --> 00:43:11
STUDENT: [INAUDIBLE]
543
00:43:11 --> 00:43:11
PROFESSOR: Pardon me?
544
00:43:11 --> 00:43:15
STUDENT: [INAUDIBLE]
545
00:43:15 --> 00:43:18
PROFESSOR: Integrals
aren't unique.
546
00:43:18 --> 00:43:21
That's part of the - but
somehow these two answers
547
00:43:21 --> 00:43:22
still have to be the same.
548
00:43:22 --> 00:43:32
STUDENT: [INAUDIBLE]
549
00:43:32 --> 00:43:36
PROFESSOR: OK.
550
00:43:36 --> 00:43:36
What do you think?
551
00:43:36 --> 00:43:38
STUDENT: If you add them
together, you just get c.
552
00:43:38 --> 00:43:40
PROFESSOR: If you add
them together you get c.
553
00:43:40 --> 00:43:44
Well, actually,
that's almost right.
554
00:43:44 --> 00:43:45
That's not what you
want to do, though.
555
00:43:45 --> 00:43:47
You don't want to add them.
556
00:43:47 --> 00:43:50
You want to subtract them.
557
00:43:50 --> 00:43:53
So let's see what happens
when you subtract them.
558
00:43:53 --> 00:43:56
I'm going to ignore the
c, for the time being.
559
00:43:56 --> 00:44:05
I get sin^2 x, 1/2 sin^2
x - (-1/2 cos^2 x).
560
00:44:05 --> 00:44:08
So the difference between
them, we hope to be 0.
561
00:44:08 --> 00:44:10
But actually of
course it's not 0.
562
00:44:10 --> 00:44:18
What it is, is it's 1/2 (sin^2
+ cos^2) which is 1/2.
563
00:44:18 --> 00:44:24
It's not 0, it's a constant.
564
00:44:24 --> 00:44:26
So what's really going on
here is that these two
565
00:44:26 --> 00:44:29
formulas are the same.
566
00:44:29 --> 00:44:31
But you have to understand
how to interpret them.
567
00:44:31 --> 00:44:34
The two constants, here's
a constant up here.
568
00:44:34 --> 00:44:37
There's a constant, c1
associated to this one.
569
00:44:37 --> 00:44:43
There's a different constant,
c2 associated to this one.
570
00:44:43 --> 00:44:46
And this family of functions
for all possible c1s and all
571
00:44:46 --> 00:44:49
possible c2s, is the same
family of functions.
572
00:44:49 --> 00:44:52
Now, what's the relationship
between c1 and c2?
573
00:44:52 --> 00:44:57
Well, if you do the
subtraction, c1 - c2 has
574
00:44:57 --> 00:44:59
to be equal to 1/2.
575
00:44:59 --> 00:45:06
They're both constants,
but they differ by 1/2.
576
00:45:06 --> 00:45:09
So this explains, when you're
dealing with families of
577
00:45:09 --> 00:45:10
things, they don't have
to look the same.
578
00:45:10 --> 00:45:13
And there are lots of trig
functions which look
579
00:45:13 --> 00:45:16
a little different.
580
00:45:16 --> 00:45:18
So there can be several
formulas that actually
581
00:45:18 --> 00:45:19
are the same.
582
00:45:19 --> 00:45:21
And it's hard to check that
they're actually the same.
583
00:45:21 --> 00:45:28
You need some trig
identities to do it.
584
00:45:28 --> 00:45:55
Let's do one more example here.
585
00:45:55 --> 00:46:06
Here's another one.
586
00:46:06 --> 00:46:14
Now, you may be thinking, and
a lot of people are, thinking
587
00:46:14 --> 00:46:22
ugh, it's got a ln in it.
588
00:46:22 --> 00:46:25
If you're experienced, you
actually can read off the
589
00:46:25 --> 00:46:26
answer just the way there were
several people who were
590
00:46:26 --> 00:46:29
shouting out the answers when
we were doing the rest
591
00:46:29 --> 00:46:31
of these problems.
592
00:46:31 --> 00:46:32
But, you do need to relax.
593
00:46:32 --> 00:46:36
Because in this case, now this
is definitely not true in
594
00:46:36 --> 00:46:37
general when we do integrals.
595
00:46:37 --> 00:46:39
But, for now, when we
do integrals, they'll
596
00:46:39 --> 00:46:40
all be manageable.
597
00:46:40 --> 00:46:42
And there's only one method.
598
00:46:42 --> 00:46:47
Which is substitution.
599
00:46:47 --> 00:46:50
And in the substitution
method, you want to go
600
00:46:50 --> 00:46:52
for the trickiest part.
601
00:46:52 --> 00:46:55
And substitute for that.
602
00:46:55 --> 00:46:59
So the substitution that I
proposed to you is that this
603
00:46:59 --> 00:47:02
should be, you should be ln x.
604
00:47:02 --> 00:47:06
And the advantage that that
has is that its differential
605
00:47:06 --> 00:47:08
is simpler then itself.
606
00:47:08 --> 00:47:15
So du = dx /x.
607
00:47:15 --> 00:47:17
Remember, we use
that in logarithmic
608
00:47:17 --> 00:47:21
differentiation, too.
609
00:47:21 --> 00:47:28
So now we can express this
using this substitution.
610
00:47:28 --> 00:47:32
And what we get is, the
integral of, so I'll divide
611
00:47:32 --> 00:47:33
the two parts here.
612
00:47:33 --> 00:47:36
It's 1 / ln x, and
then it's dx / x.
613
00:47:36 --> 00:47:43
And this part is 1 / u,
and this part is du.
614
00:47:43 --> 00:47:49
So it's the integral of du / u.
615
00:47:49 --> 00:47:58
And that is ln u + c.
616
00:47:58 --> 00:48:11
Which altogether, if I put back
in what u is, is ln (ln x) + c.
617
00:48:11 --> 00:48:14
And now we see some
uglier things.
618
00:48:14 --> 00:48:16
In fact, technically
speaking, we could take
619
00:48:16 --> 00:48:18
the absolute value here.
620
00:48:18 --> 00:48:28
And then this would be
absolute values there.
621
00:48:28 --> 00:48:33
So this is the type of example
where I really would recommend
622
00:48:33 --> 00:48:39
that you actually use the
substitution, at least for now.
623
00:48:39 --> 00:48:42
Alright, tomorrow we're
going to be doing
624
00:48:42 --> 00:48:43
differential equations.
625
00:48:43 --> 00:48:45
And we're going to
review for the test.
626
00:48:45 --> 00:48:47
I'm going to give you a handout
telling you just exactly what's
627
00:48:47 --> 00:48:48
going to be on the test.
628
00:48:48 --> 00:48:52
So, see you tomorrow.
629
00:48:52 --> 00:48:52