Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007

Jun 27, 2026 03:28 · 49:00 · English · Whisper Turbo · 2 דוברים
תעתיק זה פג היום. שדרוג לאחסון קבוע →
אינדקס הרצאה
מציג בלבד
0:00
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The following content is provided under a Creative Commons license.
0:03
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Your support will help MIT OpenCourseWare continue to offer high
0:07
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
quality educational resources for free.
0:09
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
To make a donation or to view additional materials from hundreds of MIT
0:13
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
courses, visit MIT OpenCourseWare at ocw .mit
0:18
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
.edu.
0:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're ready to begin
0:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the fifth lecture.
0:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm glad to be back.
0:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Thank you for entertaining my
0:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
colleague, Haynes Miller.
0:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So today we're going to continue where he started.
0:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
0:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what he talked about was the chain rule,
0:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is probably the most powerful technique for extending the kinds
0:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of functions that you can differentiate.
0:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now we're going to use the chain rule in some rather clever
0:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
algebraic ways today.
0:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the topic for today is what's known as
0:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
implicit differentiation.
1:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So implicit differentiation is a technique that allows
1:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you to differentiate a lot of functions you didn't even know how to find before.
1:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And it's a technique,
1:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's wait for a few people to sit down here.
1:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Physics,
1:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
huh?
1:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
1:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
More physics.
1:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We'll take a break.
1:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You can get those after class.
1:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're talking about implicit differentiation.
1:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm going to illustrate it by several examples.
1:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
1:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is one of the most important and basic formulas that
1:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we've already learned part way.
1:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
2:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of x to a power is a times
2:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the a -1.
2:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
2:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we've done so far,
2:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we've got so far is
2:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the exponents 0,
2:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus 1,
2:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus 2.
2:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Et cetera.
2:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You did the integer,
2:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
positive integer powers in the first lecture.
2:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then yesterday,
2:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Professor Miller told
2:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you about the negative powers.
2:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what we're going to do right now,
2:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
today,
2:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is we're going to consider the exponents
2:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which are rational numbers,
2:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
ratios of integers.
2:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So a is m over n.
2:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
m and n are integers.
2:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so that's our goal for right now.
2:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And we're going to use this method of implicit differentiation.
2:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In particular,
2:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's important to realize that this covers the case m equals 1.
3:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And those are the nth roots.
3:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So when we take the 1 over n power,
3:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to cover that right now.
3:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Along with many other examples.
3:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is our first example.
3:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So how do we get started?
3:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, we just write down a formula for the function.
3:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The function is y is equal to x to the power m over n.
3:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's what we're trying to deal with.
3:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now there's really only two steps.
3:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
take this equation to the nth power.
3:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So write it as y to the n is equal to x
3:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
to the m.
3:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's just the same equation,
3:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
rewritten.
3:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now,
3:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what we're going to do is we're going to differentiate.
3:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to apply d by
3:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
dx to the equation.
4:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now, why is it that we can apply it to the second equation,
4:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
not the first equation?
4:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe I should call these equations 1 and equation 2.
4:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The point is that we can apply it to equation 2.
4:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
4:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the reason is that we don't know how to differentiate
4:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the m over n.
4:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's something we just don't know yet.
4:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But we do know how to differentiate integer powers.
4:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Those are the things that we took care of before.
4:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now we're in shape to be able to do the differentiation.
4:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to write it out explicitly over here without
4:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
carrying it out just yet.
4:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's d by dx of y to the n is equal to d by dx
4:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of x to the m power.
4:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now you see
4:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this expression here requires us to do something we
4:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
couldn't do before yesterday.
4:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Namely,
4:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this y is a function of x.
5:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
rule here.
5:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the same as,
5:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is by the chain rule now,
5:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
d by dy of y to the n times
5:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
dy dx.
5:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then on the right -hand side,
5:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we can just carry it out.
5:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We know the formula.
5:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's m x to the m minus 1.
5:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
our scheme.
5:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now you'll see in a minute why we win with
5:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this.
5:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
5:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
there are two factors here.
5:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
One of them is unknown.
5:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In fact, it's what we're looking for.
5:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But the other one is going to be a known quantity.
5:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because we know how to differentiate y to the n with respect to y.
5:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same formula,
5:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
although the letter has been changed.
5:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'll write it underneath here.
5:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
n y to the n -1 dy dx is
5:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equal to m x to the m -1.
6:00
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
6:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now comes,
6:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if you like,
6:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the non -calculus part of the problem.
6:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Remember, the non -calculus part of the problem is always the messier part of the problem.
6:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we want to figure out this formula.
6:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This formula,
6:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the answer over here,
6:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which maybe I'll put in a box now.
6:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It has this expressed much more simply,
6:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
only in terms of x.
6:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And what we have to do now is to solve for dy dx using algebra,
6:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then solve all the way in terms of x.
6:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
6:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we solve for dy dx.
6:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I do that by dividing the factor on
6:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the left -hand side.
6:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I get here m x to the m -1 divided
6:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
by n y to the n -1.
6:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now I'm going to plug in.
7:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Right, this is m over n.
7:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is x to the m minus 1.
7:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now over here I'm going to put in for y x to
7:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the m over n times n minus 1.
7:11
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
7:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now we're almost done.
7:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But unfortunately we have this mess of exponents that we have to work
7:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
out.
7:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Write it one more time.
7:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I already recognize the factor a out front.
7:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's not going to be a problem for me,
7:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and that's what I'm aiming for here.
7:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But now I have to encode all of these powers.
7:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's just write it out.
7:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's m minus 1,
7:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then it's minus the quantity n minus 1 times m
7:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
over n.
7:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the law of exponents
7:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
applied to this ratio here.
7:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe just over here in the next board.
7:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have here m minus 1 minus n
8:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
minus 1 m over n is equal to
8:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
m minus 1.
8:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And if I multiply n by this,
8:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I get minus m.
8:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And if the second factor is a minus minus,
8:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's a plus.
8:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And that's plus m over n.
8:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So altogether,
8:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the two m's cancel.
8:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I have here minus 1 plus m over n.
8:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And lo and behold,
8:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's the same thing as a minus 1.
8:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just what we wanted.
8:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this equals a
8:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x to the a minus 1.
8:33
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Again,
8:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
just a bunch of arithmetic.
8:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
From this point forward,
8:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
from this substitution on,
8:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's just the arithmetic of exponents.
8:45
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
8:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we've done our first example here.
8:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I guess I want to give you a couple more examples.
9:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's just continue.
9:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The next example,
9:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'll keep it relatively simple.
9:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have example 2,
9:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is going to be the function x squared plus y
9:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
squared is equal to 1.
9:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, that's not really a function.
9:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This,
9:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
again, is a way of defining y as
9:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
a function of x implicitly.
9:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
There's the idea that I could solve for y if I wanted to.
9:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And indeed,
9:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's do that.
9:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So if you solve for y here,
9:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what happens is you get y squared is equal to 1 minus
9:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x squared.
9:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And y is equal to plus or minus square root of 1
9:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
minus x squared.
9:48
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So
9:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this,
9:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if you like,
9:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is the implicit definition.
9:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And here is the explicit.
10:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
function y is a function of x.
10:02
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now,
10:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
just for my own convenience,
10:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just going to take the positive branch.
10:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is the function,
10:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
let's just draw it right above here.
10:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's really a circle in disguise.
10:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm just going to take the top part of the circle.
10:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we'll take that top hump here.
10:23
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that means I'm erasing this minus sign.
10:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just taking the positive branch.
10:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just for my convenience,
10:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I could do it just as well with the negative branch.
10:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now I've
10:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
taken the solution and I can
10:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
differentiate this.
10:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So rather than using this dy by dx notation over
10:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
here, I'm going to switch notations over here because it's less writing.
10:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm going to write y' and change notations.
10:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I want to take the derivative of this.
11:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is a somewhat complicated function here.
11:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's the square root of .
11:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And the right way always to look at functions like
11:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this is to rewrite them using
11:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the fractional power notation.
11:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the first step in computing a
11:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
derivative of a square root.
11:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then the second step here
11:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is what?
11:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Somebody want to tell me?
11:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule.
11:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule. Yeah, that's good.
11:42
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Chain rule,
11:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's it.
11:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we have two things.
11:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We start with one and then we do something else to it.
11:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So whenever we do two things to something,
11:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we need to apply the chain rule.
11:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So 1 minus x squared,
11:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
square root.
11:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
11:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so how do we do that?
11:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well, the first factor I claim is the derivative of this thing.
12:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is a half blah to the minus 1 half.
12:06
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm doing this by the sort of advanced method now because we've already
12:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
graduated.
12:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You already did the chain rule last time.
12:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what does this mean?
12:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is an abbreviation for the derivative
12:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
with respect to blah of blah to
12:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the 1 half.
12:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Whatever it is.
12:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the first factor that we're going to use.
12:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Rather than actually write out a variable for it,
12:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and pass through as I did previously with this y and
12:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x variable here,
12:38
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just going to skip that step and I'm going to let you imagine it
12:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
as being a placeholder for that variable.
12:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this variable is now parenthesis.
12:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to multiply that by the rate of change
12:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
of what's inside with respect to x,
12:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and that is going to be .
12:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The derivative of is .
13:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now again,
13:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're glomming everything together here.
13:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We couldn't have done this example two before example one.
13:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because we needed to know that the power rule worked
13:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
not just for a integer,
13:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but also for a equals a half.
13:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We're using the case a equals a half right here.
13:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's one half times,
13:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and this minus a half here is a minus one.
13:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Right?
13:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the case a equals 1 half.
13:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
a minus 1 happens to be minus 1 half.
13:35
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
13:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so I'm putting all those things together.
13:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And you know,
13:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
within a week you have to be doing this very automatically.
13:45
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to do it at this speed now.
13:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You want to do it even faster,
13:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
ultimately.
13:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Yes?
13:50
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The question is,
13:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
could I have done it implicitly without the square roots?
13:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And the answer is yes,
13:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
that's what I'm about to do.
14:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is an illustration of what's called the explicit
14:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
solution.
14:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this guy is what's called explicit.
14:10
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I
14:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
want to contrast it with the method that we're going to now use today.
14:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it involves a lot of complications.
14:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It involves the chain rule.
14:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And as we'll see,
14:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it can get messier and messier.
14:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And then there's the implicit method,
14:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which I claim is easier.
14:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's see what happens if you do it implicitly.
14:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The implicit method involves,
14:39
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
instead of writing the function in this relatively complicated
14:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
way with a square root,
14:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it involves leaving it alone.
14:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Don't do anything to it.
14:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
In this previous case,
14:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we were left with something which was complicated,
14:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
say x to the 1 3rd or x to the 1 1 half or something complicated.
14:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
We had to simplify it.
15:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Equation 1,
15:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
which is more complicated,
15:01
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we simplified it and then differentiated it.
15:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And so that was the simpler case.
15:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well here, the simplest thing for us,
15:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the simplest equation for us to differentiate is the one we started with.
15:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because squares are practically the easiest things after first powers or maybe
15:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
0 powers to differentiate.
15:17
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we're leaving it alone.
15:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is the simplest possible form for it.
15:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now we're going to differentiate.
15:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what happens?
15:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So again,
15:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what's the method?
15:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Let me remind you.
15:27
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You're applying d by dx to the equation.
15:30
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So you have to differentiate the left side of the equation and differentiate
15:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the right side of the equation.
15:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it's this.
15:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And what you get is 2x plus 2y
15:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
y' is equal to what?
15:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
0.
15:51
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Derivative of 1 is 0.
15:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the chain rule
15:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
again. I did it a different way.
16:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm trying to get you used to many different notations at once.
16:03
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
16:04
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
really just 2.
16:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Just the prime notation and the dy dx notation.
16:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this is what I get.
16:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now,
16:14
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
all I have to do is solve for y'.
16:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So that y',
16:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
if I put the 2x on the other side,
16:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is minus 2x and then divide by 2y.
16:26
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Which is minus x over y.
16:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's compare our solutions.
16:32
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I apologize,
16:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to have to erase something to do that.
16:37
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's compare our two solutions.
16:44
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to put this underneath and simplify it.
16:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So what was our solution over here?
16:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It was 1 half 1 minus x squared to
16:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the minus 1 half times minus 2x.
16:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That was what we got over here.
16:58
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And that
17:02
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
is the same thing if I cancel the 2's and I change
17:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it back to looking like a square root.
17:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same thing as minus x divided by square root of 1 minus x squared.
17:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
formula for the derivative when I do it the explicit way.
17:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll just compare them,
17:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
these two expressions here.
17:25
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And
17:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
notice that they are the same.
17:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
They're the same because y is
17:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equal to square root of 1 minus x squared.
17:38
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Yeah,
17:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
question.
17:41
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The
17:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
question is, why did the implicit method not give the bottom half of the circle?
17:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Very good question.
17:52
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The answer to that is that it did.
17:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I just didn't mention it.
17:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Wait,
17:59
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
wait, wait, I'll explain.
18:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So suppose I'd stuck in a minus sign here.
18:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I would have gotten this with an
18:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
extra minus sign.
18:09
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But then when I compared it to what was over there,
18:12
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I would have had to have another different minus sign over here.
18:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually both places would get an extra minus sign.
18:19
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And they would still coincide.
18:20
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually the implicit method is a little better.
18:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It doesn't even notice the difference between the branches.
18:24
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It does the job on both the top and the bottom half.
18:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Another way of saying that is that you're calculating the
18:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
slopes here.
18:33
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So let's look at this picture.
18:34
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Here's a slope.
18:36
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Let's just take a look at a positive value of x and just check
18:40
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the signs to see what's happening.
18:42
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
If you take a positive value of x over here,
18:46
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x is positive,
18:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this denominator is positive,
18:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the slope is negative.
18:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You can see that it's tilting down.
18:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it's OK.
18:53
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
18:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
on the bottom side,
18:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's going to be tilting up.
18:58
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And similarly,
19:00
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
what's happening up here is that both x and y are positive.
19:03
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this x and this y are positive,
19:05
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and the slope is negative.
19:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
On the other hand,
19:07
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
on the bottom side,
19:08
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x is still positive,
19:10
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but y is negative.
19:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And it's tilting up because the
19:13
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
The denominator's negative,
19:15
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the numerator's positive,
19:16
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and then this minus sign makes it a positive slope.
19:18
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So it matches perfectly in every category.
19:22
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is complicated,
19:25
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
however, and it's easier just to keep track of one branch at a time,
19:29
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
even in advanced math.
19:31
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So we only do it one branch at a time.
19:35
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Other questions?
19:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so now I want to give you a slightly more complicated
19:47
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
example here.
19:48
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And indeed,
19:49
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
some of the,
19:50
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so here's a little more complicated example.
19:54
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's not going to be the most complicated example,
19:56
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but you know,
19:57
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it'll be a little bit.
20:11
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this
20:18
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
example,
20:19
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to give you a fourth order equation.
20:22
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So y to the fourth plus xy squared minus
20:27
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
2 is equal to 0.
20:28
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
20:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it just so happens that there's a trick to
20:36
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
solving this equation.
20:38
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So actually you can do both the explicit method and
20:42
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the non -explicit method.
20:45
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so the explicit method would say,
20:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, well I want to solve for this.
20:51
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to use the quadratic formula,
20:53
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
but on y squared.
20:55
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
This is quadratic in y squared.
20:57
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Because there's a fourth power and a second power in the first and the
21:01
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
third powers are missing.
21:14
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
times minus 2 divided by 2.
21:18
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So this x is the b,
21:22
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this minus 2 is the c,
21:24
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
and a is equal to 1 in the quadratic formula.
21:28
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And so the formula for y is
21:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
plus or minus the square root of minus
21:36
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
x plus or minus the square root of x squared plus 8
21:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
divided by 2.
21:43
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
All right?
21:44
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So now you can see this problem of branches,
21:47
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
this happens actually in a lot of cases,
21:50
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
coming up in an elaborate way.
21:53
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
You have two choices for the sign here,
21:54
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you have two choices for the sign here.
21:56
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Conceivably there are as many as four roots to this equation because it's a fourth degree
22:00
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
equation.
22:01
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
It's quite a mess,
22:02
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
you should have to check each branch separately.
22:05
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And this really is that level of complexity.
22:09
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And in general,
22:09
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
it's very difficult to figure out the
22:14
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
formulas for quartic equations.
22:17
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
But fortunately,
22:19
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're never going to use them.
22:21
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That is,
22:21
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
we're never going to need those formulas.
22:23
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the implicit method is
22:28
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
far easier
22:31
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So the implicit method just says,
22:34
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
OK, I'll leave the equation in its simplest form.
22:37
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And now differentiate.
22:40
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So when I differentiate,
22:41
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I get 4y3 y' plus,
22:46
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
now here I have to apply the product rule.
22:49
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I differentiate the x and the y squared.
22:55
S… Speaker 1 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
separately.
22:55
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
First I differentiate with respect to x,
22:57
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
so I get y squared.
22:59
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
Then I differentiate with respect to the other factor,
23:03
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
the y squared factor.
23:04
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
And I get x times 2y y' and
23:08
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
then the 0 gives me 0.
23:10
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So minus 0 equals 0.
23:13
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So there's the implicit
23:17
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
differentiation step.
23:23
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
I just want to solve for y prime.
23:26
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to factor out 4y cubed plus
23:30
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
2xy.
23:32
S… Speaker 2 (Lec 5 _ MIT 18.01 Single Variable Calculus, Fall 2007)
That's the factor on y prime.

תעתיק זה נוצר על ידי AI (הכרה אוטומטית של דיבור). ייתכן שיש בו שגיאות אשר מאשרות את השמע המקורי לשימוש קריטי. מדיניות AI

❤️ אוהבים את STT.ai? ספרו לחברים שלכם!
תקציר
לחץ לסכם כדי ליצור סיכום AI של תעתיק זה.
מסכם...
שאל את אל על התעתיק הזה.
שאל כל דבר על התמליל הזה, הבינה המלאכותית תמצא חלקים רלוונטיים ותענה.
תמלול נוסף