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U2S4V13 Recitation video.txt
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U2S4V13 Recitation video.txt
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#
# File: content-mit-18-01-1x-captions/U2S4V13 Recitation video.txt
#
# Captions for MITx 18.01.1x module [D-PIwPu15tU]
#
# This file has 158 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Welcome back to recitation.
In this video segment, we're going
to look at the chain rule, and specifically, we're
going to answer a question that I've placed on the board.
I want you to find two values for theta
so that d d theta of the function cosine
squared theta to the fourth equals 0.
At this point, I should also point out a few things.
One, theta is a variable in this case,
and if you haven't seen it before,
frequently we use it when we're dealing
with trigonometric functions.
Theta often represents the variable that measures angle.
So if you haven't seen theta before,
this is what it looks like.
That's how you say it and I'm actually
doing the same things we've done before.
I'm taking the derivative with respect
to the variable theta of a function of theta.
The other thing I want to point out,
which I think we know already, but just to be sure,
is that this squared here, cosine squared theta
to the fourth, means I'm taking cosine of theta to the fourth,
and I'm actually squaring it.
So with that knowledge, I would like
us to use the chain rule to find two values for theta
where this derivative is equal to 0.
So I'll give you a moment to take a stab
at finding the derivative, setting it equal to zero,
and finding some values for theta that
make the derivative equal zero.
And we'll come back and then I'll work it out as well.
OK, so the first thing we obviously need to do
is be able to take the derivative of this function
on the left-hand side.
And I should say it's a little more complicated
than an example you saw in the lecture because in the lecture
you were given example with just two functions.
So I want to write it a little differently just
to show very obviously what the three functions are
we're composing.
So this function, I can rewrite it as cosine of theta
to the fourth.
Does that look like a 4?
There we go.
And then I square that whole thing.
That's really what the function is.
OK, so we can see what is the outermost function here?
The outermost function is actually the quantity
of something squared.
So the outermost function is x squared.
What's the next function in in this composition?
The next function in is the cosine function.
And then the last function in is the function taking something
raising it to the fourth.
So it's very important that you understand the composition,
which is the outermost function, which is the innermost function
in order to do this chain rule.
Now, as you saw in recitation, you were given the example dy--
you had y as a function of t--
sorry, not in recitation.
In the lecture you were given y as a function of t
or a function of x, and then you had
to put one other variable in the middle.
Here we're going to have a composition of three functions.
So we need to have two other things in the middle.
So let's write this out.
First, the outermost function.
We'll call the whole thing y.
So y is equal to x squared is the outermost function.
So then this whole thing is x, now.
So then we'll write x is equal to cosine of w.
OK, and then w is equal to theta to the fourth.
OK, again, this is what you saw before in the lecture.
So you write the outermost function and then
that function is a function of cosine of another one--
cosine of w, and w is a function of theta to the fourth.
So if I put all these back together,
I have theta to the fourth in here, and then I square that,
and I come back to the function I wanted.
Now, we know from the lecture what
we need to do to find, essentially, dy d theta.
That's what we're looking for.
OK, so dy d theta, you remember from the lecture, should be dy
dx times dx dw times dw d theta.
OK, so it's slightly more complicated
than we saw before because there's one more term.
OK, so now let's work out what these are.
Well, dy dx is fairly straightforward.
dy dx is just 2x.
And then dx dw--
well, what's the derivative of the cosine function?
The derivative of cosine is negative sine.
So this is times negative sine of w times--
and what's dw d theta?
So w is the function theta to the fourth, so dw d theta
is 4 theta to the third.
Now, when you look at this, you should
remember x we sort of inserted into the problem
to make the problem easier, and w we inserted into the problem
to make the problem easier for us to follow.
So we don't want all these x's and w's.
We want everything in terms of theta, right?
But what is w?
Well, w is theta to the fourth.
So I can replace that by theta to the fourth.
And what is x?
x is cosine w, but w is theta to the fourth.
So x is actually cosine of theta to the fourth.
OK.
So I get 2 cosine of theta to the fourth times
negative sine--
w, again, is theta to the fourth--
of theta to the fourth times 4 theta to the third, and let's
make this nicer.
I'll bring the coefficient and the theta
to the third in front, and this minus sign in front.
I get negative 8 theta to the third cosine of theta
to the fourth sine of theta to the fourth.
OK, and then the problem asked to find where
the derivative is equal to 0.
Find two values.
Now, why did I ask you to find two values?
Because one is very easy.
If this thing is set equal to 0, one value
should stand out right away for theta that makes this product
0, and that is theta equals 0.
So theta equals 0 is our easiest answer.
OK, so if you didn't do that, then you
want to the more challenging stuff.
You could have done the other things also.
So what about cosine theta to the fourth?
If we want a product of three things to be 0,
then at least one of them has to be 0.
So I could have set cosine theta to the fourth equal to 0,
and for what values of an angle--
I should just say.
For what values of theta to the fourth is this going to be 0?
Well, this means that theta to the fourth
is equal to either maybe pi/2 or you could add another pi.
OK, so we could say pi-- well, let's
just say that's one example.
Theta to the fourth equals pi/2 would work because cosine
of pi//2 is equal to 0.
OK.
And so then we could have said theta
is equal to pi/2 to the 1/4.
OK, that's another example there.
We could have also done--
we could have added pi to this, and gotten another answer,
but I only asked for two.
So I guess that I'm allowed to stop there.