Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007

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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
.edu.
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S… Speaker 5 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
In
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
today's lecture,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to develop several more formulas that will allow us to reach
0:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
our goal of differentiating everything.
0:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So these are derivative formulas.
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S… Speaker 5 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
they come in two flavors.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The first kind is specific.
0:48
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So some specific function
0:52
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we're giving the derivative of.
0:55
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that would be,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
for example,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x to the n or 1 over x.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Those are the ones that we did a couple of lectures ago.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then there are general formulas.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And the general ones don't actually give you a formula for a specific function,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but tell you something like,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
if you take two functions and add them together,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
their derivative is the sum of the derivatives.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Or if you multiply by a constant,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
for example,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so c times u,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of that is c times u',
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
where c is constant.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so these kinds of formulas are very useful,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
both the specific and the general kind.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
For example,
1:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
they're needed,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so we need both kinds for
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
polynomials.
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S… Speaker 5 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
more generally, pretty much any set of formulas that we give you will give you a few functions to
2:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
start out with,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and then you'll be able to generate lots more by these general formulas.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So today,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we want to concentrate on the trig functions.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And so we'll start out with some specific formulas.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And they're going to
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
be the formulas for the derivative of the sine function and
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the cosine function.
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S… Speaker 3 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's what
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll spend the first part of the lecture on.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And at the same time,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I hope to get you very used to dealing with trig
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
functions.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
think of as a gradual process.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so in order to calculate these,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to start over here and just start
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the calculation.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So here we go.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's check what happens with the sine function.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I take sine x plus delta
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x, I subtract sine x,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and I divide by,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sorry, let's leave a little space there,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sine x and divide by delta x.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so this is the difference quotient.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And eventually I'm going to have to take the limit as delta x goes to 0.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And there's really only one thing we
3:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
can do with this to simplify it or change
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it. And that is to use the sum formula for
3:40
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the sine function.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Oh, that's not what it is,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
huh? OK.
3:56
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So what is it?
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Sine x?
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Sine delta x.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK, good, you remember.
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S… Speaker 3 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Plus?
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Cosine?
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S… Speaker 4 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
No?
4:07
S… Speaker 4 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Oh,
4:09
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
4:09
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So which is it?
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
4:14
S… Speaker 4 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
4:16
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
let's take a vote.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Is it sine sine or is it sine cosine?
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S… Speaker 3 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
4:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So is this going to be cosine?
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All
4:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
right. You better remember these formulas,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
all right?
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK, it turns out that it's sine,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine.
4:36
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine, sine.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So here we go.
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S… Speaker 4 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
No,
4:42
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you've got to do x here.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Sine delta x.
4:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So now,
4:48
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
there's lots of places to get confused here.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And you're going to need to make sure that you get it right.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to put those in parentheses here.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
b is sin a cos b
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
plus cos a sin
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
b.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now that's what I did over here,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
except it was the letter x was a,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and the letter b
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
was delta x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now that's just the first part.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's just this part.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
of the expression.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I still have to remember the minus sin x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That comes at the end.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Minus sin x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to remember the denominator,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is delta x.
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
now, the next thing
5:47
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to do is we're going to try to group the terms.
5:51
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And the difficulty with all such arguments
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the following one.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Any tricky limit is basically 0 over 0 when you
6:02
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
set delta x equal to 0.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If I set delta x equal to 0,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this is sin x minus sin x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So it's a 0 over 0 term.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Here we have various things which are 0 and various things which are non -zero.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
We must group the terms so that
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
A 0 stays over a 0.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Otherwise we're going to have no hope.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If we get some 1 over 0 term,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll get something meaningless in the limit.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I claim that the right thing to do here is to notice,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and I'll just point out this one thing,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is that when delta x goes to 0,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this cosine of 0 is 1.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So it doesn't cancel unless we throw in this extra
6:39
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sine term here.
6:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to use this common factor.
6:43
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And combine those terms.
6:44
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is really the only thing that you're going to have to check in this particular calculation.
6:48
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So we have the common factor of sin x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then that multiplies something that will cancel,
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is cos delta x minus 1 divided by delta x.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the first term.
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now what's left?
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S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
7:03
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
there's a cos x that factors out.
7:05
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the other factor is sin delta x over delta
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S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x.
7:18
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Does anyone remember from last time what this thing goes to?
7:21
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
How many people say
7:25
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
1? How many people say 0?
7:28
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
7:29
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it's 0.
7:30
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's my favorite number.
7:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
7:33
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
0.
7:33
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
It's the easiest number to deal with.
7:35
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this goes to 0.
7:37
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's what happens as delta x tends to 0.
7:41
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now
7:46
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
how about this one?
7:46
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This one goes to 1.
7:49
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
My second favorite number.
7:50
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Almost as easy to deal with as 0.
7:54
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And these things are picked for a reason.
7:56
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
They're the simplest numbers to deal with.
7:58
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So altogether,
7:58
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this thing,
8:00
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x goes to 0,
8:02
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
goes to what?
8:03
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I want a single person
8:07
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
to answer, a brave volunteer.
8:09
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, right back there.
8:10
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Cosine.
8:12
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Because this factor is 0,
8:14
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it cancels.
8:15
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And this factor has a 1,
8:16
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so it's cosine.
8:17
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So it's cosine x.
8:19
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So our conclusion over here,
8:21
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll put it in.
8:24
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
orange, is that the derivative of the sine is the
8:28
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine.
8:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
8:35
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
now I still want to label these very important limit facts here.
8:38
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This one we'll call A,
8:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and this one we're going to call B,
8:42
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
because we haven't checked them yet.
8:44
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I promised you I would do that,
8:45
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and I'll have to do that this time.
8:47
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So we're relying on those things being true.
8:52
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now I'm going to do the same thing with the cosine function,
8:55
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
except in order to do it I'm going to have to remember the sum rule for
8:59
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine.
9:00
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So we're going to do almost the same calculation here.
9:03
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
We're going to see that that will work out.
9:04
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But now you have to remember that cosine of a plus b is
9:09
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
equal to cosine,
9:11
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine.
9:12
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
No,
9:12
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it's not cosine squared because there are two different quantities here.
9:16
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Cosine a,
9:17
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine b minus,
9:19
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
here's the minus sign,
9:20
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sine a.
9:23
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right.
9:24
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So you'll have to be willing
9:29
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
to call those forth at will right
9:33
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
now. So let's do the cosine now.
9:36
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's cosine x plus delta x minus
9:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine x divided by delta x.
9:44
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
There's the difference quotient.
9:47
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
for the cosine function.
9:49
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now I'm going to do the same thing I did before,
9:51
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
except I'm going to apply the second rule,
9:53
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that is the sum rule for cosine.
9:55
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's going to give me cosine x,
9:57
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine delta x.
10:00
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
minus sine x sine delta x.
10:03
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to remember again to subtract the cosine.
10:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Divided by this delta x.
10:10
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now I'm going to regroup just the way I did before.
10:14
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And I get the common factor of cosine,
10:18
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
multiplying cosine delta x minus 1 divided
10:23
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
by delta x.
10:24
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And here I get the sine x,
10:27
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but actually it's minus sine x.
10:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have sine delta x divided by delta x.
10:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The only difference is this minus sign,
10:38
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which I stuck inside there.
10:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
10:43
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that's not the only difference,
10:44
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but it's a crucial difference.
10:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Again,
10:49
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
by a,
10:50
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we get that this is 0,
10:52
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x tends to 0.
10:54
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is 1.
10:55
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Those are the properties I called a and b.
11:01
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x tends to 0 is that
11:06
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we get negative sin x.
11:08
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the factor,
11:09
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
negative sin x.
11:10
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this guy is negative sin x.
11:14
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
11:19
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll put a little box around that too.
11:20
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
11:25
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
these formulas take a little bit of getting used to.
11:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But before I
11:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
do that, I'm going to explain to you the proofs of a and b.
11:37
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
11:38
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so we'll get ourselves started by
11:43
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
mentioning that.
11:44
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe before I do that though,
11:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to show you how A and B fit into the proofs of these
11:50
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
theorems.
11:51
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So let me just make some
11:55
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
remarks here.
11:56
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So
12:06
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this is just a remark,
12:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but it's meant to help you to frame
12:12
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
how these proofs worked.
12:15
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
12:16
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to point out that if you take the rate of change of sine
12:21
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x,
12:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
now let's start with cosine.
12:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Because it's actually a little bit less obvious.
12:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If I take the rate of change of cosine x,
12:33
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so in other words this derivative,
12:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
at x equals 0,
12:36
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
then by definition
12:40
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this is a certain limit as delta x goes to 0.
12:44
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So which one is it?
12:49
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
at 0 plus delta x,
12:51
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but that's just delta x.
12:52
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And I have to subtract cosine at 0,
12:56
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that's the base point,
12:57
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but that's just 1.
12:58
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to divide by delta x.
13:02
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And lo and behold,
13:04
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you can see that this is exactly the limit that we had over there.
13:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This is the one that we know is 0 by what we call property A.
13:12
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And similarly,
13:16
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
if I take the derivative of sin x at
13:21
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x equals 0,
13:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Then that's going to be the limit,
13:24
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x goes to 0,
13:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
of sine delta x over delta x.
13:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's because I should be subtracting sine
13:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
of 0.
13:35
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But sine of 0 is 0.
13:36
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is going to be 1
13:40
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
by our property b.
13:42
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
13:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so the remark that I want to make in addition to this
13:51
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is something about the structure of these two proofs.
13:54
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Which is the derivatives of sine
14:02
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and cosine at
14:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
x equals 0 give all
14:14
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
values of d
14:19
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
by dx, sine x,
14:21
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
d by dx,
14:23
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
cosine x.
14:25
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's really what this argument is showing us.
14:27
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Is that we just need one rate of change at one place,
14:31
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and then we work out all the rest of them.
14:38
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's really the substance of this proof.
14:40
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That, of course,
14:41
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
really then shows that it boils down to showing what this rate of change is in
14:45
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
these two cases.
14:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So now there's enough suspense that we want
14:50
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
to make sure that we know that those answers are correct.
15:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
15:08
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so let's demonstrate both of them.
15:11
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll start with b.
15:13
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I need to figure out property
15:17
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
b. Now,
15:18
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we only have one alternative as to a type of proof that
15:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we can give of this kind of result.
15:24
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's because we only have one way of describing
15:28
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sine and cosine functions.
15:31
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That is geometrically.
15:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So we have to give a geometric proof.
15:43
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
To write down a geometric proof,
15:45
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to have to draw a picture.
15:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And the first step in the proof really is to replace this
15:50
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
variable delta x,
15:51
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is going to 0,
15:53
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
with another name which is suggestive of what we're going to do,
15:57
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the letter theta for an angle.
15:59
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's draw a picture of what it is
16:03
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that we're going to do.
16:04
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's the circle.
16:06
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And here is the origin.
16:10
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's some little angle.
16:13
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, I'll draw it a little larger so that it's visible.
16:15
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's theta.
16:16
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is the unit circle.
16:20
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I won't write that down on here,
16:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but that's the unit circle.
16:23
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now sin theta is
16:28
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this vertical distance here.
16:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe I'll draw it in a different color so that we can see it all.
16:33
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's this distance.
16:38
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this distance is sin theta.
16:49
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Almost the only other thing we have to write down in this picture to have it work out
16:54
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is that we have to recognize that when theta is the angle,
16:58
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that's also the arc length of this
17:02
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
piece of the circle when measured in radians.
17:04
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this length here is also arc
17:08
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
length theta.
17:12
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That little piece in there.
17:14
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe I'll use a different color for that to indicate it.
17:18
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So there's orange,
17:19
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and that's this.
17:20
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This little chunk there.
17:22
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So those are the two
17:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
pieces.
17:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now, in order to persuade
17:31
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you now that the limit
17:35
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is what it's supposed to be,
17:37
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to extend the picture just a little bit.
17:39
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to double it,
17:40
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
just for my own linguistic sake and so that I can tell you a story.
17:44
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that you'll remember this.
17:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to take a theta angle below,
17:49
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and I'll have another copy of sin theta down here.
17:53
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now the total picture is really
17:58
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
like a bow and
18:03
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
its bow string there.
18:04
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So what we have here is a
18:08
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
length of 2 sin theta.
18:11
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe I'll write it this way.
18:12
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
2 sin theta.
18:13
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I just doubled it.
18:21
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Whoops, I got it backwards.
18:24
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Sorry about that.
18:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Trying to be fancy with the colored chalk,
18:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and I have it reversed here.
18:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is not 2 sin theta.
18:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
2 sin theta is the vertical,
18:33
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that's the green.
18:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's try that again.
18:36
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This is 2 sin theta.
18:38
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then in the denominator,
18:42
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I have the arc length,
18:44
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is theta is the first half,
18:46
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and so double it is 2 theta.
18:48
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So if you like,
18:51
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this is the bow.
18:54
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And up here we have the bow string.
18:57
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
19:04
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
of course, we can cancel the 2's.
19:07
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's equal to sin theta divided by theta.
19:10
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And so now why does this tend to 1 as theta goes
19:14
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
to 0?
19:16
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
19:18
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it's because as the angle theta gets
19:22
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
very small,
19:23
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this curved piece looks more and more like
19:27
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
a straight one.
19:28
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right?
19:29
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And if you get very,
19:30
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
very close here,
19:31
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the green segment and the orange segment would just merge.
19:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
They would be practically on top of each other.
19:36
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And they have closer and closer and closer to the same length.
19:39
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's why
19:43
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this is true.
20:00
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
curves are nearly straight.
20:04
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
20:06
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so that's the principle that we're using.
20:09
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Or short pieces of curves,
20:11
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
if you like, are nearly straight.
20:13
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So if
20:19
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you like, this is the principle.
20:20
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So short
20:24
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
pieces of curves.
20:26
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All
20:31
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
right, so now I also need to give you a proof of A.
20:34
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And
20:39
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that has to do with this cosine function here.
20:43
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This is the property A.
20:47
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And so I'm going
20:51
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
to do this by flipping it around.
20:53
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Because it turns out that this numerator is a negative number.
20:55
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If I want to interpret it as a length,
20:57
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to want a positive quantity.
21:00
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to write down 1 minus cosine theta here.
21:03
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I'm going to divide by theta there.
21:07
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Again,
21:08
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to make some kind of interpretation.
21:10
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Now this time I'm going to draw the same
21:14
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
sort of bow and arrow arrangement.
21:15
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But maybe I'll exaggerate it a little bit.
21:18
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's the vertex of the sector,
21:22
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
but we'll maybe make it a little longer.
21:24
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All
21:31
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
right, so here it is.
21:33
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And here was that middle line which was the unit,
21:36
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
whoops.
21:37
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
21:38
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
I think I'm going to have to tilt it up.
21:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Let's try from here.
21:42
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
All
21:47
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
right.
21:47
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
21:49
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you know, on your pencil and paper it'll look better than it does on my blackboard.
21:53
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
21:54
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so here we are,
21:55
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
here's this shape.
21:58
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This angle is supposed to be theta.
22:00
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And this angle is another theta.
22:02
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So here we have a length which is again theta.
22:05
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And another length which is theta over here.
22:07
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the same as in the other picture except we've exaggerated
22:11
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
it here.
22:12
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now we have this vertical line,
22:14
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which again I'm going to draw in green,
22:16
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the bowstring.
22:17
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But notice that as the
22:21
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
vertex gets farther and farther away,
22:24
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The curved line gets closer and closer to being a vertical line.
22:28
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
That's sort of the flip side by expansion of the zoom
22:32
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
in principle.
22:32
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The principle that curves are nearly straight when you zoom in.
22:36
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If you zoom out,
22:37
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
that would mean sending this vertex way,
22:39
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
way out somewhere.
22:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The curved line,
22:43
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
the piece of the circle,
22:44
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
gets more and more straight.
22:48
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And now let me show you where this numerator 1
22:52
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
minus cosine theta is on this picture.
22:55
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So where is it?
22:58
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
23:01
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this whole distance is 1.
23:02
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
But the distance from the vertex to the green
23:07
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
is cosine of theta.
23:08
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
Because this is theta,
23:11
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
so dropping down the perpendicular,
23:13
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
this distance back to the origin is cosine
23:17
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
theta.
23:19
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
bitty segment here,
23:21
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
which is basically the gap between the
23:26
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
curve and the vertical segment.
23:27
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
So the gap is equal to 1 minus cosine
23:31
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
theta.
23:32
S… Speaker 1 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And so now
23:36
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you can see that as
23:39
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
This point gets farther away.
23:40
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
If this got sent off to the status center,
23:43
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
you would hardly be able to tell the difference.
23:45
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
The bowstring would coincide with the bow,
23:47
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
and this little gap between the bowstring and the bow would be tending to 0.
23:52
S… Speaker 2 (Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's the statement that this tends to 0,

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