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U1S7V07 Derivative of cosine at 0.txt
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U1S7V07 Derivative of cosine at 0.txt
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#
# File: content-mit-18-01-1x-captions/U1S7V07 Derivative of cosine at 0.txt
#
# Captions for MITx 18.01.1x module [TyzcGQR0kLA]
#
# This file has 98 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
We have one more limit left to compute
and we want to approach this the same way.
That is, we're going to give a geometric proof.
To get started, let's go ahead and replace the quantity delta
x everywhere with theta.
So that's what I've done here.
And our limit has become the limit as theta approaches
0 of cosine theta minus 1 all over theta.
To give a geometric proof, we're going
to need a picture so let's go ahead
and start with a unit circle.
We'll draw in an angle theta.
Let's go ahead and draw in what we figured out before.
So we have that this vertical distance here in yellow
is sine theta and this arc length is theta.
What we need to find is this numerator
in our limit, cosine theta minus 1.
What we know is that the radius of our circle
is 1 because it's a unit circle.
And this horizontal distance here,
the leg of this right triangle, is cosine theta.
So the segment right here in green,
this is the quantity 1 minus cosine theta.
Observe that this is actually the negative of the numerator
that we have in our limit.
But that's OK.
We can just take a negative at the end and we'll be fine.
But it's much easier for us to visualize positive distances
rather than negative distances.
To make things a little bit easier to look at,
I'm going to go ahead and erase this 1 and this cosine theta.
We're interested in what happens as theta approaches 0.
So I'm going to shrink the angle theta.
Because this is getting hard to look at,
I'm going to enlarge the circle.
And I'm enlarging it in such a way
that the relative magnitude of sine theta is staying fixed.
Let's go ahead and compare this to the original circle
before I shrunk theta.
You can see that the sine theta is exactly the same size.
So let's keep up the process of shrinking theta and enlarging
the radius of our circle until we can't fit it on the page
anymore.
And look at what happened to 1 minus cosine theta.
Observe that now it is so tiny you can barely see it.
In the previous exercises, you saw
that when theta is very small, like it is here,
sine theta is essentially equal to theta.
And both of these quantities are going to 0.
But what we see geometrically from this picture
is that 1 minus cosine theta is going to 0 much, much faster.
What this is telling us about this limit
is that the numerator is going to 0 so much
more quickly than the denominator
that it's winning this race to 0.
So this is a tricky idea.
Both the numerator and denominator
are actually going to 0.
However, because the numerator tends
to 0 so much more quickly, this limit actually is equal to 0.
Tricky, I know.
But let's think about what this means.
Remember that we've identified this limit
as the derivative of cosine theta at the point theta
equals 0.
What this means is that if we take the graph of cosine theta
then the tangent line has a slope of 0 at theta equals 0.
And that's exactly what we see here
and that's what we should expect.
So our geometric argument does really make sense.
The slope of the tangent line is equal to 0.
Everything works out.
So now I want to do a little recap
and remind you why we were computing
these limits to begin with.
Remember that in the very first video
Professor Jerison showed us that the derivative of sine
is equal to cosine times the derivative of sine at 0
plus sine times the derivative of cosine at 0.
We just found that the derivative of cosine at 0
is equal to 0.
And previously we saw that the derivative of sine at 0
is equal to 1.
So the derivative of sine x is equal to cosine x.
Similarly, you can work this out on your own
using the exact same arguments.
We can see that the derivative of cosine x
is equal to negative sine x.
This is a pretty neat relationship
that the derivatives of these two functions
have with each other, so the next exercises are going
to explore this a little bit.
And, in other exciting news, we now
have two pretty interesting functions
that we know the derivative of.