Local linearity and differentiability

WEBVTT
Kind: captions
Language: en

00:00:00.589 --> 00:00:02.261
- [Instructor] What we're
going to do in this video

00:00:02.261 --> 00:00:06.046
is explore the relationship
between local linearity

00:00:06.046 --> 00:00:10.295
at a point and
differentiability at a point.

00:00:10.295 --> 00:00:13.407
So local linearity is this
idea that if we zoom in

00:00:13.407 --> 00:00:15.264
sufficiently on a point,

00:00:15.264 --> 00:00:17.888
that even a non-linear
function that is differentiable

00:00:17.888 --> 00:00:21.302
at that point will actually look linear.

00:00:21.302 --> 00:00:23.531
So let me show some examples of that.

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So let's say we had y

00:00:25.504 --> 00:00:26.421
is equal to

00:00:27.571 --> 00:00:28.404
x squared.

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So that's that there, clearly
a non-linear function.

00:00:32.517 --> 00:00:34.862
But we can zoom in on a point,

00:00:34.862 --> 00:00:37.486
and if we zoom sufficiently in,

00:00:37.486 --> 00:00:40.063
we will see that it looks roughly linear.

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So let's say we wanna zoom in
on the point one comma one,

00:00:43.221 --> 00:00:44.289
so let's do that.

00:00:44.289 --> 00:00:47.285
So zooming in on the point one comma one,

00:00:47.285 --> 00:00:52.068
already it is looking
roughly linear at that point.

00:00:52.068 --> 00:00:54.924
And this property of local linearity

00:00:54.924 --> 00:00:56.526
is very helpful

00:00:56.526 --> 00:00:59.824
when trying to approximate
a function around a point.

00:00:59.824 --> 00:01:01.403
So for example, we could figure out,

00:01:01.403 --> 00:01:04.119
we could take the derivative
at the point one one,

00:01:04.119 --> 00:01:06.348
use that as the slope of our tangent line,

00:01:06.348 --> 00:01:08.438
find the equation of the tangent line,

00:01:08.438 --> 00:01:11.735
and use that equation
to approximate values

00:01:11.735 --> 00:01:13.568
of our function around

00:01:14.731 --> 00:01:15.814
x equals one.

00:01:16.821 --> 00:01:18.005
And you might not need to do that

00:01:18.005 --> 00:01:20.141
for y is equal to x squared,

00:01:20.141 --> 00:01:21.859
but it could actually be very very useful

00:01:21.859 --> 00:01:24.390
for a more complex function.

00:01:24.390 --> 00:01:25.807
But the big takeaway here,

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at the point one one,

00:01:27.479 --> 00:01:31.171
it is displaying this
idea of local linearity,

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and it is also
differentiable at that point.

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Now let's look at another example

00:01:35.815 --> 00:01:38.508
of a point on a function

00:01:38.508 --> 00:01:40.273
where we aren't differentiable,

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and we also don't see the local linearity.

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So for example,

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let's do the

00:01:47.773 --> 00:01:49.440
absolute value of x,

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and let me shift it over a little bit

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just so that we don't overlap as much.

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Alright, so the absolute
value of x minus one.

00:02:01.078 --> 00:02:03.307
It actually is differentiable

00:02:03.307 --> 00:02:06.140
as long as we're not at
this corner right over here,

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as long as we're not at
the point one comma zero.

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For any other x value,
it is differentiable,

00:02:12.131 --> 00:02:14.360
but right at x equals one,

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we've talked in other videos

00:02:15.730 --> 00:02:18.052
how we aren't differentiable there.

00:02:18.052 --> 00:02:20.258
And then we can use this
local linearity idea

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to test it as well.

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And once again, this is
not rigorous mathematics,

00:02:24.158 --> 00:02:26.202
but it is to give you an intuition.

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No matter how far we zoom in,

00:02:29.638 --> 00:02:32.221
we still see this sharp corner.

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It would be hard to construct
the only tangent line,

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a unique line, that goes through
this point one comma zero.

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I can construct an actual
infinite number of lines

00:02:43.060 --> 00:02:45.265
that go through one comma zero

00:02:45.265 --> 00:02:48.563
but that do not go through
the rest of the curve.

00:02:48.563 --> 00:02:49.793
And so notice,

00:02:49.793 --> 00:02:51.930
wherever you see a hard
corner like we're seeing

00:02:51.930 --> 00:02:55.087
at one comma zero in this
absolute value function,

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that's a pretty good indication

00:02:56.736 --> 00:02:59.058
that we are not going to be differentiable

00:02:59.058 --> 00:03:00.225
at that point.

00:03:01.450 --> 00:03:03.214
Now let's zoom out a little bit,

00:03:03.214 --> 00:03:04.584
and let's take another function.

00:03:04.584 --> 00:03:06.767
Let's take a function where

00:03:06.767 --> 00:03:09.252
the differentiability or the
lack of differentiability

00:03:09.252 --> 00:03:10.761
is not because of a corner,

00:03:10.761 --> 00:03:12.944
but it's because as we zoom in,

00:03:12.944 --> 00:03:14.314
it starts to look linear,

00:03:14.314 --> 00:03:16.566
but it starts to look
like a vertical line.

00:03:16.566 --> 00:03:18.725
So a good example of that would be

00:03:18.725 --> 00:03:19.892
square root of

00:03:21.280 --> 00:03:22.742
let's say

00:03:22.742 --> 00:03:24.742
four minus x squared.

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So that's the top half of
a circle of radius two.

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And let's focus on the
point two comma zero.

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Because right over there,

00:03:34.631 --> 00:03:36.930
we actually are not differentiable,

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and if we zoom in far enough,

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we see right at two comma zero

00:03:41.086 --> 00:03:43.803
that we are approaching

00:03:43.803 --> 00:03:46.470
what looks like a vertical line.

00:03:48.795 --> 00:03:49.956
So once again,

00:03:49.956 --> 00:03:54.600
we would not be differentiable
at two comma zero.

00:03:54.600 --> 00:03:56.086
Now another thing I wanna point out,

00:03:56.086 --> 00:03:58.617
all of these, you really
didn't have to zoom in too much

00:03:58.617 --> 00:04:00.568
to appreciate that hey I got a corner here

00:04:00.568 --> 00:04:02.263
on this absolute value function,

00:04:02.263 --> 00:04:05.769
or at two comma zero, or
at negative two comma zero,

00:04:05.769 --> 00:04:07.603
something a little bit
stranger than normal

00:04:07.603 --> 00:04:10.831
is happening there, so maybe
I'm not differentiable.

00:04:10.831 --> 00:04:14.221
But there are some functions
that we don't see as typically

00:04:14.221 --> 00:04:17.077
in a algebra or precalculus
or calculus class,

00:04:17.077 --> 00:04:19.933
but it can look like a hard corner

00:04:19.933 --> 00:04:22.023
from a zoomed out perspective,

00:04:22.023 --> 00:04:24.972
but as we zoom in once again
we'll see the local linearity,

00:04:24.972 --> 00:04:28.223
and they are also
differentiable at those points.

00:04:28.223 --> 00:04:29.964
So a good example of that,

00:04:29.964 --> 00:04:32.170
let me actually get rid of some of these

00:04:32.170 --> 00:04:34.515
just so that we can really zoom in.

00:04:34.515 --> 00:04:36.598
Let's say y is equal to x

00:04:37.882 --> 00:04:41.179
to the, and I'm gonna make
a very large exponent here,

00:04:41.179 --> 00:04:42.921
so x to the 10th power.

00:04:42.921 --> 00:04:45.707
It's starting to look at
little bit like a corner there.

00:04:45.707 --> 00:04:47.727
Let's make it to the 100th power.

00:04:47.727 --> 00:04:50.142
Well now it's looking even
more like a corner there.

00:04:50.142 --> 00:04:52.580
Let me go to the 1,000th
power just for good measure.

00:04:52.580 --> 00:04:54.136
So at this scale,

00:04:54.136 --> 00:04:56.505
it looks like we have
a corner at the point

00:04:56.505 --> 00:04:57.755
one comma zero.

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Now this curve actually
does not go to the point

00:05:00.731 --> 00:05:02.588
one comma zero.

00:05:02.588 --> 00:05:03.935
If x is one,

00:05:03.935 --> 00:05:05.444
then y is going to be one,

00:05:05.444 --> 00:05:07.464
and we'll see that as we zoom in,

00:05:07.464 --> 00:05:09.624
this what looks like a hard corner

00:05:09.624 --> 00:05:11.551
is going to soften.

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And that's good because this function

00:05:13.385 --> 00:05:16.845
is actually differentiable
at every value of x.

00:05:16.845 --> 00:05:17.937
It's a little bit more exotic

00:05:17.937 --> 00:05:19.167
that what we typically see,

00:05:19.167 --> 00:05:21.420
but as we zoom in,
we'll actually see that.

00:05:21.420 --> 00:05:23.045
Let's just zoom in on what looks like

00:05:23.045 --> 00:05:25.483
a fairly hard corner,

00:05:25.483 --> 00:05:27.573
but if we zoom sufficiently enough,

00:05:27.573 --> 00:05:29.732
even at the part that
looks like the hardest part

00:05:29.732 --> 00:05:30.847
of the corner,

00:05:30.847 --> 00:05:33.169
the real corner, we'll see
that it starts to soften

00:05:33.169 --> 00:05:34.701
and it curves.

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And if we zoom in sufficiently,

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it will actually look like a line.

00:05:39.624 --> 00:05:41.783
It's hard to believe when
you're really zoomed out,

00:05:41.783 --> 00:05:44.593
and I'm going at the point that
really looked like a corner

00:05:44.593 --> 00:05:46.613
from a distance.

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But as we zoom on in,

00:05:48.448 --> 00:05:52.349
we see once again this local linearity

00:05:52.349 --> 00:05:54.276
that's a non-vertical line.

00:05:54.276 --> 00:05:56.830
And so once again, this is true
at any point on this curve,

00:05:56.830 --> 00:05:59.198
that we are going to be differentiable.

00:05:59.198 --> 00:06:00.568
So the whole point here is,

00:06:00.568 --> 00:06:02.078
sometimes you might have to zoom in a lot,

00:06:02.078 --> 00:06:04.516
a tool like Desmos which
I'm using right now

00:06:04.516 --> 00:06:06.513
is very helpful for doing that.

00:06:06.513 --> 00:06:09.369
And this isn't rigorous mathematics,

00:06:09.369 --> 00:06:11.784
but it's to give you an intuitive sense

00:06:11.784 --> 00:06:13.641
that if you zoom in sufficiently,

00:06:13.641 --> 00:06:16.242
and you start to see a
curve looking more and more

00:06:16.242 --> 00:06:17.612
like a line,

00:06:17.612 --> 00:06:19.748
good indication that
you are differentiable.

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If you keep zooming in and it still looks

00:06:21.675 --> 00:06:22.999
like a hard corner,

00:06:22.999 --> 00:06:24.462
of if you zoom in and it looks like

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the tangent might be vertical,

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well then some questions
should arise in your brain.


Differentiability and continuity

WEBVTT
Kind: captions
Language: en

00:00:00.184 --> 00:00:01.722
- [Instructor] What we're
going to do in this video

00:00:01.722 --> 00:00:05.889
is explore the notion of
differentiability at a point.

00:00:07.489 --> 00:00:10.491
And that is just a fancy way of saying

00:00:10.491 --> 00:00:14.433
does the function have a
defined derivative at a point?

00:00:14.433 --> 00:00:18.292
So let's just remind ourselves
a definition of a derivative.

00:00:18.292 --> 00:00:20.405
And there's multiple ways of writing this.

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For the sake of this
video, I'll write it as

00:00:22.548 --> 00:00:25.396
the derivative of our function at point C,

00:00:25.396 --> 00:00:27.919
this is Lagrange notation
with this F prime.

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The derivative of our function
F at C is going to be equal

00:00:31.246 --> 00:00:33.746
to the limit as X approaches Z

00:00:35.474 --> 00:00:37.474
of F of X, minus F of C,

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over X minus C.

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And at first when you see this formula,

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and we've seen it before, it
looks a little bit strange,

00:00:47.978 --> 00:00:51.184
but all it is is it's
calculating the slope,

00:00:51.184 --> 00:00:53.297
this is our change in the
value of our function,

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or you could think of
it as our change in Y,

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if Y is equal to F of X,
and this is our change in X.

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And we're just trying to
see, well, what is that slope

00:01:00.966 --> 00:01:04.536
as X gets closer and closer
to C, as our change in X

00:01:04.536 --> 00:01:06.286
gets closer and closer to zero?

00:01:06.286 --> 00:01:08.734
And we talk about that in other videos.

00:01:08.734 --> 00:01:11.013
So I'm now going to make a
few claims in this video,

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and I'm not going to
prove them rigorously.

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There's another video that
will go a little bit more

00:01:14.953 --> 00:01:16.492
into the proof direction.

00:01:16.492 --> 00:01:18.645
But this is more to get an intuition.

00:01:18.645 --> 00:01:21.573
And so the first claim
that I'm going to make

00:01:21.573 --> 00:01:23.740
is if F is differentiable,

00:01:27.537 --> 00:01:28.704
at X equals C,

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at X equals C,

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then F is continuous

00:01:39.103 --> 00:01:40.270
at X equals C.

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So I'm saying if we know
it's differentiable,

00:01:44.261 --> 00:01:45.845
if we can find this limit,

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if we can find this
derivative at X equals C,

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then our function is also
continuous at X equals C.

00:01:51.814 --> 00:01:53.726
It doesn't necessarily
mean the other way around,

00:01:53.726 --> 00:01:55.280
and actually we'll look
at a case where it's not

00:01:55.280 --> 00:01:56.873
necessarily the case the
other way around that

00:01:56.873 --> 00:01:59.836
if you're continuous, then
you're definitely differentiable.

00:01:59.836 --> 00:02:03.120
But another way to interpret
what I just wrote down is,

00:02:03.120 --> 00:02:05.312
if you are not continuous,
then you definitely

00:02:05.312 --> 00:02:07.178
will not be differentiable.

00:02:07.178 --> 00:02:08.761
If F not continuous

00:02:12.303 --> 00:02:14.053
at X equals C, then F

00:02:18.126 --> 00:02:19.959
is not differentiable,

00:02:23.410 --> 00:02:26.243
differentiable at X is equal to C.

00:02:28.886 --> 00:02:33.312
So let me give a few examples
of a non-continuous function

00:02:33.312 --> 00:02:37.573
and then think about would we
be able to find this limit.

00:02:37.573 --> 00:02:40.946
So the first is where
you have a discontinuity.

00:02:40.946 --> 00:02:44.080
Our function is defined at
C, it's equal to this value,

00:02:44.080 --> 00:02:46.514
but you can see as X
becomes larger than C,

00:02:46.514 --> 00:02:49.395
it just jumps down and
shifts right over here.

00:02:49.395 --> 00:02:52.604
So what would happen if you
were trying to find this limit?

00:02:52.604 --> 00:02:54.925
Well, remember, all this
is is a slope of a line

00:02:54.925 --> 00:02:57.850
between when X is some arbitrary value,

00:02:57.850 --> 00:03:00.289
let's say it's out here,
so that would be X,

00:03:00.289 --> 00:03:03.539
this would be the point X comma F of X,

00:03:04.508 --> 00:03:08.675
and then this is the point C
comma F of C right over here.

00:03:09.871 --> 00:03:12.038
So this is C comma F of C.

00:03:14.542 --> 00:03:18.523
So if you find the left side
of the limit right over here,

00:03:18.523 --> 00:03:21.447
you're essentially saying
okay, let's find this slope.

00:03:21.447 --> 00:03:23.757
And then let me get a little bit closer,

00:03:23.757 --> 00:03:25.703
and let's get X a little bit closer

00:03:25.703 --> 00:03:27.164
and then let's find this slope.

00:03:27.164 --> 00:03:30.127
And then let's get X even closer than that

00:03:30.127 --> 00:03:31.829
and find this slope.

00:03:31.829 --> 00:03:34.422
And in all of those
cases, it would be zero.

00:03:34.422 --> 00:03:35.926
The slope is zero.

00:03:35.926 --> 00:03:38.810
So one way to think about it,
the derivative or this limit

00:03:38.810 --> 00:03:42.629
as we approach from the left,
seems to be approaching zero.

00:03:42.629 --> 00:03:46.049
But what about if we were
to take Xs to the right?

00:03:46.049 --> 00:03:48.211
So instead of our Xs being there,

00:03:48.211 --> 00:03:52.397
what if we were to take
Xs right over here?

00:03:52.397 --> 00:03:55.480
Well, for this point, X comma F of X,

00:03:57.879 --> 00:04:01.700
our slope, if we take F of X minus F of C

00:04:01.700 --> 00:04:05.867
over X minus C, that would
be the slope of this line.

00:04:07.782 --> 00:04:10.948
If we get X to be even closer,
let's say right over here,

00:04:10.948 --> 00:04:14.448
then this would be the slope of this line.

00:04:15.290 --> 00:04:18.134
If we get even closer,
then this expression

00:04:18.134 --> 00:04:21.017
would be the slope of this line.

00:04:21.017 --> 00:04:25.184
And so as we get closer and
closer to X being equal to C,

00:04:26.213 --> 00:04:28.288
we see that our slope is actually

00:04:28.288 --> 00:04:30.158
approaching negative infinity.

00:04:30.158 --> 00:04:32.064
And most importantly, it's approaching

00:04:32.064 --> 00:04:34.382
a very different value from the right.

00:04:34.382 --> 00:04:36.535
This expression is approaching
a very different value

00:04:36.535 --> 00:04:39.868
from the right as it is from the left.

00:04:39.868 --> 00:04:44.664
And so in this case, this
limit up here won't exist.

00:04:44.664 --> 00:04:48.883
So we can clearly say this
is not differentiable.

00:04:48.883 --> 00:04:51.887
So once again, not a proof here.

00:04:51.887 --> 00:04:53.886
I'm just getting an
intuition for if something

00:04:53.886 --> 00:04:57.625
isn't continuous, it's pretty
clear, at least in this case,

00:04:57.625 --> 00:04:58.926
that it's not going to be differentiable.

00:04:58.926 --> 00:05:00.716
Let's look at another case.

00:05:00.716 --> 00:05:04.734
Let's look at a case where we
have what's sometimes called

00:05:04.734 --> 00:05:08.470
a removable discontinuity
or a point discontinuity.

00:05:08.470 --> 00:05:11.683
So once again, let's say we're
approaching from the left.

00:05:11.683 --> 00:05:15.350
This is X, this is the
point X comma F of X.

00:05:17.900 --> 00:05:21.632
Now what's interesting is
where as this expression

00:05:21.632 --> 00:05:25.573
is the slope of the line
connecting X comma F of X

00:05:25.573 --> 00:05:29.392
and C comma F of C, which is
this point, not that point,

00:05:29.392 --> 00:05:30.349
remember we have this removable

00:05:30.349 --> 00:05:32.261
discontinuity right over here,

00:05:32.261 --> 00:05:34.740
and so this would be this
expression is calculating

00:05:34.740 --> 00:05:36.657
the slope of that line.

00:05:38.115 --> 00:05:40.794
And then if X gets even closer
to C, well, then we're gonna

00:05:40.794 --> 00:05:42.507
be calculating the slope of that line.

00:05:42.507 --> 00:05:44.251
If X gets even closer to C,
we're gonna be calculating

00:05:44.251 --> 00:05:45.834
the slope of that line.

00:05:45.834 --> 00:05:49.001
And so as we approach from
the left, as X approaches C

00:05:49.001 --> 00:05:51.314
from the left, we
actually have a situation

00:05:51.314 --> 00:05:53.667
where this expression
right over here is going

00:05:53.667 --> 00:05:55.704
to approach negative infinity.

00:05:55.704 --> 00:05:57.938
And if we approach from
the right, if we approach

00:05:57.938 --> 00:06:02.250
with Xs larger than C, well,
this is our X comma F of X,

00:06:02.250 --> 00:06:05.052
so we have a positive slope
and then as we get closer,

00:06:05.052 --> 00:06:07.077
it gets more positive, more positive

00:06:07.077 --> 00:06:08.866
approaches positive infinity.

00:06:08.866 --> 00:06:11.337
But either way, it's not
approaching a finite value.

00:06:11.337 --> 00:06:13.084
And one side is approaching
positive infinity,

00:06:13.084 --> 00:06:14.864
and the other side is
approaching negative infinity.

00:06:14.864 --> 00:06:19.654
This, the limit of this
expression, is not going to exist.

00:06:19.654 --> 00:06:22.118
So once again, I'm not
doing a rigorous proof here,

00:06:22.118 --> 00:06:24.559
but try to construct a
discontinuous function

00:06:24.559 --> 00:06:26.389
where you will be able to find this.

00:06:26.389 --> 00:06:28.095
It is very, very hard.

00:06:28.095 --> 00:06:29.435
And you might say, well,
what about the situations

00:06:29.435 --> 00:06:32.685
where F is not even defined
at C, which for sure

00:06:32.685 --> 00:06:35.568
you're not gonna be continuous
if F is not defined at C.

00:06:35.568 --> 00:06:38.946
Well if F is not defined
at C, then this part

00:06:38.946 --> 00:06:40.527
of the expression
wouldn't even make sense,

00:06:40.527 --> 00:06:43.208
so you definitely wouldn't
be differentiable.

00:06:43.208 --> 00:06:45.000
But now let's ask another thing.

00:06:45.000 --> 00:06:47.120
I've just given you good
arguments for when you're

00:06:47.120 --> 00:06:50.654
not continuous, you're not
going to be differentiable,

00:06:50.654 --> 00:06:53.659
but can we make another claim
that if you are continuous,

00:06:53.659 --> 00:06:56.536
then you definitely
will be differentiable?

00:06:56.536 --> 00:06:59.420
Well, it turns out that there are for sure

00:06:59.420 --> 00:07:01.896
many functions, an infinite
number of functions,

00:07:01.896 --> 00:07:06.083
that can be continuous at
C, but not differentiable.

00:07:06.083 --> 00:07:09.414
So for example, this could be
an absolute value function.

00:07:09.414 --> 00:07:10.917
It doesn't have to be an
absolute value function,

00:07:10.917 --> 00:07:12.799
but this could be Y is equal to

00:07:12.799 --> 00:07:16.243
the absolute value of X minus C.

00:07:16.243 --> 00:07:20.393
And why is this one not
differentiable at C?

00:07:20.393 --> 00:07:21.693
Well, think about what's happening.

00:07:21.693 --> 00:07:22.585
Think about this expression.

00:07:22.585 --> 00:07:25.596
Remember, this expression
all it's doing is calculating

00:07:25.596 --> 00:07:28.482
the slope between the point X comma F of X

00:07:28.482 --> 00:07:30.675
and the point C comma F of C.

00:07:30.675 --> 00:07:34.842
So if X is, say, out here,
this is X comma F of X,

00:07:35.920 --> 00:07:38.974
it's going to be calculated,
so if we take the limit

00:07:38.974 --> 00:07:42.221
as X approaches C from the left,

00:07:42.221 --> 00:07:44.778
we'll be looking at this slope.

00:07:44.778 --> 00:07:46.937
And as we get closer, we'll
be looking at this slope

00:07:46.937 --> 00:07:48.318
which is actually going to be the same.

00:07:48.318 --> 00:07:50.430
In this case it would be a negative one.

00:07:50.430 --> 00:07:52.463
So as X approaches C from the left,

00:07:52.463 --> 00:07:54.898
this expression would be negative one.

00:07:54.898 --> 00:07:58.362
But as X approaches C from the right,

00:07:58.362 --> 00:08:00.798
this expression is going to be one.

00:08:00.798 --> 00:08:03.191
The slope of the line that
connects these points is one.

00:08:03.191 --> 00:08:05.663
The slope of the line that
connects these points is one.

00:08:05.663 --> 00:08:09.071
So the limit of this expression,
or I would say the value

00:08:09.071 --> 00:08:11.869
of this expression, is
approaching two different values

00:08:11.869 --> 00:08:14.348
as X approaches C from
the left or the right.

00:08:14.348 --> 00:08:16.501
From the left, it's
approaching negative one,

00:08:16.501 --> 00:08:19.065
or it's constantly negative
one and so it's approaching

00:08:19.065 --> 00:08:20.527
negative one, you could say.

00:08:20.527 --> 00:08:22.679
And from the right, it's one,

00:08:22.679 --> 00:08:25.273
and it's approaching one the entire time.

00:08:25.273 --> 00:08:27.915
And so we know if you're
approaching two different values

00:08:27.915 --> 00:08:30.963
from on the left side or
the right side of the limit,

00:08:30.963 --> 00:08:32.999
then this limit will not exist.

00:08:32.999 --> 00:08:36.416
So here, this is not, not differentiable.

00:08:37.299 --> 00:08:39.654
And even intuitively, we
think of the derivative

00:08:39.654 --> 00:08:42.331
as the slope of the tangent line.

00:08:42.331 --> 00:08:44.129
And you could actually
draw an infinite number

00:08:44.129 --> 00:08:45.390
of tangent lines here.

00:08:45.390 --> 00:08:46.835
That's one way to think about it.

00:08:46.835 --> 00:08:49.795
You could say, well, maybe
this is the tangent line

00:08:49.795 --> 00:08:51.908
right over there, but
then why can't I make

00:08:51.908 --> 00:08:55.681
something like this the tangent line?

00:08:55.681 --> 00:09:00.073
That only intersects at
the point C comma zero.

00:09:00.073 --> 00:09:01.819
And then you could keep
doing things like that.

00:09:01.819 --> 00:09:03.643
Why can't that be the tangent line?

00:09:03.643 --> 00:09:05.673
And you could go on and on and on.

00:09:05.673 --> 00:09:08.687
So the big takeaways here,
at least intuitively,

00:09:08.687 --> 00:09:10.962
in a future video I'm
going to prove to you

00:09:10.962 --> 00:09:15.193
that if F is differentiable at
C that it is continuous at C,

00:09:15.193 --> 00:09:16.950
which can also be interpreted
as that if you're not

00:09:16.950 --> 00:09:19.913
continuous at C, then you're
not gonna be differentiable.

00:09:19.913 --> 00:09:21.377
These two examples will hopefully give you

00:09:21.377 --> 00:09:22.923
some intuition for that.

00:09:22.923 --> 00:09:25.519
But it's not the case that
if something is continuous

00:09:25.519 --> 00:09:27.306
that it has to be differentiable.

00:09:27.306 --> 00:09:29.673
It oftentimes will be
differentiable, but it doesn't have

00:09:29.673 --> 00:09:32.066
to be differentiable, and
this absolute value function

00:09:32.066 --> 00:09:34.997
is an example of a
continuous function at C,

00:09:34.997 --> 00:09:37.174
but it is not differentiable at C.


Estimating derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.535 --> 00:00:02.301
- [Instructor] So we're
told that this table

00:00:02.301 --> 00:00:05.837
gives select values of the
differentiable function F.

00:00:05.837 --> 00:00:07.929
So it gives us the value of the function

00:00:07.929 --> 00:00:11.082
at a few values for X, in particular,

00:00:11.082 --> 00:00:12.950
five different values for X and

00:00:12.950 --> 00:00:15.813
it tells us what the
corresponding f(x) is.

00:00:15.813 --> 00:00:17.921
And they say, what is the best estimate

00:00:17.921 --> 00:00:19.762
for f'(4)?

00:00:19.762 --> 00:00:22.158
So this is the derivative
of our function F

00:00:22.158 --> 00:00:24.094
when X is equal to four.

00:00:24.094 --> 00:00:25.809
Or another way to think about it,

00:00:25.809 --> 00:00:27.402
what is the slope of the tangent line

00:00:27.402 --> 00:00:29.960
when X is equal to four for f(x)?

00:00:29.960 --> 00:00:32.454
So what is the best
estimate for f'(4)

00:00:32.454 --> 00:00:35.564
we can make based on this table?

00:00:35.564 --> 00:00:37.551
So, let's just visualize what's going on

00:00:37.551 --> 00:00:39.782
before we even look at the choices.

00:00:39.782 --> 00:00:42.192
So let me draw some axes here.

00:00:42.192 --> 00:00:44.062
And let me plot these points.

00:00:44.062 --> 00:00:46.851
We know that these would
sit on the curve of Y

00:00:46.851 --> 00:00:48.564
is equal to f(x).

00:00:48.564 --> 00:00:50.573
When X is zero, f(x) is 72.

00:00:50.573 --> 00:00:52.533
So this is the point (0,72).

00:00:52.533 --> 00:00:54.450
This is the point (3,95)

00:00:55.539 --> 00:00:58.307
clearly, two different
scales on the X and Y axes.

00:00:58.307 --> 00:01:00.011
This is the point

00:01:00.011 --> 00:01:00.844
(5,112).

00:01:01.904 --> 00:01:02.904
This is (6,77).

00:01:03.866 --> 00:01:05.808
This is (9,54), actually, let me

00:01:05.808 --> 00:01:06.831
write out

00:01:06.831 --> 00:01:07.664
the

00:01:08.607 --> 00:01:10.784
this is one, two, three,

00:01:10.784 --> 00:01:12.463
four, five, six,

00:01:12.463 --> 00:01:14.315
seven, eight, nine,

00:01:14.315 --> 00:01:15.148
and ten.

00:01:16.224 --> 00:01:18.141
Now they want us to know, they want to

00:01:18.141 --> 00:01:19.527
what is the derivative of our function

00:01:19.527 --> 00:01:20.400
when F is equal to four?

00:01:20.400 --> 00:01:21.403
Well, they haven't told us

00:01:21.403 --> 00:01:23.052
even what the value of F is at four.

00:01:23.052 --> 00:01:24.878
We don't know what that point is.

00:01:24.878 --> 00:01:26.166
But what they're trying to do is,

00:01:26.166 --> 00:01:27.838
well, we're trying to
make a best estimate.

00:01:27.838 --> 00:01:30.247
And using these points, we don't even know

00:01:30.247 --> 00:01:32.137
exactly what the curve looks like.

00:01:32.137 --> 00:01:33.820
It could look like all sorts of things.

00:01:33.820 --> 00:01:37.028
We could try to fit a
reasonably smooth curve.

00:01:37.028 --> 00:01:39.261
The curve might look something like that.

00:01:39.261 --> 00:01:40.906
But it might be wackier.

00:01:40.906 --> 00:01:44.145
It might do something like this.

00:01:44.145 --> 00:01:45.516
Let me try to do it.

00:01:45.516 --> 00:01:48.131
It might look something

00:01:48.131 --> 00:01:48.964
like this.

00:01:50.282 --> 00:01:51.724
So we don't know, for sure.

00:01:51.724 --> 00:01:52.975
All we know is that

00:01:52.975 --> 00:01:55.470
it needs to go through those points.

00:01:55.470 --> 00:01:57.570
'Cause they've just
sampled to the function

00:01:57.570 --> 00:01:58.994
at those points.

00:01:58.994 --> 00:02:00.573
But let's just, for the sake of

00:02:00.573 --> 00:02:02.856
this exercise, let's assume the simplest,

00:02:02.856 --> 00:02:04.387
let's say it's a nice smooth

00:02:04.387 --> 00:02:06.425
curve without too many twists and turns

00:02:06.425 --> 00:02:09.103
that goes through these
points, just like that.

00:02:09.103 --> 00:02:11.489
So what they're asking, okay,

00:02:11.489 --> 00:02:13.513
when X is equal to four

00:02:13.513 --> 00:02:17.027
if this yellow curve were the actual curve

00:02:17.027 --> 00:02:19.996
then what is the slope of the
tangent line, at that point?

00:02:19.996 --> 00:02:21.692
So we would be visualizing that.

00:02:21.692 --> 00:02:22.862
Now to be clear,

00:02:22.862 --> 00:02:24.654
this tangent line that I just drew

00:02:24.654 --> 00:02:25.568
this would be for

00:02:25.568 --> 00:02:27.589
this version of our function that I did

00:02:27.589 --> 00:02:28.721
connecting these points.

00:02:28.721 --> 00:02:30.627
That does not have to be

00:02:30.627 --> 00:02:31.677
the actual function.

00:02:31.677 --> 00:02:33.409
We know that the actual function has to go

00:02:33.409 --> 00:02:34.357
through those points.

00:02:34.357 --> 00:02:36.627
But I'm just doing this
for visualization purposes.

00:02:36.627 --> 00:02:39.069
One of the whole ideas here is

00:02:39.069 --> 00:02:40.917
that all we do have is the sample

00:02:40.917 --> 00:02:42.505
and we're trying to get a best estimate.

00:02:42.505 --> 00:02:43.387
We don't know

00:02:43.387 --> 00:02:44.983
if it's even gonna be a good estimate.

00:02:44.983 --> 00:02:47.178
It's just going to be a best estimate.

00:02:47.178 --> 00:02:48.664
So what we generally do

00:02:48.664 --> 00:02:50.606
when we just have some data

00:02:50.606 --> 00:02:52.861
around a point, is,
let's use a data points

00:02:52.861 --> 00:02:54.225
that are closest

00:02:54.225 --> 00:02:55.372
to that point

00:02:55.372 --> 00:02:58.061
and find slopes of secant lines

00:02:58.061 --> 00:03:00.046
pretty close around that point.

00:03:00.046 --> 00:03:02.543
And that's going to give
us our best estimate

00:03:02.543 --> 00:03:04.813
for the slope of the tangent line.

00:03:04.813 --> 00:03:07.048
So what points do we have near

00:03:07.048 --> 00:03:09.227
F of four, or near the point

00:03:09.227 --> 00:03:10.772
(4,f(4))?

00:03:10.772 --> 00:03:13.151
Well, they give us what F is equal to

00:03:13.151 --> 00:03:14.543
when X is equal to three.

00:03:14.543 --> 00:03:16.153
They give us this point, right over here.

00:03:16.153 --> 00:03:18.027
Let me do this in another color.

00:03:18.027 --> 00:03:19.527
So, (3,95)

00:03:20.796 --> 00:03:22.775
that is that right over there.

00:03:22.775 --> 00:03:24.641
And they also give us

00:03:24.641 --> 00:03:26.717
(5,112).

00:03:26.717 --> 00:03:28.680
That is that point right over there.

00:03:28.680 --> 00:03:30.128
And so what we could do

00:03:30.128 --> 00:03:31.681
we could say, well, what is the average

00:03:31.681 --> 00:03:33.643
rate of change between these two points?

00:03:33.643 --> 00:03:35.066
Another way to think about it is,

00:03:35.066 --> 00:03:37.309
what is the slope of the secant line

00:03:37.309 --> 00:03:38.898
between those two points.

00:03:38.898 --> 00:03:41.042
And, that would be our best estimate

00:03:41.042 --> 00:03:43.357
for the slope of the tangent line

00:03:43.357 --> 00:03:45.397
at X equals four.

00:03:45.397 --> 00:03:46.844
Do we know that it's a good estimate?

00:03:46.844 --> 00:03:48.187
Do we know that it's even close?

00:03:48.187 --> 00:03:49.577
No, we don't know for sure, but that would

00:03:49.577 --> 00:03:50.500
be the best estimate.

00:03:50.500 --> 00:03:52.573
It would be better than trying to take the

00:03:52.573 --> 00:03:54.372
the average rate of change between

00:03:54.372 --> 00:03:55.528
When X equals three

00:03:55.528 --> 00:03:56.405
and X equals six.

00:03:56.405 --> 00:03:58.113
Or between when X equals zero

00:03:58.113 --> 00:03:59.287
and X equals nine.

00:03:59.287 --> 00:04:01.835
These are pretty close around four.

00:04:01.835 --> 00:04:03.189
And so, let's do that.

00:04:03.189 --> 00:04:04.922
Let's find the average rate of change

00:04:04.922 --> 00:04:07.687
between when X goes from three to five.

00:04:07.687 --> 00:04:09.456
So, we can see here

00:04:09.456 --> 00:04:10.789
our change in X.

00:04:12.107 --> 00:04:13.749
Let me do this in a new color.

00:04:13.749 --> 00:04:15.658
So our change in X here

00:04:15.658 --> 00:04:17.163
is equal to

00:04:17.163 --> 00:04:18.037
plus two.

00:04:18.037 --> 00:04:19.682
And I can draw that out.

00:04:19.682 --> 00:04:21.454
My change in X here

00:04:21.454 --> 00:04:22.287
is

00:04:23.339 --> 00:04:24.956
plus two.

00:04:24.956 --> 00:04:27.021
And, my change in Y

00:04:27.021 --> 00:04:28.810
is going to be

00:04:28.810 --> 00:04:32.196
when my X increased by
two, my change in Y is

00:04:32.196 --> 00:04:34.278
plus, let's see, this is

00:04:34.278 --> 00:04:36.028
if I add ten I get to

00:04:37.484 --> 00:04:39.177
I get to 105.

00:04:39.177 --> 00:04:40.366
If I add another seven

00:04:40.366 --> 00:04:42.398
so this is plus 17.

00:04:42.398 --> 00:04:45.953
So this is plus 17 right over here.

00:04:45.953 --> 00:04:47.416
Plus 17.

00:04:47.416 --> 00:04:50.164
And so my change in Y over change in X.

00:04:50.164 --> 00:04:51.325
Change in Y

00:04:51.325 --> 00:04:52.992
over my change in X.

00:04:53.893 --> 00:04:56.289
For this secant line between

00:04:56.289 --> 00:04:57.918
when X is equaling three

00:04:57.918 --> 00:04:59.874
and X is equaling five

00:04:59.874 --> 00:05:02.943
is going to be equal to 17 over two.

00:05:02.943 --> 00:05:04.110
Seventeen over

00:05:05.984 --> 00:05:06.884
two.

00:05:06.884 --> 00:05:09.102
Which is equal to 8.5.

00:05:09.102 --> 00:05:12.086
So the slope of this
green line here is 8.5.

00:05:12.086 --> 00:05:13.545
And that would be our best estimate

00:05:13.545 --> 00:05:16.452
for the slope of the tangent
line when X equals four

00:05:16.452 --> 00:05:19.388
of the curve Y is equal to f(x).

00:05:19.388 --> 00:05:20.618
And so, lucky for us

00:05:20.618 --> 00:05:22.164
the people who wrote this question had

00:05:22.164 --> 00:05:24.888
the exact same logic, and
they did it right over there.

00:05:24.888 --> 00:05:27.098
So you wouldn't have to
graph it the way I did.

00:05:27.098 --> 00:05:29.839
I did it just to help us
visualize what's going on.

00:05:29.839 --> 00:05:32.014
In general, when you
see a question like this

00:05:32.014 --> 00:05:33.078
they're really saying, look,

00:05:33.078 --> 00:05:34.606
you don't have all the data you need

00:05:34.606 --> 00:05:37.282
to figure out exactly
what f'(4) is.

00:05:37.282 --> 00:05:40.200
But if you can find
points close to, or around

00:05:40.200 --> 00:05:41.301
f'(4)

00:05:41.301 --> 00:05:43.325
and find the secant
line, the average rate

00:05:43.325 --> 00:05:44.909
of the slope of the secant line.

00:05:44.909 --> 00:05:47.295
Or the average rate of
change between those points

00:05:47.295 --> 00:05:49.130
that's going to be our best estimate

00:05:49.130 --> 00:05:51.524
for the instantaneous rate of change

00:05:51.524 --> 00:05:52.819
when X equals four.

00:05:52.819 --> 00:05:55.494
Or the derivative when X equals four.


Secant lines & average rate of change

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.016
- [Instructor] So right over here

00:00:01.016 --> 00:00:03.623
we have the graph of y
is equal to x squared,

00:00:03.623 --> 00:00:06.496
or at least part of the graph
of y is equal to x squared.

00:00:06.496 --> 00:00:08.325
And the first thing I'd like to tackle is

00:00:08.325 --> 00:00:10.349
think about the average rate of change

00:00:10.349 --> 00:00:12.334
of y with respect to x

00:00:12.334 --> 00:00:16.505
over the interval from x
equaling 1 to x equaling 3.

00:00:16.505 --> 00:00:18.168
So let me write that down.

00:00:18.168 --> 00:00:21.668
We want to know the average rate of change

00:00:24.011 --> 00:00:26.545
of y with respect to x

00:00:26.545 --> 00:00:30.897
over the interval from
x going from 1 to 3.

00:00:30.897 --> 00:00:32.039
And that's a closed interval,

00:00:32.039 --> 00:00:35.299
where x could be 1, and
x could be equal to 3.

00:00:35.299 --> 00:00:38.147
Well we could do this even
without looking at the graph.

00:00:38.147 --> 00:00:40.275
If I were to just make a table here,

00:00:40.275 --> 00:00:44.442
where, if this is x, and this
is y is equal to x squared,

00:00:45.290 --> 00:00:49.004
when x is equal to 1, y
is equal to 1 squared,

00:00:49.004 --> 00:00:49.876
which is just 1.

00:00:49.876 --> 00:00:52.168
You see that right over there.

00:00:52.168 --> 00:00:55.068
And when x is equal to 3,

00:00:55.068 --> 00:00:58.498
y is equal to 3 squared,
which is equal to 9.

00:00:58.498 --> 00:01:01.976
And so you can see when x is equal to 3,

00:01:01.976 --> 00:01:03.559
y is equal to 9.

00:01:04.636 --> 00:01:06.743
And to figure out the
average rate of change

00:01:06.743 --> 00:01:08.476
of y with respect to x,

00:01:08.476 --> 00:01:10.610
you say, "Okay, well
what's my change in x?"

00:01:10.610 --> 00:01:11.811
Well, we could see very clearly

00:01:11.811 --> 00:01:14.171
that our change in x over this interval

00:01:14.171 --> 00:01:16.409
is equal to positive 2.

00:01:16.409 --> 00:01:19.909
Well, what's our change in
y over the same interval?

00:01:19.909 --> 00:01:22.678
Our change in y is equal to ...

00:01:22.678 --> 00:01:25.599
When x increased by 2 from 1 to 3,

00:01:25.599 --> 00:01:29.252
y increases by 8, so it's
going to be a positive 8.

00:01:29.252 --> 00:01:31.080
So what is our average rate of change?

00:01:31.080 --> 00:01:33.670
Well, it's going to be our change in y,

00:01:33.670 --> 00:01:35.503
or our change in x,

00:01:37.502 --> 00:01:41.669
which is equal to 8 over
2, which is equal to 4.

00:01:46.727 --> 00:01:49.383
So that would be our
average rate of change.

00:01:49.383 --> 00:01:52.013
Over that interval, on average,

00:01:52.013 --> 00:01:56.180
every time x increases by
1, y is increasing by 4.

00:01:57.515 --> 00:01:59.257
And how did we calculate that?

00:01:59.257 --> 00:02:01.129
We looked at our change in x,

00:02:01.129 --> 00:02:02.757
let me draw that here ...

00:02:02.757 --> 00:02:05.508
We looked at our change in x,

00:02:05.508 --> 00:02:08.258
and we looked at our change in y,

00:02:09.548 --> 00:02:12.700
which would be this right over here,

00:02:12.700 --> 00:02:15.313
and we calculated change
in y over change of x

00:02:15.313 --> 00:02:17.533
for average rate of change.

00:02:17.533 --> 00:02:20.084
Now this might be looking
fairly familiar to you,

00:02:20.084 --> 00:02:21.155
because you're used to thinking about

00:02:21.155 --> 00:02:23.419
change in y over change in x

00:02:23.419 --> 00:02:27.651
as the slope of a line
connecting two points.

00:02:27.651 --> 00:02:30.499
And that's indeed what we did calculate.

00:02:30.499 --> 00:02:34.666
If you were to draw a secant
line between these two points,

00:02:35.810 --> 00:02:37.560
we essentially just calculated

00:02:37.560 --> 00:02:40.190
the slope of that secant line.

00:02:40.190 --> 00:02:44.317
And so the average rate of
change between two points,

00:02:44.317 --> 00:02:48.484
that is the same thing as
the slope of the secant line.

00:02:49.385 --> 00:02:51.266
And by looking at the secant line,

00:02:51.266 --> 00:02:54.148
in comparison to the
curve over that interval,

00:02:54.148 --> 00:02:55.898
it hopefully gives you a visual intuition

00:02:55.898 --> 00:02:58.719
for what even average
rate of change means.

00:02:58.719 --> 00:03:01.105
Because in the beginning
part of the interval,

00:03:01.105 --> 00:03:02.464
you see that the secant line

00:03:02.464 --> 00:03:04.693
is actually increasing at a faster rate,

00:03:04.693 --> 00:03:06.696
but then as we get closer to 3,

00:03:06.696 --> 00:03:08.367
it looks like our yellow curve

00:03:08.367 --> 00:03:10.840
is increasing at a faster
rate than the secant line,

00:03:10.840 --> 00:03:12.852
and then they eventually catch up.

00:03:12.852 --> 00:03:14.872
And so that's why the
slope of the secant line

00:03:14.872 --> 00:03:16.605
is the average rate of change.

00:03:16.605 --> 00:03:18.276
Is it the exact rate of
change at every point?

00:03:18.276 --> 00:03:19.600
Absolutely not.

00:03:19.600 --> 00:03:22.447
The curve's rate of change
is constantly changing.

00:03:22.447 --> 00:03:23.910
It's at a slower rate of change

00:03:23.910 --> 00:03:25.512
in the beginning part of this interval,

00:03:25.512 --> 00:03:28.307
and then it's actually
increasing at a higher rate

00:03:28.307 --> 00:03:30.177
as we get closer and closer to three.

00:03:30.177 --> 00:03:31.910
So over the interval,

00:03:31.910 --> 00:03:36.577
the change in y over the change
in x is exactly the same.

00:03:36.577 --> 00:03:39.085
Now one question you might be wondering is

00:03:39.085 --> 00:03:41.462
why are you learning this
is in a calculus class?

00:03:41.462 --> 00:03:44.179
Couldn't you have learned
this in an algebra class?

00:03:44.179 --> 00:03:45.564
The answer is yes.

00:03:45.564 --> 00:03:47.009
But what's going to be interesting,

00:03:47.009 --> 00:03:50.818
and is really one of the
foundational ideas of calculus is

00:03:50.818 --> 00:03:52.620
well what happens as these points

00:03:52.620 --> 00:03:54.501
get closer and closer together?

00:03:54.501 --> 00:03:57.653
We found the average rate
of change between 1 and 3,

00:03:57.653 --> 00:04:01.820
or the slope of the secant
line from (1, 1) to (3, 9).

00:04:03.352 --> 00:04:06.487
But what instead if you found
the slope of the secant line

00:04:06.487 --> 00:04:10.570
between (2, 4) and (3, 9)?

00:04:10.570 --> 00:04:13.113
So what if you found this slope?

00:04:13.113 --> 00:04:15.055
But what if you wanted to get even closer?

00:04:15.055 --> 00:04:17.519
Let's say you wanted to find
the slope of the secant line

00:04:17.519 --> 00:04:18.936
between the point

00:04:20.434 --> 00:04:22.851
(2.5, 6.25)

00:04:24.230 --> 00:04:26.233
and (3, 9)?

00:04:26.233 --> 00:04:27.574
And what if you just kept getting

00:04:27.574 --> 00:04:29.838
closer and closer and closer?

00:04:29.838 --> 00:04:32.398
Well then, the slopes
of these secant lines

00:04:32.398 --> 00:04:34.078
are going to get closer and closer

00:04:34.078 --> 00:04:38.589
to the slope of the
tangent line at x equals 3.

00:04:38.589 --> 00:04:41.506
And if we can figure out the
slope of the tangent line,

00:04:41.506 --> 00:04:43.012
well then we're in business.

00:04:43.012 --> 00:04:45.651
Because then we're not talking
about average rate of change,

00:04:45.651 --> 00:04:46.678
we're going to be talking about

00:04:46.678 --> 00:04:48.933
instantaneous rate of change,

00:04:48.933 --> 00:04:50.762
which is one of the central ideas,

00:04:50.762 --> 00:04:52.495
that is the derivative,

00:04:52.495 --> 00:04:54.045
and we're going to get there soon.

00:04:54.045 --> 00:04:55.856
But it's really important to appreciate

00:04:55.856 --> 00:04:58.468
that average rate of
change between two points

00:04:58.468 --> 00:05:00.802
is the same thing as the
slope of the secant line.

00:05:00.802 --> 00:05:03.283
And as those points get
closer and closer together,

00:05:03.283 --> 00:05:05.216
and as the secant line is connecting

00:05:05.216 --> 00:05:07.384
two closer and closer points together,

00:05:07.384 --> 00:05:10.005
that distance between the points,

00:05:10.005 --> 00:05:12.618
between the x values of
the points approach 0,

00:05:12.618 --> 00:05:16.043
very interesting things
are going to happen.


Derivative as a concept

WEBVTT
Kind: captions
Language: en

00:00:00.111 --> 00:00:02.793
- [Instructor] You are likely
already familiar with the idea

00:00:02.793 --> 00:00:04.355
of a slope of a line.

00:00:04.355 --> 00:00:07.667
If you're not, I encourage you
to review it on Khan Academy,

00:00:07.667 --> 00:00:10.481
but all it is, it's
describing the rate of change

00:00:10.481 --> 00:00:11.661
of a vertical variable

00:00:11.661 --> 00:00:14.137
with respect to a horizontal variable,

00:00:14.137 --> 00:00:17.324
so for example, here I
have our classic y axis

00:00:17.324 --> 00:00:19.132
in the vertical direction and x axis

00:00:19.132 --> 00:00:20.732
in the horizontal direction,

00:00:20.732 --> 00:00:23.184
and if I wanted to figure
out the slope of this line,

00:00:23.184 --> 00:00:24.876
I could pick two points,

00:00:24.876 --> 00:00:27.496
say that point and that point.

00:00:27.496 --> 00:00:29.368
I could say, "Okay, from
this point to this point,

00:00:29.368 --> 00:00:31.036
what is my change in x?"

00:00:31.036 --> 00:00:34.868
Well, my change in x would be
this distance right over here,

00:00:34.868 --> 00:00:35.868
change in x,

00:00:36.753 --> 00:00:39.162
the Greek letter delta,
this triangle here.

00:00:39.162 --> 00:00:42.858
It's just shorthand for
"change," so change in x,

00:00:42.858 --> 00:00:46.687
and I could also
calculate the change in y,

00:00:46.687 --> 00:00:50.718
so this point going up to
that point, our change in y,

00:00:50.718 --> 00:00:54.718
would be this, right over
here, our change in y,

00:00:56.061 --> 00:01:00.337
and then, we would define
slope, or we have defined slope

00:01:00.337 --> 00:01:03.143
as change in y over change in x,

00:01:03.143 --> 00:01:06.444
so slope is equal to the rate of change

00:01:06.444 --> 00:01:08.243
of our vertical variable

00:01:08.243 --> 00:01:10.955
over the rate of change of
our horizontal variable,

00:01:10.955 --> 00:01:14.038
sometimes described as rise over run,

00:01:15.066 --> 00:01:18.765
and for any line, it's
associated with a slope

00:01:18.765 --> 00:01:22.041
because it has a constant rate of change.

00:01:22.041 --> 00:01:24.632
If you took any two points on this line,

00:01:24.632 --> 00:01:27.582
no matter how far apart or
no matter how close together,

00:01:27.582 --> 00:01:29.073
anywhere they sit on the line,

00:01:29.073 --> 00:01:31.216
if you were to do this calculation,

00:01:31.216 --> 00:01:33.622
you would get the same slope.

00:01:33.622 --> 00:01:36.057
That's what makes it a line,

00:01:36.057 --> 00:01:37.086
but what's fascinating

00:01:37.086 --> 00:01:40.055
about calculus is we're
going to build the tools

00:01:40.055 --> 00:01:43.152
so that we can think about
the rate of change not just

00:01:43.152 --> 00:01:45.784
of a line, which we've
called "slope" in the past,

00:01:45.784 --> 00:01:47.637
we can think about the rate of change,

00:01:47.637 --> 00:01:52.058
the instantaneous rate
of change of a curve,

00:01:52.058 --> 00:01:53.364
of something whose rate

00:01:53.364 --> 00:01:56.525
of change is possibly constantly changing.

00:01:56.525 --> 00:02:00.692
So for example, here's a curve
where the rate of change of y

00:02:01.731 --> 00:02:05.322
with respect to x is constantly changing,

00:02:05.322 --> 00:02:08.228
even if we wanted to use
our traditional tools.

00:02:08.228 --> 00:02:10.490
If we said, "Okay, we can
calculate the average rate

00:02:10.490 --> 00:02:14.686
of change," let's say between
this point and this point.

00:02:14.686 --> 00:02:15.867
Well, what would it be?

00:02:15.867 --> 00:02:18.224
Well, the average rate of
change between this point and

00:02:18.224 --> 00:02:19.729
this point would be the slope of the line

00:02:19.729 --> 00:02:21.189
that connects them,

00:02:21.189 --> 00:02:25.131
so it would be the slope of
this line of the secant line,

00:02:25.131 --> 00:02:26.378
but if we picked two different points,

00:02:26.378 --> 00:02:28.535
we pick this point and this point,

00:02:28.535 --> 00:02:29.428
the average rate of change

00:02:29.428 --> 00:02:32.319
between those points all of a
sudden looks quite different.

00:02:32.319 --> 00:02:34.935
It looks like it has a higher slope.

00:02:34.935 --> 00:02:38.424
So even when we take the
slopes between two points

00:02:38.424 --> 00:02:40.761
on the line, the secant lines,

00:02:40.761 --> 00:02:43.578
you can see that those
slopes are changing,

00:02:43.578 --> 00:02:45.435
but what if we wanted to ask ourselves

00:02:45.435 --> 00:02:47.768
an even more interesting question.

00:02:47.768 --> 00:02:52.184
What is the instantaneous
rate of change at a point?

00:02:52.184 --> 00:02:55.941
So for example, how fast is y changing

00:02:55.941 --> 00:02:58.584
with respect to x exactly at that point,

00:02:58.584 --> 00:03:01.711
exactly when x is equal to that value.

00:03:01.711 --> 00:03:03.781
Let's call it x one.

00:03:03.781 --> 00:03:05.908
Well, one way you could think about it is

00:03:05.908 --> 00:03:09.389
what if we could draw a
tangent line to this point,

00:03:09.389 --> 00:03:12.405
a line that just touches
the graph right over there,

00:03:12.405 --> 00:03:15.162
and we can calculate
the slope of that line?

00:03:15.162 --> 00:03:19.069
Well, that should be the
rate of change at that point,

00:03:19.069 --> 00:03:21.445
the instantaneous rate of change.

00:03:21.445 --> 00:03:22.645
So in this case,

00:03:22.645 --> 00:03:27.113
the tangent line might
look something like that.

00:03:27.113 --> 00:03:30.354
If we know the slope of this,

00:03:30.354 --> 00:03:31.187
well then we could say that

00:03:31.187 --> 00:03:34.512
that's the instantaneous
rate of change at that point.

00:03:34.512 --> 00:03:37.601
Why do I say instantaneous rate of change?

00:03:37.601 --> 00:03:40.204
Well, think about the
video on these sprinters,

00:03:40.204 --> 00:03:42.014
Usain Bolt example.

00:03:42.014 --> 00:03:44.589
If we wanted to figure out
the speed of Usain Bolt

00:03:44.589 --> 00:03:48.950
at a given instant, well maybe
this describes his position

00:03:48.950 --> 00:03:52.776
with respect to time if y
was position and x is time.

00:03:52.776 --> 00:03:55.547
Usually, you would see t as
time, but let's say x is time,

00:03:55.547 --> 00:03:58.683
so then, if were talking
about right at this time,

00:03:58.683 --> 00:04:02.129
we're talking about
the instantaneous rate,

00:04:02.129 --> 00:04:06.606
and this idea is the central
idea of differential calculus,

00:04:06.606 --> 00:04:09.289
and it's known as a derivative,

00:04:09.289 --> 00:04:13.017
the slope of the tangent line,
which you could also view

00:04:13.017 --> 00:04:15.603
as the instantaneous rate of change.

00:04:15.603 --> 00:04:17.043
I'm putting an exclamation mark

00:04:17.043 --> 00:04:20.160
because it's so
conceptually important here.

00:04:20.160 --> 00:04:23.368
So how can we denote a derivative?

00:04:23.368 --> 00:04:26.240
One way is known as Leibniz's notation,

00:04:26.240 --> 00:04:28.713
and Leibniz is one of
the fathers of calculus

00:04:28.713 --> 00:04:30.371
along with Isaac Newton,

00:04:30.371 --> 00:04:33.078
and his notation, you
would denote the slope

00:04:33.078 --> 00:04:34.463
of the tangent line

00:04:34.463 --> 00:04:36.380
as equaling dy over dx.

00:04:39.993 --> 00:04:42.139
Now why do I like this notation?

00:04:42.139 --> 00:04:45.375
Because it really comes
from this idea of a slope,

00:04:45.375 --> 00:04:47.993
which is change in y over change in x.

00:04:47.993 --> 00:04:50.087
As you'll see in future videos,

00:04:50.087 --> 00:04:51.558
one way to think about the slope

00:04:51.558 --> 00:04:53.372
of the tangent line is, well,

00:04:53.372 --> 00:04:55.722
let's calculate the slope of secant lines.

00:04:55.722 --> 00:04:57.839
Let's say between that
point and that point,

00:04:57.839 --> 00:04:58.722
but then let's get even closer,

00:04:58.722 --> 00:05:00.011
say that point and that point,

00:05:00.011 --> 00:05:00.844
and then let's get even closer

00:05:00.844 --> 00:05:01.916
and that point and that point,

00:05:01.916 --> 00:05:03.209
and then let's get even closer,

00:05:03.209 --> 00:05:06.063
and let's see what happens as the change

00:05:06.063 --> 00:05:08.063
in x approaches zero,

00:05:09.346 --> 00:05:12.124
and so using these d's instead of deltas,

00:05:12.124 --> 00:05:14.004
this was Leibniz's way of saying,

00:05:14.004 --> 00:05:16.590
"Hey, what happens if my changes

00:05:16.590 --> 00:05:20.023
in, say, x become close to zero?"

00:05:20.023 --> 00:05:20.856
So this idea,

00:05:20.856 --> 00:05:23.280
this is known as sometimes
differential notation,

00:05:23.280 --> 00:05:25.908
Leibniz's notation, is
instead of just change

00:05:25.908 --> 00:05:29.926
in y over change in x,
super small changes in y

00:05:29.926 --> 00:05:32.630
for a super small change in x,

00:05:32.630 --> 00:05:36.213
especially as the change
in x approaches zero,

00:05:36.213 --> 00:05:37.046
and as you will see,

00:05:37.046 --> 00:05:39.679
that is how we will
calculate the derivative.

00:05:39.679 --> 00:05:42.366
Now, there's other notations.

00:05:42.366 --> 00:05:46.533
If this curve is described
as y is equal to f of x.

00:05:47.694 --> 00:05:49.102
The slope of the tangent line

00:05:49.102 --> 00:05:51.066
at that point could be denoted

00:05:51.066 --> 00:05:53.733
as equaling f prime of x one.

00:05:57.073 --> 00:05:59.703
So this notation takes a little
bit of time getting used to,

00:05:59.703 --> 00:06:01.429
the Lagrange notation.

00:06:01.429 --> 00:06:05.068
It's saying f prime is
representing the derivative.

00:06:05.068 --> 00:06:07.815
It's telling us the
slope of the tangent line

00:06:07.815 --> 00:06:09.467
for a given point,

00:06:09.467 --> 00:06:13.458
so if you input an x into
this function into f,

00:06:13.458 --> 00:06:16.309
you're getting the corresponding y value.

00:06:16.309 --> 00:06:19.881
If you input an x into f prime,

00:06:19.881 --> 00:06:24.048
you're getting the slope of
the tangent line at that point.

00:06:25.017 --> 00:06:27.919
Now, another notation that
you'll see less likely

00:06:27.919 --> 00:06:31.452
in a calculus class but you
might see in a physics class

00:06:31.452 --> 00:06:35.173
is the notation y with a dot over it,

00:06:35.173 --> 00:06:38.719
so you could write this
is y with a dot over it,

00:06:38.719 --> 00:06:40.835
which also denotes the derivative.

00:06:40.835 --> 00:06:43.053
You might also see y prime.

00:06:43.053 --> 00:06:46.261
This would be more common in a math class.

00:06:46.261 --> 00:06:49.489
Now as we march forward
in our calculus adventure,

00:06:49.489 --> 00:06:52.940
we will build the tools to
actually calculate these things,

00:06:52.940 --> 00:06:54.538
and if you're already
familiar with limits,

00:06:54.538 --> 00:06:56.462
they will be very useful,
as you could imagine,

00:06:56.462 --> 00:06:58.196
'cause we're really going
to be taking the limit

00:06:58.196 --> 00:07:02.020
of our change in y over
change in x as our change

00:07:02.020 --> 00:07:04.020
in x approaches zero,

00:07:04.981 --> 00:07:06.346
and we're not just going
to be able to figure it out

00:07:06.346 --> 00:07:07.476
for a point.

00:07:07.476 --> 00:07:09.730
We're going to be able to
figure out general equations

00:07:09.730 --> 00:07:12.770
that described the derivative
for any given point,

00:07:12.770 --> 00:07:14.853
so be very, very excited.


Limits of trigonometric functions

WEBVTT
Kind: captions
Language: en

00:00:00.454 --> 00:00:02.078
- [Instructor] What we're
going to do in this video

00:00:02.078 --> 00:00:04.932
is think about limits involving
trigonometric functions.

00:00:04.932 --> 00:00:08.577
So let's just start with a
fairly straightforward one.

00:00:08.577 --> 00:00:12.744
Let's find the limit as x
approaches pi of sine of x.

00:00:14.421 --> 00:00:17.328
Pause the video and see if
you can figure this out.

00:00:17.328 --> 00:00:20.944
Well, with both sine of x and cosine of x,

00:00:20.944 --> 00:00:24.139
they are defined for all real numbers,

00:00:24.139 --> 00:00:25.789
so their domain is all real numbers.

00:00:25.789 --> 00:00:28.111
You can put any real number in here for x

00:00:28.111 --> 00:00:29.823
and it will give you an output.

00:00:29.823 --> 00:00:31.041
It is defined.

00:00:31.041 --> 00:00:34.549
And they are also continuous
over their entire domain,

00:00:34.549 --> 00:00:37.258
in fact, all of the trigonometric
functions are continuous

00:00:37.258 --> 00:00:39.576
over their entire domain.

00:00:39.576 --> 00:00:43.079
And so for sine of x,
because it's continuous,

00:00:43.079 --> 00:00:45.532
and is defined at sine of pi,

00:00:45.532 --> 00:00:48.546
we would say that this
is the same thing as

00:00:48.546 --> 00:00:49.463
sine of pi,

00:00:50.463 --> 00:00:53.034
and sine of pi, you might already know,

00:00:53.034 --> 00:00:54.451
is equal to zero.

00:00:55.610 --> 00:00:58.477
Now we could do a similar
exercise with cosine of x,

00:00:58.477 --> 00:01:01.815
so if I were to say what's
the limit as x approaches,

00:01:01.815 --> 00:01:06.120
I'll just take an arbitrary
angle, x approaches pi over four

00:01:06.120 --> 00:01:07.370
of cosine of x?

00:01:08.951 --> 00:01:09.868
Well once again,

00:01:09.868 --> 00:01:13.566
cosine of x is defined
for all real numbers,

00:01:13.566 --> 00:01:15.230
x can be any real number.

00:01:15.230 --> 00:01:16.608
It's also continuous.

00:01:16.608 --> 00:01:20.567
So for cosine of x, this
limit is just gonna be cosine

00:01:20.567 --> 00:01:21.900
of pi over four,

00:01:24.011 --> 00:01:26.472
and that is going to be equal

00:01:26.472 --> 00:01:29.115
to square root of two over two.

00:01:29.115 --> 00:01:31.994
This is one of those useful angles

00:01:31.994 --> 00:01:34.318
to know the sine and cosine of.

00:01:34.318 --> 00:01:37.161
If you're thinking degrees,
this is a 45 degree angle.

00:01:37.161 --> 00:01:40.762
And in general, if I'm dealing
with a sine or a cosine,

00:01:40.762 --> 00:01:44.345
the limit as x approaches a of sine of x

00:01:46.391 --> 00:01:48.224
is equal to sine of a.

00:01:49.934 --> 00:01:52.413
Once again, this is going
to be true for any a,

00:01:52.413 --> 00:01:54.023
any real number a.

00:01:54.023 --> 00:01:56.909
And I can make a similar
statement about cosine of x.

00:01:56.909 --> 00:02:00.326
Limit as x approaches a of cosine of x

00:02:01.997 --> 00:02:04.284
is equal to cosine of a.

00:02:04.284 --> 00:02:05.971
Now, I've been saying it over and over,

00:02:05.971 --> 00:02:08.905
that's because both of their
domains are all real numbers,

00:02:08.905 --> 00:02:12.287
they are defined for all
real numbers that you put in,

00:02:12.287 --> 00:02:15.257
and they're continuous
on their entire domain.

00:02:15.257 --> 00:02:18.590
But now, let's do slightly more involved

00:02:19.739 --> 00:02:22.672
trigonometric functions,
or ones that aren't defined

00:02:22.672 --> 00:02:25.358
for all real numbers, that
their domains are constrained

00:02:25.358 --> 00:02:27.591
just a little bit more.

00:02:27.591 --> 00:02:30.573
So let's say if we were to take the limit

00:02:30.573 --> 00:02:33.740
as x approaches pi of tangent of x.

00:02:37.361 --> 00:02:40.174
What is this going to be equal to?

00:02:40.174 --> 00:02:43.365
Well, this is the same thing as the limit

00:02:43.365 --> 00:02:45.599
as x approaches pi.

00:02:45.599 --> 00:02:49.682
Tangent of x is sine
of x over cosine of x.

00:02:52.332 --> 00:02:55.582
And so both of these are defined for pi

00:02:56.771 --> 00:02:59.938
and so we could just substitute pi in.

00:03:00.811 --> 00:03:02.913
And we just wanna ensure
that we don't get a zero

00:03:02.913 --> 00:03:05.598
in the denominator, because
that would make it undefined.

00:03:05.598 --> 00:03:07.265
So we get sine of pi

00:03:11.189 --> 00:03:15.356
over cosine of pi which is
equal to zero over negative one,

00:03:16.647 --> 00:03:17.692
which is completely fine.

00:03:17.692 --> 00:03:19.549
If it was negative one over
zero, we'd be in trouble.

00:03:19.549 --> 00:03:21.991
But this is just gonna be equal to zero.

00:03:21.991 --> 00:03:23.698
So that works out.

00:03:23.698 --> 00:03:25.784
But if I were to ask
you, what is the limit

00:03:25.784 --> 00:03:29.701
as x approaches pi over
two of tangent of x?

00:03:32.251 --> 00:03:35.809
Pause the video and try to work that out.

00:03:35.809 --> 00:03:36.852
Well, think about it.

00:03:36.852 --> 00:03:40.523
This is the limit as x
approaches pi over two

00:03:40.523 --> 00:03:43.273
of sine of x over cosine of x.

00:03:46.535 --> 00:03:49.741
Now sine of pi over two is one,

00:03:49.741 --> 00:03:51.992
but cosine of pi over two is zero.

00:03:51.992 --> 00:03:53.697
So if you were to just substitute in,

00:03:53.697 --> 00:03:56.530
this would give you one over zero.

00:03:58.180 --> 00:04:01.562
And one way to think
about it is pi over two

00:04:01.562 --> 00:04:04.391
is not in the domain of tangent of x.

00:04:04.391 --> 00:04:07.588
And so this limit actually
turns out, it doesn't exist.

00:04:07.588 --> 00:04:09.915
In general, if we're
dealing with the sine,

00:04:09.915 --> 00:04:14.055
cosine, tangent, or cosecant,
secant, or cotangent,

00:04:14.055 --> 00:04:16.551
if we're taking a limit to a point

00:04:16.551 --> 00:04:20.336
that's in their domain,
then the value of the limit

00:04:20.336 --> 00:04:22.477
is going to be the same thing as the value

00:04:22.477 --> 00:04:24.455
of the function at that point.

00:04:24.455 --> 00:04:26.162
If you're taking a limit to a point

00:04:26.162 --> 00:04:27.607
that's not in their domain,

00:04:27.607 --> 00:04:30.321
there's a good chance that
we're not going to have a limit.

00:04:30.321 --> 00:04:32.571
So here, there is no limit.

00:04:33.551 --> 00:04:36.458
And the way to do that is that pi over two

00:04:36.458 --> 00:04:39.231
is not in tangent of x's domain.

00:04:39.231 --> 00:04:41.412
If you were to graph
tan of x, you would see

00:04:41.412 --> 00:04:44.825
a vertical asymptote at pi over two.

00:04:44.825 --> 00:04:47.635
Let's do one more of these.

00:04:47.635 --> 00:04:51.802
So let's say the limit as x
approaches pi of cotangent of x,

00:04:55.665 --> 00:04:57.482
pause the video and see
if you can figure out

00:04:57.482 --> 00:04:59.889
what that's going to be.

00:04:59.889 --> 00:05:01.423
Well, one way to think about it,

00:05:01.423 --> 00:05:03.703
cotangent of x is one over tangent of x,

00:05:03.703 --> 00:05:06.620
it's cosine of x over sine of x.

00:05:08.581 --> 00:05:12.277
This is a limit as x approaches pi

00:05:12.277 --> 00:05:13.110
of this.

00:05:14.067 --> 00:05:17.567
And is pi in the domain of cotangent of x?

00:05:19.351 --> 00:05:22.137
Well, no, if you were
just to substitute pi in,

00:05:22.137 --> 00:05:25.160
you're gonna get negative one over zero.

00:05:25.160 --> 00:05:28.074
And so that is not in the
domain of cotangent of x.

00:05:28.074 --> 00:05:30.231
If you were to plot it, you
would see a vertical asymptote

00:05:30.231 --> 00:05:31.383
right over there.

00:05:31.383 --> 00:05:32.970
And so we have no limit.

00:05:32.970 --> 00:05:34.387
We have no limit.

00:05:37.252 --> 00:05:40.561
So once again, this is
not in the domain of that,

00:05:40.561 --> 00:05:43.461
and so good chance that we have no limit.

00:05:43.461 --> 00:05:45.818
When the thing we're taking the limit to

00:05:45.818 --> 00:05:48.929
is in the domain of the
trigonometric function,

00:05:48.929 --> 00:05:51.215
we're going to have a defined limit.

00:05:51.215 --> 00:05:53.972
And sine and cosine in
particular are defined

00:05:53.972 --> 00:05:56.383
for all real numbers
and they're continuous

00:05:56.383 --> 00:05:57.858
over all real numbers.

00:05:57.858 --> 00:05:59.773
So you take the limit
to anything for them,

00:05:59.773 --> 00:06:02.088
it's going to be defined
and it's going to be

00:06:02.088 --> 00:06:05.269
the value of the function at that point.


Example: Analyzing the difference in distributions

WEBVTT
Kind: captions
Language: en

00:00:00.317 --> 00:00:02.981
- [Instructor] Suppose
that men have a mean height

00:00:02.981 --> 00:00:06.008
of 178 centimeters with
a standard deviation

00:00:06.008 --> 00:00:07.997
of eight centimeters.

00:00:07.997 --> 00:00:10.818
Women have a mean height
of 170 centimeters

00:00:10.818 --> 00:00:13.830
with a standard deviation
of six centimeters.

00:00:13.830 --> 00:00:17.857
The male and female heights
are each normally distributed.

00:00:17.857 --> 00:00:22.225
We independently, randomly
select a man and a woman.

00:00:22.225 --> 00:00:24.564
What is the probability that the woman

00:00:24.564 --> 00:00:26.503
is taller than the man?

00:00:26.503 --> 00:00:29.109
So I encourage you to pause
this video and think through it,

00:00:29.109 --> 00:00:31.284
and I'll give you a hint.

00:00:31.284 --> 00:00:34.583
What if we were to define
the random variable M

00:00:34.583 --> 00:00:39.058
as equal to the height of
a randomly selected man?

00:00:39.058 --> 00:00:40.808
Height of random man.

00:00:43.642 --> 00:00:46.693
What if we define the random variable W

00:00:46.693 --> 00:00:48.776
to be equal to the height

00:00:50.141 --> 00:00:51.641
of a random woman?

00:00:53.763 --> 00:00:54.596
Woman.

00:00:55.584 --> 00:00:58.157
And we defined a third random variable

00:00:58.157 --> 00:01:00.323
in terms of these first two?

00:01:00.323 --> 00:01:03.892
So let me call this D, for difference.

00:01:03.892 --> 00:01:06.592
And it is equal to the
difference in height

00:01:06.592 --> 00:01:10.011
between a randomly selected man,

00:01:10.011 --> 00:01:12.511
and a randomly selected woman.

00:01:14.468 --> 00:01:16.713
So D, the random variable D is equal

00:01:16.713 --> 00:01:20.880
to the random variable M,
minus the random variable W.

00:01:22.027 --> 00:01:25.730
So the first two are clearly
normally distributed.

00:01:25.730 --> 00:01:27.278
They tell us that right over here.

00:01:27.278 --> 00:01:30.194
The male and female heights
are each normally distributed.

00:01:30.194 --> 00:01:33.202
And we also know, or you're about to know,

00:01:33.202 --> 00:01:35.633
that the difference of random variables

00:01:35.633 --> 00:01:37.846
that are each normally distributed

00:01:37.846 --> 00:01:40.678
is also going to be normally distributed.

00:01:40.678 --> 00:01:42.559
So given this, can you think about

00:01:42.559 --> 00:01:44.156
how to tackle this question?

00:01:44.156 --> 00:01:48.846
The probability that the
woman is taller than the man.

00:01:48.846 --> 00:01:50.406
Alright now let's work
through this together.

00:01:50.406 --> 00:01:52.666
And to help us visualize,
I'll draw the normal

00:01:52.666 --> 00:01:57.296
distribution curves for
these three random variables.

00:01:57.296 --> 00:02:01.268
So this first one is for the variable M,

00:02:01.268 --> 00:02:06.172
and so right here in the
middle, that is the mean of M.

00:02:06.172 --> 00:02:07.504
And we know that this is going to

00:02:07.504 --> 00:02:09.837
be equal to 178 centimeters.

00:02:11.529 --> 00:02:13.459
We'll assume everything is in centimeters.

00:02:13.459 --> 00:02:15.147
We also know that it
has a standard deviation

00:02:15.147 --> 00:02:16.168
of eight centimeters.

00:02:16.168 --> 00:02:18.898
So for example, if this is
one standard deviation above,

00:02:18.898 --> 00:02:20.988
this is one standard deviation below,

00:02:20.988 --> 00:02:24.622
this point right over here
would be eight centimeters more

00:02:24.622 --> 00:02:27.594
than 178, so that would be 186,

00:02:27.594 --> 00:02:30.072
and this would be eight
centimeters below that,

00:02:30.072 --> 00:02:32.256
so this would be 170 centimeters.

00:02:32.256 --> 00:02:35.112
So this is for the random variable M.

00:02:35.112 --> 00:02:37.863
Now let's think about
the random variable W.

00:02:37.863 --> 00:02:40.359
The random variable W,

00:02:40.359 --> 00:02:43.192
the mean of W they tell us is 170.

00:02:47.097 --> 00:02:50.122
And one standard deviation above the mean

00:02:50.122 --> 00:02:52.685
is going to be six
centimeters above the mean.

00:02:52.685 --> 00:02:56.024
The standard deviation
is six, six centimeters,

00:02:56.024 --> 00:02:57.888
so this would be minus six,

00:02:57.888 --> 00:03:00.701
is to go one standard
deviation below the mean.

00:03:00.701 --> 00:03:03.877
Now let's think about the
difference between the two.

00:03:03.877 --> 00:03:05.822
The random variable D.

00:03:05.822 --> 00:03:07.900
So let me think about this one a bit.

00:03:07.900 --> 00:03:09.682
The random variable D.

00:03:09.682 --> 00:03:13.407
The mean of D is going to be equal to

00:03:13.407 --> 00:03:17.164
the differences in the means
of these random variables.

00:03:17.164 --> 00:03:20.334
So it's going to be
equal to the mean of M,

00:03:20.334 --> 00:03:23.251
the mean of M, minus the mean of W.

00:03:24.328 --> 00:03:25.995
Minus the mean of W.

00:03:26.940 --> 00:03:31.381
Well we know both of these,
this is gonna be 178 minus 170,

00:03:31.381 --> 00:03:32.779
so let me write that down.

00:03:32.779 --> 00:03:35.446
This is equal to 178 centimeters

00:03:37.492 --> 00:03:39.325
minus 170 centimeters.

00:03:40.940 --> 00:03:43.310
Which is going to be equal to,

00:03:43.310 --> 00:03:44.894
I'll do it in this color,

00:03:44.894 --> 00:03:47.902
this is going to be equal
to eight centimeters.

00:03:47.902 --> 00:03:50.654
So this is eight right over here.

00:03:50.654 --> 00:03:53.800
Now what about the standard deviation?

00:03:53.800 --> 00:03:57.781
Assuming these two random
variables are independent,

00:03:57.781 --> 00:04:00.044
and they tell us that
we are independently,

00:04:00.044 --> 00:04:01.809
randomly selecting a man and a woman.

00:04:01.809 --> 00:04:02.995
The height of the man shouldn't affect

00:04:02.995 --> 00:04:05.381
the height of the woman, or vice versa.

00:04:05.381 --> 00:04:08.200
Assuming that these two
are independent variables,

00:04:08.200 --> 00:04:10.797
if you take the sum or
the difference of these,

00:04:10.797 --> 00:04:13.577
then the spread will increase.

00:04:13.577 --> 00:04:16.236
But you won't just add
the standard deviations.

00:04:16.236 --> 00:04:19.231
What you would actually
do is say the variance

00:04:19.231 --> 00:04:21.883
of the difference is going to be the sum

00:04:21.883 --> 00:04:24.344
of these two variances.

00:04:24.344 --> 00:04:26.063
So let me write that down.

00:04:26.063 --> 00:04:27.875
So I could write variance with VAR,

00:04:27.875 --> 00:04:30.853
or I could write it as the
standard deviation squared.

00:04:30.853 --> 00:04:31.686
So let me write that.

00:04:31.686 --> 00:04:35.853
The standard deviation of D,
of our difference, squared,

00:04:37.343 --> 00:04:39.362
which is the variance,
is going to be equal

00:04:39.362 --> 00:04:42.195
to the variance of our variable M,

00:04:43.903 --> 00:04:46.903
plus the variance of our variable W.

00:04:48.993 --> 00:04:50.711
Now this might be a little
bit counterintuitive.

00:04:50.711 --> 00:04:51.844
This might've made sense to you

00:04:51.844 --> 00:04:53.626
if this was plus right over here.

00:04:53.626 --> 00:04:56.500
But it doesn't matter if we
are adding or subtracting,

00:04:56.500 --> 00:04:59.617
and these are truly independent variables.

00:04:59.617 --> 00:05:02.601
Then regardless of whether
we're adding or subtracting,

00:05:02.601 --> 00:05:05.393
you would add the variances.

00:05:05.393 --> 00:05:07.227
And so we can figure this out.

00:05:07.227 --> 00:05:11.029
This is going to be equal
to the standard deviation

00:05:11.029 --> 00:05:12.946
of variable M is eight.

00:05:13.872 --> 00:05:16.789
So eight squared is going to be 64.

00:05:17.932 --> 00:05:21.142
And then we have six squared.

00:05:21.142 --> 00:05:22.320
This right over here is six.

00:05:22.320 --> 00:05:25.118
Six squared is going to be 36.

00:05:25.118 --> 00:05:26.778
You add these two together,

00:05:26.778 --> 00:05:29.928
this is going to be equal to 100.

00:05:29.928 --> 00:05:34.237
And so the variance of this
distribution right over here

00:05:34.237 --> 00:05:35.950
is going to be equal to 100.

00:05:35.950 --> 00:05:38.655
Well what's the standard
deviation of that distribution?

00:05:38.655 --> 00:05:39.488
Well it's going to be equal

00:05:39.488 --> 00:05:41.452
to the square root of the variance.

00:05:41.452 --> 00:05:45.360
So the square root of
100, which is equal to 10.

00:05:45.360 --> 00:05:48.331
So for example, one standard
deviation above the mean

00:05:48.331 --> 00:05:49.831
is going to be 18.

00:05:51.692 --> 00:05:54.147
One standard deviation below the mean

00:05:54.147 --> 00:05:57.771
is going to be equal to negative two.

00:05:57.771 --> 00:05:59.644
And so now using this distribution

00:05:59.644 --> 00:06:01.399
we can actually answer this question.

00:06:01.399 --> 00:06:03.402
What is the probability that the woman

00:06:03.402 --> 00:06:05.225
is taller than the man?

00:06:05.225 --> 00:06:07.448
Well we can rewrite
that question as saying,

00:06:07.448 --> 00:06:11.941
what is the probability that
the random variable D is,

00:06:11.941 --> 00:06:12.945
what conditions would it be?

00:06:12.945 --> 00:06:15.604
Pause the video and think about it.

00:06:15.604 --> 00:06:18.779
Well the situation's where the
woman is taller than the man,

00:06:18.779 --> 00:06:21.142
if the woman is taller than the man,

00:06:21.142 --> 00:06:22.953
then this is gonna be a negative value.

00:06:22.953 --> 00:06:25.426
Then D is gonna be less than zero.

00:06:25.426 --> 00:06:27.034
So what we really want to do is figure out

00:06:27.034 --> 00:06:30.451
the probability that D is less than zero.

00:06:31.545 --> 00:06:35.712
And so what we want to do if
we say zero is right over...

00:06:39.199 --> 00:06:43.366
If we say that zero is right
over here on our distribution,

00:06:44.302 --> 00:06:46.802
so that is D is equal to zero,

00:06:48.350 --> 00:06:50.838
we want to figure out,
well what is the area

00:06:50.838 --> 00:06:53.486
under the curve less than that?

00:06:53.486 --> 00:06:55.902
So we want to figure out this entire area.

00:06:55.902 --> 00:06:58.257
There's a couple of
ways you could do this.

00:06:58.257 --> 00:07:02.460
You could figure out the Z
score for D equaling zero,

00:07:02.460 --> 00:07:04.302
and that's pretty straightforward.

00:07:04.302 --> 00:07:08.469
You could just say, this Z is
equal to zero minus our mean

00:07:10.386 --> 00:07:14.469
of eight divided by our
standard deviation of 10.

00:07:16.063 --> 00:07:18.553
So it's negative eight over 10,

00:07:18.553 --> 00:07:22.136
which is equal to negative 8/10.

00:07:23.128 --> 00:07:24.794
So you could look up a Z table and say,

00:07:24.794 --> 00:07:27.017
what is the total area under the curve

00:07:27.017 --> 00:07:29.767
below Z is equal to negative 0.8?

00:07:31.223 --> 00:07:32.770
Another way you could do this

00:07:32.770 --> 00:07:34.898
is you could use a graphing calculator.

00:07:34.898 --> 00:07:37.003
I have a TI-84 here.

00:07:37.003 --> 00:07:41.170
Where you have a normal
cumulative distribution function.

00:07:42.754 --> 00:07:44.913
I'm gonna press second VARS,

00:07:44.913 --> 00:07:48.182
and that gets me to distribution.

00:07:48.182 --> 00:07:50.063
And so I have these various functions.

00:07:50.063 --> 00:07:53.633
I want normal cumulative
distribution function,

00:07:53.633 --> 00:07:56.243
so that is choice two.

00:07:56.243 --> 00:07:57.935
And then the lower bound.

00:07:57.935 --> 00:08:00.309
Well I want to go to negative infinity.

00:08:00.309 --> 00:08:03.037
Well calculators don't have
a negative infinity button,

00:08:03.037 --> 00:08:04.552
but you could put in a very,

00:08:04.552 --> 00:08:06.629
very, very, very negative number

00:08:06.629 --> 00:08:09.910
that for our purposes, is
equivalent to negative infinity.

00:08:09.910 --> 00:08:14.077
So we could say negative one
times 10 to the 99th power.

00:08:15.585 --> 00:08:17.033
And the way we do that is second,

00:08:17.033 --> 00:08:19.627
these two capital E's
are saying essentially,

00:08:19.627 --> 00:08:23.590
times 10 to the, and I'll say 99th power.

00:08:23.590 --> 00:08:27.016
So this is a very, very,
very negative number.

00:08:27.016 --> 00:08:31.016
The upper bound here, we
want to go delete this.

00:08:32.942 --> 00:08:34.906
The upper bound is going to be zero.

00:08:34.906 --> 00:08:37.812
We're finding the area
from negative infinity

00:08:37.812 --> 00:08:39.479
all the way to zero.

00:08:40.677 --> 00:08:44.182
The mean here, well we've
already figured that out.

00:08:44.182 --> 00:08:45.682
The mean is eight.

00:08:46.795 --> 00:08:49.373
And then the standard deviation here,

00:08:49.373 --> 00:08:53.499
we figured this out too,
this is equal to 10.

00:08:53.499 --> 00:08:55.419
And so when we pick this,

00:08:55.419 --> 00:08:58.643
we're gonna go back to
the main screen, enter.

00:08:58.643 --> 00:09:00.361
So this is we could've just typed this in

00:09:00.361 --> 00:09:01.667
directly on the main screen.

00:09:01.667 --> 00:09:05.260
This says look, we are looking
at a normal distribution.

00:09:05.260 --> 00:09:08.105
We want to find the cumulative
area between two bounds.

00:09:08.105 --> 00:09:10.137
In this case, it's from
negative infinity to zero.

00:09:10.137 --> 00:09:12.747
From negative infinity to zero,

00:09:12.747 --> 00:09:16.914
where the mean is eight, and
the standard deviation is 10.

00:09:17.904 --> 00:09:21.821
We press enter, and we
get approximately 0.212.

00:09:26.847 --> 00:09:28.764
Is approximately 0.212.

00:09:30.785 --> 00:09:32.585
Or we could say what is the probability

00:09:32.585 --> 00:09:34.959
that the woman is taller than the man?

00:09:34.959 --> 00:09:37.750
Well, 0.212, or approximately,

00:09:37.750 --> 00:09:41.475
there's a 21.2% chance of that happening.

00:09:41.475 --> 00:09:44.308
A little better than one in five.


Example: Transforming a discrete random variable

WEBVTT
Kind: captions
Language: en

00:00:01.103 --> 00:00:03.464
- [Instructor] Anush is
playing a carnival game

00:00:03.464 --> 00:00:06.453
that involves two free throws.

00:00:06.453 --> 00:00:10.683
The table below displays the
probability distribution of X,

00:00:10.683 --> 00:00:14.667
the number of shots that
Anush makes in a set of two

00:00:14.667 --> 00:00:18.244
attempts, along with
some summary statistics.

00:00:18.244 --> 00:00:20.328
So here's the random variable X

00:00:20.328 --> 00:00:22.188
it's a discrete random variable.

00:00:22.188 --> 00:00:24.303
It only takes on a finite number of value,

00:00:24.303 --> 00:00:26.438
sometimes you can say
it takes on a countable

00:00:26.438 --> 00:00:27.324
number of values.

00:00:27.324 --> 00:00:29.592
We see we can either
make zero free throws,

00:00:29.592 --> 00:00:31.509
one, or two of the two.

00:00:32.347 --> 00:00:35.159
And the probability that
he makes zero is here,

00:00:35.159 --> 00:00:36.812
one is here, two is here.

00:00:36.812 --> 00:00:39.288
And then they also give us the mean of X

00:00:39.288 --> 00:00:41.940
and the standard deviation of X.

00:00:41.940 --> 00:00:46.299
Then they tell us if the
game costs Anush $15 to play

00:00:46.299 --> 00:00:49.309
and he wins $10 per shot he makes,

00:00:49.309 --> 00:00:53.476
what are the mean and standard
deviation of his net gain

00:00:54.512 --> 00:00:57.310
from playing the game N?

00:00:57.310 --> 00:01:00.941
All right, so let's define
a new random variable N.

00:01:00.941 --> 00:01:03.524
Which is equal to his net gain.

00:01:05.063 --> 00:01:05.896
Net gain.

00:01:07.013 --> 00:01:09.513
We can define N in terms of X.

00:01:10.372 --> 00:01:13.508
What is his net gain going to be?

00:01:13.508 --> 00:01:17.014
Well let's see, N, it's
going to be equal to 10 times

00:01:17.014 --> 00:01:19.110
however many shots he makes.

00:01:19.110 --> 00:01:21.443
So it's gonna be 10 times X.

00:01:22.719 --> 00:01:26.450
And then no matter what,
he has to pay $15 to play.

00:01:26.450 --> 00:01:27.283
Minus 15.

00:01:28.520 --> 00:01:30.431
In fact, we can set up a little table here

00:01:30.431 --> 00:01:33.228
for the probability distribution of N.

00:01:33.228 --> 00:01:35.610
So, let me make it right over here.

00:01:35.610 --> 00:01:37.472
So I'll make it look just like this one.

00:01:37.472 --> 00:01:39.389
N is equal to net gain.

00:01:41.699 --> 00:01:44.496
And here we'll have the probability of N.

00:01:44.496 --> 00:01:45.893
And there's three outcomes here.

00:01:45.893 --> 00:01:49.042
The outcome that corresponds
to him making zero shots,

00:01:49.042 --> 00:01:51.291
well that would be 10 times zero minus 15.

00:01:51.291 --> 00:01:53.717
That would be a net gain of negative 15.

00:01:53.717 --> 00:01:57.217
And would have the same probability, 0.16.

00:01:58.387 --> 00:02:01.282
When he makes one shot,
the net gain is gonna be

00:02:01.282 --> 00:02:02.900
10 times one minus 15

00:02:02.900 --> 00:02:04.615
which is, negative five.

00:02:04.615 --> 00:02:06.784
Which is going to have
the same probability.

00:02:06.784 --> 00:02:09.024
He has a 48% chance of making one shot.

00:02:09.024 --> 00:02:13.612
And so it's a 48% chance
of losing five dollars.

00:02:13.612 --> 00:02:15.547
And then last, but not least,

00:02:15.547 --> 00:02:19.606
when X is two his net gain
is gonna be positive five.

00:02:19.606 --> 00:02:21.034
Plus five.

00:02:21.034 --> 00:02:23.451
And so this is a 0.36 chance.

00:02:25.852 --> 00:02:28.330
So what they want us to figure out are,

00:02:28.330 --> 00:02:29.405
what are the mean

00:02:29.405 --> 00:02:31.367
and standard deviation of his net gain?

00:02:31.367 --> 00:02:34.058
So let's first figure out the mean

00:02:34.058 --> 00:02:34.891
of N.

00:02:36.357 --> 00:02:39.511
Well, if you scale a random variable

00:02:39.511 --> 00:02:41.583
the corresponding mean is going to be

00:02:41.583 --> 00:02:43.160
scaled by the same amount.

00:02:43.160 --> 00:02:45.062
And if you shift a random variable

00:02:45.062 --> 00:02:47.362
the corresponding mean is gonna be

00:02:47.362 --> 00:02:49.544
shifted by the same amount.

00:02:49.544 --> 00:02:52.061
So the mean of N is gonna be 10 times

00:02:52.061 --> 00:02:53.928
the mean of X

00:02:53.928 --> 00:02:54.761
minus 15.

00:02:56.064 --> 00:03:00.170
Which is equal to 10 times 1.2 minus 15.

00:03:00.170 --> 00:03:01.467
1.2.

00:03:01.467 --> 00:03:05.423
So there's 12 minus 15 which is equal to

00:03:05.423 --> 00:03:06.868
negative three.

00:03:06.868 --> 00:03:09.029
Now the standard deviation of N

00:03:09.029 --> 00:03:10.469
is gonna be slightly different.

00:03:10.469 --> 00:03:13.564
For the standard
deviation, scaling matters.

00:03:13.564 --> 00:03:16.556
If you scale a random
variable by a certain value

00:03:16.556 --> 00:03:19.049
you would also scale
the standard deviation

00:03:19.049 --> 00:03:20.239
by the same value.

00:03:20.239 --> 00:03:22.078
So this is going to be equal to 10

00:03:22.078 --> 00:03:24.911
times the standard deviation of X.

00:03:25.769 --> 00:03:27.781
Now you might say, what
about the shift over here?

00:03:27.781 --> 00:03:31.244
Well, the shift should
not affect the spread

00:03:31.244 --> 00:03:32.849
of the random variable.

00:03:32.849 --> 00:03:34.907
If you're scaling the
random variable, well,

00:03:34.907 --> 00:03:36.757
your spread should grow by the amount

00:03:36.757 --> 00:03:37.924
that you're scaling it.

00:03:37.924 --> 00:03:39.830
But by shifting it, it doesn't affect

00:03:39.830 --> 00:03:42.426
how much you disperse from the mean.

00:03:42.426 --> 00:03:46.308
So standard deviation is
only affected by the scaling,

00:03:46.308 --> 00:03:48.627
but not by the shifting.

00:03:48.627 --> 00:03:51.065
So this is going to be 10 times

00:03:51.065 --> 00:03:51.898
0.69

00:03:52.842 --> 00:03:54.108
which is going to.

00:03:54.108 --> 00:03:55.344
This was an approximation.

00:03:55.344 --> 00:03:57.222
So I'll say this is approximately

00:03:57.222 --> 00:03:58.055
to 6.9.

00:04:00.705 --> 00:04:04.676
So this is our new
distribution for our net gain.

00:04:04.676 --> 00:04:06.299
This is the mean of our net gain.

00:04:06.299 --> 00:04:09.773
And this is roughly the standard
deviation of our net gain.


Example: Analyzing distribution of sum of two normally distributed random variables

WEBVTT
Kind: captions
Language: en

00:00:00.328 --> 00:00:01.869
- [Instructor] Shinji commutes to work

00:00:01.869 --> 00:00:04.300
and he worries about running out of fuel.

00:00:04.300 --> 00:00:06.378
The amount of fuel he uses follows

00:00:06.378 --> 00:00:10.467
a normal distribution for
each part of his commute,

00:00:10.467 --> 00:00:14.985
but the amount of fuel he uses
on the way home varies more.

00:00:14.985 --> 00:00:18.337
The amounts of fuel he uses
for each part of the commute

00:00:18.337 --> 00:00:21.944
are also independent of each other.

00:00:21.944 --> 00:00:24.688
Here are summary statistics for the amount

00:00:24.688 --> 00:00:28.903
of fuel Shinji uses for
each part of his commute.

00:00:28.903 --> 00:00:33.254
So when he goes to work he uses
a mean of 10 liters of fuel,

00:00:33.254 --> 00:00:36.107
with a standard deviation of 1.5 liters.

00:00:36.107 --> 00:00:39.307
And on the way home, he also
has a mean of 10 liters,

00:00:39.307 --> 00:00:41.391
but there is more variation.

00:00:41.391 --> 00:00:42.473
There is more spread.

00:00:42.473 --> 00:00:46.249
He has a standard deviation of two liters.

00:00:46.249 --> 00:00:51.122
Suppose that Shinji has 25
liters of fuel in his tank

00:00:51.122 --> 00:00:54.534
and he intends to drive
to work and back home.

00:00:54.534 --> 00:00:58.630
What is the probability that
Shinji runs out of fuel?

00:00:58.630 --> 00:01:00.277
All right, this is really interesting.

00:01:00.277 --> 00:01:02.838
We have the distributions
for the amount of fuel

00:01:02.838 --> 00:01:04.592
he uses to work and to home,

00:01:04.592 --> 00:01:07.225
and they say that these
are normal distributions.

00:01:07.225 --> 00:01:08.078
They say that right over here,

00:01:08.078 --> 00:01:10.090
follows a normal distribution.

00:01:10.090 --> 00:01:11.394
But here we're talking
about the total amount

00:01:11.394 --> 00:01:15.145
of fuel he has to go
to work and to go home.

00:01:15.145 --> 00:01:18.860
So what we wanna do is come
up with a total distribution,

00:01:18.860 --> 00:01:20.934
home and back, I guess you could say.

00:01:20.934 --> 00:01:24.996
We could say, call this work plus home.

00:01:24.996 --> 00:01:25.923
Home and back.

00:01:25.923 --> 00:01:28.052
If you have two random variables

00:01:28.052 --> 00:01:31.046
that can be described
by normal distributions

00:01:31.046 --> 00:01:34.633
and you were to define a new
random variable as their sum,

00:01:34.633 --> 00:01:36.884
the distribution of
that new random variable

00:01:36.884 --> 00:01:39.569
will still be a normal distribution

00:01:39.569 --> 00:01:41.969
and its mean will be the sum of the means

00:01:41.969 --> 00:01:43.863
of those other random variables.

00:01:43.863 --> 00:01:46.767
So the mean here, I'll say the mean

00:01:46.767 --> 00:01:50.934
of work plus home is going
to be equal to 20 liters.

00:01:52.355 --> 00:01:57.258
He will use a mean of 20
liters in the roundtrip.

00:01:57.258 --> 00:02:02.058
Now for the standard
deviation, from home plus work,

00:02:02.058 --> 00:02:04.214
you can't just add the standard deviations

00:02:04.214 --> 00:02:06.517
going and coming back.

00:02:06.517 --> 00:02:09.579
But because the amount
of fuel going to work

00:02:09.579 --> 00:02:11.316
and the amount of fuel coming home

00:02:11.316 --> 00:02:13.851
are independent random variables,

00:02:13.851 --> 00:02:16.017
because they are
independent of each other,

00:02:16.017 --> 00:02:18.474
we can add the variances.

00:02:18.474 --> 00:02:20.051
And only because they are independent

00:02:20.051 --> 00:02:22.190
can we add the variances.

00:02:22.190 --> 00:02:25.523
So what you can say is that the variance

00:02:26.599 --> 00:02:29.266
of the combined trip is equal to

00:02:30.113 --> 00:02:32.530
the variance of going to work

00:02:33.388 --> 00:02:36.055
plus the variance of going home.

00:02:37.116 --> 00:02:38.781
So what's the variance of going to work?

00:02:38.781 --> 00:02:42.948
Well, 1.5 squared is, so
this will be 1.5 squared,

00:02:44.529 --> 00:02:46.145
and what's the variance coming home?

00:02:46.145 --> 00:02:49.534
Well, this is going to be
two squared, two squared.

00:02:49.534 --> 00:02:51.951
Well, this is 2.25 plus four,

00:02:53.083 --> 00:02:55.000
which is equal to 6.25.

00:02:55.895 --> 00:02:58.562
So the variance on the roundtrip

00:03:00.097 --> 00:03:02.226
is equal to 6.25.

00:03:02.226 --> 00:03:04.217
If I were to take the square root of that,

00:03:04.217 --> 00:03:06.050
which is equal to 2.5,

00:03:07.392 --> 00:03:09.949
we can now describe
the normal distribution

00:03:09.949 --> 00:03:14.116
of the roundtrip and use
that to answer the question.

00:03:15.900 --> 00:03:18.528
So we have this normal distribution

00:03:18.528 --> 00:03:22.217
that might look something like this.

00:03:22.217 --> 00:03:24.717
We know its mean is 20 liters.

00:03:26.798 --> 00:03:29.196
So this is 20 liters.

00:03:29.196 --> 00:03:30.864
And we want to know
what is the probability

00:03:30.864 --> 00:03:32.655
that Shinji runs out of fuel.

00:03:32.655 --> 00:03:33.913
Well, to run out of fuel,

00:03:33.913 --> 00:03:37.782
he would need to require
more than 25 liters of fuel.

00:03:37.782 --> 00:03:41.975
So if 25 liters of fuel
is right over here,

00:03:41.975 --> 00:03:44.392
so this is 25 liters of fuel,

00:03:45.908 --> 00:03:48.879
the scenario where Shinji runs out of fuel

00:03:48.879 --> 00:03:50.612
is right over here,

00:03:50.612 --> 00:03:54.449
this is where he needs
more than 25 liters.

00:03:54.449 --> 00:03:56.891
He actually has 25 liters in his tank.

00:03:56.891 --> 00:04:00.893
So how do we figure out
that area right over there?

00:04:00.893 --> 00:04:02.395
Well, we could use a z-table.

00:04:02.395 --> 00:04:04.047
We could say how many standard deviations

00:04:04.047 --> 00:04:06.380
above the mean is 25 liters?

00:04:07.283 --> 00:04:10.388
Well, it is five liters above the mean,

00:04:10.388 --> 00:04:11.397
so let me write this down.

00:04:11.397 --> 00:04:14.064
So the Z here, the Z is equal to

00:04:15.436 --> 00:04:17.769
25 minus the mean, minus 20,

00:04:18.821 --> 00:04:21.665
divided by the standard deviation for,

00:04:21.665 --> 00:04:24.899
I guess you could say this
combined normal distribution.

00:04:24.899 --> 00:04:27.638
This is two standard
deviations above the mean

00:04:27.638 --> 00:04:30.171
or a z-score of plus two.

00:04:30.171 --> 00:04:32.404
So if we look at a z-table
and we look exactly

00:04:32.404 --> 00:04:35.125
two standard deviations above the mean,

00:04:35.125 --> 00:04:38.655
that will give us this
area, the cumulative area

00:04:38.655 --> 00:04:42.280
below two standard
deviations above the mean.

00:04:42.280 --> 00:04:44.438
And then if we subtract that from one,

00:04:44.438 --> 00:04:47.247
we will get the area that we care about.

00:04:47.247 --> 00:04:50.139
So let's get our z-table out.

00:04:50.139 --> 00:04:53.605
We care about a z-score of exactly two,

00:04:53.605 --> 00:04:56.438
so 2.00 is right over here, .9772.

00:05:00.858 --> 00:05:05.025
So that tells us that this
area right over here is 0.9772,

00:05:07.298 --> 00:05:08.756
and so that blue area,

00:05:08.756 --> 00:05:11.053
the probability that
Shinji runs out of fuel

00:05:11.053 --> 00:05:13.720
is going to be one minus 0.9772,

00:05:17.813 --> 00:05:19.891
and what is that going to be equal to?

00:05:19.891 --> 00:05:23.141
Let's see, this is going to be equal to

00:05:24.084 --> 00:05:24.917
0.0228.

00:05:31.670 --> 00:05:32.924
Did I do that right?

00:05:32.924 --> 00:05:34.625
I think I did that right.

00:05:34.625 --> 00:05:35.542
Yes, 0.0228

00:05:39.122 --> 00:05:41.479
is the probability that
Shinji runs out of fuel.

00:05:41.479 --> 00:05:42.956
If you want to think of it as a percent,

00:05:42.956 --> 00:05:46.539
2.28% chance that he runs out of fuel.