Worked example: area between curves

WEBVTT
Kind: captions
Language: en

00:00:00.427 --> 00:00:01.384
- [Instructor] What we're going to do

00:00:01.384 --> 00:00:03.306
using our powers of calculus

00:00:03.306 --> 00:00:07.196
is find the area of this yellow region

00:00:07.196 --> 00:00:09.233
and if at any point you get inspired,

00:00:09.233 --> 00:00:11.374
I always encourage you to pause the video

00:00:11.374 --> 00:00:14.126
and try to work through it on your own.

00:00:14.126 --> 00:00:15.963
So, the key here is you
might recognize hey,

00:00:15.963 --> 00:00:18.063
this is an area between curves.

00:00:18.063 --> 00:00:20.025
A definite integral might be useful,

00:00:20.025 --> 00:00:22.979
so I'll just set up the
definite integral sign

00:00:22.979 --> 00:00:24.761
and so, first we're gonna
think about what are left

00:00:24.761 --> 00:00:27.468
and right boundaries of our region?

00:00:27.468 --> 00:00:28.996
Well, it looks like the left boundary

00:00:28.996 --> 00:00:31.391
is where the two graphs
intersect right over here

00:00:31.391 --> 00:00:32.313
and the right boundary

00:00:32.313 --> 00:00:34.468
is where they intersect right over there.

00:00:34.468 --> 00:00:36.353
Well, what is this point of intersection?

00:00:36.353 --> 00:00:39.344
It looks like it is
negative one, negative two.

00:00:39.344 --> 00:00:40.763
We can verify that.

00:00:40.763 --> 00:00:43.492
In this red curve, if X is negative one,

00:00:43.492 --> 00:00:45.475
let's see, you square that,
you'll get one minus three.

00:00:45.475 --> 00:00:48.129
You do indeed get Y is
equal to negative two

00:00:48.129 --> 00:00:49.254
and in this blue function

00:00:49.254 --> 00:00:50.596
was X is equal to negative one,

00:00:50.596 --> 00:00:53.029
you get one minus four plus one.

00:00:53.029 --> 00:00:56.798
Once again, you do indeed get
Y is equal to negative two

00:00:56.798 --> 00:00:57.698
and the same thing is true

00:00:57.698 --> 00:01:00.234
when X is equal to one.

00:01:00.234 --> 00:01:02.710
One minus three, negative two,

00:01:02.710 --> 00:01:06.010
one minus four plus one, negative two.

00:01:06.010 --> 00:01:07.955
So, our bounds are
indeed, we're going from X

00:01:07.955 --> 00:01:11.291
equals negative one to
X equals positive one

00:01:11.291 --> 00:01:13.966
and now let's think about
our upper and lower bounds.

00:01:13.966 --> 00:01:16.606
Over that interval this blue graph

00:01:16.606 --> 00:01:18.272
is our upper bound

00:01:18.272 --> 00:01:21.018
and so, we would subtract the lower bound

00:01:21.018 --> 00:01:22.372
from the upper bound,

00:01:22.372 --> 00:01:24.319
so we would have X to the fourth,

00:01:24.319 --> 00:01:27.418
minus four X squared plus one

00:01:27.418 --> 00:01:31.480
and from that we will subtract X squared

00:01:31.480 --> 00:01:32.647
minus three DX

00:01:34.000 --> 00:01:35.640
and in many other videos we have talked

00:01:35.640 --> 00:01:37.450
about why you do this,

00:01:37.450 --> 00:01:39.677
why this makes sense to just subtract

00:01:39.677 --> 00:01:42.805
the lower graph from the upper graph

00:01:42.805 --> 00:01:45.046
when you're finding the area between them

00:01:45.046 --> 00:01:47.750
but now we just have to
evaluate this definite integral,

00:01:47.750 --> 00:01:50.386
so let's just get down to business.

00:01:50.386 --> 00:01:54.380
Alright, so we have the integral
from negative one to one,

00:01:54.380 --> 00:01:58.878
and so, we have X to the
fourth, X to the fourth,

00:01:58.878 --> 00:02:01.408
and now we have minus four X squared

00:02:01.408 --> 00:02:03.657
and then when you distribute
this negative sign,

00:02:03.657 --> 00:02:06.232
you're gonna subtract another X squared,

00:02:06.232 --> 00:02:09.867
so you're gonna have minus five X squared

00:02:09.867 --> 00:02:11.696
and then you have plus one

00:02:11.696 --> 00:02:14.273
and then you're gonna
subtract a negative three,

00:02:14.273 --> 00:02:16.150
so it's gonna be one plus three,

00:02:16.150 --> 00:02:18.650
so it's gonna be plus four DX,

00:02:20.681 --> 00:02:22.058
DX and just to be clear

00:02:22.058 --> 00:02:24.317
I should put parentheses right over there

00:02:24.317 --> 00:02:26.335
because it's really the
DX is being multiplied

00:02:26.335 --> 00:02:28.652
by this entire expression

00:02:28.652 --> 00:02:31.158
and so, let's see, let's find
the antiderivative of this.

00:02:31.158 --> 00:02:33.139
This should be pretty straightforward.

00:02:33.139 --> 00:02:34.788
We're just gonna use
the reverse power rule

00:02:34.788 --> 00:02:37.946
multiple times, so this is
going to be the antiderivative

00:02:37.946 --> 00:02:41.265
of X to the fourth is X
to the fifth over five.

00:02:41.265 --> 00:02:42.734
We just incremented the exponent

00:02:42.734 --> 00:02:45.275
and divided by that incremented exponent,

00:02:45.275 --> 00:02:46.612
minus, same idea here,

00:02:46.612 --> 00:02:49.112
five X to the third over three

00:02:50.097 --> 00:02:51.014
plus four X

00:02:52.212 --> 00:02:54.641
and then we are going to evaluate it

00:02:54.641 --> 00:02:56.926
at one and then subtract from that

00:02:56.926 --> 00:02:59.204
and evaluate it at negative one.

00:02:59.204 --> 00:03:02.745
So, let's first evaluate it at one.

00:03:02.745 --> 00:03:05.018
We're gonna get 1/5

00:03:05.018 --> 00:03:07.518
minus 5/3 plus four

00:03:11.095 --> 00:03:14.678
and now let us evaluate
it at negative one,

00:03:15.659 --> 00:03:18.464
so minus, let's see, if
this is negative one,

00:03:18.464 --> 00:03:21.955
we get a negative 1/5, negative 1/5

00:03:21.955 --> 00:03:25.812
and this is gonna be plus 5/3,

00:03:25.812 --> 00:03:30.204
plus 5/3 and then this is
going to be minus four,

00:03:30.204 --> 00:03:32.155
minus four but then when you distribute

00:03:32.155 --> 00:03:36.200
the negative sign, we're
going to distribute this

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over all of these terms,

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and so this is going to be,

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if we make this positive,
this will be positive,

00:03:42.194 --> 00:03:43.658
this one will be negative

00:03:43.658 --> 00:03:45.458
and then this one will be positive,

00:03:45.458 --> 00:03:48.035
so you have 1/5 plus 1/5

00:03:48.035 --> 00:03:51.880
which is going to be 2/5, that and that,

00:03:51.880 --> 00:03:54.135
and then minus 5/3, minus 5/3,

00:03:54.135 --> 00:03:55.968
so minus 10 over three

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and then four plus four,

00:03:58.880 --> 00:04:00.434
so plus eight

00:04:00.434 --> 00:04:02.414
and so, we just need to simplify this.

00:04:02.414 --> 00:04:04.123
This is going to be, let's see,

00:04:04.123 --> 00:04:05.746
it's going to be eight

00:04:05.746 --> 00:04:08.827
and then if I write, so plus,

00:04:08.827 --> 00:04:11.807
I'm gonna write these two
with a denominator of 15

00:04:11.807 --> 00:04:14.612
'cause that's the common
denominator of three and five.

00:04:14.612 --> 00:04:19.541
Let's see, 2/5 is 16/15,
yeah, that's right,

00:04:19.541 --> 00:04:20.963
five times three is 15.

00:04:20.963 --> 00:04:22.887
Two times three is six

00:04:22.887 --> 00:04:25.455
and then 10/3, let's see, if we multiply

00:04:25.455 --> 00:04:26.630
the denominator times fives,

00:04:26.630 --> 00:04:28.179
we have to multiply the
numerator times five,

00:04:28.179 --> 00:04:30.548
so this is going to be 50/15

00:04:30.548 --> 00:04:35.236
and so, what's 6/15 minus 50/15?

00:04:35.236 --> 00:04:38.650
So, this is going to be equal to eight

00:04:38.650 --> 00:04:41.233
minus six minus 50 is minus 44,

00:04:43.758 --> 00:04:45.091
minus 44 over 15

00:04:48.601 --> 00:04:50.341
and so what is 44 over 15?

00:04:50.341 --> 00:04:54.341
44 over 15 is equal to two and 14/15,

00:04:57.898 --> 00:04:59.167
so that's really what we're subtracting.

00:04:59.167 --> 00:05:02.405
We're gonna subtract two and 14/15,

00:05:02.405 --> 00:05:05.136
so if you subtract two from this

00:05:05.136 --> 00:05:07.969
you would get six minus 14 over 15

00:05:10.302 --> 00:05:12.462
'cause we still have to subtract the 14/15

00:05:12.462 --> 00:05:14.997
and then six minus 14/15

00:05:14.997 --> 00:05:19.080
is going to be equal to five and 1/15.

00:05:19.966 --> 00:05:22.150
So, just like that, we were indeed

00:05:22.150 --> 00:05:24.495
able to figure out this area.


Worked example: differentiating polar functions

WEBVTT
Kind: captions
Language: en

00:00:00.563 --> 00:00:02.955
- [Instructor] Let r be the
function given by r of theta

00:00:02.955 --> 00:00:06.287
is equal to three theta sine
theta for theta is between

00:00:06.287 --> 00:00:08.829
zero and two pie,
including zero and two pie.

00:00:08.829 --> 00:00:10.739
The graph of r in polar
coordinates consists

00:00:10.739 --> 00:00:14.354
of two loops, as shown
in the figure above.

00:00:14.354 --> 00:00:15.899
So let's think about why it has two loops.

00:00:15.899 --> 00:00:20.060
So as our theta, when
theta is zero, r is zero,

00:00:20.060 --> 00:00:22.220
and then as our theta increases,

00:00:22.220 --> 00:00:24.898
we start tracing out this first loop,

00:00:24.898 --> 00:00:28.791
all the way until when
theta's equal to pie.

00:00:28.791 --> 00:00:30.860
So we've traced out this first loop

00:00:30.860 --> 00:00:34.410
from theta's equal to zero
to theta's equal to pie.

00:00:34.410 --> 00:00:37.748
And then the second loop has a larger r,

00:00:37.748 --> 00:00:39.432
so these are larger r's, this is when

00:00:39.432 --> 00:00:41.914
we're going from pie to two pie.

00:00:41.914 --> 00:00:44.669
And you might say, "Well why
doesn't it show up down here?"

00:00:44.669 --> 00:00:48.067
Well, between sine of
pie and sine of two pie,

00:00:48.067 --> 00:00:50.153
this part right over here
is going to be negative,

00:00:50.153 --> 00:00:52.524
so it flips it over, the r onto this side.

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And the magnitude of the r is larger

00:00:53.933 --> 00:00:55.999
and larger because of this three theta.

00:00:55.999 --> 00:00:57.565
And so when we go from pie to two pie,

00:00:57.565 --> 00:01:01.028
we trace out the larger
circle, fair enough,

00:01:01.028 --> 00:01:03.355
that seems pretty straightforward.

00:01:03.355 --> 00:01:06.207
Point P is on the graph of r,

00:01:06.207 --> 00:01:08.892
right over there, and the y-axis.

00:01:08.892 --> 00:01:11.283
Find the rate of change
of the x-coordinate

00:01:11.283 --> 00:01:14.852
with respect to theta at the point P.

00:01:14.852 --> 00:01:16.612
Alright, so let's think
about this a little bit.

00:01:16.612 --> 00:01:18.310
They don't give us x
as a function of theta.

00:01:18.310 --> 00:01:21.594
We have to figure out that
from what they've given us.

00:01:21.594 --> 00:01:24.911
So just as a bit of a polar
coordinates refresher,

00:01:24.911 --> 00:01:28.224
if this is our theta, right over there,

00:01:28.224 --> 00:01:30.830
this is our r, and that would be a point

00:01:30.830 --> 00:01:33.700
on our curve for this theta.

00:01:33.700 --> 00:01:36.020
Now how do you convert that to x and y's?

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Well you can construct a little bit

00:01:37.847 --> 00:01:40.409
of a right triangle right over here.

00:01:40.409 --> 00:01:43.659
And we know from our basic trigonometry

00:01:44.808 --> 00:01:48.148
that the length of this
base right over here,

00:01:48.148 --> 00:01:50.776
this is going to be the hypotenuse.

00:01:50.776 --> 00:01:54.665
Let me just write that, that's
going to be our x-coordinate.

00:01:54.665 --> 00:01:56.648
Our x-coordinate right over here

00:01:56.648 --> 00:01:59.667
is going to be equal to our hypotenuse

00:01:59.667 --> 00:02:02.750
which is r times the cosine of theta.

00:02:04.788 --> 00:02:08.114
If we wanted the y-coordinate
as a function of r and theta,

00:02:08.114 --> 00:02:10.685
it'd be y is equal to r sine of theta,

00:02:10.685 --> 00:02:11.542
but they don't want us to worry about

00:02:11.542 --> 00:02:13.617
y here, just the x-coordinate.

00:02:13.617 --> 00:02:18.215
So we know this, but we want
it purely in terms of theta.

00:02:18.215 --> 00:02:19.719
So how do we get there?

00:02:19.719 --> 00:02:23.063
Well what we can do is,
take this expression for r,

00:02:23.063 --> 00:02:24.937
r itself is a function of theta,

00:02:24.937 --> 00:02:28.063
and replace it right over there.

00:02:28.063 --> 00:02:30.208
And so what we can do, is we can write,

00:02:30.208 --> 00:02:34.041
well, x of theta is
going to be equal to r,

00:02:34.895 --> 00:02:38.312
which itself is three theta sine of theta

00:02:40.852 --> 00:02:44.602
times cosine of theta,
times cosine of theta.

00:02:46.701 --> 00:02:48.583
And now, we wanna find the rate of change

00:02:48.583 --> 00:02:50.225
of the x-coordinate with respect to theta

00:02:50.225 --> 00:02:52.356
at a point, so let's
just find the derivative

00:02:52.356 --> 00:02:54.508
of x with respect to theta.

00:02:54.508 --> 00:02:57.828
So x prime of theta is equal to,

00:02:57.828 --> 00:03:00.777
well I have the product of
three expressions over here.

00:03:00.777 --> 00:03:03.308
I have this first expression, three theta,

00:03:03.308 --> 00:03:06.228
then I have sine theta, and
then I have cosine theta.

00:03:06.228 --> 00:03:08.842
So we can apply the product
rule to find the derivative.

00:03:08.842 --> 00:03:09.869
If you're using the product rule

00:03:09.869 --> 00:03:11.279
with the expression of three things,

00:03:11.279 --> 00:03:12.834
you essentially just
follow the same pattern

00:03:12.834 --> 00:03:14.452
when you're taking the
product of two things.

00:03:14.452 --> 00:03:16.065
The first term is gonna be the derivative

00:03:16.065 --> 00:03:19.134
of the first of the expressions, three,

00:03:19.134 --> 00:03:21.699
times the other two
expressions, so we're gonna have

00:03:21.699 --> 00:03:25.199
three times sine of theta cosine of theta,

00:03:27.516 --> 00:03:30.267
plus the second term is going to be

00:03:30.267 --> 00:03:33.127
the derivative of the middle term times

00:03:33.127 --> 00:03:36.710
the other two expressions,
so we're gonna have

00:03:36.710 --> 00:03:40.356
three theta and then
derivative of sine theta

00:03:40.356 --> 00:03:43.152
is cosine theta, times
another cosine theta,

00:03:43.152 --> 00:03:46.129
you're gonna have cosine squared of theta,

00:03:46.129 --> 00:03:49.416
or cosine of theta
squared, just like that.

00:03:49.416 --> 00:03:52.431
And then you're gonna have the
derivative of the last term,

00:03:52.431 --> 00:03:54.299
is going to be the
derivative of cosine theta

00:03:54.299 --> 00:03:56.596
times these other two expressions.

00:03:56.596 --> 00:03:57.911
Well the derivative of cosine theta is

00:03:57.911 --> 00:04:00.567
negative sine theta, so if you multiply

00:04:00.567 --> 00:04:03.995
negative sine theta times
three theta sine theta, you're

00:04:03.995 --> 00:04:08.162
going to have negative three
theta sine squared theta.

00:04:12.267 --> 00:04:15.539
And so, we want to
evaluate this at point P.

00:04:15.539 --> 00:04:17.972
So what is theta at point P?

00:04:17.972 --> 00:04:21.222
Well, point P does happen
on our first pass around,

00:04:21.222 --> 00:04:24.305
and so, at point P, theta is equal to

00:04:25.653 --> 00:04:29.570
theta right over here is
equal to pie over two,

00:04:31.300 --> 00:04:34.727
so pie over two, so what
we really just need to find

00:04:34.727 --> 00:04:38.647
is, well what is x prime of pie over two?

00:04:38.647 --> 00:04:42.480
Well, that is going to
be equal to three times

00:04:44.163 --> 00:04:47.080
sine of pie over two, which is one,

00:04:48.317 --> 00:04:51.984
times cosine of pie
over two, which is zero,

00:04:53.302 --> 00:04:57.469
so this whole thing is zero,
plus three times pie over two,

00:04:58.811 --> 00:05:01.561
this is three pie over two, times

00:05:02.480 --> 00:05:05.083
cosine squared of pie over two, or

00:05:05.083 --> 00:05:06.916
cosine of pie over two squared.

00:05:06.916 --> 00:05:10.453
Well that's just zero, so
so far, everything is zero,

00:05:10.453 --> 00:05:13.855
minus three times pie over
two, three pie over two

00:05:13.855 --> 00:05:16.772
times sine of pie over two squared.

00:05:18.948 --> 00:05:20.094
Well what's sine of pie over two?

00:05:20.094 --> 00:05:22.442
Well that's one, you square
it, you still get one.

00:05:22.442 --> 00:05:27.363
So all of this simplified to
negative three pie over two.

00:05:27.363 --> 00:05:29.002
Now it's always good
to get a reality check.

00:05:29.002 --> 00:05:31.176
Does this make sense
that the rate of change

00:05:31.176 --> 00:05:34.956
of x with respect to theta is
negative three pie over two?

00:05:34.956 --> 00:05:35.943
Well think about what's happening.

00:05:35.943 --> 00:05:37.843
As theta increases a little bit,

00:05:37.843 --> 00:05:40.238
x is definitely going to decrease,

00:05:40.238 --> 00:05:44.302
so it makes sense that we
have a negative out here.

00:05:44.302 --> 00:05:46.442
So, right over here, rate of change of x

00:05:46.442 --> 00:05:50.206
with respect to theta,
negative three pie over two.

00:05:50.206 --> 00:05:53.361
As theta increases, our
x for sure is decreasing,

00:05:53.361 --> 00:05:56.097
so at least it does make intuitive sense.


Worked example: Logistic model word problem

WEBVTT
Kind: captions
Language: en

00:00:00.694 --> 00:00:03.441
- [Narrator] The population
P of T of bacteria

00:00:03.441 --> 00:00:06.159
in a petry dish satisfies the logistic

00:00:06.159 --> 00:00:07.760
differential equation.

00:00:07.760 --> 00:00:10.360
The rate of change of
population with respect to time

00:00:10.360 --> 00:00:13.299
is equal to two times the population times

00:00:13.299 --> 00:00:15.513
the difference between six

00:00:15.513 --> 00:00:18.125
and the population divided by 8000,

00:00:18.125 --> 00:00:21.205
where T is measured in hours
and the initial population

00:00:21.205 --> 00:00:22.538
is 700 bacteria.

00:00:23.746 --> 00:00:27.470
What is the carrying
capacity of the population;

00:00:27.470 --> 00:00:30.074
and what is the population's size

00:00:30.074 --> 00:00:32.574
when it's growing the fastest?

00:00:33.926 --> 00:00:36.563
So in order to even attempt
to answer these questions

00:00:36.563 --> 00:00:39.214
at any point if you're inspired,
definitely pause the video

00:00:39.214 --> 00:00:41.503
and try to answer it by yourself.

00:00:41.503 --> 00:00:43.942
Let's just remind ourselves
what we're talking about

00:00:43.942 --> 00:00:44.990
or what they're talking about

00:00:44.990 --> 00:00:47.024
with the logistic differential equation

00:00:47.024 --> 00:00:49.882
and the carrying capacity.

00:00:49.882 --> 00:00:52.312
So in general a logistic
differential equation

00:00:52.312 --> 00:00:54.604
is one where we seeing
the rate of change of,

00:00:54.604 --> 00:00:56.252
and it's often referring to population,

00:00:56.252 --> 00:00:58.047
so let's just stick with population.

00:00:58.047 --> 00:01:01.546
So the rate of change of our
population with respect to time

00:01:01.546 --> 00:01:05.977
is proportional to the
product of the population

00:01:05.977 --> 00:01:07.990
and the difference between what's known

00:01:07.990 --> 00:01:11.590
as the carrying capacity
and the population.

00:01:11.590 --> 00:01:14.866
Now why is this a model
that you'll see a lot,

00:01:14.866 --> 00:01:16.512
especially why is it useful

00:01:16.512 --> 00:01:18.757
for studying things like populations.

00:01:18.757 --> 00:01:21.674
When a population is small

00:01:21.674 --> 00:01:24.590
the environment really isn't limiting it

00:01:24.590 --> 00:01:27.842
and so assuming it starts
from some none zero value,

00:01:27.842 --> 00:01:31.493
this thing grows, this
thing is not going to get

00:01:31.493 --> 00:01:35.122
much smaller and so
our population is going

00:01:35.122 --> 00:01:38.894
to a rate of change is going to increase.

00:01:38.894 --> 00:01:41.651
And so let me just draw
a little graph here

00:01:41.651 --> 00:01:43.761
to show the typical solution

00:01:43.761 --> 00:01:45.857
to a logistic differential equation.

00:01:45.857 --> 00:01:47.997
So this is our population, this is time,

00:01:47.997 --> 00:01:50.620
so when our population is low,

00:01:50.620 --> 00:01:52.500
let's say it's gonna start
from some non zero value,

00:01:52.500 --> 00:01:53.963
if it was zero what would happen,

00:01:53.963 --> 00:01:55.943
well then our rate of
change would just be zero

00:01:55.943 --> 00:01:57.811
and our population would never
grow and that makes sense.

00:01:57.811 --> 00:02:00.073
If you have no bunnies on your island

00:02:00.073 --> 00:02:02.424
then there never will be
any bunnies on your island

00:02:02.424 --> 00:02:05.340
but if you have a few
bunnies well initially,

00:02:05.340 --> 00:02:06.609
their rate of change,

00:02:06.609 --> 00:02:09.383
the rate of population is
gonna keep accelerating

00:02:09.383 --> 00:02:13.581
as this thing grows, it's
gonna keep accelerating

00:02:13.581 --> 00:02:17.321
but then at some point your
environment is going to

00:02:17.321 --> 00:02:20.764
limit how many bunnies
for example or bacteria

00:02:20.764 --> 00:02:22.390
can grow in your environment.

00:02:22.390 --> 00:02:25.504
Because once the
population gets close to A,

00:02:25.504 --> 00:02:28.090
this thing right over here
is going to approach zero

00:02:28.090 --> 00:02:29.945
and its going to make our rate of change

00:02:29.945 --> 00:02:31.806
smaller and smaller and smaller.

00:02:31.806 --> 00:02:34.742
And so you can imagine
in the limiting case

00:02:34.742 --> 00:02:36.909
as P gets very close to A,

00:02:41.982 --> 00:02:44.108
you can imagine as T approaches infinity,

00:02:44.108 --> 00:02:46.771
our rate of change is
going to approach zero.

00:02:46.771 --> 00:02:48.629
So one way to think about it is,

00:02:48.629 --> 00:02:52.796
our population would acemtote
towards the carrying capacity.

00:02:54.088 --> 00:02:58.544
That is A right over here,
that is our carrying capacity.

00:02:58.544 --> 00:02:59.942
So there's a couple of ways

00:02:59.942 --> 00:03:02.927
of answering this first question; one way

00:03:02.927 --> 00:03:05.456
is we can actually put our
logistic differential equation

00:03:05.456 --> 00:03:08.047
in this form and then we can recognize

00:03:08.047 --> 00:03:10.083
what the carrying capacity is.

00:03:10.083 --> 00:03:12.224
The other way is to think
about, well what happens

00:03:12.224 --> 00:03:13.893
as T approaches infinity.

00:03:13.893 --> 00:03:17.828
As T approaches infinity,
this thing approaches zero

00:03:17.828 --> 00:03:20.611
and so we can think from this
logistic differential equation

00:03:20.611 --> 00:03:24.194
what P values would
make this thing be zero

00:03:25.272 --> 00:03:26.598
based on this differential equation

00:03:26.598 --> 00:03:28.711
or when this thing approaches zero,

00:03:28.711 --> 00:03:31.719
what P values would this approach.

00:03:31.719 --> 00:03:33.515
So let's do it both ways.

00:03:33.515 --> 00:03:37.130
So one way, let me write it
in this form right over here,

00:03:37.130 --> 00:03:40.670
so it's close, this is
six minus P over 8000,

00:03:40.670 --> 00:03:42.477
can we make it in the form
where we just have some number

00:03:42.477 --> 00:03:43.770
minus P.

00:03:43.770 --> 00:03:46.670
Well if we multiply this times 8000

00:03:46.670 --> 00:03:48.664
and then we divided this by 8000

00:03:48.664 --> 00:03:50.885
we wouldn't be changing the
value of that expression

00:03:50.885 --> 00:03:52.038
so let's do that.

00:03:52.038 --> 00:03:54.610
If we divide this by 8000 you have

00:03:54.610 --> 00:03:58.777
DP DT is equal to 2P over
8000 is P over 4000 times

00:04:03.934 --> 00:04:07.787
and now let's multiply
this expression by 8000.

00:04:07.787 --> 00:04:11.954
So six times 8000 is 48,000
minus P over 8000 times 8000

00:04:14.960 --> 00:04:17.070
is minus P and there you have it.

00:04:17.070 --> 00:04:19.502
We've written it down
to somewhat classic form

00:04:19.502 --> 00:04:21.793
and we can see that the carrying capacity

00:04:21.793 --> 00:04:25.293
is 48,000, I guess we say 48,000 bacteria.

00:04:27.459 --> 00:04:31.367
So this thing right over here is 48,000.

00:04:31.367 --> 00:04:33.317
Another way we could think about it is,

00:04:33.317 --> 00:04:35.384
well the carrying capacity is what happens

00:04:35.384 --> 00:04:37.068
as T approaches infinity.

00:04:37.068 --> 00:04:39.813
So as T approaches infinity,
this thing right over here

00:04:39.813 --> 00:04:41.646
approaches zero, you can
see the rate of change

00:04:41.646 --> 00:04:43.369
approaches zero.

00:04:43.369 --> 00:04:47.252
So when that approaches
zero what does P approach?

00:04:47.252 --> 00:04:51.419
So we can just solve for
this; six minus P over 8000.

00:04:53.128 --> 00:04:55.038
Well there's two situations, two Ps

00:04:55.038 --> 00:04:57.385
for which our derivative is equal to zero.

00:04:57.385 --> 00:04:59.923
There's one case when
this is equal to zero,

00:04:59.923 --> 00:05:01.892
in which case our population is to zero

00:05:01.892 --> 00:05:04.975
or the other case is when
this is equal to zero.

00:05:04.975 --> 00:05:09.740
So let's do that; six minus
P over 8000 is equal to zero

00:05:09.740 --> 00:05:13.323
or we could say P over
8000 is equal to six

00:05:15.382 --> 00:05:17.811
and so we can say multiply
both sides by 8000,

00:05:17.811 --> 00:05:19.478
P is equal to 48000.

00:05:21.046 --> 00:05:24.348
Which is exactly what
we had right over there.

00:05:24.348 --> 00:05:26.173
So now let's answer the second part.

00:05:26.173 --> 00:05:30.176
What is the population's size
when it's growing the fastest?

00:05:30.176 --> 00:05:33.302
So intuitively you can see
when that is right over here,

00:05:33.302 --> 00:05:36.381
the rate of change is gonna grow

00:05:36.381 --> 00:05:39.041
but then as we approach
the carrying capacity,

00:05:39.041 --> 00:05:40.842
the rate of change is
gonna start slowing down.

00:05:40.842 --> 00:05:43.741
So your maximum rate of change,
it's growing the fastest,

00:05:43.741 --> 00:05:47.908
is right about there but how
do we figure out it exactly.

00:05:49.543 --> 00:05:50.978
Well you could go back to the logistic

00:05:50.978 --> 00:05:52.493
differential equation.

00:05:52.493 --> 00:05:54.931
You can see that it's
really our rate of change

00:05:54.931 --> 00:05:57.142
is a function, you could view it as

00:05:57.142 --> 00:06:00.090
a function of our
population right over here.

00:06:00.090 --> 00:06:02.674
And this is actually a
quadratic expression,

00:06:02.674 --> 00:06:05.470
this is a concave downward
quadratic expression,

00:06:05.470 --> 00:06:06.373
it would look like this,

00:06:06.373 --> 00:06:10.354
if you were graphing rate
of change, let me do that,

00:06:10.354 --> 00:06:13.354
if you were to graph rate of change,

00:06:14.355 --> 00:06:17.188
DP DT as a function of population,

00:06:18.396 --> 00:06:22.563
well when the population is
small, so when the population

00:06:24.340 --> 00:06:28.350
is, say 700 or that's
where we're starting.

00:06:28.350 --> 00:06:30.480
I'll just speak in generalities,

00:06:30.480 --> 00:06:34.916
when your population is
small DPT, your rate is small

00:06:34.916 --> 00:06:39.083
but then it increases and
then some point around there

00:06:40.604 --> 00:06:45.018
the rate of change starts to
decrease and it approaches zero

00:06:45.018 --> 00:06:49.702
as our population is approaching
the carrying capacity.

00:06:49.702 --> 00:06:52.730
So this right over here for
example, would actually be

00:06:52.730 --> 00:06:54.829
our carrying capacity.

00:06:54.829 --> 00:06:59.132
What is this maximum point right over here

00:06:59.132 --> 00:07:01.018
and there's a couple of
ways you could approach it

00:07:01.018 --> 00:07:03.421
with calculus and even
algebra, we have many tools

00:07:03.421 --> 00:07:04.854
for identifying this maximum point

00:07:04.854 --> 00:07:07.665
which is really just a vertex
of this downward opening,

00:07:07.665 --> 00:07:09.909
this concave downward parabola.

00:07:09.909 --> 00:07:13.017
So let's just do that and
let's find this vertex

00:07:13.017 --> 00:07:16.459
and the vertex is just
half way between the zeros

00:07:16.459 --> 00:07:18.411
of this quadratic.

00:07:18.411 --> 00:07:22.229
So let's find the P values
that make this equal to zero

00:07:22.229 --> 00:07:25.146
which we actually just figured out,

00:07:26.655 --> 00:07:30.584
the vertex is just gonna
be half way between that.

00:07:30.584 --> 00:07:34.411
So when the population is zero
our rate of change is zero

00:07:34.411 --> 00:07:38.315
when the population is A,
which we know is 48,000,

00:07:38.315 --> 00:07:42.482
our rate of change is zero and
so our maximum rate of change

00:07:44.879 --> 00:07:47.538
is gonna happen half way
between those two points

00:07:47.538 --> 00:07:48.788
which is 24,000

00:07:50.140 --> 00:07:53.150
So it is a population of 24,000.

00:07:53.150 --> 00:07:54.833
When we got that by really just saying

00:07:54.833 --> 00:07:56.802
well at one point does this quadratic,

00:07:56.802 --> 00:07:58.634
quadratic is a function of P,

00:07:58.634 --> 00:08:01.121
when does that hit a maximum point.

00:08:01.121 --> 00:08:03.607
Well that's going to be
half way between the zeros,

00:08:03.607 --> 00:08:07.398
the zeros happen when P equals
zero and P is equal to 48,000

00:08:07.398 --> 00:08:11.218
so that's gonna happen when
the population is 24,000.

00:08:11.218 --> 00:08:14.673
So this type of problem seems
very intimidating at first,

00:08:14.673 --> 00:08:16.318
logistic differential equation,

00:08:16.318 --> 00:08:18.905
how do I actually solve
this and then analyze it

00:08:18.905 --> 00:08:20.619
but the key is to; one,

00:08:20.619 --> 00:08:23.035
recognize a logistic
differential equation,

00:08:23.035 --> 00:08:25.287
see what it's talking about

00:08:25.287 --> 00:08:28.847
and then maybe think of this
in terms of our rate of change

00:08:28.847 --> 00:08:30.781
as a function of our population.

00:08:30.781 --> 00:08:33.758
And remember that the carrying
capacity is what happens

00:08:33.758 --> 00:08:38.337
as T approaches infinity
and as T approaches infinity

00:08:38.337 --> 00:08:42.074
our rate of change approaches zero.

00:08:42.074 --> 00:08:43.645
And so if that's approaching zero,

00:08:43.645 --> 00:08:46.625
what must our population be approaching.


Worked example: over- and under-estimation of Riemann sums

WEBVTT
Kind: captions
Language: en

00:00:00.753 --> 00:00:03.392
- [Instructor] The continuous
function g is graphed.

00:00:03.392 --> 00:00:05.786
We're interested in the
area under the curve

00:00:05.786 --> 00:00:08.664
between x equals negative
seven and x equals seven,

00:00:08.664 --> 00:00:11.546
and we're considering using
Riemann sums to approximate it.

00:00:11.546 --> 00:00:12.995
So this is the area that thinking about

00:00:12.995 --> 00:00:14.560
in this light blue color.

00:00:14.560 --> 00:00:18.805
Order the areas from least
on top to greatest on bottom.

00:00:18.805 --> 00:00:21.707
So this is a screenshot
from a Khan Academy exercise

00:00:21.707 --> 00:00:22.632
where you would be expected

00:00:22.632 --> 00:00:24.986
to actually click and drag these around.

00:00:24.986 --> 00:00:26.034
But it's just a screenshot.

00:00:26.034 --> 00:00:28.322
So what I'm gonna do instead
of dragging them around,

00:00:28.322 --> 00:00:30.287
I'm just gonna write numbers ordering them

00:00:30.287 --> 00:00:33.480
from least to greatest
where one would be the least

00:00:33.480 --> 00:00:36.430
and then three would be the greatest.

00:00:36.430 --> 00:00:38.293
So pause this video and
try to think about these.

00:00:38.293 --> 00:00:40.451
Which of these is the least,
which is in the middle,

00:00:40.451 --> 00:00:43.013
and which is the greatest?

00:00:43.013 --> 00:00:45.621
So let's just draw out
what a left Riemann sum,

00:00:45.621 --> 00:00:47.496
a right Riemann sum
would actually look like,

00:00:47.496 --> 00:00:50.128
and compare it to the actual area.

00:00:50.128 --> 00:00:54.467
And we could do an arbitrary
number of subdivisions.

00:00:54.467 --> 00:00:56.496
I would encourage us to do fewer

00:00:56.496 --> 00:00:59.201
because we're just trying to
get a general sense of things.

00:00:59.201 --> 00:01:01.451
And they don't even have
to be equal subdivisions.

00:01:01.451 --> 00:01:03.373
So let's start with the left Riemann sum.

00:01:03.373 --> 00:01:07.738
So we wanna start at x
equals negative seven,

00:01:07.738 --> 00:01:10.938
and we wanna go to x equals seven.

00:01:10.938 --> 00:01:12.112
Well, let's say that this is

00:01:12.112 --> 00:01:15.280
the first rectangle right over here.

00:01:15.280 --> 00:01:17.758
So this is our first subdivision.

00:01:17.758 --> 00:01:18.995
And it's a left Riemann sum,

00:01:18.995 --> 00:01:20.528
so we would use the value of the function

00:01:20.528 --> 00:01:22.234
at the left end of that subdivision,

00:01:22.234 --> 00:01:24.615
which is negative seven,
x equals negative seven.

00:01:24.615 --> 00:01:26.961
The value of the function there is 12.

00:01:26.961 --> 00:01:28.706
And so this would be our first rectangle.

00:01:28.706 --> 00:01:29.804
You already get a sense that this

00:01:29.804 --> 00:01:33.746
is gonna be an overestimate
relative to the actual area.

00:01:33.746 --> 00:01:35.957
And so the next subdivision
would start here,

00:01:35.957 --> 00:01:38.800
so this would be our
height of our rectangle.

00:01:38.800 --> 00:01:41.682
And once again, they don't
have to be equal subdivisions.

00:01:41.682 --> 00:01:44.159
They often are, but I'm just
show you unequal subdivision

00:01:44.159 --> 00:01:46.936
just to show you that this
is still a valid Riemann sum.

00:01:46.936 --> 00:01:49.879
And once again, this is an overestimate

00:01:49.879 --> 00:01:52.699
where the actual area that
we're trying to approximate

00:01:52.699 --> 00:01:55.207
is smaller than the
area of this rectangle.

00:01:55.207 --> 00:01:57.262
And then let's say the third
subdivision right over here

00:01:57.262 --> 00:01:59.729
starts right over there at x equals three.

00:01:59.729 --> 00:02:02.023
And we use the left
end of the subdivision,

00:02:02.023 --> 00:02:03.508
the value of the function there,

00:02:03.508 --> 00:02:05.081
to define the height of the rectangle.

00:02:05.081 --> 00:02:08.432
And once again, you see
it is an overestimate.

00:02:08.432 --> 00:02:12.599
So the left Riemann sum is
clearly an overestimate.

00:02:14.317 --> 00:02:16.484
And it's pretty clear why.

00:02:17.847 --> 00:02:19.677
This function never increases.

00:02:19.677 --> 00:02:20.833
It's either decreasing,

00:02:20.833 --> 00:02:23.643
or it looks like it stays
flat at certain points.

00:02:23.643 --> 00:02:26.883
And so for a function
like that, the left edge,

00:02:26.883 --> 00:02:29.018
the value of the function at the left edge

00:02:29.018 --> 00:02:31.940
is going to be just as high or higher

00:02:31.940 --> 00:02:35.440
than any other value the function takes on

00:02:36.348 --> 00:02:38.561
over that interval for the subdivision.

00:02:38.561 --> 00:02:41.635
And so you get left with
all of this extra area

00:02:41.635 --> 00:02:45.176
that is part of the overestimate.

00:02:45.176 --> 00:02:48.529
Or this area that is
larger than the actual area

00:02:48.529 --> 00:02:50.266
that you're trying to approximate.

00:02:50.266 --> 00:02:52.135
Now let's think about a right Riemann sum.

00:02:52.135 --> 00:02:53.866
And I'll do different subdivisions.

00:02:53.866 --> 00:02:55.078
Let's say the first subdivision goes

00:02:55.078 --> 00:02:57.544
from negative seven to negative five.

00:02:57.544 --> 00:03:01.254
And here we would use the right
edge to define the height.

00:03:01.254 --> 00:03:04.660
So f of negative five, or g of
negative five, I should say.

00:03:04.660 --> 00:03:06.345
So that's right over there.

00:03:06.345 --> 00:03:07.869
That's our first rectangle.

00:03:07.869 --> 00:03:12.144
Maybe our next rectangle,
the right edge is zero.

00:03:12.144 --> 00:03:15.674
So this would be it right over there.

00:03:15.674 --> 00:03:17.931
And then maybe we'll do four rectangles.

00:03:17.931 --> 00:03:20.956
Maybe our third subdivision,

00:03:20.956 --> 00:03:25.167
the right edge is at x is equal to three.

00:03:25.167 --> 00:03:28.458
So it would be right over there.

00:03:28.458 --> 00:03:30.941
And then our fourth subdivision,

00:03:30.941 --> 00:03:32.710
let's just do it x equals seven.

00:03:32.710 --> 00:03:34.702
And we're using the right
edge of the subdivisions.

00:03:34.702 --> 00:03:36.487
Remember, this is a right Riemann sum,

00:03:36.487 --> 00:03:37.673
so we use the right edge.

00:03:37.673 --> 00:03:40.942
The value of the function
there is just like that.

00:03:40.942 --> 00:03:44.168
And now you can see for any
one of these subdivisions,

00:03:44.168 --> 00:03:46.361
our rectangles are under estimates

00:03:46.361 --> 00:03:48.695
of the area under the curve.

00:03:48.695 --> 00:03:50.445
Under, underestimate.

00:03:52.243 --> 00:03:55.513
And that's because once again,
in this particular case,

00:03:55.513 --> 00:03:57.842
the function never increases.

00:03:57.842 --> 00:04:00.347
It's either decreasing or staying flat.

00:04:00.347 --> 00:04:02.676
So if you use the value of the
function at the right edge,

00:04:02.676 --> 00:04:05.481
it's going to be smaller,

00:04:05.481 --> 00:04:07.786
it's never going to be
large than the value

00:04:07.786 --> 00:04:11.936
that the function takes on in
the rest of that subdivision.

00:04:11.936 --> 00:04:14.903
And so we are continuously
underestimating.

00:04:14.903 --> 00:04:17.028
We're missing this.

00:04:17.028 --> 00:04:21.350
All of this area right over
there is not being included.

00:04:21.350 --> 00:04:22.657
So we have an underestimate.

00:04:22.657 --> 00:04:25.913
So if wanna rank these
from least to greatest,

00:04:25.913 --> 00:04:28.666
well, the right Riemann sum is the least.

00:04:28.666 --> 00:04:30.502
It is underestimating it.

00:04:30.502 --> 00:04:32.459
Then you have the actual
area of the curve,

00:04:32.459 --> 00:04:34.563
which is just the area
of the curve. (laughs)

00:04:34.563 --> 00:04:37.092
And then you have the left Riemann sum,

00:04:37.092 --> 00:04:39.259
which is the overestimate.


Over- and under-estimation of Riemann sums

WEBVTT
Kind: captions
Language: en

00:00:01.083 --> 00:00:03.521
- [Narrator] Consider the left
and right Riemann sums that

00:00:03.521 --> 00:00:06.572
would approximate the area
under y is equal to g of x

00:00:06.572 --> 00:00:10.679
between x equals two and x equals eight.

00:00:10.679 --> 00:00:14.515
We want to approximate this
light blue area right here.

00:00:14.515 --> 00:00:18.723
Are the approximations over
estimations or underestimations?

00:00:18.723 --> 00:00:20.321
So, let's just think about each of them.

00:00:20.321 --> 00:00:22.741
Let's consider the left
and right Riemann sums.

00:00:22.741 --> 00:00:23.954
First the left.

00:00:23.954 --> 00:00:25.945
I'm just gonna write left for
short but I'm talking about

00:00:25.945 --> 00:00:27.611
the left Riemann sum.

00:00:27.611 --> 00:00:30.322
They don't tell us how many
subdivisions to make for

00:00:30.322 --> 00:00:34.305
our approximation so
that's up to us to decide.

00:00:34.305 --> 00:00:36.364
Let's say we went with three subdivisions.

00:00:36.364 --> 00:00:37.927
Let's say we want to make them equal.

00:00:37.927 --> 00:00:39.964
They don't have to be,
but let's say we do.

00:00:39.964 --> 00:00:41.764
The first one would go from two to four,

00:00:41.764 --> 00:00:43.586
the next one would go from four to six,

00:00:43.586 --> 00:00:45.834
and the next one would
go from six to eight.

00:00:45.834 --> 00:00:47.320
If we do a left Riemann sum,

00:00:47.320 --> 00:00:50.437
you use the left side of
each of these subdivisions

00:00:50.437 --> 00:00:51.935
in order to find the height.

00:00:51.935 --> 00:00:54.319
You evaluate the function at
the left end of each of those

00:00:54.319 --> 00:00:58.482
subdivisions for the height of
our approximating rectangles.

00:00:58.482 --> 00:01:02.649
We would use g of two to
set the height of our first

00:01:04.648 --> 00:01:08.688
approximating rectangle, just like that.

00:01:08.688 --> 00:01:13.233
Then we would use g of four
for the next rectangle.

00:01:13.233 --> 00:01:16.227
We would be right over there.

00:01:16.227 --> 00:01:19.728
Then you would use g of six
to represent the height of

00:01:19.728 --> 00:01:24.128
our third and our final
rectangle, right over there.

00:01:24.128 --> 00:01:27.681
Now, when it's drawn out like
this, it's pretty clear that

00:01:27.681 --> 00:01:31.848
our left Riemann sum is going
to be an overestimation.

00:01:33.678 --> 00:01:35.341
Why do we know that?

00:01:35.341 --> 00:01:38.311
Because these rectangles, the
area that they're trying to

00:01:38.311 --> 00:01:40.517
approximate is contained
in the rectangles.

00:01:40.517 --> 00:01:43.869
And these rectangles have
this surplus area so they're

00:01:43.869 --> 00:01:45.758
always going to be larger than the areas

00:01:45.758 --> 00:01:47.831
that they're trying to approximate.

00:01:47.831 --> 00:01:48.783
And in general,

00:01:48.783 --> 00:01:51.442
if you have a function that's
decreasing over the interval

00:01:51.442 --> 00:01:53.229
that we care about right over here,

00:01:53.229 --> 00:01:55.731
strictly decreasing the entire time.

00:01:55.731 --> 00:01:59.103
If you use the left edge of
each subdivision to approximate,

00:01:59.103 --> 00:02:01.251
you're going to have an overestimate.

00:02:01.251 --> 00:02:02.483
Because the left edge,

00:02:02.483 --> 00:02:03.905
the value of the function there,

00:02:03.905 --> 00:02:06.044
is going to be higher than
the value of the function at

00:02:06.044 --> 00:02:08.235
any of the point in the subdivision.

00:02:08.235 --> 00:02:10.644
That's why for decreasing function,

00:02:10.644 --> 00:02:14.144
the left Riemann sum is going
to be an overestimation.

00:02:14.144 --> 00:02:17.887
Now let's think about the
right Riemann sum and you might

00:02:17.887 --> 00:02:19.985
already guess that's
going to be the opposite

00:02:19.985 --> 00:02:21.587
but let's visualize that.

00:02:21.587 --> 00:02:24.303
Let's just go with the
same three subdivisions.

00:02:24.303 --> 00:02:27.094
But now let's use the
right side of each of these

00:02:27.094 --> 00:02:30.114
subdivisions to define the height.

00:02:30.114 --> 00:02:32.515
For this first rectangle the
height is going to be defined

00:02:32.515 --> 00:02:33.598
by g of four.

00:02:35.006 --> 00:02:36.678
That's right over there.

00:02:36.678 --> 00:02:39.506
And then for the second one
it's going to be g of six.

00:02:39.506 --> 00:02:40.863
That is right over there.

00:02:40.863 --> 00:02:43.765
For the third one it's
going to be g of eight.

00:02:43.765 --> 00:02:47.286
Let me shade these in to make
it clear which rectangles

00:02:47.286 --> 00:02:48.255
we're talking about.

00:02:48.255 --> 00:02:51.957
This would be the right Riemann
sum to approximate the area.

00:02:51.957 --> 00:02:53.107
It's very clear here

00:02:53.107 --> 00:02:56.904
that this is going to be an underestimate.

00:02:56.904 --> 00:03:01.071
Underestimate because we see
in each of these intervals,

00:03:02.408 --> 00:03:06.407
the right Riemann sum or the
rectangle that we're using

00:03:06.407 --> 00:03:09.557
for the right Riemann sum
is a subset of the area

00:03:09.557 --> 00:03:11.005
that its trying to estimate.

00:03:11.005 --> 00:03:14.311
We're not able to, it doesn't
capture this extra area

00:03:14.311 --> 00:03:15.259
right over there.

00:03:15.259 --> 00:03:16.189
And once again,

00:03:16.189 --> 00:03:19.519
that is because this is a
strictly decreasing function.

00:03:19.519 --> 00:03:22.287
So if you use the right end
point of any one of these or

00:03:22.287 --> 00:03:24.859
the right side of any of
these subdivisions in order to

00:03:24.859 --> 00:03:26.208
define the height,

00:03:26.208 --> 00:03:29.667
that right value of g is going
to be the lowest value of g

00:03:29.667 --> 00:03:33.508
in that subdivision so it's
going to be a lower height than

00:03:33.508 --> 00:03:36.418
what you could say is even the
average height of the value

00:03:36.418 --> 00:03:38.607
of the function over that interval.

00:03:38.607 --> 00:03:41.423
So you're going to have an
underestimate in this situation.

00:03:41.423 --> 00:03:43.657
Now, if your function
was strictly increasing,

00:03:43.657 --> 00:03:47.157
then these two things would be
swapped around and of course,

00:03:47.157 --> 00:03:48.824
there are many functions
that are neither strictly

00:03:48.824 --> 00:03:50.322
increasing or decreasing

00:03:50.322 --> 00:03:52.754
and then it would depend on the function.

00:03:52.754 --> 00:03:53.587
Sometimes even,

00:03:53.587 --> 00:03:56.939
it would depend on the type
of subdivisions you choose to

00:03:56.939 --> 00:04:00.508
decide whether you have an
overestimate or an underestimate.


Limits from graphs

WEBVTT
Kind: captions
Language: en

00:00:00.485 --> 00:00:03.053
- [Instructor] So we have
the graph of Y equals f of x

00:00:03.053 --> 00:00:04.850
right over here and we want to figure out

00:00:04.850 --> 00:00:07.053
three different limits and like always

00:00:07.053 --> 00:00:09.043
pause this video and see
if you can figure it out

00:00:09.043 --> 00:00:11.800
on your own before we do it together.

00:00:11.800 --> 00:00:13.720
Alright now first let's
think about what's the limit

00:00:13.720 --> 00:00:15.839
of f of x it's x approaches six.

00:00:15.839 --> 00:00:19.746
So as x, I'm gonna do this
in a color you can see,

00:00:19.746 --> 00:00:22.469
as x approaches six from both sides

00:00:22.469 --> 00:00:25.309
well as we approach six
from the left hand side,

00:00:25.309 --> 00:00:27.436
from values less than six,

00:00:27.436 --> 00:00:31.565
it looks like our f of
x is approaching one

00:00:31.565 --> 00:00:35.732
and as we approach x equals
six from the right hand side

00:00:37.171 --> 00:00:41.652
it looks like our f of x is
once again approaching one

00:00:41.652 --> 00:00:43.284
and in order for this limit to exist,

00:00:43.284 --> 00:00:45.529
we need to be approaching the same value

00:00:45.529 --> 00:00:47.756
from both the left and the right hand side

00:00:47.756 --> 00:00:50.239
and so here at least graphically,

00:00:50.239 --> 00:00:53.672
so you never are sure
with a graph but this is

00:00:53.672 --> 00:00:56.838
a pretty good estimate, it looks
like we are approaching one

00:00:56.838 --> 00:00:59.972
right over there, in a darker color.

00:00:59.972 --> 00:01:02.077
Now let's do this next one.

00:01:02.077 --> 00:01:04.647
The limit of f of x is x approaches four

00:01:04.647 --> 00:01:09.007
so as we approach four
from the left hand side

00:01:09.007 --> 00:01:10.424
what is going on?

00:01:11.805 --> 00:01:14.580
Well as we approach four
from the left hand side

00:01:14.580 --> 00:01:17.387
it looks like our function,
the value of our function

00:01:17.387 --> 00:01:20.803
it looks like it is approaching three.

00:01:20.803 --> 00:01:23.578
Remember you can have a
limit exist at an x value

00:01:23.578 --> 00:01:26.405
where the function itself is not defined,

00:01:26.405 --> 00:01:28.964
the function , if you said
after four, it's not defined

00:01:28.964 --> 00:01:31.711
but it looks like when we
approach it from the left

00:01:31.711 --> 00:01:34.446
when we approach x
equals four from the left

00:01:34.446 --> 00:01:36.066
it looks like f is approaching three

00:01:36.066 --> 00:01:38.982
and then we approach four from the right,

00:01:38.982 --> 00:01:43.193
once again, it looks like our
function is approaching three

00:01:43.193 --> 00:01:48.127
so here I would say, at
least from what we can tell

00:01:48.127 --> 00:01:50.867
from the graph it looks like the limit

00:01:50.867 --> 00:01:53.834
of f of x is x approaches four is three,

00:01:53.834 --> 00:01:56.706
even though the function
itself is not defined yet.

00:01:56.706 --> 00:01:59.408
Now let's think about the
limit as x approaches two.

00:01:59.408 --> 00:02:04.306
So this is interesting the
function is defined there

00:02:04.306 --> 00:02:08.407
f of two is two, let's
see when we approach

00:02:08.407 --> 00:02:12.574
from the left hand side
it looks like our function

00:02:13.688 --> 00:02:16.271
is approaching the value of two

00:02:17.119 --> 00:02:19.923
but when we approach
from the right hand side,

00:02:19.923 --> 00:02:22.267
when we approach x equals
two from the right hand side,

00:02:22.267 --> 00:02:25.267
our function is getting
closer and closer to five

00:02:25.267 --> 00:02:28.831
it's not quite getting
to five but as we go from

00:02:28.831 --> 00:02:33.367
you know 2.1 2.01 2.001
it looks like our function

00:02:33.367 --> 00:02:36.159
the value of our function's
getting closer and closer

00:02:36.159 --> 00:02:39.418
to five and since we are
approaching two different values

00:02:39.418 --> 00:02:42.028
from the left hand side
and the right hand side

00:02:42.028 --> 00:02:44.946
as x approached two
from the left hand side

00:02:44.946 --> 00:02:47.137
and the right hand side we
would say that this limit

00:02:47.137 --> 00:02:49.887
does not exist so does not exist.

00:02:53.631 --> 00:02:54.631
Which is interesting.

00:02:54.631 --> 00:02:56.579
In this first case the
function is defined at six

00:02:56.579 --> 00:03:00.069
and the limit is equal to
the value of the function

00:03:00.069 --> 00:03:03.458
at x equals six, here the
function was not defined

00:03:03.458 --> 00:03:05.798
at x equals four, but the limit does exist

00:03:05.798 --> 00:03:10.536
here the function is defined
at f equals, x equals two

00:03:10.536 --> 00:03:13.455
but the limit does not exist
as we approach x equals two

00:03:13.455 --> 00:03:16.835
let's do another function
just to get more cases

00:03:16.835 --> 00:03:20.014
of looking at graphical limits.

00:03:20.014 --> 00:03:24.097
So here we have the graph
of Y is equal to g of x

00:03:24.983 --> 00:03:27.490
and once again pause this
video and have a go at it

00:03:27.490 --> 00:03:29.356
and see if you can figure
out these limits graphically.

00:03:29.356 --> 00:03:32.621
So first we have the
limit as x approaches five

00:03:32.621 --> 00:03:36.523
g of x so as we approach
five from the left hand side

00:03:36.523 --> 00:03:38.714
it looks like we are
approaching this value

00:03:38.714 --> 00:03:42.547
let me just draw a
straight Line that takes us

00:03:44.613 --> 00:03:47.371
so it looks like we're
approaching this value

00:03:47.371 --> 00:03:50.665
and as we approach five
from the right hand side

00:03:50.665 --> 00:03:54.031
it also looks like we are
approaching that same value.

00:03:54.031 --> 00:03:58.294
And so this value, just
eye balling it off of here

00:03:58.294 --> 00:04:02.019
looks like it's about .4
so I'll say this limit

00:04:02.019 --> 00:04:04.700
definitely exists just
when looking at a graph

00:04:04.700 --> 00:04:06.500
it's not that precise

00:04:06.500 --> 00:04:10.134
so I would say it's approximately 0.4

00:04:10.134 --> 00:04:13.288
it might be 0.41 it might be 0.41456789

00:04:13.288 --> 00:04:16.183
we don't know exactly
just looking at this graph

00:04:16.183 --> 00:04:19.227
but it looks like a value
roughly around there.

00:04:19.227 --> 00:04:21.587
Now let's think about the limit of g of x

00:04:21.587 --> 00:04:24.572
as x approaches seven so
let's do the same exercise.

00:04:24.572 --> 00:04:26.853
What happens as we approach from the left

00:04:26.853 --> 00:04:30.638
from values less than
seven 6.9, 6.99, 6.999

00:04:30.638 --> 00:04:34.666
well it looks like the
value of our function

00:04:34.666 --> 00:04:37.596
is approaching two, it doesn't matter

00:04:37.596 --> 00:04:40.445
that the actual function is
defined g of seven is five

00:04:40.445 --> 00:04:42.963
but as we approach from the left,

00:04:42.963 --> 00:04:45.752
as x goes 6.9, 6.99 and so on,

00:04:45.752 --> 00:04:48.810
it looks like our value of our function

00:04:48.810 --> 00:04:52.626
is approaching two, and as
we approach x equals seven

00:04:52.626 --> 00:04:55.514
from the right hand side it
seems like the same thing

00:04:55.514 --> 00:04:58.188
is happening it seems like
we are approaching two

00:04:58.188 --> 00:05:01.827
and so I would say that this
is going to be equal to two

00:05:01.827 --> 00:05:04.593
and so once again, the
function is defined there

00:05:04.593 --> 00:05:08.244
and the limit exists
there but the g of seven

00:05:08.244 --> 00:05:12.411
is different than the value
if the limit of g of x

00:05:13.507 --> 00:05:14.976
as x approaches seven.

00:05:14.976 --> 00:05:16.439
Now let's do one more.

00:05:16.439 --> 00:05:18.687
What's the limit as x approaches one.

00:05:18.687 --> 00:05:20.463
Well we'll do the same thing,

00:05:20.463 --> 00:05:23.244
from the left hand side,
it looks like we're going

00:05:23.244 --> 00:05:27.327
unbounded as x goes .9,
0.99, 0.999 and 0.9999

00:05:30.367 --> 00:05:33.515
it looks like we're just going
unbounded towards infinity

00:05:33.515 --> 00:05:36.007
and as we approach from
the right hand side

00:05:36.007 --> 00:05:39.724
it looks like the same thing is happening

00:05:39.724 --> 00:05:42.781
we're going unbounded to infinity.

00:05:42.781 --> 00:05:45.462
So formally, sometimes
informally people will say

00:05:45.462 --> 00:05:47.318
oh it's approaching infinity
or something like that

00:05:47.318 --> 00:05:50.288
but if we wanna be formal
about what a limit means

00:05:50.288 --> 00:05:52.736
in this context because it is unbounded

00:05:52.736 --> 00:05:55.187
we would say that it does not exist.

00:05:55.187 --> 00:05:56.437
Does not exist.


Worked example: problem involving definite integral (algebraic)

WEBVTT
Kind: captions
Language: en

00:00:00.941 --> 00:00:03.129
- [Instructor] We are told
the population of a town

00:00:03.129 --> 00:00:06.296
grows at a rate of e to the 1.2t power

00:00:08.420 --> 00:00:12.587
minus two t people per year
where t is the number of years.

00:00:14.881 --> 00:00:18.964
At t equals two years,
the town has 1,500 people.

00:00:20.533 --> 00:00:23.101
First, they ask us
approximately by how many people

00:00:23.101 --> 00:00:24.738
does the population grow

00:00:24.738 --> 00:00:27.237
between t equals two and t equals five,

00:00:27.237 --> 00:00:28.993
and then what is the town's population

00:00:28.993 --> 00:00:30.893
at t equals five years.

00:00:30.893 --> 00:00:32.328
If we actually figure
out this first question,

00:00:32.328 --> 00:00:34.384
the second question is actually
pretty straight-forward.

00:00:34.384 --> 00:00:35.855
We figure out the amount that it grows,

00:00:35.855 --> 00:00:38.432
and then add it to what we
were at at t equals two,

00:00:38.432 --> 00:00:40.147
add it to 1,500.

00:00:40.147 --> 00:00:43.549
So pause this video and see
if you can figure it out.

00:00:43.549 --> 00:00:47.337
The key here is to appreciate
that this right over here

00:00:47.337 --> 00:00:50.457
is expressing the rate of how fast

00:00:50.457 --> 00:00:52.286
the population is growing.

00:00:52.286 --> 00:00:54.946
And we've then seen in
multiple videos now,

00:00:54.946 --> 00:00:57.687
let me just draw, do a quick review

00:00:57.687 --> 00:01:00.270
of this notion of a rate curve.

00:01:01.701 --> 00:01:05.868
So those are my axes, and this
is my t-axis, my time axis.

00:01:09.108 --> 00:01:11.871
And so this is showing
me how my rate of change

00:01:11.871 --> 00:01:14.371
changes as a function of time.

00:01:15.787 --> 00:01:17.981
Let's say it's something like this.

00:01:17.981 --> 00:01:20.312
So once again, if I said at this time

00:01:20.312 --> 00:01:22.068
right over here, this is my rate,

00:01:22.068 --> 00:01:25.290
this doesn't tell me, for
example, what my population is.

00:01:25.290 --> 00:01:29.072
This tells me what is my rate
of change of a population.

00:01:29.072 --> 00:01:31.449
And we have seen in previous videos

00:01:31.449 --> 00:01:34.112
that if you wanna figure
out the change in the thing

00:01:34.112 --> 00:01:38.548
that the rate is the
rate of change of, say,

00:01:38.548 --> 00:01:41.346
the change in population,
you would find the area

00:01:41.346 --> 00:01:45.046
under the rate curve between
those two appropriate times.

00:01:45.046 --> 00:01:46.867
And why does that make sense?

00:01:46.867 --> 00:01:50.254
Well, imagine a very small
change in time right over here.

00:01:50.254 --> 00:01:52.021
If you have a very small change in time,

00:01:52.021 --> 00:01:55.226
and if you assume that your
rate is approximately constant

00:01:55.226 --> 00:01:57.705
over that very small change in time,

00:01:57.705 --> 00:02:00.064
well, then your change in,
let's say we're measuring

00:02:00.064 --> 00:02:02.511
the rate of change of population here,

00:02:02.511 --> 00:02:04.708
your accumulation, you could say,

00:02:04.708 --> 00:02:08.903
is going to be your rate
times your change in time,

00:02:08.903 --> 00:02:11.203
which would be the area of this rectangle.

00:02:11.203 --> 00:02:14.040
And so roughly that would
be the area under the curve

00:02:14.040 --> 00:02:17.627
over that very, very small change in time.

00:02:17.627 --> 00:02:20.574
What we really wanna do is
find the area under this curve

00:02:20.574 --> 00:02:23.760
from t equals two to t equals five.

00:02:23.760 --> 00:02:25.689
And we have seen multiple
times in calculus

00:02:25.689 --> 00:02:27.298
how to express that.

00:02:27.298 --> 00:02:31.215
So the definite integral
from t is equal to two

00:02:32.305 --> 00:02:36.160
to t is equal to five of this expression

00:02:36.160 --> 00:02:39.160
of e to the 1.2t minus two t dt, dt.

00:02:46.615 --> 00:02:49.266
So, if we just evaluate
that, that will be the answer

00:02:49.266 --> 00:02:51.369
to this first question.

00:02:51.369 --> 00:02:52.762
So what is this going to be?

00:02:52.762 --> 00:02:54.611
Well, let's actually work it out.

00:02:54.611 --> 00:02:58.528
So what is the antiderivative
of e to the 1.2t?

00:02:59.539 --> 00:03:01.226
Well, let me just try to do it over here.

00:03:01.226 --> 00:03:03.835
So if I am trying to calculate,

00:03:03.835 --> 00:03:08.410
let me write it as e to the
five, or actually 6/5 t.

00:03:08.410 --> 00:03:11.179
Say, 12/10 is the same thing as 6/5.

00:03:11.179 --> 00:03:14.563
6/5 t dt, so this is
an indefinite integral,

00:03:14.563 --> 00:03:17.062
I'm just trying to figure
out the antiderivative here.

00:03:17.062 --> 00:03:19.450
Well, if I had a 6/5 right over here,

00:03:19.450 --> 00:03:22.403
then use substitution, or
sometimes you would say

00:03:22.403 --> 00:03:24.756
the inverse chain rule,
would be very appropriate.

00:03:24.756 --> 00:03:26.742
Well, we could put a 6/5 there

00:03:26.742 --> 00:03:28.797
if we write a 5/6 right over here.

00:03:28.797 --> 00:03:31.331
5/6 times 6/5, then we can take constants

00:03:31.331 --> 00:03:33.267
in and out of the integral like this.

00:03:33.267 --> 00:03:35.249
Scaling constants, I should say.

00:03:35.249 --> 00:03:39.515
Well, now, so this is
going to be equal to 5/6,

00:03:39.515 --> 00:03:41.083
this 5/6 right over here.

00:03:41.083 --> 00:03:43.288
This antiderivative is
pretty straight-forward.

00:03:43.288 --> 00:03:46.153
Since I have the derivative
of 6/5 t right over here,

00:03:46.153 --> 00:03:49.149
I can find the antiderivative
with respect to 6/5 t,

00:03:49.149 --> 00:03:50.979
which is essentially I'm
doing u-substitution.

00:03:50.979 --> 00:03:53.378
If you have to do u-substitution,
you would make that

00:03:53.378 --> 00:03:57.212
right over there u, and then
that and that would be your du.

00:03:57.212 --> 00:03:58.611
But, needless to say, this would be

00:03:58.611 --> 00:04:02.778
5/6 times e to the 6/5 t,
and if you're thinking about

00:04:04.018 --> 00:04:05.542
the indefinite integral,
you would then have

00:04:05.542 --> 00:04:07.565
a plus c here, of course.

00:04:07.565 --> 00:04:09.612
And you could verify that
the derivative of this

00:04:09.612 --> 00:04:11.612
is indeed e to the 1.2t.

00:04:13.190 --> 00:04:16.247
So this is going to be equal to,

00:04:16.247 --> 00:04:18.212
so this part right over here,

00:04:18.212 --> 00:04:22.379
the antiderivative is 5/6 e to the 6/5 t,

00:04:25.480 --> 00:04:28.460
and then this part right over here,

00:04:28.460 --> 00:04:30.472
the antiderivative of two t is t squared,

00:04:30.472 --> 00:04:32.055
so minus t squared.

00:04:32.962 --> 00:04:37.154
And we are going to evaluate
that at five and two

00:04:37.154 --> 00:04:38.605
and find the difference.

00:04:38.605 --> 00:04:41.890
So let's evaluate this at
when t is equal to five.

00:04:41.890 --> 00:04:43.406
Well, you are going to, and let me

00:04:43.406 --> 00:04:44.627
color-code this a little bit.

00:04:44.627 --> 00:04:47.907
When t is equal to five you get

00:04:47.907 --> 00:04:52.074
5/6 times e to the 6/5
times five is e to the 6th

00:04:53.596 --> 00:04:56.263
minus five squared, so minus 25.

00:04:57.938 --> 00:05:00.938
And so from that I want to subtract.

00:05:02.141 --> 00:05:04.891
When we evaluate it at two we get

00:05:06.779 --> 00:05:10.946
5/6 e to the 6/5 times
two is the same thing

00:05:13.709 --> 00:05:17.626
as 12/5, or we could say that's 2.4.

00:05:18.740 --> 00:05:21.740
2.4 minus four, two squared is four.

00:05:25.262 --> 00:05:27.095
And so what do we get?

00:05:27.995 --> 00:05:30.404
Well, there's a couple of
ways that we could do this.

00:05:30.404 --> 00:05:34.101
So then we could write this
as, let me write it this way,

00:05:34.101 --> 00:05:38.620
we could write this, so
we have a 5/6 and a 5/6,

00:05:38.620 --> 00:05:42.787
so we could write this as
5/6 times e to the sixth,

00:05:44.631 --> 00:05:47.548
e to the sixth, minus e to the 2.4,

00:05:49.101 --> 00:05:52.235
minus 'cause we distribute
that negative sign,

00:05:52.235 --> 00:05:54.985
e to the 2.4, e to the 2.4 power.

00:05:56.844 --> 00:06:00.177
And then we have minus 25,
let me just get another color

00:06:00.177 --> 00:06:01.468
to keep track of it.

00:06:01.468 --> 00:06:05.444
We have minus 25, and then
you have minus negative four.

00:06:05.444 --> 00:06:07.202
So that would be plus four.

00:06:07.202 --> 00:06:09.369
So that would be minus 21.

00:06:10.748 --> 00:06:13.430
And I would need a calculator
to figure this out,

00:06:13.430 --> 00:06:15.450
so let me do that.

00:06:15.450 --> 00:06:18.950
Let me get my calculator on this computer.

00:06:19.806 --> 00:06:21.345
And, there we go.

00:06:21.345 --> 00:06:24.280
And so let's see, if we wanna find

00:06:24.280 --> 00:06:28.282
e to the sixth power, that's 403, okay,

00:06:28.282 --> 00:06:32.032
now let me figure out,
so minus, I would say,

00:06:35.157 --> 00:06:38.762
20, 2.4, that looks like
a 24, I'll correct it

00:06:38.762 --> 00:06:40.968
as soon as I get back to that screen.

00:06:40.968 --> 00:06:44.498
E to that power, e to the 2.4 power.

00:06:44.498 --> 00:06:47.525
And I get equals, so what's in parentheses

00:06:47.525 --> 00:06:50.951
is this number right here, so times 5/6.

00:06:50.951 --> 00:06:55.118
Times five divided by six
is equal to that minus 21.

00:06:57.923 --> 00:07:00.173
Minus 21, is equal to this.

00:07:01.452 --> 00:07:04.299
So, if I round to the nearest
hundredth it's going to be

00:07:04.299 --> 00:07:06.049
approximately 306.00.

00:07:08.200 --> 00:07:10.867
So this is approximately 306.00.

00:07:13.175 --> 00:07:16.053
So approximately by how many
people does the population grow

00:07:16.053 --> 00:07:17.967
between t equals two and t equals five?

00:07:17.967 --> 00:07:20.722
Well, by approximately 306 people.

00:07:20.722 --> 00:07:23.136
Let me write that down.

00:07:23.136 --> 00:07:25.647
So approximately 306 people.

00:07:25.647 --> 00:07:27.136
And they say, what is
the town's population

00:07:27.136 --> 00:07:28.735
at t equals five.

00:07:28.735 --> 00:07:32.863
Well at time two at two
years we had 1,500 people.

00:07:32.863 --> 00:07:37.030
And then we grow over this
interval by 306, so plus 306.

00:07:39.634 --> 00:07:41.727
Well, that's gonna get me to,

00:07:41.727 --> 00:07:43.506
we deserve a little bit of a drum roll,

00:07:43.506 --> 00:07:46.506
1,806 people at t equals five years.


Area between curves

WEBVTT
Kind: captions
Language: en

00:00:00.613 --> 00:00:01.797
- [Instructor] We have already covered

00:00:01.797 --> 00:00:06.511
the notion of area between
a curve and the x-axis

00:00:06.511 --> 00:00:08.624
using a definite integral.

00:00:08.624 --> 00:00:11.364
We are now going to then extend this

00:00:11.364 --> 00:00:14.336
to think about the area between curves.

00:00:14.336 --> 00:00:16.124
So let's say we care about the region

00:00:16.124 --> 00:00:19.583
from x equals a to x equals b

00:00:19.583 --> 00:00:23.229
between y equals f of x
and y is equal to g of x.

00:00:23.229 --> 00:00:25.063
So that would be

00:00:25.063 --> 00:00:26.313
this area right

00:00:27.803 --> 00:00:28.636
over here.

00:00:29.498 --> 00:00:31.124
So based on what you already know

00:00:31.124 --> 00:00:33.074
about definite integrals,

00:00:33.074 --> 00:00:35.582
how would you actually
try to calculate this?

00:00:35.582 --> 00:00:38.159
Well one natural thing that you might say

00:00:38.159 --> 00:00:39.297
is well look,

00:00:39.297 --> 00:00:41.317
if I were to take the integral

00:00:41.317 --> 00:00:42.234
from a to b

00:00:43.430 --> 00:00:44.763
of f of x dx,

00:00:46.844 --> 00:00:49.514
that would give me the entire area

00:00:49.514 --> 00:00:50.514
below f of x

00:00:51.720 --> 00:00:53.833
and above the x-axis.

00:00:53.833 --> 00:00:56.132
And then if I were to subtract from that

00:00:56.132 --> 00:00:58.848
this area right over here,

00:00:58.848 --> 00:01:00.776
which is equal to

00:01:00.776 --> 00:01:03.585
that's the definite integral from a to b

00:01:03.585 --> 00:01:04.918
of g of x dx.

00:01:07.045 --> 00:01:09.530
Well then I would net out
with the original area

00:01:09.530 --> 00:01:10.830
that I cared about.

00:01:10.830 --> 00:01:13.709
I would net out with this
area right over here.

00:01:13.709 --> 00:01:15.869
And that indeed would be the case.

00:01:15.869 --> 00:01:17.842
And we know from our
integration properties

00:01:17.842 --> 00:01:19.421
that we can rewrite this as

00:01:19.421 --> 00:01:21.348
the integral from a to b

00:01:21.348 --> 00:01:24.715
of, let me put some parentheses here,

00:01:24.715 --> 00:01:25.548
of f of x

00:01:27.409 --> 00:01:28.492
minus g of x,

00:01:29.986 --> 00:01:30.986
minus g of x

00:01:34.189 --> 00:01:35.559
dx.

00:01:35.559 --> 00:01:37.254
And now I'll make a claim to you,

00:01:37.254 --> 00:01:39.228
and we'll build a little
bit more intuition for this

00:01:39.228 --> 00:01:40.946
as we go through this video,

00:01:40.946 --> 00:01:43.245
but over an integral from a to b

00:01:43.245 --> 00:01:45.497
where f of x is greater than g of x,

00:01:45.497 --> 00:01:46.821
like this interval right over here,

00:01:46.821 --> 00:01:49.398
this is always going to be the case,

00:01:49.398 --> 00:01:51.395
that the area between the curves

00:01:51.395 --> 00:01:53.415
is going to be the integral

00:01:53.415 --> 00:01:55.830
for the x-interval that we
care about, from a to b,

00:01:55.830 --> 00:01:58.338
of f of x minus g of x.

00:01:58.338 --> 00:01:59.522
So I know what you're thinking,

00:01:59.522 --> 00:02:01.403
you're like okay well that
worked when both of them

00:02:01.403 --> 00:02:03.028
were above the x-axis,

00:02:03.028 --> 00:02:04.909
but what about the case when f of x

00:02:04.909 --> 00:02:07.115
is above the x-axis and g of x

00:02:07.115 --> 00:02:08.833
is below the x-axis?

00:02:08.833 --> 00:02:10.296
So for example,

00:02:10.296 --> 00:02:12.897
let's say that we were to
think about this interval

00:02:12.897 --> 00:02:13.918
right over here.

00:02:13.918 --> 00:02:15.822
Let's say this is the point c,

00:02:15.822 --> 00:02:17.819
and that's x equals c,

00:02:17.819 --> 00:02:20.258
this is x equals d right over here.

00:02:20.258 --> 00:02:22.765
So what if we wanted to calculate

00:02:22.765 --> 00:02:24.348
this area that I am

00:02:25.877 --> 00:02:27.850
shading in right over here?

00:02:27.850 --> 00:02:30.776
You might say well does
this actually work?

00:02:30.776 --> 00:02:32.471
Well let's think about now

00:02:32.471 --> 00:02:33.971
what the integral,

00:02:35.095 --> 00:02:37.138
let's think about what the integral

00:02:37.138 --> 00:02:40.388
from c to d of f of x dx represents.

00:02:44.035 --> 00:02:48.911
Well that would represent
this area right over here.

00:02:48.911 --> 00:02:52.161
And what would the integral from c to d

00:02:53.601 --> 00:02:55.768
of g of x dx represent?

00:02:56.899 --> 00:02:59.499
Well you might say it is

00:02:59.499 --> 00:03:01.264
this area right over here,

00:03:01.264 --> 00:03:02.541
but remember,

00:03:02.541 --> 00:03:06.349
over this interval g of
x is below the x-axis.

00:03:06.349 --> 00:03:09.205
So this would give you a negative value.

00:03:09.205 --> 00:03:11.109
But if you wanted this total area,

00:03:11.109 --> 00:03:12.944
what you could do is take this blue area,

00:03:12.944 --> 00:03:14.314
which is positive,

00:03:14.314 --> 00:03:17.286
and then subtract this negative area,

00:03:17.286 --> 00:03:21.280
and so then you would get
the entire positive area.

00:03:21.280 --> 00:03:23.811
Well this just amounted to,

00:03:23.811 --> 00:03:26.318
this is equivalent to the integral

00:03:26.318 --> 00:03:27.235
from c to d

00:03:28.547 --> 00:03:29.380
of f of x,

00:03:31.520 --> 00:03:35.119
of f of x minus g of x again,

00:03:35.119 --> 00:03:36.202
minus g of x.

00:03:37.859 --> 00:03:40.436
Let me make it clear, we've
got parentheses there,

00:03:40.436 --> 00:03:42.436
and then we have our dx.

00:03:43.803 --> 00:03:44.801
So once again,

00:03:44.801 --> 00:03:46.287
even over this interval when one of,

00:03:46.287 --> 00:03:48.145
when f of x was above the x-axis

00:03:48.145 --> 00:03:49.979
and g of x was below the x-axis,

00:03:49.979 --> 00:03:53.787
we it still boiled down to the same thing.

00:03:53.787 --> 00:03:55.297
Well let's take another scenario.

00:03:55.297 --> 00:03:57.108
Let's take the scenario when they are both

00:03:57.108 --> 00:03:59.058
below the x-axis.

00:03:59.058 --> 00:04:02.449
Let's say that we wanted to go from

00:04:02.449 --> 00:04:05.514
x equals, well I won't
use e since that is a

00:04:05.514 --> 00:04:09.206
loaded letter in mathematics,
and so is f and g.

00:04:09.206 --> 00:04:11.504
Well let's just say well
I'm kinda of running

00:04:11.504 --> 00:04:12.758
out of letters now.

00:04:12.758 --> 00:04:15.614
Let's say that I am gonna go from

00:04:15.614 --> 00:04:18.494
I don't know, let's just call this m,

00:04:18.494 --> 00:04:21.141
and let's call this n right over here.

00:04:21.141 --> 00:04:25.134
Well n is getting, let's
put n right over here.

00:04:25.134 --> 00:04:27.108
So what I care about

00:04:27.108 --> 00:04:28.191
is this area,

00:04:29.569 --> 00:04:31.497
the area once again below f.

00:04:31.497 --> 00:04:33.215
We're assuming that we're
looking at intervals

00:04:33.215 --> 00:04:34.701
where f is greater than g,

00:04:34.701 --> 00:04:36.791
so below f and greater than g.

00:04:36.791 --> 00:04:38.672
Will it still amount to this with now

00:04:38.672 --> 00:04:40.552
the endpoints being m and n?

00:04:40.552 --> 00:04:42.735
Well let's think about it a little bit.

00:04:42.735 --> 00:04:45.220
If we were to evaluate that integral

00:04:45.220 --> 00:04:46.137
from m to n

00:04:48.006 --> 00:04:51.094
of, I'll just put my dx here,

00:04:51.094 --> 00:04:51.927
of f of x

00:04:54.066 --> 00:04:54.899
minus,

00:04:56.226 --> 00:04:57.309
minus g of x,

00:04:58.780 --> 00:05:01.311
we already know from
our integral properties,

00:05:01.311 --> 00:05:03.842
this is going to be equal to the integral

00:05:03.842 --> 00:05:05.305
from m to n

00:05:05.305 --> 00:05:06.555
of f of x dx

00:05:08.115 --> 00:05:09.615
minus the integral

00:05:11.249 --> 00:05:12.166
from m to n

00:05:13.269 --> 00:05:14.602
of g of x dx.

00:05:15.847 --> 00:05:17.867
Now let's think about what
each of these represent.

00:05:17.867 --> 00:05:20.584
So this yellow integral right over here,

00:05:20.584 --> 00:05:21.834
that would give

00:05:23.115 --> 00:05:26.342
this the negative of this area.

00:05:26.342 --> 00:05:28.920
So that would give a negative value here.

00:05:28.920 --> 00:05:31.845
But the magnitude of it,
the absolute value of it,

00:05:31.845 --> 00:05:34.144
would be this area right over there.

00:05:34.144 --> 00:05:35.677
Now what would just the integral,

00:05:35.677 --> 00:05:37.256
not even thinking about
the negative sign here,

00:05:37.256 --> 00:05:38.974
what would the integral of this g of x

00:05:38.974 --> 00:05:41.040
of this blue integral give?

00:05:41.040 --> 00:05:43.571
Well that would give this the negative

00:05:43.571 --> 00:05:45.615
of this entire area.

00:05:45.615 --> 00:05:48.401
But now we're gonna take
the negative of that,

00:05:48.401 --> 00:05:51.907
and so this part right over here,

00:05:51.907 --> 00:05:54.415
this entire part including
this negative sign,

00:05:54.415 --> 00:05:56.250
would give us,

00:05:56.250 --> 00:05:59.036
would give us this entire area,

00:05:59.036 --> 00:06:00.383
the entire area.

00:06:00.383 --> 00:06:02.171
This would actually give a positive value

00:06:02.171 --> 00:06:04.168
because we're taking the
negative of a negative.

00:06:04.168 --> 00:06:06.559
But if with the area that we care about

00:06:06.559 --> 00:06:09.671
right over here, the area that
we cared about originally,

00:06:09.671 --> 00:06:12.759
we would want to subtract
out this yellow area.

00:06:12.759 --> 00:06:14.663
Well this right over here,

00:06:14.663 --> 00:06:16.219
this yellow integral from,

00:06:16.219 --> 00:06:19.377
the definite integral
from m to n of f of x dx,

00:06:19.377 --> 00:06:20.398
that's exactly that.

00:06:20.398 --> 00:06:22.674
That is the negative of that yellow area.

00:06:22.674 --> 00:06:25.716
So if you add the blue area,

00:06:25.716 --> 00:06:27.991
and so the negative of a
negative is gonna be positive,

00:06:27.991 --> 00:06:29.477
and then this is going to be the negative

00:06:29.477 --> 00:06:30.778
of the yellow area,

00:06:30.778 --> 00:06:32.775
you would net out once again to the area

00:06:32.775 --> 00:06:34.493
that we think about.

00:06:34.493 --> 00:06:36.025
So in every case we saw,

00:06:36.025 --> 00:06:37.697
if we're talking about an interval

00:06:37.697 --> 00:06:39.903
where f of x is greater than g of x,

00:06:39.903 --> 00:06:42.341
the area between the curves

00:06:42.341 --> 00:06:44.826
is just the definite
integral over that interval

00:06:44.826 --> 00:06:47.493
of f of x minus g of x dx.


Example: Describing a distribution

WEBVTT
Kind: captions
Language: en

00:00:00.537 --> 00:00:03.356
- [Instructor] Sometimes in
life, like say, on an exam,

00:00:03.356 --> 00:00:05.854
in particular, like an AP exam,

00:00:05.854 --> 00:00:10.021
you might be asked to describe
or compare a distribution.

00:00:11.065 --> 00:00:12.600
And so we're gonna get an example

00:00:12.600 --> 00:00:15.788
of doing that right over here.

00:00:15.788 --> 00:00:18.753
Sometimes in life, say on an exam,

00:00:18.753 --> 00:00:21.371
especially on something like an AP exam,

00:00:21.371 --> 00:00:26.110
you're asked to describe
or compare a distribution.

00:00:26.110 --> 00:00:28.559
And what we're gonna do in
this video is do exactly that,

00:00:28.559 --> 00:00:30.179
in fact, this one we're gonna describe

00:00:30.179 --> 00:00:33.230
and in a future video we're
going to compare distributions.

00:00:33.230 --> 00:00:36.190
Now before we even read
about this distribution

00:00:36.190 --> 00:00:40.588
or look at this distribution,
if you're asked to describe

00:00:40.588 --> 00:00:44.129
a distribution, there's
four things that you should

00:00:44.129 --> 00:00:45.411
be thinking about.

00:00:45.411 --> 00:00:49.120
You should be thinking about
the shape of the distribution.

00:00:49.120 --> 00:00:51.857
And when we're talking about shape,

00:00:51.857 --> 00:00:55.299
there could be left-skew,
there could be right-skew,

00:00:55.299 --> 00:00:57.730
and we'll see examples of these.

00:00:57.730 --> 00:00:59.625
And we've talked about them
in detail in other videos.

00:00:59.625 --> 00:01:01.478
It could be symmetric,

00:01:01.478 --> 00:01:04.185
these are the ones that we typically see,

00:01:04.185 --> 00:01:07.616
although there might be
other types of shapes,

00:01:07.616 --> 00:01:11.299
you'll have your center of distribution.

00:01:11.299 --> 00:01:12.949
And there's multiple
ways to be thinking about

00:01:12.949 --> 00:01:15.830
the center of distribution,
we've talked about this before,

00:01:15.830 --> 00:01:19.298
you have your mean, you have your median,

00:01:19.298 --> 00:01:21.533
these are the two most typical ones.

00:01:21.533 --> 00:01:23.866
You have a notion of spread.

00:01:24.718 --> 00:01:27.323
And for spread you could use range,

00:01:27.323 --> 00:01:30.625
you could use interquartile range,

00:01:30.625 --> 00:01:34.792
you could use something like
a mean absolute deviation,

00:01:37.926 --> 00:01:40.240
you could use the standard deviation,

00:01:40.240 --> 00:01:42.511
these are all measures of spread.

00:01:42.511 --> 00:01:45.267
And then, you probably
should at least comment

00:01:45.267 --> 00:01:47.284
about outliers, even
if you don't see them,

00:01:47.284 --> 00:01:49.673
it's a good idea to
comment just to make sure

00:01:49.673 --> 00:01:53.294
that you are being
relatively comprehensive.

00:01:53.294 --> 00:01:57.256
So now, given that, let's
describe the distribution

00:01:57.256 --> 00:01:58.232
right over here.

00:01:58.232 --> 00:02:00.057
It says in the state of Connecticut,

00:02:00.057 --> 00:02:02.640
the Department of Motor Vehicles, the DMV,

00:02:02.640 --> 00:02:06.713
requires 16 and 17 year
olds to take a 25 question

00:02:06.713 --> 00:02:10.868
knowledge test in order to
obtain a Learner's Permit.

00:02:10.868 --> 00:02:14.509
To pass, prospective drivers
must correctly answer

00:02:14.509 --> 00:02:17.051
at least 20 questions.

00:02:17.051 --> 00:02:21.002
On one Monday, 22 teenagers took the test.

00:02:21.002 --> 00:02:24.169
The dot plot below shows their scores.

00:02:25.076 --> 00:02:26.370
So why don't you pause this video

00:02:26.370 --> 00:02:28.579
and see if you can take a
shot at describing the shape,

00:02:28.579 --> 00:02:30.729
the center, the spread and the outliers.

00:02:30.729 --> 00:02:32.492
Some of these you might
be able to come with

00:02:32.492 --> 00:02:34.487
the actual numbers, you might be able

00:02:34.487 --> 00:02:35.808
to calculate some of these,

00:02:35.808 --> 00:02:37.330
but really just to get a sense of it,

00:02:37.330 --> 00:02:38.994
why don't you take a shot at it.

00:02:38.994 --> 00:02:40.136
Alright, now let's do this together.

00:02:40.136 --> 00:02:42.723
So first, on the shape.

00:02:42.723 --> 00:02:45.332
So what we see is, we have,

00:02:45.332 --> 00:02:48.618
in most of the distribution
is in this part

00:02:48.618 --> 00:02:52.697
between 20 and 25, but then
we have this fairly long

00:02:52.697 --> 00:02:54.551
tail to the left

00:02:54.551 --> 00:02:58.294
and so this tells us
that we have a left-skew

00:02:58.294 --> 00:03:02.576
or it is a left-skewed
distribution right over here.

00:03:02.576 --> 00:03:06.909
So we have done the shape,
it's a left-skewed distribution

00:03:06.909 --> 00:03:09.876
because the tail goes to the left.

00:03:09.876 --> 00:03:13.076
Now what about the center
of this distribution?

00:03:13.076 --> 00:03:14.868
So there's a few ways to measure center,

00:03:14.868 --> 00:03:17.433
mean or median, just for
the sake of simplicity,

00:03:17.433 --> 00:03:19.430
I'll think about the median here,

00:03:19.430 --> 00:03:21.213
I can eyeball that to some degree.

00:03:21.213 --> 00:03:22.799
You could also calculate the mean,

00:03:22.799 --> 00:03:24.525
it would take a little bit more time.

00:03:24.525 --> 00:03:27.562
I would guess that it's someplace,

00:03:27.562 --> 00:03:30.051
not even calculating it, I
would guess that it's someplace

00:03:30.051 --> 00:03:31.880
in this range right over there

00:03:31.880 --> 00:03:34.140
but let me actually calculate it.

00:03:34.140 --> 00:03:37.630
So the median, there's 22 data points,

00:03:37.630 --> 00:03:41.236
so the median is whatever number
has 11 onto the right of it

00:03:41.236 --> 00:03:44.386
and 11 to the left, half of 22.

00:03:44.386 --> 00:03:48.118
So let's see, we have one,
two, three, four, five,

00:03:48.118 --> 00:03:50.785
six, seven, eight, nine, 10, 11.

00:03:53.144 --> 00:03:54.727
So the median here,

00:03:55.706 --> 00:03:57.189
is going to be,

00:03:57.189 --> 00:03:59.022
let's see, this is 23.

00:04:00.161 --> 00:04:02.013
Because we have a bunch of 23s,

00:04:02.013 --> 00:04:05.814
one, two, three, four, five, six 23s

00:04:05.814 --> 00:04:08.201
and if we were to just
order all of the data points

00:04:08.201 --> 00:04:11.618
11 of the data points would be 23 or less

00:04:12.477 --> 00:04:14.727
and 11 would be 23 or more.

00:04:15.729 --> 00:04:19.562
So our median here, so
I could say our center,

00:04:20.855 --> 00:04:23.188
is 23, if we use the median.

00:04:24.463 --> 00:04:26.291
And actually, let me write that down.

00:04:26.291 --> 00:04:30.386
So our median is 23, that's
the measure of center

00:04:30.386 --> 00:04:32.253
that I decided to use.

00:04:32.253 --> 00:04:34.378
Now, what about spread?

00:04:34.378 --> 00:04:37.283
Well the simplest measure
of spread is just the range

00:04:37.283 --> 00:04:40.407
which is just the highest
value minus the lowest value.

00:04:40.407 --> 00:04:43.157
And so our range here would be 25

00:04:44.298 --> 00:04:45.215
minus four.

00:04:46.430 --> 00:04:49.062
25 minus four is equal to 21,

00:04:49.062 --> 00:04:51.119
so that is a measure of range.

00:04:51.119 --> 00:04:54.986
You could have others but this
one is very easy to calculate

00:04:54.986 --> 00:04:58.673
and then if we think about
outliers, well there are a few

00:04:58.673 --> 00:05:00.171
outliers I would consider

00:05:00.171 --> 00:05:02.582
and it's very subjective.

00:05:02.582 --> 00:05:05.068
People can debate, you know
if there's a dot right over

00:05:05.068 --> 00:05:07.339
there, is that an outlier or not.

00:05:07.339 --> 00:05:10.173
But I would say that these four

00:05:10.173 --> 00:05:13.649
right over here, I
would consider outliers.

00:05:13.649 --> 00:05:17.259
So I would say
approximately four outliers,

00:05:17.259 --> 00:05:18.859
but once again this is subjective.

00:05:18.859 --> 00:05:21.627
The main point of this exercise is to just

00:05:21.627 --> 00:05:24.381
get in the habit of
thinking about these things.

00:05:24.381 --> 00:05:28.165
And statistics is all
about creating engineering,

00:05:28.165 --> 00:05:31.111
one could say, different
measurements for center,

00:05:31.111 --> 00:05:32.558
for spread.

00:05:32.558 --> 00:05:36.061
Different ways to describe the shape,

00:05:36.061 --> 00:05:38.440
but the point is is to just think about

00:05:38.440 --> 00:05:39.867
these various dimensions.


Example: Comparing distributions

WEBVTT
Kind: captions
Language: en

00:00:00.652 --> 00:00:02.054
- [Instructor] What we're
going to do in this video

00:00:02.054 --> 00:00:05.389
is start to compare distributions.

00:00:05.389 --> 00:00:07.572
So, for example here, we
have two distributions

00:00:07.572 --> 00:00:10.003
that show the various temperatures

00:00:10.003 --> 00:00:13.314
different cities get during
the month of January.

00:00:13.314 --> 00:00:15.021
This is the distribution for Portland.

00:00:15.021 --> 00:00:16.910
For example, they get eight days

00:00:16.910 --> 00:00:19.592
between one and four degrees Celsius.

00:00:19.592 --> 00:00:22.439
The get 12 days between four
and seven degrees Celsius,

00:00:22.439 --> 00:00:23.698
and so forth, and so on,

00:00:23.698 --> 00:00:26.370
and then this is the
distribution for Minneapolis.

00:00:26.370 --> 00:00:29.062
Now, when we make these
comparisons, what we're going

00:00:29.062 --> 00:00:32.267
to focus on is the center
of the distributions,

00:00:32.267 --> 00:00:35.350
to compare that, and also the spread.

00:00:36.239 --> 00:00:37.269
Sometimes people will talk

00:00:37.269 --> 00:00:40.569
about the variability
of the distributions.

00:00:40.569 --> 00:00:43.451
And so, this is, these are the things

00:00:43.451 --> 00:00:44.888
that we're going to compare.

00:00:44.888 --> 00:00:46.316
And making the comparison,

00:00:46.316 --> 00:00:48.466
we're actually going to try to eyeball it.

00:00:48.466 --> 00:00:51.070
We're not going to try to
pick a measure of central

00:00:51.070 --> 00:00:53.586
tendencies, say the mean or
the median and then calculate

00:00:53.586 --> 00:00:55.535
precisely what those
numbers are for these.

00:00:55.535 --> 00:00:57.922
We might wanna do those if they're close.

00:00:57.922 --> 00:01:00.479
But if we can eyeball it,
that would be even better.

00:01:00.479 --> 00:01:02.947
Similar for the spread and variability.

00:01:02.947 --> 00:01:05.133
In either of these cases,
there are multiple measures

00:01:05.133 --> 00:01:07.196
in our statistical toolkit center.

00:01:07.196 --> 00:01:10.738
Mean, median is valuable of the center.

00:01:10.738 --> 00:01:14.894
For spread/variability, the
range, the interquartile range,

00:01:14.894 --> 00:01:17.398
the mean absolute deviation,
the standard deviation.

00:01:17.398 --> 00:01:18.620
These are all measures.

00:01:18.620 --> 00:01:22.354
But sometimes, you can just
kind of gauge it by looking.

00:01:22.354 --> 00:01:26.354
So, in this first comparison,
which distribution

00:01:27.876 --> 00:01:31.543
has a higher center,
or are they comparable?

00:01:33.743 --> 00:01:37.856
Well, if you look at the
distribution for Portland,

00:01:37.856 --> 00:01:40.564
the center of this distribution,
let's say if we were

00:01:40.564 --> 00:01:43.098
to just think about the mean,
although I think the mean

00:01:43.098 --> 00:01:46.196
and the median would be
reasonably close right over here.

00:01:46.196 --> 00:01:49.550
It seems like it would
be around, it would be

00:01:49.550 --> 00:01:53.765
around seven or maybe a
little bit lower than seven.

00:01:53.765 --> 00:01:55.434
So it would be kind of in...

00:01:55.434 --> 00:01:57.030
It would be kind of in that range.

00:01:57.030 --> 00:02:00.678
Maybe between five and seven
would be our central tendency.

00:02:00.678 --> 00:02:02.577
It would be either our mean or our median,

00:02:02.577 --> 00:02:06.162
while for Minneapolis,
it looks like our center

00:02:06.162 --> 00:02:08.958
is much closer to maybe negative two

00:02:08.958 --> 00:02:10.851
or negative three degrees Celsius.

00:02:10.851 --> 00:02:12.374
So here, even though we don't know

00:02:12.374 --> 00:02:13.853
precisely what the mean

00:02:13.853 --> 00:02:16.516
or the median is of each
of these distributions,

00:02:16.516 --> 00:02:20.516
you can say that Portland,
Portland distribution

00:02:23.678 --> 00:02:25.345
has a higher center.

00:02:28.437 --> 00:02:29.937
Has higher center.

00:02:33.793 --> 00:02:36.957
However, you wanna measure
either mean or median.

00:02:36.957 --> 00:02:39.369
Now, what about the spread or variability?

00:02:39.369 --> 00:02:42.434
Well, if you just superficially
thought about range,

00:02:42.434 --> 00:02:46.187
you see here that there's
nothing below one degree Celsius

00:02:46.187 --> 00:02:47.640
and nothing above 13.

00:02:47.640 --> 00:02:52.143
So you have about a 13-degree
range at most right over here.

00:02:52.143 --> 00:02:54.805
In fact, what might be
contributing to this first column?

00:02:54.805 --> 00:02:56.566
It might be a bunch of
things at three degrees

00:02:56.566 --> 00:02:58.246
or even 3.9 degrees,

00:02:58.246 --> 00:03:00.240
and similarly, what's
contributing to this last column

00:03:00.240 --> 00:03:02.522
might be a bunch of
things at 10.1 degrees.

00:03:02.522 --> 00:03:05.761
But at most, you have a
12-degree range right over here,

00:03:05.761 --> 00:03:09.086
while over here, it looks
like you have, well,

00:03:09.086 --> 00:03:12.936
it looks like it's
approaching a 27-degree range.

00:03:12.936 --> 00:03:15.827
So, based on that, and even
if you just eyeball it,

00:03:15.827 --> 00:03:17.941
this is just, we're using the same scales

00:03:17.941 --> 00:03:20.797
for our horizontal axes
here, the temperature axes,

00:03:20.797 --> 00:03:24.047
and this is just a much wider distribution

00:03:24.047 --> 00:03:26.188
than what you see over here.

00:03:26.188 --> 00:03:28.070
And so you would say that
the Minneapolis distribution

00:03:28.070 --> 00:03:30.592
has more spread or a higher spread

00:03:30.592 --> 00:03:32.545
or more variability.

00:03:32.545 --> 00:03:35.378
So, higher spread right over here.

00:03:36.542 --> 00:03:37.777
Let's do another example.

00:03:37.777 --> 00:03:41.944
And we'll use a different
representation for the data here.

00:03:42.820 --> 00:03:44.203
So we're told at the Olympic games,

00:03:44.203 --> 00:03:47.013
many events have several
rounds of competition.

00:03:47.013 --> 00:03:50.577
One of these events is the
men's 100-meter backstroke.

00:03:50.577 --> 00:03:53.132
The upper dot plot shows
the times in seconds

00:03:53.132 --> 00:03:54.526
of the top eight finishers

00:03:54.526 --> 00:03:57.315
in the final round of the 2012 Olympics.

00:03:57.315 --> 00:03:59.672
So that's the green right
over here, the final round.

00:03:59.672 --> 00:04:02.971
The lower dot plot shows
the times of the same

00:04:02.971 --> 00:04:06.350
eight swimmers but in
the semi-final round.

00:04:06.350 --> 00:04:08.850
So, given these distributions,

00:04:10.207 --> 00:04:12.707
which one has a higher center?

00:04:14.011 --> 00:04:16.650
Well, once again, and
here, you can actually,

00:04:16.650 --> 00:04:18.761
it's a little bit easier to eyeball here

00:04:18.761 --> 00:04:19.890
what the median might be.

00:04:19.890 --> 00:04:21.787
The mean, I would probably have to do

00:04:21.787 --> 00:04:23.237
a little bit more mathematics.

00:04:23.237 --> 00:04:24.942
But let's say the median...

00:04:24.942 --> 00:04:25.775
Let's see.

00:04:25.775 --> 00:04:26.768
There's one, two, three, four, five,

00:04:26.768 --> 00:04:28.641
six, seven, eight data points.

00:04:28.641 --> 00:04:29.802
So the median is gonna sit

00:04:29.802 --> 00:04:32.107
between the lower four and the upper four.

00:04:32.107 --> 00:04:34.156
So the central tendency right over here

00:04:34.156 --> 00:04:36.115
is for the final round, is,

00:04:36.115 --> 00:04:39.772
looks like it's around 57.1 seconds,

00:04:39.772 --> 00:04:42.322
while if we, especially if
we think about the median,

00:04:42.322 --> 00:04:45.330
while the central tendency
for the semi-final round,

00:04:45.330 --> 00:04:46.163
let's see.

00:04:46.163 --> 00:04:46.996
One, two, three, four,

00:04:46.996 --> 00:04:48.077
five, six, seven, eight.

00:04:48.077 --> 00:04:50.584
Looks like it is right about there.

00:04:50.584 --> 00:04:54.251
So this is about 57,
more than 57.3 seconds.

00:04:57.037 --> 00:04:59.664
So, the semi-final round
seems to have a higher

00:04:59.664 --> 00:05:02.540
central tendency, which is a
little bit counterintuitive.

00:05:02.540 --> 00:05:04.342
You would expect the
finalists to be running

00:05:04.342 --> 00:05:06.387
faster on average than the semi-finalists,

00:05:06.387 --> 00:05:07.957
but that's not what this data is showing.

00:05:07.957 --> 00:05:11.457
So the semi-final round has higher center.

00:05:12.351 --> 00:05:14.184
Higher, higher center.

00:05:16.854 --> 00:05:19.274
And I just eyeballed the median.

00:05:19.274 --> 00:05:20.778
And I suspect that the mean

00:05:20.778 --> 00:05:23.523
would also be higher in
this second distribution.

00:05:23.523 --> 00:05:26.090
And now what about variability?

00:05:26.090 --> 00:05:28.213
Well, once again, if you
just looked at range,

00:05:28.213 --> 00:05:30.192
these are both at the same scale.

00:05:30.192 --> 00:05:33.865
If you just visually look,
the variability here,

00:05:33.865 --> 00:05:37.185
the range for the final round is larger

00:05:37.185 --> 00:05:40.668
than the range for the semi-final round.

00:05:40.668 --> 00:05:42.143
So you would say that the final round

00:05:42.143 --> 00:05:44.060
has higher variability.

00:05:46.462 --> 00:05:47.462
Variability.

00:05:48.324 --> 00:05:49.762
It has a higher range.

00:05:49.762 --> 00:05:53.436
Eyeballing it, it looks
like it has a higher,

00:05:53.436 --> 00:05:56.316
a higher spread, and
there is of course times

00:05:56.316 --> 00:05:59.592
where one distribution
could have a higher range

00:05:59.592 --> 00:06:02.292
but then it might have a
lower standard deviation.

00:06:02.292 --> 00:06:04.693
For example, you could have
data that's like, you know,

00:06:04.693 --> 00:06:06.707
two data points that are really far apart,

00:06:06.707 --> 00:06:09.658
but then all the other
data just sits right,

00:06:09.658 --> 00:06:11.352
it's really, really closely packed.

00:06:11.352 --> 00:06:14.497
So, for example, a
distribution like this...

00:06:14.497 --> 00:06:16.248
I'll draw the horizontal axis here,

00:06:16.248 --> 00:06:19.054
just so you can imagine
it as a distribution.

00:06:19.054 --> 00:06:20.626
A distribution like this

00:06:20.626 --> 00:06:25.532
might have a higher range
but lower standard deviation

00:06:25.532 --> 00:06:28.032
than a distribution like this.

00:06:29.446 --> 00:06:30.285
Let me just...

00:06:30.285 --> 00:06:31.906
I'm just drawing a very rough example.

00:06:31.906 --> 00:06:35.406
A distribution like this has a lower range

00:06:37.035 --> 00:06:40.793
but actually might have a
higher standard deviation.

00:06:40.793 --> 00:06:42.090
Might have a higher standard deviation

00:06:42.090 --> 00:06:42.923
than the one above it.

00:06:42.923 --> 00:06:44.967
In fact, I can make that even better.

00:06:44.967 --> 00:06:46.967
A distribution like this

00:06:49.598 --> 00:06:51.982
would have a lower range

00:06:51.982 --> 00:06:55.419
but it would also have a
higher standard deviation.

00:06:55.419 --> 00:06:57.048
So you can't just look at...

00:06:57.048 --> 00:06:59.673
It's not always the case
that just by looking at one

00:06:59.673 --> 00:07:03.009
of these measures, the range
of the standard deviation,

00:07:03.009 --> 00:07:03.897
you'll know for sure,

00:07:03.897 --> 00:07:06.153
but in cases like this, it's safe to say

00:07:06.153 --> 00:07:08.025
when you're looking at it
by inspection that look,

00:07:08.025 --> 00:07:10.485
this green, the final round data

00:07:10.485 --> 00:07:13.711
does seem to have a higher
range, higher variability,

00:07:13.711 --> 00:07:15.178
and so I feel pretty good.

00:07:15.178 --> 00:07:17.669
This is a very high-level comparison.