Calculating average value of function over interval

WEBVTT
Kind: captions
Language: en

00:00:00.573 --> 00:00:02.479
- Let's say that we have the function

00:00:02.479 --> 00:00:05.458
F of X is equal to X squared

00:00:05.458 --> 00:00:07.968
plus one and what we want to do

00:00:07.968 --> 00:00:10.605
is we want to figure out the average value

00:00:10.605 --> 00:00:15.605
of our function F on the interval,

00:00:16.129 --> 00:00:19.225
on the closed interval between zero

00:00:19.225 --> 00:00:24.225
and let's say between zero and three.

00:00:24.391 --> 00:00:25.806
I encourage you to pause this video

00:00:25.948 --> 00:00:27.718
especially if you've seen the other videos

00:00:27.718 --> 00:00:29.618
on introducing the idea of an average

00:00:29.618 --> 00:00:31.031
value of a function and figure out

00:00:31.031 --> 00:00:31.787
what this is.

00:00:31.787 --> 00:00:33.865
What is the average
value of our function F

00:00:33.865 --> 00:00:34.938
over this interval?

00:00:35.621 --> 00:00:37.574
So, I'm assuming you've had a go at it.

00:00:37.574 --> 00:00:39.536
Let's just visualize what's going on

00:00:39.536 --> 00:00:42.850
and then we can actually find the average.

00:00:43.691 --> 00:00:46.141
So that's my Y axis.

00:00:46.876 --> 00:00:49.283
This is my X axis.

00:00:50.682 --> 00:00:51.774
Now over the interval between

00:00:51.774 --> 00:00:53.402
zero and three, so let's say

00:00:53.402 --> 00:00:58.402
this is the zero, this is one, two, three.

00:00:59.384 --> 00:01:00.538
It's a closed interval.

00:01:00.770 --> 00:01:03.217
When X is zero F of zero

00:01:03.217 --> 00:01:04.560
is going to be one.

00:01:04.763 --> 00:01:07.401
So, we're going to be right over here.

00:01:08.534 --> 00:01:10.408
F of one is two.

00:01:10.785 --> 00:01:14.548
So it's one, two, three.

00:01:14.573 --> 00:01:15.876
Actually, let me make my scale

00:01:15.876 --> 00:01:17.351
a little bit smaller on that.

00:01:17.351 --> 00:01:18.876
I have to go all the way up to 10.

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So this is going to be 10.

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This is going to be five.

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And then one, two, three.

00:01:27.241 --> 00:01:29.797
This is the hardest part is making

00:01:29.797 --> 00:01:30.832
this even.

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So see this is going to be in the middle.

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Pretty good, and then
let's see in the middle.

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Then we have that.

00:01:37.229 --> 00:01:38.663
Good enough. All right.

00:01:38.974 --> 00:01:40.895
So, we're going to be there.

00:01:41.370 --> 00:01:42.559
We're going to be there.

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I have obviously different scales

00:01:44.355 --> 00:01:45.475
for X and Y axis.

00:01:46.029 --> 00:01:47.787
Two squared plus one is five.

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Three squared plus one is 10.

00:01:51.077 --> 00:01:53.078
Three squared plus one is 10.

00:01:53.435 --> 00:01:55.209
So it's going to look something like this.

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This is what our function is going

00:01:56.476 --> 00:01:57.379
to look like.

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So, that's the graph of Y is

00:01:59.798 --> 00:02:01.014
equal to F of X.

00:02:01.210 --> 00:02:02.928
And we care about the average value

00:02:02.928 --> 00:02:06.179
on the closed interval
between zero and three.

00:02:06.367 --> 00:02:09.592
Between zero and three.

00:02:10.350 --> 00:02:11.728
So, one way to think about it,

00:02:11.728 --> 00:02:12.620
you could apply the formula,

00:02:12.620 --> 00:02:13.366
but it's very important to think about

00:02:13.366 --> 00:02:14.842
what does that formula actually mean?

00:02:14.842 --> 00:02:15.959
Once again, you shouldn't memorize

00:02:15.959 --> 00:02:17.064
this formula because it actually kind

00:02:17.064 --> 00:02:19.299
of falls out out of
what it actually means.

00:02:20.249 --> 00:02:23.141
So the average of our function

00:02:23.354 --> 00:02:26.861
is going to be equal to the definite

00:02:26.861 --> 00:02:28.693
integral over this interval.

00:02:28.870 --> 00:02:30.681
So, essentially the area under this curve.

00:02:30.849 --> 00:02:33.279
So, it's going to be
the definite intergral

00:02:33.279 --> 00:02:37.643
from zero to three of F of X which is

00:02:37.643 --> 00:02:41.074
X squared plus one DX.

00:02:41.333 --> 00:02:42.799
Then we're going to take this area.

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We're going to take this area

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right over here and we're going

00:02:46.041 --> 00:02:48.669
to divide it by the width of our interval

00:02:48.999 --> 00:02:50.727
to essentially come up
with the average height,

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or the average value of our function.

00:02:53.417 --> 00:02:56.091
So, we're going to divide it by B minus A,

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or three minus zero, which is just

00:02:59.527 --> 00:03:00.838
going to be three.

00:03:01.685 --> 00:03:03.194
And so now we just have to evaluate this.

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So, this is going to be equal to

00:03:04.908 --> 00:03:06.379
one third times --

00:03:07.220 --> 00:03:09.045
Let's see the antiderivative of X squared

00:03:09.045 --> 00:03:11.678
is X to the third over three.

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Antiderivative of one is X,

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and we're going to evaluate
it from zero to three.

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So, this is going to be equal to one third

00:03:22.401 --> 00:03:25.918
times when we evaluate it at three.

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Let me use another color here.

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When we evaluate it at three it's

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going to be three to the third

00:03:29.079 --> 00:03:30.413
divided by three.

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Well, that's just going to be 27 divided

00:03:31.790 --> 00:03:32.526
by three.

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That's nine plus three and then

00:03:35.965 --> 00:03:39.864
when we evaluated zero, minus zero

00:03:40.688 --> 00:03:41.970
minus zero.

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So, it's just minus.

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When you evaluated zero it's

00:03:44.963 --> 00:03:46.002
just going to be zero.

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And so, we are left with --

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I'm going to make the brackets

00:03:51.080 --> 00:03:51.912
that same color.

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This is going to be one third

00:03:53.763 --> 00:03:55.081
times 12.

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One third times 12, which is equal

00:03:59.646 --> 00:04:00.721
to four.

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Which is equal to four.

00:04:03.896 --> 00:04:05.507
So this is the average value

00:04:05.507 --> 00:04:06.338
of our function.

00:04:06.506 --> 00:04:07.873
The average value of our function

00:04:07.873 --> 00:04:12.873
over this interval is equal to four.

00:04:14.990 --> 00:04:16.969
Notice, our function actually hits

00:04:16.969 --> 00:04:19.347
that value at some point in the interval.

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At some point in the interval,

00:04:22.287 --> 00:04:24.793
something lower than two
but greater than one.

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We can maybe call that C.

00:04:27.084 --> 00:04:28.994
It looks like our
function hits that value.

00:04:29.359 --> 00:04:33.969
This is actually a generally true thing.

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This is a mean value theorem

00:04:36.260 --> 00:04:39.208
for integrals and we'll
go into more depth there.

00:04:39.348 --> 00:04:41.022
But you can see this kind of does look

00:04:41.022 --> 00:04:42.254
like it's average value.

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That if you imagine the box, if you

00:04:43.831 --> 00:04:45.520
multiplied this height, this average value

00:04:45.520 --> 00:04:47.814
times this width you would have this

00:04:47.814 --> 00:04:50.837
area right over here, and this

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area right over here is the same,

00:04:53.872 --> 00:04:56.068
this area that I'm highlighting

00:04:56.068 --> 00:04:58.067
in yellow right over here is the same

00:04:58.074 --> 00:05:00.281
as the area under the curve

00:05:00.540 --> 00:05:02.374
because we have the average height

00:05:02.374 --> 00:05:04.048
times the width is the same

00:05:04.048 --> 00:05:05.364
thing as the area under the curve.

00:05:05.406 --> 00:05:06.235
So, anyway hopefully you found

00:05:06.235 --> 00:05:07.154
that interesting.


Average value over a closed interval

WEBVTT
Kind: captions
Language: en

00:00:00.030 --> 00:00:04.410
what I want to do in this video is think
about the idea of an average value of a

00:00:04.410 --> 00:00:09.300
function over some closed interval so
what do I mean by that and how could we

00:00:09.300 --> 00:00:13.500
think about what average value of a
function even means so let's say that's

00:00:13.500 --> 00:00:18.480
my y-axis
and let's say that this is my this right

00:00:18.480 --> 00:00:24.960
over here is my x-axis and let me draw a
function here so let's say the function

00:00:24.960 --> 00:00:30.599
looks something like that that's the
graph of y is equal to Y is equal to f

00:00:30.599 --> 00:00:34.140
of X and now let's think about a closed
interval so we're going to think about

00:00:34.140 --> 00:00:38.070
the closed interval between a and B
including a and B that's what makes it

00:00:38.070 --> 00:00:42.239
closed or including our endpoints we're
going to think about this interval right

00:00:42.239 --> 00:00:47.340
over here so between x is equal to a and
X is equal to B what is the average

00:00:47.340 --> 00:00:51.660
value of this function one way to think
about it is what is the average height

00:00:51.660 --> 00:00:57.030
of this function so how could we what
would that mean well one way to think

00:00:57.030 --> 00:01:01.890
about it it would be some height so that
if we multiply it times the width of

00:01:01.890 --> 00:01:07.409
this interval we'll get the area under
the curve so another way to think about

00:01:07.409 --> 00:01:11.850
it is the area under the curve right
over here the area let me do this in a

00:01:11.850 --> 00:01:16.229
different color so the area under this
curve right over here I'll shaded in

00:01:16.229 --> 00:01:21.799
yellow we already know that we can
express this as the definite integral

00:01:21.799 --> 00:01:30.000
from A to B of f of X DX the average
value of our function over this closed

00:01:30.000 --> 00:01:37.229
interval a B let me write that over over
the closed interval between a and B

00:01:37.229 --> 00:01:42.390
including a and B we could think about
it as some height some height let me do

00:01:42.390 --> 00:01:47.220
this in a new color some value of our
function some height let me think about

00:01:47.220 --> 00:01:53.100
maybe some height right over here so
that if we multiply this height times

00:01:53.100 --> 00:01:58.590
this width we're going to get the area
of a rectangle the rectangle is would be

00:01:58.590 --> 00:02:02.670
the area of this rectangle right over
here and that rectangle is going to have

00:02:02.670 --> 00:02:07.950
the same area as the area under the
curve which is a reasonable way if you

00:02:07.950 --> 00:02:11.700
kind of remember how when you even think
about finding the area or one way to

00:02:11.700 --> 00:02:15.129
find think about the area of
a trapezoid you can multiply if you have a

00:02:15.129 --> 00:02:21.280
trapezoid if you have a trapezoid like
this you have a trapezoid like this this

00:02:21.280 --> 00:02:24.730
is you can kind of you kind of turn 90
degrees but your multiply the height

00:02:24.730 --> 00:02:31.299
times the average width of the trapezoid
and then that will give you its area so

00:02:31.299 --> 00:02:34.810
this would be the average width which in
a which in a trapezoid like this would

00:02:34.810 --> 00:02:38.739
just be halfway between this function is
not linear so it's not necessarily going

00:02:38.739 --> 00:02:44.799
to be halfway in between but it's that
same idea so how could we use this idea

00:02:44.799 --> 00:02:49.299
where this is right over here this
height this height right over here we

00:02:49.299 --> 00:02:54.730
could call this we could call this the
functions average the functions average

00:02:54.730 --> 00:02:58.239
how could we use all of this to come
back to with a formula for the the the

00:02:58.239 --> 00:03:03.280
average of a function over this closed
interval well let's just Express in math

00:03:03.280 --> 00:03:06.159
what we've already said we already said
that this function average should be

00:03:06.159 --> 00:03:10.959
some height so let's say the function
average so if I that's a height and if I

00:03:10.959 --> 00:03:16.239
multiply it times the width of this
interval so this width right over here

00:03:16.239 --> 00:03:19.690
this width right over here is just going
to be the larger value minus the smaller

00:03:19.690 --> 00:03:24.609
value so that's going to be B minus a so
the average value of the function times

00:03:24.609 --> 00:03:32.290
the width of the interval should give us
an area that is equivalent to the area

00:03:32.290 --> 00:03:38.919
under the curve so it should be equal to
the definite integral from A to B of f

00:03:38.919 --> 00:03:45.549
of X DX and so if we just if we knew all
of this other stuff we could solve for

00:03:45.549 --> 00:03:49.900
the functions average the functions
average we divide both sides by B minus

00:03:49.900 --> 00:03:55.810
a the functions average is going to be
equal to just dividing both sides by B

00:03:55.810 --> 00:04:02.620
minus a you're going to get 1 over B
minus a times the definite integral the

00:04:02.620 --> 00:04:08.799
definite integral from A to B of f of X
DX or another way to think about it

00:04:08.799 --> 00:04:11.620
you're going to figure out what the area
under the curve is over that interval

00:04:11.620 --> 00:04:15.760
you're going to divide that by the width
and then you're going to have the

00:04:15.760 --> 00:04:18.909
functions average you want me to think
about you're going to have the average

00:04:18.909 --> 00:04:23.590
height and once again I'd like to remind
you that because this is you shouldn't

00:04:23.590 --> 00:04:26.590
just sit there and try to memorize this
thing and just just get a

00:04:26.590 --> 00:04:29.770
conceptual understanding of what this is
really just trying to say area under the

00:04:29.770 --> 00:04:32.080
curve divided by the width well that's
just going to give you the average

00:04:32.080 --> 00:04:35.860
height that's going to give you the
average of the function in the next

00:04:35.860 --> 00:04:38.620
video we'll actually apply this formula
to see that it's actually

00:04:38.620 --> 00:04:43.889
straightforward to calculate if you if
you can figure out the definite integral


Ranking area estimates

WEBVTT
Kind: captions
Language: en

00:00:00.834 --> 00:00:04.300
- [Voiceover] Right over
here we have the graph of F

00:00:04.300 --> 00:00:08.167
and then we have four
different expressions.

00:00:08.167 --> 00:00:10.489
What I encourage you to
do is pause this video

00:00:10.489 --> 00:00:12.028
and see if you can
figure out which of these

00:00:12.028 --> 00:00:14.634
expressions would give
the largest quantity,

00:00:14.634 --> 00:00:16.567
the second largest quantity, the second

00:00:16.567 --> 00:00:20.811
smallest quantity and
the smallest quantity.

00:00:20.811 --> 00:00:22.767
I'm assuming you have paused the video

00:00:22.767 --> 00:00:24.079
and you have given an attempt.

00:00:24.079 --> 00:00:26.300
Now let's work through this together.

00:00:26.300 --> 00:00:28.367
This first expression right over here,

00:00:28.367 --> 00:00:31.100
we're taking the sum from
I equals zero to nine.

00:00:31.100 --> 00:00:33.967
We're actually taking
the sum of 10 things,

00:00:33.967 --> 00:00:35.586
because we're taking
the zero thing, first,

00:00:35.586 --> 00:00:37.433
second, third, all the way up to nine.

00:00:37.433 --> 00:00:38.966
So this is actually going
to be the sum of 10 things

00:00:38.966 --> 00:00:41.233
because we're starting at zero.

00:00:41.233 --> 00:00:46.233
We start at F of negative five plus zero.

00:00:50.069 --> 00:00:51.228
We're saying negative five,

00:00:51.228 --> 00:00:53.897
F of negative five plus zero.

00:00:53.897 --> 00:00:56.967
That's this height right over here.

00:00:56.967 --> 00:00:59.935
That's that height right over there.

00:00:59.935 --> 00:01:02.567
We're going to take that,
and then when I equals one

00:01:02.567 --> 00:01:04.389
it's going to be negative five plus one,

00:01:04.389 --> 00:01:05.995
which is negative four.

00:01:05.995 --> 00:01:08.033
It's that height right over there.

00:01:08.033 --> 00:01:11.966
Then negative five plus
two when I equals two.

00:01:11.966 --> 00:01:15.100
It's going to be that
height right over there.

00:01:15.100 --> 00:01:17.859
We're essentially going
to sum up all the way.

00:01:17.859 --> 00:01:20.293
This is going to be negative five,

00:01:20.293 --> 00:01:24.700
negative five all the way up to, negative

00:01:24.700 --> 00:01:28.634
five plus nine is going to
get us all the way to four.

00:01:28.634 --> 00:01:31.526
It's going to be all the way over there.

00:01:31.526 --> 00:01:33.602
All the way over there.

00:01:33.602 --> 00:01:35.700
You might be guessing,
"Well, how do I relate this?

00:01:35.700 --> 00:01:37.500
"They've already kind of made us think

00:01:37.500 --> 00:01:39.300
"that we're going to somehow relate this

00:01:39.300 --> 00:01:41.634
"to area somehow, but how do we

00:01:41.634 --> 00:01:44.166
"actually make that relationship?"

00:01:44.166 --> 00:01:47.500
Because right now, as this is written,

00:01:47.500 --> 00:01:49.535
it's just giving us essentially a bunch

00:01:49.535 --> 00:01:52.319
of the values of the
functions at different points.

00:01:52.319 --> 00:01:53.100
I guess you could say it's a lot

00:01:53.100 --> 00:01:55.700
of these heights, right over here.

00:01:55.700 --> 00:01:59.073
But one thing that might jump out at you

00:01:59.073 --> 00:02:02.166
is you could construct
rectangles, all that have

00:02:02.166 --> 00:02:05.034
width one, and so if
you multiply the height

00:02:05.034 --> 00:02:06.633
times the width, the area is going to be

00:02:06.633 --> 00:02:07.766
the same thing as the height.

00:02:07.766 --> 00:02:10.634
If we put a one times one right over here,

00:02:10.634 --> 00:02:11.708
this makes it very
clear that you're taking

00:02:11.708 --> 00:02:16.433
the height times the
width of this rectangle

00:02:16.433 --> 00:02:17.833
and then this rectangle.

00:02:17.833 --> 00:02:21.760
You essentially have
a bunch of left-handed

00:02:21.760 --> 00:02:24.719
rectangles that you could imagine are

00:02:24.719 --> 00:02:29.719
trying to estimate this bluish
area that was shaded in.

00:02:31.100 --> 00:02:32.900
It's clearly going to be an underestimate,

00:02:32.900 --> 00:02:36.817
because it's giving up these areas.

00:02:36.817 --> 00:02:38.812
It's giving up those
areas, right over there.

00:02:38.812 --> 00:02:40.966
All of these rectangles are sitting,

00:02:40.966 --> 00:02:42.168
they're either just touching or they are

00:02:42.168 --> 00:02:44.033
below the actual function.

00:02:44.033 --> 00:02:45.368
Let me just write this right over here.

00:02:45.368 --> 00:02:49.704
This is going to be an underestimate

00:02:49.704 --> 00:02:53.433
of the area of this blue area.

00:02:53.433 --> 00:02:55.659
Now let's think about
what this one is, here.

00:02:55.659 --> 00:02:59.033
This is F, it's the same thing
that we're taking the sum of.

00:02:59.033 --> 00:03:03.033
We're starting at I equals
one and we're going to 10.

00:03:03.033 --> 00:03:05.300
Once again, 10 things.

00:03:05.300 --> 00:03:08.500
Negative five plus one is negative four.

00:03:08.500 --> 00:03:12.100
F of that is this line, right over here.

00:03:12.100 --> 00:03:14.233
Is that line right over there.

00:03:14.233 --> 00:03:17.566
It looks like we're taking
right-handed rectangles,

00:03:17.566 --> 00:03:19.833
because you could say times one,

00:03:19.833 --> 00:03:22.300
times one would be the area.

00:03:22.300 --> 00:03:23.834
Obviously, if we multiply
by one, we're not

00:03:23.834 --> 00:03:26.946
changing the value of this expression.

00:03:26.946 --> 00:03:30.500
That would be the area of this
first right-handed rectangle.

00:03:30.500 --> 00:03:33.433
Then when I is equal to
two, it's going to be

00:03:33.433 --> 00:03:35.833
F of negative three, and so you're

00:03:35.833 --> 00:03:38.843
looking at this one right over here.

00:03:38.843 --> 00:03:43.833
I can, let me draw at
least this part of it.

00:03:43.833 --> 00:03:46.500
It's going to look something like this.

00:03:46.500 --> 00:03:47.765
It's going to look something like this,

00:03:47.765 --> 00:03:49.633
where now we're dealing with right-handed

00:03:49.633 --> 00:03:52.484
rectangles, and these are
clearly an overestimate.

00:03:52.484 --> 00:03:55.084
These are right-handed rectangles.

00:03:55.084 --> 00:03:57.700
You're going to go all the
way, when I is equal to 10,

00:03:57.700 --> 00:04:00.504
negative five plus 10 is five F of five.

00:04:00.504 --> 00:04:02.967
That's this line right over here,

00:04:02.967 --> 00:04:05.900
or this length right over here, F of five.

00:04:05.900 --> 00:04:08.700
Of course, we're multiplying it by one.

00:04:08.700 --> 00:04:12.233
It's going to look like that,
and we could keep going.

00:04:12.233 --> 00:04:14.146
I think you get the general idea now.

00:04:14.146 --> 00:04:16.367
These are all going to be right-handed

00:04:16.367 --> 00:04:19.433
or these are all right handed
rectangles that I've drawn.

00:04:19.433 --> 00:04:21.035
These are going to be an
overestimate of the area,

00:04:21.035 --> 00:04:23.233
because they all have this little extra,

00:04:23.233 --> 00:04:26.033
they all have this extra
region, right over here.

00:04:26.033 --> 00:04:31.033
These are going to be an overestimate.

00:04:31.168 --> 00:04:33.566
Now let's think about
this one, right over here.

00:04:33.566 --> 00:04:36.700
This is, we're going to
start at I equals one

00:04:36.700 --> 00:04:38.967
and we're going to go to 20.

00:04:38.967 --> 00:04:40.766
It looks like we're going to do rectangles

00:04:40.766 --> 00:04:45.166
instead of width one, we're
going to width one half.

00:04:45.166 --> 00:04:48.100
Once again, since we're
starting at I equals one,

00:04:48.100 --> 00:04:51.366
these look like right-handed
rectangles again.

00:04:51.366 --> 00:04:53.367
This is, that we're
going to use to estimate.

00:04:53.367 --> 00:04:55.566
Let me do this in a color
that you can actually see.

00:04:55.566 --> 00:04:57.233
I'll do this in orange.

00:04:57.233 --> 00:04:59.166
I'll do this in orange right over here.

00:04:59.166 --> 00:05:03.211
The first one is going to be
negative five plus one half.

00:05:03.211 --> 00:05:05.433
It's going to be this, F
of that, which is going

00:05:05.433 --> 00:05:08.073
to take you right over there.

00:05:08.073 --> 00:05:08.874
Then you're going to multiply

00:05:08.874 --> 00:05:12.100
it times the width, which is one half.

00:05:12.100 --> 00:05:15.500
Now we have twice as many, twice as many

00:05:15.500 --> 00:05:18.300
right-handed rectangles.

00:05:18.300 --> 00:05:21.700
It's going to look like this.

00:05:21.700 --> 00:05:22.930
Twice as, and I won't do all of them

00:05:22.930 --> 00:05:24.546
because it takes some time.

00:05:24.546 --> 00:05:26.300
We're going to have twice as many

00:05:26.300 --> 00:05:29.453
right-handed rectangles.

00:05:29.453 --> 00:05:33.869
It's still going to be an overestimate,

00:05:33.869 --> 00:05:35.468
but it's going to be
less of an overestimate

00:05:35.468 --> 00:05:37.161
than this one over here.

00:05:37.161 --> 00:05:38.633
Because this one over here, you had all

00:05:38.633 --> 00:05:41.100
of this extra green
space above the function.

00:05:41.100 --> 00:05:44.033
Now we have a lot less extra space,

00:05:44.033 --> 00:05:47.900
a lot less extra space above the function.

00:05:47.900 --> 00:05:49.367
It's a better estimate, but it's still

00:05:49.367 --> 00:05:51.233
going to be an overestimate,
because at least

00:05:51.233 --> 00:05:53.167
for this function right over here,

00:05:53.167 --> 00:05:56.301
because at least over this interval

00:05:56.301 --> 00:05:58.033
where the function is increasing,

00:05:58.033 --> 00:06:01.700
the right-handed rectangle
is giving us an overestimate.

00:06:01.700 --> 00:06:02.945
But this is a little bit more precise

00:06:02.945 --> 00:06:07.945
because we're using narrower
rectangles to estimate.

00:06:09.033 --> 00:06:13.233
Overestimate but less so, but less so

00:06:13.233 --> 00:06:15.500
than this one right over here.

00:06:15.500 --> 00:06:17.166
This right over here, this
is a definite interval,

00:06:17.166 --> 00:06:19.902
from negative five to five of F of X, DX.

00:06:19.902 --> 00:06:22.034
You can imagine, this
is essentially the limit

00:06:22.034 --> 00:06:23.766
as we take these widths to be smaller

00:06:23.766 --> 00:06:24.900
and smaller and smaller.

00:06:24.900 --> 00:06:28.100
We essentially end up having
essentially an infinite number.

00:06:28.100 --> 00:06:29.688
We approach an infinite number

00:06:29.688 --> 00:06:31.434
of these right over here.

00:06:31.434 --> 00:06:33.300
This is the actual area.

00:06:33.300 --> 00:06:35.834
This is what's actually depicted in blue.

00:06:35.834 --> 00:06:39.167
This is the actual area.

00:06:39.167 --> 00:06:42.100
If I were to list this
from largest to smallest.

00:06:42.100 --> 00:06:44.367
The biggest overestimate, this is

00:06:44.367 --> 00:06:47.766
the biggest overestimate,
right over there.

00:06:47.766 --> 00:06:49.367
This is still going to be an overestimate

00:06:49.367 --> 00:06:50.700
but it's a little bit more precise

00:06:50.700 --> 00:06:52.900
because we have more rectangles.

00:06:52.900 --> 00:06:54.700
I'll put that two.

00:06:54.700 --> 00:06:56.366
This is the actual area.

00:06:56.366 --> 00:06:57.900
I'll put that three.

00:06:57.900 --> 00:06:59.766
Then this is actually an underestimate.

00:06:59.766 --> 00:07:01.500
I would rank this four.

00:07:01.500 --> 00:07:03.766
This is the largest of the values,

00:07:03.766 --> 00:07:08.438
and this is going to be
the smallest of the values.


Heavier ball

WEBVTT
Kind: captions
Language: en

00:00:00.204 --> 00:00:02.663
- [Voiceover] So we have these
nine balls right over here.

00:00:02.663 --> 00:00:05.694
We're going to assume that
they are completely identical.

00:00:05.694 --> 00:00:07.504
At least they are identical in appearance.

00:00:07.504 --> 00:00:11.812
But one of the nine
balls is heavier, just a

00:00:11.812 --> 00:00:16.280
little bit heavier, is heavier
than the other eight balls.

00:00:16.280 --> 00:00:18.329
And my question to you is:

00:00:18.329 --> 00:00:20.387
What's the minimum number of times that

00:00:20.387 --> 00:00:23.717
we can use this scale in order to know,

00:00:23.717 --> 00:00:26.273
definitively, which is the heavier ball.

00:00:26.273 --> 00:00:29.478
So there's some number of
weighings using the scale,

00:00:29.478 --> 00:00:31.543
that after that number of weighings I know

00:00:31.543 --> 00:00:34.231
for a fact that I've
found the heavier ball.

00:00:34.231 --> 00:00:35.960
We're not going to do
something based on luck that

00:00:35.960 --> 00:00:38.579
you just happen to pick the
right ball when you weigh it.

00:00:38.579 --> 00:00:41.136
It has to be 100 percent
chance after this number

00:00:41.136 --> 00:00:43.290
of weighings that you have the ball.

00:00:43.290 --> 00:00:45.236
And so what is the minimum
number of those, what

00:00:45.236 --> 00:00:48.183
is the minimum number of
weighings using the scale?

00:00:48.183 --> 00:00:50.640
And I encourage you to pause the video and

00:00:50.640 --> 00:00:51.844
think about it as long as necessary to

00:00:51.844 --> 00:00:53.786
come up with your own conclusions.

00:00:54.985 --> 00:00:56.841
So I'm assuming you've
given a go at it, so

00:00:56.841 --> 00:00:58.401
I'll give you a couple of hints now.

00:00:58.401 --> 00:01:00.628
So my first hint is that you can do it in

00:01:00.628 --> 00:01:02.966
exactly two weighings of the scale.

00:01:02.966 --> 00:01:05.352
If I do two weighings of the scale, I know

00:01:05.352 --> 00:01:09.128
for a fact that I can find,
not just through luck,

00:01:09.128 --> 00:01:11.770
I can find definitively the heavier ball.

00:01:11.770 --> 00:01:13.138
So that's my first hint and if that

00:01:13.138 --> 00:01:14.867
helps you, pause the video.

00:01:14.867 --> 00:01:17.104
I'm about to give you another hint.

00:01:17.104 --> 00:01:20.451
So my second hint is that
with each weighing of

00:01:20.451 --> 00:01:23.188
the scale you should be able to rule out

00:01:23.188 --> 00:01:25.732
2/3 of the balls that are essentially

00:01:25.732 --> 00:01:28.404
still candidates for the heavier ball.

00:01:28.404 --> 00:01:30.994
So if that helps you, once
again, pause the video.

00:01:32.224 --> 00:01:33.958
So now I'm assuming you've had a go at it

00:01:33.958 --> 00:01:35.279
and maybe you were able to figure it out.

00:01:35.279 --> 00:01:36.437
Maybe you weren't.

00:01:36.437 --> 00:01:38.277
So now let's work through it together.

00:01:38.277 --> 00:01:40.242
So I mentioned that in each weighing

00:01:40.242 --> 00:01:42.507
you can rule out 2/3 of the balls.

00:01:42.507 --> 00:01:43.910
So how do we do that?

00:01:43.910 --> 00:01:46.526
So in the first weighing,
what we essentially do is

00:01:46.526 --> 00:01:51.526
take our nine balls and put
it into three groups of three.

00:01:51.945 --> 00:01:53.423
And we take two of those groups

00:01:53.423 --> 00:01:56.462
of three, so we take this group.

00:01:56.462 --> 00:01:58.579
Let me actually do that
in a different color.

00:01:58.579 --> 00:02:02.528
So we can take this group
of three right over here.

00:02:02.528 --> 00:02:04.705
Put those three balls on
that side of the scale.

00:02:04.705 --> 00:02:06.613
And then we can take these three balls,

00:02:06.613 --> 00:02:09.870
and put it on that side of the scale.

00:02:09.870 --> 00:02:10.710
And so you're essentially weighing

00:02:10.710 --> 00:02:14.510
three versus three balls.

00:02:14.510 --> 00:02:17.537
Now, there's a couple of outcomes here.

00:02:17.537 --> 00:02:19.829
You're either going to have a balance,

00:02:19.829 --> 00:02:22.869
you're going to have the left is heavier.

00:02:22.869 --> 00:02:24.269
So let me write that.

00:02:24.269 --> 00:02:28.965
Say it's going to tip down,
so the left is heavier.

00:02:30.150 --> 00:02:32.408
Or the right is heavier.

00:02:32.408 --> 00:02:37.408
Or the right is heavier.

00:02:37.580 --> 00:02:39.309
Now what does each of these tell you?

00:02:39.309 --> 00:02:42.735
Well if this, if they
balance, that tells you

00:02:42.735 --> 00:02:45.523
that the third group has the heavy ball.

00:02:45.523 --> 00:02:47.256
So, actually let me write it this way.

00:02:47.256 --> 00:02:51.038
If this is group one,
group two, group three,

00:02:51.038 --> 00:02:56.038
then this tells you that
group three has heavy ball.

00:02:57.116 --> 00:03:01.098
Has heavy ball.

00:03:01.098 --> 00:03:02.860
If the left is heavier, then we know

00:03:02.860 --> 00:03:05.619
group one has the heavy ball.

00:03:05.619 --> 00:03:10.151
Group one has the heavy ball.

00:03:10.151 --> 00:03:11.753
And then finally, of
course, if the right is

00:03:11.753 --> 00:03:16.753
heavier we know that group
two has the heavy ball.

00:03:19.223 --> 00:03:21.255
Now just like that with
one way, we have narrowed

00:03:21.255 --> 00:03:22.603
it down to one of the three groups.

00:03:22.603 --> 00:03:24.360
We have essentially narrowed it down.

00:03:24.360 --> 00:03:27.559
We now know that our heavy
ball is one of three balls.

00:03:27.559 --> 00:03:28.862
It's either one of these three,

00:03:28.862 --> 00:03:30.709
one of these three, or one of these three.

00:03:30.709 --> 00:03:32.113
And so I just repeat the process.

00:03:32.113 --> 00:03:35.521
But instead of doing it
with three balls at a time,

00:03:35.521 --> 00:03:37.337
I now do it with one ball at a time.

00:03:37.337 --> 00:03:39.199
So if I'm taking three balls.

00:03:39.199 --> 00:03:44.199
If I have three balls, what
I could do is, I will now...

00:03:44.718 --> 00:03:48.785
So my step two, I guess
I could say, my step two.

00:03:48.785 --> 00:03:52.719
I now weigh one versus one.

00:03:52.719 --> 00:03:54.562
And once again I have the outcomes.

00:03:54.562 --> 00:03:57.629
So if they are balanced, if they are

00:03:57.629 --> 00:04:00.610
balanced, then that means the...

00:04:00.610 --> 00:04:01.974
So once again, if we're taking, say, this

00:04:01.974 --> 00:04:03.659
group of three, we're in the balanced

00:04:03.659 --> 00:04:06.192
situation from the first weighing.

00:04:06.192 --> 00:04:10.807
And so if we put, if
we put this ball here,

00:04:10.807 --> 00:04:14.321
and this ball here, if
they are balanced then

00:04:14.321 --> 00:04:16.445
we know that this must be the heavy ball.

00:04:16.445 --> 00:04:18.251
Because these two are the same.

00:04:18.251 --> 00:04:22.807
If the left goes down,
if the left goes down

00:04:22.807 --> 00:04:24.835
then we know this is the heavy ball.

00:04:24.835 --> 00:04:27.678
And likewise, if the right goes down,

00:04:28.816 --> 00:04:31.297
if the right goes down, we know that

00:04:31.297 --> 00:04:34.700
this is going to be the heavy ball.

00:04:34.700 --> 00:04:36.119
So this is actually a little bit of

00:04:36.119 --> 00:04:37.337
a brain teaser that you see.

00:04:37.337 --> 00:04:38.446
It's a pretty common one.

00:04:38.446 --> 00:04:39.547
It's actually even, sometimes you'll

00:04:39.547 --> 00:04:40.796
hear it in some job interviews.

00:04:40.796 --> 00:04:43.176
But you can see, it comes
out of the idea that through

00:04:43.176 --> 00:04:46.533
each weighing you can
rule out 2/3 of the balls.

00:04:46.533 --> 00:04:47.917
And so you could use this principle,

00:04:47.917 --> 00:04:49.290
if you want, to drive other brain teasers.

00:04:49.290 --> 00:04:51.127
What if you had 27 balls?

00:04:51.127 --> 00:04:52.144
How many weighings would you need?

00:04:52.144 --> 00:04:53.922
What if you had 81?

00:04:53.922 --> 00:04:55.804
And sometimes when you
see this brain teaser,

00:04:55.804 --> 00:04:58.213
instead of giving you
a nice clean, I guess

00:04:58.213 --> 00:04:59.896
you could say, power of
three right over here,

00:04:59.896 --> 00:05:01.346
they might give you something off.

00:05:01.346 --> 00:05:03.143
So they might give you eight balls.

00:05:03.143 --> 00:05:05.082
But the exact same principle holds.

00:05:05.082 --> 00:05:07.706
If you had eight balls,
you could split it up

00:05:07.706 --> 00:05:10.200
into two groups of
three, and then two more.

00:05:10.200 --> 00:05:13.875
And then do the same, and then
do the exact same process.

00:05:13.875 --> 00:05:15.402
But I think people like
to do the eight balls

00:05:15.402 --> 00:05:17.251
because it takes you
a little away from the

00:05:17.251 --> 00:05:18.771
idea of maybe you have to divide it into

00:05:18.771 --> 00:05:20.664
groups of three, or something like that.

00:05:20.664 --> 00:05:24.474
Anyway, hopefully you have enjoyed this.


Riemann sums and sigma notation

WEBVTT
Kind: captions
Language: en

00:00:00.775 --> 00:00:02.803
- [Voiceover] The graph
of F is shown below.

00:00:02.935 --> 00:00:05.930
A total of 24 right hand rectangles

00:00:05.930 --> 00:00:06.978
are shown.

00:00:07.388 --> 00:00:09.417
So, what do I mean by
right hand rectangles?

00:00:09.432 --> 00:00:10.919
So, there's clearly 24 rectangles.

00:00:11.087 --> 00:00:12.108
You can count them.

00:00:12.157 --> 00:00:14.308
And right hand rectangle means that

00:00:14.308 --> 00:00:16.524
for each of these rectangles the height

00:00:16.524 --> 00:00:19.357
of the rectangle is defined by the value

00:00:19.357 --> 00:00:21.499
of the function on the right hand side

00:00:21.499 --> 00:00:22.429
of the rectangle.

00:00:22.532 --> 00:00:24.377
So you can see this is the right hand side

00:00:24.377 --> 00:00:26.313
of this first rectangle and if you take

00:00:26.313 --> 00:00:27.771
the value of the function of that point

00:00:27.771 --> 00:00:29.748
that is the height of the rectangle.

00:00:30.166 --> 00:00:33.059
A left hand rectangle would define

00:00:33.059 --> 00:00:34.852
the height of the rectangle by the value

00:00:34.852 --> 00:00:36.080
of the function on the left hand side

00:00:36.080 --> 00:00:36.827
of the rectangle.

00:00:36.827 --> 00:00:38.576
So, a left handed rectangle's height,

00:00:38.576 --> 00:00:39.927
the first rectangle's height would look

00:00:39.927 --> 00:00:40.325
like that.

00:00:40.325 --> 00:00:41.652
That's what they mean by right

00:00:41.652 --> 00:00:42.891
handed rectangle

00:00:43.123 --> 00:00:44.962
Fair enough, eight in blue.

00:00:45.022 --> 00:00:47.311
We see that. 16 in red. All right.

00:00:47.641 --> 00:00:50.847
All 24 of the rectangles
have the same width.

00:00:51.364 --> 00:00:52.780
Which of the statements below

00:00:52.780 --> 00:00:54.806
is or are true?

00:00:55.164 --> 00:00:57.198
They give us three expressions

00:00:57.198 --> 00:00:59.400
in sigma notation and they say,

00:00:59.400 --> 00:01:00.679
like this first one is the sum of the

00:01:00.679 --> 00:01:02.253
areas of the blue rectangles.

00:01:02.253 --> 00:01:03.182
This is the sum of the areas of

00:01:03.182 --> 00:01:04.101
the red rectangles.

00:01:04.188 --> 00:01:05.464
This is the sum of the areas of all

00:01:05.464 --> 00:01:06.586
of the rectangles.

00:01:07.102 --> 00:01:09.110
So, I encourage you now to pause

00:01:09.110 --> 00:01:11.625
the video and try to
determine on your own,

00:01:11.701 --> 00:01:15.242
which of the statements is or are true.

00:01:16.363 --> 00:01:19.354
So, I assume you've had a go at it.

00:01:19.495 --> 00:01:20.529
Let's just go through each of these

00:01:20.529 --> 00:01:22.720
and see whether they make sense.

00:01:22.886 --> 00:01:24.182
So this first one, the sum

00:01:24.182 --> 00:01:25.701
of the areas of the blue rectangles.

00:01:25.701 --> 00:01:27.385
Well, we know we have one, two, three,

00:01:27.385 --> 00:01:30.350
four, five, six, seven,
eight blue rectangles,

00:01:30.960 --> 00:01:32.849
and we're summing from one to eight.

00:01:32.902 --> 00:01:34.827
So it seems like we're summing eight

00:01:34.827 --> 00:01:36.177
things right over here.

00:01:36.177 --> 00:01:39.195
This is one, two, three, four, five,

00:01:39.613 --> 00:01:41.713
six, seven, eight.

00:01:41.926 --> 00:01:43.928
So, this is looking good right over here.

00:01:44.286 --> 00:01:46.272
And then we take F of something

00:01:46.272 --> 00:01:47.446
times one half.

00:01:47.783 --> 00:01:50.595
So, we're not even looking at this yet.

00:01:50.783 --> 00:01:52.339
It looks like this would be the height

00:01:52.339 --> 00:01:53.483
of each of the rectangles.

00:01:53.518 --> 00:01:54.362
Remember, we're taking the value

00:01:54.362 --> 00:01:56.021
of the function on the right hand side

00:01:56.072 --> 00:01:57.942
for the height, and this would be

00:01:57.942 --> 00:01:58.829
the width.

00:01:59.240 --> 00:02:01.166
So does it make sense that the width

00:02:01.166 --> 00:02:03.427
of each of these rectangles is one half?

00:02:04.135 --> 00:02:06.834
Well, the total distance between X

00:02:06.834 --> 00:02:09.932
equals negative five and X equals seven,

00:02:10.371 --> 00:02:11.597
that distance is 12.

00:02:11.782 --> 00:02:13.784
Five plus seven, that's 12,

00:02:14.061 --> 00:02:17.005
and we're dividing it into 24 rectangles

00:02:17.037 --> 00:02:18.094
of equal width.

00:02:18.182 --> 00:02:20.521
So, if you divide 12 divided by 24

00:02:20.529 --> 00:02:22.832
each of these are going to have a width

00:02:24.970 --> 00:02:26.304
of one half.

00:02:26.581 --> 00:02:28.893
So this is checking out that the one half.

00:02:28.893 --> 00:02:30.830
Now let's think about this part.

00:02:30.999 --> 00:02:33.150
Let's think about the F of negative five

00:02:33.150 --> 00:02:36.103
plus I over two.

00:02:36.522 --> 00:02:38.172
So, let's see.

00:02:38.172 --> 00:02:41.186
When I is equal to one, so we're going to

00:02:41.186 --> 00:02:43.518
take one half times F of negative five

00:02:43.572 --> 00:02:46.541
plus one over two.

00:02:46.719 --> 00:02:47.329
Right?

00:02:47.329 --> 00:02:49.084
I is one. So negative five plus one half

00:02:49.084 --> 00:02:50.218
is going to get us to this point right

00:02:50.218 --> 00:02:51.078
over here.

00:02:51.216 --> 00:02:54.492
F of that is going to be this

00:02:54.492 --> 00:02:57.539
distance, this height right over here.

00:02:57.993 --> 00:02:59.606
This is consistent with these being

00:02:59.606 --> 00:03:00.850
right handed rectangles.

00:03:01.305 --> 00:03:02.897
So, this is definitely the case.

00:03:02.897 --> 00:03:03.904
When I is equal to one we're

00:03:03.904 --> 00:03:08.381
definitely finding this
area right over here.

00:03:08.954 --> 00:03:10.753
When I is equal to two it's going to be

00:03:10.753 --> 00:03:13.499
negative five plus two over two.

00:03:13.977 --> 00:03:15.612
So two over two, we're going to add

00:03:15.612 --> 00:03:17.549
one, we're going to go over here.

00:03:17.745 --> 00:03:19.761
So once again we're doing one half,

00:03:19.761 --> 00:03:21.903
which is this right over here.

00:03:22.154 --> 00:03:26.130
That's the width times F of negative five

00:03:26.130 --> 00:03:27.967
plus two over two, which is F of negative

00:03:27.967 --> 00:03:30.786
four, which is this
height right over here.

00:03:30.801 --> 00:03:32.532
So, once again that is that area.

00:03:32.593 --> 00:03:33.790
You can keep following it.

00:03:33.798 --> 00:03:35.763
Everytime we're taking the function

00:03:37.578 --> 00:03:39.774
this first one is negative five

00:03:39.774 --> 00:03:42.322
plus one half, and then for each

00:03:42.322 --> 00:03:45.030
increment we're adding a half to

00:03:45.030 --> 00:03:47.620
I guess the right hand side is one way

00:03:47.620 --> 00:03:48.495
to think about it.

00:03:48.532 --> 00:03:50.545
So, this actually makes complete sense.

00:03:50.545 --> 00:03:52.194
We're doing this for the first eight

00:03:52.194 --> 00:03:55.236
so this is indeed true.

00:03:55.324 --> 00:03:58.188
This is the sum of the areas

00:03:58.869 --> 00:04:00.600
of the blue rectangles.

00:04:01.406 --> 00:04:02.833
Now let's look at this one over here.

00:04:02.847 --> 00:04:04.416
The sum of the areas
of the red rectangles.

00:04:04.958 --> 00:04:06.762
At first this looks pretty interesting.

00:04:07.354 --> 00:04:09.753
We're taking the sum of 16 things,

00:04:09.929 --> 00:04:11.740
and we do indeed have 16 things

00:04:11.836 --> 00:04:12.740
right over here.

00:04:13.069 --> 00:04:15.401
We have the width of each of those

00:04:15.435 --> 00:04:18.472
16 things or for each of those things

00:04:18.472 --> 00:04:19.713
we want to figure out an area,

00:04:19.808 --> 00:04:21.660
and it is indeed the case that each

00:04:21.660 --> 00:04:23.904
of these has a width of one half.

00:04:24.253 --> 00:04:25.872
But what happens when we take F of

00:04:25.872 --> 00:04:28.512
negative one plus I over two.

00:04:28.797 --> 00:04:32.261
So, we're starting right over here at

00:04:32.261 --> 00:04:33.802
negative one.

00:04:34.269 --> 00:04:35.983
Negative one plus I over two.

00:04:35.983 --> 00:04:36.968
When I equals one we're going

00:04:36.968 --> 00:04:38.428
to be at this point right over here,

00:04:38.660 --> 00:04:40.885
and F of that is going to be --

00:04:41.196 --> 00:04:42.034
You might say, "Hey, isn't that going

00:04:42.034 --> 00:04:43.322
to be the height of that rectangle?

00:04:43.348 --> 00:04:44.482
When I equals two isn't it going to be

00:04:44.482 --> 00:04:45.645
the height of that rectangle?

00:04:45.886 --> 00:04:47.747
And when I is equal to three isn't it

00:04:47.747 --> 00:04:49.020
going to be the height of that rectangle?"

00:04:49.566 --> 00:04:52.384
And that's where we
have to be very careful.

00:04:52.687 --> 00:04:54.352
They're going to have the same

00:04:54.352 --> 00:04:56.612
absolute value, but these are going to be

00:04:56.612 --> 00:04:58.809
these are all going to be negative values.

00:04:59.680 --> 00:05:03.120
These are all going to be negative

00:05:03.162 --> 00:05:08.162
because we see between this value of

00:05:08.910 --> 00:05:09.875
our function.

00:05:09.896 --> 00:05:11.701
So that looks like between negative

00:05:11.701 --> 00:05:14.274
one half all the way to seven our function

00:05:14.274 --> 00:05:15.601
is actually negative.

00:05:15.994 --> 00:05:17.029
So one way to think about it, you'd be

00:05:17.029 --> 00:05:18.422
having negative height so when you

00:05:18.422 --> 00:05:19.972
multiply these two things you're going

00:05:19.972 --> 00:05:21.215
to get a negative number.

00:05:21.660 --> 00:05:23.541
So this whole thing is going to be

00:05:23.541 --> 00:05:25.412
a negative number and so you're

00:05:25.412 --> 00:05:27.484
essentially going to get
the negative of the sums

00:05:27.484 --> 00:05:29.359
of the areas of the red rectangles,

00:05:29.527 --> 00:05:30.498
but that's not the same thing as the

00:05:30.498 --> 00:05:32.003
sum of the areas of the red rectangles.

00:05:32.028 --> 00:05:35.477
An area is, at least in
the traditional sense,

00:05:35.610 --> 00:05:38.066
you would expect to have, if you were

00:05:38.066 --> 00:05:39.793
just seeing what's the
area how much carpet

00:05:39.793 --> 00:05:41.882
you would need to cover
this someone would say

00:05:41.882 --> 00:05:43.278
that will be a positive value.

00:05:43.339 --> 00:05:44.778
But this is going to be a negative version

00:05:44.778 --> 00:05:45.462
of that.

00:05:45.462 --> 00:05:46.596
So that is not the sum of the areas

00:05:46.596 --> 00:05:47.958
of the areas of the red rectangles.

00:05:47.965 --> 00:05:50.574
It's the negative of the areas of the red

00:05:50.574 --> 00:05:52.466
rectangles. So we'd rule that one out.

00:05:53.165 --> 00:05:54.265
Then this last choice.

00:05:54.468 --> 00:05:55.630
So, this expression is the sum

00:05:55.630 --> 00:05:57.090
of the areas of all the rectangles.

00:05:57.573 --> 00:06:00.193
And so this one is going from I equals

00:06:00.193 --> 00:06:01.110
one to 24.

00:06:01.110 --> 00:06:02.373
So it's literally 24 things.

00:06:02.549 --> 00:06:05.505
It's starting here and
it just keeps going.

00:06:05.789 --> 00:06:08.086
And if this said from I
equals one to I equals

00:06:08.086 --> 00:06:10.246
eight it would be the first choice,

00:06:10.604 --> 00:06:12.036
but then this falls into the problem

00:06:12.036 --> 00:06:14.058
again of once we get
past, when we get to I

00:06:14.058 --> 00:06:17.464
equal nine this thing right over here

00:06:17.693 --> 00:06:19.233
will turn negative, and it's going

00:06:19.233 --> 00:06:20.551
to give the negative area.

00:06:20.781 --> 00:06:22.031
So it's essentially going to net out

00:06:22.031 --> 00:06:24.299
this positive area against this negative

00:06:24.299 --> 00:06:26.082
area right over here.

00:06:26.260 --> 00:06:28.104
So it's not the sum of the areas

00:06:28.104 --> 00:06:29.053
of all the rectangles.

00:06:29.139 --> 00:06:30.536
It's going to be this area

00:06:30.750 --> 00:06:34.244
essentially minus this
area right over there.


Tangent graph

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.700
- [Voiceover] What I
hope to do in this video

00:00:01.700 --> 00:00:06.700
is get some familiarity with
the graph of tangent of theta.

00:00:08.100 --> 00:00:11.833
And to do that, I'll set
up a little unit circle

00:00:11.833 --> 00:00:13.700
so that we can visualize what the tangent

00:00:13.700 --> 00:00:15.734
of various thetas are.

00:00:15.734 --> 00:00:19.961
Let's say that's a y-axis
and this is my x-axis.

00:00:20.566 --> 00:00:23.701
That is my x-axis, and the unit circle

00:00:23.701 --> 00:00:28.066
would look something like this.

00:00:28.233 --> 00:00:31.146
We already know, this is all a refresher

00:00:31.146 --> 00:00:33.966
of the unit circle
definition of trig functions,

00:00:33.966 --> 00:00:36.900
that if I have an angle, an angle theta,

00:00:36.900 --> 00:00:39.900
where one side is a positive x-axis

00:00:39.900 --> 00:00:42.900
and then the other side,
so this is the other side,

00:00:42.900 --> 00:00:45.700
so the angle is formed like this,

00:00:45.700 --> 00:00:48.166
that where this ray
intersects the unit circle,

00:00:48.166 --> 00:00:49.166
the coordinates of that,

00:00:49.166 --> 00:00:52.033
the x and y coordinates
are the sine of theta.

00:00:52.033 --> 00:00:54.566
Sorry, the x-coordinate
is the cosine of theta.

00:00:54.566 --> 00:00:58.261
It's the cosine of theta, sine of theta.

00:00:58.833 --> 00:01:01.900
The x-coordinate here is cosine theta.

00:01:01.900 --> 00:01:04.300
The y-coordinate there
is the sine of theta.

00:01:04.300 --> 00:01:06.500
But we're concerned
about tangent of theta.

00:01:06.500 --> 00:01:09.100
We know that tangent of
theta is the same thing

00:01:09.100 --> 00:01:13.666
as the sine of theta
over the cosine of theta.

00:01:14.100 --> 00:01:17.166
Or if you're starting from
the origin and you're going

00:01:17.166 --> 00:01:21.166
and you're taking the value of
essentially the y-coordinate,

00:01:21.166 --> 00:01:23.433
the y-coordinate over the x-coordinate,

00:01:23.433 --> 00:01:25.642
it's essentially the slope of this line.

00:01:25.950 --> 00:01:30.950
This is going to be your
change in y over change in x.

00:01:31.633 --> 00:01:33.833
This right over here is
going to be the slope,

00:01:33.833 --> 00:01:35.300
the slope, I guess you could say,

00:01:35.300 --> 00:01:37.700
of this ray right over here.

00:01:37.700 --> 00:01:39.900
That's going to help us visualize

00:01:39.900 --> 00:01:42.366
what the tangents of different thetas are.

00:01:42.366 --> 00:01:45.033
Let me clean up my unit
circle a little bit

00:01:45.033 --> 00:01:46.766
just so that we can ...

00:01:46.766 --> 00:01:48.100
All right, there we go.

00:01:48.376 --> 00:01:50.667
So now let's make a table.

00:01:50.884 --> 00:01:54.496
So let's make a table.

00:01:56.878 --> 00:01:58.900
So for various thetas, let's think about

00:01:58.900 --> 00:02:01.540
what tangent of theta is going to be,

00:02:01.540 --> 00:02:02.766
tangent of theta.

00:02:02.766 --> 00:02:05.700
So maybe the easiest one,
if theta is zero radians,

00:02:05.700 --> 00:02:08.715
so if it's zero radians, what
is the slope of this ray?

00:02:08.715 --> 00:02:10.546
That ray's, the slope is zero.

00:02:10.546 --> 00:02:13.034
As x changes, y doesn't change at all.

00:02:13.311 --> 00:02:14.832
Now let's think about,

00:02:14.832 --> 00:02:16.566
and I'm just going to pick values

00:02:16.566 --> 00:02:18.300
that are very easy for us to think about

00:02:18.300 --> 00:02:20.433
what the tangent of those values are,

00:02:20.433 --> 00:02:21.837
and they'll help us form,

00:02:21.837 --> 00:02:23.966
they'll help us think about
the shape of the graph

00:02:23.966 --> 00:02:26.266
of y is equal to tangent of theta.

00:02:26.445 --> 00:02:31.445
So let's take pi over four radians.

00:02:33.833 --> 00:02:37.700
This one right over here,
theta is equal to pi over four.

00:02:37.700 --> 00:02:38.960
Now why is that interesting?

00:02:38.960 --> 00:02:40.554
And sometimes, it's
easier thinking degrees.

00:02:40.554 --> 00:02:42.635
It's a 45-degree angle.

00:02:42.635 --> 00:02:45.833
This here, your x-coordinate
and your y-coordinate

00:02:45.833 --> 00:02:47.044
is the same.

00:02:47.044 --> 00:02:48.833
You might remember, it's
square root of two over two,

00:02:48.833 --> 00:02:49.900
square root of two over two.

00:02:49.900 --> 00:02:51.500
But the important thing is

00:02:51.500 --> 00:02:53.033
whatever you move in the x direction,

00:02:53.033 --> 00:02:55.238
you move the same in the y direction.

00:02:55.238 --> 00:02:58.566
So the slope of this ray right over here

00:02:58.566 --> 00:03:00.833
is going to be equal to one,

00:03:00.833 --> 00:03:01.966
or another way of thinking about it,

00:03:01.966 --> 00:03:04.500
tangent of theta is
going to be equal to one.

00:03:04.500 --> 00:03:06.433
Or sine of theta over cosine of theta,

00:03:06.433 --> 00:03:07.366
they're the same thing,

00:03:07.366 --> 00:03:09.433
so you're going to get one.

00:03:10.862 --> 00:03:13.032
Let me just clean that up here

00:03:13.032 --> 00:03:16.466
just because I'm going to keep
reusing the same unit circle.

00:03:16.633 --> 00:03:20.996
So if I have theta is pi over four,

00:03:20.996 --> 00:03:24.233
then the tangent of theta
is going to be equal to one.

00:03:24.233 --> 00:03:28.566
Now what if theta is equal
to negative pi over four?

00:03:28.566 --> 00:03:31.733
That is this right over here.

00:03:33.949 --> 00:03:36.866
Let me just draw a little triangle here.

00:03:36.900 --> 00:03:39.167
So when x, this x-coordinate over here,

00:03:39.167 --> 00:03:42.300
is square root of two
over two, we know that.

00:03:42.300 --> 00:03:44.202
We've seen that multiple
times, square root of two ...

00:03:44.202 --> 00:03:48.161
Actually, let me label
it a little bit better.

00:03:48.161 --> 00:03:50.166
So here, our theta is equal to

00:03:50.166 --> 00:03:52.833
negative pi over four radians.

00:03:53.109 --> 00:03:55.130
Or if you like to think in degrees,

00:03:55.130 --> 00:03:57.766
this would be negative 45 degrees.

00:03:57.766 --> 00:04:01.300
And now, your sine and
cosine of this angle

00:04:01.300 --> 00:04:03.633
are going to be the
opposites of each other.

00:04:03.633 --> 00:04:06.500
The cosine is square root of two over two.

00:04:06.500 --> 00:04:08.167
The x-coordinate of where this intersects

00:04:08.167 --> 00:04:09.700
is square root of two over two.

00:04:09.700 --> 00:04:13.433
The y-coordinate here is
negative square root of two,

00:04:13.433 --> 00:04:17.066
negative square root of two over two.

00:04:17.345 --> 00:04:18.966
So what's the tangent?

00:04:18.966 --> 00:04:21.833
Well, it's going to be
your sine over your cosine,

00:04:21.833 --> 00:04:23.766
which is going to be negative
one, and you see that.

00:04:23.766 --> 00:04:25.433
For however much you
move in the x direction,

00:04:25.433 --> 00:04:27.237
you move the opposite of that.

00:04:27.237 --> 00:04:30.900
You move the negative of
that in the y direction.

00:04:30.900 --> 00:04:32.002
Let me clean this up a little bit

00:04:32.002 --> 00:04:35.584
because I want to keep
reusing my unit circle.

00:04:35.900 --> 00:04:38.166
So there you go.

00:04:38.166 --> 00:04:40.900
And so this is going to be negative one.

00:04:41.144 --> 00:04:45.366
This is going to be negative one.

00:04:45.366 --> 00:04:48.350
Actually, let's just start
plotting a few of these points.

00:04:48.442 --> 00:04:51.689
If we assume that this is the theta axis,

00:04:51.689 --> 00:04:53.386
if you can see that,
that's the theta axis,

00:04:53.386 --> 00:04:57.366
and if this is the
y-axis, that's the y-axis,

00:04:57.366 --> 00:05:02.233
we immediately see
tangent of zero is zero.

00:05:02.233 --> 00:05:06.700
Tangent of pi over four is
one, thinking in radians.

00:05:06.700 --> 00:05:10.666
Tangent of negative pi
over four is negative one.

00:05:11.566 --> 00:05:12.340
Now let's think of it.

00:05:12.340 --> 00:05:13.925
Right now, if you just
saw that, you might say,

00:05:13.925 --> 00:05:15.079
"Oh, maybe this is some type of a line,"

00:05:15.079 --> 00:05:17.900
but we'll see very clearly it's not a line

00:05:17.900 --> 00:05:22.900
because what happens as our
angle gets closer and closer to,

00:05:24.996 --> 00:05:28.700
as our angle gets closer
and closer to pi over two,

00:05:28.700 --> 00:05:30.858
what happens to the slope of this line?

00:05:30.858 --> 00:05:32.033
So that is theta.

00:05:32.033 --> 00:05:34.433
We're getting closer and
closer to pi over two.

00:05:34.433 --> 00:05:36.633
This ray, I guess I should say,

00:05:36.633 --> 00:05:39.500
is getting closer and closer
to approaching the vertical,

00:05:39.500 --> 00:05:43.392
so its slope is getting more
and more and more positive,

00:05:43.392 --> 00:05:45.566
and if you go all the way to pi over two,

00:05:45.566 --> 00:05:47.300
the slope at that point
is really undefined

00:05:47.300 --> 00:05:48.802
but it's approaching,

00:05:48.802 --> 00:05:51.986
one way to think about it is
it is approaching infinity.

00:05:51.986 --> 00:05:53.706
So as you get closer and
closer to pi over two,

00:05:53.706 --> 00:05:55.366
so I'm going to make a ...

00:05:55.366 --> 00:05:57.689
I'm going to draw essentially
a vertical asymptote

00:05:57.689 --> 00:06:00.800
right over here at pi over two

00:06:01.319 --> 00:06:03.500
because it's not going to be ...

00:06:03.500 --> 00:06:04.500
I guess one way we could think about it,

00:06:04.500 --> 00:06:06.233
it's approaching infinity there,

00:06:06.233 --> 00:06:10.263
so this is going to be
looking something like this.

00:06:10.263 --> 00:06:12.900
It's going to be looking
something like this.

00:06:12.900 --> 00:06:14.900
The slope of the ray as
you get closer and closer

00:06:14.900 --> 00:06:18.368
to pi over two is getting
closer and closer to infinity.

00:06:18.368 --> 00:06:22.366
What happens when the angle
is getting closer and closer

00:06:22.366 --> 00:06:24.433
to negative pi over two?

00:06:24.433 --> 00:06:27.233
Is getting closer and closer
to negative pi over two?

00:06:27.233 --> 00:06:28.900
Well, then, the slope is getting

00:06:28.900 --> 00:06:31.700
more and more and more negative.

00:06:31.700 --> 00:06:33.700
It's really approaching negative.

00:06:33.700 --> 00:06:36.100
It's approaching negative infinity.

00:06:36.100 --> 00:06:37.700
So let me draw that.

00:06:37.700 --> 00:06:40.014
Once again, not quite
defined right over there,

00:06:40.014 --> 00:06:42.566
we have a vertical asymptote,

00:06:42.566 --> 00:06:47.566
and we are approaching negative infinity.

00:06:48.088 --> 00:06:50.233
We are approaching negative infinity.

00:06:50.233 --> 00:06:52.766
That's what the graph of
tangent of theta looks

00:06:52.766 --> 00:06:57.366
just over this section
of, I guess we could say

00:06:57.366 --> 00:06:59.833
the theta axis, but then
we could keep going.

00:06:59.833 --> 00:07:02.300
Then we could keep going because

00:07:02.300 --> 00:07:04.309
if our angle, right after
we cross pi over two,

00:07:04.309 --> 00:07:07.135
so let's say we've just
crossed pi over two,

00:07:07.135 --> 00:07:09.100
so we went right across it,

00:07:09.100 --> 00:07:10.633
now what is the slope?

00:07:10.633 --> 00:07:12.700
What is the slope of this thing?

00:07:12.700 --> 00:07:15.381
Well, the slope of this
thing is hugely negative.

00:07:15.381 --> 00:07:17.033
It looks almost like what
I just drew down here.

00:07:17.033 --> 00:07:18.766
It's hugely negative.

00:07:18.827 --> 00:07:21.633
So then the graph jumps back down here,

00:07:21.633 --> 00:07:23.833
and it's hugely negative again.

00:07:23.833 --> 00:07:24.900
It's hugely negative.

00:07:24.900 --> 00:07:27.366
And then as we increase our theta,

00:07:27.366 --> 00:07:31.166
as we increase our theta,
it becomes less and less

00:07:31.166 --> 00:07:35.724
and less negative all
the way to when we go to,

00:07:35.724 --> 00:07:36.767
what is this ...

00:07:36.767 --> 00:07:38.375
all the way until we go to ...

00:07:38.375 --> 00:07:40.233
let me plot this ...

00:07:40.233 --> 00:07:43.167
this angle right over here.

00:07:43.167 --> 00:07:44.725
Now what is this angle?

00:07:44.725 --> 00:07:46.366
This, well, I haven't told you yet.

00:07:46.658 --> 00:07:49.043
Let's say that this angle right over here

00:07:49.043 --> 00:07:52.000
is three pi over four.

00:07:52.366 --> 00:07:54.567
Now why did I pick three pi over four?

00:07:54.567 --> 00:07:59.567
Because that is pi over
two plus pi over four.

00:08:00.700 --> 00:08:03.371
Or you could say two pis over
for plus another pi over four

00:08:03.371 --> 00:08:04.700
is three pi over four.

00:08:04.700 --> 00:08:06.114
And the reason why this is interesting

00:08:06.114 --> 00:08:08.028
is because it is another,

00:08:08.028 --> 00:08:09.700
it's forming another,
I guess you could say

00:08:09.700 --> 00:08:14.176
pi over four- pi over
four-pi over two triangle,

00:08:14.176 --> 00:08:17.233
or a 45-45-90 triangle,

00:08:17.233 --> 00:08:20.644
where the x and y-coordinates
or the x and y distances

00:08:20.644 --> 00:08:22.101
have the same magnitude.

00:08:22.101 --> 00:08:24.033
But now, the x is going to be negative

00:08:24.033 --> 00:08:25.500
and the y is positive.

00:08:25.500 --> 00:08:27.702
So the slope here is
going to be the slope,

00:08:27.702 --> 00:08:32.100
the same slope as we had for
negative pi over four radians.

00:08:32.100 --> 00:08:34.333
We're going to have a
slope of negative one.

00:08:35.300 --> 00:08:39.642
At three pi over four, we
have a slope of negative one.

00:08:39.642 --> 00:08:43.633
Then we increase our
angle all the way to pi.

00:08:43.633 --> 00:08:46.233
Now our slope is back to zero.

00:08:46.233 --> 00:08:48.433
Our slope is back to zero.

00:08:48.433 --> 00:08:51.033
And then as we go beyond that,

00:08:51.033 --> 00:08:56.033
as we go to, as we increase
by another pi over four,

00:08:57.433 --> 00:09:00.166
our slope goes back to being positive one.

00:09:00.396 --> 00:09:02.966
Our slope goes back to being positive one.

00:09:02.966 --> 00:09:04.690
And then once again, as we approach

00:09:04.690 --> 00:09:06.433
three pi over two,

00:09:06.433 --> 00:09:09.214
our slope is becoming more
and more and more positive,

00:09:09.214 --> 00:09:14.100
getting, approaching positive infinity.

00:09:14.100 --> 00:09:16.377
This slope knows if you move a
little bit in the x direction,

00:09:16.377 --> 00:09:18.900
you're moving a lot up in the y direction.

00:09:18.900 --> 00:09:21.766
So once again, so now the graph
is going to look like this.

00:09:21.766 --> 00:09:24.566
Let me do it in a color
that you can actually see.

00:09:24.904 --> 00:09:29.904
The graph is going to
look something like this.

00:09:32.689 --> 00:09:34.833
And it will just continue to do this.

00:09:34.833 --> 00:09:39.833
It will just continue to
do this every pi radians,

00:09:41.725 --> 00:09:43.705
actually, let me do that as a dotted line,

00:09:43.705 --> 00:09:48.033
every pi radians over
and over and over again.

00:09:48.033 --> 00:09:52.748
Let me go back, pi, and I
can draw these asymptotes.

00:09:52.934 --> 00:09:55.233
I can draw these asymptotes.

00:09:55.233 --> 00:09:57.166
Let me draw that and that.

00:09:57.166 --> 00:09:59.269
And so the graph of tangent,

00:09:59.530 --> 00:10:02.196
the graph of tangent of
theta is going to look,

00:10:02.196 --> 00:10:05.833
is going to look something like this.

00:10:05.833 --> 00:10:07.166
And we could obviously, it's periodic,

00:10:07.166 --> 00:10:09.090
we could just keep doing
it on and on and on

00:10:09.090 --> 00:10:10.733
in both directions.


Triangles: using angles to categorize

WEBVTT
Kind: captions
Language: en

00:00:00.385 --> 00:00:02.733
Voiceover: We've already
seen that we can categorize

00:00:02.733 --> 00:00:07.314
triangles as being equilateral,
isoceles, or scalene

00:00:07.314 --> 00:00:09.732
based on the lengths of
the sides of the triangle.

00:00:09.732 --> 00:00:12.434
So, if none of the lengths are congruent,

00:00:12.434 --> 00:00:15.314
if you have something like this,

00:00:15.314 --> 00:00:17.139
we would consider this scalene.

00:00:17.139 --> 00:00:19.254
I am assuming that this side
is not equal to that side

00:00:19.254 --> 00:00:21.723
and neither of these
are equal to that side.

00:00:21.738 --> 00:00:23.999
So that would be scalene,
and this is all review.

00:00:24.242 --> 00:00:27.074
If I have at least two
of the sides being equal,

00:00:27.074 --> 00:00:30.053
so let's say that side is the same length

00:00:30.053 --> 00:00:32.709
as that side right over there,

00:00:33.753 --> 00:00:36.439
so I'll mark it off as
these are the same length,

00:00:36.439 --> 00:00:42.941
this would be an isosceles triangle.

00:00:43.216 --> 00:00:47.491
Then if all three of them, all
three sides, are congruent,

00:00:48.617 --> 00:00:52.664
if all three sides are the same length,

00:00:52.664 --> 00:00:55.232
we would call this equilateral.

00:00:57.153 --> 00:01:00.008
And in most circles, you could
also say this is isosceles

00:01:00.008 --> 00:01:03.507
because isosceles would be at
least two sides being equal.

00:01:03.507 --> 00:01:05.120
So this one definitely
has at least two sides,

00:01:05.120 --> 00:01:06.549
it has all three.

00:01:06.549 --> 00:01:09.485
So this is, you could say,
equilateral and isosceles.

00:01:09.601 --> 00:01:11.351
While this one, if we
assume this third side

00:01:11.351 --> 00:01:12.496
is a different length,

00:01:12.496 --> 00:01:14.441
this would just be
isosceles, not equilateral.

00:01:14.441 --> 00:01:15.488
Now all of that is review.

00:01:15.809 --> 00:01:17.328
Now, what I want to
think about in this video

00:01:17.328 --> 00:01:20.344
is well what if we're not
given the lengths of the sides

00:01:20.344 --> 00:01:22.981
and what if we're just
given a few of the angles.

00:01:23.295 --> 00:01:28.630
So, for example, let's say
we were given a triangle.

00:01:28.630 --> 00:01:30.569
Let's say that we were given a triangle

00:01:30.569 --> 00:01:34.406
where we're given a few
of the angle measures.

00:01:34.456 --> 00:01:41.268
So let's say we are told that
this angle right over here

00:01:41.268 --> 00:01:46.561
is 40 degrees and this angle
right over here is 50 degrees.

00:01:47.340 --> 00:01:50.348
Now, based on that, could
you somehow classify

00:01:50.348 --> 00:01:53.312
this as scalene,
isosceles, or equilateral?

00:01:54.037 --> 00:01:56.440
Well, the key here to
realize is if you know

00:01:56.440 --> 00:02:00.229
two interior angles of a
triangle you can always

00:02:00.229 --> 00:02:02.683
figure out the third because the three

00:02:02.683 --> 00:02:05.541
need to add up to 180 degrees.

00:02:06.041 --> 00:02:10.382
So if this is 40 and that is
50, these two add up to 90.

00:02:10.620 --> 00:02:12.205
So to add up to 180 degrees,

00:02:12.205 --> 00:02:16.038
this one must be a 90-degree angle.

00:02:16.038 --> 00:02:19.624
We can even mark it as a right angle.

00:02:20.195 --> 00:02:24.550
So if you have a triangle where
all of the interior angles

00:02:24.550 --> 00:02:26.430
are different, that means that all of the

00:02:26.430 --> 00:02:28.600
side lengths are going to be different.

00:02:28.600 --> 00:02:31.363
One way to think about it
is if this angle became

00:02:31.363 --> 00:02:41.632
wider, then this length
would have to become wider.

00:02:42.296 --> 00:02:45.255
If this angle became larger or smaller,

00:02:45.255 --> 00:02:48.337
then this side is going to have
to become larger or smaller.

00:02:48.566 --> 00:02:50.699
If this angle became larger or smaller,

00:02:50.699 --> 00:02:53.339
then this side is going to
become larger or smaller.

00:02:53.364 --> 00:02:56.189
So hopefully you
appreciate that if you have

00:02:56.189 --> 00:02:57.818
three different angles
you are going to have

00:02:57.818 --> 00:02:59.278
three different side lengths.

00:02:59.337 --> 00:03:01.290
So just based on the angles here,

00:03:01.290 --> 00:03:03.135
that we have three different angles,

00:03:03.135 --> 00:03:09.339
we can say that this is going
to be a scalene triangle.

00:03:09.819 --> 00:03:17.368
Now we can look at a
couple other examples.

00:03:17.368 --> 00:03:18.657
Let's do an interesting one.

00:03:26.564 --> 00:03:33.155
Let's say this angle right
over here is 70 degrees

00:03:33.155 --> 00:03:37.227
and let's say this angle
over here is 40 degrees.

00:03:37.647 --> 00:03:40.210
Now, based on the
information I have given you,

00:03:40.210 --> 00:03:42.145
what kind of a triangle
is this going to be?

00:03:42.145 --> 00:03:43.931
Can you even figure it out?

00:03:43.937 --> 00:03:45.582
Well, we use the same idea.

00:03:45.582 --> 00:03:48.175
The interior angles need to add up to 180.

00:03:48.175 --> 00:03:54.025
70 plus 40 is 110, so
you can say 110 plus what

00:03:54.260 --> 00:03:56.830
is equal to 180 degrees?

00:03:56.830 --> 00:03:59.680
Well, this is going to
have to be 70 degrees.

00:04:00.019 --> 00:04:02.660
So this angle right
over here is 70 degrees.

00:04:03.114 --> 00:04:06.191
So now we have a scenario
where two of the angles

00:04:06.191 --> 00:04:07.868
have the same measure.

00:04:10.376 --> 00:04:11.502
One way to think about it,

00:04:11.502 --> 00:04:13.157
based on whether this
angle is large or small,

00:04:13.157 --> 00:04:16.287
Is going to define the length of that side

00:04:16.287 --> 00:04:18.186
and this angle right over here,

00:04:18.186 --> 00:04:20.111
depending on how large or small it is,

00:04:20.111 --> 00:04:22.652
is going to define the
length on this side.

00:04:23.081 --> 00:04:26.491
So because these two angles are congruent,

00:04:26.491 --> 00:04:28.239
because they have the same measure,

00:04:28.239 --> 00:04:30.689
their opposite sides are
going to be congruent.

00:04:30.830 --> 00:04:32.902
So this is going to be the same as that.

00:04:33.023 --> 00:04:35.554
So just based on the information
I had started giving you,

00:04:35.554 --> 00:04:38.224
because you can show that two angles

00:04:38.224 --> 00:04:40.403
are going to be the same, you can say that

00:04:40.403 --> 00:04:45.814
this is going to be an isosceles triangle.

00:04:46.591 --> 00:04:48.534
Now let's do a third example,
and you could probably guess

00:04:48.534 --> 00:04:50.907
what I am going to do
in this third example.

00:04:54.449 --> 00:04:56.856
Let's say this angle is 60 degrees.

00:04:57.069 --> 00:04:59.450
What type of a triangle
is this going to be?

00:05:00.346 --> 00:05:02.135
Well, if this is 60 and this is 60,

00:05:02.135 --> 00:05:04.263
to make them add up to 180,

00:05:04.263 --> 00:05:08.616
that would have to be 60 degrees as well.

00:05:09.108 --> 00:05:12.123
If you have all of the angles congruent,

00:05:12.123 --> 00:05:14.763
that means that all of
the sides are congruent

00:05:14.842 --> 00:05:18.425
and so now you are dealing
with an equilateral.

00:05:18.432 --> 00:05:20.826
Now, as we said before,
also you can view this

00:05:20.826 --> 00:05:22.938
as a subset as isosceles
because you have at least

00:05:22.938 --> 00:05:25.656
two angles and you have
two sides being congruent,

00:05:25.656 --> 00:05:27.204
but here it is all three,

00:05:27.204 --> 00:05:29.600
so this is an equilateral triangle.


Worked example: Riemann sums in summation notation

WEBVTT
Kind: captions
Language: en

00:00:00.577 --> 00:00:01.912
Voiceover: What I want
to do in this video,

00:00:01.912 --> 00:00:05.879
is get a little bit of
practice trying to approximate

00:00:05.879 --> 00:00:08.494
the area under curves,
and also get a little bit

00:00:08.494 --> 00:00:13.295
more familiarity with the
sigma notation in this context.

00:00:13.295 --> 00:00:15.062
So what we have here, we have the graph of

00:00:15.062 --> 00:00:18.628
f of x is equal to one plus 0.1 x squared,

00:00:18.628 --> 00:00:20.546
that's this curve right over here,

00:00:20.546 --> 00:00:23.812
and then we have these rectangles that are

00:00:23.812 --> 00:00:27.263
trying to approximate
the area under the curve,

00:00:27.263 --> 00:00:32.428
the area under the function
f, between x equals zero,

00:00:32.428 --> 00:00:37.227
and x equals 8, and the
way that this diagram,

00:00:37.227 --> 00:00:39.046
or the way that we are
attempting to do it,

00:00:39.046 --> 00:00:41.963
is by splitting it into four rectangles,

00:00:41.963 --> 00:00:44.228
and so we can call this rectangle one,

00:00:44.228 --> 00:00:50.446
this is rectangle two, rectangle
three, and rectangle four.

00:00:50.446 --> 00:00:53.227
And each of their heights,
let's see the interval,

00:00:53.227 --> 00:00:55.595
it looks like they each
have a width of two,

00:00:55.595 --> 00:00:58.713
so they are equally spaced,
so we go from zero to eight,

00:00:58.713 --> 00:01:02.845
and we split into four sections,
so each has a width of two,

00:01:02.845 --> 00:01:04.695
so they're each going to be two wide,

00:01:04.695 --> 00:01:07.928
so that's two, that's two,
that's two, that's two.

00:01:07.928 --> 00:01:14.327
And their height seems to
be based on the midpoint,

00:01:14.327 --> 00:01:16.179
so between the start,
between the left side

00:01:16.179 --> 00:01:17.827
and the right side of the rectangle,

00:01:17.827 --> 00:01:20.728
you take the value of the
function at the middle value,

00:01:20.728 --> 00:01:21.846
right over here, so for example,

00:01:21.846 --> 00:01:24.277
this height right over
here looks like f of 1,

00:01:24.277 --> 00:01:27.762
this height right over
here looks like f of 3,

00:01:27.762 --> 00:01:31.212
this height of this rectangle is f of 5,

00:01:31.212 --> 00:01:34.263
this height right over here is f of 7.

00:01:34.263 --> 00:01:36.727
So given the way that
this has been constructed,

00:01:36.727 --> 00:01:38.712
and we want to take the sum of the areas

00:01:38.712 --> 00:01:41.646
of these rectangles as an
approximation as the area

00:01:41.646 --> 00:01:44.428
under this curve, how would we
write that as sigma notation?

00:01:44.428 --> 00:01:45.845
And I'll get us started,
and then I encourage you

00:01:45.845 --> 00:01:48.045
to pause the video and try to finish it.

00:01:48.045 --> 00:01:51.811
So the sum of these rectangles,

00:01:51.811 --> 00:01:54.762
we could say it's the
sum of, so we'll have

00:01:54.762 --> 00:01:58.297
n equals one to four, 'cause
we have four rectangles.

00:01:58.297 --> 00:02:02.762
And I encourage you to finish this up.

00:02:02.762 --> 00:02:06.061
Actually, just write it
in terms of the function,

00:02:06.061 --> 00:02:08.077
use function notation, you
don't have to write it out

00:02:08.077 --> 00:02:12.394
as one plus .01 times something squared.

00:02:12.394 --> 00:02:14.362
So I'm assuming you've had a go at it.

00:02:14.362 --> 00:02:19.244
So for each of these, so for
the first rectangle over here,

00:02:19.244 --> 00:02:22.678
We're gonna multiply two times the height,

00:02:22.678 --> 00:02:25.763
so the height right over
here is one, and this is

00:02:25.763 --> 00:02:28.577
the first rectangle, so
you might be tempted to say

00:02:28.577 --> 00:02:32.878
times f of n, but then that breaks down

00:02:32.878 --> 00:02:34.379
as we go into the second rectangle.

00:02:34.379 --> 00:02:36.629
The second rectangle,
the two still applies.

00:02:36.629 --> 00:02:39.744
This two is the width of the rectangle,

00:02:39.744 --> 00:02:43.595
but now we want to multiply
it times f of three,

00:02:43.595 --> 00:02:52.512
not f of two, so this f of n
isn't going to pass muster.

00:02:52.512 --> 00:02:55.678
And so let's see how we
want to think about it.

00:02:55.678 --> 00:03:04.345
So when n is one, two, three,
four, we're gonna take f of...

00:03:04.345 --> 00:03:07.128
So f of n, I shouldn't say
f of n, we're gonna take

00:03:07.128 --> 00:03:10.528
f of something, so here this
first one, we're gonna take

00:03:10.528 --> 00:03:14.477
f of one, then here, for
the second rectangle,

00:03:14.477 --> 00:03:18.346
we're taking f of three for the height,

00:03:18.346 --> 00:03:22.821
for the third rectangle
we're taking f of five,

00:03:22.821 --> 00:03:28.371
and then for the fourth rectangle,
we're taking f of seven.

00:03:28.371 --> 00:03:30.440
So what's the relationship over here?

00:03:30.440 --> 00:03:34.523
Let's see, it looks like
if you multiply by two

00:03:34.523 --> 00:03:37.391
and subtract one, so two
times one minus one is one,

00:03:37.391 --> 00:03:39.023
two times two minus one is three,

00:03:39.023 --> 00:03:42.940
two times three minus one is five,

00:03:42.940 --> 00:03:45.457
two times four minus one is seven.

00:03:45.457 --> 00:03:49.794
So this is two n minus one, so the area of

00:03:49.794 --> 00:03:52.488
each of these rectangles, the base is two,

00:03:52.488 --> 00:03:59.071
and the height is f of two and minus one.

00:03:59.071 --> 00:04:02.421
So that hopefully makes
it a little bit clear,

00:04:02.421 --> 00:04:04.787
kind of mapping between the sigma notation

00:04:04.787 --> 00:04:06.919
and what we're actually trying to do.

00:04:06.919 --> 00:04:08.770
And now let's just for fun,

00:04:08.770 --> 00:04:10.456
let's actually try to evaluate this thing.

00:04:10.456 --> 00:04:13.088
What is this thing going to evaluate to?

00:04:13.088 --> 00:04:18.337
Well, this is going to
evaluate to two times f of,

00:04:18.337 --> 00:04:21.572
when n is equal to one
this is one, f of 1,

00:04:21.572 --> 00:04:25.687
plus two times, when n
is two, this is gonna be

00:04:25.687 --> 00:04:30.738
f of, two times two minus
one is three, f of 3,

00:04:30.738 --> 00:04:35.220
when n is three, this is
gonna be two times f of 5,

00:04:35.220 --> 00:04:42.038
when n is four, this is going
to be two times f of seven,

00:04:42.038 --> 00:04:46.519
four times two minus one
is seven, f of seven.

00:04:46.519 --> 00:04:49.539
And so that is going to be,
well we're gonna have to

00:04:49.539 --> 00:04:51.870
evaluate a bunch of
these things over here,

00:04:51.870 --> 00:04:54.337
so let me actually, let me clear this out

00:04:54.337 --> 00:04:56.554
so I have a little bit more real estate.

00:04:56.554 --> 00:04:59.670
I'm feeling this might get
a little bit messy now.

00:04:59.670 --> 00:05:02.371
So this is going to be, actually
we could factor out a two.

00:05:02.371 --> 00:05:05.655
So this is going to be
equal to two times f of one

00:05:05.655 --> 00:05:09.238
is one plus 0.1 times one squared.

00:05:09.238 --> 00:05:13.121
So it's one plus 0.1, let me
color-code it a little bit

00:05:13.121 --> 00:05:14.988
so we can keep track of things.

00:05:14.988 --> 00:05:23.669
So this right over here is
1.1, so one plus 0.1 is 1.1,

00:05:23.669 --> 00:05:29.361
this right over here, f of
3, so that's one plus .1

00:05:29.361 --> 00:05:36.268
times three squared, nine,
so one plus .9, so it's 1.9.

00:05:36.268 --> 00:05:40.619
And then, let's see, this
one right over here, f of 5,

00:05:40.619 --> 00:05:44.034
is gonna be one plus,
see five squared is 25,

00:05:44.034 --> 00:05:50.068
times .1 is 2.5, so one
plus 2.5 is gonna be 3.5.

00:05:50.068 --> 00:05:57.534
And then finally, f of
seven, is going to be

00:05:57.534 --> 00:05:59.819
one plus .1 times seven squared,

00:05:59.819 --> 00:06:10.785
so this is 49 times .1 is
4.9 plus one, so plus 5.9.

00:06:10.785 --> 00:06:13.933
And so what is this going to be equal to?

00:06:13.933 --> 00:06:18.118
So let's see, 1.1 plus 1.9, these two

00:06:18.118 --> 00:06:21.285
are going to sum up to be equal to three,

00:06:21.285 --> 00:06:24.568
and then these two are
going to sum up to be,

00:06:24.568 --> 00:06:29.534
let's see, if we add
the five, we get to 8.5,

00:06:29.534 --> 00:06:35.734
and then we add the .9, we
get to 9.4, so plus 9.4.

00:06:35.734 --> 00:06:38.818
Did I do that right -
three plus five is eight,

00:06:38.818 --> 00:06:43.620
.5 plus .9 is 1.4, yep,
and so this is going to be,

00:06:43.620 --> 00:06:45.418
so once again, we have
the two times it all,

00:06:45.418 --> 00:06:50.818
so this is going to be
equal to two times 12.4,

00:06:50.818 --> 00:06:56.619
which is equal to 24.8,
which is our approximation.

00:06:56.619 --> 00:06:57.833
Once again, this is just an approximation

00:06:57.833 --> 00:07:00.917
using these rectangles of
the area under the curve

00:07:00.917 --> 00:07:04.917
between x equals zero, and x equals eight.


Worked example: problem involving definite integral (graphical)

WEBVTT
Kind: captions
Language: en

00:00:01.099 --> 00:00:03.259
Voiceover: It took 20
minutes before Jughead

00:00:03.259 --> 00:00:06.114
noticed that his hot
tub had sprung a leak.

00:00:06.147 --> 00:00:09.566
Once he realized it, he
opened the hot tub's drain

00:00:09.566 --> 00:00:12.511
and the rest of the water
rushed out in 40 minutes.

00:00:12.924 --> 00:00:15.631
The rate at which the water
drained from the hot tub

00:00:15.631 --> 00:00:18.602
in gallons per minute is shown.

00:00:18.897 --> 00:00:21.867
How many gallons of
water were in the hot tub

00:00:21.867 --> 00:00:24.408
before it started to leak?

00:00:25.115 --> 00:00:26.880
So let's see what they have over here.

00:00:26.897 --> 00:00:29.480
So they've plotted gallons per minute

00:00:29.480 --> 00:00:31.902
versus time in minutes.

00:00:32.251 --> 00:00:34.240
So we see here, this blue line,

00:00:34.240 --> 00:00:39.408
this is the rate at which
water drained from the hot tub.

00:00:39.422 --> 00:00:41.797
So at minute zero, the
water wasn't really draining

00:00:41.797 --> 00:00:45.164
from the hot tub and then
not just more drained

00:00:45.164 --> 00:00:48.623
but the rate at which the
water was draining increased.

00:00:48.826 --> 00:00:53.766
So 10 minutes into his
bath, the water was draining

00:00:53.766 --> 00:00:57.186
at a gallon per minute and
then 20 minutes into his bath

00:00:57.186 --> 00:01:01.080
water was draining at
two gallons per minute.

00:01:01.443 --> 00:01:03.974
Then he noticed it and he opens the drain.

00:01:04.014 --> 00:01:06.249
I guess he wants to accelerate
the end of his bath.

00:01:06.491 --> 00:01:08.244
So he opens the drain
and then all of a sudden

00:01:08.244 --> 00:01:10.503
water starts draining out
at a much higher rate,

00:01:10.503 --> 00:01:13.808
at 20 gallons per minute,
but then that decreases.

00:01:14.068 --> 00:01:17.171
So we could think about physically
why that might decrease.

00:01:17.171 --> 00:01:20.527
Maybe there was just less
pressure or whatever.

00:01:20.527 --> 00:01:22.056
We're not going to go
into the physics of it,

00:01:22.056 --> 00:01:24.311
but we're just going to
take this chart as fact.

00:01:24.539 --> 00:01:26.933
The rate at which the
water drains decreases

00:01:26.933 --> 00:01:28.822
all the way to the 60th minute,

00:01:28.822 --> 00:01:31.499
which is 40 minutes after
he opened the drain.

00:01:32.700 --> 00:01:33.636
At the 60th minute, all of the water

00:01:33.636 --> 00:01:36.925
was actually drained out.

00:01:37.506 --> 00:01:39.036
So given that, how do we think about

00:01:39.036 --> 00:01:41.699
how many gallons of
water were in the hot tub

00:01:41.699 --> 00:01:43.313
before it started to leak?

00:01:43.483 --> 00:01:45.018
And I encourage you to pause the video

00:01:45.018 --> 00:01:46.990
and try to see if you could
figure it out yourself

00:01:46.990 --> 00:01:48.776
before I work through it.

00:01:49.414 --> 00:01:50.779
Well, to answer this question:

00:01:50.779 --> 00:01:52.417
How many gallons of
water were in the hot tub

00:01:52.417 --> 00:01:54.010
before it started to leak?

00:01:54.010 --> 00:01:55.547
That is answering the same question as

00:01:55.547 --> 00:01:58.098
How much total hot water drained out?

00:01:58.842 --> 00:02:07.150
So how much drained out?

00:02:07.686 --> 00:02:09.615
And to think about that,
we can just kind of go back

00:02:09.615 --> 00:02:12.947
to what we knew before we
learned about calculus.

00:02:13.563 --> 00:02:16.048
If I have something
happening at a fixed rate,

00:02:16.048 --> 00:02:18.677
so let's say that this
is gallons per minute.

00:02:19.251 --> 00:02:23.544
So this is still the same
context, I guess you could say.

00:02:23.544 --> 00:02:24.984
This is time in minutes.

00:02:24.984 --> 00:02:28.154
Let's say things are draining
out at a constant rate.

00:02:28.817 --> 00:02:30.815
If you wanted to figure
out how much drains out

00:02:30.815 --> 00:02:32.468
over a certain interval of time,

00:02:32.468 --> 00:02:34.682
let's say this interval
of time right over here,

00:02:34.682 --> 00:02:37.771
let's call that delta t,

00:02:37.771 --> 00:02:44.674
you would just multiply the
rate over that interval of time,

00:02:45.597 --> 00:02:48.927
which we could represent by

00:02:48.927 --> 00:02:51.644
this orange height right over here,

00:02:51.644 --> 00:02:55.011
times the amount of time that passed by,

00:02:55.011 --> 00:02:57.875
which would give you the area

00:02:57.875 --> 00:03:00.425
under the curve over that interval.

00:03:00.425 --> 00:03:02.616
the area under the curve
over that interval.

00:03:02.766 --> 00:03:05.672
This area would tell you the gallons

00:03:05.672 --> 00:03:08.815
that drained over that delta t.

00:03:09.327 --> 00:03:13.487
And this doesn't just apply
when you have a constant rate.

00:03:13.971 --> 00:03:16.145
If your rate looked something like this,

00:03:16.145 --> 00:03:20.043
as we've seen in other videos,

00:03:20.043 --> 00:03:27.162
you could figure out the
amount that has drained

00:03:27.162 --> 00:03:28.547
in a specific interval, let's say

00:03:28.547 --> 00:03:30.258
this interval right over here,

00:03:30.258 --> 00:03:33.990
by essentially figuring out
the area under the curve

00:03:33.990 --> 00:03:38.485
over that interval,
and the units work out.

00:03:38.831 --> 00:03:43.101
If you multiply gallons
per minute times minute,

00:03:43.181 --> 00:03:52.422
this area is going to
be in terms of gallons.

00:03:52.422 --> 00:03:53.902
Another way you could think about it,

00:03:53.902 --> 00:03:55.465
and this goes back to,
well, how do you figure out

00:03:55.465 --> 00:03:57.581
the area of a trapezoid right over here.

00:03:57.581 --> 00:03:58.751
Well, the area of a trapezoid,

00:03:58.751 --> 00:04:01.835
you could find the average
height of the trapezoid,

00:04:01.835 --> 00:04:02.941
which would be the average

00:04:02.941 --> 00:04:05.543
of the beginning and the end period,

00:04:05.543 --> 00:04:07.712
you take that average over there,

00:04:07.712 --> 00:04:10.278
and this would work for a line like this,

00:04:10.278 --> 00:04:11.527
if you take that average height

00:04:11.527 --> 00:04:16.675
and multiply by the change in time,

00:04:16.675 --> 00:04:21.489
you are going to figure out that area.

00:04:21.489 --> 00:04:23.842
And that is another way
of thinking about it.

00:04:23.891 --> 00:04:26.464
You are taking the average
rate over that interval

00:04:26.464 --> 00:04:28.967
times the interval, and
that is going to give you

00:04:28.967 --> 00:04:30.518
the total number of gallons.

00:04:30.558 --> 00:04:32.385
And so we just have to
apply that idea over here.

00:04:32.385 --> 00:04:33.915
We just have to literally figure out

00:04:33.915 --> 00:04:36.023
the area under the curve
over the entire interval

00:04:36.023 --> 00:04:38.418
when the water was actually
leaking or draining.

00:04:38.477 --> 00:04:40.034
So essentially the area under the curve

00:04:40.034 --> 00:04:42.134
between zero minutes and 60 minutes.

00:04:42.482 --> 00:04:48.680
And so it is going to be this area plus

00:04:48.680 --> 00:04:54.633
all of this area under this part as well.

00:04:54.711 --> 00:04:56.023
To help us think about that,

00:04:56.023 --> 00:04:58.110
I'm going to just split
that up into some sections.

00:04:58.110 --> 00:05:01.624
So I'll have this
triangular section up here.

00:05:02.798 --> 00:05:04.365
I could just think about
this as a trapezoid,

00:05:04.365 --> 00:05:05.615
but I'm just going to split it up

00:05:05.615 --> 00:05:07.185
into a triangle and a rectangle.

00:05:07.622 --> 00:05:12.683
And then I have this section
right over here in green.

00:05:13.110 --> 00:05:15.778
So what is the area of this entire thing?

00:05:16.099 --> 00:05:17.306
Well, the area right over here,

00:05:17.306 --> 00:05:33.771
we have 20 minutes times two
gallons per minute times 1/2.

00:05:33.771 --> 00:05:35.878
This gives us the area
under this triangle.

00:05:35.878 --> 00:05:42.852
So that's going to be 20 gallons.

00:05:42.852 --> 00:05:45.355
We see that the units work out nicely.

00:05:46.181 --> 00:05:48.214
So that is essentially
how much has drained out

00:05:48.214 --> 00:05:49.882
in the first 20 minutes.

00:05:50.130 --> 00:05:55.404
And then this green area
is going to be 40 minutes

00:05:55.404 --> 00:06:00.605
times 10 gallons per minute.

00:06:00.605 --> 00:06:01.728
And actually maybe you know the units,

00:06:01.728 --> 00:06:03.299
since I'm breaking it
up in this strange way,

00:06:03.299 --> 00:06:06.541
I'll just figure out this
area in a unit-less way.

00:06:06.541 --> 00:06:10.573
So 40 times 10, which is equal to 400.

00:06:11.379 --> 00:06:15.033
And then, finally, in blue,
I have 40 times this height

00:06:15.033 --> 00:06:18.354
right over here between
10 and 20 is another 10

00:06:18.354 --> 00:06:20.315
but then I'm going to
multiply that times 1/2.

00:06:20.315 --> 00:06:24.185
So it's going to be 40 times 10 times 1/2,

00:06:24.185 --> 00:06:26.877
which is going to be 200.

00:06:26.903 --> 00:06:30.326
And so when you add all
of these areas together,

00:06:30.357 --> 00:06:33.575
you are going to get 400
plus 200 is 600 plus 20.

00:06:33.761 --> 00:06:35.887
You are going to have 620 gallons.

00:06:36.473 --> 00:06:43.802
620 gallons is how much
water in total drained out

00:06:43.802 --> 00:06:47.763
or how much water was in the
tub before it started to leak,

00:06:47.763 --> 00:06:50.289
or the hot tub, yeah this is
quite big for a regular tub,

00:06:50.289 --> 00:06:53.038
bu this could be like a Jacuzzi
or a hot tub or something.