Nightline: Learn Everything, Online

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:02.253
Bill: Well, first went the record stores

00:00:02.253 --> 00:00:04.254
then crowds thinned down at the library

00:00:04.254 --> 00:00:06.670
and now you have to wonder
if physical classrooms

00:00:06.670 --> 00:00:09.004
could become another
casualty of the Internet age.

00:00:09.004 --> 00:00:11.919
Well if Khan Academy is
any indication, the old

00:00:11.919 --> 00:00:14.886
schoolhouse will survive
but the way kids spend

00:00:14.886 --> 00:00:17.587
their time inside it could change forever.

00:00:17.587 --> 00:00:21.254
So why are millions logging on
to get lessons from one man?

00:00:21.254 --> 00:00:22.671
Could your kids benefit?

00:00:22.671 --> 00:00:25.547
My co-anchor, Terry
Moran went to find out.

00:00:25.547 --> 00:00:26.843
(school bell rings)

00:00:26.843 --> 00:00:28.231
Terry: No matter how old you are,

00:00:28.231 --> 00:00:31.142
you can probably still
remember what the bell

00:00:31.142 --> 00:00:33.495
sounded like at your school.

00:00:33.495 --> 00:00:35.305
You can still remember
pushing your way through the

00:00:35.305 --> 00:00:38.587
hallways to your next
class or the minute hand on

00:00:38.587 --> 00:00:41.047
the clock in your math
class that you swore ran

00:00:41.047 --> 00:00:45.223
slower than every other clock in
the building, but that was then.

00:00:45.253 --> 00:00:47.287
This is now.

00:00:47.287 --> 00:00:50.387
Khan: 8 meters and the
same thing along the side.

00:00:50.387 --> 00:00:53.123
Terry: Welcome to the Khan
Academy, online learning

00:00:53.123 --> 00:00:56.788
on steroids, a one world
schoolhouse that is

00:00:56.788 --> 00:01:00.491
skyrocketing in popularity,
sparking controversy and

00:01:00.491 --> 00:01:02.567
fundamentally challenging the very notion

00:01:02.567 --> 00:01:05.586
of education in the 21st century.

00:01:05.586 --> 00:01:07.973
Khan: What we're talking right
now is the learning side.

00:01:07.973 --> 00:01:09.808
How can we democratize
the information, the

00:01:09.808 --> 00:01:11.755
knowledge, so that anyone
can get to whatever level

00:01:11.755 --> 00:01:13.086
they want to.

00:01:13.086 --> 00:01:15.481
Terry: Sal Khan founded
the Khan Academy kind of

00:01:15.481 --> 00:01:18.169
by accident, more on that in a minute.

00:01:18.169 --> 00:01:20.650
First, the sheer scale of this thing.

00:01:20.650 --> 00:01:23.054
Seven million students
around the world are

00:01:23.054 --> 00:01:26.087
attending the Khan
Academy online every month

00:01:26.087 --> 00:01:30.528
and that number is
growing by 400% per year.

00:01:30.528 --> 00:01:33.464
The thousands of lessons,
everything from Algebra

00:01:33.464 --> 00:01:36.504
to Medieval History to the
Fiscal Cliff, they've been

00:01:36.504 --> 00:01:41.587
viewed more than 200 million
times and all for free.

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Terry: Are you out to dismantle
our educational system?

00:01:45.048 --> 00:01:47.213
Khan: I'm not an education radical.

00:01:47.213 --> 00:01:49.798
A lot of what I'm doing
and we're doing is really

00:01:49.798 --> 00:01:52.132
bringing back very, very
old ideas that have been

00:01:52.132 --> 00:01:54.596
proven, but somehow got lost in the fray?

00:01:54.596 --> 00:01:57.928
Terry: Sal Khan is an
unlikely education reformer.

00:01:57.928 --> 00:02:01.262
He's a former hedge fund
analyst, MIT undergrad,

00:02:01.262 --> 00:02:03.959
Harvard MBA, who just really stumbled

00:02:03.959 --> 00:02:05.819
upon his revolution when
he tried to help a young

00:02:05.819 --> 00:02:10.462
cousin with her algebra
homework back in 2004.

00:02:10.462 --> 00:02:12.325
Khan: I started tutoring her
remotely. I was in Boston,

00:02:12.325 --> 00:02:15.940
she was in New Orleans
and after that I started

00:02:15.940 --> 00:02:17.394
tutoring other family
members and eventually

00:02:17.394 --> 00:02:18.599
put the stuff on YouTube.

00:02:18.599 --> 00:02:21.128
Terry: In a few weeks, Khan
noticed that the "stuff"

00:02:21.128 --> 00:02:23.802
that he was making for his
relatives was going viral.

00:02:23.802 --> 00:02:27.904
The requests from strangers
for more, more, more.

00:02:27.904 --> 00:02:30.848
Khan: Antibiotics.

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Terry: Visually, they aren't
the kind of videos you'd

00:02:32.977 --> 00:02:36.683
expect to go viral. They're
more of a virtual blackboard,

00:02:36.683 --> 00:02:39.767
but they work and education
reformers all over the

00:02:39.767 --> 00:02:43.122
world have noticed,
including some heavy hitters.

00:02:43.122 --> 00:02:45.517
Bill: There's a new website
that I've just been using,

00:02:45.517 --> 00:02:48.895
with my kids recently called
Khan Academy, K-H-A-N,

00:02:48.895 --> 00:02:53.293
just one guy doing some
unbelievable 15-minute tutorials.

00:02:53.293 --> 00:02:55.234
Terry: Bill Gates did
more than look at the Khan

00:02:55.234 --> 00:02:59.335
Academy videos. He helped
to bankroll the project.

00:02:59.335 --> 00:03:02.375
So in 2009, Khan finally
quit his day job and

00:03:02.375 --> 00:03:05.378
partnered with the Gates
Foundation and Google to

00:03:05.378 --> 00:03:07.751
form the Khan Academy.

00:03:07.751 --> 00:03:09.216
Here's how it works.

00:03:09.216 --> 00:03:11.551
Students watch the
videos and then work on a

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problem set based on the
lesson on the website.

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If you pass, you move
onto the next lesson.

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If it looks like you don't
understand a concept,

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the site takes you back
through what you are missing.

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The goal is not to replace
the traditional classroom.

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The goal is to liberate.

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Khan: The idea of
students being grouped in

00:03:30.720 --> 00:03:33.414
age base co-horts and
then going at a set pace,

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this is the school that
we all kind of grew up in,

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this is actually not the
way students were educated

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for most of history.

00:03:40.131 --> 00:03:43.130
Terry: Instead, Khan and
his allies see classrooms

00:03:43.130 --> 00:03:46.497
thrumming with creative activity,
spurred by the fact that

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kids will have already
engaged the subject matter

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through the online lessons.

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What does your ideal classroom look like?

00:03:53.804 --> 00:03:57.192
Khan: It won't be a bunch
of chairs all positioned

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looking at someone in the
front of the classroom.

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It'll be a much more collaborative space.

00:04:01.526 --> 00:04:04.916
It's not one pace fits all
anymore and the teacher

00:04:04.916 --> 00:04:06.666
will be doing focus
interventions with them.

00:04:06.666 --> 00:04:08.583
They'll be mentoring them.
They'll be inspiring them.

00:04:08.583 --> 00:04:10.121
They won't be lecturing at them.

00:04:10.121 --> 00:04:12.582
Terry: About 25-30
schools, most of them based

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around Silicon Valley
have started implementing

00:04:15.526 --> 00:04:18.862
the Khan Academy as a pilot
program, exploring how

00:04:18.862 --> 00:04:22.399
it might change teaching and learning.

00:04:22.399 --> 00:04:25.704
There are plenty of critics,
but one big group of supporters.

00:04:26.833 --> 00:04:28.500
Khan: The single biggest
thing that happened and

00:04:28.500 --> 00:04:31.376
it wasn't obvious to us at
first, the teachers and the

00:04:31.376 --> 00:04:33.778
principal told us, is that
the students started to take

00:04:33.778 --> 00:04:34.960
ownership of their learning.

00:04:34.960 --> 00:04:37.246
They started to not
say, "Hey, I'm passive."

00:04:37.246 --> 00:04:38.348
"Tell me what to do next."

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They started to say, "These are my goals."

00:04:41.518 --> 00:04:42.875
"I'm going to seek out the information."

00:04:42.875 --> 00:04:45.888
"Teacher, you're my coach.
You're my mentor. Help me do it."

00:04:45.888 --> 00:04:49.126
Terry: They're already
online, maybe at least in some

00:04:49.126 --> 00:04:53.864
ways that's where their
classrooms can be, too.

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Bill: And Khan's book,
The One World Schoolhouse,

00:04:56.864 --> 00:04:59.266
is in stores now. Thanks to Terry.


Formal definition of limits Part 4: using the definition

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.390
In the last video, we
took our first look

00:00:02.390 --> 00:00:04.889
at the epsilon-delta definition
of limits, which essentially

00:00:04.889 --> 00:00:08.800
says if you claim that the limit
of f of x as x approaches C

00:00:08.800 --> 00:00:13.020
is equal to L, then that
must mean by the definition

00:00:13.020 --> 00:00:16.250
that if you were given any
positive epsilon that it

00:00:16.250 --> 00:00:20.010
essentially tells us how close
we want f of x to be to L.

00:00:20.010 --> 00:00:23.152
We can always find
a delta that's

00:00:23.152 --> 00:00:25.110
greater than 0, which is
essentially telling us

00:00:25.110 --> 00:00:30.050
our distance from C such that
if x is within delta of C,

00:00:30.050 --> 00:00:34.285
then f of x is going to
be within epsilon of L.

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If we can find a
delta for any epsilon,

00:00:37.530 --> 00:00:41.310
then we can say
that this is indeed

00:00:41.310 --> 00:00:43.715
the limit of f of x
as x approaches C.

00:00:43.715 --> 00:00:45.090
Now, I know what
you're thinking.

00:00:45.090 --> 00:00:47.250
This seems all very abstract.

00:00:47.250 --> 00:00:49.530
I want to somehow
use this thing.

00:00:49.530 --> 00:00:51.660
And what we will do in
this video is use it

00:00:51.660 --> 00:00:56.040
and to rigorously prove that
a limit actually exists.

00:00:56.040 --> 00:00:59.400
So, right over here, I've
defined a function, f of x.

00:00:59.400 --> 00:01:03.850
It's equal to 2x everywhere
except for x equals 5.

00:01:03.850 --> 00:01:06.960
So it's 2x everywhere for
all the other values of x,

00:01:06.960 --> 00:01:09.340
but when x is equal to
5, it's just equal to x.

00:01:09.340 --> 00:01:12.750
So I could have
really just written 5.

00:01:12.750 --> 00:01:15.430
It's equal to 5 when
x is equal to 5.

00:01:15.430 --> 00:01:16.370
It's equal to itself.

00:01:16.370 --> 00:01:18.420
And so we've drawn
the graph here.

00:01:18.420 --> 00:01:20.760
Everywhere else, it
looks just like 2x.

00:01:20.760 --> 00:01:24.360
At x is equal to 5, it's
not along the line 2x.

00:01:24.360 --> 00:01:26.580
Instead, the function
is defined to be

00:01:26.580 --> 00:01:28.610
that point right over there.

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And if I were to ask you
what is the limit of f

00:01:33.660 --> 00:01:35.799
of x as x approaches
5, you might

00:01:35.799 --> 00:01:37.090
think of it pretty intuitively.

00:01:37.090 --> 00:01:37.756
Well, let's see.

00:01:37.756 --> 00:01:41.930
The closer I get
to 5, the closer

00:01:41.930 --> 00:01:44.740
f of x seems to
be getting to 10.

00:01:50.090 --> 00:01:53.380
And so, you might fairly
intuitively make the claim

00:01:53.380 --> 00:01:58.550
that the limit of f of x
as x approaches 5 really

00:01:58.550 --> 00:01:59.890
is equal to 10.

00:01:59.890 --> 00:02:01.140
It looks that way.

00:02:01.140 --> 00:02:02.980
But what we're
going to do is use

00:02:02.980 --> 00:02:08.080
the epsilon-delta definition
to actually prove it.

00:02:08.080 --> 00:02:11.030
And the way that most of
these proofs typically go

00:02:11.030 --> 00:02:13.330
is we define delta
in the abstract.

00:02:13.330 --> 00:02:15.500
And then we essentially
try to come up with a way

00:02:15.500 --> 00:02:20.590
that given any epsilon, we can
always come up with a delta.

00:02:20.590 --> 00:02:22.110
Or another way is
we're going to try

00:02:22.110 --> 00:02:26.510
to describe our delta as
a function of epsilon,

00:02:26.510 --> 00:02:28.230
not to confuse you too much.

00:02:28.230 --> 00:02:29.810
But maybe I shouldn't
use f again.

00:02:29.810 --> 00:02:35.520
But delta equals
function of epsilon

00:02:35.520 --> 00:02:37.770
that is defined for
any positive epsilon.

00:02:37.770 --> 00:02:39.960
So you give me an
epsilon, I just

00:02:39.960 --> 00:02:42.400
put into our little formula
or little function box.

00:02:42.400 --> 00:02:44.070
And I will always
get you a delta.

00:02:44.070 --> 00:02:46.790
And if I can do that for any
epsilon that'll always give you

00:02:46.790 --> 00:02:52.600
a delta, where this is true,
that if x is within delta of C,

00:02:52.600 --> 00:02:55.180
then the corresponding f of x
is going to be within epsilon

00:02:55.180 --> 00:02:57.870
of L. Then the limit
definitely exists.

00:02:57.870 --> 00:03:00.090
So let's try to do that.

00:03:00.090 --> 00:03:05.490
So let's think about being
within delta of our C.

00:03:05.490 --> 00:03:09.070
So, let's think about this
right over here is 5 plus delta,

00:03:09.070 --> 00:03:11.312
this is 5 minus delta.

00:03:11.312 --> 00:03:13.270
So that's our range we're
going to think about.

00:03:13.270 --> 00:03:15.170
We're going to think about
it in the abstract at first.

00:03:15.170 --> 00:03:16.753
And then we're going
to try to come up

00:03:16.753 --> 00:03:19.600
with a formula for delta
in terms of epsilon.

00:03:19.600 --> 00:03:23.750
So how could we describe all of
the x's that are in this range

00:03:23.750 --> 00:03:25.216
but not equal to 5 itself?

00:03:25.216 --> 00:03:26.840
Because we really
care about the things

00:03:26.840 --> 00:03:30.310
that are within delta of 5,
but not necessarily equal to 5.

00:03:30.310 --> 00:03:32.400
This is just a
strictly less than.

00:03:32.400 --> 00:03:35.940
They're within a range
of C, but not equal to C.

00:03:35.940 --> 00:03:40.950
Well, that's going to be all of
the x's that satisfy x minus 5

00:03:40.950 --> 00:03:44.460
is less than delta.

00:03:44.460 --> 00:03:48.000
That describes all of
these x's right over here.

00:03:48.000 --> 00:03:51.250
And now, what we're going to
do, and the way these proofs

00:03:51.250 --> 00:03:53.030
typically go, is
we're going to try

00:03:53.030 --> 00:03:57.100
to manipulate the left-hand
side of the inequality,

00:03:57.100 --> 00:03:59.180
so it starts to look
something like this,

00:03:59.180 --> 00:04:01.350
or it starts to look
exactly like that.

00:04:01.350 --> 00:04:03.570
And as we do that,
the right-hand side

00:04:03.570 --> 00:04:06.560
of the inequality is going to
be expressed in terms of delta.

00:04:06.560 --> 00:04:08.730
And then we can
essentially say well, look.

00:04:08.730 --> 00:04:11.771
If the right-hand
side is in terms

00:04:11.771 --> 00:04:13.520
of delta and the
left-hand side looks just

00:04:13.520 --> 00:04:16.880
like that, that really defines
how we can express delta

00:04:16.880 --> 00:04:18.560
in terms of epsilon.

00:04:18.560 --> 00:04:20.360
If that doesn't make
sense, bear with me.

00:04:20.360 --> 00:04:21.690
I'm about to do it.

00:04:21.690 --> 00:04:26.930
So, if we want x minus 5 to
look a lot more like this, when

00:04:26.930 --> 00:04:30.780
x is not equal to 5-- in all
of this, this whole interval,

00:04:30.780 --> 00:04:35.520
x is not equal to 5--
f of x is equal to 2x,

00:04:35.520 --> 00:04:37.280
our proposed limit
is equal to 10.

00:04:37.280 --> 00:04:40.250
So if we could somehow get
this to be 2x minus 10,

00:04:40.250 --> 00:04:41.515
then we're in good shape.

00:04:41.515 --> 00:04:42.890
And the easiest
way to do that is

00:04:42.890 --> 00:04:45.730
to multiply both sides
of this inequality by 2.

00:04:50.750 --> 00:04:52.960
And 2 times the absolute
value of something,

00:04:52.960 --> 00:04:55.760
that's the same thing as the
absolute value of 2 times

00:04:55.760 --> 00:04:56.300
that thing.

00:04:56.300 --> 00:04:59.300
If I were to say 2 times
the absolute value of a,

00:04:59.300 --> 00:05:02.460
that's the same thing as
the absolute value of 2a.

00:05:02.460 --> 00:05:04.800
So on the left-hand
side right over here,

00:05:04.800 --> 00:05:08.930
this is just going to be the
absolute value of 2x minus 10.

00:05:08.930 --> 00:05:11.180
And it's going to be less
than on the right-hand side,

00:05:11.180 --> 00:05:14.010
you just end up with a 2 delta.

00:05:14.010 --> 00:05:16.140
Now, what do we have here
on the left-hand side?

00:05:16.140 --> 00:05:19.740
Well, this is f of x as
long as x does not equal 5.

00:05:19.740 --> 00:05:21.070
And this is our limit.

00:05:21.070 --> 00:05:28.570
So we can rewrite this as f of
x minus L is less than 2 delta.

00:05:28.570 --> 00:05:34.210
And this is for x
does not equal 5.

00:05:34.210 --> 00:05:37.850
This is f of x, this
literally is our limit.

00:05:37.850 --> 00:05:39.540
Now this is interesting.

00:05:39.540 --> 00:05:41.690
This statement right
over here is almost

00:05:41.690 --> 00:05:43.650
exactly what we want
right over here,

00:05:43.650 --> 00:05:46.490
except the right sides
are just different.

00:05:46.490 --> 00:05:49.730
In terms of epsilon, this
has it in terms of delta.

00:05:49.730 --> 00:05:56.050
So, how can we define delta
so that 2 delta is essentially

00:05:56.050 --> 00:05:57.854
going to be epsilon?

00:05:57.854 --> 00:05:58.895
Well, this is our chance.

00:06:03.790 --> 00:06:07.420
And this is where we're defining
delta as a function of epsilon.

00:06:07.420 --> 00:06:10.600
We're going to make 2
delta equal epsilon.

00:06:10.600 --> 00:06:12.490
Or if you divide
both sides by 2,

00:06:12.490 --> 00:06:16.535
we're going to make delta
equal to epsilon over 2.

00:06:22.310 --> 00:06:24.840
Let me switch colors just
to ease the monotony.

00:06:24.840 --> 00:06:32.132
If you make delta
equal epsilon over 2,

00:06:32.132 --> 00:06:33.590
then this statement
right over here

00:06:33.590 --> 00:06:37.530
becomes the absolute
value of f of x minus L

00:06:37.530 --> 00:06:40.150
is less than,
instead of 2 delta,

00:06:40.150 --> 00:06:42.730
it'll be less than 2
times epsilon over 2.

00:06:42.730 --> 00:06:45.360
It's just going to
be less than epsilon.

00:06:45.360 --> 00:06:46.800
So this is the key.

00:06:46.800 --> 00:06:49.510
If someone gives you any
positive number epsilon

00:06:49.510 --> 00:06:53.970
for this function, as long as
you make delta equal epsilon

00:06:53.970 --> 00:06:59.060
over 2, then any x
within that range,

00:06:59.060 --> 00:07:01.610
that corresponding
f of x is going

00:07:01.610 --> 00:07:04.615
to be within epsilon
of our limit.

00:07:07.845 --> 00:07:10.220
And remember, it has to be
true for any positive epsilon.

00:07:10.220 --> 00:07:12.020
But you could see how
the game could go.

00:07:12.020 --> 00:07:15.360
If someone gives
you the epsilon,

00:07:15.360 --> 00:07:18.270
let's say they want to be
within 0.5 of our limit.

00:07:18.270 --> 00:07:20.770
So our limit is up here,
so our epsilon is 0.5.

00:07:20.770 --> 00:07:23.790
So it would literally be the
range I want to be between 10

00:07:23.790 --> 00:07:26.160
plus epsilon would be 10.5.

00:07:26.160 --> 00:07:29.340
And then 10 minus
epsilon would be 9.5.

00:07:29.340 --> 00:07:31.270
Well, we just came
up with a formula.

00:07:31.270 --> 00:07:34.170
We just have to make
delta equal to epsilon

00:07:34.170 --> 00:07:37.380
over 2, which is equal to 0.25.

00:07:37.380 --> 00:07:44.990
So that'll give us a range
between 4.75 and 5.25.

00:07:44.990 --> 00:07:51.040
And as long as we pick an
x between 4.75 and 5.25,

00:07:51.040 --> 00:07:53.510
but not x equals 5, the
corresponding f of x

00:07:53.510 --> 00:07:57.090
will be between 9.5 and 10.5.

00:07:57.090 --> 00:07:59.120
And so, you give
me any epsilon, I

00:07:59.120 --> 00:08:02.617
can just apply this
formula right over here

00:08:02.617 --> 00:08:03.700
to come up with the delta.

00:08:03.700 --> 00:08:05.550
This would apply
for any real number.

00:08:05.550 --> 00:08:07.350
I mean, especially
any positive number.

00:08:07.350 --> 00:08:09.350
For any epsilon
you give me, I just

00:08:09.350 --> 00:08:12.730
get a delta defined
this way, and then I

00:08:12.730 --> 00:08:16.490
can go through this arc.

00:08:16.490 --> 00:08:19.170
If the absolute value of x
minus 5 is less than delta,

00:08:19.170 --> 00:08:21.990
if delta is defined
in this way, which

00:08:21.990 --> 00:08:23.880
I could define for
any epsilon, then it

00:08:23.880 --> 00:08:26.130
will be the case
that f of x will

00:08:26.130 --> 00:08:29.730
be within epsilon of our limit.


Epsilon-delta definition of limits

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:02.490
In the last video,
we tried to come up

00:00:02.490 --> 00:00:05.120
with a somewhat rigorous
definition of what a limit is,

00:00:05.120 --> 00:00:07.520
where we say when you
say that the limit of f

00:00:07.520 --> 00:00:11.090
of x as x approaches C is
equal L, you're really saying--

00:00:11.090 --> 00:00:13.740
and this is the somewhat
rigorous definition--

00:00:13.740 --> 00:00:16.540
that you can get f
of x as close as you

00:00:16.540 --> 00:00:21.870
want to L by making x
sufficiently close to C.

00:00:21.870 --> 00:00:24.120
So let's see if we can put
a little bit of meat on it.

00:00:24.120 --> 00:00:26.280
So instead of saying
as close as you want,

00:00:26.280 --> 00:00:29.285
let's call that some
positive number epsilon.

00:00:29.285 --> 00:00:32.210
So I'm just going to use the
Greek letter epsilon right over

00:00:32.210 --> 00:00:32.860
there.

00:00:32.860 --> 00:00:34.600
So it really turns into a game.

00:00:38.810 --> 00:00:39.920
So this is the game.

00:00:39.920 --> 00:00:54.450
You tell me how close you
want f of x to be to L.

00:00:54.450 --> 00:00:58.050
And you do this by giving me
a positive number that we call

00:00:58.050 --> 00:01:00.810
epsilon, which is really
how close you want

00:01:00.810 --> 00:01:10.220
f of x to be to L. So you give
a positive number epsilon.

00:01:10.220 --> 00:01:12.780
And epsilon is how
close do you want to be?

00:01:12.780 --> 00:01:14.540
How close?

00:01:14.540 --> 00:01:18.260
So for example, if
epsilon is 0.01,

00:01:18.260 --> 00:01:23.870
that says that you want f of x
to be within 0.01 of epsilon.

00:01:23.870 --> 00:01:27.790
And so what I then
do is I say well, OK.

00:01:27.790 --> 00:01:29.530
You've given me that epsilon.

00:01:29.530 --> 00:01:39.130
I'm going to find you another
positive number which we'll

00:01:39.130 --> 00:01:42.750
call delta-- the lowercase
delta, the Greek letter

00:01:42.750 --> 00:01:56.210
delta-- such that where
if x is within delta of C,

00:01:56.210 --> 00:02:10.129
then f of x will be within
epsilon of our limit.

00:02:10.129 --> 00:02:12.420
So let's see if these are
really saying the same thing.

00:02:12.420 --> 00:02:14.400
In this yellow definition
right over here,

00:02:14.400 --> 00:02:16.520
we said you can
get f of x as close

00:02:16.520 --> 00:02:21.400
as you want to L by making
x sufficiently close to C.

00:02:21.400 --> 00:02:23.042
This second definition,
which I kind of

00:02:23.042 --> 00:02:26.190
made as a little bit more of a
game, is doing the same thing.

00:02:26.190 --> 00:02:30.126
Someone is saying how close
they want f of x to be to L

00:02:30.126 --> 00:02:33.710
and the burden is then to
find a delta where as long

00:02:33.710 --> 00:02:37.240
as x is within delta
of C, then f of x

00:02:37.240 --> 00:02:40.460
will be within
epsilon of the limit.

00:02:40.460 --> 00:02:41.380
So that is doing it.

00:02:41.380 --> 00:02:44.190
It's saying look, if we are
constraining x in such a way

00:02:44.190 --> 00:02:47.210
that if x is in that
range to C, then

00:02:47.210 --> 00:02:49.825
f of x will be as
close as you want.

00:02:49.825 --> 00:02:51.450
So let's make this
a little bit clearer

00:02:51.450 --> 00:02:53.690
by diagramming right over here.

00:02:53.690 --> 00:02:57.210
You show up and you
say well, I want

00:02:57.210 --> 00:03:01.690
f of x to be within
epsilon of our limit.

00:03:05.460 --> 00:03:09.600
This point right over here
is our limit plus epsilon.

00:03:09.600 --> 00:03:17.510
And this right over here might
be our limit minus epsilon.

00:03:17.510 --> 00:03:19.010
And you say, OK, sure.

00:03:19.010 --> 00:03:25.380
I think I can get your f of x
within this range of our limit.

00:03:25.380 --> 00:03:29.430
And I can do that by
defining a range around C.

00:03:29.430 --> 00:03:32.150
And I could visually
look at this boundary.

00:03:32.150 --> 00:03:34.150
But I could even go
narrower than that boundary.

00:03:34.150 --> 00:03:37.570
I could go right over here.

00:03:37.570 --> 00:03:39.590
Says OK, I meet your challenge.

00:03:39.590 --> 00:03:42.940
I will find another
number delta.

00:03:42.940 --> 00:03:46.010
So this right over
here is C plus delta.

00:03:46.010 --> 00:03:49.170
This right over here
is C minus-- let

00:03:49.170 --> 00:03:52.770
me write this down--
is C minus delta.

00:03:52.770 --> 00:03:55.020
So I'll find you some
delta so that if you

00:03:55.020 --> 00:04:01.415
take any x in the range
C minus delta to C

00:04:01.415 --> 00:04:03.350
plus delta-- and maybe
the function's not even

00:04:03.350 --> 00:04:05.350
defined at C, so
we think of ones

00:04:05.350 --> 00:04:07.350
that maybe aren't C, but
are getting very close.

00:04:07.350 --> 00:04:11.370
If you find any x in
that range, f of those

00:04:11.370 --> 00:04:17.459
x's are going to be as close
as you want to your limit.

00:04:17.459 --> 00:04:22.686
They're going to be within
the range L plus epsilon or L

00:04:22.686 --> 00:04:24.850
minus epsilon.

00:04:24.850 --> 00:04:27.390
So what's another
way of saying this?

00:04:27.390 --> 00:04:30.350
Another way of saying this
is you give me an epsilon,

00:04:30.350 --> 00:04:32.160
then I will find you a delta.

00:04:32.160 --> 00:04:34.670
So let me write this in a
little bit more math notation.

00:04:34.670 --> 00:04:36.310
So I'll write the
same exact statements

00:04:36.310 --> 00:04:39.771
with a little bit
more math here.

00:04:39.771 --> 00:04:41.020
But it's the exact same thing.

00:04:44.939 --> 00:04:45.980
Let me write it this way.

00:04:45.980 --> 00:04:55.480
Given an epsilon
greater than 0-- so

00:04:55.480 --> 00:04:57.800
that's kind of the
first part of the game--

00:04:57.800 --> 00:05:06.330
we can find a delta
greater than 0, such

00:05:06.330 --> 00:05:13.540
that if x is within delta of C.

00:05:13.540 --> 00:05:17.304
So what's another way of saying
that x is within delta of C?

00:05:17.304 --> 00:05:18.720
Well, one way you
could say, well,

00:05:18.720 --> 00:05:22.010
what's the distance
between x and C

00:05:22.010 --> 00:05:24.030
is going to be less than delta.

00:05:24.030 --> 00:05:28.240
This statement is true for any
x that's within delta of C.

00:05:28.240 --> 00:05:31.810
The difference between the two
is going to be less than delta.

00:05:31.810 --> 00:05:36.290
So that if you pick an x that
is in this range between C

00:05:36.290 --> 00:05:38.820
minus delta and C
plus delta, and these

00:05:38.820 --> 00:05:43.480
are the x's that satisfy
that right over here, then--

00:05:43.480 --> 00:05:46.440
and I'll do this in
a new color-- then

00:05:46.440 --> 00:05:51.920
the distance between your
f of x and your limit--

00:05:51.920 --> 00:05:53.920
and this is just the
distance between the f of x

00:05:53.920 --> 00:05:57.450
and the limit, it's going
to be less than epsilon.

00:05:57.450 --> 00:06:02.770
So all this is saying is, if
the limit truly does exist,

00:06:02.770 --> 00:06:05.990
it truly is L, is if you give
me any positive number epsilon,

00:06:05.990 --> 00:06:09.580
it could be super, super small
one, we can find a delta.

00:06:09.580 --> 00:06:11.970
So we can define
a range around C

00:06:11.970 --> 00:06:15.220
so that if we take
any x value that

00:06:15.220 --> 00:06:17.770
is within delta of C,
that's all this statement

00:06:17.770 --> 00:06:19.570
is saying that the
distance between x and C

00:06:19.570 --> 00:06:20.830
is less than delta.

00:06:20.830 --> 00:06:22.330
So it's within delta
of C. So that's

00:06:22.330 --> 00:06:24.080
these points right over here.

00:06:24.080 --> 00:06:28.440
That f of those
x's, the function

00:06:28.440 --> 00:06:32.150
evaluated at those x's is
going to be within the range

00:06:32.150 --> 00:06:33.970
that you are specifying.

00:06:33.970 --> 00:06:37.130
It's going to be within
epsilon of our limit.

00:06:37.130 --> 00:06:40.810
The f of x, the difference
between f of x, and your limit

00:06:40.810 --> 00:06:42.840
will be less than epsilon.

00:06:42.840 --> 00:06:47.490
Your f of x is going to
sit some place over there.

00:06:47.490 --> 00:06:50.840
So that's all the epsilon-delta
definition is telling us.

00:06:50.840 --> 00:06:54.660
In the next video, we will
prove that a limit exists

00:06:54.660 --> 00:06:58.710
by using this
definition of limits.


Formal definition of limits Part 2: building the idea

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.330
Let's try to come up with
a mathematically rigorous

00:00:03.330 --> 00:00:05.790
definition for what
this statement means.

00:00:05.790 --> 00:00:07.590
The statement that
the limit of f

00:00:07.590 --> 00:00:11.990
of x as x approaches c
is equal to L. So let's

00:00:11.990 --> 00:00:24.490
say that this means that
you can get f of x as close

00:00:24.490 --> 00:00:29.734
to L as you want.

00:00:29.734 --> 00:00:31.400
I'll put that in
quotes right over here,

00:00:31.400 --> 00:00:32.899
because it's kind
of a little loosey

00:00:32.899 --> 00:00:34.120
goosey as how close is that.

00:00:34.120 --> 00:00:39.930
But as close as
you want by getting

00:00:39.930 --> 00:00:49.770
x sufficiently close to c.

00:00:49.770 --> 00:00:52.310
So another way of saying
this is, if you tell me, hey,

00:00:52.310 --> 00:00:57.740
I want to get my f of x to
be within 0.5 of this limit.

00:00:57.740 --> 00:01:00.160
Then you're telling me if
this limit is actually true,

00:01:00.160 --> 00:01:04.030
you should be able to
hand me a value around c.

00:01:04.030 --> 00:01:07.050
That if x is within
that range, then f of x

00:01:07.050 --> 00:01:10.840
is definitely going to be
as close to L as I desire.

00:01:10.840 --> 00:01:13.489
So let me draw that out to
make it a little bit clearer.

00:01:13.489 --> 00:01:14.530
And I'm going to zoom in.

00:01:14.530 --> 00:01:16.490
I'm going to draw
another diagram.

00:01:16.490 --> 00:01:24.770
So let's say that this right
over here is my y-axis.

00:01:24.770 --> 00:01:25.980
And I'm going to zoom in.

00:01:25.980 --> 00:01:27.570
I'm going to draw a slightly
different function, just so we

00:01:27.570 --> 00:01:29.570
can really focus on what's
going on around here.

00:01:29.570 --> 00:01:33.660
The range is around c, and the
range is around L. So that's x.

00:01:33.660 --> 00:01:36.190
This right over here is y.

00:01:36.190 --> 00:01:41.566
Let's say that this is c.

00:01:41.566 --> 00:01:43.190
And let's just zoom
in on our function.

00:01:43.190 --> 00:01:46.780
So let's say our function
looks, is doing something like,

00:01:46.780 --> 00:01:49.015
let's say it does something
like, let's see, I

00:01:49.015 --> 00:01:50.390
don't want it to
be defined at c.

00:01:50.390 --> 00:01:52.477
At least just for
the-- it could be.

00:01:52.477 --> 00:01:54.560
You can always find a limit
even where is defined.

00:01:54.560 --> 00:01:57.269
But let's say our function
looks something like that.

00:01:57.269 --> 00:01:59.560
And it can have a little kink
in it, the way I drew it.

00:01:59.560 --> 00:02:01.080
So it looks something like this.

00:02:01.080 --> 00:02:02.027
It's undefined.

00:02:02.027 --> 00:02:03.610
Let me draw it a
little bit different.

00:02:06.240 --> 00:02:10.710
So it is undefined
when x is equal to c.

00:02:10.710 --> 00:02:12.910
So this is the point
where there's a hole.

00:02:12.910 --> 00:02:15.776
It is undefined when
x is equal to c.

00:02:15.776 --> 00:02:19.420
So it even has a little
kink in it, just like that.

00:02:19.420 --> 00:02:23.040
And what we want to do
is prove that the limit,

00:02:23.040 --> 00:02:26.020
as x, the limit of f of x--
and let me make it clear,

00:02:26.020 --> 00:02:28.160
this is the graph
of y is equal to f

00:02:28.160 --> 00:02:32.420
of x-- we want to get an idea
for what this definition is

00:02:32.420 --> 00:02:33.270
saying.

00:02:33.270 --> 00:02:37.870
If we're claiming that the limit
of f of x, as x approaches c,

00:02:37.870 --> 00:02:38.640
is L.

00:02:38.640 --> 00:02:41.820
So conceptually, we
get the gist already.

00:02:41.820 --> 00:02:46.430
We already get the gist that
this right over here is L.

00:02:46.430 --> 00:02:49.650
But what is this
definition saying?

00:02:49.650 --> 00:02:53.520
Well, it's saying that you
can get f of x as close

00:02:53.520 --> 00:02:57.150
to L as you want.

00:02:57.150 --> 00:03:00.390
So if you tell
someone, I want to get

00:03:00.390 --> 00:03:02.723
f of x within a
certain range of L,

00:03:02.723 --> 00:03:05.300
then if this limit
is actually true,

00:03:05.300 --> 00:03:10.920
if the limit of f of
x as x approaches c

00:03:10.920 --> 00:03:13.870
really is equal to
L, then they should

00:03:13.870 --> 00:03:16.010
be able to find
a range around c.

00:03:16.010 --> 00:03:18.250
That as long as x is
around that range,

00:03:18.250 --> 00:03:21.197
your f of x is going to be
in the range that you want.

00:03:21.197 --> 00:03:23.030
So let me actually go
through that exercise.

00:03:23.030 --> 00:03:25.720
It really is a little
bit like a game.

00:03:25.720 --> 00:03:28.690
So someone comes up to
you and says, well, OK.

00:03:28.690 --> 00:03:30.930
I don't necessarily
believe that you're

00:03:30.930 --> 00:03:34.125
claiming the limit of f of x as
x approaches c is equal to L.

00:03:34.125 --> 00:03:36.350
I'm not really sure
if that's the case.

00:03:36.350 --> 00:03:38.350
But I agree with
this definition.

00:03:38.350 --> 00:03:42.790
So I want to get within 0.5.

00:03:42.790 --> 00:03:50.540
I want to get f of x within 0.5
of L. So this right over here

00:03:50.540 --> 00:03:54.180
would be L plus 0.5.

00:03:54.180 --> 00:04:02.110
And this right over
here is L minus 0.5.

00:04:02.110 --> 00:04:03.350
And then you say, fine.

00:04:03.350 --> 00:04:05.350
I'm going to give
you a range around c,

00:04:05.350 --> 00:04:08.290
that if you take any x within
that range, your f of x

00:04:08.290 --> 00:04:12.780
is always going to fall in
this range that you care about.

00:04:12.780 --> 00:04:15.330
And so you look at this-- and
obviously we haven't explicitly

00:04:15.330 --> 00:04:16.277
defined this function.

00:04:16.277 --> 00:04:18.860
But you can even eyeball it, the
way this function is defined.

00:04:18.860 --> 00:04:21.089
It won't be that easy
for all functions.

00:04:21.089 --> 00:04:22.890
But you look at it like this.

00:04:22.890 --> 00:04:25.206
And you say that this
value, just the way

00:04:25.206 --> 00:04:26.580
it's drawn right
over here, let's

00:04:26.580 --> 00:04:31.770
say that this is c minus 0.25.

00:04:31.770 --> 00:04:40.710
And let's say that this value
right over here is c plus 0.25.

00:04:40.710 --> 00:04:42.450
And so you tell
them, look, as long

00:04:42.450 --> 00:04:46.200
as you get x within
0.25 of c, so as long

00:04:46.200 --> 00:04:49.110
as your x's are sitting
someplace over here,

00:04:49.110 --> 00:04:54.240
the corresponding f of x is
going to sit in the range

00:04:54.240 --> 00:04:56.170
that you care about.

00:04:56.170 --> 00:04:57.470
And you say, OK, fine.

00:04:57.470 --> 00:04:59.214
You won that round.

00:04:59.214 --> 00:05:00.380
Let me make it even tighter.

00:05:00.380 --> 00:05:05.130
Maybe instead of saying within
the 0.5, I want to get within

00:05:05.130 --> 00:05:07.361
is 0.05.

00:05:07.361 --> 00:05:09.610
And then you'd have to do
this exercise again and find

00:05:09.610 --> 00:05:10.850
another range.

00:05:10.850 --> 00:05:14.600
And in order for
this to be true,

00:05:14.600 --> 00:05:18.739
you would have to be able
to do this for any range

00:05:18.739 --> 00:05:19.530
that they give you.

00:05:19.530 --> 00:05:22.015
For any range around
L that they give you,

00:05:22.015 --> 00:05:24.950
you have to be able to get
f of x within that range

00:05:24.950 --> 00:05:27.410
by finding a range around c.

00:05:27.410 --> 00:05:29.480
That as long as x is
that range around c,

00:05:29.480 --> 00:05:32.434
f of x is going to
sit within that range.

00:05:32.434 --> 00:05:34.350
So I'll let you think
about that a little bit.

00:05:34.350 --> 00:05:35.558
There's a lot to think about.

00:05:35.558 --> 00:05:37.202
But hopefully this made sense.

00:05:37.202 --> 00:05:39.160
We did it for the particular
example of someone

00:05:39.160 --> 00:05:43.830
hands you the 0.5, I want
f of x within the 0.5 of L,

00:05:43.830 --> 00:05:48.279
and you say, well, as long
as x is within 0.25 of c,

00:05:48.279 --> 00:05:49.320
you're going to match it.

00:05:49.320 --> 00:05:51.153
You need to be able to
do that for any range

00:05:51.153 --> 00:05:54.190
they give you around
L. And then this limit

00:05:54.190 --> 00:05:55.960
will definitely be true.

00:05:55.960 --> 00:05:59.010
So in the next video, we
will now generalize that.

00:05:59.010 --> 00:06:02.950
And that will really bring us
to the famous epsilon delta

00:06:02.950 --> 00:06:05.470
definition of limits.


Formal definition of limits Part 1: intuition review

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.520
Let's review our intuition
of what a limit even is.

00:00:04.520 --> 00:00:07.580
So let me draw some axes here.

00:00:07.580 --> 00:00:13.260
So let's say this is my y-axis,
so try to draw a vertical line.

00:00:13.260 --> 00:00:17.050
So that right over
there is my y-axis.

00:00:17.050 --> 00:00:18.761
And then let's say
this is my x-axis.

00:00:18.761 --> 00:00:20.135
I'll focus on the
first quadrant,

00:00:20.135 --> 00:00:22.000
although I don't have to.

00:00:22.000 --> 00:00:24.940
So let's say this right
over here is my x-axis.

00:00:24.940 --> 00:00:26.780
And then let me draw a function.

00:00:26.780 --> 00:00:30.000
So let's say my function
looks something like that,

00:00:30.000 --> 00:00:32.280
could look like anything,
but that seems suitable.

00:00:32.280 --> 00:00:36.297
So this is the function
y is equal to f of x.

00:00:36.297 --> 00:00:38.380
And just for the sake of
conceptual understanding,

00:00:38.380 --> 00:00:39.900
I'm going to say it's
not defined at a point.

00:00:39.900 --> 00:00:40.800
I didn't have to do this.

00:00:40.800 --> 00:00:43.020
You can find the limit as
x approaches a point where

00:00:43.020 --> 00:00:44.722
the function
actually is defined,

00:00:44.722 --> 00:00:47.180
but it becomes that much more
interesting, at least for me,

00:00:47.180 --> 00:00:49.513
or you start to understand
why a limit might be relevant

00:00:49.513 --> 00:00:51.690
where a function is not
defined at some point.

00:00:51.690 --> 00:00:53.430
So the way I've drawn
it, this function

00:00:53.430 --> 00:01:02.320
is not defined when
x is equal to c.

00:01:02.320 --> 00:01:04.989
Now, the way that we've
thought about a limit

00:01:04.989 --> 00:01:09.310
is what does f of x
approach as x approaches c?

00:01:09.310 --> 00:01:11.270
So let's think about
that a little bit.

00:01:11.270 --> 00:01:14.582
When x is a reasonable
bit lower than c,

00:01:14.582 --> 00:01:19.840
f of x, for our function that we
just drew, is right over here.

00:01:19.840 --> 00:01:23.020
That's what f of x is going to
be equal. y is equal to f of x.

00:01:23.020 --> 00:01:27.490
When x gets a little bit
closer, then our f of x

00:01:27.490 --> 00:01:31.560
is right over there.

00:01:31.560 --> 00:01:34.070
When x gets even closer,
maybe really almost

00:01:34.070 --> 00:01:38.900
at c, but not quite at c, then
our f of x is right over here.

00:01:38.900 --> 00:01:41.750
And the way we see it,
we see that our f of x

00:01:41.750 --> 00:01:47.300
seems to be-- as x gets
closer and closer to c

00:01:47.300 --> 00:01:50.240
it looks like our f
of x is getting closer

00:01:50.240 --> 00:01:55.790
and closer to some
value right over there.

00:01:55.790 --> 00:01:57.992
Maybe I'll draw it
with a more solid line.

00:01:57.992 --> 00:01:59.450
And that was actually
only the case

00:01:59.450 --> 00:02:01.840
when x was getting closer
to c from the left,

00:02:01.840 --> 00:02:04.290
from values of x less than c.

00:02:04.290 --> 00:02:07.650
But what happens as we
get closer and closer

00:02:07.650 --> 00:02:10.490
to c from values of x
that are larger than c?

00:02:10.490 --> 00:02:14.970
Well, when x is over here,
f of x is right over here.

00:02:14.970 --> 00:02:18.790
And so that's what f of x
is, is right over there.

00:02:18.790 --> 00:02:21.810
When x gets a little bit
closer to c, our f of x

00:02:21.810 --> 00:02:24.140
is right over there.

00:02:24.140 --> 00:02:29.710
When x is just very slightly
larger than c, then our f of x

00:02:29.710 --> 00:02:31.770
is right over there.

00:02:31.770 --> 00:02:33.360
And you see, once
again, it seems

00:02:33.360 --> 00:02:37.510
to be approaching
that same value.

00:02:37.510 --> 00:02:40.140
And we call that value,
that value that f of x

00:02:40.140 --> 00:02:42.460
seems to be approaching
as x approaches

00:02:42.460 --> 00:02:46.150
c, we call that value
L, or the limit.

00:02:46.150 --> 00:02:47.700
And so the way we
would denote it

00:02:47.700 --> 00:02:49.124
is we would call that the limit.

00:02:49.124 --> 00:02:50.790
We don't have to call
it L all the time,

00:02:50.790 --> 00:02:52.716
but it is referred
to as the limit.

00:02:52.716 --> 00:02:54.090
And the way that
we would kind of

00:02:54.090 --> 00:03:00.560
write that mathematically is
we would say the limit of f

00:03:00.560 --> 00:03:08.120
of x as x approaches
c is equal to L.

00:03:08.120 --> 00:03:10.655
And this is a fine conceptual
understanding of limits,

00:03:10.655 --> 00:03:12.280
and it really will
take you pretty far,

00:03:12.280 --> 00:03:13.880
and you're ready to
progress and start thinking

00:03:13.880 --> 00:03:15.410
about taking a lot of limits.

00:03:15.410 --> 00:03:17.940
But this isn't a very
mathematically-rigorous

00:03:17.940 --> 00:03:19.800
definition of limits.

00:03:19.800 --> 00:03:22.180
And so this sets us
up for the intuition.

00:03:22.180 --> 00:03:24.470
In the next few videos,
we will introduce

00:03:24.470 --> 00:03:26.610
a mathematically-rigorous
definition

00:03:26.610 --> 00:03:29.120
of limits that will
allow us to do things

00:03:29.120 --> 00:03:32.360
like prove that the
limit as x approaches c

00:03:32.360 --> 00:03:35.820
truly is equal to L.


Differentiating polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.431 --> 00:00:02.430
Now that we know the power
rule, and we saw that

00:00:02.430 --> 00:00:04.670
in the last video, that
the derivative with respect

00:00:04.670 --> 00:00:10.190
to x, of x to the n, is going to
be equal to n times x to the n

00:00:10.190 --> 00:00:14.590
minus 1 for n not equal 0.

00:00:14.590 --> 00:00:18.347
I thought I would expose you
to a few more rules or concepts

00:00:18.347 --> 00:00:20.680
or properties of derivatives
that essentially will allow

00:00:20.680 --> 00:00:22.860
us to take the derivative
of any polynomial.

00:00:22.860 --> 00:00:25.214
So this is powerful
stuff going on.

00:00:25.214 --> 00:00:26.880
So the first thing I
want to think about

00:00:26.880 --> 00:00:31.420
is, why this little special
case for n not equaling 0?

00:00:31.420 --> 00:00:33.750
What happens if n equals 0?

00:00:33.750 --> 00:00:35.350
So let's just think
of the situation.

00:00:35.350 --> 00:00:37.880
Let's try to take the
derivative with respect

00:00:37.880 --> 00:00:42.860
to x of x to the 0 power.

00:00:42.860 --> 00:00:46.690
Well, what is x to the
0 power going to be?

00:00:46.690 --> 00:00:49.430
And we can assume that x for
this case right over here

00:00:49.430 --> 00:00:50.640
is not equal to 0.

00:00:50.640 --> 00:00:53.800
0 to the 0, weird things
happen at that point.

00:00:53.800 --> 00:00:55.600
But if x does not
equal 0, what is

00:00:55.600 --> 00:00:57.512
x to the 0 power going to be?

00:00:57.512 --> 00:00:59.970
Well, this is the same thing
as the derivative with respect

00:00:59.970 --> 00:01:02.350
to x of 1.

00:01:02.350 --> 00:01:05.069
x to the 0 power is
just going to be 1.

00:01:05.069 --> 00:01:10.050
And so what is the derivative
with respect to x of 1?

00:01:10.050 --> 00:01:13.020
And to answer that question,
I'll just graph it.

00:01:13.020 --> 00:01:16.030
I'll just graph f of
x equals 1 to make

00:01:16.030 --> 00:01:17.170
it a little bit clearer.

00:01:17.170 --> 00:01:19.540
So that's my y-axis.

00:01:19.540 --> 00:01:23.480
This is my x-axis.

00:01:23.480 --> 00:01:26.520
And let me graph y equals
1, or f of x equals 1.

00:01:26.520 --> 00:01:28.230
So that's 1 right
over there. f of x

00:01:28.230 --> 00:01:31.350
equals 1 is just
a horizontal line.

00:01:31.350 --> 00:01:33.120
So that right over
there is the graph,

00:01:33.120 --> 00:01:38.870
y is equal to f of x,
which is equal to 1.

00:01:38.870 --> 00:01:41.260
Now, remember the derivative,
one way to conceptualize

00:01:41.260 --> 00:01:44.810
is just the slope of the
tangent line at any point.

00:01:44.810 --> 00:01:48.040
So what is the slope of the
tangent line at this point?

00:01:48.040 --> 00:01:50.370
And actually, what's the
slope at every point?

00:01:50.370 --> 00:01:52.710
Well, this is a line, so
the slope doesn't change.

00:01:52.710 --> 00:01:53.710
It has a constant slope.

00:01:53.710 --> 00:01:55.293
And it's a completely
horizontal line.

00:01:55.293 --> 00:01:57.500
It has a slope of 0.

00:01:57.500 --> 00:01:59.570
So the slope at every
point over here,

00:01:59.570 --> 00:02:02.760
slope is going to be equal to 0.

00:02:02.760 --> 00:02:06.420
So the slope of this
line at any point

00:02:06.420 --> 00:02:09.470
is just going to be equal to 0.

00:02:09.470 --> 00:02:11.800
And that's actually going
to be true for any constant.

00:02:11.800 --> 00:02:14.720
The derivative, if I had a
function, let's say that f of x

00:02:14.720 --> 00:02:16.150
is equal to 3.

00:02:16.150 --> 00:02:19.580
Let's say that's
y is equal to 3.

00:02:19.580 --> 00:02:22.720
What's the derivative
of y with respect

00:02:22.720 --> 00:02:24.080
to x going to be equal to?

00:02:24.080 --> 00:02:27.210
And I'm intentionally showing
you all the different ways

00:02:27.210 --> 00:02:29.440
of the notation for derivatives.

00:02:29.440 --> 00:02:31.620
So what's the derivative
of y with respect to x?

00:02:31.620 --> 00:02:33.730
It can also be
written as y prime.

00:02:33.730 --> 00:02:36.610
What's that going
to be equal to?

00:02:36.610 --> 00:02:39.290
Well, it's the slope
at any given point.

00:02:39.290 --> 00:02:41.660
And you see that no matter
what x you're looking at,

00:02:41.660 --> 00:02:43.580
the slope here is going to be 0.

00:02:43.580 --> 00:02:45.140
So it's going to be 0.

00:02:45.140 --> 00:02:46.570
So it's not just x to the 0.

00:02:46.570 --> 00:02:48.400
If you take the derivative
of any constant,

00:02:48.400 --> 00:02:50.095
you're going to get 0.

00:02:50.095 --> 00:02:50.970
So let me write that.

00:02:50.970 --> 00:02:54.600
Derivative with respect
to x of any constant--

00:02:54.600 --> 00:03:00.790
so let's say of a where this
is just a constant, that's

00:03:00.790 --> 00:03:02.720
going to be equal to 0.

00:03:02.720 --> 00:03:05.010
So pretty straightforward idea.

00:03:05.010 --> 00:03:08.040
Now let's explore a
few more properties.

00:03:08.040 --> 00:03:11.370
Let's say I want to take
the derivative with respect

00:03:11.370 --> 00:03:15.380
to x of-- let's use
the same A. Let's say

00:03:15.380 --> 00:03:18.465
I have some constant
times some function.

00:03:21.040 --> 00:03:23.090
Well, derivatives
work out quite well.

00:03:23.090 --> 00:03:26.070
You can actually take this
little scalar multiplier,

00:03:26.070 --> 00:03:28.500
this little constant, and
take it out of the derivative.

00:03:28.500 --> 00:03:34.590
This is going to be
equal to A. I didn't

00:03:34.590 --> 00:03:36.230
want to do that magenta color.

00:03:36.230 --> 00:03:41.790
It's going to be equal to A
times the derivative of f of x.

00:03:47.150 --> 00:03:50.470
Let me do that blue color.

00:03:50.470 --> 00:03:52.860
And the other way to denote
the derivative of f of x

00:03:52.860 --> 00:03:55.110
is to just say that
this is the same thing.

00:03:55.110 --> 00:03:59.270
This is equal to A times
this thing right over here

00:03:59.270 --> 00:04:04.020
is the exact same
thing as f prime of x.

00:04:04.020 --> 00:04:06.890
Now this might all look
like really fancy notation,

00:04:06.890 --> 00:04:08.390
but I think if I
gave you an example

00:04:08.390 --> 00:04:09.950
it might make some sense.

00:04:09.950 --> 00:04:14.380
So what about if I were to ask
you the derivative with respect

00:04:14.380 --> 00:04:28.210
to x of 2 times x
to the fifth power?

00:04:28.210 --> 00:04:30.420
Well, this property
that I just articulated

00:04:30.420 --> 00:04:32.560
says, well, this is going
to be the same thing as 2

00:04:32.560 --> 00:04:40.010
times the derivative of x to the
fifth, 2 times the derivative

00:04:40.010 --> 00:04:43.330
with respect to x
of x to the fifth.

00:04:43.330 --> 00:04:45.740
Essentially, I could just
take this scalar multiplier

00:04:45.740 --> 00:04:48.160
and put it in front
of the derivative.

00:04:48.160 --> 00:04:50.060
So this right here,
this is the derivative

00:04:50.060 --> 00:04:53.180
with respect to x
of x to the fifth.

00:04:53.180 --> 00:04:55.390
And we know how to do
that using the power rule.

00:04:55.390 --> 00:04:59.350
This is going to be equal to
2 times-- let me write that.

00:04:59.350 --> 00:05:01.600
I want to keep it
consistent with the colors.

00:05:01.600 --> 00:05:05.540
This is going to be 2 times the
derivative of x to the fifth.

00:05:05.540 --> 00:05:07.400
Well, the power rule
tells us, n is 5.

00:05:07.400 --> 00:05:10.990
It's going to be 5x
to the 5 minus 1 or 5x

00:05:10.990 --> 00:05:12.620
to the fourth power.

00:05:12.620 --> 00:05:17.130
So it's going to be 5x to
the fourth power, which

00:05:17.130 --> 00:05:23.230
is going to be equal to 2
times 5 is 10, x to the fourth.

00:05:23.230 --> 00:05:24.980
So 2x to the fifth,
you can literally just

00:05:24.980 --> 00:05:26.771
say, OK, the power rule
tells me derivative

00:05:26.771 --> 00:05:28.160
of that is 5x to the fourth.

00:05:28.160 --> 00:05:30.180
5 times 2 is 10.

00:05:30.180 --> 00:05:32.710
So that simplifies
our life a good bit.

00:05:32.710 --> 00:05:35.126
We can now, using the power
rule and this one property,

00:05:35.126 --> 00:05:36.500
take the derivative
anything that

00:05:36.500 --> 00:05:39.910
takes the form Ax
to the n power.

00:05:39.910 --> 00:05:41.900
Now let's think about
another very useful

00:05:41.900 --> 00:05:42.770
derivative property.

00:05:42.770 --> 00:05:44.644
And these don't just
apply to the power rule,

00:05:44.644 --> 00:05:45.940
they apply to any derivative.

00:05:45.940 --> 00:05:47.610
But they are especially
useful for the power rule

00:05:47.610 --> 00:05:50.130
because it allows us to
construct polynomials and take

00:05:50.130 --> 00:05:51.570
the derivatives of them.

00:05:51.570 --> 00:05:53.070
But if I were to
take the derivative

00:05:53.070 --> 00:05:56.120
of the sum of two functions--
so the derivative of,

00:05:56.120 --> 00:05:58.870
let's say one function
is f of x and then

00:05:58.870 --> 00:06:02.620
the other function is g of x.

00:06:02.620 --> 00:06:04.700
It's lucky for us
that this ends up

00:06:04.700 --> 00:06:06.630
being the same thing
as the derivative of f

00:06:06.630 --> 00:06:09.220
of x plus the
derivative of g of x.

00:06:09.220 --> 00:06:12.130
So this is the same
thing as f-- actually,

00:06:12.130 --> 00:06:14.980
let me use that derivative
operator just to make it clear.

00:06:14.980 --> 00:06:17.890
It's the same thing as the
derivative with respect

00:06:17.890 --> 00:06:21.760
to x of f of x plus the
derivative with respect

00:06:21.760 --> 00:06:24.250
to x of g of x.

00:06:24.250 --> 00:06:27.340
So we'll put f of
x right over here

00:06:27.340 --> 00:06:32.270
and put g of x right over there.

00:06:32.270 --> 00:06:33.870
And so with the
other notation, we

00:06:33.870 --> 00:06:36.260
can say this is going
to be the same thing.

00:06:36.260 --> 00:06:40.440
Derivative with respect to x
of f of x, we can write as f

00:06:40.440 --> 00:06:41.790
prime of x.

00:06:41.790 --> 00:06:45.810
And the derivative with
respect to x of g of x,

00:06:45.810 --> 00:06:49.650
we can write as g prime of x.

00:06:49.650 --> 00:06:52.641
Now, once again, this might
look like kind of fancy notation

00:06:52.641 --> 00:06:53.140
to you.

00:06:53.140 --> 00:06:56.740
But when you see an example,
it'll make it pretty clear.

00:06:56.740 --> 00:07:00.470
If I want to take the
derivative with respect

00:07:00.470 --> 00:07:06.180
to x of let's say x
to the third power

00:07:06.180 --> 00:07:12.600
plus x to the negative
4 power, this just

00:07:12.600 --> 00:07:15.330
tells us that the
derivative of the sum

00:07:15.330 --> 00:07:16.960
is just the sum of
the derivatives.

00:07:16.960 --> 00:07:20.200
So we can take the derivative of
this term using the power rule.

00:07:20.200 --> 00:07:23.660
So it's going to be 3x squared.

00:07:23.660 --> 00:07:26.140
And to that, we can add the
derivative of this thing right

00:07:26.140 --> 00:07:27.460
over here.

00:07:27.460 --> 00:07:29.580
So it's going to
be plus-- that's

00:07:29.580 --> 00:07:33.830
a different shade of blue--
and over here is negative 4.

00:07:33.830 --> 00:07:40.720
So it's plus negative 4 times
x to the negative 4 minus 1,

00:07:40.720 --> 00:07:44.010
or x to the negative 5 power.

00:07:44.010 --> 00:07:46.490
So we have-- and I could
just simplify a little bit.

00:07:46.490 --> 00:07:51.020
This is going to be
equal to 3x squared

00:07:51.020 --> 00:07:56.060
minus 4x to the negative 5.

00:07:56.060 --> 00:08:00.050
And so now we have all the
tools we need in our toolkit

00:08:00.050 --> 00:08:03.290
to essentially take the
derivative of any polynomial.

00:08:03.290 --> 00:08:05.730
So let's give ourselves
a little practice there.

00:08:05.730 --> 00:08:11.820
So let's say that I have--
and I'll do it in white.

00:08:11.820 --> 00:08:15.400
Let's say that f
of x is equal to 2x

00:08:15.400 --> 00:08:25.970
to the third power minus 7x
squared plus 3x minus 100.

00:08:25.970 --> 00:08:27.810
What is f prime of x?

00:08:27.810 --> 00:08:32.017
What is the derivative of f
with respect to x going to be?

00:08:32.017 --> 00:08:34.100
Well, we can use the
properties that we just said.

00:08:34.100 --> 00:08:35.599
The derivative of
this is just going

00:08:35.599 --> 00:08:38.209
to be 2 times the derivative
of x to the third.

00:08:38.209 --> 00:08:40.500
Derivative of x to the third
is going to be 3x squared,

00:08:40.500 --> 00:08:44.840
so it's just going to
be 2 times 3x squared.

00:08:44.840 --> 00:08:46.950
What's the derivative
of negative 7x

00:08:46.950 --> 00:08:47.940
squared going to be?

00:08:47.940 --> 00:08:50.430
Well, it's just going
to be negative 7 times

00:08:50.430 --> 00:08:55.080
the derivative of x
squared, which is 2x.

00:08:55.080 --> 00:08:57.837
What is the derivative
of 3x going to be?

00:08:57.837 --> 00:09:00.170
Well, it's just going to be
3 times the derivative of x,

00:09:00.170 --> 00:09:02.710
or 3 times the derivative
of x to the first.

00:09:02.710 --> 00:09:04.720
The derivative of x to
the first is just 1.

00:09:04.720 --> 00:09:07.890
So this is just going
to be plus 3 times--

00:09:07.890 --> 00:09:11.000
we could say 1x to the
0-- but that's just 1.

00:09:11.000 --> 00:09:14.500
And then finally, what's
the derivative of a constant

00:09:14.500 --> 00:09:15.240
going to be?

00:09:15.240 --> 00:09:17.890
Let me do that in
a different color.

00:09:17.890 --> 00:09:20.402
What's the derivative of
a constant going to be?

00:09:20.402 --> 00:09:22.610
Well, we covered that at
the beginning of this video.

00:09:22.610 --> 00:09:24.390
The derivative of
any constant is just

00:09:24.390 --> 00:09:27.990
going to be 0, so plus 0.

00:09:27.990 --> 00:09:30.960
And so now we are
ready to simplify.

00:09:30.960 --> 00:09:33.610
The derivative of
f is going to be

00:09:33.610 --> 00:09:36.920
2 times 3x squared
is just 6x squared.

00:09:36.920 --> 00:09:41.900
Negative 7 times 2x is
negative 14x plus 3.

00:09:41.900 --> 00:09:43.720
And we don't have to
write the 0 there.

00:09:43.720 --> 00:09:44.630
And we're done.

00:09:44.630 --> 00:09:47.404
We now have all the
properties in our tool belt

00:09:47.404 --> 00:09:49.070
to find the derivative
of any polynomial

00:09:49.070 --> 00:09:52.490
and actually things that
even go beyond polynomials.


Justifying the power rule

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:02.000
What I want to do
in this video is

00:00:02.000 --> 00:00:04.140
to see whether the
power rule is giving us

00:00:04.140 --> 00:00:06.055
results that at least
seem reasonable.

00:00:06.055 --> 00:00:08.480
This is by no means a
proof of the power rule,

00:00:08.480 --> 00:00:12.720
but at least we'll feel a little
bit more comfortable using it.

00:00:12.720 --> 00:00:18.190
So let's say that f
of x is equal to x.

00:00:18.190 --> 00:00:20.440
The power rule tells
us that f prime

00:00:20.440 --> 00:00:23.270
of x is going to
be equal to what?

00:00:23.270 --> 00:00:25.990
Well, x is the same thing
as x to the first power.

00:00:25.990 --> 00:00:28.194
So n is implicitly
1 right over here.

00:00:28.194 --> 00:00:29.360
So we bring the 1 out front.

00:00:29.360 --> 00:00:32.740
It'll be 1 times x to
the 1 minus 1 power.

00:00:32.740 --> 00:00:36.200
So it's going to be 1 times
x to the 0 power. x to the 0

00:00:36.200 --> 00:00:36.730
is just 1.

00:00:36.730 --> 00:00:38.920
So it's just going
to be equal to 1.

00:00:38.920 --> 00:00:40.560
Now, does that makes
conceptual sense

00:00:40.560 --> 00:00:43.100
if we actually try to
visualize these functions?

00:00:43.100 --> 00:00:47.370
So let me actually try
to graph these functions.

00:00:47.370 --> 00:00:50.700
So that's my y-axis.

00:00:50.700 --> 00:00:52.230
This is my x-axis.

00:00:52.230 --> 00:00:55.050
And let me graph y equals x.

00:00:55.050 --> 00:00:57.320
So y is equal to f of x here.

00:00:57.320 --> 00:00:58.960
So y is equal to x.

00:00:58.960 --> 00:01:01.180
So it looks something like that.

00:01:01.180 --> 00:01:04.209
So y is equal to x.

00:01:04.209 --> 00:01:06.000
Or this is f of x
is equal to x, or y

00:01:06.000 --> 00:01:08.560
is equal to this f of
x right over there.

00:01:08.560 --> 00:01:12.540
Now, actually, let me
just call that f of x just

00:01:12.540 --> 00:01:13.910
to not confuse you.

00:01:13.910 --> 00:01:15.930
So this right over
here is f of x

00:01:15.930 --> 00:01:17.870
is equal to x that I
graphed right over here.

00:01:17.870 --> 00:01:21.330
y is equal to f of x,
which is equal to x.

00:01:21.330 --> 00:01:23.280
And now, let me
graph the derivative.

00:01:23.280 --> 00:01:24.850
Let me graph f prime of x.

00:01:24.850 --> 00:01:26.370
That's saying it's 1.

00:01:26.370 --> 00:01:27.850
That's saying it's 1 for all x.

00:01:27.850 --> 00:01:30.821
Regardless of what x is,
it's going to be equal to 1.

00:01:30.821 --> 00:01:32.320
Is this consistent
with what we know

00:01:32.320 --> 00:01:35.380
about derivatives and
slopes and all the rest?

00:01:35.380 --> 00:01:36.890
Well, let's look
at our function.

00:01:36.890 --> 00:01:41.560
What is the slope of the tangent
line right at this point?

00:01:41.560 --> 00:01:45.130
Well, right over here, this
has slope 1 continuously.

00:01:45.130 --> 00:01:47.060
Or it has a constant slope of 1.

00:01:47.060 --> 00:01:49.220
Slope is equal to 1
no matter what x is.

00:01:49.220 --> 00:01:50.212
It's a line.

00:01:50.212 --> 00:01:51.795
And for a line, the
slope is constant.

00:01:51.795 --> 00:01:54.840
So over here, the
slope is indeed 1.

00:01:54.840 --> 00:01:57.390
If you go to this point over
here, the slope is indeed 1.

00:01:57.390 --> 00:02:00.890
If you go over here,
the slope is indeed 1.

00:02:00.890 --> 00:02:05.260
So we've got a pretty
valid response there.

00:02:05.260 --> 00:02:08.100
Now, let's try something
where the slope might change.

00:02:08.100 --> 00:02:13.740
So let's say I have g of
x is equal to x squared.

00:02:13.740 --> 00:02:16.610
The power rule tells
us that g prime of x

00:02:16.610 --> 00:02:18.550
would be equal to what?

00:02:18.550 --> 00:02:19.700
Well, n is equal to 2.

00:02:19.700 --> 00:02:23.080
So it's going to be 2
times x to the 2 minus 1.

00:02:23.080 --> 00:02:25.660
Or it's going to be equal
to 2 x to the first power.

00:02:25.660 --> 00:02:27.860
It's going to be equal to 2x.

00:02:27.860 --> 00:02:31.067
So let's see if this
makes a reasonable sense.

00:02:31.067 --> 00:02:33.400
And I'm going to try to graph
this one a little bit more

00:02:33.400 --> 00:02:33.899
precisely.

00:02:38.470 --> 00:02:42.360
Let's see how precisely
I can graph it.

00:02:42.360 --> 00:02:44.717
So this is the x-axis, y-axis.

00:02:44.717 --> 00:02:46.050
Let me mark some stuff off here.

00:02:46.050 --> 00:02:50.990
So this is 1, 2, 3, 4, 5.

00:02:50.990 --> 00:02:55.322
This is 1, 2, 3, 4.

00:02:55.322 --> 00:02:58.550
1, 2, 3, 4.

00:02:58.550 --> 00:03:01.210
So g of x.

00:03:01.210 --> 00:03:02.930
When x is 0, it's 0.

00:03:02.930 --> 00:03:05.390
When x is 1, it is 1.

00:03:05.390 --> 00:03:06.780
When x is negative 1, it's 1.

00:03:06.780 --> 00:03:09.350
When x is 2, it is 4.

00:03:09.350 --> 00:03:11.849
So that puts us right
over there-- 1, 2, 3, 4.

00:03:11.849 --> 00:03:12.890
Puts us right over there.

00:03:12.890 --> 00:03:15.170
When x is negative
2, you get to 4.

00:03:15.170 --> 00:03:15.917
It's a parabola.

00:03:15.917 --> 00:03:17.250
You've seen this for many years.

00:03:20.575 --> 00:03:22.200
I put that point a
little bit too high.

00:03:22.200 --> 00:03:23.660
It looks something like this.

00:03:26.270 --> 00:03:29.020
Actually, the last two points I
graphed are a little bit weird.

00:03:29.020 --> 00:03:31.840
So this might be
right over here.

00:03:31.840 --> 00:03:34.690
So it looks something like this.

00:03:34.690 --> 00:03:35.980
It looks something like that.

00:03:35.980 --> 00:03:38.550
And then, when you
come over here,

00:03:38.550 --> 00:03:41.810
it looks something like that.

00:03:41.810 --> 00:03:42.490
It's symmetric.

00:03:42.490 --> 00:03:45.250
So I'm trying my best
to draw it reasonably.

00:03:45.250 --> 00:03:46.130
So there you go.

00:03:46.130 --> 00:03:51.860
That's the graph of g of x.
g of x is equal to x squared.

00:03:51.860 --> 00:03:54.360
Now, let's graph g prime of
x or what the power rule is

00:03:54.360 --> 00:03:56.950
telling us a g prime of x is.

00:03:56.950 --> 00:03:59.240
So g prime of x is equal to 2x.

00:03:59.240 --> 00:04:03.230
So that's just a line that goes
through the origin of slope 2.

00:04:03.230 --> 00:04:04.920
So it looks something like that.

00:04:04.920 --> 00:04:07.400
When x is equal to
1, y is equal to 2.

00:04:07.400 --> 00:04:11.670
When x is equal to 2, y
or g of x is equal to 4.

00:04:11.670 --> 00:04:16.540
So it looks something like this.

00:04:16.540 --> 00:04:18.720
Let me try my best to
draw a straight line.

00:04:18.720 --> 00:04:20.839
It looks something like this.

00:04:20.839 --> 00:04:25.010
Now, does this make sense?

00:04:25.010 --> 00:04:26.800
Well, if you just
eyeball it really fast,

00:04:26.800 --> 00:04:28.670
if you look at this
point right over here--

00:04:28.670 --> 00:04:33.190
and you want to think about
the slope of the tangent line.

00:04:33.190 --> 00:04:36.470
Let me do this in a color that
pops out a little bit more.

00:04:36.470 --> 00:04:39.490
So the tangent line would
look something like this.

00:04:39.490 --> 00:04:43.650
So it looks like it has a
reasonably high negative slope.

00:04:43.650 --> 00:04:44.150
Yeah.

00:04:44.150 --> 00:04:45.190
It's definitely
a negative slope,

00:04:45.190 --> 00:04:46.814
and it's a pretty
steep negative slope.

00:04:52.590 --> 00:04:58.460
For x is equal to negative
2, g prime of negative 2

00:04:58.460 --> 00:05:04.140
is equal to 2 times negative 2,
which is equal to negative 4.

00:05:04.140 --> 00:05:06.621
So this is claiming that
the slope at this point-- so

00:05:06.621 --> 00:05:08.120
this right over
here is negative 4--

00:05:08.120 --> 00:05:09.703
is saying that the
slope of this point

00:05:09.703 --> 00:05:12.220
is negative 4. m is
equal to negative 4.

00:05:12.220 --> 00:05:13.740
That looks about right.

00:05:13.740 --> 00:05:16.830
It's a fairly steep
negative slope.

00:05:16.830 --> 00:05:19.370
Now, what happens if
you go right over here

00:05:19.370 --> 00:05:21.410
when x is equal to 0?

00:05:21.410 --> 00:05:24.090
Well, our derivative--
if you say g prime of 0--

00:05:24.090 --> 00:05:27.730
is telling us that the slope of
our original function, g, at x

00:05:27.730 --> 00:05:31.710
is equal to 0 is 2 times 0 is 0.

00:05:31.710 --> 00:05:32.874
Well, does that make sense?

00:05:32.874 --> 00:05:34.540
Well, if we go to our
original parabola,

00:05:34.540 --> 00:05:36.289
it does indeed make sense.

00:05:36.289 --> 00:05:37.830
That's the slope of
the tangent line.

00:05:37.830 --> 00:05:39.720
The tangent line looks
something like this.

00:05:39.720 --> 00:05:40.520
We're at a minimum point.

00:05:40.520 --> 00:05:41.353
We're at the vertex.

00:05:41.353 --> 00:05:44.430
The slope really
does look to be 0.

00:05:44.430 --> 00:05:47.340
And what if you
go right over here

00:05:47.340 --> 00:05:51.120
to x equals 2, the slope
of the tangent line?

00:05:51.120 --> 00:05:54.780
Well, over here, the tangent
line looks something like this.

00:05:54.780 --> 00:05:59.040
It looks like a fairly
steep positive slope.

00:05:59.040 --> 00:06:02.672
What is our derivative telling
us based on the power rule?

00:06:02.672 --> 00:06:04.130
So this is essentially
saying, hey,

00:06:04.130 --> 00:06:06.160
tell me what the slope
of the tangent line for g

00:06:06.160 --> 00:06:09.630
is when x is equal to 2.

00:06:09.630 --> 00:06:10.630
Well, we figured it out.

00:06:10.630 --> 00:06:11.870
It's going to be 2 times x.

00:06:11.870 --> 00:06:14.170
It's going to be 2 times
2, which is equal to 4.

00:06:14.170 --> 00:06:17.580
It's telling us that the
slope over here is 4.

00:06:17.580 --> 00:06:19.800
And I'm just using m.
m is often the letter

00:06:19.800 --> 00:06:21.089
used to denote slope.

00:06:21.089 --> 00:06:23.130
They're saying that the
slope of the tangent line

00:06:23.130 --> 00:06:26.430
there is 4, which seems
completely, completely

00:06:26.430 --> 00:06:27.310
reasonable.

00:06:27.310 --> 00:06:29.260
So I encourage you to
try this out yourself.

00:06:29.260 --> 00:06:32.700
I encourage you to try
to estimate the slopes

00:06:32.700 --> 00:06:34.729
by calculating, by
taking points closer

00:06:34.729 --> 00:06:36.020
and closer around those points.

00:06:36.020 --> 00:06:37.980
And you'll see that
the power rule really

00:06:37.980 --> 00:06:41.860
does give you results
that actually make sense.


Power rule

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.650
In this video, we will
cover the power rule,

00:00:03.650 --> 00:00:05.550
which really simplifies
our life when

00:00:05.550 --> 00:00:07.540
it comes to taking
derivatives, especially

00:00:07.540 --> 00:00:09.630
derivatives of polynomials.

00:00:09.630 --> 00:00:11.030
You are probably
already familiar

00:00:11.030 --> 00:00:14.090
with the definition
of a derivative,

00:00:14.090 --> 00:00:16.560
limit is delta x
approaches 0 of f

00:00:16.560 --> 00:00:21.820
of x plus delta x minus f of
x, all of that over delta x.

00:00:21.820 --> 00:00:23.460
And it really just
comes out of trying

00:00:23.460 --> 00:00:26.674
to find the slope of a tangent
line at any given point.

00:00:26.674 --> 00:00:28.590
But we're going to see
what the power rule is.

00:00:28.590 --> 00:00:29.710
It simplifies our life.

00:00:29.710 --> 00:00:32.890
We won't have to take these
sometimes complicated limits.

00:00:32.890 --> 00:00:35.060
And we're not going to
prove it in this video,

00:00:35.060 --> 00:00:37.499
but we'll hopefully get
a sense of how to use it.

00:00:37.499 --> 00:00:40.040
And in future videos, we'll get
a sense of why it makes sense

00:00:40.040 --> 00:00:41.530
and even prove it.

00:00:41.530 --> 00:00:44.910
So the power rule just tells us
that if I have some function,

00:00:44.910 --> 00:00:49.670
f of x, and it's equal
to some power of x, so x

00:00:49.670 --> 00:00:55.710
to the n power, where
n does not equal 0.

00:00:55.710 --> 00:00:57.210
So n can be anything.

00:00:57.210 --> 00:01:00.290
It can be positive, a
negative, it could be-- it

00:01:00.290 --> 00:01:01.960
does not have to be an integer.

00:01:01.960 --> 00:01:07.150
The power rule tells us that
the derivative of this, f prime

00:01:07.150 --> 00:01:10.460
of x, is just going
to be equal to n,

00:01:10.460 --> 00:01:17.490
so you're literally bringing
this out front, n times x,

00:01:17.490 --> 00:01:20.040
and then you just decrement
the power, times x

00:01:20.040 --> 00:01:23.889
to the n minus 1 power.

00:01:23.889 --> 00:01:25.430
So let's do a couple
of examples just

00:01:25.430 --> 00:01:28.640
to make sure that that
actually makes sense.

00:01:28.640 --> 00:01:32.360
So let's ask ourselves,
well let's say that f of x

00:01:32.360 --> 00:01:34.420
was equal to x squared.

00:01:34.420 --> 00:01:36.490
Based on the power
rule, what is f

00:01:36.490 --> 00:01:40.820
prime of x going to be equal to?

00:01:40.820 --> 00:01:43.550
Well, in this
situation, our n is 2.

00:01:43.550 --> 00:01:45.440
So we bring the 2 out front.

00:01:45.440 --> 00:01:49.910
2 times x to the
2 minus 1 power.

00:01:49.910 --> 00:01:52.800
So that's going to be 2 times
x to the first power, which

00:01:52.800 --> 00:01:54.600
is just equal to 2x.

00:01:54.600 --> 00:01:56.480
That was pretty straightforward.

00:01:56.480 --> 00:01:58.210
Let's think about
the situation where,

00:01:58.210 --> 00:02:03.150
let's say we have g of x is
equal to x to the third power.

00:02:03.150 --> 00:02:08.280
What is g prime of x going
to be in this scenario?

00:02:08.280 --> 00:02:12.236
Well, n is 3, so we just
literally pattern match here.

00:02:12.236 --> 00:02:13.610
This is-- you're
probably finding

00:02:13.610 --> 00:02:15.260
this shockingly straightforward.

00:02:15.260 --> 00:02:20.740
So this is going to be 3 times
x to the 3 minus 1 power,

00:02:20.740 --> 00:02:24.995
or this is going to be
equal to 3x squared.

00:02:24.995 --> 00:02:25.620
And we're done.

00:02:25.620 --> 00:02:26.590
In the next video
we'll think about

00:02:26.590 --> 00:02:28.015
whether this
actually makes sense.

00:02:28.015 --> 00:02:29.640
Let's do one more
example, just to show

00:02:29.640 --> 00:02:31.620
it doesn't have to
necessarily apply

00:02:31.620 --> 00:02:34.660
to only these kind
of positive integers.

00:02:34.660 --> 00:02:36.390
We could have a
scenario where maybe we

00:02:36.390 --> 00:02:44.470
have h of x. h of x is equal
to x to the negative 100 power.

00:02:44.470 --> 00:02:46.450
The power rule tells
us that h prime of x

00:02:46.450 --> 00:02:48.770
would be equal to what?

00:02:48.770 --> 00:02:52.030
Well n is negative 100,
so it's negative 100x

00:02:52.030 --> 00:02:54.820
to the negative
100 minus 1, which

00:02:54.820 --> 00:02:59.450
is equal to negative
100x to the negative 101.

00:02:59.450 --> 00:03:00.890
Let's do one more.

00:03:00.890 --> 00:03:03.260
Let's say we had z of x.

00:03:03.260 --> 00:03:11.013
z of x is equal to x
to the 2.571 power.

00:03:11.013 --> 00:03:16.180
And we are concerned with
what is z prime of x?

00:03:16.180 --> 00:03:18.890
Well once again, power
rule simplifies our life,

00:03:18.890 --> 00:03:21.000
n it's 2.571, so
it's going to be

00:03:21.000 --> 00:03:27.760
2.571 times x to the
2.571 minus 1 power.

00:03:27.760 --> 00:03:29.632
So it's going to
be equal to-- let

00:03:29.632 --> 00:03:32.480
me make sure I'm not falling
off the bottom of the page--

00:03:32.480 --> 00:03:40.734
2.571 times x to
the 1.571 power.

00:03:40.734 --> 00:03:41.900
Hopefully, you enjoyed that.

00:03:41.900 --> 00:03:44.175
And in the next few
videos, we will not only

00:03:44.175 --> 00:03:46.050
expose you to more
properties of derivatives,

00:03:46.050 --> 00:03:48.341
we'll get a sense for why
the power rule at least makes

00:03:48.341 --> 00:03:49.080
intuitive sense.

00:03:49.080 --> 00:03:53.660
And then also prove the
power rule for a few cases.


Part 2 of shell method with 2 functions of y

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.710
In the last video, we set
up this definite integral

00:00:02.710 --> 00:00:05.290
to evaluate the volume of
the solid of revolution

00:00:05.290 --> 00:00:07.350
that we set up using
the shell method.

00:00:07.350 --> 00:00:09.770
So now, let's just
evaluate this thing.

00:00:09.770 --> 00:00:12.760
So the main thing is just
simplifying this expression.

00:00:12.760 --> 00:00:17.150
I'll start off by trying to
simplify this part of it.

00:00:17.150 --> 00:00:19.980
So that's going to be y plus 1.

00:00:19.980 --> 00:00:22.604
I just ate an apple,
so something weird just

00:00:22.604 --> 00:00:23.520
happened in my throat.

00:00:23.520 --> 00:00:24.960
But anyway, that's done with.

00:00:24.960 --> 00:00:29.930
y plus 1 minus y
squared minus 2y plus 1.

00:00:29.930 --> 00:00:33.250
I just expanded
out this binomial.

00:00:33.250 --> 00:00:37.470
And then that would simplify
to-- another apple in my throat

00:00:37.470 --> 00:00:41.880
moment-- so that's going to
be y plus 1 minus y squared

00:00:41.880 --> 00:00:45.245
plus 2y minus 1.

00:00:45.245 --> 00:00:48.877
So this 1 and this
negative 1 cancel out.

00:00:48.877 --> 00:00:49.460
And let's see.

00:00:49.460 --> 00:00:54.850
You get negative
y squared plus 3y.

00:00:54.850 --> 00:01:01.342
And then we're going to
multiply that times y plus 2.

00:01:01.342 --> 00:01:03.050
So when you multiply
y plus 2 times this,

00:01:03.050 --> 00:01:06.120
so you have y times
negative y squared,

00:01:06.120 --> 00:01:10.580
it gets us negative
y to the third power.

00:01:10.580 --> 00:01:16.560
y times 3y is going
to be plus 3y squared.

00:01:16.560 --> 00:01:22.880
2 times negative y squared
is negative 2y squared.

00:01:22.880 --> 00:01:26.790
And then 2 times 3y is plus 6y.

00:01:26.790 --> 00:01:28.440
So then you go all
the way down here.

00:01:28.440 --> 00:01:30.800
This thing can simplify,
too, because you have 3y

00:01:30.800 --> 00:01:32.070
squared minus 2y squared.

00:01:32.070 --> 00:01:35.130
So this going to be
negative y to the third

00:01:35.130 --> 00:01:37.310
plus-- this part right
over here simplifies

00:01:37.310 --> 00:01:42.160
to just y squared--
y squared plus 6y.

00:01:42.160 --> 00:01:47.090
So that's this entire part
simplified to this down here.

00:01:47.090 --> 00:01:50.420
We can take the 2 pi out
of the integral sign.

00:01:50.420 --> 00:01:52.000
So let's do that.

00:01:52.000 --> 00:01:53.715
We're integrating
from y is equal to 0

00:01:53.715 --> 00:01:56.240
to y is equal to 3 dy.

00:01:56.240 --> 00:01:58.790
And I took the 2 pi
out here, and that

00:01:58.790 --> 00:02:00.129
is equal to our volume.

00:02:00.129 --> 00:02:01.920
And so now, we're
essentially ready to take

00:02:01.920 --> 00:02:02.880
the antiderivative.

00:02:02.880 --> 00:02:07.940
This is going to be equal to
2 pi times the antiderivative

00:02:07.940 --> 00:02:14.090
of this business evaluated
at 3 minus evaluated at 0.

00:02:14.090 --> 00:02:15.620
And I'll color code it.

00:02:15.620 --> 00:02:17.140
I found this useful.

00:02:17.140 --> 00:02:18.640
The antiderivative
of y to the third

00:02:18.640 --> 00:02:20.135
is y to the fourth
over 4, so this

00:02:20.135 --> 00:02:24.040
is negative y to
the fourth over 4.

00:02:24.040 --> 00:02:28.780
Antiderivative of y squared is
y to the third over 2-- or y

00:02:28.780 --> 00:02:30.240
to the third over
3, I should say.

00:02:32.980 --> 00:02:37.070
And then finally,
I'll do it in yellow.

00:02:37.070 --> 00:02:44.340
Antiderivative of 6y is 3y
squared, so plus 3y squared.

00:02:44.340 --> 00:02:49.260
And we are going to evaluate
all of this business at 0 and 3.

00:02:49.260 --> 00:02:50.240
So this simplifies.

00:02:50.240 --> 00:02:54.380
This going to be equal to 2
pi times-- well, let's see.

00:02:54.380 --> 00:02:56.110
Let me do it in the same colors.

00:02:56.110 --> 00:02:58.150
3 to the fourth power is 81.

00:02:58.150 --> 00:03:03.835
So it's negative
81 over 4, plus-- 3

00:03:03.835 --> 00:03:08.330
to the third is 27
divided by 3 is 9, plus 9.

00:03:08.330 --> 00:03:17.874
And then 3 squared is 9
times 3 is 27, plus 27.

00:03:17.874 --> 00:03:20.040
And then when you evaluate
all of these things at 0,

00:03:20.040 --> 00:03:20.810
you just get 0.

00:03:20.810 --> 00:03:22.185
So you're just
subtracting out 0,

00:03:22.185 --> 00:03:25.560
so we really don't have to
do anything else with the 0.

00:03:25.560 --> 00:03:27.465
And now we are
ready to simplify.

00:03:30.570 --> 00:03:31.170
Let me see.

00:03:33.960 --> 00:03:36.050
Actually, let's just
add them all up.

00:03:36.050 --> 00:03:42.060
So this is going to
be 9 plus 27 is 36.

00:03:42.060 --> 00:03:44.960
So that is 36.

00:03:44.960 --> 00:03:47.160
And if we want to add it
to negative 81 over 4,

00:03:47.160 --> 00:03:49.150
we just have to find
a common denominator.

00:03:49.150 --> 00:03:53.900
So all of this business is going
to be equal to 2 pi times--

00:03:53.900 --> 00:03:59.130
and so our common denominator
can be 4, times something

00:03:59.130 --> 00:04:00.610
over 4.

00:04:00.610 --> 00:04:08.590
We have negative 81 over 4,
and then 36 times 4 is 144.

00:04:08.590 --> 00:04:09.420
Is that right?

00:04:09.420 --> 00:04:10.640
Yeah, that's 144.

00:04:10.640 --> 00:04:15.780
So 36 times 4, so it's plus 144.

00:04:15.780 --> 00:04:19.740
30 times 4 is 120 plus
another 24 is 144.

00:04:19.740 --> 00:04:22.780
So you have 144,
essentially, minus 81.

00:04:22.780 --> 00:04:26.871
So this is going to be
equal to 2 pi times--

00:04:26.871 --> 00:04:28.370
and actually, I can
even simplify it

00:04:28.370 --> 00:04:31.990
little bit more, because we
have a 2 here and a 4 there.

00:04:31.990 --> 00:04:34.210
So divide the numerator
and denominator by 2

00:04:34.210 --> 00:04:35.670
so you get it over 2.

00:04:35.670 --> 00:04:41.120
So you're going to have pi
times-- this is going to be 44.

00:04:41.120 --> 00:04:41.620
Let's see.

00:04:41.620 --> 00:04:43.456
If this was an 80,
this would be 64.

00:04:46.635 --> 00:04:47.593
So it's going to be 63.

00:04:49.659 --> 00:04:50.700
Let me write it this way.

00:04:50.700 --> 00:04:57.460
It's going to be 63 pi over 2.

00:04:57.460 --> 00:04:58.630
Did I do that right?

00:04:58.630 --> 00:05:01.780
60 plus 81 is 141.

00:05:01.780 --> 00:05:03.610
Add another 3, you get a 144.

00:05:03.610 --> 00:05:04.110
Yep.

00:05:04.110 --> 00:05:04.735
And we're done.

00:05:04.735 --> 00:05:07.740
We figured out the
volume of our front

00:05:07.740 --> 00:05:11.060
of jet engine-looking shape.


Shell method with two functions of y

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:05.500
I'm going to take the region
in between the two curves

00:00:05.500 --> 00:00:08.570
here, between the yellow curve--
defined as a function of y

00:00:08.570 --> 00:00:12.560
as x is equal to y minus 1
squared-- and this bluish-green

00:00:12.560 --> 00:00:15.420
looking line-- where y
is equal to x minus 1.

00:00:15.420 --> 00:00:18.350
So I'm going to take this
region right over here,

00:00:18.350 --> 00:00:23.510
and I'm going to rotate
it around the line,

00:00:23.510 --> 00:00:26.890
y equals negative 2,
to get this shape that

00:00:26.890 --> 00:00:30.520
looks like the front of a jet
engine or something like that.

00:00:30.520 --> 00:00:32.900
And we're going to want to
figure out what its volume is.

00:00:32.900 --> 00:00:34.316
And you can actually
approach this

00:00:34.316 --> 00:00:36.741
with either the disk
method or the shell method.

00:00:36.741 --> 00:00:38.240
In the disk method,
you would create

00:00:38.240 --> 00:00:42.770
disks that look like this.

00:00:42.770 --> 00:00:46.294
And you would be doing
integrating with respect to x.

00:00:46.294 --> 00:00:48.460
You will have to break up
the problem appropriately,

00:00:48.460 --> 00:00:50.815
because you have a
different lower boundary.

00:00:50.815 --> 00:00:52.940
You would have to break
this up into two functions,

00:00:52.940 --> 00:00:55.010
an upper function
and a lower boundary

00:00:55.010 --> 00:00:55.980
for this interval in x.

00:00:55.980 --> 00:00:57.980
And then a different one
for that interval in x.

00:00:57.980 --> 00:00:59.524
But you could use
the disk method.

00:00:59.524 --> 00:01:00.940
But instead of
that, we don't feel

00:01:00.940 --> 00:01:02.780
like breaking up the functions
and doing all of that.

00:01:02.780 --> 00:01:04.614
So we're going to do
it in the shell method,

00:01:04.614 --> 00:01:07.238
especially because we've already
expressed one of our functions

00:01:07.238 --> 00:01:08.170
as a function of y.

00:01:08.170 --> 00:01:09.967
And this one won't
be too hard to do.

00:01:09.967 --> 00:01:11.550
So what we're going
to do, once again,

00:01:11.550 --> 00:01:18.215
is imagine constructing
these little rectangles that

00:01:18.215 --> 00:01:20.780
have height dy.

00:01:20.780 --> 00:01:23.300
And what we're going to do
is rotate those rectangles

00:01:23.300 --> 00:01:25.140
around the line y
equals negative 2.

00:01:25.140 --> 00:01:27.910
So let me draw that same
rectangle over here.

00:01:27.910 --> 00:01:30.860
And when you do that,
you construct a shell.

00:01:30.860 --> 00:01:32.020
So let me do that.

00:01:32.020 --> 00:01:33.870
So, go between these two points.

00:01:33.870 --> 00:01:39.090
And then this is what it would
look like when it's down here.

00:01:39.090 --> 00:01:41.700
And then, let me make it
clear that this constructs

00:01:41.700 --> 00:01:50.720
a shell of thickness
dy or of depth dy.

00:01:50.720 --> 00:01:51.900
So let me do it like that.

00:01:51.900 --> 00:01:54.930
So that's my shell, and it
has some thickness to it.

00:01:54.930 --> 00:01:59.250
And that thickness is dy.

00:01:59.250 --> 00:02:01.910
So that's the
thickness of my shell.

00:02:01.910 --> 00:02:04.320
And let me shade
it in a little bit.

00:02:04.320 --> 00:02:08.214
Make sure you can see the
3-dimensionality of my shell.

00:02:08.214 --> 00:02:10.130
So like we've done with
all of these problems,

00:02:10.130 --> 00:02:13.650
our goal is to really
just figure out

00:02:13.650 --> 00:02:16.220
what the volume
of each shell is.

00:02:16.220 --> 00:02:19.520
And then we can enter it for
a specific y in our interval.

00:02:19.520 --> 00:02:23.110
And then we integrate along
all the y's of our interval.

00:02:23.110 --> 00:02:24.885
And we've done this
many times before.

00:02:24.885 --> 00:02:26.260
The first thing
we think about is

00:02:26.260 --> 00:02:30.744
what's the radius
one of these shells?

00:02:30.744 --> 00:02:32.410
So what I'm doing
right here in magenta,

00:02:32.410 --> 00:02:35.120
what is the radius of
something like that?

00:02:35.120 --> 00:02:36.850
Well, it's essentially
going to be

00:02:36.850 --> 00:02:42.370
the distance between y
is equal to negative 2

00:02:42.370 --> 00:02:45.270
and our y value for
that specific y.

00:02:49.540 --> 00:02:54.720
So this distance
right over here is y.

00:02:54.720 --> 00:02:58.340
And then we're going
to have another 2.

00:02:58.340 --> 00:03:01.604
So the whole distance
is going to be y plus 2.

00:03:01.604 --> 00:03:03.270
Another way of thinking
about it is this

00:03:03.270 --> 00:03:05.460
is essentially y
minus negative 2

00:03:05.460 --> 00:03:08.890
to get the distance, which
is going to be y plus 2.

00:03:08.890 --> 00:03:11.660
So the radius of one
of our little shells

00:03:11.660 --> 00:03:14.900
is going to be y plus 2.

00:03:14.900 --> 00:03:16.960
If the radius is
y plus 2, then we

00:03:16.960 --> 00:03:20.420
know that the circumference
of this circle right over here

00:03:20.420 --> 00:03:25.790
is going to be 2
pi times y plus 2.

00:03:25.790 --> 00:03:28.550
And then the surface area,
the outside surface area,

00:03:28.550 --> 00:03:33.810
of the shell, the
stuff out here,

00:03:33.810 --> 00:03:38.290
is just going to be that
circumference times I guess

00:03:38.290 --> 00:03:41.600
you could say the width
of this shell times

00:03:41.600 --> 00:03:44.350
this distance right over here.

00:03:44.350 --> 00:03:47.830
Or we could say times this
distance right over here.

00:03:47.830 --> 00:03:49.070
And what is that distance?

00:03:49.070 --> 00:03:51.770
Remember, we want everything
expressed as a function of y.

00:03:51.770 --> 00:03:55.520
Well, it's going to be the upper
function as a function of y

00:03:55.520 --> 00:03:57.447
minus the lower function.

00:03:57.447 --> 00:03:59.030
And we think about
the upper function,

00:03:59.030 --> 00:04:00.446
it's the function
that's giving us

00:04:00.446 --> 00:04:02.120
higher x values
in that interval.

00:04:02.120 --> 00:04:05.150
And so this blue function
is the upper function

00:04:05.150 --> 00:04:06.980
when we think in terms of y.

00:04:06.980 --> 00:04:08.900
But we have to express
it as a function of y.

00:04:08.900 --> 00:04:10.560
So let me do that.

00:04:10.560 --> 00:04:13.100
So we can rewrite this
as add 1 to both sides,

00:04:13.100 --> 00:04:16.100
you get x is equal to y plus 1.

00:04:16.100 --> 00:04:18.370
So that is our upper function.

00:04:18.370 --> 00:04:19.870
And then this is
our lower function.

00:04:19.870 --> 00:04:21.622
If you were to tilt
your head to the right

00:04:21.622 --> 00:04:23.080
and look at it that
way, you'll see

00:04:23.080 --> 00:04:24.880
that this will be
the upper function,

00:04:24.880 --> 00:04:26.700
and this will be
the lower function

00:04:26.700 --> 00:04:28.750
for the same value of y.

00:04:28.750 --> 00:04:31.750
This gives a higher value
of x than this one does

00:04:31.750 --> 00:04:33.380
for the same value of y.

00:04:33.380 --> 00:04:36.630
It gives the upper x values.

00:04:36.630 --> 00:04:39.890
So, the area is going
to be the circumference

00:04:39.890 --> 00:04:41.550
times this dimension.

00:04:44.061 --> 00:04:44.810
Let me write this.

00:04:44.810 --> 00:04:51.760
The area of one of those shells
is going to be 2 pi times y

00:04:51.760 --> 00:04:57.430
plus 2 times the distance
between the upper function.

00:04:57.430 --> 00:05:02.660
So the distance between the
upper function y plus 1,

00:05:02.660 --> 00:05:05.930
x is equal to y plus 1,
and the lower function, x

00:05:05.930 --> 00:05:09.885
is equal to y minus 1 squared.

00:05:13.530 --> 00:05:16.280
I'll put the parentheses
in that same color.

00:05:16.280 --> 00:05:18.799
And then if we want the
volume of that shell,

00:05:18.799 --> 00:05:21.340
so we've got the outside surface
area of the shell right now.

00:05:21.340 --> 00:05:24.480
We just multiply it by its
depth, which is just dy.

00:05:24.480 --> 00:05:27.670
So that sets up our integral.

00:05:27.670 --> 00:05:29.960
The volume of one shell--
I'll do it all in one color

00:05:29.960 --> 00:05:40.620
now-- is 2 pi times y plus 2
times y plus 1 minus y minus 1

00:05:40.620 --> 00:05:41.326
squared.

00:05:41.326 --> 00:05:44.960
Then we multiply that times
the depth of each shell, dy.

00:05:44.960 --> 00:05:47.700
And then we integrate
over the interval.

00:05:47.700 --> 00:05:49.310
So the volume is
going to be this.

00:05:49.310 --> 00:05:50.317
And what's the interval?

00:05:50.317 --> 00:05:51.650
You might be able to eyeball it.

00:05:51.650 --> 00:05:53.470
But we can actually
solve that explicitly.

00:05:53.470 --> 00:05:55.980
When do these two
things equal each other?

00:05:55.980 --> 00:05:57.560
Well, you could
just set y plus 1

00:05:57.560 --> 00:06:00.910
to be equal to y minus 1
squared, so let's do that.

00:06:00.910 --> 00:06:07.150
So if we set y plus 1,
this, to be equal to that,

00:06:07.150 --> 00:06:10.347
it's going to be equal
to y minus 1 squared.

00:06:10.347 --> 00:06:11.430
So let me expand that out.

00:06:11.430 --> 00:06:15.762
That's going to be y
squared minus 2y plus 1.

00:06:15.762 --> 00:06:16.540
And let's see.

00:06:16.540 --> 00:06:18.240
I could subtract
y from both sides

00:06:18.240 --> 00:06:20.550
and I could subtract
1 from both sides.

00:06:20.550 --> 00:06:22.950
Minus y, minus 1.

00:06:22.950 --> 00:06:26.790
And then I am left with,
on the left hand side, 0.

00:06:26.790 --> 00:06:30.832
And on the right hand side,
I have y squared minus 3 y.

00:06:34.010 --> 00:06:38.190
And that's it, plus 0.

00:06:38.190 --> 00:06:42.110
So 0 is equal to
y times y minus 3.

00:06:42.110 --> 00:06:43.790
So the zeros of
this, when these are

00:06:43.790 --> 00:06:47.967
equal are when y is equal
to 0 or y is equal to 3.

00:06:47.967 --> 00:06:49.300
And we see that right over here.

00:06:49.300 --> 00:06:53.480
When y is equal to 0, these
two functions intersect.

00:06:53.480 --> 00:06:57.520
And when y is equal to 3,
these two functions intersect.

00:06:57.520 --> 00:07:00.430
So our interval is going
to be from y is equal to 0

00:07:00.430 --> 00:07:02.290
to y is equal to 3.

00:07:02.290 --> 00:07:04.080
So using the shell
method, we have

00:07:04.080 --> 00:07:06.550
been able to set up
our definite integral.

00:07:06.550 --> 00:07:11.030
And now we can think about how
we can evaluate this thing.


Calculating integral with shell method

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.600
In the last video we
were able to set up

00:00:02.600 --> 00:00:07.080
this definite integral using
the shell or the hollow cylinder

00:00:07.080 --> 00:00:09.280
method in order to
figure out the volume

00:00:09.280 --> 00:00:12.100
of this solid of revolution.

00:00:12.100 --> 00:00:14.260
And so now let's just
evaluate this thing.

00:00:14.260 --> 00:00:16.250
And really the main
thing we have to do here

00:00:16.250 --> 00:00:19.130
is just to multiply
what we have here out.

00:00:19.130 --> 00:00:20.860
So multiply this expression out.

00:00:20.860 --> 00:00:23.210
So this is going to be equal
to-- I'll take the 2 pi out

00:00:23.210 --> 00:00:24.260
of the integral.

00:00:24.260 --> 00:00:26.614
2 pi times the
integral from 0 to 1.

00:00:26.614 --> 00:00:28.030
Let's see, 2 times
the square root

00:00:28.030 --> 00:00:31.210
of x is 2-- I'll write it
as 2 square roots of x.

00:00:31.210 --> 00:00:32.920
But I'll write it
as 2x to the 1/2.

00:00:32.920 --> 00:00:33.830
It'll make it a
little bit easier

00:00:33.830 --> 00:00:35.996
to take the antiderivative
conceptually, or at least

00:00:35.996 --> 00:00:36.810
in our brain.

00:00:36.810 --> 00:00:39.730
So two times the square
root of x is 2x to the 1/2.

00:00:39.730 --> 00:00:45.000
2 times negative x squared
is negative 2 x squared.

00:00:45.000 --> 00:00:49.090
And then we have negative x
times the square root of x.

00:00:49.090 --> 00:00:51.200
Well, that's x to the
first times x to the 1/2.

00:00:51.200 --> 00:00:55.830
That's going to be negative
x to the 3/2 power.

00:00:55.830 --> 00:00:58.260
And then we have negative
x times negative x

00:00:58.260 --> 00:01:01.960
squared that's going to be
positive x to the third power.

00:01:01.960 --> 00:01:04.420
And all of that dx.

00:01:04.420 --> 00:01:06.900
And so now we're ready to
take the antiderivative.

00:01:06.900 --> 00:01:10.080
So this is going to be equal to
2 pi times the antiderivative

00:01:10.080 --> 00:01:15.410
of all of this business
evaluated at 1 and at 0.

00:01:15.410 --> 00:01:21.260
So the antiderivative
of 2 times x to the 1/2

00:01:21.260 --> 00:01:23.600
is going to be 2-- it's
going to be-- let's see.

00:01:23.600 --> 00:01:25.950
We're going to take x
to the 3/2 times 2/3.

00:01:25.950 --> 00:01:31.120
So it's going to be
4/3 x to the 3/2.

00:01:31.120 --> 00:01:32.950
And then for this
term right over here

00:01:32.950 --> 00:01:36.280
it's going to be negative
2/3 x to the third.

00:01:39.357 --> 00:01:40.940
And you could take
the derivative here

00:01:40.940 --> 00:01:43.271
to verify that you
actually do get this.

00:01:43.271 --> 00:01:45.770
And then right over here, let's
see, if we incremented this,

00:01:45.770 --> 00:01:48.680
you get x to the 5/2.

00:01:48.680 --> 00:01:51.840
And so we're going to
want to multiply by 2/5.

00:01:51.840 --> 00:01:55.726
So minus-- let me do
this in another color.

00:01:55.726 --> 00:01:58.520
Let's see, so this one
right over here, it's

00:01:58.520 --> 00:02:05.380
going to be minus 2/5
x to the 5/2 power.

00:02:05.380 --> 00:02:06.770
Yep, that works out.

00:02:06.770 --> 00:02:11.270
And then finally you're going
to have x to the fourth over 4

00:02:11.270 --> 00:02:13.890
plus-- let me do that
in a different color--

00:02:13.890 --> 00:02:21.390
plus x to the fourth over 4.

00:02:21.390 --> 00:02:24.127
That's this term
right over here.

00:02:24.127 --> 00:02:25.960
And now we just have
to evaluate at 1 and 0.

00:02:25.960 --> 00:02:29.230
And 0, luckily, all of these
terms end up being a 0.

00:02:29.230 --> 00:02:31.450
So that's nice and cancels out.

00:02:31.450 --> 00:02:33.661
And so we are just
left with-- we're

00:02:33.661 --> 00:02:34.910
just-- [INAUDIBLE] cancel out.

00:02:34.910 --> 00:02:36.030
It just evaluates to 0.

00:02:36.030 --> 00:02:39.110
So this is just 2 pi times when
you evaluate all this business

00:02:39.110 --> 00:02:40.100
at 1.

00:02:40.100 --> 00:02:54.010
So that's going to be 4/3
minus 2/3 minus 2/5 plus 1/4.

00:02:58.480 --> 00:03:00.630
And the least common
multiple right over here

00:03:00.630 --> 00:03:03.790
looks like 60, so
we're going to want

00:03:03.790 --> 00:03:05.730
to put all this over
a denominator of 60.

00:03:05.730 --> 00:03:09.110
So it's going to be 2 pi
times all of this business

00:03:09.110 --> 00:03:11.570
over a denominator of 60.

00:03:11.570 --> 00:03:18.060
And 4/3 is same thing as 80/60.

00:03:18.060 --> 00:03:22.320
Negative 2/3 is the same
thing as negative 40/60.

00:03:22.320 --> 00:03:28.180
Negative 2/5 is the same
thing as negative 24/60.

00:03:28.180 --> 00:03:31.700
And then 1/4 is the
same thing as 15/60.

00:03:31.700 --> 00:03:33.917
So this is equal to--
and actually this

00:03:33.917 --> 00:03:35.500
will cancel over
here, and you'll just

00:03:35.500 --> 00:03:37.139
get a 30 in your denominator.

00:03:37.139 --> 00:03:38.680
So in your denominator,
you get a 30.

00:03:38.680 --> 00:03:42.070
And up here 80 minus 40 is 40.

00:03:42.070 --> 00:03:46.275
40 minus 24 gets us to 16.

00:03:46.275 --> 00:03:49.510
16 plus 15 is 31.

00:03:49.510 --> 00:03:55.970
So we get 31 times pi
over 30 for the volume

00:03:55.970 --> 00:03:59.170
of the figure right over there.