Power rule introduction (old)

WEBVTT
Kind: captions
Language: en

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Welcome back.

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In the last presentation I
showed you that if I had the

00:00:04.820 --> 00:00:13.530
function f of x is equal to x
squared, that the derivative of

00:00:13.530 --> 00:00:17.980
this function, which is denoted
by f-- look at that, my pen

00:00:17.980 --> 00:00:19.835
is already malfunctioning.

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The derivative of that
function, f prime of

00:00:26.190 --> 00:00:28.770
x, is equal to 2x.

00:00:28.770 --> 00:00:32.050
And I used the limit
definition of a derivative.

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I used, let me write
it down here.

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This pen is horrible.

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I need to really figure out
some other tool to use.

00:00:42.520 --> 00:00:47.160
The limit as h approaches 0 --
sometimes you'll see delta x

00:00:47.160 --> 00:00:51.440
instead of h, but it's the same
thing-- of f of x plus

00:00:51.440 --> 00:00:57.900
h minus f of x over h.

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And I used this definition of a
derivative, which is really

00:01:00.890 --> 00:01:03.860
just the slope at any given
point along the curve,

00:01:03.860 --> 00:01:04.670
to figure this out.

00:01:04.670 --> 00:01:06.940
That if f of x is equal
to x squared, that

00:01:06.940 --> 00:01:08.970
the derivative is 2x.

00:01:08.970 --> 00:01:11.360
And you could actually
use this to do others.

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And I won't do it now,
maybe I'll do it in a

00:01:12.940 --> 00:01:13.710
future presentation.

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But it turns out that if you
have f of x is equal to x to

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the third, that the derivative
is f prime of x is

00:01:26.630 --> 00:01:32.700
equal to 3x squared.

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If f of x is equal to x to
the fourth, well then the

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derivative is equal
to 4x to the third.

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I think you're starting
to see a pattern here.

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If I actually wrote up here
that if f of x -- let me see

00:01:51.812 --> 00:01:55.080
if I have space to
write it neatly.

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If I wrote f of x -- I hope
you can see this -- f

00:01:57.510 --> 00:01:59.340
of x is equal to x.

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Well you know this.

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I mean, y equals x, what's
the slope of y equals x?

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That's just 1, right?

00:02:05.600 --> 00:02:07.330
y equals x, that's
a slope of 1.

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You didn't need to know
calculus to know that.

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f prime of x is
just equal to 1.

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And then you can probably
guess what the next one is.

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If f of x is equal to x to the
fifth, then the derivative is--

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I think you could guess--
5 x to the fourth.

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So in general, for any
expression within a polynomial,

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or any degree x to whatever
power-- let's say f of x is

00:02:42.090 --> 00:02:47.000
equal to-- this pen
drives me nuts.

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f of x is equal to
x to the n, right?

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Where n could be any exponent.

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Then f prime of x is equal
to nx to the n minus 1.

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And you see this is what
the case was in all

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these situations.

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That 1 didn't show up.

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n minus 1.

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So if n was 25, x to the
25th power, the derivative

00:03:13.110 --> 00:03:16.110
would be 25 x to the 24th.

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So I'm going to use this rule
and then I'm going to show

00:03:18.826 --> 00:03:19.690
you a couple of other ones.

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And then now we can figure out
the derivative of pretty much

00:03:21.920 --> 00:03:23.270
any polynomial function.

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So just another
couple of rules.

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This might be a little
intuitive for you, and if you

00:03:32.180 --> 00:03:33.830
use that limit definition of a
derivative, you could

00:03:33.830 --> 00:03:36.630
actually prove it.

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But if I want to figure out the
derivative of, let's say, the

00:03:42.170 --> 00:03:51.200
derivative of-- So another way
of-- this is kind of, what is

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the change with respect to x?

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This is another notation.

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I think this is what Leibniz
uses to figure out the

00:03:57.530 --> 00:03:58.400
derivative operator.

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So if I wanted to find the
derivative of A f of x, where A

00:04:08.855 --> 00:04:09.910
is just some constant number.

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It could be 5 times f of x.

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This is the same thing
as saying A times the

00:04:15.850 --> 00:04:21.460
derivative of f of x.

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And what does that tell us?

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Well, this tells us that,
let's say I had f of x.

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f of x is equal to-- and this
only works with the constants--

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f of x is equal to 5x squared.

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Right?

00:04:36.460 --> 00:04:40.590
Well this is the same thing
as 5 times x squared.

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I know I'm stating the obvious.

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So we can just say that the
derivative of this is just 5

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times the derivative
of x squared.

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So f prime of x is equal to
5 times, and what's the

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derivative of x squared?

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Right, it's 2x.

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So it equals 10x.

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Right?

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Similarly, let's say I
had g of x, just using

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a different letter.

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g of x is equal to-- and my
pen keeps malfunctioning.

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g of x is equal to, let's
say, 3x to the 12th.

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Then g prime of x, or the
derivative of g, is equal

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to 3 times the derivative
of x to the 12th.

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Well we know what that is.

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It's 12 x to the 11th.

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Which you would have seen.

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12x to the 11th.

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This equals 36x to the 11th.

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Pretty straightforward, right?

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You just multiply the
constant times whatever the

00:05:44.540 --> 00:05:45.250
derivative would have been.

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I think you get that.

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Now one other thing.

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If I wanted to apply the
derivative operator-- let me

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change colors just to mix
things up a little bit.

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Let's say if I wanted to apply
the derivative of operator-- I

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think this is called
the addition rule.

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It might be a little
bit obvious.

00:06:06.650 --> 00:06:13.020
f of x plus g of x.

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This is the same thing as the
derivative of f of x plus

00:06:26.060 --> 00:06:29.110
the derivative3 of g of x.

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That might seem a little
complicated to you, but all

00:06:30.640 --> 00:06:33.290
it's saying is that you can
find the derivative of each of

00:06:33.290 --> 00:06:35.920
the parts when you're adding
up, and then that's the

00:06:35.920 --> 00:06:37.860
derivative of the whole thing.

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I'll do a couple of examples.

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So what does this tell us?

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This is also the same
thing, of course.

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This is, I believe,
Leibniz's notation.

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And then Lagrange's notation
is-- of course these were the

00:06:50.000 --> 00:06:53.530
founding fathers of calculus.

00:06:53.530 --> 00:06:57.335
That's the same thing as f
prime of x plus g prime of x.

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And let me apply this, because
whenever you apply it, I think

00:06:59.910 --> 00:07:01.140
it starts to seem a
lot more obvious.

00:07:01.140 --> 00:07:19.500
So let's say f of x is equal
to 3x squared plus 5x plus 3.

00:07:19.500 --> 00:07:22.080
Well, if we just want to figure
out the derivative, we say f

00:07:22.080 --> 00:07:25.445
prime of x, we just find the
derivative of each

00:07:25.445 --> 00:07:26.080
of these terms.

00:07:26.080 --> 00:07:29.070
Well, this is 3 times the
derivative of x squared.

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The derivative of x squared,
we already figured

00:07:30.200 --> 00:07:31.930
out, is 2x, right?

00:07:31.930 --> 00:07:34.230
So this becomes 6x.

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Really you just take the 2,
multiply it by the 3, and

00:07:36.140 --> 00:07:38.580
then decrement the 2 by 1.

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So it's really 6x to the first,
which is the same thing as 6x.

00:07:42.120 --> 00:07:45.182
Plus the derivative of 5x is 5.

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And you know that because if I
just had a line that's y equals

00:07:48.440 --> 00:07:50.920
5x, the slope is 5, right?

00:07:50.920 --> 00:07:53.950
Plus, what's the derivative
of a constant function?

00:07:53.950 --> 00:07:55.410
What's the derivative of 3?

00:07:55.410 --> 00:07:57.410
Well, I'll give you a hint.

00:07:57.410 --> 00:08:00.950
Graph y equals 3 and tell
me what the slope is.

00:08:00.950 --> 00:08:05.240
Right, the derivative
of a constant is 0.

00:08:05.240 --> 00:08:08.230
I'll show other times why that
might be more intuitive.

00:08:08.230 --> 00:08:09.500
Plus 0.

00:08:09.500 --> 00:08:10.720
You can just ignore that.

00:08:10.720 --> 00:08:13.770
f prime of x is
equal to 6x plus 5.

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Let's do some more.

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I think the more examples
we do, the better.

00:08:24.470 --> 00:08:27.790
And I want to keep switching
notations, so you don't get

00:08:27.790 --> 00:08:29.340
daunted whenever you see
it in a different way.

00:08:29.340 --> 00:08:39.540
Let's say y equals 10x to
the fifth minus 7x to the

00:08:39.540 --> 00:08:46.350
third plus 4x plus 1.

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So here we're going to apply
the derivative operator.

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So we say dy-- this
is I think Leibniz's

00:08:55.760 --> 00:08:59.040
notation-- dy over dx.

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And that's just the change
in y over the change in x,

00:09:01.170 --> 00:09:02.960
over very small changes.

00:09:02.960 --> 00:09:06.040
That's kind of how I view this
d, like a very small delta.

00:09:06.040 --> 00:09:17.500
Is equal to 5 times 10 is 50
x to the fourth minus 21 --

00:09:17.500 --> 00:09:24.530
right, 3 times 7--
x squared plus 4.

00:09:24.530 --> 00:09:27.160
And then the 1, the
derivative of 1 is just 0.

00:09:27.160 --> 00:09:27.650
So there it is.

00:09:27.650 --> 00:09:29.340
We figured out the
derivative of this very

00:09:29.340 --> 00:09:30.210
complicated function.

00:09:30.210 --> 00:09:31.510
And it was pretty
straightforward.

00:09:31.510 --> 00:09:34.050
I think you'll find that
derivatives of polynomials are

00:09:34.050 --> 00:09:36.570
actually more straightforward
than a lot of concepts that you

00:09:36.570 --> 00:09:39.170
learned a lot earlier
in mathematics.

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That's all the time I have
now for this presentation.

00:09:41.160 --> 00:09:43.820
In the next couple I'll just do
a bunch of more examples, and

00:09:43.820 --> 00:09:46.100
I'll show you some more rules
for solving even more

00:09:46.100 --> 00:09:47.175
complicated derivatives.

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See you in the next
presentation.


Calculus: Derivatives 2

WEBVTT
Kind: captions
Language: en

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In the last presentation, I
hopefully gave you a little

00:00:02.730 --> 00:00:05.290
bit of an intuition of
what a derivative is.

00:00:05.290 --> 00:00:07.920
It's really just a way to
find the slope at a given

00:00:07.920 --> 00:00:10.100
point along the curve.

00:00:10.100 --> 00:00:12.210
Now we'll actually apply
it to some functions.

00:00:12.210 --> 00:00:18.200
So let's say I had
the function f of x.

00:00:18.200 --> 00:00:22.150
f of x is equal to x squared.

00:00:22.150 --> 00:00:27.325
And I want to know what is
the slope of this curve.

00:00:40.410 --> 00:00:47.220
What is the slope at x is equal
to-- let's say at x equals 3.

00:00:47.220 --> 00:00:48.520
What is the slope of x?

00:00:48.520 --> 00:00:51.720
Let's draw out what I'm asking.

00:00:51.720 --> 00:00:53.300
Coordinate axis.

00:00:53.300 --> 00:00:57.310
x-coordinate, that's
the y-coordinate.

00:00:57.310 --> 00:00:59.260
And then if I were to draw--
let me pick a different color.

00:01:05.910 --> 00:01:10.690
So we want to say what is the
slope when x is equal to 3.

00:01:16.310 --> 00:01:19.590
This is x equals 3.

00:01:19.590 --> 00:01:23.240
And of course when x equals
3, f of x is equal to 9.

00:01:23.240 --> 00:01:25.010
We know that, right?

00:01:28.830 --> 00:01:32.000
So what we do is we take a
point, maybe a little bit

00:01:32.000 --> 00:01:33.120
further along the curve.

00:01:33.120 --> 00:01:37.880
Let's say this point
right here is 3 plus h.

00:01:37.880 --> 00:01:39.780
And I keep it abstract as h
because as you know we're

00:01:39.780 --> 00:01:41.720
going to take the limit
as h approaches 0.

00:01:41.720 --> 00:01:45.230
And at this point
right here is what?

00:01:45.230 --> 00:01:48.390
It's 3 plus h squared, right?

00:01:48.390 --> 00:01:52.000
Because the function is f of
x is equal to x squared.

00:01:52.000 --> 00:02:08.700
So this point right here is
3 plus h, 3 plus h squared.

00:02:08.700 --> 00:02:10.909
Because we just take the 3 plus
h and put it into x squared

00:02:10.909 --> 00:02:12.360
and we get 3 plus h squared.

00:02:12.360 --> 00:02:17.000
And this point here
is of course 3, 9.

00:02:17.000 --> 00:02:19.340
What we want to do is we
want to find the slope

00:02:19.340 --> 00:02:20.540
between these two point.

00:02:23.600 --> 00:02:25.310
I really have to
find a better tool.

00:02:25.310 --> 00:02:28.890
This one keeps freezing, I
think it's too CPU intensive.

00:02:28.890 --> 00:02:29.500
But anyway.

00:02:29.500 --> 00:02:31.480
So we want to find the slope
between these two points.

00:02:31.480 --> 00:02:34.500
So what's the slope? so it's a
change in y, so it's 3 plus h

00:02:34.500 --> 00:02:50.240
squared minus this y minus
9 over the change in x.

00:02:50.240 --> 00:03:03.000
Well that's 3 plus h minus 3.

00:03:03.000 --> 00:03:05.780
So if we simplify this top
part or we multiply it out,

00:03:05.780 --> 00:03:06.720
what's 3 plus h squared?

00:03:06.720 --> 00:03:19.260
That's 9 plus 6h plus h
squared, and then get the minus

00:03:19.260 --> 00:03:26.520
9, and all of that is over--
well this 3 and this minus

00:03:26.520 --> 00:03:29.730
3 cancel out, so all
you're left is with h.

00:03:29.730 --> 00:03:31.890
And even if we simplify
this, this 9 minus

00:03:31.890 --> 00:03:34.892
9, they cancel out.

00:03:34.892 --> 00:03:37.005
So let me go up here.

00:03:40.700 --> 00:03:49.210
We're left with-- this pen
keeps freezing-- it's 6h

00:03:49.210 --> 00:03:53.040
plus h squared over h.

00:03:53.040 --> 00:03:55.500
And now we would simplify this,
right, because we can divide

00:03:55.500 --> 00:03:57.160
the top and the bottom, that
numerator and the

00:03:57.160 --> 00:03:58.260
denominator by h.

00:03:58.260 --> 00:04:05.200
And you get 6 plus h squared.

00:04:05.200 --> 00:04:08.230
So that's the slope
between these two points.

00:04:08.230 --> 00:04:10.240
It's 6 plus h squared.

00:04:10.240 --> 00:04:12.980
So if we want to find the
instantaneous slope at the

00:04:12.980 --> 00:04:17.560
point x equals 3, f of x is
equal to 9, or the point 3,9,

00:04:17.560 --> 00:04:20.450
we just have to find the limit
as h approaches 0 here.

00:04:20.450 --> 00:04:27.670
So we'll just take the
limit as h approaches 0.

00:04:27.670 --> 00:04:29.240
Well this is an easy
limit problem, right?

00:04:29.240 --> 00:04:32.620
What's the limit of 6 plus h
squared as h approaches 0?

00:04:32.620 --> 00:04:34.340
Well it equals 6.

00:04:34.340 --> 00:04:38.300
So we now know that the
slope of this curve at the

00:04:38.300 --> 00:04:41.970
point x equals 3 is 6.

00:04:45.860 --> 00:04:49.600
So if you actually did a
traditional rise over run,

00:04:49.600 --> 00:04:52.390
the slope, this change in
y over change in x is 6.

00:04:52.390 --> 00:04:54.830
So we have the instantaneous
slope at exactly the

00:04:54.830 --> 00:04:57.140
point x is equal to 3.

00:04:57.140 --> 00:05:00.070
So that's useful.

00:05:00.070 --> 00:05:05.960
You know if this was a graph of
someone's position, we would

00:05:05.960 --> 00:05:08.570
then know kind of the
instantaneous velocity, which

00:05:08.570 --> 00:05:09.820
is-- well I won't go into that.

00:05:09.820 --> 00:05:11.710
I'll do a separate
module on physics.

00:05:11.710 --> 00:05:13.620
But this was useful, but
let's see if we can do more

00:05:13.620 --> 00:05:16.440
generalized version where we
don't have to know ahead of

00:05:16.440 --> 00:05:18.550
time what point we want
to find the slope at.

00:05:18.550 --> 00:05:22.290
If we can get a generalized
formula for the slope at any

00:05:22.290 --> 00:05:26.870
point along the graph f of
x is equal to x squared.

00:05:26.870 --> 00:05:30.140
So let me clear this.

00:05:30.140 --> 00:05:42.470
So we're going to stick with f
of x is equal to x squared.

00:05:42.470 --> 00:05:46.180
And we know that the slope at
any point of this is just going

00:05:46.180 --> 00:06:03.050
to be the limit as h approaches
0 of f of x plus h

00:06:03.050 --> 00:06:07.530
minus f of access.

00:06:07.530 --> 00:06:11.650
All of that over h.

00:06:11.650 --> 00:06:14.590
This part right here, this is
just the slope formula that

00:06:14.590 --> 00:06:16.080
you learned years ago.

00:06:16.080 --> 00:06:18.610
It's just change in
y over change in x.

00:06:18.610 --> 00:06:21.450
And all we're doing is we're
seeing what happens as the

00:06:21.450 --> 00:06:23.860
change in x gets smaller and
smaller and smaller as it

00:06:23.860 --> 00:06:24.860
actually approaches 0.

00:06:24.860 --> 00:06:27.090
And that's why we can get
the instantaneous change at

00:06:27.090 --> 00:06:28.250
that point in the curve.

00:06:28.250 --> 00:06:30.610
So let's apply this
definition of a derivative

00:06:30.610 --> 00:06:33.490
to this function.

00:06:33.490 --> 00:06:37.040
And actually if you want to
know the notation, I think

00:06:37.040 --> 00:06:39.190
this is the notation
Lagrange came up with.

00:06:39.190 --> 00:06:43.950
This is equal to f prime of x.

00:06:43.950 --> 00:06:45.210
Don't take my word
on it on Lagrange.

00:06:45.210 --> 00:06:46.970
You might want to look
it up on Wikipedia.

00:06:46.970 --> 00:06:48.250
But this [UNINTELLIGIBLE]

00:06:48.250 --> 00:06:51.980
derivative of f of
x is f prime of x.

00:06:51.980 --> 00:06:53.630
Let's apply it to x squared.

00:06:53.630 --> 00:06:59.490
So we're going to say the
limit as h approaches

00:06:59.490 --> 00:07:02.790
0 of f of x plus h.

00:07:02.790 --> 00:07:08.210
Well, f of x plus h is
just-- this pen driving me

00:07:08.210 --> 00:07:13.580
crazy-- x plus h squared.

00:07:13.580 --> 00:07:16.690
I just took the x plus h
and put it into f of x.

00:07:16.690 --> 00:07:24.730
Minus f of x--well that's
just x squared-- over h.

00:07:24.730 --> 00:07:30.490
And this is equal to the
limit as h approaches 0.

00:07:30.490 --> 00:07:32.240
Just multiply this out of.

00:07:32.240 --> 00:07:43.710
x squared plus 2xh plus h
squared minus x squared--

00:07:43.710 --> 00:07:48.350
running out of space--
all of that over h.

00:07:48.350 --> 00:07:49.310
Let's simplify this.

00:07:49.310 --> 00:07:53.160
This x squared cancels out
with this minus x squared.

00:07:53.160 --> 00:07:56.260
And then we can divide the
numerator and the denominator

00:07:56.260 --> 00:08:07.005
by h, and we're left with the
limit as h approaches 0.

00:08:07.005 --> 00:08:14.180
Numerator and denominator
by h of 2x plus h.

00:08:14.180 --> 00:08:16.470
Well this is easy.

00:08:16.470 --> 00:08:19.520
This goes to 0, this
is just equal to 2x.

00:08:19.520 --> 00:08:20.640
So there we have it.

00:08:20.640 --> 00:08:24.520
The limit as h approaches
0 is equal to 2x.

00:08:24.520 --> 00:08:27.650
And this is equal to f prime of
x, so the derivative of f of

00:08:27.650 --> 00:08:32.415
x, which is the denoted by f
prime of x is equal to 2x.

00:08:32.415 --> 00:08:33.660
Well what does it tell us?

00:08:33.660 --> 00:08:35.270
What have we done
for ourselves?

00:08:35.270 --> 00:08:38.930
Well now I can give you any
point along the curve.

00:08:38.930 --> 00:08:43.464
Let's say we want to know the
slope at the point 16, right.

00:08:46.260 --> 00:08:51.200
When at the point 16,256.

00:08:51.200 --> 00:08:53.870
That's a point along f
of x equals x squared.

00:08:53.870 --> 00:08:55.870
It's just 16 and
then 16 squared.

00:08:55.870 --> 00:08:57.250
What's the slope at that point?

00:08:57.250 --> 00:08:59.746
Well we now know the
slope is 2 times 16.

00:09:04.640 --> 00:09:09.340
So the slope is equal to 32.

00:09:09.340 --> 00:09:12.545
Whatever the x value is you
just put into this f prime of

00:09:12.545 --> 00:09:15.750
x function or the derivative
function, and you'll get

00:09:15.750 --> 00:09:17.060
the slope at that point.

00:09:17.060 --> 00:09:19.270
I think that's pretty neat and
I'll show you how in future

00:09:19.270 --> 00:09:22.100
presentations how we can
apply this to physics and

00:09:22.100 --> 00:09:24.090
optimization problems and a
whole other set of things.

00:09:24.090 --> 00:09:26.290
And I'm also going to show you
how to find the derivatives for

00:09:26.290 --> 00:09:28.180
a whole set of other functions.

00:09:28.180 --> 00:09:29.030
I'll see in the
next presentation.


Calculus: Derivatives 1

WEBVTT
Kind: captions
Language: en

00:00:01.080 --> 00:00:03.790
Welcome to the presentation
on derivatives.

00:00:03.790 --> 00:00:06.290
I think you're going to find
that this is when math starts

00:00:06.290 --> 00:00:10.530
to become a lot more fun than
it was just a few topics ago.

00:00:10.530 --> 00:00:11.990
Well let's get started
with our derivatives.

00:00:11.990 --> 00:00:13.150
I know it sounds
very complicated.

00:00:13.150 --> 00:00:16.560
Well, in general, if I have a
straight line-- let me see if I

00:00:16.560 --> 00:00:20.990
can draw a straight line
properly-- if I had a straight

00:00:20.990 --> 00:00:28.020
line-- that's my coordinate
axes, which aren't straight--

00:00:28.020 --> 00:00:29.120
this is a straight line.

00:00:31.700 --> 00:00:34.630
But when I have a straight line
like that, and I ask you to

00:00:34.630 --> 00:00:37.150
find the slope-- I think you
already know how to do this--

00:00:37.150 --> 00:00:39.540
it's just the change in y
divided by the change in x.

00:00:39.540 --> 00:00:43.800
If I wanted to find the slope--
really I mean the slope is the

00:00:43.800 --> 00:00:46.140
same, because it is a straight
line, the slope is the same

00:00:46.140 --> 00:00:50.040
across the whole line, but if I
want to find the slope at any

00:00:50.040 --> 00:00:51.840
point in this line, what I
would do is I would pick a

00:00:51.840 --> 00:00:56.000
point x-- say I'd
pick this point.

00:00:56.000 --> 00:01:00.280
We'd pick a different color--
I'd take this point, I'd pick

00:01:00.280 --> 00:01:02.840
this point-- it's pretty
arbitrary, I could pick any two

00:01:02.840 --> 00:01:05.990
points, and I would figure out
what the change in y is-- this

00:01:05.990 --> 00:01:09.860
is the change in y, delta y,
that's just another way of

00:01:09.860 --> 00:01:15.860
saying change in y-- and
this is the change in x.

00:01:15.860 --> 00:01:16.920
delta x.

00:01:16.920 --> 00:01:22.140
And we figured out that the
slope is defined really as

00:01:22.140 --> 00:01:30.250
change in y divided
by change in x.

00:01:33.540 --> 00:01:37.640
And another way of saying that
is delta-- it's that triangle--

00:01:37.640 --> 00:01:41.180
delta y divided by delta x.

00:01:41.180 --> 00:01:42.660
Very straightforward.

00:01:42.660 --> 00:01:45.100
Now what happens, though,
if we're not dealing

00:01:45.100 --> 00:01:45.860
with a straight line?

00:01:45.860 --> 00:01:49.780
Let me see if I have
space to draw that.

00:01:49.780 --> 00:01:52.920
Another coordinate axes.

00:01:52.920 --> 00:01:55.780
Still pretty messy, but I
think you'll get the point.

00:02:00.073 --> 00:02:02.960
Now let's say, instead of just
a regular line like this, this

00:02:02.960 --> 00:02:04.690
follows the standard
y equals mx plus b.

00:02:04.690 --> 00:02:09.640
Let's just say I had the
curve y equals x squared.

00:02:09.640 --> 00:02:12.150
Let me draw it in a
different color.

00:02:12.150 --> 00:02:15.940
So y equals x squared looks
something like this.

00:02:15.940 --> 00:02:19.320
It's a curve, you're probably
pretty familiar with it by now.

00:02:19.320 --> 00:02:20.650
And what I'm going to
ask you is, what is the

00:02:20.650 --> 00:02:23.180
slope of this curve?

00:02:23.180 --> 00:02:24.440
And think about that.

00:02:24.440 --> 00:02:26.930
What does it mean to take
the slope of a curve now?

00:02:26.930 --> 00:02:29.150
Well, in this line, the slope
was the same throughout

00:02:29.150 --> 00:02:30.400
the whole line.

00:02:30.400 --> 00:02:31.780
But if you look at this
curve, doesn't the

00:02:31.780 --> 00:02:32.960
slope change, right?

00:02:32.960 --> 00:02:36.560
Here it's almost flat, and it
gets steeper steeper steeper

00:02:36.560 --> 00:02:38.865
steeper steeper until
gets pretty steep.

00:02:38.865 --> 00:02:41.000
And if you go really far out,
it gets extremely steep.

00:02:41.000 --> 00:02:42.630
So you're probably saying,
well, how do you figure out

00:02:42.630 --> 00:02:45.910
the slope of a curve whose
slope keeps changing?

00:02:45.910 --> 00:02:48.170
Well there is no slope
for the entire curve.

00:02:48.170 --> 00:02:50.900
For a line, there is a slope
for the entire line, because

00:02:50.900 --> 00:02:52.250
the slope never changes.

00:02:52.250 --> 00:02:54.380
But what we could try to
do is figure out what the

00:02:54.380 --> 00:02:56.720
slope is at a given point.

00:02:56.720 --> 00:02:59.540
And the slope at a given point
would be the same as the

00:02:59.540 --> 00:03:00.890
slope of a tangent line.

00:03:00.890 --> 00:03:08.260
For example-- let me pick a
green-- the slope at this point

00:03:08.260 --> 00:03:17.970
right here would be the same
as the slope of this line.

00:03:17.970 --> 00:03:19.180
Right?

00:03:19.180 --> 00:03:20.550
Because this line
is tangent to it.

00:03:20.550 --> 00:03:24.350
So it just touches that curve,
and at that exact point, they

00:03:24.350 --> 00:03:27.820
would have-- this blue curve, y
equals x squared, would have

00:03:27.820 --> 00:03:30.650
the same slope as
this green line.

00:03:30.650 --> 00:03:33.050
But if we go to a point back
here, even though this is a

00:03:33.050 --> 00:03:36.940
really badly drawn graph,
the slope would be

00:03:36.940 --> 00:03:38.700
something like this.

00:03:38.700 --> 00:03:40.015
The tangent slope.

00:03:40.015 --> 00:03:42.520
The slope would be a negative
slope, and here it's a positive

00:03:42.520 --> 00:03:47.940
slope, but if we took a
point here, the slope would

00:03:47.940 --> 00:03:50.630
be even more positive.

00:03:50.630 --> 00:03:52.030
So how are we going
to figure this out?

00:03:52.030 --> 00:03:55.900
How are we going to figure out
what the slope is at any point

00:03:55.900 --> 00:03:58.850
along the curve y
equals x squared?

00:03:58.850 --> 00:04:01.590
That's where the derivative
comes into use, and now for the

00:04:01.590 --> 00:04:03.290
first time you'll actually see
why a limit is actually

00:04:03.290 --> 00:04:06.010
a useful concept.

00:04:06.010 --> 00:04:09.130
So let me try to
redraw the curve.

00:04:09.130 --> 00:04:15.750
OK, I'll draw my axes, that's
the y-axis-- I'll just do it in

00:04:15.750 --> 00:04:22.630
the first quadrant-- and this
is-- I really have to find a

00:04:22.630 --> 00:04:29.470
better tool to do my-- this is
x coordinate, and then let

00:04:29.470 --> 00:04:31.620
me draw my curve in yellow.

00:04:34.135 --> 00:04:37.790
So y equals x squared looks
something like this.

00:04:37.790 --> 00:04:40.520
I'm really concentrating
to draw this at

00:04:40.520 --> 00:04:41.700
least decently good.

00:04:41.700 --> 00:04:42.800
OK.

00:04:42.800 --> 00:04:46.890
So let's say we want to find
the slope at this point.

00:04:53.560 --> 00:05:00.040
Let's call this point a.

00:05:00.040 --> 00:05:02.070
At this point, x equals a.

00:05:02.070 --> 00:05:06.845
And of course this is f of a.

00:05:11.340 --> 00:05:13.190
So what we could try to do
is, we could try to find

00:05:13.190 --> 00:05:15.410
the slope of a secant line.

00:05:15.410 --> 00:05:19.920
A line between-- we take
another point, say, somewhat

00:05:19.920 --> 00:05:26.550
close, to this point on the
graph, let's say here, and if

00:05:26.550 --> 00:05:29.530
we could figure out the slope
of this line, it would be a

00:05:29.530 --> 00:05:34.000
bit of an approximation of
the slope of the curve

00:05:34.000 --> 00:05:35.090
exactly at this point.

00:05:35.090 --> 00:05:37.740
So let me draw
that secant line.

00:05:44.460 --> 00:05:45.100
Something like that.

00:05:45.100 --> 00:05:46.690
Secant line looks
something like that.

00:05:46.690 --> 00:05:56.050
And let's say that this point
right here is a plus h, where

00:05:56.050 --> 00:05:59.860
this distance is just h, this
is a plus h, we're just going

00:05:59.860 --> 00:06:05.160
h away from a, and then
this point right here

00:06:05.160 --> 00:06:09.060
is f of a plus h.

00:06:11.640 --> 00:06:13.103
My pen is malfunctioning.

00:06:17.730 --> 00:06:19.550
So this would be an
approximation for what the

00:06:19.550 --> 00:06:21.220
slope is at this point.

00:06:21.220 --> 00:06:24.980
And the closer that h gets,
the closer this point gets to

00:06:24.980 --> 00:06:27.390
this point, the better our
approximation is going to be,

00:06:27.390 --> 00:06:30.520
all the way to the point that
if we could actually get the

00:06:30.520 --> 00:06:34.140
slope where h equals 0, that
would actually be the slope,

00:06:34.140 --> 00:06:37.050
the instantaneous slope, at
that point in the curve.

00:06:37.050 --> 00:06:41.440
But how can we figure out what
the slope is when h equals 0?

00:06:44.800 --> 00:06:46.670
So right now, we're saying that
the slope between these two

00:06:46.670 --> 00:06:49.900
points, it would be the
change in y, so what's

00:06:49.900 --> 00:06:51.040
the change in y?

00:06:51.040 --> 00:06:57.170
It's this, so that this point
right here is-- the x

00:06:57.170 --> 00:07:00.710
coordinate is-- my thing just
keeps messing up-- the x

00:07:00.710 --> 00:07:11.330
coordinate is a plus h, and the
y coordinate is f of a plus h.

00:07:15.180 --> 00:07:22.050
And this point right here, the
coordinate is a and f of a.

00:07:22.050 --> 00:07:25.370
So if we just use the standard
slope formula, like before, we

00:07:25.370 --> 00:07:27.610
would say change in
y over change in x.

00:07:27.610 --> 00:07:29.100
Well, what's the change in y?

00:07:29.100 --> 00:07:37.680
It's f of a plus h-- this
y coordinate minus this y

00:07:37.680 --> 00:07:46.850
coordinate-- minus f of
a over the change in x.

00:07:46.850 --> 00:07:53.010
Well that change in x is this
x coordinate, a plus h, minus

00:07:53.010 --> 00:07:55.720
this x coordinate, minus a.

00:07:55.720 --> 00:07:58.480
And of course this a
and this a cancel out.

00:07:58.480 --> 00:08:01.490
So it's f of a plus h,
minus f of a, all over h.

00:08:01.490 --> 00:08:05.400
This is just the slope
of this secant line.

00:08:05.400 --> 00:08:08.810
And if we want to get the slope
of the tangent line, we would

00:08:08.810 --> 00:08:11.980
just have to find what happens
as h gets smaller and

00:08:11.980 --> 00:08:12.780
smaller and smaller.

00:08:12.780 --> 00:08:14.470
And I think you know
where I'm going.

00:08:14.470 --> 00:08:16.840
Really, we just want to, if we
want to find the slope of this

00:08:16.840 --> 00:08:19.140
tangent line, we just have
to find the limit of this

00:08:19.140 --> 00:08:28.780
value as h approaches 0.

00:08:28.780 --> 00:08:32.700
And then, as h approaches 0,
this secant line is going to

00:08:32.700 --> 00:08:36.710
get closer and closer to the
slope of the tangent line.

00:08:36.710 --> 00:08:40.590
And then we'll know the exact
slope at the instantaneous

00:08:40.590 --> 00:08:41.900
point along the curve.

00:08:41.900 --> 00:08:44.150
And actually, it turns out
that this is the definition

00:08:44.150 --> 00:08:46.800
of the derivative.

00:08:46.800 --> 00:08:50.780
And the derivative is nothing
more than the slope of a

00:08:50.780 --> 00:08:53.010
curve at an exact point.

00:08:53.010 --> 00:08:56.310
And this is super useful,
because for the first time,

00:08:56.310 --> 00:08:58.570
everything we've talked
about to this point is

00:08:58.570 --> 00:08:59.560
the slope of a line.

00:08:59.560 --> 00:09:03.050
But now we can take any
continuous curve, or most

00:09:03.050 --> 00:09:06.800
continuous curves, and find
the slope of that curve

00:09:06.800 --> 00:09:08.340
at an exact point.

00:09:08.340 --> 00:09:11.940
So now that I've given you the
definition of what a derivative

00:09:11.940 --> 00:09:13.690
is, and maybe hopefully a
little bit of intuition, in the

00:09:13.690 --> 00:09:17.070
next presentation I'm going to
use this definition to actually

00:09:17.070 --> 00:09:20.040
apply it to some functions,
like x squared and others, and

00:09:20.040 --> 00:09:21.930
give you some more problems.

00:09:21.930 --> 00:09:24.070
I'll see you in the
next presentation


Limit examples (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.330
Let's do some more
limit examples.

00:00:03.330 --> 00:00:05.250
So let's get another problem.

00:00:05.250 --> 00:00:19.860
If I had the limit as x
approaches 3 of, let's say,

00:00:19.860 --> 00:00:31.190
x squared minus 6x plus 9
over x squared minus 9.

00:00:31.190 --> 00:00:33.450
So the first thing I like to do
whenever I see any of these

00:00:33.450 --> 00:00:35.150
limits problems is just
substitute the number in and

00:00:35.150 --> 00:00:36.413
see if I get something
that makes sense, and

00:00:36.413 --> 00:00:37.670
then we'd be done.

00:00:37.670 --> 00:00:39.170
Well, usually we'd be done.

00:00:39.170 --> 00:00:41.490
I don't want to make these
sweeping statements.

00:00:41.490 --> 00:00:43.880
If the function is
continuous, we'd be done.

00:00:43.880 --> 00:00:47.250
But if we put the 3 in the
numerator, we get 3 squared,

00:00:47.250 --> 00:00:51.040
which is 9, minus 18 plus 9.

00:00:51.040 --> 00:00:52.660
So that equals 0.

00:00:52.660 --> 00:00:55.510
And the denominator also--
let's see, 3 squared minus

00:00:55.510 --> 00:00:56.400
9, that also equals 0.

00:00:56.400 --> 00:01:00.970
So we don't like having 0/0.

00:01:00.970 --> 00:01:03.840
My pen tool is
malfunctioning again.

00:01:03.840 --> 00:01:07.290
So we don't like getting 0, 0,
0, so is there any way we can

00:01:07.290 --> 00:01:12.010
simplify this expression to
maybe get it to an expression

00:01:12.010 --> 00:01:15.110
that, when we evaluate it at x
equals 3, we actually get

00:01:15.110 --> 00:01:16.640
something that makes sense?

00:01:16.640 --> 00:01:19.180
Well, whenever I see two of
these polynomials here, and

00:01:19.180 --> 00:01:22.220
they look, just by inspecting
them, relatively easy to

00:01:22.220 --> 00:01:25.220
factor, I like to factor them
out because maybe there's the

00:01:25.220 --> 00:01:27.250
same factor in the numerator
and the denominator, and

00:01:27.250 --> 00:01:28.610
then we can simplify it.

00:01:28.610 --> 00:01:32.110
So let's say that this is the
same thing as-- that looks

00:01:32.110 --> 00:01:39.690
like it's x plus 3--
no, no, no, x minus 3.

00:01:39.690 --> 00:01:42.450
This is x minus 3.

00:01:42.450 --> 00:01:44.280
It actually looks like it's x
minus 3 squared, but we're

00:01:44.280 --> 00:01:46.840
just going to write x minus 3
times x minus 3, which is, of

00:01:46.840 --> 00:01:48.820
course, x minus 3 squared.

00:01:48.820 --> 00:01:51.940
And then in the denominator,
you know how to factor these,

00:01:51.940 --> 00:02:01.310
this is x plus 3 times
x minus 3, all right?

00:02:01.310 --> 00:02:03.990
So the limit as x approaches 3
of this expression is the same

00:02:03.990 --> 00:02:07.960
thing as the limit as
x approaches 3 of

00:02:07.960 --> 00:02:08.760
this expression.

00:02:08.760 --> 00:02:12.660
And, of course, there's nothing
we can do to change the fact

00:02:12.660 --> 00:02:16.150
that this function, or this
expression, is undefined

00:02:16.150 --> 00:02:17.360
at x equals 3.

00:02:17.360 --> 00:02:19.500
But if we can simplify
it, we can figure out

00:02:19.500 --> 00:02:21.320
what it approaches.

00:02:21.320 --> 00:02:26.010
Well, if we assume that x is
any number but 3, we can cross

00:02:26.010 --> 00:02:28.330
out these two terms because
then they wouldn't be 0, right?

00:02:28.330 --> 00:02:31.020
It only is 0 when x is equal to
3 because-- so in the numerator

00:02:31.020 --> 00:02:32.990
and the denominator, we
can cross this out.

00:02:32.990 --> 00:02:36.250
And we can say-- and I'm not
being very rigorous here, but

00:02:36.250 --> 00:02:38.610
this is kind of how it's
taught, and I think you get the

00:02:38.610 --> 00:02:43.180
intuition-- that this is the
same thing as the limit as x

00:02:43.180 --> 00:02:49.200
approaches 3 of x minus
3 over x plus 3.

00:02:49.200 --> 00:02:52.330
Now let's just try to stick
the x in and see what we get.

00:02:52.330 --> 00:02:54.010
Well, in the numerator,
we get 3 minus 3.

00:02:54.010 --> 00:02:56.030
We still get 0.

00:02:56.030 --> 00:02:58.190
But in the denominator
here, we get 6, right?

00:02:58.190 --> 00:02:59.520
3 plus 3 is 6.

00:02:59.520 --> 00:03:00.770
So now we get a good number.

00:03:00.770 --> 00:03:04.680
0 or 6, well, that's a
real number, so it's 0.

00:03:04.680 --> 00:03:05.710
0/6 is 0.

00:03:05.710 --> 00:03:07.170
So that was interesting.

00:03:07.170 --> 00:03:09.660
The first time we did it,
we got the answer 0/0.

00:03:09.660 --> 00:03:12.760
And now we get the answer
0 by simplifying.

00:03:12.760 --> 00:03:15.590
But, of course, it's very
important to remember that

00:03:15.590 --> 00:03:18.810
this expression is not
defined at x equals 3.

00:03:18.810 --> 00:03:21.280
It's defined everywhere but,
but if we were to graph it, and

00:03:21.280 --> 00:03:24.070
I encourage you to do so, you
would see that as you get

00:03:24.070 --> 00:03:26.980
closer and closer to x
equals 3, the value of this

00:03:26.980 --> 00:03:28.960
expression will equal 0.

00:03:28.960 --> 00:03:30.060
And I know what
you're thinking.

00:03:30.060 --> 00:03:31.350
Well, this was 0/0.

00:03:31.350 --> 00:03:38.190
Is every time I get 0/0 going
to end up just becoming 0 when

00:03:38.190 --> 00:03:39.300
I evaluate the expression?

00:03:39.300 --> 00:03:42.240
Well, let's explore that.

00:03:42.240 --> 00:03:45.570
Let me clear this.

00:03:45.570 --> 00:03:56.080
Let's say what is-- pen is not
working-- the limit as x

00:03:56.080 --> 00:04:17.860
approaches 1 of x squared
minus x minus 2.

00:04:21.650 --> 00:04:23.640
No, let's say x squared
plus x minus 2.

00:04:23.640 --> 00:04:26.110
As you can see, I do all
this in my head, and

00:04:26.110 --> 00:04:27.450
I'm prone to mistakes.

00:04:27.450 --> 00:04:32.370
And all of that over x minus 1.

00:04:32.370 --> 00:04:33.940
Well, once again, if we just
evaluate it, let's see what

00:04:33.940 --> 00:04:34.930
happens when x equals 1.

00:04:34.930 --> 00:04:38.200
You get 1 squared plus
1, so it's 2 minus 2.

00:04:38.200 --> 00:04:40.010
You get 0/0.

00:04:40.010 --> 00:04:44.640
So once again, we get 0/0, and
we have to do something to

00:04:44.640 --> 00:04:46.960
this maybe to simplify it.

00:04:46.960 --> 00:04:48.620
Well, let's factor the top.

00:04:48.620 --> 00:04:54.530
So that's the same thing as
the limit as x approaches 1.

00:04:54.530 --> 00:05:01.450
Well, that's x minus 1
times x plus 2, right?

00:05:07.020 --> 00:05:09.840
And I think you'll often
discover when you see a lot of

00:05:09.840 --> 00:05:13.720
limit problems that even if
this top factor, if this top

00:05:13.720 --> 00:05:16.710
expression, is hard to factor,
chances are, one of the things

00:05:16.710 --> 00:05:19.390
in the denominator that are
making this expression

00:05:19.390 --> 00:05:21.770
undefined is probably
a factor up here.

00:05:21.770 --> 00:05:24.390
So sometimes you might get a
more complex thing that isn't

00:05:24.390 --> 00:05:27.510
as easy to factor as this, but
a good starting point is to

00:05:27.510 --> 00:05:30.840
guess that one of the factors
is going to be in the bottom

00:05:30.840 --> 00:05:33.360
expression because that's kind
of the trick of these problems,

00:05:33.360 --> 00:05:35.750
to just simplify
the expression.

00:05:35.750 --> 00:05:39.390
So once again, if we assume
that x does not equal 1, and

00:05:39.390 --> 00:05:43.440
this expression would not be
0 and this would not be 0,

00:05:43.440 --> 00:05:47.230
then these two could
be canceled out.

00:05:47.230 --> 00:05:51.010
And we get that this is just
the same thing as the limit as

00:05:51.010 --> 00:05:55.070
x approaches 1 of x plus 2.

00:05:55.070 --> 00:05:56.030
Well, now this is pretty easy.

00:05:56.030 --> 00:05:58.850
What's the limit as x
approaches 1 of x plus 2?

00:05:58.850 --> 00:06:03.180
Well, you just stick 1 in
there, and you get 3.

00:06:03.180 --> 00:06:04.010
So it's interesting.

00:06:04.010 --> 00:06:07.840
When we just tried to
evaluate the expression at

00:06:07.840 --> 00:06:10.580
x equals 1, we got 0/0.

00:06:10.580 --> 00:06:14.840
And in the previous example, we
saw that it evaluated out when

00:06:14.840 --> 00:06:17.870
you simplified it to 0, and in
this example, it came out to 3.

00:06:17.870 --> 00:06:19.760
And I really encourage you, if
you have a graphing calculator,

00:06:19.760 --> 00:06:23.110
graph these functions that
we're doing and see and show

00:06:23.110 --> 00:06:26.080
yourself visually that it's
true, that the limit as you

00:06:26.080 --> 00:06:30.190
approach, say, x equals 1
actually does approach the

00:06:30.190 --> 00:06:31.950
limits that were solving for.

00:06:31.950 --> 00:06:34.600
And make up your own problems.

00:06:34.600 --> 00:06:36.940
Hell, that's what I'm doing.

00:06:36.940 --> 00:06:38.650
So you could prove
it to yourself.

00:06:38.650 --> 00:06:40.320
So let's do another.

00:06:40.320 --> 00:06:42.280
Let's do one that I think
is pretty interesting.

00:06:48.020 --> 00:06:57.770
Let's say what's the limit
as x approaches infinity?

00:06:57.770 --> 00:07:04.300
The limit as x approaches
infinity of, let's say, x

00:07:04.300 --> 00:07:17.650
squared plus 3 over
x to the third.

00:07:17.650 --> 00:07:19.960
So the way I think about these
problems as they approach

00:07:19.960 --> 00:07:21.910
infinity, just think about what
happens when you get

00:07:21.910 --> 00:07:24.960
really, really, really
large values of x.

00:07:24.960 --> 00:07:28.030
And kind of a cheating way of
doing this is, if you have a

00:07:28.030 --> 00:07:29.840
calculator, even if you don't
have a calculator, put

00:07:29.840 --> 00:07:31.120
in huge numbers here.

00:07:31.120 --> 00:07:34.610
See what happens when x is a
million, see what happens when

00:07:34.610 --> 00:07:36.900
x is a billion, see what
happens when x is a trillion,

00:07:36.900 --> 00:07:38.030
and I think you'll
get the point.

00:07:38.030 --> 00:07:40.270
You'll see what-- if there
is a limit here, you'll

00:07:40.270 --> 00:07:41.500
see what it's going to.

00:07:41.500 --> 00:07:44.090
But the way I think about it
is, in the numerator, kind of

00:07:44.090 --> 00:07:48.180
the fastest-growing term here
is the x squared term, right?

00:07:48.180 --> 00:07:50.970
This is the fastest-growing
term here.

00:07:50.970 --> 00:07:52.820
In the denominator, what's
the fastest-growing term?

00:07:52.820 --> 00:07:54.475
Well, in the denominator,
the fastest-growing term

00:07:54.475 --> 00:07:56.440
is this x to the third.

00:07:56.440 --> 00:07:58.160
Well, what's going to
grow faster, x to the

00:07:58.160 --> 00:08:00.000
third or x squared?

00:08:00.000 --> 00:08:01.840
Well, yeah, x to the third's
going to grow a lot

00:08:01.840 --> 00:08:03.130
faster than x squared.

00:08:03.130 --> 00:08:06.310
So this denominator, as you get
larger and larger and larger

00:08:06.310 --> 00:08:10.230
values of x, is going to grow a
lot faster than that numerator.

00:08:10.230 --> 00:08:12.700
So you could imagine if the
denominator's growing much,

00:08:12.700 --> 00:08:14.905
much, much faster than the
numerator, as you get larger

00:08:14.905 --> 00:08:17.130
and larger numbers, you're
going to get a smaller and

00:08:17.130 --> 00:08:18.630
smaller and smaller
fraction, right?

00:08:18.630 --> 00:08:20.280
It's going to approach 0.

00:08:20.280 --> 00:08:26.590
And so as you go to
infinity, it approaches 0.

00:08:26.590 --> 00:08:29.530
I know that I kind of just
hand waved, but that's really

00:08:29.530 --> 00:08:30.320
how you think about it.

00:08:30.320 --> 00:08:32.830
Another way you could do
it is you could actually

00:08:32.830 --> 00:08:34.940
divide this fraction.

00:08:34.940 --> 00:08:37.470
You could actually divide this
rational expression, and you'll

00:08:37.470 --> 00:08:39.750
get something like 1/x plus
something, something,

00:08:39.750 --> 00:08:42.620
something, and then you'd also
see, oh, well, the limit as x

00:08:42.620 --> 00:08:45.270
approaches infinity
of 1/x is also 0.

00:08:45.270 --> 00:08:46.900
Let's do one more.

00:08:46.900 --> 00:08:49.190
I'll do this fast so
I can confuse you.

00:08:49.190 --> 00:09:00.490
The limit as x approaches
infinity of 3x squared plus

00:09:00.490 --> 00:09:05.670
x over 4x squared minus 5.

00:09:08.170 --> 00:09:10.680
These problems kind of look
confusing sometimes, but

00:09:10.680 --> 00:09:11.480
they're really easy.

00:09:11.480 --> 00:09:13.330
You just have to think about
what happens as you get

00:09:13.330 --> 00:09:14.700
really large values of x.

00:09:14.700 --> 00:09:19.320
Well, as you get really large
values of x, these small terms,

00:09:19.320 --> 00:09:21.670
these ones that don't grow as
fast as these large terms,

00:09:21.670 --> 00:09:23.640
kind of don't matter anymore,
right, because you're getting

00:09:23.640 --> 00:09:25.270
really large values of x.

00:09:25.270 --> 00:09:28.210
And this case, these don't
matter anymore, and then

00:09:28.210 --> 00:09:32.210
these two x terms grow
at the same pace, right?

00:09:32.210 --> 00:09:34.070
And they'll always be
kind of growing in

00:09:34.070 --> 00:09:35.000
this ratio of 3 to 4.

00:09:35.000 --> 00:09:37.870
So the limit here is
actually that easy.

00:09:37.870 --> 00:09:39.880
It's 3/4.

00:09:39.880 --> 00:09:41.320
So what you do is you just
figure out what's the

00:09:41.320 --> 00:09:43.940
fastest-growing term on the
top, what's the fastest-growing

00:09:43.940 --> 00:09:47.000
term on the bottom, and then
figure out what it approaches.

00:09:47.000 --> 00:09:49.680
If they're the same term, then
they kind of cancel out, and

00:09:49.680 --> 00:09:51.700
you say the limit
approaches 3/4.

00:09:51.700 --> 00:09:54.410
It's a very nonrigorous way
of doing it, but it gets

00:09:54.410 --> 00:09:55.570
you the right answer.

00:09:55.570 --> 00:09:57.370
See you in the next
presentation.


Limit examples (part 1)

WEBVTT
Kind: captions
Language: en

00:00:01.280 --> 00:00:02.440
Welcome back.

00:00:02.440 --> 00:00:04.410
Now that we hopefully have a
little bit of an intuition of

00:00:04.410 --> 00:00:07.360
what a limit is, or finding
the limit of a function is,

00:00:07.360 --> 00:00:08.150
let's do some problems.

00:00:08.150 --> 00:00:12.150
Some of these you might
actually see on your exams or

00:00:12.150 --> 00:00:14.610
when you're actually trying to
solve a general limit problem.

00:00:14.610 --> 00:00:20.440
So let's say what is the
limit-- once again, my

00:00:20.440 --> 00:00:23.020
pen is not working.

00:00:23.020 --> 00:00:31.740
What is the limit as
x approaches-- let's

00:00:31.740 --> 00:00:34.960
say negative 1.

00:00:34.960 --> 00:00:40.170
And let me see, what's a good--
let's say my expression is--

00:00:40.170 --> 00:00:42.480
I'll put it in parentheses
so it's cleaner.

00:00:42.480 --> 00:00:53.490
It's 2x plus 2 over x plus 1.

00:00:53.490 --> 00:00:56.070
So the first thing I would
always try to do is just say

00:00:56.070 --> 00:00:58.930
what happens if I just stick x
straight into this expression?

00:00:58.930 --> 00:00:59.810
What happens?

00:00:59.810 --> 00:01:03.360
Well, what's 2x plus 2 when
x is equal to negative 1?

00:01:03.360 --> 00:01:04.790
2 times negative 1.

00:01:04.790 --> 00:01:12.090
2 times negative 1 plus 2
over negative 1 plus 1.

00:01:12.090 --> 00:01:17.030
Well, the numerator is negative
2 plus 2-- that equals 0--

00:01:17.030 --> 00:01:18.540
over-- what's the denominator?

00:01:18.540 --> 00:01:19.930
Negative 1 plus 1.

00:01:19.930 --> 00:01:21.150
Over 0.

00:01:21.150 --> 00:01:23.510
And do we know
what 0 over 0 is?

00:01:23.510 --> 00:01:24.020
Well, no.

00:01:24.020 --> 00:01:26.250
It's undefined, right?

00:01:26.250 --> 00:01:29.320
So here's a case, just like
what we saw in that first

00:01:29.320 --> 00:01:33.940
video, where the limit actually
can't equal what the expression

00:01:33.940 --> 00:01:36.770
equals when you substitute x
for the number you're trying to

00:01:36.770 --> 00:01:40.560
find the limit of because you
get an undefined answer.

00:01:40.560 --> 00:01:42.750
So let's see if, using the
limit, we can come up

00:01:42.750 --> 00:01:45.080
with a better answer for
what it's approaching.

00:01:45.080 --> 00:01:47.120
Well, since we're just starting
with these limit problems,

00:01:47.120 --> 00:01:48.580
let me draw a graph.

00:01:48.580 --> 00:01:50.700
And I think this is going
to give you the intuition

00:01:50.700 --> 00:01:52.090
for what we're doing.

00:01:52.090 --> 00:01:53.560
It'll probably give
you the answer.

00:01:53.560 --> 00:01:57.180
But then I'll show you how
to solve this analytical.

00:01:57.180 --> 00:02:06.120
So if I draw a graph,
these are the axes.

00:02:06.120 --> 00:02:09.710
Actually, I'll do the
graphical and the analytical

00:02:09.710 --> 00:02:11.130
at the same time.

00:02:11.130 --> 00:02:13.450
So I want to rewrite this
expression in a way that

00:02:13.450 --> 00:02:15.020
maybe I can simplify it.

00:02:15.020 --> 00:02:16.840
So 2x plus 2.

00:02:16.840 --> 00:02:22.805
Isn't that the same thing
as 2 times x plus 1?

00:02:25.390 --> 00:02:27.960
2 times x plus 1, right?

00:02:27.960 --> 00:02:31.250
2x plus 2 is the same thing as
2 times x plus one, and then

00:02:31.250 --> 00:02:39.190
all of that is over x plus 1.

00:02:39.190 --> 00:02:44.440
So as long as this expression
and this expression don't equal

00:02:44.440 --> 00:02:48.040
0, it actually turns out that
this function-- let's say

00:02:48.040 --> 00:02:51.260
this is f of x, right?

00:02:51.260 --> 00:02:52.940
This function.

00:02:52.940 --> 00:02:56.660
Well, for every value other
than x is equal to negative

00:02:56.660 --> 00:02:59.770
1, you could actually
cancel this and this out.

00:02:59.770 --> 00:03:06.360
And so really, we see that f of
x is equal to-- I need to find

00:03:06.360 --> 00:03:15.750
a better tool-- f of x is equal
to 2 when x does not

00:03:15.750 --> 00:03:18.010
equal negative 1.

00:03:18.010 --> 00:03:22.510
And we saw when x is equal to
negative 1, it's undefined.

00:03:22.510 --> 00:03:32.050
So undefined when
equals negative 1.

00:03:32.050 --> 00:03:33.470
So how would we graph that?

00:03:33.470 --> 00:03:36.090
We showed that f of x is equal
to 2 when x does not equal

00:03:36.090 --> 00:03:38.240
negative 1 and f of x
is undefined when x

00:03:38.240 --> 00:03:39.630
equals negative 1.

00:03:39.630 --> 00:03:43.240
And once again, all I did is
kind of rewrite this exact

00:03:43.240 --> 00:03:44.520
same function, right?

00:03:44.520 --> 00:03:47.380
I showed that I could simplify
and I could divide the

00:03:47.380 --> 00:03:50.930
numerator and denominator by x
plus 1 as long as x does not

00:03:50.930 --> 00:03:53.200
equal negative 1, and that
otherwise, it's undefined.

00:03:53.200 --> 00:03:54.310
So let me graph this.

00:03:54.310 --> 00:03:56.200
I'm going to get a
different color.

00:03:56.200 --> 00:03:58.890
Maybe I'll go with red.

00:03:58.890 --> 00:04:00.280
So this is 2.

00:04:00.280 --> 00:04:03.660
So we see that x is-- and let
me say this is negative 1.

00:04:06.190 --> 00:04:10.080
So for every other value other
than negative 1, the value of

00:04:10.080 --> 00:04:13.910
this, of f of x, is equal to 2.

00:04:13.910 --> 00:04:22.170
This is 1, this is 2,
this is 3, and so on.

00:04:22.170 --> 00:04:24.950
At negative 1, the
graph is undefined.

00:04:24.950 --> 00:04:26.160
So there's a hole there.

00:04:26.160 --> 00:04:29.940
And then we keep going
to the left-hand side.

00:04:29.940 --> 00:04:33.480
So if we're going to do the
limit, we can just visually

00:04:33.480 --> 00:04:37.560
say, well, as x-- let me
do another color now.

00:04:37.560 --> 00:04:44.400
As x comes from the left-hand
side, what does f of x equal?

00:04:44.400 --> 00:04:47.660
Well, f of x is 2, 2, 2, 2,
2. f of x is equal to 2

00:04:47.660 --> 00:04:51.230
until we get to exactly
negative 1, right?

00:04:51.230 --> 00:04:53.700
And similarly, when we go
from the other hand, it's

00:04:53.700 --> 00:04:54.940
the exact same thing.

00:04:54.940 --> 00:04:59.040
f of x is 2, 2, 2 until
we get to negative 1.

00:04:59.040 --> 00:05:02.310
So you'll see, and I'll make
sure you see it visually here,

00:05:02.310 --> 00:05:09.180
that the limit as approaches
negative 1 of 2x plus 2 over

00:05:09.180 --> 00:05:11.900
x plus 1, it equals 2.

00:05:11.900 --> 00:05:13.500
Let me draw a line here
so you don't get messed

00:05:13.500 --> 00:05:15.520
up with all of it.

00:05:15.520 --> 00:05:21.090
And I'm not formally, I guess,
proving here that the limit is

00:05:21.090 --> 00:05:23.460
2, but I'm showing you kind of
an analytical way, and this is

00:05:23.460 --> 00:05:26.040
actually how it tends to be
done in algebra class, is that

00:05:26.040 --> 00:05:29.730
you tend to simplify the
expression so that you say, oh,

00:05:29.730 --> 00:05:32.630
if there wasn't a hole here,
what would the f of

00:05:32.630 --> 00:05:33.760
x equal, right?

00:05:33.760 --> 00:05:35.910
And then you'd just
evaluate it at that point.

00:05:35.910 --> 00:05:39.790
I think this might give you a
little intuition, but this

00:05:39.790 --> 00:05:42.050
isn't a formal solution.

00:05:42.050 --> 00:05:46.710
But unless you're asked to,
you tend not to be asked

00:05:46.710 --> 00:05:47.800
for a formal solution.

00:05:47.800 --> 00:05:49.800
You actually just tend to ask
what the limit is, and this is

00:05:49.800 --> 00:05:50.690
the way you could solve it.

00:05:50.690 --> 00:05:54.430
And actually another way that
you could-- I mean, I often

00:05:54.430 --> 00:05:57.310
used to check my answers when I
used to do it is you could take

00:05:57.310 --> 00:06:01.620
a calculator and try in-- what
happens when: what is f

00:06:01.620 --> 00:06:08.035
of minus 1.001, right?

00:06:08.035 --> 00:06:15.730
And you can also try what is
f of negative 0.999, right?

00:06:15.730 --> 00:06:17.210
Because what you want to do is
you want to say, well, what

00:06:17.210 --> 00:06:20.010
does the function equal when I
get really close to negative 1?

00:06:20.010 --> 00:06:23.410
And then you could keep going
closer and closer to negative

00:06:23.410 --> 00:06:25.490
1 and see what the function
approaches, and in this

00:06:25.490 --> 00:06:27.750
case, you'll see that
it approaches 2.

00:06:27.750 --> 00:06:28.740
So let's do another problem.

00:06:32.240 --> 00:06:46.160
Well, let's say, what is
the limit as x approaches

00:06:46.160 --> 00:06:51.570
0 of 1 over x?

00:06:51.570 --> 00:06:55.360
I think here it might be useful
to draw this graph because

00:06:55.360 --> 00:06:58.520
it'll give you a visual reason,
a visual represent-- actually,

00:06:58.520 --> 00:06:59.730
let's do it both ways.

00:06:59.730 --> 00:07:01.970
Let's say-- let's do it the
picking-numbers method because

00:07:01.970 --> 00:07:03.580
I think that'll give you an
intuition and maybe it'll

00:07:03.580 --> 00:07:04.500
help us draw the graph.

00:07:04.500 --> 00:07:09.410
So let's say that
this is f of x.

00:07:09.410 --> 00:07:12.580
f of x-- you can tell my
presentation is very

00:07:12.580 --> 00:07:17.690
unplanned-- f of x is
equal to 1 over x.

00:07:17.690 --> 00:07:19.520
And we want to find the
limit as x approaches 0.

00:07:19.520 --> 00:07:26.870
So what is f of-- actually,
let's make a table.

00:07:26.870 --> 00:07:27.760
f of x.

00:07:36.670 --> 00:07:41.000
So clearly when x is equal
to 0, we don't know.

00:07:41.000 --> 00:07:42.310
It's undefined.

00:07:42.310 --> 00:07:43.340
1 over 0 is undefined.

00:07:51.960 --> 00:07:59.170
But what happens when
x equals minus 0.01?

00:07:59.170 --> 00:08:04.050
Well, with minus 0.01, 1 over
minus 0.01, that is equal

00:08:04.050 --> 00:08:07.750
to negative 100, right?

00:08:07.750 --> 00:08:13.070
What happens when x is equal
to minus 0.001, right?

00:08:13.070 --> 00:08:14.880
So we're getting closer
and closer to 0 from

00:08:14.880 --> 00:08:16.240
the negative direction.

00:08:16.240 --> 00:08:20.430
Well, here it equals--
make sure my pen is

00:08:20.430 --> 00:08:24.900
working, color right.

00:08:24.900 --> 00:08:28.540
Something's wrong with my tool.

00:08:28.540 --> 00:08:30.250
Now my computer's
breaking down.

00:08:30.250 --> 00:08:31.740
Let's see what's going on.

00:08:38.060 --> 00:08:40.260
I think my computer just froze.

00:08:40.260 --> 00:08:43.330
Well, I'm going to try to solve
this, and in the very next

00:08:43.330 --> 00:08:46.730
video, I'm going to continue
on with this problem.

00:08:46.730 --> 00:08:48.650
So I'll actually see you in the
next presentation while I

00:08:48.650 --> 00:08:52.770
figure out why my pen isn't
working, and then we'll

00:08:52.770 --> 00:08:53.780
continue with this problem.

00:08:53.780 --> 00:08:56.290
See you very soon.


Limit examples (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.240
OK, hopefully, my
tool is working now.

00:00:03.240 --> 00:00:07.490
But anyway, so we were saying
when x is equal to minus 0.001,

00:00:07.490 --> 00:00:09.590
so we're getting closer and
closer to 0 from the negative

00:00:09.590 --> 00:00:13.140
side, f of x is equal
to minus 1,000, right?

00:00:13.140 --> 00:00:14.750
You can just evaluate
it yourself, right?

00:00:14.750 --> 00:00:18.800
And as you see, as x approaches
0 from the negative direction,

00:00:18.800 --> 00:00:22.590
we get larger and larger-- or I
guess you could say smaller and

00:00:22.590 --> 00:00:24.110
smaller negative
numbers, right?

00:00:24.110 --> 00:00:29.690
You get-- you know, if it's
minus 0.0001, you'd get minus

00:00:29.690 --> 00:00:32.730
10,000, and then minus 100,000,
and then minus 1 million, you

00:00:32.730 --> 00:00:35.040
could imagine the closer and
closer you get to zero.

00:00:35.040 --> 00:00:38.440
Similarly, when you go from the
other direction, when you say

00:00:38.440 --> 00:00:45.650
what is-- when x is 0.01, there
you get positive 100, right?

00:00:45.650 --> 00:00:54.890
When x is point-- the thing is
frozen again-- when it's 0.001,

00:00:54.890 --> 00:00:58.420
you get positive 1,000.

00:00:58.420 --> 00:01:03.270
So as you see, as you approach
0 from the negative direction,

00:01:03.270 --> 00:01:06.210
you get larger and larger
negative values, or I guess

00:01:06.210 --> 00:01:07.420
smaller and smaller
negative values.

00:01:07.420 --> 00:01:11.440
And as you go from the positive
direction, you get larger

00:01:11.440 --> 00:01:12.190
and larger values.

00:01:12.190 --> 00:01:14.700
Let me graph this just to give
you a sense of what this graph

00:01:14.700 --> 00:01:16.190
looks like because this is
actually a good graph to know

00:01:16.190 --> 00:01:17.320
what it looks like
just generally.

00:01:20.040 --> 00:01:25.510
So let's say I have the x-axis.

00:01:25.510 --> 00:01:26.530
This is the y-axis.

00:01:29.200 --> 00:01:31.570
Change my color.

00:01:31.570 --> 00:01:37.090
So when x is a negative number,
as x gets really, really,

00:01:37.090 --> 00:01:40.320
really negative, as x is like
negative infinity, this

00:01:40.320 --> 00:01:41.890
is approaching zero, but
it's still going to be a

00:01:41.890 --> 00:01:43.330
slightly negative number.

00:01:43.330 --> 00:01:47.040
And then as we see from what we
drew, as we approach x is equal

00:01:47.040 --> 00:01:50.040
to 0, we asymptote, and we
approach negative

00:01:50.040 --> 00:01:53.090
infinity, right?

00:01:53.090 --> 00:01:56.850
And similarly, from positive
numbers, if you go out to

00:01:56.850 --> 00:01:59.210
the right really far, it
approaches 0, but it's

00:01:59.210 --> 00:02:00.980
still going to be positive.

00:02:00.980 --> 00:02:04.400
And as we gets closer and
closer to 0, it spikes up, and

00:02:04.400 --> 00:02:05.450
it goes to positive infinity.

00:02:05.450 --> 00:02:08.770
You never quite get
x is equal to 0.

00:02:08.770 --> 00:02:13.260
So in this situation, you
actually have as x approaches--

00:02:13.260 --> 00:02:16.320
so let me give you a different
notation, which you'll

00:02:16.320 --> 00:02:17.380
probably see eventually.

00:02:17.380 --> 00:02:19.800
I might actually do a separate
presentation on this.

00:02:19.800 --> 00:02:28.940
The limit as x approaches 0
from the positive direction,

00:02:28.940 --> 00:02:35.180
that's this notation
here, of 1/x, right?

00:02:35.180 --> 00:02:38.260
So this is as x approaches 0
from the positive direction,

00:02:38.260 --> 00:02:43.585
from the right-hand side, well,
this is equal to infinity.

00:02:46.550 --> 00:02:56.120
And then the limit as x-- this
pen, this pen-- the limit as x

00:02:56.120 --> 00:03:01.340
approaches 0 from the
negative side of 1/x.

00:03:01.340 --> 00:03:03.460
This notation just says
the limit as I approach

00:03:03.460 --> 00:03:04.420
from the negative side.

00:03:04.420 --> 00:03:09.120
So as I approach x equal 0 from
this direction, right, from

00:03:09.120 --> 00:03:10.560
this direction, what happens?

00:03:10.560 --> 00:03:13.530
Well, that is equal
to minus infinity.

00:03:16.550 --> 00:03:19.110
So since I'm approaching
a different value when I

00:03:19.110 --> 00:03:21.500
approach from one side
or the other, this limit

00:03:21.500 --> 00:03:23.130
is actually undefined.

00:03:23.130 --> 00:03:25.625
I mean, we could say that from
the positive side, it's

00:03:25.625 --> 00:03:27.660
positive infinity, or from the
negative side, it's negative

00:03:27.660 --> 00:03:30.310
infinity, but they have to
equal the same thing for

00:03:30.310 --> 00:03:31.800
this limit to be defined.

00:03:31.800 --> 00:03:34.390
So this is equal to undefined.

00:03:39.740 --> 00:03:43.590
So let's do another problem,
and I think this should

00:03:43.590 --> 00:03:44.810
be interesting now.

00:03:44.810 --> 00:03:48.350
So let's say, just keeping that
last problem we had in mind,

00:03:48.350 --> 00:04:03.160
what's the limit as x
approaches 0 of 1/x squared?

00:04:03.160 --> 00:04:06.270
So in this situation,
I'll draw the graph.

00:04:09.080 --> 00:04:12.620
That's my x-axis.

00:04:12.620 --> 00:04:14.140
That's my y-axis.

00:04:14.140 --> 00:04:17.400
So here, no matter what value
we put into x, we get a

00:04:17.400 --> 00:04:18.560
positive value, right?

00:04:18.560 --> 00:04:19.370
Because you're going
to square it.

00:04:19.370 --> 00:04:25.360
If you put minus-- you could
actually-- oh, let me do it.

00:04:25.360 --> 00:04:28.830
It'll be instructive, I think.

00:04:28.830 --> 00:04:30.940
Once again, obviously you
can't just put x equal to 0.

00:04:30.940 --> 00:04:33.330
You'll get 1/0,
which is undefined.

00:04:33.330 --> 00:04:35.220
But let's say 1 over x squared.

00:04:35.220 --> 00:04:37.400
What does 1 over x
squared evaluate to?

00:04:37.400 --> 00:04:46.250
So when x is 0.1, 0.1 squared
is 0.01, so 1/x is 100.

00:04:46.250 --> 00:04:52.600
Similarly, if I do minus 0.1,
minus 0.1 squared is positive

00:04:52.600 --> 00:04:56.440
0.01, so then 1 over that
is still 100, right?

00:04:56.440 --> 00:04:58.630
So regardless of whether we put
a negative or positive number

00:04:58.630 --> 00:05:01.380
here, we get a positive value.

00:05:01.380 --> 00:05:07.070
And similarly, if I put-- if we
say x is 0.01, if you evaluate

00:05:07.070 --> 00:05:14.540
it, you'll get 10,000, and if
we put minus 0.01, you'll get

00:05:14.540 --> 00:05:15.920
positive 10,000 as well, right?

00:05:15.920 --> 00:05:17.420
Because we square it.

00:05:17.420 --> 00:05:19.240
So in this graph, if you were
to draw it, and if you have a

00:05:19.240 --> 00:05:22.060
graphing calculator, you
should experiment, it

00:05:22.060 --> 00:05:24.720
looks something like this.

00:05:24.720 --> 00:05:26.480
I can see this dark blue.

00:05:26.480 --> 00:05:29.680
So from the negative side, it
approaches infinity, right?

00:05:29.680 --> 00:05:30.180
You can see that.

00:05:30.180 --> 00:05:33.000
As we get to smaller and
smaller-- as we get closer and

00:05:33.000 --> 00:05:35.860
closer to 0 from the negative
side, it approaches infinity.

00:05:35.860 --> 00:05:43.280
As we go from the positive
side-- these are actually

00:05:43.280 --> 00:05:45.050
symmetric, although I didn't
draw it that symmetric-- it

00:05:45.050 --> 00:05:46.390
also approaches infinity.

00:05:46.390 --> 00:05:51.780
So this is a case in which
the limit-- oh, that's

00:05:51.780 --> 00:05:52.380
not too bright.

00:05:52.380 --> 00:05:58.720
I don't know if you can see --
the limit as x approaches 0

00:05:58.720 --> 00:06:03.850
from the negative side of 1
over x squared is equal to

00:06:03.850 --> 00:06:11.240
infinity, and the limit as x
approaches 0 from the positive

00:06:11.240 --> 00:06:16.070
side of 1 over x squared is
also equal to infinity.

00:06:16.070 --> 00:06:18.550
So when you go from the
left-hand side, it

00:06:18.550 --> 00:06:19.560
equals infinity, right?

00:06:19.560 --> 00:06:21.860
It goes to infinity
as you approach 0.

00:06:21.860 --> 00:06:23.360
And as you go from the
right-hand side, it

00:06:23.360 --> 00:06:26.030
also goes to infinity.

00:06:26.030 --> 00:06:29.800
And so the limit in general
is equal to infinity.

00:06:29.800 --> 00:06:34.520
And this is why I got excited
when I first started

00:06:34.520 --> 00:06:35.100
learning limits.

00:06:35.100 --> 00:06:38.770
Because for the first time,
infinity is a legitimate answer

00:06:38.770 --> 00:06:41.950
to your problem, which, I don't
know, on some metaphysical

00:06:41.950 --> 00:06:43.660
level got me kind of excited.

00:06:43.660 --> 00:06:47.750
But anyway, I will do more
problems in the next

00:06:47.750 --> 00:06:49.950
presentation because you can
never do enough limit problems.

00:06:49.950 --> 00:06:51.610
And in a couple of
presentations, I actually give

00:06:51.610 --> 00:06:54.670
you the formal, kind of
rigorous mathematical

00:06:54.670 --> 00:06:56.890
definition of the limits.


Introduction to limits 2

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.610
Welcome to the
presentation on limits.

00:00:03.610 --> 00:00:06.290
Let's get started with some--
well, first an explanation

00:00:06.290 --> 00:00:07.530
before I do any problems.

00:00:07.530 --> 00:00:11.110
So let's say I had-- let me
make sure I have the right

00:00:11.110 --> 00:00:12.070
color and my pen works.

00:00:12.070 --> 00:00:17.480
OK, let's say I had the limit,
and I'll explain what a

00:00:17.480 --> 00:00:18.840
limit is in a second.

00:00:18.840 --> 00:00:22.780
But the way you write it is you
say the limit-- oh, my color is

00:00:22.780 --> 00:00:26.860
on the wrong-- OK, let me
use the pen and yellow.

00:00:26.860 --> 00:00:38.181
OK, the limit as x
approaches 2 of x squared.

00:00:42.550 --> 00:00:46.340
Now, all this is saying is what
value does the expression x

00:00:46.340 --> 00:00:49.680
squared approach as
x approaches 2?

00:00:49.680 --> 00:00:51.130
Well, this is pretty easy.

00:00:51.130 --> 00:00:53.065
If we look at-- let me
at least draw a graph.

00:00:53.065 --> 00:00:57.010
I'll stay in this yellow color.

00:00:57.010 --> 00:00:59.420
So let me draw.

00:00:59.420 --> 00:01:03.813
x squared looks something
like-- let me use

00:01:03.813 --> 00:01:05.610
a different color.

00:01:05.610 --> 00:01:07.900
x square looks something
like this, right?

00:01:10.730 --> 00:01:20.040
And when x is equal to 2, y,
or the expression-- because

00:01:20.040 --> 00:01:21.320
we don't say what
this is equal to.

00:01:21.320 --> 00:01:23.570
It's just the expression-- x
squared is equal to 4, right?

00:01:27.650 --> 00:01:33.100
So a limit is saying, as x
approaches 2, as x approaches 2

00:01:33.100 --> 00:01:37.920
from both sides, from numbers
left than 2 and from numbers

00:01:37.920 --> 00:01:41.940
right than 2, what does
the expression approach?

00:01:41.940 --> 00:01:44.630
And you might, I think, already
see where this is going and be

00:01:44.630 --> 00:01:46.920
wondering why we're even going
to the trouble of learning this

00:01:46.920 --> 00:01:50.040
new concept because it seems
pretty obvious, but as x-- as

00:01:50.040 --> 00:01:53.320
we get to x closer and closer
to 2 from this direction, and

00:01:53.320 --> 00:01:55.530
as we get to x closer
and closer to 2 to this

00:01:55.530 --> 00:01:58.940
direction, what does
this expression equal?

00:01:58.940 --> 00:02:01.670
Well, it essentially
equals 4, right?

00:02:01.670 --> 00:02:04.690
The expression is equal to 4.

00:02:04.690 --> 00:02:08.090
The way I think about it is as
you move on the curve closer

00:02:08.090 --> 00:02:12.390
and closer to the expression's
value, what does the

00:02:12.390 --> 00:02:13.500
expression equal?

00:02:13.500 --> 00:02:14.660
In this case, it equals 4.

00:02:14.660 --> 00:02:17.850
You're probably saying, Sal,
this seems like a useless

00:02:17.850 --> 00:02:20.690
concept because I could have
just stuck 2 in there, and I

00:02:20.690 --> 00:02:23.850
know that if this is-- say this
is f of x, that if f of x is

00:02:23.850 --> 00:02:27.590
equal to x squared, that f of 2
is equal to 4, and that would

00:02:27.590 --> 00:02:29.050
have been a no-brainer.

00:02:29.050 --> 00:02:33.840
Well, let me maybe give you one
wrinkle on that, and hopefully

00:02:33.840 --> 00:02:36.970
now you'll start to see what
the use of a limit is.

00:02:36.970 --> 00:02:51.620
Let me to define-- let me say
f of x is equal to x squared

00:02:51.620 --> 00:02:59.390
when, if x does not equal 2,
and let's say it equals

00:02:59.390 --> 00:03:04.270
3 when x equals 2.

00:03:04.270 --> 00:03:04.560
Interesting.

00:03:04.560 --> 00:03:08.600
So it's a slight variation on
this expression right here.

00:03:08.600 --> 00:03:09.490
So this is our new f of x.

00:03:09.490 --> 00:03:10.670
So let me ask you a question.

00:03:10.670 --> 00:03:16.300
What is-- my pen still works--
what is the limit-- I used

00:03:16.300 --> 00:03:23.180
cursive this time-- what is the
limit as x-- that's an x--

00:03:23.180 --> 00:03:25.580
as x approaches 2 of f of x?

00:03:29.590 --> 00:03:30.210
That's an x.

00:03:30.210 --> 00:03:31.120
It says x approaches 2.

00:03:31.120 --> 00:03:31.800
It's just like that.

00:03:31.800 --> 00:03:33.000
I just-- I don't know.

00:03:33.000 --> 00:03:35.460
For some reason, my brain
is working functionally.

00:03:35.460 --> 00:03:39.250
OK, so let me graph this now.

00:03:39.250 --> 00:03:43.360
So that's an equally
neat-looking graph as

00:03:43.360 --> 00:03:46.100
the one I just drew.

00:03:46.100 --> 00:03:46.790
Let me draw.

00:03:46.790 --> 00:03:50.100
So now it's almost the same as
this curve, except something

00:03:50.100 --> 00:03:52.330
interesting happens
at x equals 2.

00:03:52.330 --> 00:03:53.715
So it's just like this.

00:03:53.715 --> 00:03:58.480
It's like an x squared
curve like that.

00:03:58.480 --> 00:04:02.710
But at x equals 2 and
f of x equals 4, we

00:04:02.710 --> 00:04:04.890
draw a little hole.

00:04:04.890 --> 00:04:08.050
We draw a hole because it's
not defined at x equals 2.

00:04:08.050 --> 00:04:10.320
This is x equals 2.

00:04:10.320 --> 00:04:11.960
This is 2.

00:04:11.960 --> 00:04:12.680
This is 4.

00:04:12.680 --> 00:04:14.690
This is the f of x
axis, of course.

00:04:14.690 --> 00:04:19.820
And when x is equal to
2-- let's say this is 3.

00:04:19.820 --> 00:04:23.860
When x is equal to 2,
f of x is equal to 3.

00:04:23.860 --> 00:04:25.240
This is actually
right below this.

00:04:25.240 --> 00:04:27.610
I should-- it doesn't look
completely right below it,

00:04:27.610 --> 00:04:29.730
but I think you got
to get the picture.

00:04:29.730 --> 00:04:31.400
See, this graph is x squared.

00:04:31.400 --> 00:04:35.983
It's exactly x squared until
we get to x equals 2.

00:04:35.983 --> 00:04:40.030
At x equals 2, We have a
grap-- No, not a grap.

00:04:40.030 --> 00:04:42.770
We have a gap in the
graph, which maybe

00:04:42.770 --> 00:04:44.720
could be called a grap.

00:04:44.720 --> 00:04:49.540
We have a gap in the graph, and
then we keep-- and then after x

00:04:49.540 --> 00:04:51.350
equals 2, we keep moving on.

00:04:51.350 --> 00:04:54.300
And that gap, and that gap
is defined right here, what

00:04:54.300 --> 00:04:55.350
happens when x equals 2?

00:04:55.350 --> 00:04:57.300
Well, then f of x
is equal to 3.

00:04:57.300 --> 00:05:02.960
So this graph kind of goes--
it's just like x squared, but

00:05:02.960 --> 00:05:08.470
instead of f of 2 being 4, f
of 2 drops down to 3, but

00:05:08.470 --> 00:05:10.260
then we keep on going.

00:05:10.260 --> 00:05:12.400
So going back to the limit
problem, what is the

00:05:12.400 --> 00:05:14.880
limit as x approaches 2?

00:05:14.880 --> 00:05:17.050
Now, well, let's think
about the same thing.

00:05:17.050 --> 00:05:19.200
We're going to go-- this
is how I visualize it.

00:05:19.200 --> 00:05:21.000
I go along the curve.

00:05:21.000 --> 00:05:23.130
Let me pick a different color.

00:05:23.130 --> 00:05:28.050
So as x approaches 2 from this
side, from the left-hand side

00:05:28.050 --> 00:05:35.200
or from numbers less than 2, f
of x is approaching values

00:05:35.200 --> 00:05:40.520
approaching 4, right? f of x
is approaching 4 as x

00:05:40.520 --> 00:05:41.540
approaches 2, right?

00:05:41.540 --> 00:05:42.200
I think you see that.

00:05:42.200 --> 00:05:46.820
If you just follow along the
curve, as you approach f of 2,

00:05:46.820 --> 00:05:48.420
you get closer and closer to 4.

00:05:48.420 --> 00:05:53.010
Similarly, as you go from the
right-hand side-- make sure

00:05:53.010 --> 00:05:54.370
my thing's still working.

00:05:54.370 --> 00:05:57.780
As you go from the right-hand
side, you go along the

00:05:57.780 --> 00:06:05.200
curve, and f of x is also
slowly approaching 4.

00:06:05.200 --> 00:06:07.120
So, as you can see, as we
go closer and closer and

00:06:07.120 --> 00:06:11.610
closer to x equals 2, f of
whatever number that is

00:06:11.610 --> 00:06:13.540
approaches 4, right?

00:06:13.540 --> 00:06:15.860
So, in this case, the
limit as x approaches

00:06:15.860 --> 00:06:21.130
2 is also equal to 4.

00:06:21.130 --> 00:06:24.120
Well, this is interesting
because, in this case, the

00:06:24.120 --> 00:06:35.980
limit as x approaches 2 of f
of x does not equal f of 2.

00:06:35.980 --> 00:06:38.170
Now, normally, this
would be on this line.

00:06:38.170 --> 00:06:41.300
In this case, the limit as you
approach the expression is

00:06:41.300 --> 00:06:44.030
equal to evaluating the
expression of that value.

00:06:44.030 --> 00:06:46.450
In this case, the limit isn't.

00:06:46.450 --> 00:06:49.220
I think now you're starting to
see why the limit is a slightly

00:06:49.220 --> 00:06:51.260
different concept than just
evaluating the function at

00:06:51.260 --> 00:06:53.770
that point because you have
functions where, for whatever

00:06:53.770 --> 00:06:57.530
reason at a certain point,
either the function might not

00:06:57.530 --> 00:07:01.640
be defined or the function kind
of jumps up or down, but as you

00:07:01.640 --> 00:07:05.370
approach that point, you still
approach a value different than

00:07:05.370 --> 00:07:06.700
the function at that point.

00:07:06.700 --> 00:07:08.110
Now, that's my introduction.

00:07:08.110 --> 00:07:12.020
I think this will give you
intuition for what a limit is.

00:07:12.020 --> 00:07:14.110
In another presentation, I'll
give you the more formal

00:07:14.110 --> 00:07:16.460
mathematical, you know,
the delta-epsilon

00:07:16.460 --> 00:07:17.890
definition of a limit.

00:07:17.890 --> 00:07:20.190
And actually, in the very next
module, I'm now going to

00:07:20.190 --> 00:07:23.210
do a bunch of problems
involving the limit.

00:07:23.210 --> 00:07:25.510
I think as you do more and more
problems, you'll get more and

00:07:25.510 --> 00:07:28.380
more of an intuition as
to what a limit is.

00:07:28.380 --> 00:07:30.490
And then as we go into drill
derivatives and integrals,

00:07:30.490 --> 00:07:33.350
you'll actually understand why
people probably even invented

00:07:33.350 --> 00:07:34.910
limits to begin with.

00:07:34.910 --> 00:07:36.930
We'll see you in the
next presentation.


Domain of a function

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.070
Welcome to my presentation
on domain of a function.

00:00:04.070 --> 00:00:05.060
So what's is the domain?

00:00:05.060 --> 00:00:07.720
The domain of a function,
you'll often hear it combined

00:00:07.720 --> 00:00:09.090
with domain and range.

00:00:09.090 --> 00:00:12.830
But the domain of a function is
just what values can I put into

00:00:12.830 --> 00:00:15.150
a function and get
a valid output.

00:00:15.150 --> 00:00:16.360
So let's start with
something examples.

00:00:16.360 --> 00:00:32.880
Let's say I had f of x is equal
to, let's say, x squared.

00:00:35.410 --> 00:00:36.270
So let me ask you a question.

00:00:36.270 --> 00:00:39.440
What values of x can I put
in here so I get a valid

00:00:39.440 --> 00:00:41.460
answer for x squared?

00:00:41.460 --> 00:00:44.820
Well, I can really put anything
in here, any real number.

00:00:44.820 --> 00:00:53.980
So here I'll say that the
domain is the set of x's

00:00:53.980 --> 00:00:58.620
such that x is a member
of the real numbers.

00:00:58.620 --> 00:01:02.020
So this is just a fancy way of
saying that OK, this r with

00:01:02.020 --> 00:01:04.370
this kind of double backbone
here, that just means real

00:01:04.370 --> 00:01:06.210
numbers, and I think you're
familiar with real numbers now.

00:01:06.210 --> 00:01:10.420
That's pretty much every number
outside of the complex numbers.

00:01:10.420 --> 00:01:11.640
And if you don't know
what complex numbers

00:01:11.640 --> 00:01:12.910
are, that's fine.

00:01:12.910 --> 00:01:14.980
You probably won't need
to know it right now.

00:01:14.980 --> 00:01:18.210
The real numbers are every
number that most people are

00:01:18.210 --> 00:01:20.290
familiar with, including
irrational numbers, including

00:01:20.290 --> 00:01:23.180
transcendental numbers,
including fractions -- every

00:01:23.180 --> 00:01:24.470
number is a real number.

00:01:24.470 --> 00:01:27.820
So the domain here is x --
x just has to be a member

00:01:27.820 --> 00:01:28.730
of the real numbers.

00:01:28.730 --> 00:01:31.840
And this little backwards
looking e or something, this

00:01:31.840 --> 00:01:35.150
just means x is a member
of the real numbers.

00:01:35.150 --> 00:01:37.200
So let's do another one
in a slight variation.

00:01:42.445 --> 00:01:51.310
So let's say I had f of x is
equal to 1 over x squared.

00:01:51.310 --> 00:01:52.620
So is this same thing now?

00:01:52.620 --> 00:01:54.890
Can I still put any x
value in here and get

00:01:54.890 --> 00:01:56.970
a reasonable answer?

00:01:56.970 --> 00:01:57.950
Well what's f of 0?

00:02:08.370 --> 00:02:12.880
f of zero is equal to 1 over 0.

00:02:12.880 --> 00:02:14.870
And what's 1 over 0?

00:02:14.870 --> 00:02:17.860
I don't know what it is,
so this is undefined.

00:02:22.470 --> 00:02:25.780
No one ever took the trouble to
define what 1 over 0 should be.

00:02:25.780 --> 00:02:27.810
And they probably didn't do, so
some people probably thought

00:02:27.810 --> 00:02:31.170
about what should be, but they
probably couldn't find out with

00:02:31.170 --> 00:02:33.450
a good definition for 1 over
0 that's consistent with

00:02:33.450 --> 00:02:34.250
the rest of mathematics.

00:02:34.250 --> 00:02:35.710
So 1 over 0 stays undefined.

00:02:35.710 --> 00:02:37.930
So f of 0 is undefined.

00:02:37.930 --> 00:02:42.700
So we can't put 0 in and get
a valid answer for f of 0.

00:02:42.700 --> 00:02:50.630
So here we say the domain is
equal to -- do little brackets,

00:02:50.630 --> 00:02:52.900
that shows kind of the
set of what x's apply.

00:02:52.900 --> 00:02:54.303
That's those little curly
brackets, I didn't

00:02:54.303 --> 00:02:56.400
draw it that well.

00:02:56.400 --> 00:03:02.030
x is a member of the real
numbers still, such that

00:03:02.030 --> 00:03:05.560
x does not equal 0.

00:03:05.560 --> 00:03:10.230
So here I just made a slight
variation on what I had before.

00:03:10.230 --> 00:03:12.925
Before we said when f of x is
equal to x squared that x

00:03:12.925 --> 00:03:15.050
is just any real number.

00:03:15.050 --> 00:03:20.250
Now we're saying that x is any
real number except for 0.

00:03:20.250 --> 00:03:22.400
This is just a fancy way of
saying it, and then these curly

00:03:22.400 --> 00:03:24.290
brackets just mean a set.

00:03:24.290 --> 00:03:26.390
Let's do a couple more ones.

00:03:26.390 --> 00:03:44.360
Let's say f of x is equal to
the square root of x minus 3.

00:03:44.360 --> 00:03:48.170
So up here we said, well this
function isn't defined when we

00:03:48.170 --> 00:03:49.680
get a 0 in the denominator.

00:03:49.680 --> 00:03:52.090
But what's interesting
about this function?

00:03:52.090 --> 00:03:55.600
Can we take a square root
of a negative number?

00:03:55.600 --> 00:03:58.000
Well until we learn about
imaginary and complex

00:03:58.000 --> 00:03:59.030
numbers we can't.

00:03:59.030 --> 00:04:03.280
So here we say well, any x is
valid here except for the x's

00:04:03.280 --> 00:04:07.220
that make this expression under
the radical sign negative.

00:04:07.220 --> 00:04:11.510
So we have to say that x minus
3 has to be greater than or

00:04:11.510 --> 00:04:13.690
equal to 0, right, because you
could have the square to 0,

00:04:13.690 --> 00:04:15.235
that's fine, it's just 0.

00:04:15.235 --> 00:04:20.070
So x minus 3 has to be greater
than or equal to 0, so x has to

00:04:20.070 --> 00:04:22.700
be greater than or equal to 3.

00:04:22.700 --> 00:04:35.130
So here our domain is x is a
member of the real numbers,

00:04:35.130 --> 00:04:41.165
such that x is greater
than or equal to 3.

00:04:45.840 --> 00:04:50.140
Let's do a slightly
more difficult one.

00:04:50.140 --> 00:05:01.260
What if I said f of x is equal
to the square root of the

00:05:01.260 --> 00:05:06.200
absolute value of x minus 3.

00:05:06.200 --> 00:05:08.320
So now it's getting a little
bit more complicated.

00:05:08.320 --> 00:05:11.020
Well just like this time
around, this expression of

00:05:11.020 --> 00:05:13.180
the radical still has to be
greater than or equal to 0.

00:05:13.180 --> 00:05:19.400
So you can just say that the
absolute value of x minus 3 is

00:05:19.400 --> 00:05:21.910
greater than or equal to 0.

00:05:21.910 --> 00:05:25.580
So we have the absolute value
of x has to be greater

00:05:25.580 --> 00:05:27.830
than or equal to 3.

00:05:27.830 --> 00:05:30.480
And if order for the absolute
value of something to be

00:05:30.480 --> 00:05:35.440
greater than or equal to
something, then that means that

00:05:35.440 --> 00:05:46.290
x has to be less than or equal
to negative 3, or x has to be

00:05:46.290 --> 00:05:48.530
greater than or equal to 3.

00:05:48.530 --> 00:05:52.460
It makes sense because x
can't be negative 2, right?

00:05:52.460 --> 00:05:55.240
Because negative 2 has an
absolute value less than 3.

00:05:55.240 --> 00:05:57.700
So x has to be less
than negative 3.

00:05:57.700 --> 00:06:00.930
It has to be further in the
negative direction than

00:06:00.930 --> 00:06:03.220
negative 3, or it has to be
further in the positive

00:06:03.220 --> 00:06:04.790
direction than positive 3.

00:06:04.790 --> 00:06:10.490
So, once again, x has to be
less than negative 3 or x

00:06:10.490 --> 00:06:13.000
has to be greater than 3,
so we have our domain.

00:06:13.000 --> 00:06:15.920
So we have it as x is
a member of the reals

00:06:15.920 --> 00:06:19.960
-- I always forget.

00:06:19.960 --> 00:06:21.460
Is that the line?

00:06:21.460 --> 00:06:23.080
I forget, it's either
a colon or a line.

00:06:23.080 --> 00:06:25.510
I'm rusty, it's been
years since I've done

00:06:25.510 --> 00:06:26.400
this kind of stuff.

00:06:26.400 --> 00:06:29.050
But anyway, I think
you get the point.

00:06:29.050 --> 00:06:32.800
It could be any real number
here, as long as x is less

00:06:32.800 --> 00:06:37.650
than negative 3, less than or
equal to negative 3, or x is

00:06:37.650 --> 00:06:39.980
greater than or equal to 3.

00:06:39.980 --> 00:06:41.650
Let me ask a question now.

00:06:41.650 --> 00:06:50.970
What if instead of this it was
-- that was the denominator,

00:06:50.970 --> 00:06:53.270
this is all a separate
problem up here.

00:06:53.270 --> 00:06:56.500
So now we have 1 over the
square root of the absolute

00:06:56.500 --> 00:06:59.160
value of x minus 3.

00:06:59.160 --> 00:07:00.730
So now how does this
change the situation?

00:07:00.730 --> 00:07:03.650
So not only does this
expression in the denominator,

00:07:03.650 --> 00:07:06.170
not only does this have to be
greater than or equal to

00:07:06.170 --> 00:07:08.360
0, can it be 0 anymore?

00:07:08.360 --> 00:07:10.960
Well no, because then you would
get the square root of 0, which

00:07:10.960 --> 00:07:13.560
is 0 and you would get a
0 in the denominator.

00:07:13.560 --> 00:07:15.310
So it's kind of like
this problem plus this

00:07:15.310 --> 00:07:16.600
problem combined.

00:07:16.600 --> 00:07:20.210
So now when you have 1 over the
square root of the absolute

00:07:20.210 --> 00:07:25.430
value of x minus 3, now it's no
longer greater than or equal to

00:07:25.430 --> 00:07:28.690
0, it's just a greater
than 0, right?

00:07:28.690 --> 00:07:30.420
it's just greater than 0.

00:07:30.420 --> 00:07:32.040
Because we can't have a 0
here in the denominator.

00:07:32.040 --> 00:07:37.080
So if it's greater than 0, then
we just say greater than 3.

00:07:37.080 --> 00:07:40.510
And essentially just get rid of
the equal signs right here.

00:07:40.510 --> 00:07:41.790
Let me erase it properly.

00:07:44.310 --> 00:07:45.750
It's a slightly different
color, but maybe

00:07:45.750 --> 00:07:47.640
you won't notice.

00:07:47.640 --> 00:07:50.130
So there you go.

00:07:50.130 --> 00:07:52.450
Actually, we should do another
example since we have time.

00:07:58.470 --> 00:08:01.180
Let me erase this.

00:08:01.180 --> 00:08:01.810
OK.

00:08:01.810 --> 00:08:35.140
Now let's say that f of x is
equal to 2, if x is even,

00:08:35.140 --> 00:08:53.970
and 1 over x minus 2 times
x minus 1, if x is odd.

00:08:53.970 --> 00:08:55.970
So what's the domain here?

00:08:55.970 --> 00:08:57.620
What is a valid x I
can put in here.

00:08:57.620 --> 00:08:59.820
So immediately we
have two clauses.

00:08:59.820 --> 00:09:13.270
If x is even we use this
clause, so f of 4 -- well,

00:09:13.270 --> 00:09:16.030
that's just equal to 2 because
we used this clause here.

00:09:16.030 --> 00:09:19.300
But this clause applies
when x is odd.

00:09:19.300 --> 00:09:21.370
Just like we did in the last
example, what are the

00:09:21.370 --> 00:09:24.270
situations where this
kind of breaks down?

00:09:24.270 --> 00:09:26.200
Well, when the
denominator is 0.

00:09:26.200 --> 00:09:29.850
Well the denominator is 0
when x is equal to 2, or

00:09:29.850 --> 00:09:32.870
x is equal to 1, right?

00:09:32.870 --> 00:09:34.970
But this clause only
applies when x is odd.

00:09:34.970 --> 00:09:37.350
So x is equal to 2 won't
apply to this clause.

00:09:37.350 --> 00:09:41.370
So only x is equal to 1
would apply to this clause.

00:09:41.370 --> 00:09:49.010
So the domain is x is a member
of the reals, such that

00:09:49.010 --> 00:09:52.320
x does not equal 1.

00:09:52.320 --> 00:09:55.160
I think that's all the
time I have for now.

00:09:55.160 --> 00:09:58.210
Have fun practicing
these domain problems.