Basic derivative rules (Part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.761 --> 00:00:03.485
- [Voiceover] So these are
both ways that you will see

00:00:03.485 --> 00:00:06.864
limit-based definitions of derivatives.

00:00:06.864 --> 00:00:09.567
This is usually if you're
thinking about the derivative

00:00:09.567 --> 00:00:11.263
at a point, here if you're thinking about

00:00:11.263 --> 00:00:16.004
the derivative in general,
but these are both equivalent.

00:00:16.004 --> 00:00:19.030
They're both based on the
slope of a tangent line,

00:00:19.030 --> 00:00:22.289
or the instantaneous rate of change,

00:00:22.289 --> 00:00:25.356
and using these, I wanna establish some of

00:00:25.356 --> 00:00:28.429
the core properties of derivatives for us.

00:00:28.429 --> 00:00:30.973
And the first one that
I'm going to do will seem

00:00:30.973 --> 00:00:34.545
like common sense, or maybe
it will once we talk about it

00:00:34.545 --> 00:00:38.752
a little bit, so if f of x,
if our function is equal to

00:00:38.752 --> 00:00:42.773
a constant value, well then, f prime of x

00:00:42.773 --> 00:00:45.190
is going to be equal to zero.

00:00:46.655 --> 00:00:49.279
Now why does that make intuitive sense?

00:00:49.279 --> 00:00:53.317
Well, we could graph
it, we could graph it,

00:00:53.317 --> 00:00:56.734
so if that's my y-axis, that's my x-axis.

00:00:58.589 --> 00:01:00.265
If I wanted to graph y equals f of x,

00:01:00.265 --> 00:01:02.944
it's gonna look like that,
where this is at the value

00:01:02.944 --> 00:01:07.027
y is equal to k, so this
is y is equal to f of x.

00:01:07.975 --> 00:01:11.760
Notice no matter what you
change x, y does not change.

00:01:11.760 --> 00:01:14.151
The slope of the tangent
line here, well frankly,

00:01:14.151 --> 00:01:17.362
is the same line, it has a slope of zero.

00:01:17.362 --> 00:01:20.086
No matter how, y is
just not changing here,

00:01:20.086 --> 00:01:21.918
and we could use either
of these definitions

00:01:21.918 --> 00:01:24.682
to establish that even
further, establish it using

00:01:24.682 --> 00:01:28.310
these limit definitions, so let's see,

00:01:28.310 --> 00:01:32.477
the limit as h approaches
zero of f of x plus h,

00:01:35.330 --> 00:01:37.452
well no matter what we
input into our function,

00:01:37.452 --> 00:01:41.804
we get k, so f of x plus
h would be k minus f of x.

00:01:41.804 --> 00:01:43.686
Well, no matter what we
put into that function,

00:01:43.686 --> 00:01:48.174
we get k over h, well,
this is just going to be

00:01:48.174 --> 00:01:52.341
zero over h so this limit is
just going to be equal to zero.

00:01:53.870 --> 00:01:58.820
So f prime of x for any, for
any x, the derivative is zero,

00:01:58.820 --> 00:02:00.652
and you see that here, that
the slope of the tangent line

00:02:00.652 --> 00:02:03.034
for any x is equal to zero.

00:02:03.034 --> 00:02:04.751
So if someone walks up to
you on the street and says,

00:02:04.751 --> 00:02:08.584
"Okay, h of x, h of x,
h of x, is equal to pi,

00:02:10.552 --> 00:02:12.883
"what is h prime of x?"

00:02:12.883 --> 00:02:15.086
Well, you say, well, pi,
that's just a constant value

00:02:15.086 --> 00:02:16.966
that the value of our
function is not changing

00:02:16.966 --> 00:02:19.799
as we change our x, the slope
of the tangent line there,

00:02:19.799 --> 00:02:21.382
the instantaneous rate of change,

00:02:21.382 --> 00:02:24.049
it is going to be equal to zero.


Differentiability at a point: algebraic (function isn't differentiable)

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:02.483
- [Voiceover] Is the function
given below continuous

00:00:02.483 --> 00:00:04.840
slash differentiable at x equals one?

00:00:04.840 --> 00:00:07.362
And they define the function g piece wise

00:00:07.362 --> 00:00:08.798
right over here, and then they

00:00:08.798 --> 00:00:09.972
give us a bunch of choices.

00:00:09.972 --> 00:00:11.133
Continuous but not differentiable.

00:00:11.133 --> 00:00:12.817
Differentiable but not continuous.

00:00:12.817 --> 00:00:13.841
Both continuous and differentiable.

00:00:13.841 --> 00:00:16.662
Neither continuous nor differentiable.

00:00:16.662 --> 00:00:18.171
And, like always, pause this video

00:00:18.171 --> 00:00:19.863
and see if you could figure this out.

00:00:19.863 --> 00:00:21.072
So let's do step by step.

00:00:21.072 --> 00:00:23.433
So first let's think about continuity.

00:00:23.433 --> 00:00:26.605
So for continuity, for g to
be continuous at x equals one

00:00:26.605 --> 00:00:30.217
that means that g of
one, that means g of one

00:00:30.217 --> 00:00:34.193
must be equal to the
limit as x approaches one

00:00:34.193 --> 00:00:35.443
of g of g of x.

00:00:36.925 --> 00:00:39.313
Well g of one, what is that going to be?

00:00:39.313 --> 00:00:43.135
G of one we're going
to fall into this case.

00:00:43.135 --> 00:00:46.884
One minus one squared is going to be zero.

00:00:46.884 --> 00:00:49.057
So if we can show that the limit of g of x

00:00:49.057 --> 00:00:51.776
as x approaches one is
the same as g of one

00:00:51.776 --> 00:00:54.468
is equal to zero than we
know we're continuous there.

00:00:54.468 --> 00:00:56.625
Well let's do the left and
right handed limits here.

00:00:56.625 --> 00:00:58.446
So if we do the left handed limit, limit,

00:00:58.446 --> 00:01:00.532
and that's especially
useful 'cause we're in

00:01:00.532 --> 00:01:02.272
these different clauses
here as we approach

00:01:02.272 --> 00:01:04.254
from the left and the right hand side.

00:01:04.254 --> 00:01:06.583
So as x approaches one from

00:01:06.583 --> 00:01:09.000
the left hand side of g of x.

00:01:10.784 --> 00:01:12.961
Well we're going to be falling
into this situation here

00:01:12.961 --> 00:01:15.722
as we approach from the
left as x is less than one.

00:01:15.722 --> 00:01:19.905
So this is going to be
the same thing as that.

00:01:19.905 --> 00:01:21.986
That's what g of x is equal to when we

00:01:21.986 --> 00:01:24.932
are less than one as we're
approaching from the left.

00:01:24.932 --> 00:01:28.349
Well this thing is defined,
and it's continuous

00:01:28.349 --> 00:01:29.687
for all real numbers.

00:01:29.687 --> 00:01:31.948
So we could just substitute one in for x,

00:01:31.948 --> 00:01:34.501
and we get this is equal to zero.

00:01:34.501 --> 00:01:36.709
So so far so good, let's
do one more of these.

00:01:36.709 --> 00:01:38.897
Let's approach from the right hand side.

00:01:38.897 --> 00:01:43.756
As x approaches one from the
right hand side of g of x.

00:01:43.756 --> 00:01:45.486
Well now we're falling into this case

00:01:45.486 --> 00:01:47.958
so g of x if we're to the right of one

00:01:47.958 --> 00:01:50.188
if values are greater or equal to one

00:01:50.188 --> 00:01:53.098
it's gonna be x minus one squared.

00:01:53.098 --> 00:01:54.898
Well once again x minus one squared

00:01:54.898 --> 00:01:57.076
that is defined for all real numbers.

00:01:57.076 --> 00:01:58.343
It's continuous for all real numbers,

00:01:58.343 --> 00:02:00.513
so we could just pop that one in there.

00:02:00.513 --> 00:02:01.793
You get one minus one squared.

00:02:01.793 --> 00:02:03.636
Well that's just zero again,

00:02:03.636 --> 00:02:05.976
so the left hand limit,
the right hand limit

00:02:05.976 --> 00:02:07.450
are both equal zero, which means that the

00:02:07.450 --> 00:02:11.825
limit of g of x as x approaches
one is equal to zero.

00:02:11.825 --> 00:02:13.336
Which is the same thing as g of one,

00:02:13.336 --> 00:02:15.430
so we are good with continuity.

00:02:15.430 --> 00:02:16.814
So we can rule out all of the ones

00:02:16.814 --> 00:02:19.181
that are saying that it's not continuous.

00:02:19.181 --> 00:02:20.841
So we can rule out that one,

00:02:20.841 --> 00:02:25.242
and we can rule out that
one right over there.

00:02:25.242 --> 00:02:28.259
So now let's think about
whether it is differentiable.

00:02:28.259 --> 00:02:30.009
So differentiability.

00:02:31.209 --> 00:02:35.376
So differentiability, I'll write
differentiability, ability.

00:02:38.312 --> 00:02:39.671
Did I, let's see, that's a long word.

00:02:39.671 --> 00:02:41.921
Differentiability, alright.

00:02:43.097 --> 00:02:45.561
Differentiability, what
needs to be true here?

00:02:45.561 --> 00:02:47.773
Well we have to have a defined limit

00:02:47.773 --> 00:02:50.523
as x approaches one for f of x

00:02:51.361 --> 00:02:53.028
minus f of one over,

00:02:54.613 --> 00:02:57.480
oh let me be careful, it's not f it's g.

00:02:57.480 --> 00:03:01.437
So we need to have a
defined limit for g of x

00:03:01.437 --> 00:03:02.604
minus g of one

00:03:03.511 --> 00:03:05.178
over x minus one.

00:03:07.276 --> 00:03:09.575
And so let's just try
to evaluate this limit

00:03:09.575 --> 00:03:11.179
from the left and right hand sides,

00:03:11.179 --> 00:03:12.296
and we can simplify it.

00:03:12.296 --> 00:03:14.275
We already know that g of one is zero.

00:03:14.275 --> 00:03:15.860
So that's just going to be zero.

00:03:15.860 --> 00:03:17.348
So we just need to find the limit

00:03:17.348 --> 00:03:19.770
as x approaches one of
g of x over x minus one

00:03:19.770 --> 00:03:21.615
or see if we can find the limit.

00:03:21.615 --> 00:03:23.915
So let's first think about the limit

00:03:23.915 --> 00:03:26.292
as we approach from the left hand side

00:03:26.292 --> 00:03:27.709
of g of x over

00:03:30.140 --> 00:03:30.973
x minus.

00:03:31.989 --> 00:03:34.755
G of x over x minus one.

00:03:34.755 --> 00:03:36.353
Well as we approach
from the left hand side,

00:03:36.353 --> 00:03:38.351
g of x is that right over there.

00:03:38.351 --> 00:03:39.654
So we could write this.

00:03:39.654 --> 00:03:40.487
Instead of writing g of x,

00:03:40.487 --> 00:03:43.181
we could write this as x minus one.

00:03:43.181 --> 00:03:45.801
X minus one over x minus one,

00:03:45.801 --> 00:03:48.765
and as long as we aren't equal to one,

00:03:48.765 --> 00:03:50.572
this thing is going to be equal

00:03:50.572 --> 00:03:52.959
as long as x does not equal one.

00:03:52.959 --> 00:03:56.300
X minus one over x minus
one is just going to be one.

00:03:56.300 --> 00:03:58.299
So this limit is going to be one.

00:03:58.299 --> 00:04:00.266
So that one worked out.

00:04:00.266 --> 00:04:02.310
Now let's think about the limit

00:04:02.310 --> 00:04:06.498
as x approaches one
from the right hand side

00:04:06.498 --> 00:04:09.532
of, once again, I could
write g of x of g of one,

00:04:09.532 --> 00:04:10.723
but g of one is just zero,

00:04:10.723 --> 00:04:15.378
so I'll just write g
of x over x minus one.

00:04:15.378 --> 00:04:17.161
Well what's g of x now?

00:04:17.161 --> 00:04:19.267
Well it's x minus one squared.

00:04:19.267 --> 00:04:20.887
So instead of writing g of x,

00:04:20.887 --> 00:04:24.554
I could write this as x minus one squared

00:04:25.724 --> 00:04:27.391
over x minus one,

00:04:28.900 --> 00:04:31.533
and so as long as x does not equal one,

00:04:31.533 --> 00:04:32.660
we're just doing the limit.

00:04:32.660 --> 00:04:35.149
We're saying as we approach
one from the right hand side.

00:04:35.149 --> 00:04:36.845
Well, this expression right over here

00:04:36.845 --> 00:04:39.915
you have x minus one squared
divided by x minus one,

00:04:39.915 --> 00:04:42.828
well, that's just going
to give us x minus one.

00:04:42.828 --> 00:04:44.837
X minus one squared divided by x minus one

00:04:44.837 --> 00:04:48.443
is just going to be x minus one,

00:04:48.443 --> 00:04:52.042
and this limit, well this
expression right over here

00:04:52.042 --> 00:04:56.265
is going to be continuous
and defined for sure all

00:04:56.265 --> 00:04:58.432
x's that are not equaling one.

00:04:58.432 --> 00:05:01.074
Actually, let me, let me, well,

00:05:01.074 --> 00:05:02.839
it was before it was this,

00:05:02.839 --> 00:05:06.350
x minus one squared over x minus one.

00:05:06.350 --> 00:05:08.251
This thing over here, as
I said, is not defined

00:05:08.251 --> 00:05:11.344
for x equals one, but it
is defined for anything

00:05:11.344 --> 00:05:15.299
for x does not equal one, and
we're just approaching one.

00:05:15.299 --> 00:05:17.628
And, if we wanted to
simplify this expression,

00:05:17.628 --> 00:05:20.536
it would get, this would just be

00:05:20.536 --> 00:05:21.369
I think I just did this,

00:05:21.369 --> 00:05:23.573
but I'm making sure I'm doing it right.

00:05:23.573 --> 00:05:25.167
This is going to be the same this as that

00:05:25.167 --> 00:05:27.270
for x not being equal to one.

00:05:27.270 --> 00:05:29.303
Well this is just going to be zero.

00:05:29.303 --> 00:05:31.852
We could just evaluate when
x is equal to one here.

00:05:31.852 --> 00:05:33.632
This is going to be equal to zero.

00:05:33.632 --> 00:05:36.034
And so notice, you get a different limit

00:05:36.034 --> 00:05:39.263
for this definition of the
derivative as we approach

00:05:39.263 --> 00:05:42.167
from the left hand side
or the right hand side,

00:05:42.167 --> 00:05:43.499
and that makes sense.

00:05:43.499 --> 00:05:45.680
This graph is gonna look something like,

00:05:45.680 --> 00:05:48.076
we have a slope of one, so it's gonna look

00:05:48.076 --> 00:05:49.466
something like this.

00:05:49.466 --> 00:05:53.335
And then right when x is equal to one

00:05:53.335 --> 00:05:56.280
and the value of our function is zero

00:05:56.280 --> 00:06:00.208
it looks something like this,
it looks something like this.

00:06:00.208 --> 00:06:02.259
And so the graph is continuous

00:06:02.259 --> 00:06:04.546
the graph for sure is continuous,

00:06:04.546 --> 00:06:07.325
but our slope coming
into that point is one,

00:06:07.325 --> 00:06:10.929
and our slope right when we
leave that point is zero.

00:06:10.929 --> 00:06:14.536
So it is not differentiable over there.

00:06:14.536 --> 00:06:18.703
So it is continuous, continuous,
but not differentiable.


Differentiability at a point: algebraic (function is differentiable)

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:01.495
- [Voiceover] Is the function given

00:00:01.495 --> 00:00:04.695
below continuous slash
differentiable at x equals three?

00:00:04.695 --> 00:00:06.269
They've defined it piece-wise,

00:00:06.269 --> 00:00:07.923
and we have some choices.

00:00:07.923 --> 00:00:09.453
Continuous, not differentiable.

00:00:09.453 --> 00:00:11.002
Differentiable, not continuous.

00:00:11.002 --> 00:00:12.476
Both continuous and differentiable.

00:00:12.476 --> 00:00:14.310
Neither continuous not differentiable.

00:00:14.310 --> 00:00:16.617
Now one of these we can knock
out right from the get go.

00:00:16.617 --> 00:00:19.973
In order to be differentiable
you need to be continuous.

00:00:19.973 --> 00:00:23.623
You cannot have differentiable
but not continuous.

00:00:23.623 --> 00:00:25.571
So let's just rule that one out.

00:00:25.571 --> 00:00:28.070
And now let's think about continuity.

00:00:28.070 --> 00:00:31.040
So let's first think about continuity.

00:00:31.040 --> 00:00:32.612
And frankly, if it isn't continuous,

00:00:32.612 --> 00:00:36.105
then it's not going to be differentiable.

00:00:36.105 --> 00:00:39.108
So let's think about it a little bit.

00:00:39.108 --> 00:00:42.674
So in order to be continuous, f of ...

00:00:42.674 --> 00:00:44.133
Using a darker color.

00:00:44.133 --> 00:00:46.716
F of three needs to be equal to

00:00:47.935 --> 00:00:51.935
the limit of f of x as x approaches three.

00:00:53.646 --> 00:00:55.815
Now what is f of three?

00:00:55.815 --> 00:01:00.038
Well let's see, we've fallen
to this case right over here,

00:01:00.038 --> 00:01:03.704
because x is equal to
three, so six times three

00:01:03.704 --> 00:01:07.287
is 18, minus nine is
nine, so this is nine.

00:01:09.009 --> 00:01:11.460
So the limit of f of x
as x approaches three

00:01:11.460 --> 00:01:13.379
needs to be equal to nine.

00:01:13.379 --> 00:01:15.161
So let's first think about the limit

00:01:15.161 --> 00:01:18.827
as we approach from the left hand side.

00:01:18.827 --> 00:01:21.822
The limit as x approaches three.

00:01:21.822 --> 00:01:25.950
X approaches three from the
left hand side of f of x.

00:01:25.950 --> 00:01:27.377
Well when x is less than three

00:01:27.377 --> 00:01:30.044
we fall into this case, so f of x

00:01:30.044 --> 00:01:33.544
is just going to be equal to x squared.

00:01:35.350 --> 00:01:39.124
And so this is defined and continuous

00:01:39.124 --> 00:01:40.780
for all real numbers, so we can just

00:01:40.780 --> 00:01:42.280
substitute the three in there.

00:01:42.280 --> 00:01:44.323
So this is going to be equal to nine.

00:01:44.323 --> 00:01:48.190
Now what's the limit of as we approach

00:01:48.190 --> 00:01:51.607
three from the right hand side of f of x?

00:01:52.780 --> 00:01:54.183
Well as we approach from the right,

00:01:54.183 --> 00:01:57.457
this one right over here is f of x

00:01:57.457 --> 00:01:59.799
is equal to six x minus nine.

00:01:59.799 --> 00:02:02.392
So we just write six x minus nine.

00:02:02.392 --> 00:02:04.092
And once again six x minus nine is defined

00:02:04.092 --> 00:02:06.262
and continuous for all real numbers,

00:02:06.262 --> 00:02:07.872
so we could just pop a three in

00:02:07.872 --> 00:02:10.086
there and you get 18 minus nine.

00:02:10.086 --> 00:02:11.863
Well this is also equal to nine,

00:02:11.863 --> 00:02:14.658
so the right and left hand, the left

00:02:14.658 --> 00:02:17.356
and right hand limits both equal nine,

00:02:17.356 --> 00:02:19.218
which is equal to the value
of the function there,

00:02:19.218 --> 00:02:21.821
so it is definitely continuous.

00:02:21.821 --> 00:02:25.821
So we can rule out this
choice right over there.

00:02:26.953 --> 00:02:29.619
And now let's think
about differentiablity.

00:02:29.619 --> 00:02:32.618
So in order to be differentiable ...

00:02:32.618 --> 00:02:36.535
So differentiable, I'll
just diff-er-ent-iable.

00:02:39.483 --> 00:02:42.234
In order to be differentiable the limit

00:02:42.234 --> 00:02:46.401
as x approaches three of
f of x minus f of three

00:02:49.075 --> 00:02:52.772
over x minus three needs to exist.

00:02:52.772 --> 00:02:55.042
So let's see if we can evaluate this.

00:02:55.042 --> 00:02:57.292
So first of all we know
what f of three is.

00:02:57.292 --> 00:02:59.500
F of three, we already evaluated this.

00:02:59.500 --> 00:03:01.371
This is going to be nine.

00:03:01.371 --> 00:03:04.071
And let's see if we can
evaluate this limit,

00:03:04.071 --> 00:03:06.147
or let's see what the
limit is as we approach

00:03:06.147 --> 00:03:07.844
from the left hand side
or the right hand side,

00:03:07.844 --> 00:03:09.140
and if they are approaching the same thing

00:03:09.140 --> 00:03:12.856
then that same thing that they
are approaching is a limit.

00:03:12.856 --> 00:03:14.706
So let's first think about the limit

00:03:14.706 --> 00:03:18.279
as x approaches three
from the left hand side.

00:03:18.279 --> 00:03:22.446
So it's over x minus three,
and we have f of x minus nine.

00:03:24.109 --> 00:03:25.761
But as we approach from
the left hand side,

00:03:25.761 --> 00:03:28.195
this is f of x, as x is less than three,

00:03:28.195 --> 00:03:29.830
f of x is equal to x squared.

00:03:29.830 --> 00:03:33.030
So this would be instead of f of x minus 9

00:03:33.030 --> 00:03:37.628
I'll write x squared minus
nine, and x squared minus nine.

00:03:37.628 --> 00:03:39.534
This is a difference of squares,

00:03:39.534 --> 00:03:43.063
so this is x plus three
times x minus three,

00:03:43.063 --> 00:03:46.063
x plus three times x minus three.

00:03:47.337 --> 00:03:50.413
And so these would cancel out.

00:03:50.413 --> 00:03:52.824
We can say that is
equivalent to x plus three

00:03:52.824 --> 00:03:54.557
as long as x does not equal three.

00:03:54.557 --> 00:03:56.389
That's okay because we're
approaching from the left,

00:03:56.389 --> 00:03:57.846
and as we approach from the left

00:03:57.846 --> 00:04:01.460
x plus three is defined
for all real numbers,

00:04:01.460 --> 00:04:02.943
it's continuous for all real numbers,

00:04:02.943 --> 00:04:04.941
so we can just substitute
the three in there.

00:04:04.941 --> 00:04:07.529
So we would get a six.

00:04:07.529 --> 00:04:09.015
So now let's try to evaluate the limit

00:04:09.015 --> 00:04:12.348
as we approach from the right hand side.

00:04:14.212 --> 00:04:16.531
So once again it's f of
x, but as we approach

00:04:16.531 --> 00:04:21.293
from the right hand side,
f of x is six x minus nine.

00:04:21.293 --> 00:04:22.495
That's our f of x.

00:04:22.495 --> 00:04:25.634
And then we have minus f
of three, which is nine.

00:04:25.634 --> 00:04:28.344
So it's six x minus 18.

00:04:28.344 --> 00:04:29.844
Six x minus 18.

00:04:31.109 --> 00:04:35.601
Well that's the same thing
as six times x minus three,

00:04:35.601 --> 00:04:37.965
and as we approach from the right,

00:04:37.965 --> 00:04:41.465
well that's just going to be equal to six.

00:04:42.858 --> 00:04:46.912
So it looks like our
derivative exists there,

00:04:46.912 --> 00:04:49.345
and it is equal to limit
as x approaches three

00:04:49.345 --> 00:04:51.033
of all of this because
this is equal to six,

00:04:51.033 --> 00:04:52.608
because the limit is
approached from the left

00:04:52.608 --> 00:04:54.447
and the right is also equal to six.

00:04:54.447 --> 00:04:55.975
So this looks like we are both

00:04:55.975 --> 00:04:58.475
continuous and differentiable.


Differentiability at a point: graphical

WEBVTT
Kind: captions
Language: en

00:00:00.415 --> 00:00:02.706
- [Voiceover] The graph of
function f is given below.

00:00:02.706 --> 00:00:06.207
It has a vertical tangent at
the point three comma zero.

00:00:06.207 --> 00:00:09.853
Three comma zero has a vertical
tangent, let me draw that.

00:00:09.853 --> 00:00:13.429
It has a vertical
tangent right over there,

00:00:13.429 --> 00:00:15.093
and a horizontal tangent at the point

00:00:15.093 --> 00:00:17.110
zero comma negative three.

00:00:17.110 --> 00:00:18.540
Zero comma negative three,

00:00:18.540 --> 00:00:23.189
so it has a horizontal
tangent right over there,

00:00:23.189 --> 00:00:26.890
and also has a horizontal
tangent at six comma three.

00:00:26.890 --> 00:00:29.261
Six comma three, let me draw

00:00:29.261 --> 00:00:32.511
the horizontal tangent, just like that.

00:00:33.931 --> 00:00:38.632
Select all the x-values for
which f is not differentiable.

00:00:38.632 --> 00:00:41.056
Select all that apply.

00:00:41.056 --> 00:00:44.889
F prime, f prime, I'll
write it in short hand.

00:00:46.014 --> 00:00:47.931
We say no f prime under

00:00:49.257 --> 00:00:51.026
it's going to happen
under three conditions.

00:00:51.026 --> 00:00:52.217
The first condition you could say

00:00:52.217 --> 00:00:54.980
well we have a vertical tangent.

00:00:54.980 --> 00:00:56.442
Vertical tangent.

00:00:56.442 --> 00:00:58.889
Why is a vertical tangent a place where

00:00:58.889 --> 00:01:00.738
it's hard to define our derivative?

00:01:00.738 --> 00:01:02.497
Well, remember, our derivative is

00:01:02.497 --> 00:01:04.805
we're really trying to find
our rate of change of y

00:01:04.805 --> 00:01:06.458
with respect to x,

00:01:06.458 --> 00:01:08.683
but when you have a vertical tangent,

00:01:08.683 --> 00:01:11.291
you change your x a very small amount,

00:01:11.291 --> 00:01:14.055
you have an infinite change in y,

00:01:14.055 --> 00:01:16.272
either in the positive or
the negative direction.

00:01:16.272 --> 00:01:19.667
That's one situation where
you have no derivative.

00:01:19.667 --> 00:01:21.971
They tell us where we have
a vertical tangent in here,

00:01:21.971 --> 00:01:23.618
where x is equal to three.

00:01:23.618 --> 00:01:24.785
We have no ...

00:01:26.151 --> 00:01:28.171
F is not differentiable at x equals three

00:01:28.171 --> 00:01:29.460
because of the vertical tangent.

00:01:29.460 --> 00:01:31.247
You might say what about
horizontal tangents?

00:01:31.247 --> 00:01:33.005
No, horizontal tangents
are completely fine.

00:01:33.005 --> 00:01:35.478
Horizontal tangents are
places where the derivative

00:01:35.478 --> 00:01:36.625
is equal to zero.

00:01:36.625 --> 00:01:38.644
F prime of six is equal to zero.

00:01:38.644 --> 00:01:41.394
F prime of zero is equal to zero.

00:01:43.472 --> 00:01:45.176
What are other scenarios?

00:01:45.176 --> 00:01:47.176
Well another scenario
where you're not gonna have

00:01:47.176 --> 00:01:51.450
a defined derivative is where
the graph is not continuous.

00:01:51.450 --> 00:01:52.700
Not continuous.

00:01:54.216 --> 00:01:57.952
We see right over here at
x equals negative three,

00:01:57.952 --> 00:02:00.285
our graph is not continuous.

00:02:01.143 --> 00:02:03.610
X equals negative three
it's not continuous.

00:02:03.610 --> 00:02:07.052
Those are thee only places
where f is not differentiable

00:02:07.052 --> 00:02:08.392
that they're giving us options on.

00:02:08.392 --> 00:02:09.915
We don't know what the graph is doing

00:02:09.915 --> 00:02:12.243
to the left or the right.

00:02:12.243 --> 00:02:14.802
These there I guess would
be interesting cases.

00:02:14.802 --> 00:02:17.737
They haven't given us those choices here.

00:02:17.737 --> 00:02:21.193
We already said, at x equals
0, the derivative is zero.

00:02:21.193 --> 00:02:22.026
It's defined.

00:02:22.026 --> 00:02:22.859
It's differentiable there.

00:02:22.859 --> 00:02:25.416
At x equals six, the derivative is zero.

00:02:25.416 --> 00:02:27.333
We have a flat tangent.

00:02:28.176 --> 00:02:30.299
Once again it's defined there as well.

00:02:30.299 --> 00:02:32.924
Let's do another one of these.

00:02:32.924 --> 00:02:34.437
Actually, I didn't include, I think that

00:02:34.437 --> 00:02:36.103
this takes care of this problem,

00:02:36.103 --> 00:02:37.429
but there's a third scenario

00:02:37.429 --> 00:02:41.453
in which we have, I'll
call it a sharp turn.

00:02:41.453 --> 00:02:42.536
A sharp turn.

00:02:43.584 --> 00:02:46.629
This isn't the most mathy
definition right over here,

00:02:46.629 --> 00:02:48.320
but it's easy to recognize.

00:02:48.320 --> 00:02:49.880
A sharp turn is something like that,

00:02:49.880 --> 00:02:53.995
or like, well no, that
doesn't look too sharp,

00:02:53.995 --> 00:02:55.078
or like this.

00:02:56.007 --> 00:02:58.512
The reason why where you
have these sharp bends

00:02:58.512 --> 00:03:00.045
or sharp turns as opposed to something

00:03:00.045 --> 00:03:02.065
that looks more smooth like that.

00:03:02.065 --> 00:03:04.146
The reason why we're not
differentiable there is

00:03:04.146 --> 00:03:06.273
as we approach this point,

00:03:06.273 --> 00:03:08.878
as we approach this
point from either side,

00:03:08.878 --> 00:03:10.112
we have different slopes.

00:03:10.112 --> 00:03:13.228
Notice our slope is
positive right over here,

00:03:13.228 --> 00:03:15.367
as x increases, y is increasing,

00:03:15.367 --> 00:03:16.897
While the slope is negative here.

00:03:16.897 --> 00:03:20.032
As you're trying to find
the limit of our slope

00:03:20.032 --> 00:03:21.188
as we approach this point,

00:03:21.188 --> 00:03:22.665
it's not going to exist
because it's different

00:03:22.665 --> 00:03:24.533
on the left hand side
and the right hand side.

00:03:24.533 --> 00:03:27.198
That's why the sharp turns, I
don't see any sharp turns here

00:03:27.198 --> 00:03:28.978
so it doesn't apply to this example.

00:03:28.978 --> 00:03:30.850
Let's do one more examples.

00:03:30.850 --> 00:03:33.156
Actually this one does
have some sharp turns.

00:03:33.156 --> 00:03:35.412
This could be interesting.

00:03:35.412 --> 00:03:38.540
The graph of function f is
given to the left right here.

00:03:38.540 --> 00:03:41.747
It has a vertical asymptote
at x equals negative three,

00:03:41.747 --> 00:03:42.834
we see that,

00:03:42.834 --> 00:03:46.417
and horizontal asymptotes
at y equals zero.

00:03:47.776 --> 00:03:51.460
This end of the curve as x
approaches negative infinity

00:03:51.460 --> 00:03:54.654
it looks like y is approaching zero.

00:03:54.654 --> 00:03:57.028
It has another horizontal
asymptote at y equals four.

00:03:57.028 --> 00:03:58.945
As x approaches infinity, it looks like

00:03:58.945 --> 00:04:02.530
our graph is trending down
to y is equal to four.

00:04:02.530 --> 00:04:06.350
Select the x values for which
f is not differentiable.

00:04:06.350 --> 00:04:10.517
First of all, we could think
about vertical tangents.

00:04:11.405 --> 00:04:13.173
Doesn't seem to have
any vertical tangents.

00:04:13.173 --> 00:04:15.491
Then we could think about
where we are not continuous.

00:04:15.491 --> 00:04:16.883
Well, we're definitely not continuous

00:04:16.883 --> 00:04:19.241
where we have this vertical
asymptote right over here.

00:04:19.241 --> 00:04:21.779
We're not continuous at
x equals negative three.

00:04:21.779 --> 00:04:25.195
We're also not continuous
at x is equal to one.

00:04:25.195 --> 00:04:27.612
Then the last situation where

00:04:28.735 --> 00:04:30.464
we are not going to be differentiable

00:04:30.464 --> 00:04:32.843
is where we have a sharp turn,

00:04:32.843 --> 00:04:36.433
or you could kind of view it
as a sharp point, on our graph.

00:04:36.433 --> 00:04:39.208
I see a sharp point right over there.

00:04:39.208 --> 00:04:41.302
Notice as we approach
from the left hand side,

00:04:41.302 --> 00:04:44.438
the slope looks like a
constant, I don't know,

00:04:44.438 --> 00:04:46.664
it's like a positive three halves,

00:04:46.664 --> 00:04:48.672
while as we go to the right side of that

00:04:48.672 --> 00:04:50.642
it looks like our slope turns negative.

00:04:50.642 --> 00:04:53.298
If you were to try to find
the limit of the slope

00:04:53.298 --> 00:04:54.915
as we approach from either side,

00:04:54.915 --> 00:04:56.041
which is essentially
what you're trying to do

00:04:56.041 --> 00:04:58.107
when you try to find the derivative,

00:04:58.107 --> 00:04:59.682
well it's not going to be defined

00:04:59.682 --> 00:05:02.909
because it's different from either side.

00:05:02.909 --> 00:05:06.367
F is also not differentiable
at the x value

00:05:06.367 --> 00:05:10.406
that gives us that little
sharp point right over there.

00:05:10.406 --> 00:05:11.896
If you were to graph the derivative,

00:05:11.896 --> 00:05:13.623
which we will do in future videos,

00:05:13.623 --> 00:05:15.706
you will see that the derivative is not

00:05:15.706 --> 00:05:17.261
continuous at that point.

00:05:17.261 --> 00:05:19.011
Let me mark that off.

00:05:20.045 --> 00:05:22.465
Then we can check x equals zero.

00:05:22.465 --> 00:05:24.077
X equals zero's completely cool.

00:05:24.077 --> 00:05:27.301
We're at a point that our tangent line

00:05:27.301 --> 00:05:28.951
is definitely not vertical.

00:05:28.951 --> 00:05:30.479
We're definitely continuous there.

00:05:30.479 --> 00:05:33.204
We definitely do not have
a sharp point or edge.

00:05:33.204 --> 00:05:36.704
We're completely cool at x equals zero.


Example of derivative as limit of average rate of change

WEBVTT
Kind: captions
Language: en

00:00:00.282 --> 00:00:01.513
- [Voiceover] Stacy wants to find

00:00:01.513 --> 00:00:05.415
the derivative of f of x
equals x squared plus one

00:00:05.415 --> 00:00:07.813
at the point x equals two.

00:00:07.813 --> 00:00:10.794
Her table below shows the
average rate of change

00:00:10.794 --> 00:00:14.339
of f over the intervals from x to two

00:00:14.339 --> 00:00:16.998
or from two to x, and
these are closed intervals,

00:00:16.998 --> 00:00:21.440
for x-values that get
increasingly closer to two.

00:00:21.440 --> 00:00:22.979
so they get-- so we're talking about

00:00:22.979 --> 00:00:24.761
the average rate of change of f

00:00:24.761 --> 00:00:26.881
over these closed intervals

00:00:26.881 --> 00:00:30.857
for x-values that get
increasingly closer to two.

00:00:30.857 --> 00:00:31.980
It looks like we're going to be dealing

00:00:31.980 --> 00:00:33.786
with some type of a limit,

00:00:33.786 --> 00:00:35.608
or we're trying to calculate
some type of a limit,

00:00:35.608 --> 00:00:37.760
or approximate some type of a limit.

00:00:37.760 --> 00:00:39.520
Let's read this data here.

00:00:39.520 --> 00:00:42.976
These are the x values
and she's trying to find

00:00:42.976 --> 00:00:44.786
the average rate of change between each

00:00:44.786 --> 00:00:47.248
of these x-values and two,

00:00:47.248 --> 00:00:49.638
or the average rate of
change of the function

00:00:49.638 --> 00:00:51.317
between when x is--

00:00:51.317 --> 00:00:54.584
one of these x values and two,

00:00:54.584 --> 00:00:56.532
and then she has the
average rate of change

00:00:56.532 --> 00:00:57.780
that she precalculated,

00:00:57.780 --> 00:00:59.014
so we don't have to get a calculator out

00:00:59.014 --> 00:01:00.097
or anything like that,

00:01:00.097 --> 00:01:03.170
and just as a reminder, how
did she calculate this 3.9?

00:01:03.170 --> 00:01:04.038
Well, they tell us.

00:01:04.038 --> 00:01:06.230
She took f of 1.9,

00:01:06.230 --> 00:01:09.190
what does the function
equal when x is 1.9?

00:01:09.190 --> 00:01:11.017
From that, she subtracted

00:01:11.017 --> 00:01:13.936
what is the value of the
function when f is equal to two,

00:01:13.936 --> 00:01:16.851
so that's really our change in f,

00:01:16.851 --> 00:01:18.601
and she divided it by

00:01:19.538 --> 00:01:21.205
the x, which is 1.9,

00:01:22.868 --> 00:01:23.701
minus two,

00:01:24.716 --> 00:01:27.383
so change in f over change in x.

00:01:28.245 --> 00:01:30.397
What is the average rate of change

00:01:30.397 --> 00:01:33.355
of our function over that interval?

00:01:33.355 --> 00:01:36.483
She did it between 1.9
and two, she got 3.9.

00:01:36.483 --> 00:01:38.475
Then she gets closer to two,

00:01:38.475 --> 00:01:41.697
so now she's doing it between 1.99 and two

00:01:41.697 --> 00:01:43.581
and it becomes 3.99,

00:01:43.581 --> 00:01:45.415
it looks like it's getting closer to four.

00:01:45.415 --> 00:01:47.740
She gets even closer to two

00:01:47.740 --> 00:01:51.835
and the average rate of change
gets even closer to four,

00:01:51.835 --> 00:01:54.415
and then she goes on
the other side of two,

00:01:54.415 --> 00:01:56.498
you could view it as this is approaching,

00:01:56.498 --> 00:02:00.897
this is-- this is approaching--
this is x approaching two

00:02:00.897 --> 00:02:02.897
from the left hand side,

00:02:03.899 --> 00:02:07.221
and this is x approaching
two from the right hand side.

00:02:07.221 --> 00:02:10.549
When it's 2.1, the average
rate of change is 4.1.

00:02:10.549 --> 00:02:12.029
When it's 2.01, once again,

00:02:12.029 --> 00:02:13.762
we're getting closer to two,

00:02:13.762 --> 00:02:15.344
we're getting closer to two,

00:02:15.344 --> 00:02:17.686
the average rate of change
is getting closer to four.

00:02:17.686 --> 00:02:19.044
The closer we get to two,

00:02:19.044 --> 00:02:21.723
the closer the average rate
of change gets to four.

00:02:21.723 --> 00:02:24.373
What this data is really
helping us approximate

00:02:24.373 --> 00:02:25.670
it's really saying, okay,

00:02:25.670 --> 00:02:29.112
the average rate of
change we know is f of x

00:02:29.112 --> 00:02:31.102
minus f of two,

00:02:31.102 --> 00:02:32.769
over x minus two,

00:02:33.751 --> 00:02:35.229
but what we're really thinking about is,

00:02:35.229 --> 00:02:39.189
well what is the limit as x approaches two

00:02:39.189 --> 00:02:40.384
right over here?

00:02:40.384 --> 00:02:42.754
That's what this data
is helping us to get at,

00:02:42.754 --> 00:02:46.806
and it looks like this
limit is equal to four.

00:02:46.806 --> 00:02:47.952
They give us it in here, it says,

00:02:47.952 --> 00:02:50.119
"Look, the closer that x gets to two

00:02:50.119 --> 00:02:52.600
"from either the left hand
side or the right hand side,

00:02:52.600 --> 00:02:55.706
"the closer that this
expression right over here,

00:02:55.706 --> 00:02:58.932
"which is this number, gets to four."

00:02:58.932 --> 00:03:00.604
You might recognize,

00:03:00.604 --> 00:03:04.396
this is one of the
definitions of a derivative.

00:03:04.396 --> 00:03:07.144
This is one of the
definitions of a derivative.

00:03:07.144 --> 00:03:10.776
This right over here
would be f prime of two,

00:03:10.776 --> 00:03:12.341
the derivative at x equals two

00:03:12.341 --> 00:03:14.510
is equal to the limit as x approaches two

00:03:14.510 --> 00:03:16.409
of all of this business.

00:03:16.409 --> 00:03:18.633
There's other ways to express
a derivative as a limit

00:03:18.633 --> 00:03:20.125
but this is one of them.

00:03:20.125 --> 00:03:21.716
There you go, from the table,

00:03:21.716 --> 00:03:22.683
what does the derivative

00:03:22.683 --> 00:03:24.817
of f of x equals x squared plus one

00:03:24.817 --> 00:03:26.588
at x equals two appear to be?

00:03:26.588 --> 00:03:28.346
Well, the derivative at x equals two

00:03:28.346 --> 00:03:29.736
appears to be

00:03:29.736 --> 00:03:31.383
equal to four,

00:03:31.383 --> 00:03:32.633
and we're done.


Secant line with arbitrary point (with simplification)

WEBVTT
Kind: captions
Language: en

00:00:00.534 --> 00:00:02.554
- [Voiceover] A secant
line intersects the graph

00:00:02.554 --> 00:00:06.047
of f of x is equal to
x squared plus five x

00:00:06.047 --> 00:00:09.669
at two points with x
coordinates three and t

00:00:09.669 --> 00:00:12.551
where t does not equal three.

00:00:12.551 --> 00:00:15.585
What is the slope of the
secant line in terms of t?

00:00:15.585 --> 00:00:19.404
Your answer must be fully
expanded and simplified.

00:00:19.404 --> 00:00:21.015
And my apologies ahead
of time if I'm a little

00:00:21.015 --> 00:00:22.139
out of breath.

00:00:22.139 --> 00:00:24.261
I just tried to do some
exercises in my office

00:00:24.261 --> 00:00:26.331
to get some blood moving
and I think I'm still

00:00:26.331 --> 00:00:27.469
a little out of breath.

00:00:27.469 --> 00:00:31.117
Anyway, so we want to
find the slope of the

00:00:31.117 --> 00:00:32.723
secant line

00:00:32.723 --> 00:00:34.477
and they essentially give us two points

00:00:34.477 --> 00:00:35.830
on the secant line.

00:00:35.830 --> 00:00:39.256
They tell us what x is at
each of those two points

00:00:39.256 --> 00:00:40.995
and if we know what an x is,

00:00:40.995 --> 00:00:42.731
we are able to figure out what f of x is

00:00:42.731 --> 00:00:44.170
at each of those points.

00:00:44.170 --> 00:00:47.161
So we could make a little table here.

00:00:47.161 --> 00:00:49.828
We know x and we know f of x.

00:00:51.201 --> 00:00:54.040
So when x is equal to three

00:00:54.040 --> 00:00:55.357
what is f of x?

00:00:55.357 --> 00:00:57.713
Well it's going to be three squared

00:00:57.713 --> 00:00:59.546
plus five times three.

00:01:00.815 --> 00:01:03.956
Well this is going to be nine plus 15

00:01:03.956 --> 00:01:04.956
which is 24.

00:01:05.837 --> 00:01:07.920
So this is going to be 24

00:01:09.308 --> 00:01:11.558
and when x is equal to t

00:01:12.919 --> 00:01:14.990
what is f of t?

00:01:14.990 --> 00:01:18.740
Well it is going to be
t squared plus five t.

00:01:20.306 --> 00:01:22.859
And so we have two points
now that are on this line,

00:01:22.859 --> 00:01:24.027
this is the secant line.

00:01:24.027 --> 00:01:25.979
It intersects our function twice

00:01:25.979 --> 00:01:27.795
so it has these two points on it.

00:01:27.795 --> 00:01:30.042
So we just have to find
our change in y between

00:01:30.042 --> 00:01:31.690
these two points.

00:01:31.690 --> 00:01:35.988
Change in y and our
change in x, change in x.

00:01:35.988 --> 00:01:39.061
And I'm assuming that
y is equal to f of x.

00:01:39.061 --> 00:01:41.894
So our slope of our secant line is

00:01:42.927 --> 00:01:44.538
change in y

00:01:44.538 --> 00:01:46.070
over change in x.

00:01:46.070 --> 00:01:48.831
Our change in y, if we
view this as our end point,

00:01:48.831 --> 00:01:50.831
the second one with the
t in it is our end point

00:01:50.831 --> 00:01:53.858
is going to be that minus that.

00:01:53.858 --> 00:01:57.275
So it's going to be t squared plus five t

00:01:58.502 --> 00:01:59.335
minus 24

00:02:01.378 --> 00:02:05.330
and then in our denominator our ending x

00:02:05.330 --> 00:02:10.096
minus our starting x is
going to be t minus three.

00:02:10.096 --> 00:02:11.561
Now they tell us our answer must be fully

00:02:11.561 --> 00:02:13.072
expanded and simplified.

00:02:13.072 --> 00:02:15.367
So maybe there's a way to
simplify this a little bit.

00:02:15.367 --> 00:02:17.813
Let's see, can I factor
the top into something

00:02:17.813 --> 00:02:19.490
that involves

00:02:19.490 --> 00:02:20.823
a t minus three?

00:02:23.582 --> 00:02:25.165
All right,

00:02:25.165 --> 00:02:26.448
so in the numerator, let's see,

00:02:26.448 --> 00:02:27.537
negative three

00:02:27.537 --> 00:02:29.488
times positive eight is negative 24.

00:02:29.488 --> 00:02:31.404
Negative three plus
positive eight is five.

00:02:31.404 --> 00:02:33.639
So we can rewrite this as

00:02:33.639 --> 00:02:34.639
t plus eight

00:02:36.078 --> 00:02:36.911
times t

00:02:39.109 --> 00:02:40.776
times t minus three.

00:02:43.253 --> 00:02:46.153
And so we could say this
is going to be equal to

00:02:46.153 --> 00:02:48.349
if we cancel out the t
minus three or we divide

00:02:48.349 --> 00:02:50.892
the numerator into the
denominator by t minus three

00:02:50.892 --> 00:02:54.700
it's going to be equal to t plus eight.

00:02:54.700 --> 00:02:57.535
Now if we wanted to be really strict,

00:02:57.535 --> 00:02:59.410
mathematically strict,

00:02:59.410 --> 00:03:02.787
this expression isn't
exactly the same as our

00:03:02.787 --> 00:03:04.807
original expression right over here.

00:03:04.807 --> 00:03:06.382
What makes them different?

00:03:06.382 --> 00:03:08.204
Well they're going to
be true for all of t's

00:03:08.204 --> 00:03:10.176
except where t equals three.

00:03:10.176 --> 00:03:12.835
This thing right over here
is defined at t equals three.

00:03:12.835 --> 00:03:15.674
In fact when t equals three
this expression is equal to 11.

00:03:15.674 --> 00:03:19.798
But this thing up here was
not defined at t equals three

00:03:19.798 --> 00:03:21.878
so if you wanted to be particular about it

00:03:21.878 --> 00:03:24.987
you want this expression
to be the exact same thing

00:03:24.987 --> 00:03:26.070
you would say

00:03:27.161 --> 00:03:29.494
four t does not equal three.

00:03:30.731 --> 00:03:34.528
Now this can take the
same inputs as this one

00:03:34.528 --> 00:03:35.945
right over there.

00:03:37.340 --> 00:03:40.392
But I'm assuming where
t does not equal three.

00:03:40.392 --> 00:03:43.526
So you could view this
as maybe a little bit

00:03:43.526 --> 00:03:44.359
redundant.

00:03:44.359 --> 00:03:45.253
But this would be,

00:03:45.253 --> 00:03:49.420
this is the slope of the
secant line in terms of t.


Secant line with arbitrary difference (with simplification)

WEBVTT
Kind: captions
Language: en

00:00:00.537 --> 00:00:02.058
- [Voiceover] A secant line intersects

00:00:02.058 --> 00:00:06.772
the curve y = 2x squared + 1 at two points

00:00:06.772 --> 00:00:10.939
with x-coordinates 4 and 4 +
h, where h does not equal 0.

00:00:12.045 --> 00:00:15.902
What is the slope of the
secant line in terms of h?

00:00:15.902 --> 00:00:19.493
Your answer must be fully
expanded and simplified.

00:00:19.493 --> 00:00:21.837
We know the two points that
are on the secant line.

00:00:21.837 --> 00:00:23.661
It might not be obvious
from how they wrote it

00:00:23.661 --> 00:00:25.213
but let's make a little table here

00:00:25.213 --> 00:00:28.446
to make that a little bit clearer.

00:00:28.446 --> 00:00:32.613
We have x and then we
have y = 2x squared + 1.

00:00:35.912 --> 00:00:38.829
So we know that when x = 4,

00:00:40.049 --> 00:00:41.696
what is y going to be equal to?

00:00:41.696 --> 00:00:45.863
It's gonna be 2(4 squared) + 1

00:00:47.565 --> 00:00:51.732
which is the same thing as 2(16) +1

00:00:54.138 --> 00:00:57.712
which is the same thing as 32 + 1

00:00:57.712 --> 00:00:59.712
so it is going to be 33.

00:01:01.903 --> 00:01:05.153
What about when x = 4 + h?

00:01:07.402 --> 00:01:11.569
Well it's going to be 2(4 +h)squared + 1.

00:01:15.711 --> 00:01:17.852
That's going to be 2 times, let's see,

00:01:17.852 --> 00:01:22.019
(4 + h)squared is going to be 16 +

00:01:25.162 --> 00:01:26.745
8h + h squared

00:01:29.898 --> 00:01:31.498
and then we have our + 1 still

00:01:31.498 --> 00:01:32.963
and if we distribute the 2

00:01:32.963 --> 00:01:37.130
that's going to get us to
32 + 16h + 2h squared + 1

00:01:41.827 --> 00:01:44.090
and then we add the 32 to the 1

00:01:44.090 --> 00:01:45.810
and actually I'm gonna
switch the order a little bit

00:01:45.810 --> 00:01:47.490
so I have the highest degree term first

00:01:47.490 --> 00:01:50.907
so it's going to be 2h squared + 16h

00:01:52.332 --> 00:01:55.165
and then + 32 + 1 is 33.

00:01:56.964 --> 00:01:58.158
We have these two points.

00:01:58.158 --> 00:02:01.421
We have one point (4, 33)

00:02:01.421 --> 00:02:05.588
and we have the other point
(4+h, 2h squared + 16h + 33)

00:02:11.731 --> 00:02:12.893
and we just have to find the slope

00:02:12.893 --> 00:02:13.884
between these two points

00:02:13.884 --> 00:02:16.853
because the secant line
contains both of these points.

00:02:16.853 --> 00:02:18.716
How do we find the slope of a line?

00:02:18.716 --> 00:02:22.299
We do change in y / change in x.

00:02:23.416 --> 00:02:24.608
What's our change in y?

00:02:24.608 --> 00:02:25.944
If we view this as the end point

00:02:25.944 --> 00:02:27.000
and this as the starting point,

00:02:27.000 --> 00:02:30.593
our change in y is going
to be this minus that.

00:02:30.593 --> 00:02:34.760
It's going to be 2h
squared + 16h + 33 - 33,

00:02:39.241 --> 00:02:43.423
those two are going to
cancel each other out,

00:02:43.423 --> 00:02:47.317
and then over, what's our change in x?

00:02:47.317 --> 00:02:49.887
If we ended at 4 + h

00:02:49.887 --> 00:02:51.061
but then we started at 4

00:02:51.061 --> 00:02:53.978
so it's gonna be 4 + h - 4.

00:02:56.037 --> 00:02:57.951
These two cancel each with each other

00:02:57.951 --> 00:03:00.048
and we are left with

00:03:00.048 --> 00:03:03.131
2h squared + 16h / h.

00:03:08.650 --> 00:03:09.768
We can divide everything in

00:03:09.768 --> 00:03:11.991
the numerator and denominator by h

00:03:11.991 --> 00:03:13.471
and what are we going to get?

00:03:13.471 --> 00:03:14.951
This is going to be 1.

00:03:14.951 --> 00:03:16.216
That's just a 1.

00:03:16.216 --> 00:03:18.151
This is just an h.

00:03:18.151 --> 00:03:21.679
We have 2h + 16 / 1.

00:03:21.679 --> 00:03:23.429
Or just 2h + 16.

00:03:28.347 --> 00:03:29.683
And we're done.

00:03:29.683 --> 00:03:33.308
This is the slope of the
secant line in terms of h.

00:03:33.308 --> 00:03:34.829
Once again we just have to think about

00:03:34.829 --> 00:03:38.883
well the secant line
contains the point (4,f(4))

00:03:38.883 --> 00:03:41.467
or 2 times 4 squared + 1 right over here

00:03:41.467 --> 00:03:43.772
and, well I didn't call this f(x)

00:03:43.772 --> 00:03:46.045
but I think you get the idea,

00:03:46.045 --> 00:03:47.598
and then when x is 4 + h

00:03:47.598 --> 00:03:48.621
this is going to be y

00:03:48.621 --> 00:03:52.788
and we just found the slope
between these two points.


The derivative & tangent line equations

WEBVTT
Kind: captions
Language: en

00:00:00.414 --> 00:00:03.023
- [Voiceover] We're told that
the tangent line to the graph

00:00:03.023 --> 00:00:05.822
of function at the the
point two comma three

00:00:05.822 --> 00:00:09.372
passes through the point seven comma six.

00:00:09.372 --> 00:00:11.039
Find f prime of two.

00:00:12.752 --> 00:00:14.642
So whenever you see something like this,

00:00:14.642 --> 00:00:16.250
it doesn't hurt to try to visualize it.

00:00:16.250 --> 00:00:17.883
You might want to draw it
out or just visualize it

00:00:17.883 --> 00:00:18.877
in your head

00:00:18.877 --> 00:00:21.459
but since you can't get in my head,

00:00:21.459 --> 00:00:23.297
I will draw it out.

00:00:23.297 --> 00:00:24.659
So let me

00:00:24.659 --> 00:00:27.616
draw the information
that they are giving us.

00:00:27.616 --> 00:00:30.809
So that's x axis and that is the y axis.

00:00:30.809 --> 00:00:33.166
Let's see the relevant points
here at two comma three

00:00:33.166 --> 00:00:35.108
and seven comma six.

00:00:35.108 --> 00:00:36.687
So let me go,

00:00:36.687 --> 00:00:38.520
one, two, three, four,

00:00:39.915 --> 00:00:42.998
five, six, seven, along the x axis

00:00:44.329 --> 00:00:47.829
and I'm going to go one, two, three, four,

00:00:49.486 --> 00:00:52.153
five, and six, along the y axis.

00:00:53.607 --> 00:00:54.845
And now this point,

00:00:54.845 --> 00:00:57.782
so we have the point two comma three,

00:00:57.782 --> 00:01:00.066
so let me mark that,

00:01:00.066 --> 00:01:03.316
so two comma three is right over there,

00:01:04.439 --> 00:01:08.239
so that's two comma three and we also have

00:01:08.239 --> 00:01:10.406
the point seven comma six.

00:01:11.424 --> 00:01:15.424
Seven comma six is going
to be right over there.

00:01:18.054 --> 00:01:20.070
Seven comma six.

00:01:20.070 --> 00:01:21.498
Let us remind ourselves
what they're saying.

00:01:21.498 --> 00:01:24.735
They're saying the tangent
line to the graph of function f

00:01:24.735 --> 00:01:26.285
at this point

00:01:26.285 --> 00:01:29.141
passes through the point seven comma six.

00:01:29.141 --> 00:01:31.370
So if it's the tangent line
to the graph at that point,

00:01:31.370 --> 00:01:33.368
it must go through two comma three,

00:01:33.368 --> 00:01:36.474
that's the only place where
it intersects our graph

00:01:36.474 --> 00:01:38.117
and it goes through seven comma six.

00:01:38.117 --> 00:01:40.271
We only need two points to define a line

00:01:40.271 --> 00:01:43.813
and so the tangent line
is going to look like,

00:01:43.813 --> 00:01:44.927
it's going to look like,

00:01:44.927 --> 00:01:46.469
let me see if I can,

00:01:46.469 --> 00:01:47.897
no that's not right.

00:01:47.897 --> 00:01:49.031
Let me draw it

00:01:49.031 --> 00:01:51.039
like it's going to look.

00:01:51.039 --> 00:01:52.813
Oh that's not exactly right.

00:01:52.813 --> 00:01:54.734
Let me try one more time.

00:01:54.734 --> 00:01:56.029
Okay, there you go.

00:01:56.029 --> 00:01:59.363
So the tangent line is
going to look like that.

00:01:59.363 --> 00:02:02.472
It goes, it's tangent two
f right at two comma three

00:02:02.472 --> 00:02:05.779
and it goes through the
point seven comma six

00:02:05.779 --> 00:02:07.928
and so we don't know anything other than f

00:02:07.928 --> 00:02:10.410
but we can imagine what f looks like.

00:02:10.410 --> 00:02:12.949
Our function f could,

00:02:12.949 --> 00:02:14.342
so our function f,

00:02:14.342 --> 00:02:15.863
it could look something like this.

00:02:15.863 --> 00:02:17.981
It just has to be tangent

00:02:17.981 --> 00:02:20.181
so that line has to be
tangent to our function

00:02:20.181 --> 00:02:21.730
right at that point.

00:02:21.730 --> 00:02:26.137
So our function f could
look something like that.

00:02:26.137 --> 00:02:29.125
So when they say, find f prime of two,

00:02:29.125 --> 00:02:30.143
they're really saying,

00:02:30.143 --> 00:02:31.441
what is the slope of the

00:02:31.441 --> 00:02:34.950
tangent line when x is equal to two?

00:02:34.950 --> 00:02:37.397
So when x is equal to two,

00:02:37.397 --> 00:02:38.951
well the slope of the tangent line

00:02:38.951 --> 00:02:40.468
is the slope of this line.

00:02:40.468 --> 00:02:43.433
They gave us, they gave
us the two points that sit

00:02:43.433 --> 00:02:44.899
on the tangent line.

00:02:44.899 --> 00:02:47.197
So we just have to figure out its slope

00:02:47.197 --> 00:02:50.120
because that is going to
be the rate of change of

00:02:50.120 --> 00:02:51.146
that function right over there,

00:02:51.146 --> 00:02:51.979
its derivative.

00:02:51.979 --> 00:02:53.926
It's going to be the
slope of the tangent line

00:02:53.926 --> 00:02:55.275
because this is the tangent line.

00:02:55.275 --> 00:02:56.108
So let's do that.

00:02:56.108 --> 00:02:57.212
So as we know,

00:02:57.212 --> 00:03:00.070
slope is change in y over change in x.

00:03:00.070 --> 00:03:02.492
So if we change our,

00:03:02.492 --> 00:03:06.030
to go from two comma
three to seven comma six,

00:03:06.030 --> 00:03:07.363
our change in x,

00:03:08.212 --> 00:03:09.717
change in x,

00:03:09.717 --> 00:03:12.168
we go from x equals two to x equals 7

00:03:12.168 --> 00:03:14.280
so our change in x is equal to five.

00:03:14.280 --> 00:03:15.947
And our change in y,

00:03:17.238 --> 00:03:18.913
our change in y,

00:03:18.913 --> 00:03:22.740
we go from y equals three to y equals six.

00:03:22.740 --> 00:03:25.847
So our change in y is equal to three.

00:03:25.847 --> 00:03:28.764
So our change in y over change in x

00:03:29.814 --> 00:03:32.523
is going to be three over five

00:03:32.523 --> 00:03:34.961
which is the slope of this line,

00:03:34.961 --> 00:03:38.830
which is the derivative
of the function at two

00:03:38.830 --> 00:03:42.828
because this is the tangent
line at x equals two.

00:03:42.828 --> 00:03:45.328
Let's do another one of these.

00:03:46.705 --> 00:03:48.827
For a function g,

00:03:48.827 --> 00:03:52.746
we are given that g of
negative one equals three

00:03:52.746 --> 00:03:57.682
and g prime of negative one
is equal to negative two.

00:03:57.682 --> 00:04:01.084
What is the equation of the
tangent line to the graph of g

00:04:01.084 --> 00:04:03.511
at x equals negative one?

00:04:03.511 --> 00:04:04.845
Alright, so once again

00:04:04.845 --> 00:04:08.722
I think it will be helpful to graph this.

00:04:08.722 --> 00:04:10.298
So

00:04:10.298 --> 00:04:11.881
we have our y axis,

00:04:13.087 --> 00:04:14.837
we have our x axis

00:04:17.452 --> 00:04:18.597
and let's see.

00:04:18.597 --> 00:04:21.880
We say for function g we are
given that g of negative one

00:04:21.880 --> 00:04:23.222
is equal to three.

00:04:23.222 --> 00:04:26.941
So the point negative one
comma three is on our function.

00:04:26.941 --> 00:04:30.066
So this is negative one and then we have,

00:04:30.066 --> 00:04:31.357
one,

00:04:31.357 --> 00:04:32.629
two,

00:04:32.629 --> 00:04:34.137
and three.

00:04:34.137 --> 00:04:36.749
So that's that right over there.

00:04:36.749 --> 00:04:38.374
That is the point.

00:04:38.374 --> 00:04:40.680
That is the point
negative one comma three,

00:04:40.680 --> 00:04:42.255
it's going to be on our function.

00:04:42.255 --> 00:04:44.690
And we also know that
g prime of negative one

00:04:44.690 --> 00:04:46.307
is equal to negative two.

00:04:46.307 --> 00:04:48.338
So the slope of the tangent line

00:04:48.338 --> 00:04:50.240
right at that point on our function

00:04:50.240 --> 00:04:51.342
is going to be negative two.

00:04:51.342 --> 00:04:52.342
That's what that tells us.

00:04:52.342 --> 00:04:53.715
The slope of the tangent line,

00:04:53.715 --> 00:04:57.420
when x is equal to negative
one is equal to negative two.

00:04:57.420 --> 00:04:59.385
So I could use that information

00:04:59.385 --> 00:05:01.866
to actually draw the tangent line.

00:05:01.866 --> 00:05:04.125
So let me see if I can,

00:05:04.125 --> 00:05:06.482
let me see if I can do this.

00:05:06.482 --> 00:05:08.437
So it will look,

00:05:08.437 --> 00:05:10.020
so I think it will,

00:05:10.910 --> 00:05:12.688
let me just draw it like this.

00:05:12.688 --> 00:05:14.753
So it's going to go,

00:05:14.753 --> 00:05:17.326
so it's a slope of negative two

00:05:17.326 --> 00:05:20.409
is going to look something like that.

00:05:22.268 --> 00:05:26.435
So as we can see if we move
positive one in the x direction,

00:05:27.613 --> 00:05:29.141
we go

00:05:29.141 --> 00:05:31.688
down two in the y direction.

00:05:31.688 --> 00:05:34.277
So that is a slope of negative two.

00:05:34.277 --> 00:05:37.726
And so you might say well, where is g?

00:05:37.726 --> 00:05:39.865
Well we could draw what g could look like.

00:05:39.865 --> 00:05:42.899
G might look something like this.

00:05:42.899 --> 00:05:46.816
Might look something like
that right over there

00:05:48.196 --> 00:05:51.362
where that is the tangent line

00:05:51.362 --> 00:05:54.281
or you can make g do all sorts
of crazy things after that

00:05:54.281 --> 00:05:56.550
but all we really care about is equation

00:05:56.550 --> 00:05:58.217
for this green line.

00:05:59.149 --> 00:06:02.685
And there's a couple of
ways that you could do this.

00:06:02.685 --> 00:06:05.179
You could say, well look
a line is generally,

00:06:05.179 --> 00:06:07.735
there's a bunch of different
ways where you can define

00:06:07.735 --> 00:06:09.266
the equation for a line.

00:06:09.266 --> 00:06:10.359
You could say

00:06:10.359 --> 00:06:13.776
a line has a form y is equal to mx plus b

00:06:14.651 --> 00:06:17.530
where m is the slope and
b is the y intercept.

00:06:17.530 --> 00:06:19.724
Well we already know what
the slope of this line is.

00:06:19.724 --> 00:06:21.081
It is negative two.

00:06:21.081 --> 00:06:25.115
So we could say y is
equal to negative two.

00:06:25.115 --> 00:06:26.865
Negative two times x.

00:06:28.165 --> 00:06:30.173
Times x plus b.

00:06:30.173 --> 00:06:31.054
And then to solve for b,

00:06:31.054 --> 00:06:33.765
we know that the point
negative one comma three

00:06:33.765 --> 00:06:35.104
is on this line

00:06:35.104 --> 00:06:37.168
and this goes back to
some of your Algebra I

00:06:37.168 --> 00:06:39.075
that you might have
learned a few years ago.

00:06:39.075 --> 00:06:42.910
So let's substitute negative
one and three for x and y.

00:06:42.910 --> 00:06:44.111
So when y

00:06:44.111 --> 00:06:45.611
is equal to three,

00:06:47.062 --> 00:06:47.895
so three,

00:06:48.897 --> 00:06:50.397
three is equal to,

00:06:51.993 --> 00:06:54.076
is equal to negative two,

00:06:55.089 --> 00:06:56.839
negative two times x.

00:06:58.264 --> 00:07:00.217
Times negative one,

00:07:00.217 --> 00:07:02.384
times negative one plus b.

00:07:03.232 --> 00:07:04.718
Plus b.

00:07:04.718 --> 00:07:05.551
And so let's see,

00:07:05.551 --> 00:07:09.241
this is negative two times
negative one is positive two.

00:07:09.241 --> 00:07:11.393
And so if you subtract
two from both sides,

00:07:11.393 --> 00:07:13.282
you get one is equal to b.

00:07:13.282 --> 00:07:14.722
And there you have it.

00:07:14.722 --> 00:07:16.763
That is equation of our line.

00:07:16.763 --> 00:07:20.304
Y is equal to negative two x plus one.

00:07:20.304 --> 00:07:21.885
And there's other ways that
you could have done this,

00:07:21.885 --> 00:07:25.838
you could have written the
line in point-slope form

00:07:25.838 --> 00:07:26.945
or you could have done it this way.

00:07:26.945 --> 00:07:28.393
You could have written it in standard form

00:07:28.393 --> 00:07:31.880
but at least this is the way
my brain likes to process it.


Secant line with arbitrary difference

WEBVTT
Kind: captions
Language: en

00:00:00.667 --> 00:00:02.866
- [Voiceover] A secant
line intersects the curve

00:00:02.866 --> 00:00:05.013
y equals the natural log of x

00:00:05.013 --> 00:00:06.124
at two points,

00:00:06.124 --> 00:00:09.177
with x-coordinates two and two plus h.

00:00:09.177 --> 00:00:12.003
What is the slope of the secant line?

00:00:12.003 --> 00:00:14.226
Well, they're giving us
two points on this line.

00:00:14.226 --> 00:00:16.717
It might not be immediately
obvious, but they're giving us

00:00:16.717 --> 00:00:19.480
the points when x is equal to two,

00:00:19.480 --> 00:00:21.621
when x is equal to two, what is y?

00:00:21.621 --> 00:00:25.800
Well y, they tell us, y is
equal to the natural log of x,

00:00:25.800 --> 00:00:29.783
so in this case it is going
to be the natural log of two,

00:00:29.783 --> 00:00:32.687
and when x is equal to

00:00:32.687 --> 00:00:34.377
two plus h,

00:00:34.377 --> 00:00:35.606
what is y?

00:00:35.606 --> 00:00:37.371
Well, y is always going
to be the natural log

00:00:37.371 --> 00:00:38.510
of whatever x is.

00:00:38.510 --> 00:00:43.405
So it's going to be the
natural log of two plus h.

00:00:43.405 --> 00:00:46.273
And so these are two points
that sit on the secant line.

00:00:46.273 --> 00:00:48.681
This happens to be where
the secant line intersects

00:00:48.681 --> 00:00:51.012
our curve, but these are
two points on the line,

00:00:51.012 --> 00:00:52.009
and if you know two points on a line,

00:00:52.009 --> 00:00:55.732
you will then be able to figure
out the slope of that line.

00:00:55.732 --> 00:00:57.252
Now we can just remind ourselves that

00:00:57.252 --> 00:01:01.002
a slope is just change
in y over change in x,

00:01:02.822 --> 00:01:04.645
and so what is this going to be?

00:01:04.645 --> 00:01:06.886
Well if we view the second
one as our endpoint,

00:01:06.886 --> 00:01:11.053
our change in y going from ln
of two to ln of two plus h,

00:01:12.717 --> 00:01:15.732
so our change in y is
going to be our endpoint.

00:01:15.732 --> 00:01:20.581
So, natural log of two plus
h minus our starting point,

00:01:20.581 --> 00:01:23.550
or our end y-value minus
our starting y-value.

00:01:23.550 --> 00:01:25.050
Natural log of two

00:01:25.893 --> 00:01:27.376
and then our change in x,

00:01:27.376 --> 00:01:31.543
our change in x is going
to be our ending x-value,

00:01:34.742 --> 00:01:39.613
two plus h, minus our starting
x-value, minus two, and

00:01:39.613 --> 00:01:43.697
of course these twos cancel
out, and if we look here

00:01:43.697 --> 00:01:46.473
it looks like we have a
choice that directly matches

00:01:46.473 --> 00:01:48.056
what we just wrote.

00:01:49.266 --> 00:01:50.099
This right over here,

00:01:50.099 --> 00:01:51.218
natural log of two plus h

00:01:51.218 --> 00:01:53.633
minus natural log of two over h.

00:01:53.633 --> 00:01:56.010
Now, if you wanna visualize
this a little bit more,

00:01:56.010 --> 00:02:00.138
we could draw a little bit,
I'm gonna clear this out

00:02:00.138 --> 00:02:02.971
so I have space to draw the graph,

00:02:04.364 --> 00:02:07.826
just so you can really visualize
that this is a secant line.

00:02:07.826 --> 00:02:09.909
So let me draw my y-axis,

00:02:11.393 --> 00:02:13.560
and let me draw my x-axis,

00:02:15.698 --> 00:02:19.227
and y equals the natural
log of x is going to look,

00:02:19.227 --> 00:02:21.819
so let me underline that,

00:02:21.819 --> 00:02:23.387
that is going to look something like this.

00:02:23.387 --> 00:02:24.971
I'm obviously hand drawing it,

00:02:24.971 --> 00:02:28.304
so not a great drawing, right over here.

00:02:30.260 --> 00:02:32.677
And so when we have the point

00:02:34.691 --> 00:02:37.108
two comma natural log of two,

00:02:38.307 --> 00:02:41.492
which would be, lets say it's over,

00:02:41.492 --> 00:02:42.992
so if this is two,

00:02:44.952 --> 00:02:47.228
then this right over here is
the natural log of two, so

00:02:47.228 --> 00:02:51.395
that's the points two comma
natural log of two, and then

00:02:52.340 --> 00:02:55.899
we have some other, we just
noted the abstract two plus h,

00:02:55.899 --> 00:02:57.437
so it's two plus something.

00:02:57.437 --> 00:03:00.512
So let's say that is two plus h.

00:03:00.512 --> 00:03:01.602
And so this is going to be the point

00:03:01.602 --> 00:03:03.247
where we sit on the graph.

00:03:03.247 --> 00:03:06.979
That's going to be two plus
h comma the natural log of

00:03:06.979 --> 00:03:11.534
two plus h, and the exercise
we just did is finding

00:03:11.534 --> 00:03:14.802
the slope of the line
that connects these two.

00:03:14.802 --> 00:03:18.635
So the line will look
something like that, and

00:03:20.158 --> 00:03:22.295
and the way that we did
this is we figured out,

00:03:22.295 --> 00:03:24.734
okay what is our change in y?

00:03:24.734 --> 00:03:28.264
So our change in, so let's
see, we are going from

00:03:28.264 --> 00:03:31.289
y equals natural log of two to y equals

00:03:31.289 --> 00:03:33.491
natural log of two plus h.

00:03:33.491 --> 00:03:35.074
So our change in y,

00:03:37.105 --> 00:03:38.605
our change in y is

00:03:40.448 --> 00:03:42.453
our natural log of two plus h

00:03:42.453 --> 00:03:44.536
minus natural log of two.

00:03:46.616 --> 00:03:50.187
Minus natural log of
two, and our change in x?

00:03:50.187 --> 00:03:53.530
Well we're going from two to two plus h,

00:03:53.530 --> 00:03:57.697
we're going from two to two
plus h, so our change in x,

00:03:58.723 --> 00:04:00.377
we just increased by h.

00:04:00.377 --> 00:04:01.876
We're going from two to two plus h,

00:04:01.876 --> 00:04:04.383
so our change in x is equal to h.

00:04:04.383 --> 00:04:06.173
So the slope of the secant line,

00:04:06.173 --> 00:04:07.938
the slope of the secant line,

00:04:07.938 --> 00:04:10.411
a secant that intersects
our graph in two points

00:04:10.411 --> 00:04:11.530
is going to be change
in y over change in x,

00:04:11.530 --> 00:04:15.697
which is once again exactly
what we have over there.


Derivative as slope of curve

WEBVTT
Kind: captions
Language: en

00:00:00.255 --> 00:00:01.407
- [Voiceover] What I
wanna do in this video

00:00:01.407 --> 00:00:04.906
is a few examples that test our intuition

00:00:04.906 --> 00:00:08.091
of the derivative as a rate of change

00:00:08.091 --> 00:00:10.632
or the steepness of a curve

00:00:10.632 --> 00:00:12.899
or the slope of a curve

00:00:12.899 --> 00:00:15.169
or the slope of a tangent line of a curve

00:00:15.169 --> 00:00:17.696
depending on how you actually
want to think about it.

00:00:17.696 --> 00:00:20.279
So here it says F prime of five

00:00:21.463 --> 00:00:23.552
so this notation, prime

00:00:23.552 --> 00:00:26.423
this is another way of saying
well what's the derivative

00:00:26.423 --> 00:00:29.430
let's estimate the derivative
of our function at five.

00:00:29.430 --> 00:00:34.028
And when we say F prime
of five this is the slope

00:00:34.028 --> 00:00:35.778
slope of tangent line

00:00:38.589 --> 00:00:40.256
tangent line at five

00:00:42.011 --> 00:00:44.344
or you could view it as the

00:00:45.500 --> 00:00:48.750
you could view it as the rate of change

00:00:51.075 --> 00:00:52.908
of Y with respect to X

00:00:54.828 --> 00:00:57.848
which is really how we define slope

00:00:57.848 --> 00:01:00.431
respect to X of our function F.

00:01:02.170 --> 00:01:04.589
So let's think about that a little bit.

00:01:04.589 --> 00:01:06.627
We see they put the point

00:01:06.627 --> 00:01:10.118
the point five comma F
of five right over here

00:01:10.118 --> 00:01:12.684
and so if we want to estimate
the slope of the tangent line

00:01:12.684 --> 00:01:16.381
if we want to estimate the
steepness of this curve

00:01:16.381 --> 00:01:18.434
we could try to draw a line

00:01:18.434 --> 00:01:20.587
that is tangent right at that point.

00:01:20.587 --> 00:01:22.619
So let me see if I can do that.

00:01:22.619 --> 00:01:26.862
So if I were to draw a line starting there

00:01:26.862 --> 00:01:29.125
if I just wanted to make a tangent

00:01:29.125 --> 00:01:31.754
it looks like it would
do something like that.

00:01:31.754 --> 00:01:33.699
Right at that point that looks to be

00:01:33.699 --> 00:01:35.811
about how steep that curve is

00:01:35.811 --> 00:01:38.267
now what makes this an
interesting thing in non-linear

00:01:38.267 --> 00:01:40.076
is that it's constantly changing

00:01:40.076 --> 00:01:43.660
the steepness it's very low
here and it gets steeper

00:01:43.660 --> 00:01:46.492
and steeper and steeper
as we move to the right

00:01:46.492 --> 00:01:48.227
for larger and larger X values.

00:01:48.227 --> 00:01:49.852
But if we look at the point in question

00:01:49.852 --> 00:01:53.676
when X is equal to five
remember F prime of five

00:01:53.676 --> 00:01:55.186
would be if you were estimating it

00:01:55.186 --> 00:01:57.427
this would be the slope of this line here.

00:01:57.427 --> 00:01:59.122
And the slope of this line it looks like

00:01:59.122 --> 00:02:01.925
for every time we move
one in the X direction

00:02:01.925 --> 00:02:05.483
we're moving two in the Y direction.

00:02:05.483 --> 00:02:09.650
Delta Y is equal to two when
delta X is equal to one.

00:02:10.538 --> 00:02:12.038
So our change in Y

00:02:13.013 --> 00:02:14.686
with respect to X

00:02:14.686 --> 00:02:16.691
at least for this tangent line here

00:02:16.691 --> 00:02:18.291
which would represent our change in Y

00:02:18.291 --> 00:02:20.844
with respect to X right at that point

00:02:20.844 --> 00:02:24.491
is going to be equal to
two over one, or two.

00:02:24.491 --> 00:02:28.515
And it's almost estimated,
but all of these are way off.

00:02:28.515 --> 00:02:31.534
Having a negative two derivative

00:02:31.534 --> 00:02:35.701
would mean that as we increase
our X our Y is decreasing.

00:02:36.636 --> 00:02:39.267
So if our curve looks something like this

00:02:39.267 --> 00:02:41.531
we would have a slope of negative two.

00:02:41.531 --> 00:02:43.814
If having slopes in this

00:02:43.814 --> 00:02:45.742
a positive of point one

00:02:45.742 --> 00:02:47.724
that would be very flat
something down here

00:02:47.724 --> 00:02:51.131
we might have a slope closer to point one.

00:02:51.131 --> 00:02:54.459
Negative point one that
might be closer on this side

00:02:54.459 --> 00:02:56.875
now we're sloping but very close to flat.

00:02:56.875 --> 00:03:00.427
A slope of zero, that would be
right over here at the bottom

00:03:00.427 --> 00:03:03.081
where right at that moment as we change X

00:03:03.081 --> 00:03:05.248
Y is not increasing or decreasing

00:03:05.248 --> 00:03:07.977
the slope of the tangent line
right at that bottom point

00:03:07.977 --> 00:03:10.935
would have a slope of zero.

00:03:10.935 --> 00:03:12.815
So I feel really good about that response.

00:03:12.815 --> 00:03:15.184
Let's do one more of these.

00:03:15.184 --> 00:03:18.851
So alright, so they're
telling us to compare

00:03:19.993 --> 00:03:23.451
the derivative of G at four
to the derivative of G at six

00:03:23.451 --> 00:03:25.751
and which of these is greater

00:03:25.751 --> 00:03:27.091
and like always, pause the video

00:03:27.091 --> 00:03:29.160
and see if you can figure this out.

00:03:29.160 --> 00:03:31.047
Well this is just an exercise

00:03:31.047 --> 00:03:33.155
let's see if we were to

00:03:33.155 --> 00:03:37.200
if we were to make a line
that indicates the slope there

00:03:37.200 --> 00:03:39.608
you can do this as a tangent line

00:03:39.608 --> 00:03:41.533
let me try to do that.

00:03:41.533 --> 00:03:44.953
So now that wouldn't,
that doesn't do a good job

00:03:44.953 --> 00:03:47.536
so right over here at

00:03:47.536 --> 00:03:49.067
that looks like a

00:03:49.067 --> 00:03:52.003
I think I can do a better job than that

00:03:52.003 --> 00:03:54.336
no that's too shallow to see

00:03:54.336 --> 00:03:57.024
not shallow's not the
word, that's too flat.

00:03:57.024 --> 00:03:59.392
So let me try to really

00:03:59.392 --> 00:04:01.747
okay, that looks pretty good.

00:04:01.747 --> 00:04:03.536
So that line that I just drew

00:04:03.536 --> 00:04:05.715
seems to be indicative of

00:04:05.715 --> 00:04:08.194
the rate of change of Y with respect to X

00:04:08.194 --> 00:04:10.082
or the slope of that curve

00:04:10.082 --> 00:04:12.715
or that line you can
view it as a tangent line

00:04:12.715 --> 00:04:15.066
so we could think about what
its slope is going to be

00:04:15.066 --> 00:04:17.877
and then if we go further down over here

00:04:17.877 --> 00:04:20.544
this one is, it looks like it is

00:04:21.842 --> 00:04:24.925
steeper but in the negative direction

00:04:25.897 --> 00:04:29.746
so it looks like it is steeper for sure

00:04:29.746 --> 00:04:31.713
but it's in the negative direction.

00:04:31.713 --> 00:04:34.480
As we increase, think of it this way

00:04:34.480 --> 00:04:36.563
as we increase X one here

00:04:37.893 --> 00:04:41.810
it looks like we are
decreasing Y by about one.

00:04:42.690 --> 00:04:45.514
So it looks like G prime of four

00:04:45.514 --> 00:04:49.426
G prime of four, the derivative
when X is equal to four

00:04:49.426 --> 00:04:52.343
is approximately, I'm estimating it

00:04:53.425 --> 00:04:54.705
negative one

00:04:54.705 --> 00:04:58.295
while the derivative
here when we increase X

00:04:58.295 --> 00:05:00.511
if we increase X by

00:05:00.511 --> 00:05:02.428
if we increase X by one

00:05:03.398 --> 00:05:07.565
it looks like we're
decreasing Y by close to three

00:05:09.118 --> 00:05:10.535
so G prime of six

00:05:11.409 --> 00:05:14.826
looks like it's closer to negative three.

00:05:15.681 --> 00:05:18.348
So which one of these is larger?

00:05:20.163 --> 00:05:23.019
Well, this one is less negative

00:05:23.019 --> 00:05:26.209
so it's going to be
greater than the other one

00:05:26.209 --> 00:05:28.148
and you could have done this intuitively

00:05:28.148 --> 00:05:29.458
if you just look at the curve

00:05:29.458 --> 00:05:31.758
this is some type of a sinusoid here

00:05:31.758 --> 00:05:35.362
you have right over here the curve is flat

00:05:35.362 --> 00:05:38.134
you have right at that moment

00:05:38.134 --> 00:05:40.194
you have no change in Y with respect to X

00:05:40.194 --> 00:05:42.443
then it starts to decrease

00:05:42.443 --> 00:05:45.058
then it decreases at an even faster rate

00:05:45.058 --> 00:05:47.226
then it decreases at a faster rate

00:05:47.226 --> 00:05:48.914
then it starts, it's still decreasing

00:05:48.914 --> 00:05:52.791
but it's decreasing at
slower and slower rates

00:05:52.791 --> 00:05:55.775
decreasing at slower rates
and right at that moment

00:05:55.775 --> 00:05:59.731
you have your slope of
your tangent line is zero

00:05:59.731 --> 00:06:03.163
then it starts to increase,
increase, so on and so forth

00:06:03.163 --> 00:06:05.213
and it just keeps happening
over and over again.

00:06:05.213 --> 00:06:08.868
So you can also think about
this in a more intuitive way.


Introduction to chemistry

WEBVTT
Kind: captions
Language: en

00:00:00.202 --> 00:00:01.235
- [Voiceover] Here's some pictures

00:00:01.235 --> 00:00:05.761
of what most people associate
when they think of chemistry.

00:00:05.761 --> 00:00:08.905
They think of scientists
working on a bench

00:00:08.905 --> 00:00:11.808
with the different vials
of different chemicals.

00:00:11.808 --> 00:00:14.386
They might think of a mad scientist.

00:00:14.386 --> 00:00:16.862
Some of them boiling and changing colors.

00:00:16.862 --> 00:00:19.595
They might associate chemistry
with chemical equations.

00:00:19.595 --> 00:00:22.704
Thinking about how different
things will react together

00:00:22.704 --> 00:00:24.530
to form other things.

00:00:24.530 --> 00:00:28.413
They might think about models
of the different molecules

00:00:28.413 --> 00:00:30.354
that can be depicted different ways.

00:00:30.354 --> 00:00:33.782
They might associate it with
the periodic table of elements.

00:00:33.782 --> 00:00:38.000
And all of these things are
a big part of chemistry.

00:00:38.000 --> 00:00:39.992
But I want you to do in this video

00:00:39.992 --> 00:00:44.159
is appreciate what at its
essence chemistry is all about.

00:00:45.424 --> 00:00:47.990
And chemistry is one of the sciences

00:00:47.990 --> 00:00:50.845
that really just helps us understand

00:00:50.845 --> 00:00:55.611
and make models and make
predictions about our reality.

00:00:55.611 --> 00:00:57.967
And even something like the
periodic table of elements,

00:00:57.967 --> 00:01:00.986
which you'll see at the front
of any chemistry classroom,

00:01:00.986 --> 00:01:02.485
you take it for granted.

00:01:02.485 --> 00:01:05.953
But this is the product of, frankly,

00:01:05.953 --> 00:01:08.650
thousands of years of human beings

00:01:08.650 --> 00:01:10.485
trying to get to an understanding

00:01:10.485 --> 00:01:13.542
of all of the different
complexity in the world.

00:01:13.542 --> 00:01:15.369
If you look at the world around us,

00:01:15.369 --> 00:01:17.563
and it doesn't even have to be our planet,

00:01:17.563 --> 00:01:19.719
it could be the universe around us,

00:01:19.719 --> 00:01:21.876
you see all these different substances

00:01:21.876 --> 00:01:24.359
that seem to be different in certain ways.

00:01:24.359 --> 00:01:28.180
You see things like
fire and rock and water.

00:01:28.180 --> 00:01:29.928
Even in the planets,

00:01:29.928 --> 00:01:32.492
you see meteorological patterns.

00:01:32.492 --> 00:01:35.395
In life, you see all of this complexity

00:01:35.395 --> 00:01:36.511
and all of these different things

00:01:36.511 --> 00:01:39.326
and it looks like there's
just like a infinite spectrum

00:01:39.326 --> 00:01:40.739
of differentness out of there.

00:01:40.739 --> 00:01:42.229
Of different substances.

00:01:42.229 --> 00:01:43.679
Even in things like our human brain.

00:01:43.679 --> 00:01:47.244
The complexity and the
electrochemical interactions.

00:01:47.244 --> 00:01:50.426
And you could imagine as a species,

00:01:50.426 --> 00:01:51.752
this is kind of overwhelming.

00:01:51.752 --> 00:01:53.993
How do you make the sense of all of this?

00:01:53.993 --> 00:01:56.326
And it was not an easy path,

00:01:57.767 --> 00:01:59.462
but over thousands of years,

00:01:59.462 --> 00:02:01.572
we did start to make sense of it.

00:02:01.572 --> 00:02:03.979
And why it's very lucky for all of us

00:02:03.979 --> 00:02:05.472
to be born when we are now

00:02:05.472 --> 00:02:07.459
or to be around when we are now.

00:02:07.459 --> 00:02:09.406
To be able to learn
chemistry where we are now

00:02:09.406 --> 00:02:10.941
is that we get the answer.

00:02:10.941 --> 00:02:13.630
And it's a partial answer,
which is also exciting,

00:02:13.630 --> 00:02:15.245
cause we don't want the full answer.

00:02:15.245 --> 00:02:17.439
But it's a partial answer
that takes us a long way.

00:02:17.439 --> 00:02:19.964
We realize that the
periodic table of elements,

00:02:19.964 --> 00:02:23.567
that all of this complexity
that we're seeing before,

00:02:23.567 --> 00:02:24.772
that at the end of the day,

00:02:24.772 --> 00:02:27.054
things are made of basic building blocks.

00:02:27.054 --> 00:02:29.544
Kind of you could imagine the legos

00:02:29.544 --> 00:02:31.127
that really make up everything.

00:02:31.127 --> 00:02:33.283
And there aren't an
infinite number of legos.

00:02:33.283 --> 00:02:34.813
There's actually a finite number of them.

00:02:34.813 --> 00:02:36.598
We're discovering more all of the time,

00:02:36.598 --> 00:02:37.431
well not all of the time,

00:02:37.431 --> 00:02:39.942
now new elements are not
discovered that frequently,

00:02:39.942 --> 00:02:41.562
but there's a few of these elements

00:02:41.562 --> 00:02:44.050
that are disproportionately showing up

00:02:44.050 --> 00:02:45.789
in a lot of what we see here.

00:02:45.789 --> 00:02:48.727
These things that seem so different.

00:02:48.727 --> 00:02:51.586
Well a lot of this is different
compositions of elements

00:02:51.586 --> 00:02:54.652
like carbon and oxygen and hydrogen.

00:02:54.652 --> 00:02:56.936
And even the elements
themselves are made of things

00:02:56.936 --> 00:03:00.212
like protons and electrons and neutrons

00:03:00.212 --> 00:03:02.284
that are just rearranged in different ways

00:03:02.284 --> 00:03:04.024
to give us these elements

00:03:04.024 --> 00:03:07.691
that have all of these
different properties.

00:03:08.761 --> 00:03:10.920
So when you think about chemistry,

00:03:10.920 --> 00:03:15.562
yes, it might visually
look something like this.

00:03:15.562 --> 00:03:17.426
These are obviously much older pictures.

00:03:17.426 --> 00:03:20.778
But at its essence, it's
how do we create models

00:03:20.778 --> 00:03:23.224
and understand the models that describe

00:03:23.224 --> 00:03:27.192
a lot of the complexity
in the universe around us?

00:03:27.192 --> 00:03:29.767
And just to put chemistry in,

00:03:29.767 --> 00:03:31.630
I guess you could say, in context

00:03:31.630 --> 00:03:34.188
with some of the other sciences,

00:03:34.188 --> 00:03:37.291
many people would say at the purest level,

00:03:37.291 --> 00:03:40.019
you would have mathematics.

00:03:40.019 --> 00:03:42.769
That math, you're studying ideas,

00:03:43.956 --> 00:03:46.141
which could even be independent,

00:03:46.141 --> 00:03:48.412
you're seeing logical ideas
that could be even independent

00:03:48.412 --> 00:03:51.941
of anything that you've ever
observed or experienced.

00:03:51.941 --> 00:03:54.462
And a lot of folks that
say if we ever communicate

00:03:54.462 --> 00:03:56.201
with another intelligent species

00:03:56.201 --> 00:03:58.111
that could be completely
different than us,

00:03:58.111 --> 00:04:00.517
math might be that common language.

00:04:00.517 --> 00:04:03.541
Because even if we perceive
the world differently

00:04:03.541 --> 00:04:05.157
and think differently in certain ways,

00:04:05.157 --> 00:04:07.765
math might be that common language.

00:04:07.765 --> 00:04:10.669
But on top of math, we start to say,

00:04:10.669 --> 00:04:13.931
well how is our reality
actually structured?

00:04:13.931 --> 00:04:15.469
At the most basic level,

00:04:15.469 --> 00:04:17.168
what are the constituents of matter

00:04:17.168 --> 00:04:19.567
and what are the mathematical properties

00:04:19.567 --> 00:04:22.380
that describe how they react together?

00:04:22.380 --> 00:04:24.831
And then, or interact with each other?

00:04:24.831 --> 00:04:26.986
Then you go one level above that,

00:04:26.986 --> 00:04:30.921
you get to the topic of this
video, which is chemistry.

00:04:30.921 --> 00:04:34.228
Which is very closely related to physics.

00:04:34.228 --> 00:04:36.345
When we talk about
these chemical equations

00:04:36.345 --> 00:04:38.870
and we create these molecular structures,

00:04:38.870 --> 00:04:41.929
the interactions between these atoms,

00:04:41.929 --> 00:04:45.293
these are quantum mechanical interactions

00:04:45.293 --> 00:04:49.640
which we do not fully understand
at the deepest level yet.

00:04:49.640 --> 00:04:52.343
But with chemistry, we
can start to make use

00:04:52.343 --> 00:04:55.201
of the math and they physics
to start to think about

00:04:55.201 --> 00:04:56.976
how all of these different building blocks

00:04:56.976 --> 00:05:00.584
can interact to explain all
sorts of different phenomena.

00:05:00.584 --> 00:05:02.406
This chemical equation you see right here,

00:05:02.406 --> 00:05:03.653
this is combustion.

00:05:03.653 --> 00:05:05.929
This is hydrogen combusting with oxygen

00:05:05.929 --> 00:05:07.793
to produce a lot of energy.

00:05:07.793 --> 00:05:09.293
To produce energy.

00:05:10.705 --> 00:05:12.784
When we imagine combustion,
we think of fire.

00:05:12.784 --> 00:05:16.343
But what even is fire at
its most fundamental level?

00:05:16.343 --> 00:05:17.378
How do we get,

00:05:17.378 --> 00:05:19.941
why do we perceive this thing here?

00:05:19.941 --> 00:05:21.400
And chemistry is super important

00:05:21.400 --> 00:05:25.335
because on top of that, we build biology.

00:05:25.335 --> 00:05:26.752
We build biology.

00:05:27.737 --> 00:05:29.892
And as you'll see as you
study all of these things,

00:05:29.892 --> 00:05:32.752
there's points where these
things start to bleed together.

00:05:32.752 --> 00:05:34.992
But the biology in, say, a human being,

00:05:34.992 --> 00:05:36.942
or really in any species,

00:05:36.942 --> 00:05:39.297
it's based on molecular interactions.

00:05:39.297 --> 00:05:42.230
Interactions between
molecules, between atoms,

00:05:42.230 --> 00:05:45.378
which, at the end of the
day, is all about chemistry.

00:05:45.378 --> 00:05:47.948
As I speak, the only reason
why I'm able to speak

00:05:47.948 --> 00:05:50.516
is because of really,

00:05:50.516 --> 00:05:53.991
hard to imagine the number
of chemical interactions

00:05:53.991 --> 00:05:56.776
happening in me right now

00:05:56.776 --> 00:05:58.560
to create this soundness.

00:05:58.560 --> 00:06:00.841
To create this thing that thinks it exists

00:06:00.841 --> 00:06:03.575
that wants to make a
video about how awesome

00:06:03.575 --> 00:06:05.642
and amazing chemistry is.

00:06:05.642 --> 00:06:06.759
And then from biology,

00:06:06.759 --> 00:06:08.826
you can build out on
all of everything else.

00:06:08.826 --> 00:06:11.812
So sciences like psychology
and economics, which of course,

00:06:11.812 --> 00:06:14.749
these things also leverage
math and other things.

00:06:14.749 --> 00:06:17.742
But this gives you kind of a sense

00:06:17.742 --> 00:06:22.087
of how we build up and how we
explain the reality around us.

00:06:22.087 --> 00:06:25.025
And not one of these is more
important than the other.

00:06:25.025 --> 00:06:27.961
These are all studying
incredibly fascinating things

00:06:27.961 --> 00:06:31.607
that as humans beings
first became thoughtful

00:06:31.607 --> 00:06:33.268
about their environment,

00:06:33.268 --> 00:06:35.384
said, "Gee, why are we here?

00:06:35.384 --> 00:06:36.417
"What is this place?

00:06:36.417 --> 00:06:37.905
"Why do we exist?

00:06:37.905 --> 00:06:39.852
"How do we exist?"

00:06:39.852 --> 00:06:42.253
And chemistry builds models

00:06:42.253 --> 00:06:45.602
for us to understand interactions
at a scale and a speed

00:06:45.602 --> 00:06:47.338
that we can't directly observe,

00:06:47.338 --> 00:06:50.397
but nonetheless, we can to
start to make predictions.

00:06:50.397 --> 00:06:52.253
So that's what's really cool about this.

00:06:52.253 --> 00:06:53.287
When you study chemistry,

00:06:53.287 --> 00:06:56.276
you should not view this
as some type of a chore

00:06:56.276 --> 00:07:00.294
that the school system
is forcing you through.

00:07:00.294 --> 00:07:04.100
There are people who would've
done anything 100 years ago

00:07:04.100 --> 00:07:07.544
to get the answers that are
in your chemistry book today

00:07:07.544 --> 00:07:09.488
or that you can learn from
your chemistry teacher

00:07:09.488 --> 00:07:11.891
or that you can learn
from a Khan Academy video.

00:07:11.891 --> 00:07:14.583
There are people in the
world in the past and today

00:07:14.583 --> 00:07:16.872
who'd do anything to be
able to understand deeply

00:07:16.872 --> 00:07:17.822
what this is.

00:07:17.822 --> 00:07:19.393
That they consider it a privilege

00:07:19.393 --> 00:07:22.226
to be able to learn at this level.

00:07:23.098 --> 00:07:25.030
And then to think about
where this could go

00:07:25.030 --> 00:07:27.017
because none of these fields are complete.

00:07:27.017 --> 00:07:30.861
We have very partial knowledge
of all of these fields.

00:07:30.861 --> 00:07:33.542
Arguably, there's an infinite
more that we could learn

00:07:33.542 --> 00:07:35.201
relative to what we know.

00:07:35.201 --> 00:07:37.980
But what's exciting is that
we have such a strong start.

00:07:37.980 --> 00:07:39.630
We're starting to make sense of it.

00:07:39.630 --> 00:07:43.380
To really describe
everything in our reality.