Inflection points (algebraic)

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:02.694
- [Voiceover] Let G of X equal
one fourth X to the fourth,

00:00:02.694 --> 00:00:05.991
minus four X to the third
power, plus 24 X squared.

00:00:05.991 --> 00:00:09.589
For what values of X does the graph of G

00:00:09.589 --> 00:00:12.013
have an inflection point?

00:00:12.013 --> 00:00:13.776
Or have a point of inflection?

00:00:13.776 --> 00:00:15.275
So let's just remind ourselves

00:00:15.275 --> 00:00:17.451
what a point of inflection is.

00:00:17.451 --> 00:00:21.718
A point of inflection is
where we go from being con,

00:00:21.718 --> 00:00:23.284
where we change our concavity.

00:00:23.284 --> 00:00:25.549
Or you can say where our second derivative

00:00:25.549 --> 00:00:28.279
G prime of X switches signs.

00:00:28.279 --> 00:00:30.362
Switches, switches signs.

00:00:31.735 --> 00:00:33.722
So let's study our second derivative.

00:00:33.722 --> 00:00:35.143
In order to study or
secondary, let's find it.

00:00:35.143 --> 00:00:38.893
So we know that G of X
is equal to one fourth

00:00:41.786 --> 00:00:45.332
X to the fourth, minus
four X to the third power,

00:00:45.332 --> 00:00:46.832
plus 24 X squared.

00:00:47.758 --> 00:00:51.638
So given that, let's
now find G prime of X.

00:00:51.638 --> 00:00:53.869
G prime of X is going to be equal to,

00:00:53.869 --> 00:00:55.908
I'm just going to apply the
power rule multiple times.

00:00:55.908 --> 00:00:57.920
Four times one fourth is just one.

00:00:57.920 --> 00:01:00.206
I'm not going to write that one down,

00:01:00.206 --> 00:01:04.154
it's going to be one times X
to the four minus one power.

00:01:04.154 --> 00:01:06.379
So four to the third power,

00:01:06.379 --> 00:01:08.796
minus three times four is 12,

00:01:10.254 --> 00:01:12.704
X to three minus one power,

00:01:12.704 --> 00:01:16.454
X to the second power,
plus two times 24, 48.

00:01:19.624 --> 00:01:22.239
X to the two minus one,
or X to the first power,

00:01:22.239 --> 00:01:24.259
I can just write that as X.

00:01:24.259 --> 00:01:25.216
So there you have it.

00:01:25.216 --> 00:01:26.133
I have our first derivative,

00:01:26.133 --> 00:01:27.485
now we want to find our second derivative.

00:01:27.485 --> 00:01:29.584
G prime prime of X is just the derivative

00:01:29.584 --> 00:01:31.330
of the first derivative
of the first derivative

00:01:31.330 --> 00:01:33.392
with respect to X, and some
more of the power rule,

00:01:33.392 --> 00:01:35.559
three X squared minus 24 X

00:01:41.717 --> 00:01:45.258
to the first, or just 24 X, plus 48.

00:01:45.258 --> 00:01:49.544
So let's think about where
this switch is signed.

00:01:49.544 --> 00:01:51.529
And this is, this is
a continuous function,

00:01:51.529 --> 00:01:53.577
is going to be defined for all Xs,

00:01:53.577 --> 00:01:55.226
so the only potential candidates

00:01:55.226 --> 00:01:57.113
of where it could switch signs

00:01:57.113 --> 00:01:59.586
are when this thing equals zero.

00:01:59.586 --> 00:02:02.193
So let's see where it equals zero.

00:02:02.193 --> 00:02:04.303
So let's set that equal to zero.

00:02:04.303 --> 00:02:07.136
Three X squared minus 24 X plus 48

00:02:08.113 --> 00:02:10.945
is equal to zero, let's see,

00:02:10.945 --> 00:02:12.259
everything is divisible by three

00:02:12.259 --> 00:02:13.659
so let's divide everything by three.

00:02:13.659 --> 00:02:16.655
So you get X squared minus eight X,

00:02:16.655 --> 00:02:19.488
plus 16, plus 16 is equal to zero.

00:02:24.563 --> 00:02:26.343
And let's see, can I factor this?

00:02:26.343 --> 00:02:28.838
Yeah, this would be X minus
four X times X minus 4.

00:02:28.838 --> 00:02:32.780
Or just you do this as
X minus four squared

00:02:32.780 --> 00:02:37.496
is equal to zero, or X
minus four is equal to zero.

00:02:37.496 --> 00:02:39.746
So, or where X equals four.

00:02:41.048 --> 00:02:44.456
So G prime prime of four is equal to zero.

00:02:44.456 --> 00:02:47.227
So let's see what's happening
on either side of that.

00:02:47.227 --> 00:02:49.175
Let's see if G, if we're actually,

00:02:49.175 --> 00:02:51.648
if we're actually switching signs or not.

00:02:51.648 --> 00:02:53.772
So let me draw a number line here.

00:02:53.772 --> 00:02:56.939
And so this is, so this is two, three,

00:02:59.724 --> 00:03:02.641
four, five, and I could keep going.

00:03:04.299 --> 00:03:06.422
And so we know that something interesting

00:03:06.422 --> 00:03:08.097
is happening right over here,

00:03:08.097 --> 00:03:10.090
G prime prime of four is equal to zero,

00:03:10.090 --> 00:03:13.096
G prime prime of four is equal to zero.

00:03:13.096 --> 00:03:15.901
So let's think about what
the second derivative is

00:03:15.901 --> 00:03:17.672
when we are less than four.

00:03:17.672 --> 00:03:21.612
And so, actually let me just
try G prime prime of zero,

00:03:21.612 --> 00:03:23.407
since that will be easy to evaluate.

00:03:23.407 --> 00:03:25.202
G prime prime of zero,

00:03:25.202 --> 00:03:27.750
well it's just going to be equal to 48.

00:03:27.750 --> 00:03:30.291
So when we are less than four,

00:03:30.291 --> 00:03:33.568
our second derivative, G
prime, the second derivative,

00:03:33.568 --> 00:03:35.318
is greater than zero.

00:03:36.466 --> 00:03:40.131
So we're actually going
to be concave upwards

00:03:40.131 --> 00:03:44.061
over this interval to the left of four.

00:03:44.061 --> 00:03:46.717
Now let's think about
to the right of four.

00:03:46.717 --> 00:03:49.572
Two, use a different color.

00:03:49.572 --> 00:03:52.496
So what about to the right of four?

00:03:52.496 --> 00:03:54.604
And so let me just evaluate,

00:03:54.604 --> 00:03:56.123
what would be the easy thing to evaluate?

00:03:56.123 --> 00:03:59.240
Well I could evaluate G prime,

00:03:59.240 --> 00:04:02.573
why not do, or of the second derivative,

00:04:03.665 --> 00:04:05.783
G prime prime, I should say, of 10?

00:04:05.783 --> 00:04:08.924
So, G, I'll do it right over here,

00:04:08.924 --> 00:04:10.242
Let me do it, well,

00:04:10.242 --> 00:04:12.767
I'm running a little bit of
space, so I'll just scroll down.

00:04:12.767 --> 00:04:16.955
So G prime prime of 10 is going
to be equal to three times

00:04:16.955 --> 00:04:20.455
10 squared, so it's 300 minus 24 times 10.

00:04:21.585 --> 00:04:24.238
So minus 240, plus 48.

00:04:24.238 --> 00:04:26.579
So let's see, this is 60.

00:04:26.579 --> 00:04:29.912
This is, so 300 minus 240 is 60 plus 48,

00:04:31.975 --> 00:04:33.824
so this is equal to 108.

00:04:33.824 --> 00:04:35.655
So it still positive.

00:04:35.655 --> 00:04:40.135
So on either side of
four, G prime prime of X

00:04:40.135 --> 00:04:42.453
is greater than zero.

00:04:42.453 --> 00:04:44.591
So even though even though
the second derivative

00:04:44.591 --> 00:04:48.130
at X equals four is equal to zero,

00:04:48.130 --> 00:04:51.331
on either side we are concave upwards.

00:04:51.331 --> 00:04:53.667
On either side, the second
derivative is positive.

00:04:53.667 --> 00:04:57.153
And so, and that was the
only potential candidate.

00:04:57.153 --> 00:05:00.502
So there are no, there are no values of X

00:05:00.502 --> 00:05:04.121
for which G has a point of inflection.

00:05:04.121 --> 00:05:05.987
X equals four would have been a value of X

00:05:05.987 --> 00:05:07.449
at which G had a point of inflection,

00:05:07.449 --> 00:05:09.396
if we switch, if the secondary derivative

00:05:09.396 --> 00:05:11.919
switched signs here, if it
went from positive to negative,

00:05:11.919 --> 00:05:12.957
or negative to positive,

00:05:12.957 --> 00:05:16.107
but it's just staying
from positive to positive.

00:05:16.107 --> 00:05:17.869
So this second derivative was positive,

00:05:17.869 --> 00:05:19.122
it just touches zero right here,

00:05:19.122 --> 00:05:20.587
and then it goes positive again.

00:05:20.587 --> 00:05:22.342
So we're going back to the question,

00:05:22.342 --> 00:05:24.505
for what X values does the graph of G

00:05:24.505 --> 00:05:25.798
have a point of inflection?

00:05:25.798 --> 00:05:26.798
No X values.

00:05:28.698 --> 00:05:32.865
I'll put an exclamation
mark there just more drama.


Planar motion example: acceleration vector

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.260
- [Voiceover] A particle
moves in the xy plane so that

00:00:02.260 --> 00:00:04.615
at any time t is greater
than or equal to zero

00:00:04.615 --> 00:00:07.584
its position vector is,
and they give us the

00:00:07.584 --> 00:00:09.417
x component and the y component

00:00:09.417 --> 00:00:12.108
of our position vectors, and
they're both functions of time.

00:00:12.108 --> 00:00:14.328
What is the particle's acceleration

00:00:14.328 --> 00:00:16.775
vector at time t equals three?

00:00:16.775 --> 00:00:19.170
Alright, so our position, let's denote that

00:00:19.170 --> 00:00:20.924
it's a vector valued function,

00:00:20.924 --> 00:00:22.512
it's gonna be a function of time.

00:00:22.512 --> 00:00:24.016
It is a vector.

00:00:24.016 --> 00:00:26.169
And they already told
us that the x component

00:00:26.169 --> 00:00:29.867
of our position is negative
three t to the third power

00:00:29.867 --> 00:00:33.360
plus four t-squared, and the y component

00:00:33.360 --> 00:00:36.110
is t to the third power plus two.

00:00:37.103 --> 00:00:38.975
And so you give me any time greater than

00:00:38.975 --> 00:00:41.128
or equal to zero, I put it in here

00:00:41.128 --> 00:00:43.930
and I can give you the
corresponding x and y components.

00:00:43.930 --> 00:00:45.389
And this is one form of notation

00:00:45.389 --> 00:00:48.889
for a vector, another way of writing this,

00:00:50.305 --> 00:00:52.498
you might be familiar
with engineering notation.

00:00:52.498 --> 00:00:53.841
It might be written like, or sometimes

00:00:53.841 --> 00:00:55.672
people write this as unit vector notation.

00:00:55.672 --> 00:00:59.213
Negative three to the
third plus four t-squared

00:00:59.213 --> 00:01:02.180
times the unit vector in
the horizontal direction,

00:01:02.180 --> 00:01:06.648
plus t to the third plus
two, times the unit vector

00:01:06.648 --> 00:01:09.090
in the vertical direction.

00:01:09.090 --> 00:01:11.446
This is just denoting the same thing.

00:01:11.446 --> 00:01:13.273
This is the x component,
this is the y component.

00:01:13.273 --> 00:01:14.896
This is the component in
the horizontal direction,

00:01:14.896 --> 00:01:17.092
this is the component in
the vertical direction,

00:01:17.092 --> 00:01:19.166
or the y component.

00:01:19.166 --> 00:01:20.910
Now the key realization is if you have

00:01:20.910 --> 00:01:23.677
the position vector, well
the velocity vector's

00:01:23.677 --> 00:01:25.994
just going to be the derivative of that.

00:01:25.994 --> 00:01:30.161
So V of t is just going to
be equal to r prime of t,

00:01:37.423 --> 00:01:39.782
which is going to be equal to,

00:01:39.782 --> 00:01:41.531
well you just have to
take the corresponding

00:01:41.531 --> 00:01:43.894
derivatives of each of the components.

00:01:43.894 --> 00:01:45.684
So let's do that.

00:01:45.684 --> 00:01:47.475
So if we wanna take the derivative

00:01:47.475 --> 00:01:50.033
of the x component here
with respect to time,

00:01:50.033 --> 00:01:51.867
we're just gonna use
the power rule a bunch.

00:01:51.867 --> 00:01:53.494
So it's three times negative three,

00:01:53.494 --> 00:01:56.077
so it's negative nine t-squared

00:01:56.912 --> 00:01:59.231
and then plus two times four is eight,

00:01:59.231 --> 00:02:01.792
so plus eight t to the first.

00:02:01.792 --> 00:02:03.500
So plus eight t.

00:02:03.500 --> 00:02:07.399
And then over here for the y component.

00:02:07.399 --> 00:02:10.361
So the derivative of t to the third

00:02:10.361 --> 00:02:12.594
with respect to t is three t-squared.

00:02:12.594 --> 00:02:14.344
Three t-squared, and then the derivative

00:02:14.344 --> 00:02:16.341
of two is just a zero, so I actually

00:02:16.341 --> 00:02:18.865
have space to write that
three t-squared even bigger.

00:02:18.865 --> 00:02:21.752
Three t-squared, alright.

00:02:21.752 --> 00:02:24.599
And if we wanna find the
acceleration function,

00:02:24.599 --> 00:02:27.079
or the vector valued function

00:02:27.079 --> 00:02:29.963
that gives us acceleration
as a function of time,

00:02:29.963 --> 00:02:31.384
well that's just going to be

00:02:31.384 --> 00:02:34.225
the derivative of the velocity function

00:02:34.225 --> 00:02:35.894
with respect to time.

00:02:35.894 --> 00:02:37.402
So this is going to be equal to,

00:02:37.402 --> 00:02:39.819
this is going to be equal to,

00:02:40.819 --> 00:02:42.647
let me give myself some space.

00:02:42.647 --> 00:02:45.003
And so the x component, well I just take

00:02:45.003 --> 00:02:48.011
the derivative of the x component again,

00:02:48.011 --> 00:02:50.249
and let me find a color
I haven't used yet,

00:02:50.249 --> 00:02:52.117
I'll use this green, so let's see.

00:02:52.117 --> 00:02:55.855
Two times negative nine, negative 18

00:02:55.855 --> 00:02:58.996
times t to the first power plus eight,

00:02:58.996 --> 00:03:01.123
derivative of eight t is just eight

00:03:01.123 --> 00:03:01.956
if we're taking the derivative

00:03:01.956 --> 00:03:03.066
with respect to t.

00:03:03.066 --> 00:03:06.281
And then here in the orange, derivative of

00:03:06.281 --> 00:03:08.602
three t-squared, so it's
two power rule here,

00:03:08.602 --> 00:03:09.783
over and over again.

00:03:09.783 --> 00:03:14.742
Two times three is six t to
the first power, just six t.

00:03:14.742 --> 00:03:16.937
So this is, we've just been able

00:03:16.937 --> 00:03:19.054
to by taking the derivative of

00:03:19.054 --> 00:03:22.470
this position vector
valued function twice,

00:03:22.470 --> 00:03:25.760
I'm able to find the
acceleration function.

00:03:25.760 --> 00:03:28.768
And now I just have to
evaluate it at t equals three.

00:03:28.768 --> 00:03:32.793
So our acceleration at t is equal to three

00:03:32.793 --> 00:03:36.210
is equal to, so in green it's going to be

00:03:37.753 --> 00:03:41.170
negative eighteen times three plus eight,

00:03:43.727 --> 00:03:46.394
comma, and then we're gonna have

00:03:47.629 --> 00:03:50.379
six times three, six times three.

00:03:52.631 --> 00:03:54.951
And so what does this simplify to?

00:03:54.951 --> 00:03:56.089
Well this is going to be equal to,

00:03:56.089 --> 00:03:59.585
let's see, negative 18 times three is

00:03:59.585 --> 00:04:02.502
negative 54, negative 54 plus eight

00:04:04.425 --> 00:04:05.675
is negative 46.

00:04:06.946 --> 00:04:11.183
Negative 46, and then six
times three is eighteen.

00:04:11.183 --> 00:04:12.690
Did I do that arithmetic right?

00:04:12.690 --> 00:04:16.189
So this is negative 54 plus eight.

00:04:16.189 --> 00:04:19.599
So negative 54 plus four
would be negative 50,

00:04:19.599 --> 00:04:21.549
plus another four would be negative 46.

00:04:21.549 --> 00:04:22.889
Yep, there you have it.

00:04:22.889 --> 00:04:25.222
Negative 46, comma eighteen.

00:04:26.749 --> 00:04:29.332
That is its acceleration vector

00:04:30.208 --> 00:04:32.208
at t is equal to three.


Inflection points (graphical)

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:01.653
- [Voiceover] We're told let g be

00:00:01.653 --> 00:00:03.525
a differentiable function defined

00:00:03.525 --> 00:00:06.615
over the closed interval
from negative four to four.

00:00:06.615 --> 00:00:10.277
The graph of g is given
right over here, given below.

00:00:10.277 --> 00:00:13.854
How many inflection points
does the graph of g have?

00:00:13.854 --> 00:00:17.506
So let's just remind ourselves,
what are inflection points?

00:00:17.506 --> 00:00:20.715
So inflection points are
where we change concavity.

00:00:20.715 --> 00:00:22.548
So we go from concave,

00:00:23.513 --> 00:00:27.323
concave upwards, upwards,
actually let me just

00:00:27.323 --> 00:00:28.931
draw it graphically.

00:00:28.931 --> 00:00:33.098
We're going from concave
upwards to concave downwards,

00:00:34.274 --> 00:00:37.607
or concave downwards to concave upwards.

00:00:40.359 --> 00:00:42.823
Or another way you could think about it,

00:00:42.823 --> 00:00:47.463
you could say we're going
from our slope increasing,

00:00:47.463 --> 00:00:49.380
increasing, increasing,

00:00:50.338 --> 00:00:52.748
to our slope decreasing.

00:00:52.748 --> 00:00:54.748
To our slope decreasing,

00:00:56.228 --> 00:00:57.496
or the other way around.

00:00:57.496 --> 00:01:00.395
Any points where your
slope goes from decreasing,

00:01:00.395 --> 00:01:04.396
our slope goes from
decreasing, to increasing.

00:01:04.396 --> 00:01:06.312
To increasing.

00:01:06.312 --> 00:01:07.824
So let's think about that.

00:01:07.824 --> 00:01:11.297
So as we start off right over here,

00:01:11.297 --> 00:01:13.357
at the extreme left it's seems like

00:01:13.357 --> 00:01:15.048
we have a very high slope.

00:01:15.048 --> 00:01:16.889
It's a very steep curve,

00:01:16.889 --> 00:01:20.209
and then it stays increasing
but it's getting less positive.

00:01:20.209 --> 00:01:21.861
So it's getting a little bit,

00:01:21.861 --> 00:01:23.258
it's getting a little bit flatter,

00:01:23.258 --> 00:01:26.743
so our slope is at a very high
level but it's decreasing.

00:01:26.743 --> 00:01:29.274
It's decreasing, decreasing, decreasing,

00:01:29.274 --> 00:01:33.304
slope is decreasing, decreasing even more.

00:01:33.304 --> 00:01:36.185
It's even more and then
it's actually going to zero,

00:01:36.185 --> 00:01:38.058
our slope is zero and
then it becomes negative.

00:01:38.058 --> 00:01:40.205
So our slope is still decreasing.

00:01:40.205 --> 00:01:45.117
Then it's becoming more
and more and more negative,

00:01:45.117 --> 00:01:49.331
and then right around, and
then right around here,

00:01:49.331 --> 00:01:53.323
it looks like it starts
becoming less negative,

00:01:53.323 --> 00:01:55.200
or starts increasing.

00:01:55.200 --> 00:01:57.741
So our slope is increasing increasing,

00:01:57.741 --> 00:01:59.987
it's really just becoming
less and less negative,

00:01:59.987 --> 00:02:02.547
and then it's going close to zero,

00:02:02.547 --> 00:02:04.334
approaching zero, it looks like our slope

00:02:04.334 --> 00:02:06.489
is zero right over here,

00:02:06.489 --> 00:02:07.705
but then it looks like right over there

00:02:07.705 --> 00:02:10.474
our slope begins decreasing again.

00:02:10.474 --> 00:02:14.444
So it looks like our
slope is decreasing again.

00:02:14.444 --> 00:02:16.915
So it looks like our slope is decreasing.

00:02:16.915 --> 00:02:20.800
It's becoming more and more
and more and more negative,

00:02:20.800 --> 00:02:22.758
and so it looks like
something interesting happened

00:02:22.758 --> 00:02:24.774
right over there, we
got a transition point,

00:02:24.774 --> 00:02:27.471
and then right around here,
it looks like it starts,

00:02:27.471 --> 00:02:30.096
the slope starts increasing again.

00:02:30.096 --> 00:02:33.720
So it looks like the
slope starts increasing.

00:02:33.720 --> 00:02:35.492
It's negative but it's becoming less

00:02:35.492 --> 00:02:38.017
and less and less negative
and then it becomes zero

00:02:38.017 --> 00:02:39.418
and then it becomes positive and then

00:02:39.418 --> 00:02:42.835
more and more and more and more positive.

00:02:44.063 --> 00:02:46.153
So inflection points are where we go

00:02:46.153 --> 00:02:49.653
from slope increasing to slope decreasing.

00:02:51.193 --> 00:02:54.087
So concave upwards to concave downwards,

00:02:54.087 --> 00:02:57.377
and so slope increasing was
here to slope decreasing,

00:02:57.377 --> 00:02:59.162
so this was an inflection point,

00:02:59.162 --> 00:03:02.126
and also from slope decreasing
to slope increasing.

00:03:02.126 --> 00:03:05.089
So that's slope decreasing
to slope increasing,

00:03:05.089 --> 00:03:08.582
and this is also slope
decreasing to slope increasing.

00:03:08.582 --> 00:03:11.810
So how many inflection points
does the graph of g have

00:03:11.810 --> 00:03:13.986
that we can see on this graph?

00:03:13.986 --> 00:03:17.112
Well it has three over the interval

00:03:17.112 --> 00:03:19.195
that at least we can see.


Worked example: Inflection points from first derivative

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:01.323
- [Voiceover] So we're told

00:00:01.323 --> 00:00:02.892
let g be a differentiable function

00:00:02.892 --> 00:00:06.253
defined over the closed interval
from negative six to six.

00:00:06.253 --> 00:00:08.435
The graph of its derivative,

00:00:08.435 --> 00:00:10.871
so they're giving the
graphing the derivative of g,

00:00:10.871 --> 00:00:12.174
g prime is given below.

00:00:12.174 --> 00:00:13.471
So this isn't the graph of g.

00:00:13.471 --> 00:00:15.539
This is the graph of g prime.

00:00:15.539 --> 00:00:19.706
What is the x value of the
left-most inflection point,

00:00:20.602 --> 00:00:23.331
inflection point in the graph of g.

00:00:23.331 --> 00:00:25.452
So they want, they don't
want to know the x value

00:00:25.452 --> 00:00:27.750
of the inflection points
in the graph of g prime,

00:00:27.750 --> 00:00:29.029
in this graph.

00:00:29.029 --> 00:00:30.362
They want to know the inflection points,

00:00:30.362 --> 00:00:34.121
the x values of the inflection
points, in the graph of g.

00:00:34.121 --> 00:00:36.107
And we have to figure
out the left-most one.

00:00:36.107 --> 00:00:37.900
So, let me just make a little table here,

00:00:37.900 --> 00:00:41.197
to think about what is
happening at inflection points

00:00:41.197 --> 00:00:44.398
in our second derivative,
our first derivative,

00:00:44.398 --> 00:00:46.970
and our actual function.

00:00:46.970 --> 00:00:49.747
So, this is g prime, prime.

00:00:49.747 --> 00:00:50.748
This is g prime.

00:00:50.748 --> 00:00:52.467
And this is our actual, I
guess you could call it,

00:00:52.467 --> 00:00:54.058
the original function.

00:00:54.058 --> 00:00:55.918
So an inflection point are points

00:00:55.918 --> 00:00:58.908
where our second derivative
is switching sides.

00:00:58.908 --> 00:01:01.947
It's going from positive
negative or negative to positive.

00:01:01.947 --> 00:01:03.778
So, let's consider that first scenario.

00:01:03.778 --> 00:01:07.826
So g, so going from positive to negative.

00:01:07.826 --> 00:01:10.493
Positive, positive, to negative.

00:01:14.026 --> 00:01:14.918
So if g prime prime,

00:01:14.918 --> 00:01:17.298
if the second derivative's
going from positive to negative,

00:01:17.298 --> 00:01:19.719
what is the first derivative doing?

00:01:19.719 --> 00:01:21.069
Well, remember, the second derivative,

00:01:21.069 --> 00:01:24.128
is the derivative of the first derivative.

00:01:24.128 --> 00:01:27.268
So, where the second
derivative is positive,

00:01:27.268 --> 00:01:28.717
where the second derivative is positive,

00:01:28.717 --> 00:01:32.226
that means that the first
derivative is increasing.

00:01:32.226 --> 00:01:35.767
So, if second derivative's
going from positive to negative,

00:01:35.767 --> 00:01:36.600
that means first derivative

00:01:36.600 --> 00:01:39.850
is going from increasing to decreasing.

00:01:41.107 --> 00:01:43.197
From increasing to decreasing.

00:01:43.197 --> 00:01:44.957
And the function itself,

00:01:44.957 --> 00:01:47.478
well when the second
derivative is positive,

00:01:47.478 --> 00:01:50.488
we are going to be, that means,

00:01:50.488 --> 00:01:54.999
that means that the slope
is constantly increasing.

00:01:54.999 --> 00:01:57.797
And so that means that
we are concave upwards.

00:01:57.797 --> 00:01:59.464
So, concave upwards.

00:02:02.377 --> 00:02:04.127
Upwards to downwards.

00:02:05.549 --> 00:02:08.707
To concave, to concave downwards.

00:02:08.707 --> 00:02:11.478
But they've given us the graph of g prime.

00:02:11.478 --> 00:02:14.004
So let's focus on what are the points

00:02:14.004 --> 00:02:17.626
where g prime is going from
increasing to decreasing.

00:02:17.626 --> 00:02:18.459
So let's see.

00:02:18.459 --> 00:02:20.848
G prime is increasing,
increasing, increasing,

00:02:20.848 --> 00:02:23.388
increasing, increasing at a slower rate,

00:02:23.388 --> 00:02:24.729
and then it starts decreasing.

00:02:24.729 --> 00:02:25.864
So, right over there

00:02:25.864 --> 00:02:27.966
it's going from increasing to decreasing.

00:02:27.966 --> 00:02:29.924
So then it's decreasing,
decreasing, decreasing.

00:02:29.924 --> 00:02:31.846
Then it goes increasing,
increasing, increasing,

00:02:31.846 --> 00:02:34.113
increasing, and then decreasing again.

00:02:34.113 --> 00:02:35.535
So that's another point where we're going

00:02:35.535 --> 00:02:37.833
from increasing to decreasing.

00:02:37.833 --> 00:02:38.666
And those are the only ones

00:02:38.666 --> 00:02:41.174
that look like we're going
from increasing to decreasing.

00:02:41.174 --> 00:02:42.765
But we're not done yet.

00:02:42.765 --> 00:02:44.564
Because it's not just about going from

00:02:44.564 --> 00:02:46.773
the second derivative going
from positive to negative,

00:02:46.773 --> 00:02:47.853
it's also the other way around.

00:02:47.853 --> 00:02:50.952
Any time the second
derivative is switching signs.

00:02:50.952 --> 00:02:52.232
So, it's also the situation

00:02:52.232 --> 00:02:55.899
where we're going from
negative to positive.

00:02:56.992 --> 00:03:01.933
Or, for the first derivative
is going from decreasing,

00:03:01.933 --> 00:03:04.792
decreasing to increasing.

00:03:04.792 --> 00:03:06.751
Decreasing to increasing.

00:03:06.751 --> 00:03:08.223
Well let's see we're
decreasing, decreasing,

00:03:08.223 --> 00:03:10.624
decreasing, and then
we're increasing, alright.

00:03:10.624 --> 00:03:12.076
So it's right there.

00:03:12.076 --> 00:03:14.567
And then we're increasing, decreasing,

00:03:14.567 --> 00:03:17.225
decreasing, decreasing,
and then we're increasing.

00:03:17.225 --> 00:03:18.254
So right over there.

00:03:18.254 --> 00:03:19.852
So these are the inflection points

00:03:19.852 --> 00:03:22.523
that I've just figured out visually.

00:03:22.523 --> 00:03:24.404
So, if you look at the choices,

00:03:24.404 --> 00:03:26.636
if we want to answer
the original question,

00:03:26.636 --> 00:03:30.488
well the left-most one is it
x is equal to negative three?

00:03:30.488 --> 00:03:32.377
It's x equals negative three.

00:03:32.377 --> 00:03:35.677
X equals negative one is indeed a x value,

00:03:35.677 --> 00:03:37.484
where we have an inflection point.

00:03:37.484 --> 00:03:39.467
And let's see, x equals two is one,

00:03:39.467 --> 00:03:40.386
and so is x equals four.

00:03:40.386 --> 00:03:43.466
So they actually listed, all
of these are inflection points.

00:03:43.466 --> 00:03:46.716
And they just wanted the left-most one.


Motion problems: finding the maximum acceleration

WEBVTT
Kind: captions
Language: en

00:00:00.305 --> 00:00:03.131
- [Voiceover] A particle
moves along the x-axis

00:00:03.131 --> 00:00:06.269
so that at any time T
greater than or equal to zero

00:00:06.269 --> 00:00:08.722
its velocity is given
by V of T is equal to

00:00:08.722 --> 00:00:12.889
negative T to the third power
plus six T squared plus two T.

00:00:14.089 --> 00:00:16.124
At what value of T does the particle

00:00:16.124 --> 00:00:19.128
obtain its maximum acceleration?

00:00:19.128 --> 00:00:20.459
So we want to figure
out when does it obtain

00:00:20.459 --> 00:00:22.355
its maximum acceleration.

00:00:22.355 --> 00:00:23.503
So let's just review what they gave us.

00:00:23.503 --> 00:00:26.944
They gave us velocity
as a function of time.

00:00:26.944 --> 00:00:28.440
So let's just remind ourselves.

00:00:28.440 --> 00:00:32.392
If we have let's say our
position is a function of time,

00:00:32.392 --> 00:00:35.943
so let's say X of T is
position as a function of time,

00:00:35.943 --> 00:00:38.213
then we if were to take
the derivative of that,

00:00:38.213 --> 00:00:39.546
so X prime of T,

00:00:41.417 --> 00:00:43.737
well that's going to be the
rate of change of position

00:00:43.737 --> 00:00:47.715
with respect to time or the
velocity as a function of time

00:00:47.715 --> 00:00:51.213
and if we were to take the
derivative of our velocity,

00:00:51.213 --> 00:00:53.439
then that's going to be the
rate of change of velocity

00:00:53.439 --> 00:00:54.734
with respect to time.

00:00:54.734 --> 00:00:58.019
Well that's going to be
acceleration as a function of time.

00:00:58.019 --> 00:00:59.514
So they gave us velocity.

00:00:59.514 --> 00:01:02.318
So from velocity, we can
figure out acceleration.

00:01:02.318 --> 00:01:04.054
So let me just rewrite that.

00:01:04.054 --> 00:01:07.178
So we know that V of T is equal to

00:01:07.178 --> 00:01:11.345
negative T to the third power
plus six T squared plus two T

00:01:14.361 --> 00:01:16.844
and so from that we can figure out

00:01:16.844 --> 00:01:19.381
the acceleration as a function of time,

00:01:19.381 --> 00:01:20.503
which is just going to be the derivative

00:01:20.503 --> 00:01:22.746
with respect to T of the velocity.

00:01:22.746 --> 00:01:24.561
So just use the power rule a bunch.

00:01:24.561 --> 00:01:28.113
So that's going to be, this
is a third power right there.

00:01:28.113 --> 00:01:30.363
So negative three T squared

00:01:32.172 --> 00:01:34.505
plus two times six is twelve

00:01:35.803 --> 00:01:37.803
T to the first plus two.

00:01:39.061 --> 00:01:41.945
So that's our acceleration
as a function of time

00:01:41.945 --> 00:01:43.868
and we want to figure out when we obtain

00:01:43.868 --> 00:01:45.710
our maximum acceleration

00:01:45.710 --> 00:01:48.834
and just inspecting this
acceleration function here,

00:01:48.834 --> 00:01:50.650
we see it's a quadratic,

00:01:50.650 --> 00:01:52.786
it has a second degree polynomial

00:01:52.786 --> 00:01:54.415
and we have a negative coefficient

00:01:54.415 --> 00:01:56.552
out in front of the highest degree term,

00:01:56.552 --> 00:01:59.008
in front of the second degree term.

00:01:59.008 --> 00:02:01.652
So it is going to be a
downward opening parabola.

00:02:01.652 --> 00:02:03.380
So it is going to be a downward opening,

00:02:03.380 --> 00:02:04.688
let me draw it in the same color,

00:02:04.688 --> 00:02:07.145
so it is going to have that general shape

00:02:07.145 --> 00:02:09.762
and so it will indeed take on,

00:02:09.762 --> 00:02:13.287
it will indeed take on a maximum value.

00:02:13.287 --> 00:02:15.690
But how do we figure
out that maximum value?

00:02:15.690 --> 00:02:17.612
Well that maximum value's going to happen

00:02:17.612 --> 00:02:19.731
when the acceleration value,

00:02:19.731 --> 00:02:23.243
when the slope of its
tangent line is equal to,

00:02:23.243 --> 00:02:26.474
when the slope of its
tangent line is equal to zero

00:02:26.474 --> 00:02:30.186
and we could also verify
that it is concave downwards

00:02:30.186 --> 00:02:32.482
at that point using the
second derivative test

00:02:32.482 --> 00:02:35.820
by showing that the second
derivative is negative there.

00:02:35.820 --> 00:02:37.155
So let's do that, let's look at the first

00:02:37.155 --> 00:02:41.322
and second derivatives of
our acceleration function.

00:02:42.896 --> 00:02:44.472
So and I'll switch colors.

00:02:44.472 --> 00:02:46.341
That one's actually a
little bit hard to see.

00:02:46.341 --> 00:02:51.041
So the first derivative, the
rate of change of acceleration

00:02:51.041 --> 00:02:53.444
is going to be equal to,

00:02:53.444 --> 00:02:56.277
so this is negative six T plus 12.

00:02:58.330 --> 00:03:01.401
Now let's think about when
does this thing equal zero?

00:03:01.401 --> 00:03:04.205
Well if we subtract 12 from both sides,

00:03:04.205 --> 00:03:07.516
we get negative six T
is equal to negative 12,

00:03:07.516 --> 00:03:09.599
divide both sides by negative six,

00:03:09.599 --> 00:03:11.766
you get T is equal to two.

00:03:12.990 --> 00:03:14.753
So a couple of things.

00:03:14.753 --> 00:03:16.302
You could just say alright look,

00:03:16.302 --> 00:03:19.542
I know that this is a downward opening

00:03:19.542 --> 00:03:20.624
parabola right over here.

00:03:20.624 --> 00:03:23.801
I have a negative coefficient
on my second degree term.

00:03:23.801 --> 00:03:26.044
I know that the slope
of the tangent line here

00:03:26.044 --> 00:03:28.180
is zero at T equals two.

00:03:28.180 --> 00:03:29.916
So that's gonna be my maximum point.

00:03:29.916 --> 00:03:31.358
Or you could go a little bit further.

00:03:31.358 --> 00:03:34.189
You could take the second derivative.

00:03:34.189 --> 00:03:36.058
Let's do that just for kicks.

00:03:36.058 --> 00:03:37.820
So we could take the second derivative

00:03:37.820 --> 00:03:40.117
of our acceleration function.

00:03:40.117 --> 00:03:44.122
So this is going to be equal
to negative six, right.

00:03:44.122 --> 00:03:46.045
The derivative of
-6t is -6

00:03:46.045 --> 00:03:47.887
the derivative constant is just zero.

00:03:47.887 --> 00:03:51.839
So this thing, the second
derivative is always negative.

00:03:51.839 --> 00:03:55.256
So we are always, always concave downward

00:04:00.438 --> 00:04:03.775
and so by the second derivative
test at T equals two,

00:04:03.775 --> 00:04:05.324
well at T equals two our second derivative

00:04:05.324 --> 00:04:07.407
of our acceleration function's
going to be negative

00:04:07.407 --> 00:04:11.088
and so we know that this
is our maximum value.

00:04:11.088 --> 00:04:13.505
Our max at T is equal to two.

00:04:14.933 --> 00:04:16.536
So at what value of T does the particle

00:04:16.536 --> 00:04:18.294
obtain its maximum acceleration?

00:04:18.294 --> 00:04:20.044
At T is equal to two.


Worked example: finding relative extrema

WEBVTT
Kind: captions
Language: en

00:00:00.757 --> 00:00:02.557
- [Voiceover] So we have g(x) being equal

00:00:02.557 --> 00:00:04.735
to x to the fourth minus x to the fifth,

00:00:04.735 --> 00:00:07.452
and what we wanna do
without having to graph g,

00:00:07.452 --> 00:00:09.528
we want to figure out at what x values

00:00:09.528 --> 00:00:12.111
does g have a relative maximum?

00:00:13.451 --> 00:00:16.612
And just to remind us what's
going on in a relative maximum,

00:00:16.612 --> 00:00:21.560
so let me draw a hypothetical
function right over here,

00:00:21.560 --> 00:00:23.884
so a relative maximum is going to happen,

00:00:23.884 --> 00:00:25.990
so you can visually inspect this,

00:00:25.990 --> 00:00:27.998
okay that looks like a relative maximum.

00:00:27.998 --> 00:00:30.806
That's kind of a top of a
mountain or top of a hill,

00:00:30.806 --> 00:00:33.861
these all look like relative
maximum and what's in common?

00:00:33.861 --> 00:00:36.987
Well the graph, the function is going from

00:00:36.987 --> 00:00:40.282
increasing to decreasing
at each of those points.

00:00:40.282 --> 00:00:42.948
It's going from increasing to decreasing.

00:00:42.948 --> 00:00:46.855
Increasing to decreasing
at either of the points,

00:00:46.855 --> 00:00:49.114
or you could say that the first derivative

00:00:49.114 --> 00:00:52.014
is going from positive to negative.

00:00:52.014 --> 00:00:54.219
So if you look at this
interval right over here,

00:00:54.219 --> 00:00:57.246
g prime is greater than zero,

00:00:57.246 --> 00:00:59.314
and then over the next interval
when you're decreasing,

00:00:59.314 --> 00:01:01.981
g prime would be less than zero.

00:01:02.851 --> 00:01:04.281
So what we really need to think about

00:01:04.281 --> 00:01:07.281
is when does g prime, so let me see,

00:01:09.116 --> 00:01:13.761
relative, we care about
relative maximum point,

00:01:13.761 --> 00:01:15.860
and so that's essentially asking

00:01:15.860 --> 00:01:17.277
when does g prime

00:01:20.601 --> 00:01:23.018
go from positive to negative?

00:01:25.350 --> 00:01:26.608
From,

00:01:26.608 --> 00:01:27.441
from,

00:01:28.640 --> 00:01:30.131
I wrote fror.

00:01:30.131 --> 00:01:32.714
From g prime greater than zero,

00:01:34.110 --> 00:01:36.277
to g prime less than zero,

00:01:37.308 --> 00:01:40.192
and the values that we
could look at or the points

00:01:40.192 --> 00:01:42.852
are our critical points
and critical points

00:01:42.852 --> 00:01:47.019
are where g prime is either
zero or it is undefined.

00:01:48.525 --> 00:01:49.744
So let's think about it,

00:01:49.744 --> 00:01:52.389
where is g prime of x equal to zero?

00:01:52.389 --> 00:01:54.497
g prime of x is equal to zero

00:01:54.497 --> 00:01:57.747
when well let's just take g prime of x.

00:01:57.747 --> 00:02:00.137
We're gonna leverage the
power rule right here,

00:02:00.137 --> 00:02:03.941
four x to the third power,
four x to the third,

00:02:03.941 --> 00:02:06.581
minus five x to the fourth.

00:02:06.581 --> 00:02:10.498
Minus five x to the
fourth is equal to zero.

00:02:11.914 --> 00:02:14.323
Let's see, we can factor
out an x to the third.

00:02:14.323 --> 00:02:16.800
So we have x to the third times four

00:02:16.800 --> 00:02:19.792
minus five x is equal to zero.

00:02:19.792 --> 00:02:24.754
So this is going to happen
when x is equal to zero.

00:02:24.754 --> 00:02:26.377
Let me not skip steps.

00:02:26.377 --> 00:02:27.792
So this is going to happen when

00:02:27.792 --> 00:02:30.188
x to the third is equal to zero,

00:02:30.188 --> 00:02:33.944
or four minus five x is equal to zero.

00:02:33.944 --> 00:02:35.613
For x to the third equaling zero,

00:02:35.613 --> 00:02:38.122
that's only gonna happen
when x is equal to zero,

00:02:38.122 --> 00:02:39.808
and four minus five x equaling zero,

00:02:39.808 --> 00:02:41.421
we'll add five x to both sides,

00:02:41.421 --> 00:02:43.444
you get four is equal to five x,

00:02:43.444 --> 00:02:47.611
divide both sides by five,
you get 4/5 is equal to x.

00:02:48.811 --> 00:02:52.115
So here these are the two places,

00:02:52.115 --> 00:02:54.994
where our derivative is equal to zero.

00:02:54.994 --> 00:02:58.245
Now are there any places where
our derivative is undefined?

00:02:58.245 --> 00:03:00.500
Well our function right over here

00:03:00.500 --> 00:03:02.801
is just a straight up polynomial.

00:03:02.801 --> 00:03:04.764
Our derivative is another polynomial,

00:03:04.764 --> 00:03:07.543
it is defined for all real numbers.

00:03:07.543 --> 00:03:10.325
So these are our two critical points,

00:03:10.325 --> 00:03:12.636
or we could even say critical values.

00:03:12.636 --> 00:03:15.382
Now let's think of what g
prime is doing on either side

00:03:15.382 --> 00:03:17.124
of these critical values,

00:03:17.124 --> 00:03:18.733
and I'll draw a little number line here

00:03:18.733 --> 00:03:21.566
to help us visualize this, and so.

00:03:23.359 --> 00:03:25.963
So there we go, little
bit of a number line.

00:03:25.963 --> 00:03:29.122
Let's see we care about
zero and we care about 4/5.

00:03:29.122 --> 00:03:32.512
So let's say this is negative 1,

00:03:32.512 --> 00:03:34.679
this is zero, this is one,

00:03:36.828 --> 00:03:40.376
and so we have one critical point at,

00:03:40.376 --> 00:03:42.068
let me do this in magenta,

00:03:42.068 --> 00:03:46.072
we have one critical point
here at x equals zero,

00:03:46.072 --> 00:03:48.145
and then we have another critical point.

00:03:48.145 --> 00:03:50.949
I will do this at x equals 4/5.

00:03:50.949 --> 00:03:53.655
So 4/5 is right around there.

00:03:53.655 --> 00:03:57.011
So that is 4/5 and let's just think about

00:03:57.011 --> 00:04:00.428
what g prime is doing in these intervals,

00:04:01.435 --> 00:04:03.632
and these critical points
are the only places

00:04:03.632 --> 00:04:07.642
where g prime might switch
sides, switch signs.

00:04:07.642 --> 00:04:10.550
So let's first think about this,

00:04:10.550 --> 00:04:12.516
let me pick some colors
I haven't used yet.

00:04:12.516 --> 00:04:13.894
So let's think about the interval

00:04:13.894 --> 00:04:16.755
from negative infinity to zero.

00:04:16.755 --> 00:04:20.630
So this is the open interval
from negative infinity to zero,

00:04:20.630 --> 00:04:22.435
and we could just plug in a value,

00:04:22.435 --> 00:04:24.463
let's try negative one,
negative one is pretty

00:04:24.463 --> 00:04:26.605
straight forward to evaluate.

00:04:26.605 --> 00:04:29.005
So let's see you have four,
you're gonna have four

00:04:29.005 --> 00:04:31.232
times negative one to the third power.

00:04:31.232 --> 00:04:33.829
So that's gonna be four
times negative one,

00:04:33.829 --> 00:04:37.996
minus five times negative
one to the fourth power.

00:04:39.276 --> 00:04:41.347
So that's just gonna be one.

00:04:41.347 --> 00:04:45.514
So let's see this is going to
be negative four minus five.

00:04:46.503 --> 00:04:50.709
Which is negative nine,
so right over here,

00:04:50.709 --> 00:04:52.714
g prime is equal to negative nine,

00:04:52.714 --> 00:04:54.192
and so we know over this whole interval,

00:04:54.192 --> 00:04:56.341
since it's to the left
of this critical point,

00:04:56.341 --> 00:05:00.258
we know that g prime
of x is less than zero,

00:05:02.235 --> 00:05:04.753
and so our function itself is decreasing

00:05:04.753 --> 00:05:08.113
over this interval and
so we know we need to go

00:05:08.113 --> 00:05:10.156
from increasing to decreasing,

00:05:10.156 --> 00:05:12.903
so you can already say well
we can't go from increasing

00:05:12.903 --> 00:05:15.055
to decreasing at this critical point,

00:05:15.055 --> 00:05:17.688
because we're already
decreasing to the left of it,

00:05:17.688 --> 00:05:18.712
but anyway let's just think about

00:05:18.712 --> 00:05:20.634
what's happening at the other intervals.

00:05:20.634 --> 00:05:24.669
So in the interval between zero and 4/5,

00:05:24.669 --> 00:05:26.209
so that interval right over there,

00:05:26.209 --> 00:05:29.482
so it's between zero and 4/5,

00:05:29.482 --> 00:05:31.366
well let's just sample a number there.

00:05:31.366 --> 00:05:35.231
Let's say the number, I don't know, 1/2?

00:05:35.231 --> 00:05:37.493
Might be fairly straightforward.

00:05:37.493 --> 00:05:41.660
So we can evaluate g prime
of 1/2, g prime of 1/2,

00:05:43.462 --> 00:05:46.661
is equal to four times
1/2 to the third power.

00:05:46.661 --> 00:05:48.499
1/2 to the third power is 1/8.

00:05:48.499 --> 00:05:52.666
So it's 4/8, or it's just 1/2 minus five

00:05:53.789 --> 00:05:57.956
times 1/2 to the fourth so
that's 5/16 minus 5/16 and

00:06:00.749 --> 00:06:03.666
so this is equal to 8/16

00:06:04.802 --> 00:06:08.461
minus 5/16 which is equal to 3/16,

00:06:08.461 --> 00:06:11.562
but the important thing is
it's equal to a positive value.

00:06:11.562 --> 00:06:14.422
So in this blue interval right over there,

00:06:14.422 --> 00:06:16.520
and actually let me put
4/5 in a different color

00:06:16.520 --> 00:06:19.237
so we see that it's not
part of that interval.

00:06:19.237 --> 00:06:23.027
So in this light blue interval right here

00:06:23.027 --> 00:06:26.349
between zero and 4/5, g prime,

00:06:26.349 --> 00:06:29.837
g prime of x is greater than zero.

00:06:29.837 --> 00:06:31.834
So we know our function is increasing,

00:06:31.834 --> 00:06:34.134
and so let's see what's
happening to the right of this,

00:06:34.134 --> 00:06:39.087
and the easiest value to
try out would just be one.

00:06:39.087 --> 00:06:42.182
So let's try out x equals one.

00:06:42.182 --> 00:06:43.745
It's in that interval.

00:06:43.745 --> 00:06:46.096
So when x equals one, I'll just write

00:06:46.096 --> 00:06:49.679
g prime of one is equal
to four minus five.

00:06:50.806 --> 00:06:53.732
Four minus five, which
is equal to negative one.

00:06:53.732 --> 00:06:56.509
So g prime of x is less than zero,

00:06:56.509 --> 00:06:59.996
g prime of x is less than zero.

00:06:59.996 --> 00:07:04.163
So our function, so we could
say g is increasing here.

00:07:05.541 --> 00:07:08.951
It is decreasing, oh
sorry let me be careful,

00:07:08.951 --> 00:07:11.381
g is decreasing here.

00:07:11.381 --> 00:07:12.853
The function itself is decreasing

00:07:12.853 --> 00:07:14.741
cause our derivative is negative.

00:07:14.741 --> 00:07:16.879
Then our function is increasing here

00:07:16.879 --> 00:07:18.391
cause our derivative is positive,

00:07:18.391 --> 00:07:20.739
and then our function is decreasing here.

00:07:20.739 --> 00:07:24.645
So at what critical
point are we going from

00:07:24.645 --> 00:07:26.629
increasing to decreasing?

00:07:26.629 --> 00:07:29.121
Well we're doing that at x equals 4/5.

00:07:29.121 --> 00:07:33.288
So we have a relative
maximum at x equals 4/5.

00:07:35.211 --> 00:07:37.453
4/5, if they said, "Well where do we have

00:07:37.453 --> 00:07:38.977
"a relative minimum point?"

00:07:38.977 --> 00:07:41.038
Well that's going to be
happen at x equals zero.

00:07:41.038 --> 00:07:43.592
We're going from decreasing to increasing,

00:07:43.592 --> 00:07:44.701
but we've answered their question

00:07:44.701 --> 00:07:47.878
of where do we find a
relative maximum point.


Linear approximation of a rational function

WEBVTT
Kind: captions
Language: en

00:00:00.442 --> 00:00:01.535
- [Voiceover] So there are situations

00:00:01.535 --> 00:00:03.204
where you have some type of a function,

00:00:03.204 --> 00:00:05.117
this is clearly a nonlinear function.

00:00:05.117 --> 00:00:07.936
F of x is equal to one over x minus one,

00:00:07.936 --> 00:00:10.404
this is its graph or at
least part of its graph

00:00:10.404 --> 00:00:11.360
right over here.

00:00:11.360 --> 00:00:14.193
But where you wanna approximate
it with a linear function

00:00:14.193 --> 00:00:17.377
especially around a certain value,

00:00:17.377 --> 00:00:18.544
and so what we're going to do is,

00:00:18.544 --> 00:00:21.476
we wanna find an approximation,
let me write this down,

00:00:21.476 --> 00:00:24.309
I wanna find an approximation for,

00:00:25.231 --> 00:00:26.342
actually meant to be clear, I wanna find

00:00:26.342 --> 00:00:27.825
a linear approximation

00:00:27.825 --> 00:00:29.862
so I'm gonna approximate it with a line.

00:00:29.862 --> 00:00:34.029
I wanna find a linear
approximation, approximation

00:00:36.549 --> 00:00:39.882
of f, of f, around, and you need to know

00:00:41.732 --> 00:00:43.683
where you're going to be approximating it,

00:00:43.683 --> 00:00:46.350
around x equals negative one.

00:00:47.600 --> 00:00:49.186
So what do we mean by that?

00:00:49.186 --> 00:00:50.754
Well let's look at this graph over here,

00:00:50.754 --> 00:00:53.739
on this curve, when x is
equal to negative one,

00:00:53.739 --> 00:00:57.906
f of negative one is
negative, is negative 1/2,

00:01:01.217 --> 00:01:06.210
which sticks us right over
there, let me use a better color,

00:01:06.210 --> 00:01:08.754
so it's right over there
and what we wanna do

00:01:08.754 --> 00:01:10.764
is approximate it with a line around that.

00:01:10.764 --> 00:01:12.296
And what we're essentially going to do

00:01:12.296 --> 00:01:13.130
is we're gonna approximate it

00:01:13.130 --> 00:01:16.106
with the equation of the tangent line.

00:01:16.106 --> 00:01:19.233
The tangent line is
going to look something,

00:01:19.233 --> 00:01:22.933
something like that and as we can see,

00:01:22.933 --> 00:01:24.598
as we get further and further from,

00:01:24.598 --> 00:01:26.551
from x equals negative one,

00:01:26.551 --> 00:01:28.571
the approximation gets worse and worse

00:01:28.571 --> 00:01:31.348
but if we stay around
x equals negative one,

00:01:31.348 --> 00:01:34.108
what's a decent, it is
a, as good as you can get

00:01:34.108 --> 00:01:38.241
for a linear approximation
or at least in this example

00:01:38.241 --> 00:01:41.533
is a very good linear approximation.

00:01:41.533 --> 00:01:44.104
So when people say, hey,
find the linear approximation

00:01:44.104 --> 00:01:46.586
of f around x equals negative one

00:01:46.586 --> 00:01:49.589
or they say, what is the following
is the best approximation

00:01:49.589 --> 00:01:51.977
and all of your choices are, are lines,

00:01:51.977 --> 00:01:53.712
well essentially, they're
asking you to find

00:01:53.712 --> 00:01:57.878
the equation of the tangent
line at x equals negative one.

00:01:57.878 --> 00:01:59.389
So let's do that.

00:01:59.389 --> 00:02:02.374
So in order to find the
equation of the tangent line,

00:02:02.374 --> 00:02:06.834
the equation of a line is
y is equal to mx plus b

00:02:06.834 --> 00:02:09.820
where m is a slope and
b is the y intercept.

00:02:09.820 --> 00:02:12.165
There's other ways that
you could think about it.

00:02:12.165 --> 00:02:14.903
You could think about it,
in terms of point slope

00:02:14.903 --> 00:02:18.903
where you could say, y
minus some y that sits on

00:02:20.232 --> 00:02:24.232
that line is equal to
the slope times x minus

00:02:25.547 --> 00:02:29.604
the corresponding x one,
so x one comma y one

00:02:29.604 --> 00:02:31.431
sits on that line some place.

00:02:31.431 --> 00:02:33.585
Actually, I like to write
this point slope form

00:02:33.585 --> 00:02:37.752
like this sometimes, y minus
y one over x minus x one

00:02:39.714 --> 00:02:42.094
is equal to b because this comes as

00:02:42.094 --> 00:02:42.927
straight out of the idea.

00:02:42.927 --> 00:02:46.573
Look, if x one and y one are on the line,

00:02:46.573 --> 00:02:50.143
the slope between any
other point on the line

00:02:50.143 --> 00:02:54.049
and that point is going to
be your slope of your line.

00:02:54.049 --> 00:02:56.825
So we could think about
it, any of these ways.

00:02:56.825 --> 00:02:59.733
So let's first find the
slope of the tangent line

00:02:59.733 --> 00:03:02.085
and that's where the derivative is useful.

00:03:02.085 --> 00:03:05.525
So f, actually let me
just write f of x again,

00:03:05.525 --> 00:03:08.232
so I'm gonna write it as x minus one

00:03:08.232 --> 00:03:10.739
to the negative one power, that makes it

00:03:10.739 --> 00:03:13.358
a little bit clearer that
we can use the power rule

00:03:13.358 --> 00:03:15.225
and a little bit of a chain rule,

00:03:15.225 --> 00:03:19.648
so the derivative of f with
respect to x is equal to,

00:03:19.648 --> 00:03:22.376
so the derivative of x minus
one of the negative one

00:03:22.376 --> 00:03:25.569
with respect to x minus one,
well that's just going to be,

00:03:25.569 --> 00:03:27.647
I'm just gonna use the power rule here,

00:03:27.647 --> 00:03:32.150
it's gonna be negative
one times x minus one

00:03:32.150 --> 00:03:34.606
to the negative two and then
we're gonna multiply that

00:03:34.606 --> 00:03:39.197
times the derivative of x
minus one with respect to x,

00:03:39.197 --> 00:03:41.166
well that's just going to be one, right.

00:03:41.166 --> 00:03:43.197
The derivative of x with
respect of x is one,

00:03:43.197 --> 00:03:45.382
the derivative of negative
one with respect to x is zero,

00:03:45.382 --> 00:03:48.607
so we could say times one here if we like

00:03:48.607 --> 00:03:51.045
or we could just not write that

00:03:51.045 --> 00:03:53.001
'cause it don't change the value.

00:03:53.001 --> 00:03:57.915
And so let's evaluate that when
x is equal to negative one.

00:03:57.915 --> 00:04:01.165
So f prime of negative one is equal to,

00:04:03.609 --> 00:04:05.707
I could just write this is, negative,

00:04:05.707 --> 00:04:10.685
all right like this way,
negative one over negative one

00:04:10.685 --> 00:04:14.852
minus one squared and so this
is going to be negative two

00:04:16.273 --> 00:04:20.440
down here, so this is equal
to negative, negative 1/4.

00:04:22.659 --> 00:04:26.297
So, the slope of our tangent line is,

00:04:26.297 --> 00:04:31.112
so I could write this way,
m is equal to negative one,

00:04:31.112 --> 00:04:35.279
negative 1/4 and so now,
we just have to write

00:04:36.130 --> 00:04:37.910
its equation down.

00:04:37.910 --> 00:04:40.363
So we already know, an x one and a y one

00:04:40.363 --> 00:04:43.262
that sits on the line,
in fact we want to use

00:04:43.262 --> 00:04:45.817
the point when x is equals negative one

00:04:45.817 --> 00:04:50.239
so we know that the
point negative one comma,

00:04:50.239 --> 00:04:52.367
we could just input it right over here,

00:04:52.367 --> 00:04:56.132
f of negative one is negative 1/2,

00:04:56.132 --> 00:04:59.204
one over negative one
minus one, negative 1/2.

00:04:59.204 --> 00:05:03.566
So we know that this negative
one comma negative 1/2,

00:05:03.566 --> 00:05:06.869
that that is on our curve
and it is on our line,

00:05:06.869 --> 00:05:08.488
that's the point at which the tangent

00:05:08.488 --> 00:05:10.813
and the curve actually intersect.

00:05:10.813 --> 00:05:12.752
And so we can use any of these,

00:05:12.752 --> 00:05:15.114
to now write the equation of our line,

00:05:15.114 --> 00:05:18.718
we could say, y, under right here,

00:05:18.718 --> 00:05:22.635
y minus y one, so minus negative 1/2

00:05:25.629 --> 00:05:28.805
is going to be equal
to, is going to be equal

00:05:28.805 --> 00:05:31.758
to our slope, negative 1/4,

00:05:31.758 --> 00:05:35.550
I'm just using the point's
slope version of our equation,

00:05:35.550 --> 00:05:39.550
is equal to our slope times x minus x one,

00:05:41.123 --> 00:05:44.511
so x minus or x squared
that we know sits on this,

00:05:44.511 --> 00:05:46.390
so minus negative one.

00:05:46.390 --> 00:05:48.563
And so let me now write all
of this in a neutral color,

00:05:48.563 --> 00:05:52.730
this will be y plus 1/2
is equal to, and I can,

00:05:53.920 --> 00:05:56.950
so this is gonna be plus
one right over there,

00:05:56.950 --> 00:05:59.138
so I can distribute the negative 1/4.

00:05:59.138 --> 00:06:03.305
So it's negative 1/4 x minus 1/4 minus 1/4

00:06:06.436 --> 00:06:10.080
and then I can subtract
1/2 from both sides

00:06:10.080 --> 00:06:14.723
so I'm gonna get y is
equal to negative 1/4 x

00:06:14.723 --> 00:06:17.213
and then if I already am subtracting 1/4

00:06:17.213 --> 00:06:21.302
and I subtract another half
is gonna be negative 3/4

00:06:21.302 --> 00:06:24.385
so minus three, minus 3/4.

00:06:25.600 --> 00:06:28.122
So, and that's actually pretty
close to what I drew up here,

00:06:28.122 --> 00:06:32.289
this should be intersecting
the y-axis at negative 3/4.

00:06:33.657 --> 00:06:36.824
So there you have it, this, this line,

00:06:37.759 --> 00:06:39.797
or you could even say, this equation,

00:06:39.797 --> 00:06:43.399
is going to be a very
good linear approximation,

00:06:43.399 --> 00:06:45.681
about as good as you can get
for a linear approximation

00:06:45.681 --> 00:06:49.848
for that nonlinear function
around x equals negative one.

00:06:50.874 --> 00:06:53.012
You might say, well why
didn't they just ask me

00:06:53.012 --> 00:06:55.616
to find the equation of the tangent line

00:06:55.616 --> 00:06:57.837
at x equals negative
one, well they could have

00:06:57.837 --> 00:07:00.088
but there's a little bit of
extra cognitive processing here,

00:07:00.088 --> 00:07:02.636
we say, okay, I can
actually use the equation

00:07:02.636 --> 00:07:05.236
of the tangent line at to approximate

00:07:05.236 --> 00:07:09.069
this function around
x equals negative one.


Finding increasing interval given the derivative

WEBVTT
Kind: captions
Language: en

00:00:00.304 --> 00:00:02.174
- [Voiceover] Let g be a function defined

00:00:02.174 --> 00:00:03.757
for all real numbers.

00:00:03.757 --> 00:00:06.962
Also let g prime, the derivative of g,

00:00:06.962 --> 00:00:08.761
be defined as g prime of x

00:00:08.761 --> 00:00:11.918
is equal to x squared over x
minus two to the third power.

00:00:11.918 --> 00:00:14.835
On which intervals is g increasing?

00:00:15.969 --> 00:00:17.937
Well, at first you might say,
they don't even give us g.

00:00:17.937 --> 00:00:20.893
How do we figure out when g is increasing?

00:00:20.893 --> 00:00:24.451
Well, the answer is all we need is g prime

00:00:24.451 --> 00:00:26.237
which they do give us.

00:00:26.237 --> 00:00:29.030
And saying on which
intervals is g increasing,

00:00:29.030 --> 00:00:29.970
that's equivalent to saying,

00:00:29.970 --> 00:00:34.212
on which intervals is the first
derivative with respect to x

00:00:34.212 --> 00:00:37.537
on which intervals is that
going to be greater than zero?

00:00:37.537 --> 00:00:39.048
If your rate of change with respect to x

00:00:39.048 --> 00:00:40.969
is greater than zero, if it's positive,

00:00:40.969 --> 00:00:44.755
then your function itself
is going to be increasing.

00:00:44.755 --> 00:00:47.587
And so there's a couple of ways
that we could approach this.

00:00:47.587 --> 00:00:49.101
You might just want to inspect

00:00:49.101 --> 00:00:51.100
kind of the structure of this
expression and think about,

00:00:51.100 --> 00:00:53.360
well, when is that going
to be greater than zero

00:00:53.360 --> 00:00:55.490
or we could do it a little
bit more methodically.

00:00:55.490 --> 00:00:57.493
We could say, well, let's
look at the critical points

00:00:57.493 --> 00:01:00.516
or the critical values for g.

00:01:00.516 --> 00:01:01.516
So critical,

00:01:02.844 --> 00:01:04.094
critical points

00:01:05.342 --> 00:01:06.175
for g

00:01:07.385 --> 00:01:10.288
and just to remind ourselves
what critical points are,

00:01:10.288 --> 00:01:14.528
that is when g prime of x is equal to zero

00:01:14.528 --> 00:01:17.195
or g prime of x is undefined,

00:01:18.809 --> 00:01:20.409
is undefined,

00:01:20.409 --> 00:01:23.667
and we have videos on critical
points or critical values

00:01:23.667 --> 00:01:26.706
and why those are relevant
is those are the places,

00:01:26.706 --> 00:01:30.665
those are possible places
where the sign could change,

00:01:30.665 --> 00:01:32.957
the sign of g prime could change.

00:01:32.957 --> 00:01:35.817
So when is g prime of x equal to zero?

00:01:35.817 --> 00:01:37.952
Well, the way to get g
prime of x equal to zero

00:01:37.952 --> 00:01:39.591
is getting the numerator equal to zero

00:01:39.591 --> 00:01:40.491
and that's only going to happen

00:01:40.491 --> 00:01:44.114
if x squared is equal to zero
or if x is equal to zero.

00:01:44.114 --> 00:01:46.279
So that's the only
place where g prime of x

00:01:46.279 --> 00:01:47.601
is equal to zero

00:01:47.601 --> 00:01:50.071
and where is g prime of x undefined?

00:01:50.071 --> 00:01:50.904
Well, it's going to be undefined

00:01:50.904 --> 00:01:52.815
if the denominator becomes undefined.

00:01:52.815 --> 00:01:57.434
The denominator becomes undefined
if the denominator is zero

00:01:57.434 --> 00:01:58.267
and so that's going to happen

00:01:58.267 --> 00:02:00.785
if x minus two is equal to zero,

00:02:00.785 --> 00:02:05.152
x minus two is equal to
zero or x is equal to two.

00:02:05.152 --> 00:02:08.750
So we have two critical
points or critical values here

00:02:08.750 --> 00:02:10.862
and what I'm going to do
is I want to graph them.

00:02:10.862 --> 00:02:12.223
Let's put them on a number line

00:02:12.223 --> 00:02:15.914
and let's just think about
what g prime is doing

00:02:15.914 --> 00:02:18.932
in the intervals between
the critical points.

00:02:18.932 --> 00:02:20.849
So let's start at zero,

00:02:22.410 --> 00:02:23.243
one, two,

00:02:25.342 --> 00:02:28.592
three and then let's go to negative one

00:02:29.477 --> 00:02:31.648
and we have a critical point at,

00:02:31.648 --> 00:02:33.239
let me do that in magenta,

00:02:33.239 --> 00:02:37.238
we have a critical point at x
equals zero right over there

00:02:37.238 --> 00:02:40.668
and we have a critical point at x equals,

00:02:40.668 --> 00:02:43.301
at x equals two right over there.

00:02:43.301 --> 00:02:46.140
And so let's think about
what g prime is doing

00:02:46.140 --> 00:02:49.043
in the intervals between
the critical values

00:02:49.043 --> 00:02:51.290
or on either side of the critical values.

00:02:51.290 --> 00:02:52.718
So let's think about,

00:02:52.718 --> 00:02:54.970
let's first think about this interval.

00:02:54.970 --> 00:02:57.825
Let me do it in this purple color.

00:02:57.825 --> 00:03:00.512
Let's think about the interval between,

00:03:00.512 --> 00:03:03.429
between negative infinity and zero.

00:03:04.759 --> 00:03:06.429
So if we think about this interval,

00:03:06.429 --> 00:03:08.883
so negative infinity and zero,

00:03:08.883 --> 00:03:10.773
that open interval,

00:03:10.773 --> 00:03:13.106
well, if we look at g prime,

00:03:13.991 --> 00:03:16.038
the numerator is still
going to be positive.

00:03:16.038 --> 00:03:17.650
If you take any negative value squared,

00:03:17.650 --> 00:03:18.530
you're going to get a positive value

00:03:18.530 --> 00:03:21.009
so this is going to be positive.

00:03:21.009 --> 00:03:22.288
Now, what about the denominator?

00:03:22.288 --> 00:03:24.863
You take a negative number,
you subtract two from it,

00:03:24.863 --> 00:03:26.506
you're still going to
get a negative number

00:03:26.506 --> 00:03:28.074
and then you take it to the third power.

00:03:28.074 --> 00:03:29.271
Well, a negative number to the third power

00:03:29.271 --> 00:03:30.812
is going to be a negative number

00:03:30.812 --> 00:03:32.797
so that right over there
is going to be negative.

00:03:32.797 --> 00:03:35.064
So you're going to have a
positive divided by a negative

00:03:35.064 --> 00:03:38.029
so g prime is going to be negative

00:03:38.029 --> 00:03:39.740
so let me write that down.

00:03:39.740 --> 00:03:43.138
So on this interval, on this interval,

00:03:43.138 --> 00:03:44.243
I'll write it like this.

00:03:44.243 --> 00:03:47.307
g prime of x is less than zero

00:03:47.307 --> 00:03:49.758
or if we cared or if we want
to know when it's decreasing,

00:03:49.758 --> 00:03:53.109
we would know it's definitely
decreasing over that interval.

00:03:53.109 --> 00:03:57.744
Now, let's take the interval
between zero and two

00:03:57.744 --> 00:03:59.077
right over here.

00:03:59.994 --> 00:04:02.324
So this is the interval

00:04:02.324 --> 00:04:05.324
from zero to two, the open interval.

00:04:06.163 --> 00:04:09.350
So what's going to go on
with g prime of x here?

00:04:09.350 --> 00:04:12.451
Well, once again, x squared,

00:04:12.451 --> 00:04:14.534
anything greater than a zero

00:04:14.534 --> 00:04:16.653
and it says we're not including
zero in this interval.

00:04:16.653 --> 00:04:20.206
Well, this is for sure
going to be positive

00:04:20.206 --> 00:04:22.576
and so let's see, if we have x minus two

00:04:22.576 --> 00:04:26.089
where x is greater than
zero but less than two.

00:04:26.089 --> 00:04:28.336
So if x, we could just say
for example, if x was one,

00:04:28.336 --> 00:04:30.422
one minus two is negative one.

00:04:30.422 --> 00:04:32.304
We're still going to get negative values

00:04:32.304 --> 00:04:34.552
in this denominator right over here.

00:04:34.552 --> 00:04:36.392
So since we're still going
to get negative values

00:04:36.392 --> 00:04:37.292
in this denominator,

00:04:37.292 --> 00:04:38.854
the denominator is still going to be,

00:04:38.854 --> 00:04:41.181
you take a negative
value to the third power,

00:04:41.181 --> 00:04:42.937
well, you're going to
still get a negative value

00:04:42.937 --> 00:04:46.333
so this is going to be negative.

00:04:46.333 --> 00:04:50.784
So you're still going to have
g prime as less than zero

00:04:50.784 --> 00:04:52.605
so let me write that down.

00:04:52.605 --> 00:04:55.105
So you still have g prime of x

00:04:55.980 --> 00:04:57.480
is less than zero.

00:04:58.471 --> 00:05:01.374
And then let's take the interval above.

00:05:01.374 --> 00:05:05.604
Let's take the interval
from two to infinity.

00:05:05.604 --> 00:05:07.459
Two to infinity.

00:05:07.459 --> 00:05:08.973
Well, the numerator is positive.

00:05:08.973 --> 00:05:10.529
It's always going to be positive

00:05:10.529 --> 00:05:13.677
for any x not being equal to zero

00:05:13.677 --> 00:05:15.369
and this denominator,

00:05:15.369 --> 00:05:17.115
you're taking values greater than two,

00:05:17.115 --> 00:05:18.169
subtracting two from it

00:05:18.169 --> 00:05:19.968
which is still going to
give you a positive value.

00:05:19.968 --> 00:05:22.523
You take the third power,
it's all going to be positive.

00:05:22.523 --> 00:05:24.628
It is all going to be positive.

00:05:24.628 --> 00:05:28.589
So this is the interval where g prime of x

00:05:28.589 --> 00:05:30.339
is greater than zero.

00:05:31.362 --> 00:05:33.466
So on which intervals is g increasing?

00:05:33.466 --> 00:05:37.544
Well, that's where g prime
of x is greater than zero

00:05:37.544 --> 00:05:40.714
so it's going to be from two,

00:05:40.714 --> 00:05:44.686
from two to infinity or we
could just write it like this.

00:05:44.686 --> 00:05:48.470
We could write x is greater than two.

00:05:48.470 --> 00:05:49.533
Either way,

00:05:49.533 --> 00:05:52.646
for either of these, g prime
of x is greater than zero

00:05:52.646 --> 00:05:56.479
and your function g is
going to be increasing.


Absolute minima & maxima (entire domain)

WEBVTT
Kind: captions
Language: en

00:00:00.653 --> 00:00:01.812
- [Voiceover] So we have the function

00:00:01.812 --> 00:00:04.014
g of x is equal to x
squared times the natural

00:00:04.014 --> 00:00:05.229
log of x.

00:00:05.229 --> 00:00:06.513
And what I wanna do in this video is see

00:00:06.513 --> 00:00:09.723
if you can figure out the absolute extrema

00:00:09.723 --> 00:00:10.873
for g of x.

00:00:10.873 --> 00:00:14.113
So are there x values where
g takes on an absolute

00:00:14.113 --> 00:00:17.754
maximum value, or an
absolute minimum value.

00:00:17.754 --> 00:00:19.839
Sometimes you might call
them a global maximum,

00:00:19.839 --> 00:00:21.574
or a global minimum.

00:00:21.574 --> 00:00:23.829
So the first thing I like
to think about is well,

00:00:23.829 --> 00:00:28.588
what's the domain for which
g is actually defined?

00:00:28.588 --> 00:00:31.123
And we know that in the natural log of x

00:00:31.123 --> 00:00:35.290
the input into natural log, it
has to be greater than zero.

00:00:36.497 --> 00:00:39.132
So the domain, the domain
is all real numbers

00:00:39.132 --> 00:00:40.685
greater than zero.

00:00:40.685 --> 00:00:43.037
So x has to be greater than zero.

00:00:43.037 --> 00:00:46.212
Anything lateral log
of zero is not defined,

00:00:46.212 --> 00:00:48.100
there is no power that you could take e to

00:00:48.100 --> 00:00:51.030
to get to zero, and natural
log of negative numbers

00:00:51.030 --> 00:00:52.964
is not defined.

00:00:52.964 --> 00:00:55.594
So that is the domain.

00:00:55.594 --> 00:00:57.692
The domain is all real numbers such that

00:00:57.692 --> 00:01:02.050
all real numbers access so
that x is greater than zero.

00:01:02.050 --> 00:01:06.217
So our absolute extrema have
to be within that domain.

00:01:07.077 --> 00:01:08.796
So to find these, let's
see if we can find some

00:01:08.796 --> 00:01:11.904
local extrema and see if any
of them are good candidates

00:01:11.904 --> 00:01:13.654
for absolute extrema.

00:01:14.497 --> 00:01:15.824
And we could find our local extrema

00:01:15.824 --> 00:01:19.800
by looking at critical
points, or critical values.

00:01:19.800 --> 00:01:22.725
So let's take the derivative of g.

00:01:22.725 --> 00:01:26.770
So g prime, using a new
color just for kicks.

00:01:26.770 --> 00:01:30.511
All right, so g prime of x is equal to,

00:01:30.511 --> 00:01:32.303
we could use the product rule here.

00:01:32.303 --> 00:01:34.989
So derivative of x squared which is two x

00:01:34.989 --> 00:01:37.156
times the natural log of x

00:01:38.036 --> 00:01:40.807
plus x squared times the derivative

00:01:40.807 --> 00:01:43.774
of natural log of x,
so that is one over x,

00:01:43.774 --> 00:01:44.992
and I can just rewrite that.

00:01:44.992 --> 00:01:47.553
X squared times one over x.

00:01:47.553 --> 00:01:50.661
And we're gonna assume that x is positive.

00:01:50.661 --> 00:01:52.578
So that is going to be,

00:01:53.558 --> 00:01:55.561
that is going to be just,

00:01:55.561 --> 00:01:57.033
and actually I didn't even
have to make that assumption

00:01:57.033 --> 00:01:58.053
for what I'm about to do,

00:01:58.053 --> 00:02:01.917
x squared divided by x
is just going to be x.

00:02:01.917 --> 00:02:02.982
All right.

00:02:02.982 --> 00:02:04.899
And so that is g prime.

00:02:05.764 --> 00:02:08.535
So now let's think about
the critical points.

00:02:08.535 --> 00:02:10.256
Critical points are where the derivative

00:02:10.256 --> 00:02:12.430
they're points in the domain,

00:02:12.430 --> 00:02:15.381
so they're gonna have to
satisfy x is greater than zero,

00:02:15.381 --> 00:02:18.464
such that g prime is either undefined

00:02:19.721 --> 00:02:22.101
or it is equal to zero.

00:02:22.101 --> 00:02:25.854
So let's first think about
when g prime is equal to zero.

00:02:25.854 --> 00:02:27.591
So let's set it equal to zero.

00:02:27.591 --> 00:02:31.758
Two x natural log of x
plus x is equal to zero.

00:02:33.455 --> 00:02:37.319
Well we can subtract, we could
subtract x from both sides

00:02:37.319 --> 00:02:41.236
of that and so we get
two x natural log of x

00:02:42.946 --> 00:02:44.863
is equal to negative x.

00:02:46.226 --> 00:02:48.976
See, if we divide both sides by x

00:02:50.160 --> 00:02:52.285
and we can do that, we
know x isn't gonna be zero,

00:02:52.285 --> 00:02:54.402
our domain is x is greater than zero.

00:02:54.402 --> 00:02:56.738
So this is going to be,
actually let's divide

00:02:56.738 --> 00:02:58.181
both sides by two x.

00:02:58.181 --> 00:03:01.497
So that we get the natural
log of x is equal to

00:03:01.497 --> 00:03:02.997
negative one half.

00:03:04.097 --> 00:03:06.523
Negative x divided by two
x is negative one half.

00:03:06.523 --> 00:03:08.398
Or we could say that x is equal to

00:03:08.398 --> 00:03:10.080
e to the negative one half

00:03:10.080 --> 00:03:10.989
is equal to x.

00:03:10.989 --> 00:03:13.422
Remember, natural log is just log base e.

00:03:13.422 --> 00:03:16.058
So e to the negative one half,

00:03:16.058 --> 00:03:17.598
which we could also write like that,

00:03:17.598 --> 00:03:19.768
e to the negative one half,

00:03:19.768 --> 00:03:21.698
or one over the square root of e.

00:03:21.698 --> 00:03:24.115
So that's a point at which g,

00:03:25.201 --> 00:03:27.497
at which our derivative I should say

00:03:27.497 --> 00:03:30.939
is equal to zero, it is a
critical point or critical value

00:03:30.939 --> 00:03:33.657
for our original function g.

00:03:33.657 --> 00:03:37.184
So and that's the only place
where g prime is equal to zero.

00:03:37.184 --> 00:03:41.496
Are there any other points
where g prime is undefined?

00:03:41.496 --> 00:03:43.666
And they're have to be
points within the domain.

00:03:43.666 --> 00:03:45.319
So let's see, what would
make this undefined?

00:03:45.319 --> 00:03:48.291
The two x and the x, that
you can evaluate for any x.

00:03:48.291 --> 00:03:52.011
Natural log of x, once again,
is only going to be defined

00:03:52.011 --> 00:03:54.772
for x greater than zero.

00:03:54.772 --> 00:03:56.548
But that's, we've already
restricted ourselves

00:03:56.548 --> 00:03:58.945
to that domain, so within the domain

00:03:58.945 --> 00:04:01.184
any point in the domain our derivative

00:04:01.184 --> 00:04:03.715
is actually going to be defined.

00:04:03.715 --> 00:04:08.043
So given that let's see what's
happening on either side

00:04:08.043 --> 00:04:10.774
of this critical point.

00:04:10.774 --> 00:04:12.748
On either side of this critical point.

00:04:12.748 --> 00:04:14.314
And I could draw a little
number line here to

00:04:14.314 --> 00:04:17.281
really help us visualize this.

00:04:17.281 --> 00:04:20.781
So, if this is negative one, this is zero,

00:04:23.708 --> 00:04:26.987
this is let's see, e to
the, this is gonna be like

00:04:26.987 --> 00:04:29.392
one over, oh boy this is,

00:04:29.392 --> 00:04:33.309
this is going to be a
little bit less than one,

00:04:35.621 --> 00:04:38.296
so let's see, why don't we plot one here,

00:04:38.296 --> 00:04:41.681
and then two here, and so
we have a critical point

00:04:41.681 --> 00:04:44.066
at one over the square root of e,

00:04:44.066 --> 00:04:45.562
and we'll put it right over there.

00:04:45.562 --> 00:04:48.041
One over the square root of e.

00:04:48.041 --> 00:04:50.547
And we know that we're only defined from,

00:04:50.547 --> 00:04:53.233
for all x's greater than zero.

00:04:53.233 --> 00:04:55.330
So let's think about the interval between

00:04:55.330 --> 00:04:57.533
zero and this critical point.

00:04:57.533 --> 00:04:58.987
Right over here.

00:04:58.987 --> 00:05:01.803
So the open interval,
from zero to one over

00:05:01.803 --> 00:05:04.221
the square root of e,
let's think about whether

00:05:04.221 --> 00:05:06.326
g prime is positive or negative there.

00:05:06.326 --> 00:05:09.076
And then let's think about it for

00:05:10.096 --> 00:05:13.412
x greater than one over
the square root of e.

00:05:13.412 --> 00:05:15.917
So that's the interval from
one over square root of e

00:05:15.917 --> 00:05:16.917
to infinity.

00:05:18.605 --> 00:05:20.888
So over that yellow interval,
we let's just try out

00:05:20.888 --> 00:05:22.971
a value that is in there.

00:05:23.859 --> 00:05:27.704
So let's just try g
prime of, I don't know,

00:05:27.704 --> 00:05:29.787
let's try g prime of 0.1.

00:05:31.304 --> 00:05:34.412
G prime of 0.1 is definitely
going to be in this interval.

00:05:34.412 --> 00:05:36.693
And so it's going to
be equal to two point,

00:05:36.693 --> 00:05:38.860
two times 0.1 is equal to,

00:05:41.722 --> 00:05:45.889
is equal to 0.2 times the
natural log of 0.1 plus 0.1.

00:05:51.565 --> 00:05:52.398
And let's see.

00:05:52.398 --> 00:05:56.297
This right over here, this is
going to be a negative value,

00:05:56.297 --> 00:05:58.662
in fact it's going to be quite,

00:05:58.662 --> 00:06:02.080
it's definitely going to be
greater than negative one.

00:06:02.080 --> 00:06:06.082
Cuz e to the negative
one only gets you to,

00:06:06.082 --> 00:06:09.933
let's see, e to the
negative one is one over e

00:06:09.933 --> 00:06:14.100
so that's one over 2.7, so
one over 2.7 is going to be,

00:06:17.155 --> 00:06:20.572
so this is going to be around 0.3 or 0.4.

00:06:23.823 --> 00:06:25.091
So in order to get point one,

00:06:25.091 --> 00:06:28.525
so this'll be around 0.3 to 0.4.

00:06:28.525 --> 00:06:32.008
So in order to get to 0.1 you
have to be even more negative.

00:06:32.008 --> 00:06:36.175
So this is going to be, I could
say less than negative one.

00:06:37.356 --> 00:06:39.167
So if this is less than negative one,

00:06:39.167 --> 00:06:41.097
and I'm multiplying it times 0.2,

00:06:41.097 --> 00:06:44.234
I'm gonna get a negative
value that is less than,

00:06:44.234 --> 00:06:49.062
less than negative 0.2, and
if I'm adding 0.1 to it,

00:06:49.062 --> 00:06:51.764
well, I'm still going
to get a negative value.

00:06:51.764 --> 00:06:55.931
So for this yellow interval, g
prime of x is less than zero.

00:06:56.778 --> 00:06:58.570
And it would be, I should
have gotten a calculator,

00:06:58.570 --> 00:06:59.997
or I could have gotten a calculator out,

00:06:59.997 --> 00:07:02.211
I could have just evaluated a lot easier.

00:07:02.211 --> 00:07:04.661
So g prime of x is less
than zero in this interval.

00:07:04.661 --> 00:07:07.250
Now let's see in this blue
interval what's going on.

00:07:07.250 --> 00:07:08.818
And this'll be easier,
we could just try out

00:07:08.818 --> 00:07:10.554
the value one.

00:07:10.554 --> 00:07:14.552
So g prime of one is equal
to two times the natural log

00:07:14.552 --> 00:07:15.885
of one plus one.

00:07:16.852 --> 00:07:18.855
Natural log of one is just zero.

00:07:18.855 --> 00:07:20.956
So all of this just simplifies to one.

00:07:20.956 --> 00:07:23.346
So over this blue interval,
I sampled a point there,

00:07:23.346 --> 00:07:26.429
g prime of x is greater than zero.

00:07:27.746 --> 00:07:32.574
So it looks like our
function is decreasing from

00:07:32.574 --> 00:07:35.122
zero to one over square root of e,

00:07:35.122 --> 00:07:37.560
and then we increase after that.

00:07:37.560 --> 00:07:40.611
And we increase for all
x's that are greater than

00:07:40.611 --> 00:07:42.499
one over the square root of e.

00:07:42.499 --> 00:07:45.158
And so our function is going to hit,

00:07:45.158 --> 00:07:46.894
if we're decreasing into that and then

00:07:46.894 --> 00:07:50.453
increasing after that, we're
hitting a global minimum point,

00:07:50.453 --> 00:07:53.379
or a absolute minimum
point at x equals one

00:07:53.379 --> 00:07:55.730
over the square root of e.

00:07:55.730 --> 00:07:57.272
So let me write this down.

00:07:57.272 --> 00:08:00.189
We hit a, we hit a absolute minimum

00:08:05.939 --> 00:08:08.373
at x equals one over

00:08:08.373 --> 00:08:11.395
the square root of e, and
there is no absolute maximum.

00:08:11.395 --> 00:08:13.427
As we get above one over
the square root of e

00:08:13.427 --> 00:08:14.884
we are just going to think about

00:08:14.884 --> 00:08:16.237
what's going to be happening here.

00:08:16.237 --> 00:08:17.306
We're just going to one,

00:08:17.306 --> 00:08:18.968
we know that our function
just keeps on increasing

00:08:18.968 --> 00:08:21.198
and increasing and increasing forever.

00:08:21.198 --> 00:08:22.499
And you could look at even this,

00:08:22.499 --> 00:08:25.441
x squared is just gonna get
unbounded towards infinity,

00:08:25.441 --> 00:08:28.086
and natural log of x is gonna
grow slower than x squared,

00:08:28.086 --> 00:08:30.324
but it's still gonna go
unbounded towards infinity.

00:08:30.324 --> 00:08:34.663
So there's no global,
or no absolute maximum.

00:08:34.663 --> 00:08:36.830
No absolute maximum point.

00:08:39.651 --> 00:08:40.911
And now let's look at the graph of this

00:08:40.911 --> 00:08:43.402
to feel good about what
we just did analytically,

00:08:43.402 --> 00:08:45.320
without looking at it graphically.

00:08:45.320 --> 00:08:47.701
And I looked at it ahead of time.

00:08:47.701 --> 00:08:49.393
So let me copy and paste it.

00:08:49.393 --> 00:08:53.121
And so this is the graph of our function.

00:08:53.121 --> 00:08:56.954
So as can see when this
point right over here,

00:08:58.598 --> 00:09:02.765
this is when, this is one
over the square root of e,

00:09:04.181 --> 00:09:05.495
it's not obvious from looking at it

00:09:05.495 --> 00:09:06.900
that it's that point.

00:09:06.900 --> 00:09:08.815
X equals one over the square root of e.

00:09:08.815 --> 00:09:12.914
And we can see that it is
indeed an absolute minimum point

00:09:12.914 --> 00:09:16.820
here and there is no
absolute maximum point.

00:09:16.820 --> 00:09:19.403
There's arbitrarily high values

00:09:20.347 --> 00:09:22.847
that our function can take on.


Finding decreasing interval given the function

WEBVTT
Kind: captions
Language: en

00:00:00.189 --> 00:00:01.698
- [Voiceover] So we
have the function f of x

00:00:01.698 --> 00:00:04.878
is equal to x to the sixth
minus three x to the fifth

00:00:04.878 --> 00:00:09.121
and we want to know over what
intervals is f decreasing

00:00:09.121 --> 00:00:11.104
and we're going to do
it without even having

00:00:11.104 --> 00:00:13.104
to graph y equals f of x

00:00:13.972 --> 00:00:16.116
and the way we do that is
we look at the derivative

00:00:16.116 --> 00:00:17.555
of f with respect to x

00:00:17.555 --> 00:00:19.623
and think about, well, when
is that less than zero?

00:00:19.623 --> 00:00:22.774
If the rate of change
of f with respect to x

00:00:22.774 --> 00:00:24.945
is less than zero, well
over those intervals

00:00:24.945 --> 00:00:26.537
it will be decreasing.

00:00:26.537 --> 00:00:28.737
So let's first take the derivative.

00:00:28.737 --> 00:00:32.582
So f prime of x is going to be equal to,

00:00:32.582 --> 00:00:34.149
just use the power rule here,

00:00:34.149 --> 00:00:37.801
it's going to be six x to the fifth power

00:00:37.801 --> 00:00:41.218
minus five times three is 15 x to the,

00:00:42.373 --> 00:00:44.414
let's decrease that exponent by one,

00:00:44.414 --> 00:00:47.029
x to the fourth power
and so let's think about

00:00:47.029 --> 00:00:50.011
when this is going to be less than zero.

00:00:50.011 --> 00:00:52.636
Over what intervals is six x to the fifth

00:00:52.636 --> 00:00:55.803
minus 15 x to the fourth less than,

00:00:57.282 --> 00:00:58.532
less than zero?

00:00:59.487 --> 00:01:03.654
So let's see, we could factor
out a three x to the fourth,

00:01:04.630 --> 00:01:07.463
so three x to the fourth times,

00:01:09.677 --> 00:01:11.729
see if we factor out a
three x to the fourth here,

00:01:11.729 --> 00:01:15.479
we'll just be left with a two x minus five

00:01:16.858 --> 00:01:18.358
is less than zero.

00:01:19.570 --> 00:01:20.839
Did I do that right?

00:01:20.839 --> 00:01:22.025
Let's see, if I were to distribute it,

00:01:22.025 --> 00:01:24.154
three times two is six,
x to the fourth times x

00:01:24.154 --> 00:01:27.254
is x to the fifth, three times five is 15

00:01:27.254 --> 00:01:28.848
and x to the fourth, yep, that's right

00:01:28.848 --> 00:01:31.837
so if I'm getting the
product of two things

00:01:31.837 --> 00:01:34.326
and I want it to be less than zero,

00:01:34.326 --> 00:01:36.978
well, there's only way for that to happen

00:01:36.978 --> 00:01:38.188
is that one or I guess you could say

00:01:38.188 --> 00:01:39.631
there's two ways for that to happen.

00:01:39.631 --> 00:01:41.063
Either the first thing is positive

00:01:41.063 --> 00:01:42.591
and the second is negative

00:01:42.591 --> 00:01:45.561
or the first is negative
and the second is positive

00:01:45.561 --> 00:01:46.685
and so let's do that.

00:01:46.685 --> 00:01:50.834
So either, either three x to the fourth

00:01:50.834 --> 00:01:51.751
is negative

00:01:53.531 --> 00:01:54.364
and,

00:01:55.497 --> 00:01:58.414
and two x minus five is positive

00:02:00.346 --> 00:02:01.531
or,

00:02:01.531 --> 00:02:05.213
let me just put the or
in a separate color here,

00:02:05.213 --> 00:02:07.463
or three x to the fourth

00:02:09.279 --> 00:02:10.196
is positive

00:02:11.259 --> 00:02:12.092
and

00:02:13.385 --> 00:02:16.052
two x minus five is negative.

00:02:17.834 --> 00:02:19.075
So let's see.

00:02:19.075 --> 00:02:21.424
This is x to the fourth,
three x to the fourth

00:02:21.424 --> 00:02:22.530
being less than zero.

00:02:22.530 --> 00:02:25.021
Well, if we divide both sides by three,

00:02:25.021 --> 00:02:27.092
this is just going to be x to the fourth

00:02:27.092 --> 00:02:29.796
needs to be less than zero.

00:02:29.796 --> 00:02:32.015
And is there any way for something to be

00:02:32.015 --> 00:02:35.184
to the fourth power to be less than zero?

00:02:35.184 --> 00:02:37.393
Well, we're assuming we're
dealing with real numbers here

00:02:37.393 --> 00:02:39.601
and any real number to the fourth power

00:02:39.601 --> 00:02:42.581
is going to be greater
than or equal to zero

00:02:42.581 --> 00:02:43.738
so it's actually impossible

00:02:43.738 --> 00:02:45.810
for something to the fourth
power to be less than zero.

00:02:45.810 --> 00:02:47.191
This thing is never,

00:02:47.191 --> 00:02:50.013
this thing is never going
to be less than zero

00:02:50.013 --> 00:02:53.244
so we can actually rule
out this first case.

00:02:53.244 --> 00:02:56.877
So we can rule out that
first case right over there.

00:02:56.877 --> 00:02:59.088
And so we're only going
to worry about this case.

00:02:59.088 --> 00:03:03.655
So three x to the fourth
being greater than zero,

00:03:03.655 --> 00:03:05.820
well, that's going to happen as long as

00:03:05.820 --> 00:03:07.737
x is not equal to zero.

00:03:09.046 --> 00:03:10.643
So this is,

00:03:10.643 --> 00:03:13.007
because for any other x,
this is going to be true.

00:03:13.007 --> 00:03:14.068
x could be negative.

00:03:14.068 --> 00:03:15.704
You take it to the fourth
power, multiply it by three,

00:03:15.704 --> 00:03:17.662
it's going to be greater than zero

00:03:17.662 --> 00:03:19.324
so this is really just the condition

00:03:19.324 --> 00:03:22.521
that x cannot be equal to zero.

00:03:22.521 --> 00:03:24.116
And let's see the second one,

00:03:24.116 --> 00:03:25.628
two x minus five less than zero,

00:03:25.628 --> 00:03:29.796
that means two x is going
to be less than five

00:03:29.796 --> 00:03:31.308
and then x

00:03:31.308 --> 00:03:33.391
is less than five halves.

00:03:35.265 --> 00:03:37.634
So as long as x is less than five halves

00:03:37.634 --> 00:03:39.893
and x is not equal to zero,

00:03:39.893 --> 00:03:42.617
this thing will be decreasing.

00:03:42.617 --> 00:03:44.660
So if we wanted to write it in terms of,

00:03:44.660 --> 00:03:47.606
it we wanted to write it
in terms of intervals,

00:03:47.606 --> 00:03:49.356
we could say x is,

00:03:50.768 --> 00:03:53.358
let me do this in a new color
just to ease the monotony.

00:03:53.358 --> 00:03:55.108
So we could say that,

00:03:56.423 --> 00:03:58.309
having trouble picking a color,

00:03:58.309 --> 00:04:01.686
so we could say that x is less than zero

00:04:01.686 --> 00:04:03.519
or zero is less than x

00:04:05.571 --> 00:04:07.820
is less than five halves.

00:04:07.820 --> 00:04:10.763
So x is less than zero, this
is all the negative values

00:04:10.763 --> 00:04:13.417
and then we're essentially
just excluding zero

00:04:13.417 --> 00:04:15.770
and then going all the way to,

00:04:15.770 --> 00:04:17.736
all the way to five halves

00:04:17.736 --> 00:04:19.031
and remember, all I did is say,

00:04:19.031 --> 00:04:21.667
well, when is our first
derivative negative

00:04:21.667 --> 00:04:23.645
because if the first
derivative is negative

00:04:23.645 --> 00:04:26.611
then the rate of change of f
with respect to x is negative

00:04:26.611 --> 00:04:30.028
or f is decreasing as x is increasing.


BONUS: "FANBOYS," a mnemonic song

WEBVTT
Kind: captions
Language: en

00:00:00.455 --> 00:00:03.958
♫ Fanboys Fanboys

00:00:03.958 --> 00:00:07.520
♫ The boys who carry the fan

00:00:07.520 --> 00:00:11.178
♫ Fanboys Fanboys

00:00:11.178 --> 00:00:14.694
♫ The boys who had a plan

00:00:14.694 --> 00:00:16.371
♫ For the way was long

00:00:16.371 --> 00:00:18.606
♫ And the day was hot

00:00:18.606 --> 00:00:21.790
♫ The boys were always prepared

00:00:21.790 --> 00:00:23.648
♫ Neither sand nor heat

00:00:23.648 --> 00:00:25.725
♫ Would assail their feet

00:00:25.725 --> 00:00:29.247
♫ They did what no others had dared

00:00:29.247 --> 00:00:32.859
♫ Fanboys Fanboys

00:00:32.859 --> 00:00:36.493
♫ The boys who carry the fan

00:00:36.493 --> 00:00:39.326
♫ Fanboys Fanboys

00:00:40.390 --> 00:00:44.125
♫ The boys who had a plan

00:00:44.125 --> 00:00:47.625
♫ The Fanboys were on a train to Laredo

00:00:47.625 --> 00:00:50.692
♫ When bandits made an assault

00:00:50.692 --> 00:00:51.525
♫ But

00:00:51.525 --> 00:00:54.683
♫ But the Fanboys fan
made a wind so mighty

00:00:54.683 --> 00:00:58.138
♫ The banditry came to a halt

00:00:58.138 --> 00:01:00.971
♫ Fanboys Fanboys

00:01:01.819 --> 00:01:05.219
♫ The boys who carry the fan

00:01:05.219 --> 00:01:08.052
♫ Fanboys Fanboys

00:01:08.952 --> 00:01:12.552
♫ The boys who had a plan

00:01:12.552 --> 00:01:16.221
♫ The magistrate said
we're all very grateful

00:01:16.221 --> 00:01:19.677
♫ But it's time for an end to your fun

00:01:19.677 --> 00:01:21.354
♫ He said to the boys

00:01:21.354 --> 00:01:23.344
♫ It's your fan or your freedom

00:01:23.344 --> 00:01:26.926
♫ And you may only have one

00:01:26.926 --> 00:01:29.759
♫ Fanboys Fanboys

00:01:30.787 --> 00:01:34.246
♫ The boys who carry the fan

00:01:34.246 --> 00:01:37.079
♫ Fanboys Fanboys

00:01:37.985 --> 00:01:41.761
♫ The boys who had a plan

00:01:41.761 --> 00:01:45.427
♫ The boys saw the land
was crying for justice

00:01:45.427 --> 00:01:48.992
♫ Yet they knew that this wasn't the way

00:01:48.992 --> 00:01:50.803
♫ So they put down their fan

00:01:50.803 --> 00:01:53.152
♫ And they picked up their notebooks

00:01:53.152 --> 00:01:57.319
♫ And this is what they had to say ♫

00:01:59.583 --> 00:02:01.879
- [Voiceover] Though we
lay down our fan this day,

00:02:01.879 --> 00:02:04.968
you too can have the power
to unite phrases, words

00:02:04.968 --> 00:02:09.135
and clauses with the coordinating
conjunctions of English.

00:02:10.769 --> 00:02:11.602
Which are:

00:02:12.905 --> 00:02:14.624
- [All] For.

00:02:14.624 --> 00:02:15.457
And.

00:02:16.435 --> 00:02:17.268
Nor.

00:02:18.171 --> 00:02:19.004
But.

00:02:19.954 --> 00:02:20.787
Or.

00:02:21.687 --> 00:02:23.320
Yet.

00:02:23.320 --> 00:02:24.153
So.

00:02:25.564 --> 00:02:28.583
- [Voiceover] You too
can harness the power

00:02:28.583 --> 00:02:29.833
of the Fanboys.

00:02:32.319 --> 00:02:33.624
♫ Fanboys

00:02:33.624 --> 00:02:35.922
♫ They're the Fanboys

00:02:35.922 --> 00:02:38.404
♫ The boys who carry the fan

00:02:38.404 --> 00:02:39.456
♫ That's their whole deal

00:02:39.456 --> 00:02:40.289
♫ Fanboys

00:02:40.289 --> 00:02:43.186
♫ We're all Fanboys

00:02:43.186 --> 00:02:47.353
♫ Remember the boys with the fan ♫


Second derivatives (parametric functions)

WEBVTT
Kind: captions
Language: en

00:00:00.206 --> 00:00:03.086
- [Voiceover] So here we have
a set of parametric equations

00:00:03.086 --> 00:00:06.209
where x and y are both
defined in terms of t.

00:00:06.209 --> 00:00:08.647
So if you input all the
possible t's that you can

00:00:08.647 --> 00:00:11.073
into these functions and then plot

00:00:11.073 --> 00:00:14.115
the corresponding x and y's for each t,

00:00:14.115 --> 00:00:16.504
this will plot a curve in the x-y plain.

00:00:16.504 --> 00:00:19.429
What I wanna do in this
video is figure out

00:00:19.429 --> 00:00:22.936
the first derivative
of y with respect to x

00:00:22.936 --> 00:00:27.046
and the second derivative
of y with respect to x.

00:00:27.046 --> 00:00:29.751
And in both cases it's
going to be in terms of t.

00:00:29.751 --> 00:00:31.608
So let's get to it.

00:00:31.608 --> 00:00:33.663
So first lets find the first derivative

00:00:33.663 --> 00:00:35.580
of y with respect to x.

00:00:37.340 --> 00:00:39.166
First derivative of y with respect to x.

00:00:39.166 --> 00:00:42.568
And we've seen this
before in other videos,

00:00:42.568 --> 00:00:44.344
where this is going to be the derivative

00:00:44.344 --> 00:00:46.806
of y with respect to t

00:00:46.806 --> 00:00:50.639
over the derivative of
x with respect to t.

00:00:51.786 --> 00:00:54.817
And so this is going to be equal to,

00:00:54.817 --> 00:00:58.996
well, what is the derivative
of y with respect to t?

00:00:58.996 --> 00:00:59.829
Dy, dt

00:01:01.411 --> 00:01:02.851
is equal to.

00:01:02.851 --> 00:01:06.264
Let's see, the derivative of either the 3t

00:01:06.264 --> 00:01:09.514
with respect to 3t is just e to the 3t.

00:01:10.452 --> 00:01:14.156
And then the derivative
of 3t with respect to t

00:01:14.156 --> 00:01:15.421
is going to be three,

00:01:15.421 --> 00:01:17.639
so I can say times three like that,

00:01:17.639 --> 00:01:20.681
or I can put that three out front.

00:01:20.681 --> 00:01:23.188
And then the derivative of negative one,

00:01:23.188 --> 00:01:25.174
well, a constant doesn't change

00:01:25.174 --> 00:01:26.903
no matter what you do to your t,

00:01:26.903 --> 00:01:28.540
so that's just going to be zero.

00:01:28.540 --> 00:01:30.349
So that's dy, dt.

00:01:30.349 --> 00:01:31.740
So it's going to be equal to

00:01:31.740 --> 00:01:32.823
3e to the 3t,

00:01:36.539 --> 00:01:38.431
all of that over,

00:01:38.431 --> 00:01:42.007
well, what's the derivative
of x with respect to t?

00:01:42.007 --> 00:01:46.090
Derivative of x with
respect to t is equal to,

00:01:47.104 --> 00:01:49.124
well, we're gonna have
the three out front,

00:01:49.124 --> 00:01:53.106
and so the derivative of e
to the 2t with respect to 2t

00:01:53.106 --> 00:01:55.939
is going to be e to the 2t

00:01:55.939 --> 00:01:58.981
and then we're going to
take the derivative of 2t

00:01:58.981 --> 00:02:01.733
with respect to t, which is just two,

00:02:01.733 --> 00:02:04.483
so this is gonna be 6e to the 2t.

00:02:07.965 --> 00:02:09.607
6e to the 2t.

00:02:09.607 --> 00:02:12.648
And lets see, we could
simplify this a little bit.

00:02:12.648 --> 00:02:14.448
I'll now go to a neutral color.

00:02:14.448 --> 00:02:15.865
This is equal to,

00:02:17.748 --> 00:02:19.916
so this is gonna be one half,

00:02:19.916 --> 00:02:21.844
that's three over six,

00:02:21.844 --> 00:02:23.594
e to the 3t minus 2t.

00:02:25.781 --> 00:02:26.781
3t minus 2t.

00:02:28.060 --> 00:02:30.649
And I'm just using exponent
properties right over here.

00:02:30.649 --> 00:02:33.211
But if I have three t's and
I take away two of those t's,

00:02:33.211 --> 00:02:34.895
I'm just gonna have a t.

00:02:34.895 --> 00:02:39.132
So this is just going to
simplify to a t right over here.

00:02:39.132 --> 00:02:43.161
So now that we've now figured
out the first derivative of y

00:02:43.161 --> 00:02:46.168
with respect to x in terms of t,

00:02:46.168 --> 00:02:48.676
now how do we find the second derivative?

00:02:48.676 --> 00:02:51.184
How do we find the second derivative

00:02:51.184 --> 00:02:53.684
of y with respect to x now?

00:02:56.896 --> 00:02:58.045
And I'll give you a hint!

00:02:58.045 --> 00:03:00.100
We're going to use this same idea.

00:03:00.100 --> 00:03:01.400
If you wanna find the rate of change

00:03:01.400 --> 00:03:03.746
of something with respect to x,

00:03:03.746 --> 00:03:05.348
you find the rate of
change of that something

00:03:05.348 --> 00:03:07.937
with respect to t and divide it by

00:03:07.937 --> 00:03:10.201
the rate of change of x with respect to t.

00:03:10.201 --> 00:03:12.534
So what this is going to be,

00:03:14.058 --> 00:03:18.989
we wanna find the derivative
of the first derivative

00:03:18.989 --> 00:03:23.517
with respect to t, so
let me write this down.

00:03:23.517 --> 00:03:25.140
We wanna take the derivative,

00:03:25.140 --> 00:03:27.113
with respect to T in the numerator,

00:03:27.113 --> 00:03:29.946
of the first derivative,

00:03:29.946 --> 00:03:32.408
which I will put in blue now,

00:03:32.408 --> 00:03:33.241
of dy, dx,

00:03:40.513 --> 00:03:42.513
all of that over dx, dt.

00:03:46.486 --> 00:03:49.546
If it doesn't jump out at you
why this is the same thing

00:03:49.546 --> 00:03:51.357
that we did before, I encourage you

00:03:51.357 --> 00:03:53.784
to pause the video and think about it.

00:03:53.784 --> 00:03:56.001
Think about what we did
over here the first time.

00:03:56.001 --> 00:03:57.894
When we wanted to find the derivative

00:03:57.894 --> 00:03:59.811
of y with respect to x,

00:04:01.516 --> 00:04:05.127
we found the derivative
of y with respect to t

00:04:05.127 --> 00:04:06.601
and then divided that by the derivative

00:04:06.601 --> 00:04:08.540
of x with respect to t.

00:04:08.540 --> 00:04:11.957
Here, we wanna find the second derivative

00:04:13.231 --> 00:04:14.563
of y with respect to x.

00:04:14.563 --> 00:04:16.118
Actually, let me just write it down

00:04:16.118 --> 00:04:17.732
out here a little bit clearer.

00:04:17.732 --> 00:04:19.984
What we really wanna do is we wanna find

00:04:19.984 --> 00:04:24.129
the derivative with respect,
let me write it this way:

00:04:24.129 --> 00:04:26.428
When we wanted to find the derivative

00:04:26.428 --> 00:04:28.595
with respect to x of y,

00:04:31.362 --> 00:04:34.729
that was equal to derivative
of y with respect to t

00:04:34.729 --> 00:04:38.468
over the derivative of
x with respect to t.

00:04:38.468 --> 00:04:43.100
Now we wanna find the
derivative with respect to x

00:04:43.100 --> 00:04:46.600
of the first derivative with respect to x.

00:04:48.092 --> 00:04:50.391
And so everywhere we saw a y here,

00:04:50.391 --> 00:04:52.980
replace that with the first derivative.

00:04:52.980 --> 00:04:55.407
So this is going to be equal to,

00:04:55.407 --> 00:04:57.055
in the numerator the derivative

00:04:57.055 --> 00:04:58.727
with respect to t

00:04:58.727 --> 00:04:59.560
of dy, dx.

00:05:03.368 --> 00:05:06.491
Notice this was derivative
with respect to t of y.

00:05:06.491 --> 00:05:09.161
In fact, let me write it that
way just so you can see it.

00:05:09.161 --> 00:05:11.078
So if I clear this out,

00:05:13.109 --> 00:05:16.069
if I clear that out we're gonna get,

00:05:16.069 --> 00:05:19.902
this is the derivative
with respect to t of y.

00:05:21.363 --> 00:05:23.871
So hopefully you see,
before we had a y there,

00:05:23.871 --> 00:05:25.621
now we have a dy, dx.

00:05:26.472 --> 00:05:27.305
Dx, dt.

00:05:28.643 --> 00:05:31.522
Now this might seem really
daunting and complicated

00:05:31.522 --> 00:05:32.970
except for the fact
that these are actually

00:05:32.970 --> 00:05:35.827
fairly straight things to evaluate.

00:05:35.827 --> 00:05:37.521
Taking the derivative with respect to t

00:05:37.521 --> 00:05:39.832
of the first derivative,
well, that's just taking

00:05:39.832 --> 00:05:41.922
the derivative with respect to t of this,

00:05:41.922 --> 00:05:42.987
and this is pretty easy.

00:05:42.987 --> 00:05:46.597
This is the derivative,
it's just gonna be one half.

00:05:46.597 --> 00:05:48.487
And the derivative with
respect to t of e to the t

00:05:48.487 --> 00:05:50.890
is just e to the t.

00:05:50.890 --> 00:05:52.411
And so that's going to be over

00:05:52.411 --> 00:05:54.068
the derivative of x with respect to t,

00:05:54.068 --> 00:05:57.179
which we already saw is 6e to the 2t.

00:05:57.179 --> 00:05:58.262
6e to the 2t.

00:06:03.475 --> 00:06:05.930
We can write this as,

00:06:05.930 --> 00:06:09.749
one half divided by six is one over 12,

00:06:09.749 --> 00:06:12.166
and then e to the t minus 2t,

00:06:14.010 --> 00:06:16.982
which is equal to, we could write this

00:06:16.982 --> 00:06:21.092
as 1/12th e to the negative t,

00:06:21.092 --> 00:06:22.810
or we could write this as

00:06:22.810 --> 00:06:24.727
one over 12 e to the t.

00:06:27.211 --> 00:06:28.461
And we're done.