Adding decimals with ones, tenths and hundredths

WEBVTT
Kind: captions
Language: en

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- Let's do some more involved
examples using decimals.

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So, let's say we wanna
add four and 22 hundredths

00:00:10.680 --> 00:00:12.680
to 61 and 37 hundredths.

00:00:16.299 --> 00:00:18.424
Like always, I encourage
you to pause the video

00:00:18.424 --> 00:00:21.230
and try to figure out on your own.

00:00:21.230 --> 00:00:23.974
Well the way that my
brain tries to tackle it,

00:00:23.974 --> 00:00:27.705
is it says, alright, I have
four and 22 hundredths,

00:00:27.705 --> 00:00:29.542
and 61 and 37 hundredths,

00:00:29.542 --> 00:00:33.045
how about I add the four plus the 61,

00:00:33.045 --> 00:00:35.415
so it's gonna be four plus 61,

00:00:35.415 --> 00:00:39.447
and then separately let
me add the 22 hundredths

00:00:39.447 --> 00:00:42.011
to the 37 hundredths,

00:00:42.011 --> 00:00:46.178
so then I'm gonna add 22
hundredths plus 37 hundredths,

00:00:48.917 --> 00:00:52.834
and so four plus 61 is
going to be equal to 65,

00:00:54.472 --> 00:00:58.490
and then 22 hundredths plus 37 hundredths,

00:00:58.490 --> 00:01:00.928
well, what's 22 plus 37?

00:01:00.928 --> 00:01:03.261
Let's see, 20 plus 30 is 50,

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and then two plus seven is nine,

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and so it's going to be 59 hundredths.

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So 59 hundredths,

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so it's gonna be 65 and 59 hundredths,

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which we can write as
65 and 59 hundredths.

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Now like I've mentioned
in previous videos,

00:01:26.854 --> 00:01:29.423
there's many different
ways of adding decimals,

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and in future videos we'll learn

00:01:31.367 --> 00:01:33.360
arguably faster ways of doing it,

00:01:33.360 --> 00:01:35.413
but I wanna show you multiple ways

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so you get a sense of ways to approach it,

00:01:38.352 --> 00:01:40.707
and this is the way
that I do it in my head

00:01:40.707 --> 00:01:44.776
if someone said hey,
something cost $61.37,

00:01:44.776 --> 00:01:47.919
and then you had to pay $4.22 in tax,

00:01:47.919 --> 00:01:49.387
how much did you have to pay in total,

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this is how I would
actually do it in my head,

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I would get $65.59.

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Let's do another example.

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So let's say that we have
three and 53 hundredths,

00:02:05.083 --> 00:02:09.083
and we wanna add that to
four and 72 hundredths,

00:02:11.466 --> 00:02:13.337
like always, pause this video

00:02:13.337 --> 00:02:16.417
and see if you can figure
it out on your own.

00:02:16.417 --> 00:02:17.810
So like we did in the previous example,

00:02:17.810 --> 00:02:21.007
we can say alright, we could add the ones,

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three ones plus four ones,

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so it's gonna be three plus four,

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and then we could add the hundredths,

00:02:27.687 --> 00:02:30.687
and we could say plus 53 hundredths,

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plus 53 hundredths,

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plus 72 hundredths,

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plus 72 hundredths,

00:02:42.013 --> 00:02:44.486
now the three plus four
is pretty straightforward,

00:02:44.486 --> 00:02:46.150
that's just going to be ...

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Yup, blue color,

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this is just going to be equal to seven,

00:02:51.072 --> 00:02:55.324
but what is 53 hundredths
plus 72 hundredths?

00:02:55.324 --> 00:02:57.324
Well, what's 53 plus 72?

00:03:02.123 --> 00:03:03.225
I'm going to write it this way,

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this is 53 hundredths, hundredths,

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and this is 72 hundredths,

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well, three plus two is
going to be equal to five,

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and then 50 plus 70 is equal to 120,

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so this is going to be 125 hundredths,

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125 hundredths,

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so how we write 125
hundredths as a decimal

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or just as a number?

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Well, there's a couple of
ways to think about it,

00:03:40.423 --> 00:03:43.590
you could view this as 100 hundredths,

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hundredths,

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plus 25 hundredths, plus 25 hundredths,

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and the reason why I
picked 100 hundredths,

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is because 100 hundredths make a whole,

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plus 25 hundredths,

00:03:58.748 --> 00:04:01.692
and so 100 hundredths
is just going to be one,

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so this is going to be
one plus 25 hundredths,

00:04:04.513 --> 00:04:07.513
so I could write this one plus 0.25,

00:04:08.496 --> 00:04:10.405
and remember, I'm adding
that to the seven here,

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so it's gonna be seven,

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plus the one, plus the 25 hundredths,

00:04:18.265 --> 00:04:21.990
so when we added the 53
hundredths plus the 72 hundredths,

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we got 125 hundredths,

00:04:24.046 --> 00:04:27.227
which is the same thing
as one and 25 hundredths,

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now, the one, we could add in
the ones to the other ones,

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and so this is going to be seven plus one,

00:04:34.645 --> 00:04:38.812
is equal to eight, eight
and 25 hundredths, so 8.25.

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Now, I'm going to
introduce you a little bit

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to how people might tackle
this more systematically,

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and I want you to think about

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how this is the exact same
thing that we just did.

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So what you're going to see in the future

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is people are gonna say oh
you line up the decimals,

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so 3.53 plus 4.72,

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and the reason why people
say line up the decimals

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is so that you're adding the right places

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to the right places, 4.72,

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and then they say alright,

00:05:07.211 --> 00:05:10.533
we'll just go place by
place and add it up,

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which is essentially what we just did

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without doing it as
organized of a way here,

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and I did it this way
so that you really get

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the understanding of what's going on.

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But you could say alright,

00:05:21.365 --> 00:05:23.957
I have three hundredths
plus two hundredths,

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so I'm adding in the hundredths place,

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so three hundredths plus two
hundredths is five hundredths,

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and then you could say okay, then I have

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50 hundredths plus 70 hundredths,

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which I did over here,

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or you could view this as
five tenths plus seven tenths,

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and five tenths and seven tenths

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is going to be equal to 12 tenths,

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but you can't write 12
tenths in the tenths place,

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but what you can do,

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so five plus seven is
going to be 12 tenths,

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but 12 tenths is the same
thing as one and two tenths,

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so you can write the two
tenths in the tenths place,

00:06:03.371 --> 00:06:04.993
and then you take the one

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and you put it in the ones place,

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so five tenths plus seven tenths

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is one and two tenths,

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and then you add the ones up,

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and so you get one plus
three plus four is eight,

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so it's eight and two
tenths and five hundredths,

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or eight and 25 hundredths, or 8.25,

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the exact same thing any way
that you try to tackle it,

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but hopefully it makes sense
what's actually going on,

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you're adding hundredths to hundredths,

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tenths to tenths, ones to ones,

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and when, say you add tenths to tenths,

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and you get something
that's larger than one,

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well, that part that is a whole number,

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you can put in the ones place,

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and we'll get much much
more practice with this

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in future videos.


Adding decimals with ones and tenths parts

WEBVTT
Kind: captions
Language: en

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- [Instructor] Last video, we
got a little bit of practice

00:00:02.300 --> 00:00:05.063
adding decimals that involved tenths.

00:00:05.063 --> 00:00:08.738
Now let's do slightly
more complicated examples.

00:00:08.738 --> 00:00:11.905
So let's say we wanna add four to 5.7,

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or we could read the second
number as five and 7/10.

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Pause this video, and
see if you can do this.

00:00:22.223 --> 00:00:25.533
The way that my brain
tries to tackle this is,

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I try to separate the whole
numbers from the tenths,

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so you can view this
as being the same thing

00:00:30.846 --> 00:00:34.096
as four plus five plus 7/10.

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All I did here is I broke
up the five and 7/10

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into five plus 7/10, and the
reason why my brain likes

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to do that is because I can
then say okay four plus five,

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that's just going to be equal to nine,

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and then I just have to add the 7/10.

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So it's gonna be nine and
7/10 which I can rewrite,

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this is going to be
equal to nine and 7/10.

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Nine and 7/10 I could write as 9.7.

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Even though in future videos
we're going to learn other ways

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of adding decimals, especially larger,

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more complicated decimals, this is still

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how my brain adds four plus 5.7.

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Especially if I need to do it in my head.

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I say okay, four plus five
is nine, and then I have

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that 7/10, so it's going
to be nine and 7/10 or 9.7.

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Now let's do another example where

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both numbers involve a decimal.

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Let's say I want to add

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6.3 to 7.4.

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So 6.3 plus 7.4.

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Once again, pause this video and try

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to work through it on your own.

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Well my brain does it the same way.

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I break up the whole
numbers and the decimals.

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Once again, there's many
different ways of adding decimals,

00:02:00.516 --> 00:02:02.696
but this is just one
way that seems to work.

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Especially for decimals like this.

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So we could view this as six and 3/10,

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so I'm breaking up the
6.3, the six and 3/10,

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into six plus 3/10

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plus seven and 4/10.

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Seven plus 4/10, and then
this you can view as,

00:02:22.476 --> 00:02:26.643
so you could view this as six
plus seven, six plus seven,

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plus, plus 3/10, plus 3/10

00:02:34.196 --> 00:02:38.129
plus 4/10, plus 4/10.

00:02:38.129 --> 00:02:41.857
If you add the ones here, you
have six ones and seven ones,

00:02:41.857 --> 00:02:46.024
that's going to be equal to
13, and then 3/10 and 4/10.

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If you have three of something
and then you add four

00:02:49.040 --> 00:02:51.551
of that, that's going to be 7/10,

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and we would write 7/10 as 0.7.

00:02:54.471 --> 00:02:56.373
Seven in the tenths place.

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Then what's 13 plus 7/10?

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Well that is going to be 13.

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This is going to be equal to 13.7.

00:03:06.705 --> 00:03:08.538
13.7, and we are done.

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Let me do one more example
that will get a little bit,

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a little bit more involved.

00:03:17.922 --> 00:03:20.582
So let me delete all of these.

00:03:20.582 --> 00:03:23.332
Let's say I wanted to add 6.3 to,

00:03:29.740 --> 00:03:32.990
and I'm gonna add that to 2 point, 2.9.

00:03:35.903 --> 00:03:39.629
Pause the video and see if
you can figure this out.

00:03:39.629 --> 00:03:41.255
Let's do the same thing.

00:03:41.255 --> 00:03:45.422
This is going to be six
and 3/10, so six plus 3/10,

00:03:46.816 --> 00:03:49.399
plus two, plus 9/10,

00:03:52.519 --> 00:03:55.370
or you could view this as six plus two,

00:03:55.370 --> 00:03:57.094
so I'll put all my ones together.

00:03:57.094 --> 00:04:01.241
Six plus two, and then I'll
put my tenths together,

00:04:01.241 --> 00:04:04.739
plus 3/10, plus 3/10.

00:04:04.739 --> 00:04:08.645
Plus 9/10, plus 9/10.

00:04:08.645 --> 00:04:11.576
And so the six plus two
is pretty straightforward.

00:04:11.576 --> 00:04:13.998
That is going to be equal to eight.

00:04:13.998 --> 00:04:17.911
Now what's 3/10 plus 9/10?

00:04:17.911 --> 00:04:20.932
This is gonna get a
little bit interesting.

00:04:20.932 --> 00:04:24.543
3/10 plus 9/10, and I could write it out.

00:04:24.543 --> 00:04:28.710
I could say this is three
tenths, this is nine tenths.

00:04:30.053 --> 00:04:34.178
Well 3/10 plus 9/10 is equal to 12/10.

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This is going to be 12/10,

00:04:36.450 --> 00:04:39.329
but how do we write 12/10 as a number?

00:04:39.329 --> 00:04:43.496
Well 12/10 is the same
thing as 10/10 plus 2/10.

00:04:46.800 --> 00:04:50.488
The reason why I broke it up
this way is 10/10 is one whole,

00:04:50.488 --> 00:04:53.287
so this is going to be equal to one.

00:04:53.287 --> 00:04:56.056
When you add these two
together, it's 12/10

00:04:56.056 --> 00:04:59.930
which is the same thing as one and 2/10.

00:04:59.930 --> 00:05:04.651
So one plus 2/10 or, well let
me just write it that way.

00:05:04.651 --> 00:05:08.818
This I can rewrite as plus one plus 2/10,

00:05:10.384 --> 00:05:12.755
and then I think you
see where this is going.

00:05:12.755 --> 00:05:16.999
I could add the eight and the
one, and I get nine and 2/10.

00:05:16.999 --> 00:05:19.332
So nine and 2/10.

00:05:20.346 --> 00:05:22.668
So it's going to be 9.2.

00:05:22.668 --> 00:05:25.280
The reason why this one was a
little bit more interesting is

00:05:25.280 --> 00:05:28.993
I added the ones, I got
six plus two is eight,

00:05:28.993 --> 00:05:30.769
but then when I added the tenths,

00:05:30.769 --> 00:05:34.158
I got something that
was more than a whole.

00:05:34.158 --> 00:05:38.325
I got 12/10 which is one and 2/10,

00:05:39.401 --> 00:05:42.474
and so I added one more whole
to the eight to get nine,

00:05:42.474 --> 00:05:45.003
and then I had those 2/10 leftover.

00:05:45.003 --> 00:05:46.857
This is really good to understand

00:05:46.857 --> 00:05:48.781
because in the future when
you're adding decimals,

00:05:48.781 --> 00:05:51.789
you'll be doing stuff like
carrying from one place

00:05:51.789 --> 00:05:54.364
to another, and this is
essentially what we did.

00:05:54.364 --> 00:05:58.521
When we added the 3/10 plus
the 9/10, we got 12/10,

00:05:58.521 --> 00:06:00.239
and so we added an extra whole,

00:06:00.239 --> 00:06:02.032
and then we had the leftover 2/10.

00:06:02.032 --> 00:06:03.876
Hopefully, that makes some sense.


Common chain rule misunderstandings

WEBVTT
Kind: captions
Language: en

00:00:00.025 --> 00:00:01.305
What we're gonna do in this video

00:00:01.305 --> 00:00:05.485
is focus on key misunderstandings
that folks often have,

00:00:05.485 --> 00:00:07.779
and we actually got
these misunderstandings

00:00:07.779 --> 00:00:10.680
from the folks who write the AP exams,

00:00:10.680 --> 00:00:13.338
from the actual College Board.

00:00:13.338 --> 00:00:16.316
So let's say that we are trying to take

00:00:16.316 --> 00:00:19.470
the derivative of the expression.

00:00:19.470 --> 00:00:21.125
So let's say we're taking the derivative

00:00:21.125 --> 00:00:23.066
of the expression,

00:00:23.066 --> 00:00:25.483
the natural log of sine of x.

00:00:29.409 --> 00:00:32.265
So the first key misconception
or misunderstanding

00:00:32.265 --> 00:00:34.285
that many people have

00:00:34.285 --> 00:00:35.283
is when you're dealing

00:00:35.283 --> 00:00:36.932
with transcendental functions like this,

00:00:36.932 --> 00:00:38.836
and transcendental functions
is just a fancy word

00:00:38.836 --> 00:00:40.810
for these functions like
trigonometric functions,

00:00:40.810 --> 00:00:42.040
logarithmic functions,

00:00:42.040 --> 00:00:45.570
that don't use standard
algebraic operations.

00:00:45.570 --> 00:00:47.613
But when you see transcendental
functions like this

00:00:47.613 --> 00:00:49.192
or compositions of them,

00:00:49.192 --> 00:00:53.000
many people confuse this with
the product of functions.

00:00:53.000 --> 00:00:54.718
So at first when they look at this,

00:00:54.718 --> 00:00:56.344
they might see this as being the same

00:00:56.344 --> 00:00:59.130
as the derivative with respect to x

00:00:59.130 --> 00:01:00.797
of natural log of x,

00:01:02.567 --> 00:01:05.400
natural log of x, times sine of x.

00:01:06.607 --> 00:01:08.511
And you can see just the
way that it's written,

00:01:08.511 --> 00:01:10.136
they look very similar,

00:01:10.136 --> 00:01:12.830
but this is the product of two functions.

00:01:12.830 --> 00:01:16.847
If you said natural log of x is f of x,

00:01:16.847 --> 00:01:19.703
and sine of x is g of x,

00:01:19.703 --> 00:01:22.582
this is the product of sine and g of x,

00:01:22.582 --> 00:01:26.112
sorry this is the product
of f of x and g of x,

00:01:26.112 --> 00:01:29.037
and here you would use the product rule.

00:01:29.037 --> 00:01:31.104
So to actually compute this,

00:01:31.104 --> 00:01:35.330
you would use the
product, the product rule.

00:01:35.330 --> 00:01:37.513
But this is a composition.

00:01:37.513 --> 00:01:39.974
Here you have f of g of x,

00:01:39.974 --> 00:01:42.224
not f of x times g of x.

00:01:43.503 --> 00:01:44.836
So here you have

00:01:46.731 --> 00:01:49.541
that is our g of x, it equals sine of x,

00:01:49.541 --> 00:01:52.072
and then our f of g of x

00:01:52.072 --> 00:01:54.138
is the natural log of sine of x.

00:01:54.138 --> 00:01:56.055
So this is f of g of x,

00:01:58.387 --> 00:02:00.663
f of g of x just like that.

00:02:00.663 --> 00:02:02.497
If someone asks you just what f of x was,

00:02:02.497 --> 00:02:03.983
well that would be natural log of x,

00:02:03.983 --> 00:02:07.675
but f of g of x is
natural log of our g of x,

00:02:07.675 --> 00:02:09.974
which is natural log of sine of x.

00:02:09.974 --> 00:02:11.623
So that's the key first thing,

00:02:11.623 --> 00:02:14.293
always make sure whether you're gonna use,

00:02:14.293 --> 00:02:16.522
especially with these
transcendental functions,

00:02:16.522 --> 00:02:18.798
that hey if this is a
composition you've gotta use

00:02:18.798 --> 00:02:20.980
the chain rule, not the product rule.

00:02:20.980 --> 00:02:22.234
It's not the product.

00:02:22.234 --> 00:02:23.628
Now sometimes you have a combination,

00:02:23.628 --> 00:02:24.789
you have a product of compositions,

00:02:24.789 --> 00:02:26.809
and then things get a
little bit more involved.

00:02:26.809 --> 00:02:29.061
But pay close attention to make sure

00:02:29.061 --> 00:02:33.310
that you're not dealing
with a composition.

00:02:33.310 --> 00:02:36.190
Now the next misconception students have

00:02:36.190 --> 00:02:37.513
is even if they recognize,

00:02:37.513 --> 00:02:38.837
okay I've gotta use the chain rule,

00:02:38.837 --> 00:02:42.204
sometimes it doesn't
go fully to completion.

00:02:42.204 --> 00:02:44.526
So let's continue using this example.

00:02:44.526 --> 00:02:45.988
The chain rule here says,

00:02:45.988 --> 00:02:49.007
look we have to take the
derivative of the outer function

00:02:49.007 --> 00:02:51.817
with respect to the inner function.

00:02:51.817 --> 00:02:53.140
So if I were to say,

00:02:53.140 --> 00:02:56.182
in this case, f of x is natural log of x,

00:02:56.182 --> 00:02:58.550
f of g of x is this expression here.

00:02:58.550 --> 00:03:00.617
So if I wanna do this first part,

00:03:00.617 --> 00:03:02.117
f prime of g of x,

00:03:05.423 --> 00:03:07.304
f prime of g of x,

00:03:07.304 --> 00:03:10.602
well the derivative of
the natural log of x

00:03:10.602 --> 00:03:11.769
is one over x.

00:03:13.458 --> 00:03:16.894
So the natural log,
derivative of natural log of x

00:03:16.894 --> 00:03:18.705
is one over x,

00:03:18.705 --> 00:03:21.027
but we don't want the
derivative where the input is x.

00:03:21.027 --> 00:03:23.721
We want the derivative
when the input is g of x.

00:03:23.721 --> 00:03:25.811
So instead of it being one over x,

00:03:25.811 --> 00:03:28.876
it's gonna be one over g of x.

00:03:28.876 --> 00:03:30.209
One over g of x,

00:03:31.198 --> 00:03:34.634
and we know that g of x
is equal to sine of x.

00:03:34.634 --> 00:03:36.801
That's equal to sine of x.

00:03:38.419 --> 00:03:40.416
Now one key misunderstanding
that the folks

00:03:40.416 --> 00:03:41.972
of the College Board told us about

00:03:41.972 --> 00:03:44.781
is many students stop right there.

00:03:44.781 --> 00:03:46.407
They just do this first part,

00:03:46.407 --> 00:03:49.379
and then they forget to
multiply this second part.

00:03:49.379 --> 00:03:51.562
So here we are not done.

00:03:51.562 --> 00:03:55.021
We need to take this and
multiply it times g prime of x.

00:03:55.021 --> 00:03:56.322
And let me write this down.

00:03:56.322 --> 00:03:58.760
g prime of x, what would that be?

00:03:58.760 --> 00:04:02.684
Well the derivative of sine
of x with respect to x,

00:04:02.684 --> 00:04:05.285
well that's just going to be cosine of x,

00:04:05.285 --> 00:04:06.285
cosine of x.

00:04:08.768 --> 00:04:11.113
So in this example right over here,

00:04:11.113 --> 00:04:13.829
the derivative is going to be,

00:04:13.829 --> 00:04:15.687
let's see if I can
squeeze it in over here,

00:04:15.687 --> 00:04:18.218
it's going to be one over
sine of x which is this part,

00:04:18.218 --> 00:04:19.718
times cosine of x.

00:04:20.703 --> 00:04:22.235
So let me write it down.

00:04:22.235 --> 00:04:25.318
It is going to be one over sine of x,

00:04:26.624 --> 00:04:28.574
we'll do that in that other color,

00:04:28.574 --> 00:04:30.157
one over sine of x,

00:04:32.196 --> 00:04:34.936
and then times cosine of x.

00:04:34.936 --> 00:04:35.958
So once again,

00:04:35.958 --> 00:04:37.560
just to make sure that you don't fall into

00:04:37.560 --> 00:04:39.604
one of these misconceptions.

00:04:39.604 --> 00:04:42.320
Let me box this off so it's a little bit,

00:04:42.320 --> 00:04:45.037
it's a little bit cleaner.

00:04:45.037 --> 00:04:46.570
So to just make sure that you don't fall

00:04:46.570 --> 00:04:49.333
into one of these misconceptions here,

00:04:49.333 --> 00:04:51.515
recognize the composition,

00:04:51.515 --> 00:04:53.837
that this is not the
product of natural log of x

00:04:53.837 --> 00:04:54.836
and sine of x.

00:04:54.836 --> 00:04:56.694
It's natural log of sine of x.

00:04:56.694 --> 00:04:59.178
And then when you're actually
applying the chain rule,

00:04:59.178 --> 00:05:01.477
derivative of the outside
with respect to the inside,

00:05:01.477 --> 00:05:03.868
so the derivative of natural
log of x is one over x,

00:05:03.868 --> 00:05:06.678
so that applied when the input is g of x

00:05:06.678 --> 00:05:09.023
is one over sine of x.

00:05:09.023 --> 00:05:11.043
And then multiply that
times the derivative

00:05:11.043 --> 00:05:13.273
of the inner function.

00:05:13.273 --> 00:05:16.314
So don't forget to do
this right over here.

00:05:16.314 --> 00:05:19.078
Now another misconception
that students have,

00:05:19.078 --> 00:05:21.934
is instead of doing what we just did,

00:05:21.934 --> 00:05:23.629
instead of applying the
chain rule like this,

00:05:23.629 --> 00:05:26.787
they take the derivative
of the outer function

00:05:26.787 --> 00:05:29.712
with respect to the derivative
of the inner function.

00:05:29.712 --> 00:05:31.663
So for example,

00:05:31.663 --> 00:05:33.799
they would compute this,

00:05:33.799 --> 00:05:35.799
f prime of g prime of x,

00:05:38.606 --> 00:05:40.606
f prime of g prime of x.

00:05:41.856 --> 00:05:46.570
Which in this case, f
prime of x is one over x,

00:05:46.570 --> 00:05:48.288
but if the input is g prime of x,

00:05:48.288 --> 00:05:50.726
g prime of x is cosine of x.

00:05:50.726 --> 00:05:53.118
So many students end up doing this

00:05:53.118 --> 00:05:56.787
where they take the
derivative of the outside,

00:05:56.787 --> 00:06:00.386
and they apply the input into that,

00:06:00.386 --> 00:06:03.103
they use the derivative
of the inside function.

00:06:03.103 --> 00:06:04.603
This is not right.

00:06:06.353 --> 00:06:08.002
Be very careful that
you're not doing that.

00:06:08.002 --> 00:06:09.581
You do the derivative
of the outside function

00:06:09.581 --> 00:06:11.392
with respect to the inside function,

00:06:11.392 --> 00:06:12.762
not taking it's derivative,

00:06:12.762 --> 00:06:14.898
and then multiply, don't
forget to multiply,

00:06:14.898 --> 00:06:18.981
times the derivative of
the inside function here.

00:06:19.891 --> 00:06:21.470
So hopefully that helps a little bit.

00:06:21.470 --> 00:06:23.443
If all of this looks
completely foreign to you,

00:06:23.443 --> 00:06:25.115
I encourage you to watch the whole series

00:06:25.115 --> 00:06:27.066
of chain rule introductory videos

00:06:27.066 --> 00:06:28.552
and worked examples we have.

00:06:28.552 --> 00:06:30.897
This is just a topping on top of that

00:06:30.897 --> 00:06:33.010
to make sure that you don't
fall into these misconceptions

00:06:33.010 --> 00:06:35.146
of applying the product rule

00:06:35.146 --> 00:06:36.980
when you really need to
be applying the chain rule

00:06:36.980 --> 00:06:39.419
or forgetting to do
part of the chain rule,

00:06:39.419 --> 00:06:41.392
multiplying by g prime of x,

00:06:41.392 --> 00:06:44.559
or evaluating f prime of g prime of x.

00:06:45.456 --> 00:06:47.456
So hopefully that helps.


Identifying composite functions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.623
- [Instructor] We're going
to do in this video is review

00:00:02.623 --> 00:00:05.339
the notion of composite functions

00:00:05.339 --> 00:00:07.940
and then build some skills recognizing

00:00:07.940 --> 00:00:10.838
how functions can actually be composed.

00:00:10.838 --> 00:00:13.604
If you've never heard of
the term composite functions

00:00:13.604 --> 00:00:15.575
or if the first few minutes of this video

00:00:15.575 --> 00:00:16.866
look unfamiliar to you,

00:00:16.866 --> 00:00:19.197
I encourage you to
watch the algebra videos

00:00:19.197 --> 00:00:21.620
on composite functions on Khan Academy.

00:00:21.620 --> 00:00:24.096
The goal of this one is
to really be a little

00:00:24.096 --> 00:00:26.551
bit of a practice before
we get into some skills

00:00:26.551 --> 00:00:28.047
that are necessary in calculus,

00:00:28.047 --> 00:00:30.087
and particularly the chain rule.

00:00:30.087 --> 00:00:34.100
So let's just review what
a composite function is.

00:00:34.100 --> 00:00:36.183
So let's say that I have,

00:00:37.115 --> 00:00:40.050
let's say that I have f of x,

00:00:40.050 --> 00:00:43.050
f of x being equal to one plus x.

00:00:44.760 --> 00:00:47.510
And let's say that we have g of x

00:00:48.456 --> 00:00:52.206
is equal to, let's say g of x is

00:00:52.206 --> 00:00:54.373
equal to cosine of x.

00:00:55.556 --> 00:00:58.223
So what would f of g of x be,

00:00:59.208 --> 00:01:00.208
f of g of x?

00:01:02.478 --> 00:01:03.951
And I encourage you to pause this video

00:01:03.951 --> 00:01:06.549
and try to work it out on your own.

00:01:06.549 --> 00:01:08.768
Well one way to think about it is

00:01:08.768 --> 00:01:11.799
the input into f of x is no longer x,

00:01:11.799 --> 00:01:13.385
it is g of x.

00:01:13.385 --> 00:01:15.406
So everywhere where we see an x

00:01:15.406 --> 00:01:16.817
in the definition of f of x,

00:01:16.817 --> 00:01:18.290
we would replace with the g of x.

00:01:18.290 --> 00:01:20.500
So this is gonna be equal to,

00:01:20.500 --> 00:01:23.417
this is gonna be equal to one plus.

00:01:24.807 --> 00:01:26.197
Instead of the input being x,

00:01:26.197 --> 00:01:28.061
the input is g of x, so the output

00:01:28.061 --> 00:01:29.644
is one plus g of x.

00:01:31.414 --> 00:01:34.726
And g of x, of course, is cosine of x.

00:01:34.726 --> 00:01:36.096
So instead of writing g of x there,

00:01:36.096 --> 00:01:38.263
I could write cosine of x.

00:01:40.891 --> 00:01:43.887
And one way to visualize this is,

00:01:43.887 --> 00:01:47.032
I'm putting my x into g of x first,

00:01:47.032 --> 00:01:50.142
so x goes into the function g,

00:01:50.142 --> 00:01:51.975
and it outputs g of x.

00:01:52.953 --> 00:01:54.935
And then we're gonna
take that output, g of x,

00:01:54.935 --> 00:01:57.168
and then input it into f of x,

00:01:57.168 --> 00:01:59.822
or input it into the
function f, I should say.

00:01:59.822 --> 00:02:01.754
We input into the function f,

00:02:01.754 --> 00:02:05.152
and then that is going to output f of

00:02:05.152 --> 00:02:08.989
whatever the input was,
and the input is g of x,

00:02:08.989 --> 00:02:09.822
g of x.

00:02:10.694 --> 00:02:12.827
So now with that review out of the way,

00:02:12.827 --> 00:02:14.887
let's see if we can go
the other way around.

00:02:14.887 --> 00:02:16.706
Let's see if we can look at some type

00:02:16.706 --> 00:02:18.012
of a function definition and say,

00:02:18.012 --> 00:02:21.317
hey, can we express that as a composition

00:02:21.317 --> 00:02:23.170
of other functions.

00:02:23.170 --> 00:02:26.310
So let's start with, let's say

00:02:26.310 --> 00:02:27.977
that I have a g of x

00:02:30.274 --> 00:02:32.024
is equal to cosine of

00:02:33.964 --> 00:02:35.797
sine of x plus one.

00:02:37.782 --> 00:02:39.866
And I also wanna state,

00:02:39.866 --> 00:02:42.193
there's oftentimes more than one way

00:02:42.193 --> 00:02:45.011
to compose, or to construct a function

00:02:45.011 --> 00:02:46.423
based on the composition of other ones.

00:02:46.423 --> 00:02:48.892
But with that said,
pause this video and say,

00:02:48.892 --> 00:02:50.444
hey, can I express g of x

00:02:50.444 --> 00:02:52.523
as a composition of two other functions,

00:02:52.523 --> 00:02:54.940
let's say an f and an h of x?

00:02:55.968 --> 00:02:58.436
So there's a couple of ways
that you could think about it.

00:02:58.436 --> 00:03:00.290
You could say, alright, well let's see,

00:03:00.290 --> 00:03:03.750
I have this sine of x right over here.

00:03:03.750 --> 00:03:06.658
So what if I called that an f of x?

00:03:06.658 --> 00:03:08.572
So let's say I called that,

00:03:08.572 --> 00:03:10.369
well actually let me
use a different variable

00:03:10.369 --> 00:03:11.499
so we don't get confused here.

00:03:11.499 --> 00:03:14.306
Let me call this u of x,

00:03:14.306 --> 00:03:16.002
the sine of x right over there.

00:03:16.002 --> 00:03:19.498
So this would be cosine
of u of x plus one.

00:03:19.498 --> 00:03:21.209
And so if we then divided,

00:03:21.209 --> 00:03:25.442
if we then defined
another function as v of x

00:03:25.442 --> 00:03:29.275
being equal to cosine
of whatever its input is

00:03:30.343 --> 00:03:33.375
plus one, well then this
looks like the composition

00:03:33.375 --> 00:03:35.088
of v and u of x.

00:03:35.088 --> 00:03:39.000
Instead of v of x, if we did v of u of x,

00:03:39.000 --> 00:03:42.543
then this would be cosine
of u of x plus one,

00:03:42.543 --> 00:03:43.948
let me write that down.

00:03:43.948 --> 00:03:46.198
So if we wrote v of u of x,

00:03:48.733 --> 00:03:52.209
which is sine of x, if we did v of u of x,

00:03:52.209 --> 00:03:54.918
that is going to be equal to

00:03:54.918 --> 00:03:58.430
cosine of, instead of an x plus one,

00:03:58.430 --> 00:04:00.782
it's going to be a u of x plus one.

00:04:00.782 --> 00:04:02.588
And u of x, let me write this here,

00:04:02.588 --> 00:04:05.255
u of x is equal to sine of x.

00:04:06.128 --> 00:04:08.050
That's how we set this up.

00:04:08.050 --> 00:04:11.011
So we can either write
cosine of u of x plus one,

00:04:11.011 --> 00:04:13.928
or cosine of sine of x plus one,

00:04:15.178 --> 00:04:17.320
which was exactly what we had before.

00:04:17.320 --> 00:04:19.576
And so this function, g of x,

00:04:19.576 --> 00:04:22.435
is we say u of x is equal to sine of x,

00:04:22.435 --> 00:04:24.778
if we say u of x is equal to sine of x,

00:04:24.778 --> 00:04:29.366
and v of x is equal to
cosine of x plus one,

00:04:29.366 --> 00:04:32.721
then we could write g
of x as the composition

00:04:32.721 --> 00:04:35.334
of these two functions.

00:04:35.334 --> 00:04:38.947
Now you could even make it a
composition of three functions.

00:04:38.947 --> 00:04:42.154
We could keep u of x to
be equal to sine of x.

00:04:42.154 --> 00:04:45.154
We could define, let's say, a w of x

00:04:46.822 --> 00:04:49.239
to be equal to x plus one.

00:04:50.298 --> 00:04:52.037
And so then, let's think about it.

00:04:52.037 --> 00:04:52.870
W of x,

00:04:54.554 --> 00:04:57.232
w of u of x, I should say,

00:04:57.232 --> 00:04:59.815
w of u, I'll do the same color,

00:05:02.170 --> 00:05:05.420
w of u of x is going to be equal to.

00:05:07.444 --> 00:05:10.153
Now my input is no longer x, it's u of x,

00:05:10.153 --> 00:05:12.990
so it's going to be a u of x plus one,

00:05:12.990 --> 00:05:15.731
or just sine of x plus one.

00:05:15.731 --> 00:05:18.902
So that's sine of x plus one.

00:05:18.902 --> 00:05:21.350
And then if we define a third function,

00:05:21.350 --> 00:05:24.670
let's say, let's see, I'll call it,

00:05:24.670 --> 00:05:27.421
let's call it h; we're
running out of variables,

00:05:27.421 --> 00:05:28.876
although there are a lot of letters left.

00:05:28.876 --> 00:05:31.089
So if I say h of x is
just equal to the cosine

00:05:31.089 --> 00:05:34.354
of whatever I input, so it's
equal to the cosine of x,

00:05:34.354 --> 00:05:37.300
well then h of w of u of x

00:05:37.300 --> 00:05:38.760
is gonna be g of x.

00:05:38.760 --> 00:05:39.891
Let me write that down.

00:05:39.891 --> 00:05:41.308
H of w of u of x,

00:05:47.335 --> 00:05:50.002
u of x, is going to be equal to,

00:05:51.410 --> 00:05:53.887
remember, h of x takes the cosine

00:05:53.887 --> 00:05:55.372
of whatever its input is,

00:05:55.372 --> 00:05:57.341
so it's gonna take the cosine.

00:05:57.341 --> 00:06:00.401
Now its input is w of u of x.

00:06:00.401 --> 00:06:02.751
We already figured out w of u of x

00:06:02.751 --> 00:06:04.589
is going to be this business.

00:06:04.589 --> 00:06:08.089
So it's going to be sine of x plus one,

00:06:10.993 --> 00:06:12.873
where the u of x is sine of x,

00:06:12.873 --> 00:06:14.949
but then we input that into w,

00:06:14.949 --> 00:06:16.986
so we got sine of x plus one,

00:06:16.986 --> 00:06:19.822
and then we inputed that into h

00:06:19.822 --> 00:06:21.584
to get cosine of that,

00:06:21.584 --> 00:06:23.132
which is our original expression,

00:06:23.132 --> 00:06:25.215
which is equal to g of x.

00:06:27.780 --> 00:06:30.444
So the whole point here is to appreciate

00:06:30.444 --> 00:06:33.259
how to recognize
compositions of functions.

00:06:33.259 --> 00:06:35.685
Now I wanna stress, it's not always

00:06:35.685 --> 00:06:37.688
going to be a composition of a function.

00:06:37.688 --> 00:06:41.131
For example, if I have some function,

00:06:41.131 --> 00:06:42.797
let me just clear this out,

00:06:42.797 --> 00:06:45.629
if I had some function f of x

00:06:45.629 --> 00:06:49.212
is equal to cosine of x times sine of x,

00:06:52.444 --> 00:06:53.989
it would be hard to express this as

00:06:53.989 --> 00:06:56.269
a composition of functions,

00:06:56.269 --> 00:06:59.274
but I can represent it as
the product of functions.

00:06:59.274 --> 00:07:01.879
For example, I could say cosine of x,

00:07:01.879 --> 00:07:04.461
I could say u of x is
equal to cosine of x.

00:07:04.461 --> 00:07:07.870
And I could say v of x,
it's a different color,

00:07:07.870 --> 00:07:09.370
I could say v of x

00:07:11.009 --> 00:07:12.657
is equal to sine of x.

00:07:12.657 --> 00:07:15.026
And so here, f of x
wouldn't be the composition

00:07:15.026 --> 00:07:17.563
of u and v, it would be the product.

00:07:17.563 --> 00:07:19.896
F of x is equal to u of x

00:07:22.013 --> 00:07:23.096
times v of x.

00:07:24.956 --> 00:07:26.696
If we were take the composition,

00:07:26.696 --> 00:07:29.196
if we were to say u of v of x,

00:07:30.553 --> 00:07:32.129
pause the video, think about what that is,

00:07:32.129 --> 00:07:33.945
and that's a little bit of review.

00:07:33.945 --> 00:07:37.195
Well this is going to be, I take u of x

00:07:38.030 --> 00:07:41.058
takes the cosine of whatever is input,

00:07:41.058 --> 00:07:45.331
and now the input is v of
x, which would be sine of x;

00:07:45.331 --> 00:07:46.164
sine of x.

00:07:47.284 --> 00:07:49.097
And then if you did v of u of x,

00:07:49.097 --> 00:07:50.206
well that'd be the other way around.

00:07:50.206 --> 00:07:52.308
It would be sine of cosine of x.

00:07:52.308 --> 00:07:54.159
But anyway, this is once again, just to

00:07:54.159 --> 00:07:55.801
help us recognize, hey, do I have,

00:07:55.801 --> 00:07:58.095
when I look at an expression
or a function definition,

00:07:58.095 --> 00:08:00.156
am I looking at products of functions,

00:08:00.156 --> 00:08:02.441
am I looking at compositions of functions?

00:08:02.441 --> 00:08:04.701
Sometimes you're looking
at products of compositions

00:08:04.701 --> 00:08:06.103
or quotients of compositions,

00:08:06.103 --> 00:08:08.021
all sorts of different combinations of how

00:08:08.021 --> 00:08:11.864
you can put functions together
to create new functions.


Example exercise using limit flow chart

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.221
- [Instructor] in a previous video,

00:00:01.221 --> 00:00:03.580
we introduced this flowchart
that helps us think

00:00:03.580 --> 00:00:05.758
about what strategies to use

00:00:05.758 --> 00:00:08.984
when trying to determine
a limit of a function

00:00:08.984 --> 00:00:11.056
as it approaches a point.

00:00:11.056 --> 00:00:12.292
What we're going to do in this video

00:00:12.292 --> 00:00:16.661
is now try to apply that in
several example exercises.

00:00:16.661 --> 00:00:18.490
So the first question,

00:00:18.490 --> 00:00:21.898
they say based on the
process in the flowchart,

00:00:21.898 --> 00:00:26.016
let's try to find the limit as
x approaches three of f of x

00:00:26.016 --> 00:00:28.801
for f of x is equal to this business.

00:00:28.801 --> 00:00:30.841
What are the steps that
we should go through?

00:00:30.841 --> 00:00:32.425
And they have these various steps,

00:00:32.425 --> 00:00:35.783
and you might say, "Well what
is A and B and C, D and A?"

00:00:35.783 --> 00:00:40.198
Well if you go back to
the original flowchart,

00:00:40.198 --> 00:00:42.664
you see this is A, this is B,

00:00:42.664 --> 00:00:45.649
this is C, this is D,
et cetera, et cetera.

00:00:45.649 --> 00:00:46.482
So let's,

00:00:47.732 --> 00:00:50.734
let's just first try to substitute,

00:00:50.734 --> 00:00:54.276
see if we can get a value for f of three.

00:00:54.276 --> 00:00:57.484
If f of three works, then
we are essentially done

00:00:57.484 --> 00:01:00.855
because this is not some
strange exotic function,

00:01:00.855 --> 00:01:03.263
it seems like it's continuous,

00:01:03.263 --> 00:01:04.604
around x equals three,

00:01:04.604 --> 00:01:06.997
if f of three exists, let's try it out.

00:01:06.997 --> 00:01:07.830
So let me,

00:01:09.592 --> 00:01:12.007
so we're going to have f of three

00:01:12.007 --> 00:01:14.750
is equal to the square root of,

00:01:14.750 --> 00:01:17.633
let's see six minus five,

00:01:17.633 --> 00:01:19.749
two times three minus five,

00:01:19.749 --> 00:01:21.266
minus one

00:01:21.266 --> 00:01:23.183
over three minus three.

00:01:26.163 --> 00:01:29.508
So in the numerator here we get zero,

00:01:29.508 --> 00:01:31.172
'cause this is one minus one,

00:01:31.172 --> 00:01:35.012
so this is going to be
equal to zero over zero.

00:01:35.012 --> 00:01:36.476
So the first thing that happens

00:01:36.476 --> 00:01:38.462
is we fall into the,

00:01:38.462 --> 00:01:40.086
in that we tried to evaluate it,

00:01:40.086 --> 00:01:42.319
we fell into indeterminate form.

00:01:42.319 --> 00:01:46.065
So the first step is going to be C.

00:01:46.065 --> 00:01:47.537
Now the next thing that I would do

00:01:47.537 --> 00:01:50.286
because we have a radical expression here,

00:01:50.286 --> 00:01:53.745
is to see if I can take a conjugate.

00:01:53.745 --> 00:01:56.790
So it's going to likely be,

00:01:56.790 --> 00:01:59.110
we're definitely gonna have C first.

00:01:59.110 --> 00:02:01.099
Then we're going to try E.

00:02:01.099 --> 00:02:02.971
And frankly that's the
only choice that has that,

00:02:02.971 --> 00:02:07.732
but let's actually see if
that actually works, CEB.

00:02:07.732 --> 00:02:12.000
So let me multiply this
expression times a conjugate.

00:02:12.000 --> 00:02:15.250
So the square root of two x minus five,

00:02:15.250 --> 00:02:18.622
minus one over x minus three

00:02:18.622 --> 00:02:22.648
times the square root of two x minus five

00:02:22.648 --> 00:02:25.471
plus one, it's the conjugate here,

00:02:25.471 --> 00:02:27.554
over square root of two x

00:02:28.987 --> 00:02:31.228
minus five plus one.

00:02:31.228 --> 00:02:34.197
And this is going to be
equal to, see this numerator.

00:02:34.197 --> 00:02:36.962
The square root of two
x minus five-squared

00:02:36.962 --> 00:02:39.806
is going to be two x minus five.

00:02:39.806 --> 00:02:42.725
And then you're going to have negative,

00:02:42.725 --> 00:02:44.719
you have a minus one-squared

00:02:44.719 --> 00:02:46.496
which is going to be minus one.

00:02:46.496 --> 00:02:49.746
All of that over x minus three times

00:02:50.602 --> 00:02:53.852
the square root of two x minus five,

00:02:54.701 --> 00:02:55.534
plus one.

00:02:56.896 --> 00:02:59.984
Let's see this stuff up here, this is,

00:02:59.984 --> 00:03:02.830
you can rewrite as two x minus six.

00:03:02.830 --> 00:03:04.739
Two x minus six,

00:03:04.739 --> 00:03:09.480
which is the same thing as
two times x minus three.

00:03:09.480 --> 00:03:12.234
And so those cancel out nicely.

00:03:12.234 --> 00:03:14.007
And so let's see this
is going to be equal to

00:03:14.007 --> 00:03:16.424
two over square root of two x

00:03:18.007 --> 00:03:20.409
minus five plus one.

00:03:20.409 --> 00:03:23.589
Now let's try to evaluate
this when x equals three.

00:03:23.589 --> 00:03:27.937
Well if we do that, we're
going to get two over,

00:03:27.937 --> 00:03:31.262
so this is going to be six
minus five, the square root,

00:03:31.262 --> 00:03:33.095
which is one plus one.

00:03:34.679 --> 00:03:37.897
So it's two over two
which is equal to one.

00:03:37.897 --> 00:03:39.644
So then we were able to evaluate it,

00:03:39.644 --> 00:03:41.511
and we actually got a value,

00:03:41.511 --> 00:03:43.604
so we feel pretty good that we're done.

00:03:43.604 --> 00:03:47.648
So we went from finding
indeterminate form,

00:03:47.648 --> 00:03:51.440
taking the conjugate and then
being able to evaluate it.

00:03:51.440 --> 00:03:53.903
Let me just show that right over here.

00:03:53.903 --> 00:03:55.495
When we immediately try to evaluate it,

00:03:55.495 --> 00:03:57.202
we got indeterminate form.

00:03:57.202 --> 00:03:59.025
We recognize that it was a radical,

00:03:59.025 --> 00:04:01.760
that it was a rational
expression with a radical on top.

00:04:01.760 --> 00:04:03.368
And so say, let's
multiply by the conjugate.

00:04:03.368 --> 00:04:05.142
We did that and then after we were able

00:04:05.142 --> 00:04:06.293
to simplify it with the conjugate,

00:04:06.293 --> 00:04:08.783
we tried to evaluate it again,

00:04:08.783 --> 00:04:11.572
and we were able to get a real number.

00:04:11.572 --> 00:04:15.739
So we feel pretty good that
that is going to be the limit.


Strategy in finding limits

WEBVTT
Kind: captions
Language: en

00:00:00.169 --> 00:00:02.272
- [Instructor] Multiple
videos and exercises

00:00:02.272 --> 00:00:04.273
we cover the various techniques

00:00:04.273 --> 00:00:06.370
for finding limits.

00:00:06.370 --> 00:00:08.936
But sometimes, it's helpful
to think about strategies

00:00:08.936 --> 00:00:12.067
for determining which technique to use.

00:00:12.067 --> 00:00:14.500
And that's what we're going
to cover in this video.

00:00:14.500 --> 00:00:16.237
What you see here is a flowchart

00:00:16.237 --> 00:00:19.076
developed by the team at Khan Academy,

00:00:19.076 --> 00:00:21.323
and I'm essentially going to
work through that flowchart.

00:00:21.323 --> 00:00:23.731
It looks a little bit
complicated at first,

00:00:23.731 --> 00:00:24.886
but hopefully it will make sense

00:00:24.886 --> 00:00:26.751
as we talk it through.

00:00:26.751 --> 00:00:28.124
So the goal is, hey,

00:00:28.124 --> 00:00:29.593
we want to find the limit of f of x

00:00:29.593 --> 00:00:31.343
as x approaches a.

00:00:32.992 --> 00:00:35.284
So what this is telling us to do is,

00:00:35.284 --> 00:00:36.312
well the first thing,

00:00:36.312 --> 00:00:40.071
just try to substitute what
happens when x equals a.

00:00:40.071 --> 00:00:41.904
Let's evaluate f of a.

00:00:42.925 --> 00:00:44.317
And this flowchart says,

00:00:44.317 --> 00:00:47.561
if f of a is equal to a real number,

00:00:47.561 --> 00:00:49.758
it's saying we're done.

00:00:49.758 --> 00:00:51.463
But then there's this little caveat here.

00:00:51.463 --> 00:00:52.753
Probably.

00:00:52.753 --> 00:00:55.684
And the reason why is that
the limit is a different thing

00:00:55.684 --> 00:00:57.574
than the value of the function.

00:00:57.574 --> 00:00:59.507
Sometimes they happen to be the same.

00:00:59.507 --> 00:01:02.285
In fact, that's the definition
of a continuous function

00:01:02.285 --> 00:01:04.213
which we talk about in previous videos,

00:01:04.213 --> 00:01:06.614
but sometimes, they aren't the same.

00:01:06.614 --> 00:01:08.613
This will not necessarily be true

00:01:08.613 --> 00:01:10.414
if you're dealing with some function

00:01:10.414 --> 00:01:11.512
that has a

00:01:11.512 --> 00:01:12.393
point

00:01:12.393 --> 00:01:13.657
discontinuity like that

00:01:13.657 --> 00:01:15.369
or a jump discontinuity,

00:01:15.369 --> 00:01:18.830
or a function that looks like this.

00:01:18.830 --> 00:01:20.924
This would not necessarily be the case.

00:01:20.924 --> 00:01:22.482
But if at that point

00:01:22.482 --> 00:01:26.327
you're trying to find the limit towards,

00:01:26.327 --> 00:01:29.318
as you approach this
point right over here,

00:01:29.318 --> 00:01:31.208
the function is continuous,

00:01:31.208 --> 00:01:33.398
it's behaving somewhat normally,

00:01:33.398 --> 00:01:36.150
then this is a good thing to keep in mind.

00:01:36.150 --> 00:01:37.063
You could just say, hey,

00:01:37.063 --> 00:01:38.551
can I just evaluate the function

00:01:38.551 --> 00:01:39.384
at that

00:01:40.390 --> 00:01:41.495
at that

00:01:41.495 --> 00:01:42.625
a over there?

00:01:42.625 --> 00:01:44.074
So in general, if you're dealing with

00:01:44.074 --> 00:01:47.357
pretty plain vanilla
functions like an x squared

00:01:47.357 --> 00:01:50.594
or if you're dealing with
rational expressions like this

00:01:50.594 --> 00:01:52.363
or trigonometric expressions,

00:01:52.363 --> 00:01:54.847
and if you're able to
just evaluate the function

00:01:54.847 --> 00:01:56.794
and it gives you a real number,

00:01:56.794 --> 00:01:58.314
you are probably done.

00:01:58.314 --> 00:02:00.543
If you're dealing with
some type of a function

00:02:00.543 --> 00:02:03.224
that has all sorts of special cases

00:02:03.224 --> 00:02:05.134
and it's piecewise defined

00:02:05.134 --> 00:02:06.730
as we've seen in previous other videos,

00:02:06.730 --> 00:02:08.690
I would be a little bit more skeptical.

00:02:08.690 --> 00:02:12.132
Or if you know visually around that point,

00:02:12.132 --> 00:02:13.441
there's some type of jump

00:02:13.441 --> 00:02:15.006
or some type of discontinuity,

00:02:15.006 --> 00:02:16.653
you've got to be a
little bit more careful.

00:02:16.653 --> 00:02:17.541
But in general,

00:02:17.541 --> 00:02:20.069
this is a pretty good rule of thumb.

00:02:20.069 --> 00:02:22.630
If you're dealing with
plain vanilla functions

00:02:22.630 --> 00:02:24.648
that are continuous,

00:02:24.648 --> 00:02:26.761
if you evaluate at x equals a

00:02:26.761 --> 00:02:27.770
and you got a real number,

00:02:27.770 --> 00:02:30.459
that's probably going to be the limit.

00:02:30.459 --> 00:02:33.198
But I always think about
the other scenarios.

00:02:33.198 --> 00:02:34.782
What happens if you evaluate it

00:02:34.782 --> 00:02:38.115
and you get some number divided by zero?

00:02:39.105 --> 00:02:40.190
Well, that case,

00:02:40.190 --> 00:02:44.698
you are probably dealing
with a vertical asymptote.

00:02:44.698 --> 00:02:46.256
And what do we mean by vertical asymptote?

00:02:46.256 --> 00:02:48.114
Well, look at this
example right over here.

00:02:48.114 --> 00:02:50.072
Where we just say the limit

00:02:50.072 --> 00:02:51.752
put that in a darker color.

00:02:51.752 --> 00:02:53.156
So if we're talking about

00:02:53.156 --> 00:02:54.408
the limit

00:02:54.408 --> 00:02:56.822
as x approaches one

00:02:56.822 --> 00:02:58.450
of one over

00:02:58.450 --> 00:03:00.011
x minus one,

00:03:00.011 --> 00:03:02.380
if you just try to
evaluate this expression

00:03:02.380 --> 00:03:03.798
at x equals one,

00:03:03.798 --> 00:03:06.256
you would get one over one minus one

00:03:06.256 --> 00:03:08.855
which is equal to one over zero.

00:03:08.855 --> 00:03:09.688
It says, okay,

00:03:09.688 --> 00:03:10.521
I'm throwing it,

00:03:10.521 --> 00:03:13.363
I'm falling into this
vertical asymptote case.

00:03:13.363 --> 00:03:14.506
And at that point,

00:03:14.506 --> 00:03:18.025
if you wanted to just understand
what was going on there

00:03:18.025 --> 00:03:20.490
or even verify that it's
a vertical asymptote,

00:03:20.490 --> 00:03:22.348
well then you can try out some numbers,

00:03:22.348 --> 00:03:23.976
you can try to plot it,

00:03:23.976 --> 00:03:25.004
you can say, alright,

00:03:25.004 --> 00:03:26.843
I probably have a vertical asymptote here

00:03:26.843 --> 00:03:28.184
at x equals one.

00:03:28.184 --> 00:03:30.464
So that's my vertical asymptote.

00:03:30.464 --> 00:03:31.747
And you can try out some values.

00:03:31.747 --> 00:03:32.580
Well, let's see.

00:03:32.580 --> 00:03:34.372
If x is greater than one,

00:03:34.372 --> 00:03:38.267
the denominator is going to be positive,

00:03:38.267 --> 00:03:39.806
and so, my graph

00:03:39.806 --> 00:03:42.143
and you would get this from
trying out a bunch of values.

00:03:42.143 --> 00:03:43.746
Might look something like this

00:03:43.746 --> 00:03:46.738
and then for values less than negative one

00:03:46.738 --> 00:03:48.609
or less than one I should say,

00:03:48.609 --> 00:03:50.576
you're gonna get negative values

00:03:50.576 --> 00:03:52.089
and so, your graph might look

00:03:52.089 --> 00:03:52.922
like something like that

00:03:52.922 --> 00:03:55.385
until you have this vertical asymptote.

00:03:55.385 --> 00:03:57.173
That's probably what you have.

00:03:57.173 --> 00:03:58.290
Now, there are cases,

00:03:58.290 --> 00:03:59.803
very special cases,

00:03:59.803 --> 00:04:02.568
where you won't necessarily
have the vertical asymptote.

00:04:02.568 --> 00:04:04.925
One example of that
would be something like

00:04:04.925 --> 00:04:05.758
one over x

00:04:07.153 --> 00:04:08.583
minus x.

00:04:08.583 --> 00:04:12.281
This one here is actually
undefined for any x you give it.

00:04:12.281 --> 00:04:13.717
So, it would be very,

00:04:13.717 --> 00:04:15.493
you will not have a vertical asymptote.

00:04:15.493 --> 00:04:17.191
But this is a very special case.

00:04:17.191 --> 00:04:18.309
Most times,

00:04:18.309 --> 00:04:20.952
you do have a vertical asymptote there.

00:04:20.952 --> 00:04:23.449
But let's say we don't fall
into either of those situations.

00:04:23.449 --> 00:04:25.313
What if when we evaluate the function,

00:04:25.313 --> 00:04:27.714
we get zero over zero?

00:04:27.714 --> 00:04:29.177
And here is an example of that.

00:04:29.177 --> 00:04:31.105
Limit is x approaches negative one

00:04:31.105 --> 00:04:33.327
of this rational expression.

00:04:33.327 --> 00:04:34.323
Let's try to evaluate it.

00:04:34.323 --> 00:04:36.456
You get negative one squared which is one

00:04:36.456 --> 00:04:39.119
minus negative one which is plus one

00:04:39.119 --> 00:04:40.070
minus two.

00:04:40.070 --> 00:04:42.720
So you get zero the numerator.

00:04:42.720 --> 00:04:44.981
And the denominator you
have negative one squared

00:04:44.981 --> 00:04:45.843
which is one

00:04:45.843 --> 00:04:47.267
minus two times negative one

00:04:47.267 --> 00:04:48.633
so plus two

00:04:48.633 --> 00:04:52.215
minus three which is equal to zero.

00:04:52.215 --> 00:04:55.338
Now this is known as indeterminate form.

00:04:55.338 --> 00:04:56.424
And so on our flowchart,

00:04:56.424 --> 00:04:58.773
we then continue to the right side of it

00:04:58.773 --> 00:05:00.287
and so here's a bunch of techniques

00:05:00.287 --> 00:05:04.527
for trying to tackle something
in indeterminate form.

00:05:04.527 --> 00:05:05.504
And

00:05:05.504 --> 00:05:06.940
likely in a few weeks

00:05:06.940 --> 00:05:08.594
you will learn another technique

00:05:08.594 --> 00:05:10.114
that involves a little more calculus

00:05:10.114 --> 00:05:12.055
called L'hospital's Rule
that we don't tackle here

00:05:12.055 --> 00:05:13.371
because that involves calculus

00:05:13.371 --> 00:05:15.484
while all of these techniques can be done

00:05:15.484 --> 00:05:16.768
with things before calculus.

00:05:16.768 --> 00:05:17.917
Some algebraic techniques

00:05:17.917 --> 00:05:20.024
and some trigonometric techniques.

00:05:20.024 --> 00:05:21.448
So the first thing that you might want to

00:05:21.448 --> 00:05:22.521
try to do

00:05:22.521 --> 00:05:25.356
especially if you're dealing
with a rational expression

00:05:25.356 --> 00:05:26.189
like this

00:05:26.189 --> 00:05:27.444
and you're getting indeterminate form,

00:05:27.444 --> 00:05:29.781
is try to factor it.

00:05:29.781 --> 00:05:30.829
Try to see

00:05:30.829 --> 00:05:33.006
if you can simplify this expression.

00:05:33.006 --> 00:05:34.066
And this expression here,

00:05:34.066 --> 00:05:35.247
you can factor it.

00:05:35.247 --> 00:05:37.463
This is the same thing as

00:05:37.463 --> 00:05:38.296
x

00:05:38.296 --> 00:05:39.213
x minus two

00:05:41.550 --> 00:05:43.133
times x plus one

00:05:44.340 --> 00:05:45.173
over

00:05:46.135 --> 00:05:47.309
x

00:05:47.309 --> 00:05:50.611
this would be x minus three

00:05:50.611 --> 00:05:52.239
times x plus one

00:05:52.239 --> 00:05:55.317
if what I just did seems
completely foreign to you

00:05:55.317 --> 00:05:57.941
I encourage you to watch
the videos on factoring

00:05:57.941 --> 00:05:59.959
factoring polynomials

00:05:59.959 --> 00:06:01.894
or factoring quadratics.

00:06:01.894 --> 00:06:02.935
And so,

00:06:02.935 --> 00:06:04.014
you can see here, alright.

00:06:04.014 --> 00:06:05.097
If I make the

00:06:06.070 --> 00:06:07.577
I can simplify this 'cause

00:06:07.577 --> 00:06:10.623
as long as x does not equal negative one,

00:06:10.623 --> 00:06:12.063
these two things are going to cancel out.

00:06:12.063 --> 00:06:15.499
So I can say that this is going
to be equal to x minus two

00:06:15.499 --> 00:06:17.721
over x minus three

00:06:17.721 --> 00:06:20.275
for x does not equal negative one.

00:06:20.275 --> 00:06:22.670
Sometimes people forget to do this part.

00:06:22.670 --> 00:06:24.898
This is if you're really
being mathematically precise.

00:06:24.898 --> 00:06:27.472
This entire expression
is the same as this one.

00:06:27.472 --> 00:06:29.630
Because this entire expression
is still not defined

00:06:29.630 --> 00:06:31.444
if x equals negative one.

00:06:31.444 --> 00:06:34.601
Although you can substitute
x equals negative one here

00:06:34.601 --> 00:06:36.140
and now get a value.

00:06:36.140 --> 00:06:38.024
So if you substitute x
equals negative one here

00:06:38.024 --> 00:06:40.891
even if it's formally taking it away to be

00:06:40.891 --> 00:06:42.347
mathematically equivalent,

00:06:42.347 --> 00:06:43.733
this would be negative one

00:06:43.733 --> 00:06:45.386
minus two

00:06:45.386 --> 00:06:46.976
which would be

00:06:46.976 --> 00:06:48.643
which would be negative three

00:06:48.643 --> 00:06:50.667
over negative one minus three

00:06:50.667 --> 00:06:51.869
which should be negative four

00:06:51.869 --> 00:06:53.829
which is equal to three fourths.

00:06:53.829 --> 00:06:56.581
So if this condition wasn't here,

00:06:56.581 --> 00:06:58.618
you can just evaluate it straight up and

00:06:58.618 --> 00:07:00.470
this is a pretty plain vanilla function.

00:07:00.470 --> 00:07:05.119
Wouldn't expect to see
anything crazy happening here.

00:07:05.119 --> 00:07:07.845
And if I can just evaluate
it at x equals negative one

00:07:07.845 --> 00:07:09.493
I feel pretty good.

00:07:09.493 --> 00:07:10.859
I feel pretty good.

00:07:10.859 --> 00:07:12.354
So once again, we're
now going in factoring.

00:07:12.354 --> 00:07:13.187
We're able to factor.

00:07:13.187 --> 00:07:14.371
We have valued,

00:07:14.371 --> 00:07:15.482
we simplify it.

00:07:15.482 --> 00:07:17.649
We evaluate the expression

00:07:18.643 --> 00:07:19.971
the simplified expression now,

00:07:19.971 --> 00:07:21.498
and now we were able to get a value.

00:07:21.498 --> 00:07:23.005
We were able to get three fourths,

00:07:23.005 --> 00:07:25.610
and so we can feel pretty
good that the limit here

00:07:25.610 --> 00:07:28.688
in this situation is three fourths.

00:07:28.688 --> 00:07:29.521
Now, let's

00:07:29.521 --> 00:07:31.223
and I would categorize what we've seen

00:07:31.223 --> 00:07:32.762
so far is

00:07:32.762 --> 00:07:34.984
the bulk of the limit exercises

00:07:34.984 --> 00:07:36.389
that you will likely encounter.

00:07:36.389 --> 00:07:37.238
Now the next two,

00:07:37.238 --> 00:07:40.916
I would call slightly fancier techniques.

00:07:40.916 --> 00:07:42.934
So if you get indeterminate form

00:07:42.934 --> 00:07:45.354
especially you'll sometimes
see it with radical expressions

00:07:45.354 --> 00:07:46.187
like this.

00:07:46.187 --> 00:07:48.355
Rational radical expressions.

00:07:48.355 --> 00:07:51.158
You might want to multiply by conjugate.

00:07:51.158 --> 00:07:53.202
So for example, in this
situation right here,

00:07:53.202 --> 00:07:56.114
if you just try to
evaluate it x equals four,

00:07:56.114 --> 00:07:58.854
you get the square root of four minus two

00:07:58.854 --> 00:08:00.521
over four minus four

00:08:01.395 --> 00:08:03.145
which is zero over zero.

00:08:03.145 --> 00:08:04.946
So it's that indeterminate form.

00:08:04.946 --> 00:08:07.308
And the technique here,
because we're seeing

00:08:07.308 --> 00:08:09.907
this radical and a rational expression

00:08:09.907 --> 00:08:12.602
let's say, maybe we can
somehow get rid of that radical

00:08:12.602 --> 00:08:14.900
or simplify it somehow.

00:08:14.900 --> 00:08:16.496
So let me rewrite.

00:08:16.496 --> 00:08:18.476
Square root of x minus two

00:08:18.476 --> 00:08:19.919
over

00:08:19.919 --> 00:08:20.966
x

00:08:20.966 --> 00:08:22.671
minus four.

00:08:22.671 --> 00:08:23.629
Let's say a conjugate,

00:08:23.629 --> 00:08:26.885
let's multiply it by the
square root of x plus two

00:08:26.885 --> 00:08:30.723
over the square root of x plus two.

00:08:30.723 --> 00:08:31.556
Once again,

00:08:31.556 --> 00:08:33.373
it's the same expression
over the same expression.

00:08:33.373 --> 00:08:36.604
So I'm not fundamentally
changing its value.

00:08:36.604 --> 00:08:38.641
And so this is going to be equal to,

00:08:38.641 --> 00:08:41.655
well if I have a plus b times a minus b

00:08:41.655 --> 00:08:43.047
I'm gonna get a difference of squares.

00:08:43.047 --> 00:08:45.569
So it's gonna be square root of x squared

00:08:45.569 --> 00:08:46.902
which is,

00:08:46.902 --> 00:08:49.505
it's going to be square root of x squared

00:08:49.505 --> 00:08:50.338
minus four

00:08:51.759 --> 00:08:52.592
over

00:08:52.592 --> 00:08:54.108
well square root of x squared

00:08:54.108 --> 00:08:56.184
is just going to be

00:08:56.184 --> 00:08:57.267
x minus four.

00:08:58.112 --> 00:09:00.845
So let me rewrite it that way.

00:09:00.845 --> 00:09:02.531
So that's x minus four

00:09:02.531 --> 00:09:03.533
over

00:09:03.533 --> 00:09:04.533
x minus four

00:09:05.372 --> 00:09:07.735
times square root of x plus two,

00:09:07.735 --> 00:09:10.775
square root of x plus two.

00:09:10.775 --> 00:09:12.218
Well, this was useful because now

00:09:12.218 --> 00:09:14.968
I can cancel out x equals four

00:09:15.870 --> 00:09:18.552
or x minus four right over here.

00:09:18.552 --> 00:09:19.385
And once again, if I wanted it

00:09:19.385 --> 00:09:21.949
mathematically to be the
exact same expression,

00:09:21.949 --> 00:09:24.531
I'd say well, now this
is going to be equal to

00:09:24.531 --> 00:09:25.738
one

00:09:25.738 --> 00:09:27.807
one over the square root of x plus two

00:09:27.807 --> 00:09:29.921
for x does not equal four,

00:09:29.921 --> 00:09:31.626
but we can definitely see what

00:09:31.626 --> 00:09:33.605
this function is approaching

00:09:33.605 --> 00:09:35.961
if we just now substitute x equals four

00:09:35.961 --> 00:09:38.420
into this simplified expression.

00:09:38.420 --> 00:09:41.581
And so, that's just gonna be one over

00:09:41.581 --> 00:09:43.784
so if we just substitute

00:09:43.784 --> 00:09:46.523
if we just substitute x equals four here

00:09:46.523 --> 00:09:49.135
you'd get one over square root of four

00:09:49.135 --> 00:09:50.246
plus two

00:09:50.246 --> 00:09:52.398
which is equal to one fourth.

00:09:52.398 --> 00:09:54.601
And once again, you can feel pretty good

00:09:54.601 --> 00:09:56.586
that this is going to be

00:09:56.586 --> 00:09:57.419
your limit.

00:09:57.419 --> 00:09:59.395
We've gone back into the green zone.

00:09:59.395 --> 00:10:00.673
If you're actually to plot

00:10:00.673 --> 00:10:02.039
this original function,

00:10:02.039 --> 00:10:04.855
you would have a point discontinuity.

00:10:04.855 --> 00:10:07.058
You would have a gap at x equals

00:10:07.058 --> 00:10:07.891
four

00:10:08.770 --> 00:10:10.768
but then when you do that simplification

00:10:10.768 --> 00:10:12.499
and factoring out that x minus

00:10:12.499 --> 00:10:14.408
or canceling out that x minus four,

00:10:14.408 --> 00:10:15.877
that gap would disappear.

00:10:15.877 --> 00:10:17.071
So that's essentially what you're doing.

00:10:17.071 --> 00:10:19.069
You're trying to find
the limit as we approach

00:10:19.069 --> 00:10:21.260
that gap which we got right there.

00:10:21.260 --> 00:10:23.111
Now, this final one.

00:10:23.111 --> 00:10:24.638
This is dealing with trig identities.

00:10:24.638 --> 00:10:26.215
And in order to do these,

00:10:26.215 --> 00:10:29.433
you have to be pretty adept
at your trig identities.

00:10:29.433 --> 00:10:32.115
So if we're saying the limit,

00:10:32.115 --> 00:10:34.497
I'll do that at a darker color.

00:10:34.497 --> 00:10:36.240
So if we're saying the limit

00:10:36.240 --> 00:10:38.877
as x approaches zero

00:10:38.877 --> 00:10:39.877
of sine of x

00:10:40.889 --> 00:10:41.722
over sine

00:10:42.862 --> 00:10:44.465
of two x

00:10:44.465 --> 00:10:46.227
well, sine of zero zero,

00:10:46.227 --> 00:10:47.060
sine of zero zero,

00:10:47.060 --> 00:10:47.900
you're gonna be at zero, zero.

00:10:47.900 --> 00:10:49.100
Once you get indeterminate form,

00:10:49.100 --> 00:10:50.789
we fall into this category,

00:10:50.789 --> 00:10:53.142
and now you might recognize
this is going to be

00:10:53.142 --> 00:10:54.981
equal to the limit

00:10:54.981 --> 00:10:58.437
as x approaches zero of sine of x

00:10:58.437 --> 00:11:00.161
we can rewrite sine of two x

00:11:00.161 --> 00:11:01.078
as two sine

00:11:02.204 --> 00:11:03.037
x cosine x

00:11:04.114 --> 00:11:07.780
and then those two can
cancel out for all x's not

00:11:07.780 --> 00:11:08.700
equaling

00:11:08.700 --> 00:11:11.318
for all x's not equaling zero

00:11:11.318 --> 00:11:13.326
if you want to be really
mathematically precise.

00:11:13.326 --> 00:11:16.388
And so, there would've
been a gap there for sure

00:11:16.388 --> 00:11:18.080
on the original graph

00:11:18.080 --> 00:11:20.193
if you were to graph y equals this.

00:11:20.193 --> 00:11:21.538
But now, for the limit purposes,

00:11:21.538 --> 00:11:23.748
you can say this is this limit

00:11:23.748 --> 00:11:24.884
is

00:11:24.884 --> 00:11:27.081
this limit is going to be the limit

00:11:27.081 --> 00:11:29.724
as x approaches zero

00:11:29.724 --> 00:11:32.057
of one over two cosine of x.

00:11:33.058 --> 00:11:35.141
And now we can go back to

00:11:36.397 --> 00:11:38.562
this green condition right over here,

00:11:38.562 --> 00:11:40.478
because we can evaluate
this at x equals zero.

00:11:40.478 --> 00:11:42.693
It's going to be one over
two times cosine of zero.

00:11:42.693 --> 00:11:44.437
Cosine of zero is one.

00:11:44.437 --> 00:11:47.987
So this is going to be equal to one half.

00:11:47.987 --> 00:11:50.637
Now in general, none of
these techniques work,

00:11:50.637 --> 00:11:52.508
and you will encounter
few other techniques

00:11:52.508 --> 00:11:54.711
further on once you learn more calculus,

00:11:54.711 --> 00:11:57.955
then you fall on the base line.

00:11:57.955 --> 00:11:59.276
Approximation.

00:11:59.276 --> 00:12:02.099
And approximation, you
can do it numerically.

00:12:02.099 --> 00:12:04.749
Try values really really really close

00:12:04.749 --> 00:12:07.297
to the number you're trying
to find the limit on.

00:12:07.297 --> 00:12:09.634
If you're trying to find the
limit as x approaches zero

00:12:09.634 --> 00:12:11.134
try 0.00000000001.

00:12:13.331 --> 00:12:15.164
Try negative 0.0000001

00:12:16.760 --> 00:12:18.593
if you're trying to find the
limit is x approaches four

00:12:18.593 --> 00:12:19.760
try 4.0000001.

00:12:21.019 --> 00:12:22.352
Try 3.9999999999

00:12:24.193 --> 00:12:25.878
and see what happens.

00:12:25.878 --> 00:12:27.781
But that's kind of the last ditch.

00:12:27.781 --> 00:12:29.614
The last ditch effort.


Sal Khan chats with Google CEO Sundar Pichai

WEBVTT
Kind: captions
Language: en

00:00:06.251 --> 00:00:08.258
- It's a huge treat to have Sundar Pichai,

00:00:08.258 --> 00:00:09.913
CEO of Google here

00:00:09.913 --> 00:00:12.060
and you know, I will give a little bit

00:00:12.060 --> 00:00:13.735
of a preamble more than I normally do

00:00:13.735 --> 00:00:15.652
and I think a lot of the team knows this,

00:00:15.652 --> 00:00:17.504
but it's always worth reminding the team,

00:00:17.504 --> 00:00:19.397
we wouldn't be here on many levels,

00:00:19.397 --> 00:00:22.001
if it wasn't for Google.

00:00:22.001 --> 00:00:25.383
Clearly, Khan Academy certainly got

00:00:25.383 --> 00:00:27.579
it's first attention on YouTube.

00:00:27.579 --> 00:00:29.081
I don't think I would have been able to do

00:00:29.081 --> 00:00:32.045
what I did had YouTube
not existed when it did.

00:00:32.045 --> 00:00:34.410
Some of you all might
not realize, but in 2005

00:00:34.410 --> 00:00:37.200
we were on Java Servlet Webhosting

00:00:37.200 --> 00:00:38.399
and that wasn't scaling.

00:00:38.399 --> 00:00:39.232
(audience laughs)

00:00:39.232 --> 00:00:40.992
And so in January, I was like

00:00:40.992 --> 00:00:42.655
hey there's this thing called App Engine

00:00:42.655 --> 00:00:43.733
and we went on there

00:00:43.733 --> 00:00:45.985
and we have been ever since.

00:00:45.985 --> 00:00:47.770
So even from a platform point of view,

00:00:47.770 --> 00:00:50.695
but most importantly, in the early days,

00:00:50.695 --> 00:00:53.846
when, you know, we were
kind of not even clear

00:00:53.846 --> 00:00:55.210
that we could become an organization.

00:00:55.210 --> 00:00:59.083
Google and Google.org was one
of the very first to step up

00:00:59.083 --> 00:01:01.018
and allow us to become
a real organization.

00:01:01.018 --> 00:01:02.826
And ever since then, through Learn Storm

00:01:02.826 --> 00:01:05.445
and even existing grants,
continue to support this.

00:01:05.445 --> 00:01:06.678
So first of all, just thank you.

00:01:06.678 --> 00:01:08.577
- Oh, well there's are a
lot of people are behind it.

00:01:08.577 --> 00:01:11.345
I am glad we are part
of something this great.

00:01:11.345 --> 00:01:12.178
- Thank you.

00:01:12.178 --> 00:01:13.377
So let's just start.

00:01:13.377 --> 00:01:14.294
You know...

00:01:15.708 --> 00:01:18.705
Well you're not new to the
job as CEO of Google anymore,

00:01:18.705 --> 00:01:20.331
it's been almost two years now.

00:01:20.331 --> 00:01:21.164
A year and a half?

00:01:21.164 --> 00:01:22.385
- Haven't counted, by something like that.

00:01:22.385 --> 00:01:25.282
- So what's your view of the you know,

00:01:25.282 --> 00:01:28.455
of the world, what you're
hoping to do at Google?

00:01:28.455 --> 00:01:31.423
- First of all, thanks for having me here.

00:01:31.423 --> 00:01:32.825
You know, I have to confess,

00:01:32.825 --> 00:01:36.130
there are times when I have
to speak at things like this

00:01:36.130 --> 00:01:38.269
and on topics which I
don't know much about.

00:01:38.269 --> 00:01:39.451
I may sneak into Khan Academy

00:01:39.451 --> 00:01:40.972
and see a video here and there,

00:01:40.972 --> 00:01:44.622
so I'm a user of the platform as well.

00:01:44.622 --> 00:01:47.530
You know, from a Google
standpoint, in some ways

00:01:47.530 --> 00:01:50.530
I think we share a lot of, you know,

00:01:51.416 --> 00:01:53.317
what your mission is as well.

00:01:53.317 --> 00:01:56.124
Our stated mission is to bring information

00:01:56.124 --> 00:01:59.683
and knowledge, you know, for everyone.

00:01:59.683 --> 00:02:02.114
We take the for everyone part seriously.

00:02:02.114 --> 00:02:06.281
And, you know, we are, we've
been working on it for a while,

00:02:07.163 --> 00:02:11.002
but we again have this
renewed sense of excitement

00:02:11.002 --> 00:02:14.072
within the company because
of machine learning and AI.

00:02:14.072 --> 00:02:16.770
We think over the next 10
years, we can, you know,

00:02:16.770 --> 00:02:21.158
take a bunch of stuff we've
been doing to a different level

00:02:21.158 --> 00:02:23.741
and we are still in early days.

00:02:24.657 --> 00:02:27.200
You know computers are
just beginning to kinda

00:02:27.200 --> 00:02:29.359
get better at things
like speech recognition,

00:02:29.359 --> 00:02:31.228
image recognition, that's what a lot of

00:02:31.228 --> 00:02:33.278
the excitement is about, when we say

00:02:33.278 --> 00:02:35.639
you know, we are making a lot

00:02:35.639 --> 00:02:37.909
of progress in machine learning.

00:02:37.909 --> 00:02:41.946
So using that thoughtfully
across everything we do

00:02:41.946 --> 00:02:44.001
and then taking things to the next level

00:02:44.001 --> 00:02:46.720
is a big part of our focus.

00:02:46.720 --> 00:02:49.373
Closely related to that is,

00:02:49.373 --> 00:02:52.572
you know, building computing
platforms for everyone too.

00:02:52.572 --> 00:02:55.723
I think, you know I was
personally very inspired

00:02:55.723 --> 00:02:57.627
by the OLPC project at the time.

00:02:57.627 --> 00:03:01.279
And I think that notion
of being able to make

00:03:01.279 --> 00:03:04.895
computing accessible to
everyone is very near and dear

00:03:04.895 --> 00:03:07.187
to a lot of us, particularly to me.

00:03:07.187 --> 00:03:09.302
You know, I didn't have
access to computing growing up

00:03:09.302 --> 00:03:12.808
and you know, I literally, mainly got it

00:03:12.808 --> 00:03:14.515
through the extent of my college

00:03:14.515 --> 00:03:17.682
and so I have a very discreet sense of

00:03:20.485 --> 00:03:23.533
how it changed what I
was able to understand

00:03:23.533 --> 00:03:25.239
and what I was able to get out of it.

00:03:25.239 --> 00:03:29.307
So I think we're in the spot
where it's very exciting to see

00:03:29.307 --> 00:03:32.456
computing, which you know,
historically worked for

00:03:32.456 --> 00:03:34.162
a small percentage of the world.

00:03:34.162 --> 00:03:37.413
Now is, that penetration is
getting better and better.

00:03:37.413 --> 00:03:40.582
You know, mobile use is
at three billion users now

00:03:40.582 --> 00:03:44.249
and I see signs we can
scale it beyond that,

00:03:45.730 --> 00:03:50.214
so we are big with platforms
like Chrome, Android.

00:03:50.214 --> 00:03:53.100
We've always cared about making
computing more affordable

00:03:53.100 --> 00:03:55.557
and reaching everyone.

00:03:55.557 --> 00:03:58.573
I think there are others
working on connectivity as well.

00:03:58.573 --> 00:04:00.089
So the combination of all that,

00:04:00.089 --> 00:04:02.418
I think we can get to a stage where,

00:04:02.418 --> 00:04:04.909
pretty much, sometime
within the next 10 years,

00:04:04.909 --> 00:04:06.845
you know for most people in the world,

00:04:06.845 --> 00:04:08.998
I think you'll be able
to give them something

00:04:08.998 --> 00:04:10.681
which with connectivity, you know

00:04:10.681 --> 00:04:13.952
you have that access of
everything in your hands.

00:04:13.952 --> 00:04:16.959
And so, I think that's a
pretty profound transformation.

00:04:16.959 --> 00:04:20.500
And so, you know, broadly that excites us.

00:04:20.500 --> 00:04:22.288
And it serves our mission well

00:04:22.288 --> 00:04:25.443
and we will do well as
a company too, meanwhile

00:04:25.443 --> 00:04:27.327
as we go through this.

00:04:27.327 --> 00:04:29.788
So a lot of it is, you
know, actually viewing

00:04:29.788 --> 00:04:32.455
these advances in machine and AI

00:04:33.363 --> 00:04:37.126
as an opportunity to bring it
in an equal way to everyone.

00:04:37.126 --> 00:04:41.464
And I think that's
largely what excites me.

00:04:41.464 --> 00:04:42.812
- And yeah, it's incredible.

00:04:42.812 --> 00:04:45.195
I mean, Google's one of the
few organizations in the world

00:04:45.195 --> 00:04:46.599
that are in the position to do this.

00:04:46.599 --> 00:04:48.359
But on top of kind of
all this, I would say,

00:04:48.359 --> 00:04:50.728
deep technical work
that you all are doing,

00:04:50.728 --> 00:04:55.101
education I've heard is a
big interest area for you.

00:04:55.101 --> 00:04:57.534
Google has several
initiatives in education,

00:04:57.534 --> 00:04:59.513
some of which we've been interfacing with.

00:04:59.513 --> 00:05:02.170
What's your view of Google's role in

00:05:02.170 --> 00:05:05.664
education, either globally or in the U.S.?

00:05:05.664 --> 00:05:08.359
- I think culturally we have
always identified, you know,

00:05:08.359 --> 00:05:11.470
pretty deeply with Khan Academy,
with education in general.

00:05:11.470 --> 00:05:15.918
And so we see this as a
big area, where, you know

00:05:15.918 --> 00:05:19.932
I think, you know I said our
mission kind of, you know,

00:05:19.932 --> 00:05:23.332
is complimentary to it
and so we wanna play a

00:05:23.332 --> 00:05:26.188
role to the extent we can.

00:05:26.188 --> 00:05:29.068
We've been fortunate in
the sense that, you know,

00:05:29.068 --> 00:05:30.978
our products have gotten very popular

00:05:30.978 --> 00:05:34.682
in educational institutions
all the way from K through 12,

00:05:34.682 --> 00:05:36.756
largely with google appls first

00:05:36.756 --> 00:05:38.870
and you know we have done
other things like classroom

00:05:38.870 --> 00:05:40.740
and Chrome Books and so on.

00:05:40.740 --> 00:05:45.057
We'll also do, you know,
not for profit initiatives

00:05:45.057 --> 00:05:48.974
in education as well, so
we are investing more.

00:05:50.023 --> 00:05:53.533
I think when I think
about, you know, I think

00:05:53.533 --> 00:05:55.878
education is obviously,

00:05:55.878 --> 00:05:59.437
most of you here understand
it better than me, but its

00:05:59.437 --> 00:06:02.758
complex and you know we
just think technology

00:06:02.758 --> 00:06:05.018
can play an enabling part.

00:06:05.018 --> 00:06:07.153
You know, I was talking
with Sal about this a bit,

00:06:07.153 --> 00:06:10.822
but when I look at
approaching many problems

00:06:10.822 --> 00:06:13.793
in the context of Google,
we try to do things,

00:06:13.793 --> 00:06:18.043
we try to iterate, we see
what works, what doesn't work

00:06:18.043 --> 00:06:21.243
and then we do these things that way.

00:06:21.243 --> 00:06:24.637
That concept is hard to do
in education, rightfully so,

00:06:24.637 --> 00:06:26.823
because you know, you don't
wanna get things wrong.

00:06:26.823 --> 00:06:28.255
But I think it's important to

00:06:28.255 --> 00:06:31.155
figure out how to innovate more

00:06:31.155 --> 00:06:34.004
and in that context, I think
technology can play a part.

00:06:34.004 --> 00:06:38.491
And so play a part in
hopefully helping people,

00:06:38.491 --> 00:06:41.603
figure out what, you
know, many things to try.

00:06:41.603 --> 00:06:45.770
When things work, be able to
do more of it, scale it more.

00:06:46.928 --> 00:06:50.687
You know, how do you bring
educators and children

00:06:50.687 --> 00:06:52.350
all in a virtual cycle together

00:06:52.350 --> 00:06:54.172
which is what you all are working on.

00:06:54.172 --> 00:06:56.386
So for me, when I look at it,

00:06:56.386 --> 00:06:58.046
it looks like a complex ecosystem.

00:06:58.046 --> 00:07:01.259
So we wanna see where we
can enable these things.

00:07:01.259 --> 00:07:04.864
So from a standpoint of
providing technology.

00:07:04.864 --> 00:07:06.915
But a lot of people are
interested in Google,

00:07:06.915 --> 00:07:08.754
beyond just the products we build

00:07:08.754 --> 00:07:10.333
or the technology we provide.

00:07:10.333 --> 00:07:12.049
You know Google is so passionate,

00:07:12.049 --> 00:07:13.754
they want to volunteer a lot in education.

00:07:13.754 --> 00:07:17.016
For Google.org it's one
of the important pillars

00:07:17.016 --> 00:07:18.838
on which we invest.

00:07:18.838 --> 00:07:21.416
So I think that's what makes it

00:07:21.416 --> 00:07:22.861
a bigger thing for us.

00:07:22.861 --> 00:07:25.385
- Yeah and you know here at Khan Academy,

00:07:25.385 --> 00:07:27.047
you know you dabble in it a little bit,

00:07:27.047 --> 00:07:28.531
you were just telling
me your daughter uses it

00:07:28.531 --> 00:07:30.557
through her school sometimes.
- Yeah.

00:07:30.557 --> 00:07:33.597
- Where do you see ways
that we can work together?

00:07:33.597 --> 00:07:36.316
How we can support each other's goals?

00:07:36.316 --> 00:07:38.429
- It's been interesting
to see, which you know,

00:07:38.429 --> 00:07:39.898
from the first time I saw Khan Academy

00:07:39.898 --> 00:07:42.214
to actually, you know, my daughter,

00:07:42.214 --> 00:07:45.046
you know I see her sometimes
come when she has homework,

00:07:45.046 --> 00:07:47.450
but it's part of like,
doing it on Khan Academy.

00:07:47.450 --> 00:07:49.302
I would once in a while lean over

00:07:49.302 --> 00:07:51.142
and make sure she was not
just using YouTube generally.

00:07:51.142 --> 00:07:52.535
(crowd laughs)

00:07:52.535 --> 00:07:53.822
I would say that's a small thing.

00:07:53.822 --> 00:07:56.134
It's pretty amazing to see that you know,

00:07:56.134 --> 00:07:57.805
teachers are using it

00:07:57.805 --> 00:08:00.639
and it's been great to see.

00:08:00.639 --> 00:08:02.690
- So you too are upset
when you're children

00:08:02.690 --> 00:08:03.816
are hooked on YouTube.

00:08:03.816 --> 00:08:05.809
- Sometimes.
(crowd laughs)

00:08:05.809 --> 00:08:07.535
- It's a sign that something's working.

00:08:07.535 --> 00:08:09.080
- Yes
(crowd laughs)

00:08:09.080 --> 00:08:12.224
No it's amazing, the kind of
stuff you learn on YouTube.

00:08:12.224 --> 00:08:13.682
I almost miss it because,

00:08:13.682 --> 00:08:15.614
they don't come to me with any questions.

00:08:15.614 --> 00:08:17.987
So like you know they are,
they kind of figure out,

00:08:17.987 --> 00:08:20.223
they get their answers
from YouTube and leave.

00:08:20.223 --> 00:08:21.901
And later I'm like, oh you
could have asked me, you know.

00:08:21.901 --> 00:08:22.850
(crowd laughs)

00:08:22.850 --> 00:08:25.807
Maybe I would have known the answer.

00:08:25.807 --> 00:08:28.619
You know, in terms of us working together,

00:08:28.619 --> 00:08:30.389
you know I do mean it when I say

00:08:30.389 --> 00:08:32.375
we think of our OLS enablers,

00:08:32.375 --> 00:08:35.347
you know, we part has
been in many ecosystems

00:08:35.347 --> 00:08:38.354
and you know, in that
way that we approach it,

00:08:38.354 --> 00:08:41.349
but I do think that our
areas where we care about,

00:08:41.349 --> 00:08:44.455
you know when I mention
the for everyone part,

00:08:44.455 --> 00:08:46.839
I think it's important, we think,

00:08:46.839 --> 00:08:48.768
one of the things that
attracted me to Google

00:08:48.768 --> 00:08:52.489
and that Google Search
worked globally, right?

00:08:52.489 --> 00:08:55.324
And you know it's kind of
like, if you had access

00:08:55.324 --> 00:08:57.707
to computing and connectivity,
and that's a big if,

00:08:57.707 --> 00:09:00.272
but if you had access to
computing and connectivity,

00:09:00.272 --> 00:09:01.957
it actually really didn't matter

00:09:01.957 --> 00:09:03.710
where you were accessing Google from.

00:09:03.710 --> 00:09:06.698
You kinda got the same thing.

00:09:06.698 --> 00:09:10.308
That equality of access
I think is important.

00:09:10.308 --> 00:09:12.695
So hopefully, you know, I'm interested in

00:09:12.695 --> 00:09:15.958
you know, our last round of efforts,

00:09:15.958 --> 00:09:18.557
you know we were excited
about your efforts

00:09:18.557 --> 00:09:20.943
to localize what you're
doing more broadly.

00:09:20.943 --> 00:09:22.929
I think that's important to us.

00:09:22.929 --> 00:09:25.598
So getting these things to
work, even in places like

00:09:25.598 --> 00:09:29.244
Brazil, Indonesia, India and
like how do you scale it up

00:09:29.244 --> 00:09:31.316
I think is something we deeply care about

00:09:31.316 --> 00:09:33.002
and feel actually we
can play a part in that.

00:09:33.002 --> 00:09:34.419
We wanna do that.

00:09:35.737 --> 00:09:38.006
You know, the other part
is, you know I think

00:09:38.006 --> 00:09:41.117
given we are working on things
like not just our products,

00:09:41.117 --> 00:09:44.018
classroom, you know,
connection via feedback

00:09:44.018 --> 00:09:46.353
on how we can evolve, what we do better.

00:09:46.353 --> 00:09:50.066
You know, better to make all
this work more effectively

00:09:50.066 --> 00:09:52.456
for teachers and students and for you all.

00:09:52.456 --> 00:09:54.500
We are openly trading
our products that way.

00:09:54.500 --> 00:09:56.465
You were mentioning that you know you guys

00:09:56.465 --> 00:09:58.244
are plugging into classroom and stuff.

00:09:58.244 --> 00:10:00.409
That's, you know,
encouraging for me to hear

00:10:00.409 --> 00:10:03.149
and we're also in early days of setting up

00:10:03.149 --> 00:10:05.912
teams internally at
Google to do other things,

00:10:05.912 --> 00:10:09.336
which have nothing to do with
our products on education.

00:10:09.336 --> 00:10:11.183
So hopefully we can align better

00:10:11.183 --> 00:10:12.959
and make some of them more complimentary.

00:10:12.959 --> 00:10:15.576
- Yeah and just a, you know,

00:10:15.576 --> 00:10:17.192
unfortunately we could talk for hours,

00:10:17.192 --> 00:10:20.431
but just kind of a, you
know, an ending question,

00:10:20.431 --> 00:10:23.534
what advice, you know,
you've seen Google grow

00:10:23.534 --> 00:10:25.608
from a relatively small stage

00:10:25.608 --> 00:10:28.805
to now it's this thing that, you know

00:10:28.805 --> 00:10:30.955
actually delivering on it's mission,

00:10:30.955 --> 00:10:33.484
and we have a huge
mission a free world class

00:10:33.484 --> 00:10:35.099
education for anyone, anywhere
and we'd like to think

00:10:35.099 --> 00:10:36.893
we're already making a dent in that,

00:10:36.893 --> 00:10:39.167
but we're serious about delivering.

00:10:39.167 --> 00:10:40.515
What advice do you have to us,

00:10:40.515 --> 00:10:42.160
you know as a relatively small team,

00:10:42.160 --> 00:10:44.611
as we try to go and try to do that

00:10:44.611 --> 00:10:46.392
over the next five or ten years.

00:10:46.392 --> 00:10:49.204
- You know, you're doing this partly,

00:10:49.204 --> 00:10:51.537
but I wouldn't underestimate

00:10:52.738 --> 00:10:55.187
you know, the change you
do at the tip of the tree,

00:10:55.187 --> 00:10:57.354
how fast it can flow down.

00:11:00.339 --> 00:11:02.929
You know, I think you've already, you know

00:11:02.929 --> 00:11:05.762
the fact that you guys are approaching

00:11:05.762 --> 00:11:09.158
a space like education and have
shown that you can innovate

00:11:09.158 --> 00:11:10.377
in the context of that space,

00:11:10.377 --> 00:11:12.559
I think that thing is big thing.

00:11:12.559 --> 00:11:13.929
You know, we take it for granted

00:11:13.929 --> 00:11:15.240
being in technology companies,

00:11:15.240 --> 00:11:16.762
like any time we face a problem,

00:11:16.762 --> 00:11:19.920
we think we can break it,
we fail by the way a lot,

00:11:19.920 --> 00:11:22.086
but at least our attitude
is that, you know

00:11:22.086 --> 00:11:24.936
you can approach anything,
however complex it is,

00:11:24.936 --> 00:11:26.601
with an engineering mindset.

00:11:26.601 --> 00:11:29.594
You know, you try and understand you know,

00:11:29.594 --> 00:11:31.142
what are the various components of it,

00:11:31.142 --> 00:11:33.032
how you can iterate
and make things better.

00:11:33.032 --> 00:11:36.526
It's kind of the default
assumption a lot of us have.

00:11:36.526 --> 00:11:39.843
But I think it doesn't always
translate to, you know,

00:11:39.843 --> 00:11:42.338
things like, you know,
many, many outside things,

00:11:42.338 --> 00:11:44.048
be it education or health care.

00:11:44.048 --> 00:11:46.321
You know when I deal with
this, you know I realize,

00:11:46.321 --> 00:11:47.601
well we take this for granted

00:11:47.601 --> 00:11:48.886
and like that's not how it works.

00:11:48.886 --> 00:11:50.173
I don't mean it in a negative way.

00:11:50.173 --> 00:11:52.192
There are good reasons
why those systems are

00:11:52.192 --> 00:11:54.473
the way they are and you know,

00:11:54.473 --> 00:11:56.566
the cost of a mistake
is exceptionally high

00:11:56.566 --> 00:11:58.994
in something like health care and so on.

00:11:58.994 --> 00:12:01.827
But I think figuring out how to,

00:12:01.827 --> 00:12:05.519
you know innovate, you know it just like

00:12:05.519 --> 00:12:06.831
I think an imperative

00:12:06.831 --> 00:12:09.498
and I think, hopefully, you know

00:12:11.057 --> 00:12:14.176
you continue pushing what
you guys are doing already

00:12:14.176 --> 00:12:17.427
and I think you've changed
already the conversation

00:12:17.427 --> 00:12:19.754
at least where all your name comes up

00:12:19.754 --> 00:12:22.275
and so you know figuring
out how to do that,

00:12:22.275 --> 00:12:24.324
I think because for
every one thing you do,

00:12:24.324 --> 00:12:27.365
there is a multiplicative
effect of how many other people

00:12:27.365 --> 00:12:29.048
get excited by it.

00:12:29.048 --> 00:12:31.069
So you know hundred
people organization here

00:12:31.069 --> 00:12:33.751
effectively over time,
you're motivating thousands

00:12:33.751 --> 00:12:36.743
and tens of thousands of people
to change their approaches

00:12:36.743 --> 00:12:37.977
and so on.

00:12:37.977 --> 00:12:41.654
I think I've always been, you know,

00:12:41.654 --> 00:12:44.172
the leveraged effect on top of a platform

00:12:44.172 --> 00:12:46.808
is always very counterintuitive to see.

00:12:46.808 --> 00:12:49.078
Like I can give thousands of examples,

00:12:49.078 --> 00:12:51.324
but the fact that everybody
starts using phones

00:12:51.324 --> 00:12:54.772
and phones have GPS and location on them,

00:12:54.772 --> 00:12:56.975
you know, you wouldn't
assume it would change

00:12:56.975 --> 00:12:58.775
transportation one day.

00:12:58.775 --> 00:13:01.204
But you kinda then like see, you know,

00:13:01.204 --> 00:13:03.410
ride sharing take off and things happen.

00:13:03.410 --> 00:13:06.404
Like you know, none of the
people who were working on phones

00:13:06.404 --> 00:13:07.987
ever had any remote

00:13:08.925 --> 00:13:11.265
sense that like people would use phones

00:13:11.265 --> 00:13:14.931
and hence because of that how
you get car changes, right?

00:13:14.931 --> 00:13:17.357
And so anything you do at a platform level

00:13:17.357 --> 00:13:19.800
has these nonlinear effects.

00:13:19.800 --> 00:13:23.828
You know extraordinarily
leveraged ways to change things.

00:13:23.828 --> 00:13:26.010
And you know, I had a lot of experience

00:13:26.010 --> 00:13:27.742
with this personally.

00:13:27.742 --> 00:13:30.110
Early on when we were working on the web,

00:13:30.110 --> 00:13:33.419
you know, few people working on browsers

00:13:33.419 --> 00:13:35.873
made changes in the platform so that

00:13:35.873 --> 00:13:38.053
you can asynchronously, you know,

00:13:38.053 --> 00:13:40.343
what is later called as AJAX.

00:13:40.343 --> 00:13:42.595
So that for example, if
you were doing email,

00:13:42.595 --> 00:13:44.755
just because you want to
change something in the email

00:13:44.755 --> 00:13:45.996
or what you're seeing, you didn't have to

00:13:45.996 --> 00:13:47.904
reload the entire page.

00:13:47.904 --> 00:13:49.909
And that small change led to things

00:13:49.909 --> 00:13:51.282
like Google Maps.

00:13:51.282 --> 00:13:54.022
You can drag and look at maps

00:13:54.022 --> 00:13:56.051
and you know, how maybe YouTube works

00:13:56.051 --> 00:13:59.022
and how literally everything
became interactive on the web.

00:13:59.022 --> 00:14:02.172
So anytime I think you
work at a platform level

00:14:02.172 --> 00:14:04.084
I think you end up impacting a lot more

00:14:04.084 --> 00:14:05.841
over time than you think.

00:14:05.841 --> 00:14:08.674
So I think trying to
platformize what you do,

00:14:08.674 --> 00:14:12.705
you know, so that others
can take it, use it

00:14:12.705 --> 00:14:15.178
over time, rethink it, you know

00:14:15.178 --> 00:14:17.228
I think is very, very powerful.

00:14:17.228 --> 00:14:20.425
So hopefully that's a way you impact

00:14:20.425 --> 00:14:22.783
a lot of how education works as well.

00:14:22.783 --> 00:14:23.788
- Awesome.

00:14:23.788 --> 00:14:24.896
Well thank you so much for visiting

00:14:24.896 --> 00:14:26.359
and let's do this again.

00:14:26.359 --> 00:14:27.915
- All right, thank you guys.

00:14:27.915 --> 00:14:30.665
(audience claps)