Manipulating functions before differentiation

WEBVTT
Kind: captions
Language: en

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- [Instructor] What I have
listed here is several

00:00:02.643 --> 00:00:06.209
of the derivative rules that
we've used in previous videos.

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If these things look unfamiliar
to you I encourage you

00:00:08.563 --> 00:00:11.619
to maybe to not watch this
video because in this video

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we're going to think about
when do we apply these rules?

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What strategies and can
we algebraically convert

00:00:17.006 --> 00:00:19.474
expressions so that we
can use a simpler rule?

00:00:19.474 --> 00:00:21.474
But this is a quick review,
this is of course the power

00:00:21.474 --> 00:00:26.002
rule right over here, very
handy for taking derivatives

00:00:26.002 --> 00:00:28.169
of X raised to some power.

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It's also we can use that
with the derivative properties

00:00:31.774 --> 00:00:35.536
of sums of derivatives or
differences of derivatives

00:00:35.536 --> 00:00:38.287
to take derivatives of polynomials.

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This right over here is the product rule.

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If I have an expression that
I want to take the derivative

00:00:43.763 --> 00:00:46.086
of and I can think of it as
the product of two functions,

00:00:46.086 --> 00:00:48.674
well then the derivative is
going to be the derivative

00:00:48.674 --> 00:00:51.128
of the first function times
the second function plus

00:00:51.128 --> 00:00:55.303
the first function times the
derivative of the second.

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Once again if this looks
completely unfamiliar to you

00:00:57.314 --> 00:00:59.742
or you're a little shaky,
go watch the videos,

00:00:59.742 --> 00:01:02.703
do the practice on the power
rule and the product rule,

00:01:02.703 --> 00:01:05.891
or in this case the quotient rule.

00:01:05.891 --> 00:01:09.108
And the quotient rule is
a little bit more involved

00:01:09.108 --> 00:01:11.852
and we have practicing videos
on that and I always have

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mixed feelings about it
because if you don't remember

00:01:13.999 --> 00:01:16.736
the quotient rule, you can
usually or you can always convert

00:01:16.736 --> 00:01:20.531
a quotient into a product
by expressing this thing

00:01:20.531 --> 00:01:24.698
at the bottom as F of X times
G of X to the negative one.

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So you could take the
derivative with a combination

00:01:28.982 --> 00:01:32.575
of the products and this
fourth rule over here,

00:01:32.575 --> 00:01:34.189
the chain rule.

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And if any of this is looking
unfamiliar again don't watch

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this video, this video is
for folks who are familiar

00:01:41.697 --> 00:01:44.464
with each of these derivative
rules or derivative

00:01:44.464 --> 00:01:46.751
techniques and now want to
think about well what are

00:01:46.751 --> 00:01:50.471
strategies for deciding
when to apply which.

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So let's do that.

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Let's say that I have the
expression, let's say I'm

00:01:55.835 --> 00:02:00.002
interested in taking the
derivative of X squared plus X

00:02:04.226 --> 00:02:06.476
minus two over X minus one.

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Which of these rules or
techniques would you use?

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Well you might immediately
say hey look this looks like

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a rational expression, I
could say this is my F of X

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right over here, I could
say this is my G of X right

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over here and I could
apply the quotient rule,

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this looks like a quotient
of two expressions.

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And you could do that and if
you do all the mathematics

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correctly, you will
get the correct answer.

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But in this case it's good
to just take a little time

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to realize well can I
simplify this algebraically so

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maybe I can do a little bit less work?

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And if you look at it that
way, you might realize

00:02:49.728 --> 00:02:52.281
wait what if I factored this
numerator I can factor it

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as X plus two times X minus one.

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And then I could cancel these
two characters out and I can

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say hey you know what this
is going to be the same thing

00:03:04.241 --> 00:03:07.712
as the derivative with
respect to X of X plus two.

00:03:07.712 --> 00:03:11.879
Derivative with respect to X
of X plus two, which is much

00:03:13.970 --> 00:03:17.662
much much much more straightforward
than trying to apply

00:03:17.662 --> 00:03:19.334
the quotient rule.

00:03:19.334 --> 00:03:21.478
Here you would just take
the derivative with respect

00:03:21.478 --> 00:03:25.156
to X of X which is just going
to be one and the derivative

00:03:25.156 --> 00:03:27.263
with respect to X of two
is just going to be zero

00:03:27.263 --> 00:03:29.423
and so all of this is
going to simplify to one.

00:03:29.423 --> 00:03:31.812
If we're taking the derivative
of that, you're essentially

00:03:31.812 --> 00:03:33.971
just using the power rule.

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And so once again just a
simple algebraic recognition

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things become much more simple.

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Let's do another example.

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So let's say that you were to
see, or someone were to ask

00:03:47.314 --> 00:03:51.314
you to take the derivative
with respect to X of,

00:03:52.689 --> 00:03:56.856
let me see, so let's say
you had X squared plus two X

00:03:58.292 --> 00:03:59.792
minus five over X.

00:04:03.811 --> 00:04:07.104
So once again you might be
tempted to use the quotient

00:04:07.104 --> 00:04:09.987
rule, this looks like the
quotient of two expressions.

00:04:09.987 --> 00:04:12.515
But then you might realize
there's some algebraic

00:04:12.515 --> 00:04:14.852
manipulations I could
do to make this simpler.

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You could express this as a
product, you could say that

00:04:17.906 --> 00:04:21.395
this is the same thing as,
and I'm just going to focus on

00:04:21.395 --> 00:04:23.699
what's inside the parentheses
or inside the brackets,

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this is the same thing as X
to the negative one times X

00:04:26.771 --> 00:04:30.442
squared plus 2x minus five
and then you might want to

00:04:30.442 --> 00:04:32.359
apply the product rule.

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But there's even a better
simplification here.

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You could just divide each of
these terms by X or one way

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to think about it distribute
this one over X across

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all the terms, X to the
negative one is the same thing

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as one over X and if you do
that X squared divided by X

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is going to be X.

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2x divided by X is going to
be two and then negative five

00:04:53.026 --> 00:04:56.497
divided by X, well you could
write that as negative five

00:04:56.497 --> 00:05:01.025
over X or negative five
X to the negative one.

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And now I'm taking the
derivative of this with respect

00:05:04.692 --> 00:05:07.882
to X is much easier than
using either the quotient

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or the power rule.

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This is going to be, let's
see the derivative of that

00:05:13.365 --> 00:05:16.049
is going to be one, derivative
of two is going to be zero

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and here even though you
have a negative exponent,

00:05:18.498 --> 00:05:20.596
it might look a little
intimidating, this is just taken

00:05:20.596 --> 00:05:22.631
using the power rule.

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So negative one times
negative five is positive five

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X to the, if we take one
less than negative one we're

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going to go the negative two power.

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So once again making this
algebraic recognition simplified

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things a good bit.

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Let's do a few more examples
of just starting to recognize

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when we might be able to
simplify things to do things

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a little bit easier.

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So let's say that someone said
hey you take the derivative

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with respect to X and I'm using
X as our variable that we're

00:05:58.193 --> 00:06:00.515
taking the derivative of
with respect to but obviously

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this works for any
variables that we are using.

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So let's say we're saying
square root of X over X squared.

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Pause this video and think
about how would you approach

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this if you want to take the
derivative with respect to X

00:06:12.468 --> 00:06:15.718
of the square root of X over X squared.

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Well once again you might
say this is a quotient of two

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expressions, might try to
apply the quotient rule,

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or you might recognize well
look this is the same thing,

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let me just focus on what's
inside the brackets, you could

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view this as X to the negative
two times the square root

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of X and then you might
want to use the product rule

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but you could simplify this even better.

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You could say this is the same
thing as X to the negative

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two times X to the one half
power and now just using our

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exponent properties, negative
two plus one half is negative

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three halves, so this is the
derivative X to the negative

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three halves power.

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And so here once again we
took something that we thought

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we might have to use the
quotient rule or use the product

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rule and now this just becomes
a straightforward using

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the power rule.

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So this is just going to be
equal to, so bring the negative

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three halves out front,
negative three halves,

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X to the negative three halves
minus one is negative five

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halves power.

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So once again just before you,
especially if you're about

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to apply the quotient rule
and sometimes even the product

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rule, just see is there an
algebraic simplification,

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sometimes a trigonometric
simplification that you can make

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that eases your job that
makes things less hairy?

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As a general tip I can't say
this is going to be always

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true but if you're taking some
type of exam and you're going

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down some really hairy route
which the quotient rule

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will often take you, it's a
good sign that hey take a pause

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before trying to run
through all of that algebra

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to apply the quotient
rule and see if you can

00:07:54.386 --> 00:07:55.719
simplify things.

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So let's give another example.

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In this one there's not an
obvious way and it really

00:08:02.662 --> 00:08:06.970
depends on what folks'
preferences are, but let's say you

00:08:06.970 --> 00:08:10.303
want to take the derivative
with respect to X of one

00:08:10.303 --> 00:08:14.470
over 2x to the negative five,
sorry, one over 2x minus five

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I should say.

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Well here you could immediately
apply the quotient rule here

00:08:22.165 --> 00:08:25.872
the numerator you view that as F of X.

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You could view this as the
same thing as the derivative

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with respect to X.

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Instead of 2x minus five, let
me do that in the blue color.

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2x minus five to the negative one power.

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In this situation, you would
use a combination of the power

00:08:46.247 --> 00:08:48.247
rule and the chain rule.

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You would say okay my G of X
is 2x minus five and F of G

00:08:54.305 --> 00:08:58.403
of X is going to be this whole expression.

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And so if you applied the
chain rule, this is going to be

00:09:00.917 --> 00:09:05.084
the derivative of the
outside function, our F of X

00:09:06.019 --> 00:09:08.353
with respect to the inside function.

00:09:08.353 --> 00:09:11.220
The derivative of F of G of
X with respect to G of X.

00:09:11.220 --> 00:09:14.785
So it's going to be negative,
we'll bring that negative

00:09:14.785 --> 00:09:16.812
out front, so we're
essentially just going to use

00:09:16.812 --> 00:09:18.148
the power rule here.

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Negative 2x minus five to
the negative two and then we

00:09:21.936 --> 00:09:26.103
multiply that times the
derivative of the inside function.

00:09:27.495 --> 00:09:29.715
So the inside function's
derivative, the derivative

00:09:29.715 --> 00:09:32.373
of 2x is two, the derivative
of negative five is zero

00:09:32.373 --> 00:09:34.498
so it's going to be times two
and of course you can simplify

00:09:34.498 --> 00:09:38.353
so it's a negative two
times all of this business.

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Let me do one more example
here just to hit the point home

00:09:41.186 --> 00:09:44.529
and once again there isn't a
must way, there isn't a way

00:09:44.529 --> 00:09:47.919
that you have to do this,
but just let you appreciate

00:09:47.919 --> 00:09:50.450
that there's multiple ways
to approach these types

00:09:50.450 --> 00:09:51.700
of derivatives.

00:09:53.048 --> 00:09:54.996
So let's say someone said
take the derivative of 2x

00:09:54.996 --> 00:09:56.413
plus one squared.

00:09:57.540 --> 00:09:59.971
Pause the video and think
about how you would do that.

00:09:59.971 --> 00:10:01.988
Well one way to do it is
just to apply the chain rule

00:10:01.988 --> 00:10:04.081
just like we just did.

00:10:04.081 --> 00:10:06.084
So you could say alright here's
going to be the derivative

00:10:06.084 --> 00:10:07.956
of the outside with respect to the inside.

00:10:07.956 --> 00:10:11.821
So it's going to be two times
2x plus one to the first

00:10:11.821 --> 00:10:15.424
power, taking one less than
that times the derivative

00:10:15.424 --> 00:10:18.883
of the inside which is just
going to be two and so this

00:10:18.883 --> 00:10:21.587
is going to be equal to
four times 2x plus one,

00:10:21.587 --> 00:10:26.279
which is equal to, if we
want to distribute the four,

00:10:26.279 --> 00:10:28.862
we could say it's 8x plus four.

00:10:29.855 --> 00:10:32.432
That's a completely
legitimate way of doing it.

00:10:32.432 --> 00:10:35.102
Now there are other ways of doing it.

00:10:35.102 --> 00:10:37.715
You could expand out
to X plus one squared.

00:10:37.715 --> 00:10:40.723
You could say hey this is the
same thing as the derivative

00:10:40.723 --> 00:10:44.890
with respect to X of 2x squared
is going to be 4x squared

00:10:47.829 --> 00:10:51.109
and then two times the product
of these terms is going

00:10:51.109 --> 00:10:53.480
to be plus 4x plus one.

00:10:53.480 --> 00:10:57.147
And now you would just
apply the power rule.

00:10:58.115 --> 00:10:59.636
It's a little bit of extra
algebra up front but you can

00:10:59.636 --> 00:11:01.505
just go straightforward with
the power rule and you're

00:11:01.505 --> 00:11:03.665
going to get this exact same thing.

00:11:03.665 --> 00:11:05.987
So the whole takeaway here is
pause look at your expression.

00:11:05.987 --> 00:11:09.412
See if there's a way to
simplify it and it's especially

00:11:09.412 --> 00:11:11.989
a good thing if you can get
out of using the quotient rule

00:11:11.989 --> 00:11:14.566
'cause that sometimes is
just hard to know or remember

00:11:14.566 --> 00:11:17.174
and even when you do remember
it, it can get quite hairy

00:11:17.174 --> 00:11:18.091
quite fast.


Approximating asymptotic limit from table

WEBVTT
Kind: captions
Language: en

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- [Instructor] The function f is defined

00:00:01.708 --> 00:00:03.719
over the real numbers.

00:00:03.719 --> 00:00:06.839
This table gives a few values of f.

00:00:06.839 --> 00:00:10.171
So when x is equal to negative 4.1,

00:00:10.171 --> 00:00:12.926
f of x is negative five.

00:00:12.926 --> 00:00:14.426
F of negative 4.01

00:00:15.387 --> 00:00:16.625
is negative 55,

00:00:16.625 --> 00:00:18.121
so they give us a bunch of values

00:00:18.121 --> 00:00:20.762
for different x's of what f of x would be.

00:00:20.762 --> 00:00:24.661
What is a reasonable
estimate for the limit

00:00:24.661 --> 00:00:28.244
as x approaches negative four of f of x.

00:00:29.092 --> 00:00:30.837
So let's think about what's happening.

00:00:30.837 --> 00:00:33.559
To think of a reasonable
estimate for the limit

00:00:33.559 --> 00:00:35.707
as x approaches negative four,

00:00:35.707 --> 00:00:37.140
we'd want to think about well,

00:00:37.140 --> 00:00:39.631
what does it look like the
function is approaching

00:00:39.631 --> 00:00:42.070
as x approaches negative four

00:00:42.070 --> 00:00:44.504
from values less than negative four,

00:00:44.504 --> 00:00:47.038
so that's what we have here on the table.

00:00:47.038 --> 00:00:48.264
And then we'd also want to think about

00:00:48.264 --> 00:00:51.297
well, what does it look like
our function is approaching

00:00:51.297 --> 00:00:53.495
as x approaches negative four

00:00:53.495 --> 00:00:56.743
from values greater than
negative four from the right.

00:00:56.743 --> 00:00:57.984
So this is approaching,

00:00:57.984 --> 00:00:59.797
this is what is f of x approaches,

00:00:59.797 --> 00:01:01.855
x approaches negative four from the left?

00:01:01.855 --> 00:01:03.484
This is what is f of x approaching

00:01:03.484 --> 00:01:07.686
as x approaches negative
four from the right?

00:01:07.686 --> 00:01:09.749
So let's think about it.

00:01:09.749 --> 00:01:13.778
When x is negative 4.1,
we're at negative five.

00:01:13.778 --> 00:01:16.703
Then we're at negative 4.01,

00:01:16.703 --> 00:01:19.056
we go to negative 55.

00:01:19.056 --> 00:01:22.803
And then when we get to negative 4.001,

00:01:22.803 --> 00:01:26.051
so we're only 1/1000
away from negative four,

00:01:26.051 --> 00:01:27.968
we get to negative 555.

00:01:30.807 --> 00:01:34.601
So this just seems to get
larger and larger and larger

00:01:34.601 --> 00:01:38.518
and then at negative
four, it's negative 5,555.

00:01:39.943 --> 00:01:41.784
So this is interesting.

00:01:41.784 --> 00:01:45.229
There could be a scenario
where this is getting more and,

00:01:45.229 --> 00:01:47.106
this is getting infinitely negative

00:01:47.106 --> 00:01:48.978
as we get close to negative four.

00:01:48.978 --> 00:01:52.031
And that negative four then jumps back up

00:01:52.031 --> 00:01:53.531
to negative 5,555.

00:01:55.875 --> 00:02:00.422
Or this is a situation where
we are at negative four.

00:02:00.422 --> 00:02:03.228
We're just kind of approaching
this value right over here.

00:02:03.228 --> 00:02:05.986
It's unclear based on the
information in the table.

00:02:05.986 --> 00:02:08.059
Now let's think about what's happening

00:02:08.059 --> 00:02:10.542
as we approach from the right.

00:02:10.542 --> 00:02:12.618
So at negative 3.9, we're at five.

00:02:12.618 --> 00:02:15.056
Negative 3.99, we're at 55.

00:02:15.056 --> 00:02:17.473
Negative 3.999, we go to 555.

00:02:19.248 --> 00:02:21.139
So this seems to be getting larger

00:02:21.139 --> 00:02:23.004
and as we get closer and
closer to negative four

00:02:23.004 --> 00:02:25.852
as x gets closer and closer to
negative four from the right,

00:02:25.852 --> 00:02:28.778
it looks like f of x is getting larger

00:02:28.778 --> 00:02:29.749
and larger and larger.

00:02:29.749 --> 00:02:31.982
It's not going more and more negative.

00:02:31.982 --> 00:02:35.312
So as I mentioned if you
wanted to think about scenarios

00:02:35.312 --> 00:02:37.005
of what might be happening here,

00:02:37.005 --> 00:02:38.197
we don't know for sure because remember

00:02:38.197 --> 00:02:41.081
when we're using a table
we're just taking samples.

00:02:41.081 --> 00:02:45.265
So let me let me draw what
could be happening here.

00:02:45.265 --> 00:02:49.245
Once again, we don't know
for sure just by sampling.

00:02:49.245 --> 00:02:50.678
But once again, they just want us to get

00:02:50.678 --> 00:02:52.230
a reasonable estimate.

00:02:52.230 --> 00:02:54.122
So that's the x-axis

00:02:54.122 --> 00:02:56.039
and that is the y-axis.

00:02:58.244 --> 00:03:01.330
And we care about x equals negative four.

00:03:01.330 --> 00:03:03.264
So let's say that this is

00:03:03.264 --> 00:03:06.731
this is x equals negative
four right over here,

00:03:06.731 --> 00:03:08.483
x equals negative four.

00:03:08.483 --> 00:03:10.104
So there's a couple of scenarios.

00:03:10.104 --> 00:03:14.271
There's one scenario where
it's doing something like this,

00:03:15.258 --> 00:03:17.245
where it's doing something like this,

00:03:17.245 --> 00:03:20.736
where it's just going to negative 5,555

00:03:20.736 --> 00:03:22.976
and so this isn't at the same scale.

00:03:22.976 --> 00:03:26.795
So there's this scenario
where this right over here

00:03:26.795 --> 00:03:28.295
is negative 5,555.

00:03:30.930 --> 00:03:32.477
But then from the right,

00:03:32.477 --> 00:03:34.267
you're doing something
completely different,

00:03:34.267 --> 00:03:35.323
or not completely different,

00:03:35.323 --> 00:03:36.740
you're going in the opposite direction.

00:03:36.740 --> 00:03:38.536
So from the right, it looks like this.

00:03:38.536 --> 00:03:40.481
You're just getting larger
and larger and larger,

00:03:40.481 --> 00:03:44.260
just getting more and
more and more and more.

00:03:44.260 --> 00:03:46.494
I guess as you get closer
and closer to negative four

00:03:46.494 --> 00:03:47.509
from the right

00:03:47.509 --> 00:03:49.539
maybe you're going to positive infinity

00:03:49.539 --> 00:03:51.366
or maybe you're going to,

00:03:51.366 --> 00:03:53.262
well we don't know based on this data.

00:03:53.262 --> 00:03:54.270
It looks like it could just be

00:03:54.270 --> 00:03:57.278
going unbounded to positive infinity

00:03:57.278 --> 00:03:58.793
and then right at negative four

00:03:58.793 --> 00:04:00.876
you go to negative 5,555.

00:04:01.753 --> 00:04:04.232
In this situation, you would have no limit

00:04:04.232 --> 00:04:05.773
at x equals negative four.

00:04:05.773 --> 00:04:07.603
Even though the function is defined there,

00:04:07.603 --> 00:04:08.699
you would have no limit,

00:04:08.699 --> 00:04:10.283
because when you approach from the right,

00:04:10.283 --> 00:04:11.784
you're going to positive infinity.

00:04:11.784 --> 00:04:13.003
When you approach from the left,

00:04:13.003 --> 00:04:14.749
you're either going to negative infinity

00:04:14.749 --> 00:04:18.567
or you're going to the
value negative 5,555.

00:04:18.567 --> 00:04:20.476
Another scenario that might be happening

00:04:20.476 --> 00:04:21.577
as we approach from the left

00:04:21.577 --> 00:04:23.253
might be something like this.

00:04:23.253 --> 00:04:25.253
It might be approaching,

00:04:26.324 --> 00:04:30.259
it might be approaching negative infinity.

00:04:30.259 --> 00:04:32.537
And then you just jump back up

00:04:32.537 --> 00:04:34.159
to this value right over here.

00:04:34.159 --> 00:04:35.539
So let me erase this
one just so you can see

00:04:35.539 --> 00:04:37.116
what I'm talking about.

00:04:37.116 --> 00:04:39.287
So this is another scenario

00:04:39.287 --> 00:04:40.781
that right at negative four,

00:04:40.781 --> 00:04:43.479
so from either side it looks like you are,

00:04:43.479 --> 00:04:45.739
you have an asymptote going on.

00:04:45.739 --> 00:04:47.797
So from either side you have

00:04:47.797 --> 00:04:49.999
this vertical asymptote right over here.

00:04:49.999 --> 00:04:52.021
So as you approach from the left,

00:04:52.021 --> 00:04:53.107
you're going to negative infinity.

00:04:53.107 --> 00:04:54.012
As you approach from the right,

00:04:54.012 --> 00:04:55.029
you're going to positive infinity,

00:04:55.029 --> 00:04:56.004
but right at negative four,

00:04:56.004 --> 00:04:57.314
you're defined at this value.

00:04:57.314 --> 00:04:58.796
So this is another possibility.

00:04:58.796 --> 00:05:01.758
But in this case as well,
there's no limit here.

00:05:01.758 --> 00:05:04.369
You're not approaching
the same finite value

00:05:04.369 --> 00:05:05.989
from both sides.

00:05:05.989 --> 00:05:08.006
Now one thing that you'll sometimes see is

00:05:08.006 --> 00:05:09.370
if people think that okay,

00:05:09.370 --> 00:05:11.005
if you're approaching the same,

00:05:11.005 --> 00:05:13.918
either from both sides you're
approaching positive infinity

00:05:13.918 --> 00:05:16.272
or from both sides you're
approaching negative infinity.

00:05:16.272 --> 00:05:20.382
Sometimes people say, "Oh
my limit is infinity."

00:05:20.382 --> 00:05:22.008
Or "My limit is negative infinity."

00:05:22.008 --> 00:05:24.287
So let's say you have
a situation like this

00:05:24.287 --> 00:05:26.454
where from, at some value,

00:05:28.243 --> 00:05:30.165
and you're think about
the limit as you approach,

00:05:30.165 --> 00:05:31.782
as x approaches that value,

00:05:31.782 --> 00:05:32.755
as you approach from either side,

00:05:32.755 --> 00:05:35.573
you're going to positive infinity.

00:05:35.573 --> 00:05:39.070
Well formerly you still wouldn't
say the limit is infinity,

00:05:39.070 --> 00:05:41.746
because a limit formally is a finite value

00:05:41.746 --> 00:05:43.860
that you are actually approaching.

00:05:43.860 --> 00:05:45.078
So in any of these scenarios,

00:05:45.078 --> 00:05:47.011
especially the scenario that we see here

00:05:47.011 --> 00:05:48.502
where you're going to
positive infinity here,

00:05:48.502 --> 00:05:49.543
negative infinity,

00:05:49.543 --> 00:05:53.501
or you're going to negative 5,555 here,

00:05:53.501 --> 00:05:55.317
the limit does not exist.

00:05:55.317 --> 00:05:57.650
So let me just circle that.


Limits from tables for oscillating functions

WEBVTT
Kind: captions
Language: en

00:00:00.153 --> 00:00:01.749
- [Instructor] The function h is defined

00:00:01.749 --> 00:00:03.492
over the real numbers.

00:00:03.492 --> 00:00:05.803
This table gives a few values of h,

00:00:05.803 --> 00:00:07.496
so they give us for different x values,

00:00:07.496 --> 00:00:09.891
What is the value of h of x?

00:00:09.891 --> 00:00:11.271
What is a reasonable estimate

00:00:11.271 --> 00:00:15.261
for the limit of h of
x as x approaches one?

00:00:15.261 --> 00:00:18.218
So with the table we
can estimate the limit.

00:00:18.218 --> 00:00:20.247
We won't know 100% for sure,

00:00:20.247 --> 00:00:22.987
but it's a good way to estimate a limit.

00:00:22.987 --> 00:00:26.906
And so what they have here
is these first three entries

00:00:26.906 --> 00:00:29.626
show what's happening as we approach

00:00:29.626 --> 00:00:32.240
x equals one from the left,

00:00:32.240 --> 00:00:35.637
from values less than x equals one.

00:00:35.637 --> 00:00:38.561
So we can see, well what is
h of x for those x values

00:00:38.561 --> 00:00:41.278
as we get closer and
closer to x equals one?

00:00:41.278 --> 00:00:44.440
And we see, let's see
what when we're at 0.9,

00:00:44.440 --> 00:00:47.357
it's 0.003, it's 3/1000.

00:00:48.254 --> 00:00:50.258
Then we get even closer to one,

00:00:50.258 --> 00:00:54.227
then the value of our
function is negative 4/100.

00:00:54.227 --> 00:00:56.005
Then when we get even closer to one,

00:00:56.005 --> 00:00:59.477
the value of h is 5/10.

00:00:59.477 --> 00:01:02.639
So we have this oscillation in sine

00:01:02.639 --> 00:01:04.834
and it seems like the absolute value's

00:01:04.834 --> 00:01:07.035
getting larger and larger.

00:01:07.035 --> 00:01:10.101
Now if we approach from the right,

00:01:10.101 --> 00:01:14.268
we approach x equals one
from values larger than one,

00:01:15.686 --> 00:01:18.003
so here's we're
approaching from the right.

00:01:18.003 --> 00:01:20.239
At 1.1 we're at 3/1000.

00:01:20.239 --> 00:01:23.998
At 1.01, we are at negative 4/100.

00:01:23.998 --> 00:01:27.253
At 1.001 or one and 1/1000,

00:01:27.253 --> 00:01:29.481
we are at 5/10.

00:01:29.481 --> 00:01:31.040
So this is interesting.

00:01:31.040 --> 00:01:34.497
When you have this oscillation of sine,

00:01:34.497 --> 00:01:37.572
you say, "Well maybe we
might be approaching zero."

00:01:37.572 --> 00:01:40.596
You know, maybe something
is getting closer and closer

00:01:40.596 --> 00:01:42.834
and closer and closer to zero.

00:01:42.834 --> 00:01:45.105
Maybe something like that is happening.

00:01:45.105 --> 00:01:46.324
But what's troubling here,

00:01:46.324 --> 00:01:49.007
the reason why I would say
that it's not reasonable,

00:01:49.007 --> 00:01:50.985
it's reasonable to think
that the limit doesn't exist,

00:01:50.985 --> 00:01:53.637
is here as we're getting
closer and closer,

00:01:53.637 --> 00:01:55.460
here as we're getting closer and closer,

00:01:55.460 --> 00:01:57.254
we're getting for the oscillations

00:01:57.254 --> 00:01:59.779
are getting more and more significant.

00:01:59.779 --> 00:02:01.524
So actually think of it this way.

00:02:01.524 --> 00:02:04.118
As we're getting from the left,

00:02:04.118 --> 00:02:05.580
as we're getting closer and closer,

00:02:05.580 --> 00:02:08.055
the oscillations are
getting more significant.

00:02:08.055 --> 00:02:09.522
And as we're going from the right

00:02:09.522 --> 00:02:10.539
as we get closer and closer,

00:02:10.539 --> 00:02:14.028
the oscillations are
getting more significant.

00:02:14.028 --> 00:02:17.359
Now it looks like there's a
little bit of symmetry here.

00:02:17.359 --> 00:02:21.710
Let me see if I can draw
symmetry reasonably well.

00:02:21.710 --> 00:02:24.070
It looks like something
like this is happening.

00:02:24.070 --> 00:02:25.126
But this is implying as we get,

00:02:25.126 --> 00:02:29.438
as we go to 0.9999, then
we're going to oscillate again

00:02:29.438 --> 00:02:32.001
and we're going to get even
further away from zero.

00:02:32.001 --> 00:02:34.279
And so we're oscillating around zero,

00:02:34.279 --> 00:02:36.515
but where the oscillations are
swinging wilder and wilder.

00:02:36.515 --> 00:02:38.466
So we're definitely not approaching zero.

00:02:38.466 --> 00:02:41.352
We're not getting closer because
the absolute value of these

00:02:41.352 --> 00:02:43.063
aren't getting closer and closer to zero.

00:02:43.063 --> 00:02:45.781
So I would say that the
limit doesn't exist.

00:02:45.781 --> 00:02:49.562
So a takeaway here is when
things start to oscillate,

00:02:49.562 --> 00:02:50.785
you're probably already say,

00:02:50.785 --> 00:02:52.574
"Okay something fishy is going on."

00:02:52.574 --> 00:02:54.851
But that doesn't mean that
the limit doesn't always,

00:02:54.851 --> 00:02:57.251
it doesn't mean that the limit
necessarily doesn't exist

00:02:57.251 --> 00:02:58.988
just because of the oscillation.

00:02:58.988 --> 00:03:01.430
But if it's oscillating around a value

00:03:01.430 --> 00:03:02.773
and as it's getting closer and closer

00:03:02.773 --> 00:03:05.823
the oscillations are swinging wilder,

00:03:05.823 --> 00:03:08.017
further and further away from that value,

00:03:08.017 --> 00:03:09.569
then you have to worry.

00:03:09.569 --> 00:03:12.778
And that then the limit
probably does not exist.

00:03:12.778 --> 00:03:14.861
Let's do another example.

00:03:16.025 --> 00:03:20.533
The function h is defined
over the real numbers.

00:03:20.533 --> 00:03:22.635
This table gives a few values of h,

00:03:22.635 --> 00:03:25.090
so it's a similar situation.

00:03:25.090 --> 00:03:27.610
What is a reasonable
estimate for the limit

00:03:27.610 --> 00:03:30.527
as x approaches three of h of x.

00:03:32.356 --> 00:03:36.912
So let's see as x approaches
three from the left,

00:03:36.912 --> 00:03:39.559
as x approaches three from the left,

00:03:39.559 --> 00:03:41.104
it looks like, let's see 2.9,

00:03:41.104 --> 00:03:42.523
we're at 2/10.

00:03:42.523 --> 00:03:46.308
But at 2.99 we go to negative 2/100.

00:03:46.308 --> 00:03:50.694
Then a 2.999, we get to positive 2/1000.

00:03:50.694 --> 00:03:53.095
So even though it's oscillating,

00:03:53.095 --> 00:03:56.062
magnitude-wise, we're getting
closer and closer to zero.

00:03:56.062 --> 00:03:58.254
So this is an oscillation that you know,

00:03:58.254 --> 00:03:59.820
it might look like this.

00:03:59.820 --> 00:04:01.384
We're oscillating in sine,

00:04:01.384 --> 00:04:03.578
but it looks like we're
getting closer and closer

00:04:03.578 --> 00:04:05.578
and closer to zero here.

00:04:06.758 --> 00:04:08.438
So this looks like we might be,

00:04:08.438 --> 00:04:10.764
looks like our function
might be approaching zero,

00:04:10.764 --> 00:04:13.815
as x approaches three from the left.

00:04:13.815 --> 00:04:16.655
Now let's think about from the right.

00:04:16.655 --> 00:04:19.274
Well from the right, we
see something similar here.

00:04:19.274 --> 00:04:21.247
3.1, we're at negative 2/10.

00:04:21.247 --> 00:04:24.170
3.01, positive 2/100.

00:04:24.170 --> 00:04:28.337
3.001, we're even closer
to three, negative 2/1000.

00:04:29.245 --> 00:04:32.817
And so it looks like we
are definitely oscillating,

00:04:32.817 --> 00:04:36.022
but the oscillations are
getting less and less wild.

00:04:36.022 --> 00:04:38.863
Their absolute values are
getting smaller and smaller,

00:04:38.863 --> 00:04:40.693
so it actually seems reasonable,

00:04:40.693 --> 00:04:42.199
we're oscillating around zero.

00:04:42.199 --> 00:04:44.716
And with every swing as
we get closer and closer

00:04:44.716 --> 00:04:45.773
to x equals three,

00:04:45.773 --> 00:04:47.978
the oscillations are
getting smaller and smaller.

00:04:47.978 --> 00:04:49.509
So once again, we don't know for sure.

00:04:49.509 --> 00:04:51.823
We can't prove it mathematically.

00:04:51.823 --> 00:04:55.388
But from this table it
is reasonable to estimate

00:04:55.388 --> 00:04:57.721
that the limit here is zero.

00:04:59.288 --> 00:05:02.297
Now these other choices are,

00:05:02.297 --> 00:05:04.293
I would call them distractors.

00:05:04.293 --> 00:05:05.501
This one down here is interesting

00:05:05.501 --> 00:05:06.942
because the function is actually defined

00:05:06.942 --> 00:05:09.855
at x equals three and it equals four.

00:05:09.855 --> 00:05:11.761
But this might be a situation
where the limit is approached,

00:05:11.761 --> 00:05:12.808
we're getting closer and closer,

00:05:12.808 --> 00:05:15.423
but the function is
defined up here at four.

00:05:15.423 --> 00:05:17.697
And this reinforces this idea that look,

00:05:17.697 --> 00:05:20.059
the limit can approach a
value that is different

00:05:20.059 --> 00:05:22.619
from the value that the
function is defined.

00:05:22.619 --> 00:05:24.730
That's actually one of
the points of limits,

00:05:24.730 --> 00:05:27.768
is to study what is a function approach

00:05:27.768 --> 00:05:30.177
even when the function might
be defined at a different value

00:05:30.177 --> 00:05:31.997
or even when the function
isn't defined at that value,

00:05:31.997 --> 00:05:35.725
what is it approaching as
x approaches that value?

00:05:35.725 --> 00:05:39.743
Now as I said, for this
situation it's not,

00:05:39.743 --> 00:05:41.738
it's reasonable to think
that the limit exists.

00:05:41.738 --> 00:05:44.178
I don't have any good
information to think otherwise.

00:05:44.178 --> 00:05:46.502
We don't know 100% sure.

00:05:46.502 --> 00:05:48.032
they put the three here,

00:05:48.032 --> 00:05:50.062
but we're thinking about
what is h approaching

00:05:50.062 --> 00:05:52.010
as x approaches three?

00:05:52.010 --> 00:05:53.516
Now there's no information
here that would imply

00:05:53.516 --> 00:05:55.509
that h is approaching three.

00:05:55.509 --> 00:05:56.509
And at 0.22,

00:05:58.317 --> 00:06:00.758
they're just playing with
the digits right over here.

00:06:00.758 --> 00:06:03.268
But this is getting further from zero

00:06:03.268 --> 00:06:05.543
than what seems like what
we're trying to approach,

00:06:05.543 --> 00:06:07.494
so I'd rule that one out as well.

00:06:07.494 --> 00:06:09.974
But the big takeaway here is

00:06:09.974 --> 00:06:13.762
if you start to see, especially
on a table or anything

00:06:13.762 --> 00:06:15.391
that as you get closer
and closer to a value

00:06:15.391 --> 00:06:16.515
that the function is oscillating

00:06:16.515 --> 00:06:19.330
that should kind of put your
spider senses on notice.

00:06:19.330 --> 00:06:20.794
But that doesn't necessarily mean

00:06:20.794 --> 00:06:23.021
that the limit doesn't exist.

00:06:23.021 --> 00:06:25.952
If the swings are
getting wilder and wilder

00:06:25.952 --> 00:06:28.795
as we approach the value that
we're oscillating around,

00:06:28.795 --> 00:06:30.742
then the limit probably doesn't exist.

00:06:30.742 --> 00:06:33.342
But if the swings, the oscillations,

00:06:33.342 --> 00:06:35.334
are getting smaller in magnitude

00:06:35.334 --> 00:06:38.218
as we approach the thing that
we're oscillating around,

00:06:38.218 --> 00:06:39.514
in this case zero,

00:06:39.514 --> 00:06:41.145
then is actually is reasonable to think

00:06:41.145 --> 00:06:42.972
that the thing that
we're oscillating around

00:06:42.972 --> 00:06:44.805
actually is the limit.


Example reasoning about limits from tables

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:02.772
- [Instructor] Told the
function g is defined

00:00:02.772 --> 00:00:05.094
over the real numbers.

00:00:05.094 --> 00:00:08.473
This table gives a few values of g.

00:00:08.473 --> 00:00:11.143
So for a few values of x,

00:00:11.143 --> 00:00:13.592
they give us what g of x is equal to.

00:00:13.592 --> 00:00:15.799
And they say what is a reasonable estimate

00:00:15.799 --> 00:00:19.966
for the limit of g of x as x
approaches 9 from the left?

00:00:22.425 --> 00:00:24.026
Or as x approaches nine,

00:00:24.026 --> 00:00:26.213
remember, when we say x approaches nine

00:00:26.213 --> 00:00:28.918
with this little superscript,
that little negative sign,

00:00:28.918 --> 00:00:30.585
that doesn't necessarily mean

00:00:30.585 --> 00:00:34.698
that we're approaching nine
from negative values of x.

00:00:34.698 --> 00:00:36.889
It means that we're approaching nine

00:00:36.889 --> 00:00:39.139
from values less than nine.

00:00:39.987 --> 00:00:41.153
And so to think about this,

00:00:41.153 --> 00:00:44.596
what is a reasonable
value for the left-sided,

00:00:44.596 --> 00:00:46.690
or the left-handed limit?

00:00:46.690 --> 00:00:49.141
We'd say, all right,
let's pick some x values

00:00:49.141 --> 00:00:51.379
that get closer that are less than nine,

00:00:51.379 --> 00:00:53.492
but keep getting closer
and closer to nine.

00:00:53.492 --> 00:00:55.410
So I would look at these x values,

00:00:55.410 --> 00:00:58.858
which they seem to have
chosen to exactly that.

00:00:58.858 --> 00:01:00.368
They're starting at 8.9,

00:01:00.368 --> 00:01:01.842
and they get even closer, 8.99,

00:01:01.842 --> 00:01:04.744
then they get even closer to 8.999.

00:01:04.744 --> 00:01:06.820
And, once again, they
only want us to figure out

00:01:06.820 --> 00:01:09.365
whether we can estimate
the left-handed limit.

00:01:09.365 --> 00:01:11.173
If we wanted to know the actual limit,

00:01:11.173 --> 00:01:13.255
we would wanna figure
out the left-handed limit

00:01:13.255 --> 00:01:14.922
and the right-handed limit

00:01:14.922 --> 00:01:16.806
and to feel good that they seem

00:01:16.806 --> 00:01:18.375
to be approaching the same thing.

00:01:18.375 --> 00:01:20.673
But here we're only talking
about the left-handed.

00:01:20.673 --> 00:01:23.587
So when I look at this at 8.9

00:01:23.587 --> 00:01:25.894
g of x is equal to negative 5.01.

00:01:25.894 --> 00:01:30.851
As we get even closer to
nine, we get to negative 4.83.

00:01:30.851 --> 00:01:35.778
As we get even closer to
nine, we get to negative 4.81.

00:01:35.778 --> 00:01:40.050
So just looking at
these three values here,

00:01:40.050 --> 00:01:43.678
for these three x's, it looks
like we're getting close

00:01:43.678 --> 00:01:46.081
to about negative 4.8.

00:01:46.081 --> 00:01:49.681
So it's approximately negative 4.8.

00:01:49.681 --> 00:01:51.153
Now we don't know that for sure.

00:01:51.153 --> 00:01:54.784
It oftentimes a limit might
end up being a nice number

00:01:54.784 --> 00:01:57.968
like negative 4.8, that's
not that nice of a number.

00:01:57.968 --> 00:02:01.385
But the limit might be negative 4.8002438

00:02:04.722 --> 00:02:05.840
that we are approaching.

00:02:05.840 --> 00:02:09.882
It doesn't always have to go
to a nice to look at number.

00:02:09.882 --> 00:02:11.422
But something interesting happens.

00:02:11.422 --> 00:02:13.151
When we get exactly to nine,

00:02:13.151 --> 00:02:16.476
all of a sudden it seems
like we get further away

00:02:16.476 --> 00:02:18.147
from negative 4.8.

00:02:18.147 --> 00:02:20.064
We jump to negative 5.3

00:02:20.958 --> 00:02:22.511
So what does that tell us?

00:02:22.511 --> 00:02:25.277
Well, this is actually trying
to tackle a key misconception

00:02:25.277 --> 00:02:27.773
that folks have when trying
to think about limits.

00:02:27.773 --> 00:02:31.312
A limit is the value
that you are approaching,

00:02:31.312 --> 00:02:33.297
a value that the function is approaching

00:02:33.297 --> 00:02:35.907
as x approaches a certain value.

00:02:35.907 --> 00:02:38.442
It isn't necessarily going to be

00:02:38.442 --> 00:02:40.892
what the function is
defined at that value.

00:02:40.892 --> 00:02:43.524
So you can be approaching
a different value

00:02:43.524 --> 00:02:45.776
than what the function is defined at.

00:02:45.776 --> 00:02:48.555
And it looks, at least looking
at inspecting this table,

00:02:48.555 --> 00:02:50.250
it looks like that's
what's happening here.

00:02:50.250 --> 00:02:52.034
So what is a reasonable limit?

00:02:52.034 --> 00:02:54.569
What is a reasonable estimate
for the left-handed limit

00:02:54.569 --> 00:02:56.016
as x approach nine of g of x?

00:02:56.016 --> 00:02:59.259
Well, I would rule out negative 5.3,

00:02:59.259 --> 00:03:01.837
even though that is what g of nine is.

00:03:01.837 --> 00:03:03.903
This is equal to g of nine.

00:03:03.903 --> 00:03:05.631
As we approach from the left,

00:03:05.631 --> 00:03:09.890
it does not look like we're
approaching negative 5.3.

00:03:09.890 --> 00:03:12.271
In fact, it looks like
we're getting further

00:03:12.271 --> 00:03:13.324
and further away.

00:03:13.324 --> 00:03:15.181
Or it looks like we're
approaching negative 4.8.

00:03:15.181 --> 00:03:17.724
And we're actually, as
we get closer and closer,

00:03:17.724 --> 00:03:18.675
we're getting a little bit further

00:03:18.675 --> 00:03:21.043
from something like negative 5.3.

00:03:21.043 --> 00:03:22.498
So I would rule this one out.

00:03:22.498 --> 00:03:25.714
Negative 4.8 I liked, even
before I looked at the choices

00:03:25.714 --> 00:03:28.275
it looked like that's
what we were approaching.

00:03:28.275 --> 00:03:30.180
So that seems like a reasonable estimate.

00:03:30.180 --> 00:03:31.507
I like that.

00:03:31.507 --> 00:03:32.340
So 4.7.

00:03:34.325 --> 00:03:37.908
So 4.7, let's see,
okay, so this is tricky.

00:03:40.406 --> 00:03:42.134
At first I was gonna say, oh, maybe.

00:03:42.134 --> 00:03:44.074
But I realized, this isn't negative 4.7.

00:03:44.074 --> 00:03:45.194
If it was negative 4.7,

00:03:45.194 --> 00:03:47.829
well, maybe that would be reasonable,

00:03:47.829 --> 00:03:49.002
we don't know for sure.

00:03:49.002 --> 00:03:50.919
But it's not, it's 4.7.

00:03:51.974 --> 00:03:53.159
So there's no reason why we would

00:03:53.159 --> 00:03:54.599
be approaching negative 4.8

00:03:54.599 --> 00:03:57.165
and then all of a sudden jump to 4.7

00:03:57.165 --> 00:03:59.053
and that'd be the limit, that's
not what we're approaching.

00:03:59.053 --> 00:04:01.181
We're not approaching a positive 4.7.

00:04:01.181 --> 00:04:02.558
So definitely rule that out.

00:04:02.558 --> 00:04:05.646
And we're definitely not approaching nine.

00:04:05.646 --> 00:04:07.470
Here, they're just trying to confuse you.

00:04:07.470 --> 00:04:10.412
The x is approaching nine, from the left,

00:04:10.412 --> 00:04:12.153
from values less than nine.

00:04:12.153 --> 00:04:14.575
But that's not what the
function itself is approaching.

00:04:14.575 --> 00:04:17.374
The function itself seems to
be getting closer and closer

00:04:17.374 --> 00:04:19.405
to negative 4.8.

00:04:19.405 --> 00:04:20.925
So rule that one out.

00:04:20.925 --> 00:04:23.342
And the limit seems to exist.

00:04:24.344 --> 00:04:27.385
Once again, we are estimating
when we have these tables.

00:04:27.385 --> 00:04:30.506
It's completely possible
that this is some type

00:04:30.506 --> 00:04:31.949
of weird function that begins

00:04:31.949 --> 00:04:33.993
to do some type of weird oscillation

00:04:33.993 --> 00:04:36.758
as we get really close, so it oscillates

00:04:36.758 --> 00:04:39.254
between negative 4.8 and negative 4.7.

00:04:39.254 --> 00:04:41.227
So we don't know 100% sure.

00:04:41.227 --> 00:04:43.798
But it's not reasonable to think,

00:04:43.798 --> 00:04:47.351
based on this information,
that the limit doesn't exist.

00:04:47.351 --> 00:04:48.454
So I would rule that out.

00:04:48.454 --> 00:04:50.773
And I like this choice right over here.

00:04:50.773 --> 00:04:53.273
Now let's do one more example.

00:04:55.347 --> 00:04:57.825
A student created a
table to help the reason

00:04:57.825 --> 00:05:00.798
about the limit as x
approaches three of f of x.

00:05:00.798 --> 00:05:02.804
And so here we want the actual limits.

00:05:02.804 --> 00:05:04.431
We want the limit as we
approach from the left

00:05:04.431 --> 00:05:05.461
and from the right,

00:05:05.461 --> 00:05:07.449
and see if we're
approaching the same thing.

00:05:07.449 --> 00:05:10.016
Based on the table, what can
you reason about the limit?

00:05:10.016 --> 00:05:14.080
So here we see x approaching,
we could say from below,

00:05:14.080 --> 00:05:16.943
even though (laughs) visually
it looks like from above.

00:05:16.943 --> 00:05:18.673
We're approaching x from the left.

00:05:18.673 --> 00:05:21.359
We're approaching x from
values, we're approaching,

00:05:21.359 --> 00:05:24.076
I should say we're
approaching three from values

00:05:24.076 --> 00:05:27.199
to the left of three, if
we're looking at the x-axis.

00:05:27.199 --> 00:05:30.682
So we're approaching three
from values less than three.

00:05:30.682 --> 00:05:33.724
And it looks like, see 2.9, negative 1.28,

00:05:33.724 --> 00:05:37.307
2.99, negative 1.42,
2.9999, negative 1.49.

00:05:39.181 --> 00:05:41.630
So it looks like we're
getting closer and closer

00:05:41.630 --> 00:05:44.130
to negative 1.5, negative 1.5.

00:05:45.740 --> 00:05:49.907
And even though the function
is undefined as x equals three,

00:05:50.974 --> 00:05:52.958
the limit could exist.

00:05:52.958 --> 00:05:55.549
In fact, that's one of the
main values of a limit.

00:05:55.549 --> 00:05:57.002
They say what is the function approaching

00:05:57.002 --> 00:06:00.055
even in cases where the
function is not defined there?

00:06:00.055 --> 00:06:02.399
So it looks like it's
approaching negative 1.5.

00:06:02.399 --> 00:06:06.566
Now if we approach three
from the positive direction,

00:06:07.532 --> 00:06:08.783
from values greater than three.

00:06:08.783 --> 00:06:12.368
So we get 3.1, then our
function is negative 1.8.

00:06:12.368 --> 00:06:14.671
3.01, our function is negative 1.63.

00:06:14.671 --> 00:06:16.421
3.001, negative 1.51.

00:06:18.943 --> 00:06:20.162
So even from the right,

00:06:20.162 --> 00:06:21.870
or even from values larger than three,

00:06:21.870 --> 00:06:23.983
it still looks like we're
approaching negative 1.5.

00:06:23.983 --> 00:06:27.198
So this could be a reasonable estimate,

00:06:27.198 --> 00:06:29.626
a reasonable guess based on this table

00:06:29.626 --> 00:06:31.262
of what the limit actually is.

00:06:31.262 --> 00:06:33.245
Now we have to be very, very careful.

00:06:33.245 --> 00:06:36.127
You don't know definitively
just by trying out these numbers

00:06:36.127 --> 00:06:38.366
that the limit is definitely negative 1.5.

00:06:38.366 --> 00:06:41.533
It might be negative 1.5187 repeating.

00:06:44.960 --> 00:06:46.251
It might be that number

00:06:46.251 --> 00:06:47.744
We don't know for sure.

00:06:47.744 --> 00:06:49.978
Now in a typical calculus
class in a typical exam,

00:06:49.978 --> 00:06:52.032
you don't typically see numbers like that.

00:06:52.032 --> 00:06:54.735
The questions are usually engineered

00:06:54.735 --> 00:06:56.431
to give you nicer numbers.

00:06:56.431 --> 00:06:57.757
But I just wanna make it clear.

00:06:57.757 --> 00:07:01.565
You might be approaching
an irrational number.

00:07:01.565 --> 00:07:05.230
You could be approaching
a pi divided by two

00:07:05.230 --> 00:07:06.230
or something like that.

00:07:06.230 --> 00:07:08.158
Not necessarily in this
case, but in general.

00:07:08.158 --> 00:07:10.302
But this is a reasonable estimate.

00:07:10.302 --> 00:07:11.712
So let's see what we can reason.

00:07:11.712 --> 00:07:13.855
So limit x approaches three of f of x

00:07:13.855 --> 00:07:15.578
is approximately negative 1.5.

00:07:15.578 --> 00:07:18.607
Yeah, I think this is a reasonable thing.

00:07:18.607 --> 00:07:20.659
So I would check that off.

00:07:20.659 --> 00:07:23.438
Now we don't know this for sure.

00:07:23.438 --> 00:07:24.991
In fact, there's even a scenario

00:07:24.991 --> 00:07:27.228
where this limit still might not exist,

00:07:27.228 --> 00:07:28.174
even if it looks like it.

00:07:28.174 --> 00:07:29.550
Maybe as we get closer and closer

00:07:29.550 --> 00:07:31.054
this is some type of weird function

00:07:31.054 --> 00:07:33.132
that starts oscillating
in some strange way

00:07:33.132 --> 00:07:36.011
or just stops getting closer
and closer to a value,

00:07:36.011 --> 00:07:38.366
even though it looks like
it's getting closer and closer

00:07:38.366 --> 00:07:39.366
right over here.

00:07:39.366 --> 00:07:41.071
But this is a reasonable,

00:07:41.071 --> 00:07:43.313
what can you reasonably reason?

00:07:43.313 --> 00:07:44.602
I'll put that here.

00:07:44.602 --> 00:07:48.666
What can you reasonably
reason about the limit?

00:07:48.666 --> 00:07:51.313
There appears to be an
asymptote at x equals three.

00:07:51.313 --> 00:07:54.459
Well, no an asymptote
would be the situation

00:07:54.459 --> 00:07:56.479
if we were a vertical asymptote,

00:07:56.479 --> 00:07:59.533
especially as if we seem
to be approaching infinity

00:07:59.533 --> 00:08:00.624
or negative infinity.

00:08:00.624 --> 00:08:02.453
And that's definitely
not the case the here.

00:08:02.453 --> 00:08:03.791
So I'd rule that out.

00:08:03.791 --> 00:08:05.071
As we approach x equals three,

00:08:05.071 --> 00:08:07.680
the values of f of x seem
to approach negative 1.5.

00:08:07.680 --> 00:08:10.655
But we can't say for sure
if that's the limit exactly.

00:08:10.655 --> 00:08:13.439
Yeah, that's the point I
was making right over here.

00:08:13.439 --> 00:08:15.808
It seems like it's
getting to negative 1.5.

00:08:15.808 --> 00:08:19.839
And most questions on most
tests are kinda engineered

00:08:19.839 --> 00:08:21.344
to give you numbers that aren't crazy.

00:08:21.344 --> 00:08:23.793
But even with this data,
even with this information,

00:08:23.793 --> 00:08:25.448
it might be approaching something

00:08:25.448 --> 00:08:27.700
a little bit more interesting.

00:08:27.700 --> 00:08:30.471
Or this could be some kinda crazy function

00:08:30.471 --> 00:08:33.572
that as we get within
a thousandth of three,

00:08:33.572 --> 00:08:36.514
maybe it's not defined even around three.

00:08:36.514 --> 00:08:39.155
Maybe it's not even defined at 2.99999,

00:08:39.155 --> 00:08:43.477
or 3.00001, or maybe it
doesn't get closer anymore.

00:08:43.477 --> 00:08:45.507
So, once again, you don't know for sure.

00:08:45.507 --> 00:08:49.668
But this does seem to be
a reasonable thing to say.

00:08:49.668 --> 00:08:53.030
The values of f of x appear to oscillate,

00:08:53.030 --> 00:08:55.205
jump back and forth,
near x equals to three,

00:08:55.205 --> 00:08:56.602
so it's hard to say what the limit is

00:08:56.602 --> 00:08:57.657
or if it even exists.

00:08:57.657 --> 00:09:00.413
Well, we're not seeing that
behavior right over here.

00:09:00.413 --> 00:09:02.038
So I wouldn't pick that.

00:09:02.038 --> 00:09:03.311
That's not a reasonable thing,

00:09:03.311 --> 00:09:04.824
'cause you're not
observing it in the data.

00:09:04.824 --> 00:09:07.530
Now there are some crazy
functions out there

00:09:07.530 --> 00:09:10.618
where all of a sudden if
you start trying out values

00:09:10.618 --> 00:09:12.334
that are much closer than
what we're seeing here,

00:09:12.334 --> 00:09:14.100
maybe some if this
behavior starts to happen,

00:09:14.100 --> 00:09:15.330
but we're not seeing it in the data,

00:09:15.330 --> 00:09:17.879
so I definitely wouldn't pick that.


Introduction to frames of reference

WEBVTT
Kind: captions
Language: en

00:00:00.055 --> 00:00:01.324
- [Instructor] What I'd
like to do in this video

00:00:01.324 --> 00:00:05.324
is talk about the notion
of a frame of reference

00:00:07.316 --> 00:00:08.722
and this is an introductory video.

00:00:08.722 --> 00:00:11.880
In future videos, we'll
go into a lot more depth,

00:00:11.880 --> 00:00:14.224
but a frame of reference
is really the idea

00:00:14.224 --> 00:00:16.987
it's a point of view from
which you are measuring things

00:00:16.987 --> 00:00:18.929
and as we'll see, many
of the quantities that

00:00:18.929 --> 00:00:23.155
we might measure in physics,
like velocity or displacement,

00:00:23.155 --> 00:00:26.161
they could be different
depending on our point of view,

00:00:26.161 --> 00:00:30.250
depending on which frame of
reference we are measuring from

00:00:30.250 --> 00:00:33.151
and to get this an intuitive grasp of it,

00:00:33.151 --> 00:00:35.638
I'm going to draw the exact same scenario

00:00:35.638 --> 00:00:38.174
from three different frames of reference.

00:00:38.174 --> 00:00:39.303
There's the first one,

00:00:39.303 --> 00:00:40.659
this is the second one,

00:00:40.659 --> 00:00:42.993
and this is the third one.

00:00:42.993 --> 00:00:46.504
So in this first frame of
reference, this first scenario,

00:00:46.504 --> 00:00:50.681
we're gonna talk about the frame
of reference of the ground.

00:00:50.681 --> 00:00:54.904
So if you are a stationary
observer on the ground,

00:00:54.904 --> 00:00:57.108
so you could imagine this is you here

00:00:57.108 --> 00:00:59.701
and you're the person doing the measuring

00:00:59.701 --> 00:01:02.431
of let's say we want
to measure velocities.

00:01:02.431 --> 00:01:04.195
So from your point of view,

00:01:04.195 --> 00:01:06.570
since you're stationary
relative to the ground,

00:01:06.570 --> 00:01:09.165
what does the ground's velocity look like?

00:01:09.165 --> 00:01:12.862
Well, you and the ground
appear to be stationary,

00:01:12.862 --> 00:01:15.397
appear to not be moving.

00:01:15.397 --> 00:01:17.374
Now, what if you take out your instruments

00:01:17.374 --> 00:01:21.147
for measuring velocity
or you see a change in,

00:01:21.147 --> 00:01:23.905
you see what the displacement
is over a certain time

00:01:23.905 --> 00:01:25.364
for the plane and the car

00:01:25.364 --> 00:01:27.181
and you're able to see okay, look,

00:01:27.181 --> 00:01:30.348
this plane has a velocity to the right

00:01:32.261 --> 00:01:34.344
of 250 meters per second,

00:01:35.939 --> 00:01:38.179
250 meters per second,

00:01:38.179 --> 00:01:39.703
and let's say this car

00:01:39.703 --> 00:01:42.370
that is moving quite fast by car standards

00:01:42.370 --> 00:01:46.707
is moving to the left
at 50 meters per second.

00:01:46.707 --> 00:01:49.326
So this should be 1/5 of that length.

00:01:49.326 --> 00:01:51.277
So let me draw a little bit.

00:01:51.277 --> 00:01:54.844
So let's say this is moving to the left

00:01:54.844 --> 00:01:57.565
at 50 meters per second.

00:01:57.565 --> 00:01:58.956
Well, none of this seems crazy.

00:01:58.956 --> 00:02:01.132
You might be able to go outside
next to the highway and see,

00:02:01.132 --> 00:02:03.396
well 50 meters per second
would be quite fast,

00:02:03.396 --> 00:02:08.163
but anyway, you could observe
this type of thing happening

00:02:08.163 --> 00:02:10.203
and it seems completely reasonably.

00:02:10.203 --> 00:02:12.565
But what if we were to change
our frame of reference,

00:02:12.565 --> 00:02:15.649
change the point of view from
which we are measuring things.

00:02:15.649 --> 00:02:19.243
So let's take the frame
of reference of the car.

00:02:19.243 --> 00:02:20.409
Well in this frame of reference,

00:02:20.409 --> 00:02:21.872
let's say you're sitting in this car

00:02:21.872 --> 00:02:23.924
and I don't recommend you
doing this while driving,

00:02:23.924 --> 00:02:25.270
let's say someone else is driving

00:02:25.270 --> 00:02:27.496
or it's an autonomous vehicle of some kind

00:02:27.496 --> 00:02:29.325
and you take out your physics instruments

00:02:29.325 --> 00:02:31.875
with the stopwatch and you
see what the displacement is

00:02:31.875 --> 00:02:35.639
of the ground and the
plane over, say, a second

00:02:35.639 --> 00:02:38.225
and you are able to first
say, from your point of view,

00:02:38.225 --> 00:02:40.640
you're like well the car is stationary,

00:02:40.640 --> 00:02:43.677
the car has a velocity of zero,

00:02:43.677 --> 00:02:45.961
the car is stationary

00:02:45.961 --> 00:02:47.513
and from your point of view,

00:02:47.513 --> 00:02:50.123
you would actually measure
the ground to be moving.

00:02:50.123 --> 00:02:53.249
You would see the trees
move past you to the right,

00:02:53.249 --> 00:02:55.909
or behind you if you're
moving to the left,

00:02:55.909 --> 00:02:57.590
and so from your point of view

00:02:57.590 --> 00:03:00.996
the ground would actually
look like it's moving

00:03:00.996 --> 00:03:04.623
in this direction, in that direction,

00:03:04.623 --> 00:03:07.349
at 50 meters per second.

00:03:07.349 --> 00:03:09.996
It would look like it's moving behind you

00:03:09.996 --> 00:03:11.935
or in this case, the
way we're looking at it,

00:03:11.935 --> 00:03:15.112
to the right at 50 meters per second.

00:03:15.112 --> 00:03:18.206
Now, what would the plane look like?

00:03:18.206 --> 00:03:20.098
Well, the plane not
only would it look like

00:03:20.098 --> 00:03:24.251
it's moving to the right
at 250 meters per second,

00:03:24.251 --> 00:03:27.593
not only would it be just
that 250 meters per second,

00:03:27.593 --> 00:03:30.476
but relative to you it'd look
like it's going even faster

00:03:30.476 --> 00:03:32.062
'cause you're going past it,

00:03:32.062 --> 00:03:35.367
you are going to the
left from the stationary,

00:03:35.367 --> 00:03:38.237
from the ground's point of
view at 50 meters per second.

00:03:38.237 --> 00:03:40.175
So the plane, to you,

00:03:40.175 --> 00:03:44.342
is gonna look like it's going
250 plus 50 meters per second.

00:03:47.875 --> 00:03:50.653
So the vector would look like this

00:03:50.653 --> 00:03:54.621
and so it would look like it's
going to the right at 300,

00:03:54.621 --> 00:03:57.222
let me write that in that orange color,

00:03:57.222 --> 00:03:59.305
at 300 meters per second.

00:04:01.559 --> 00:04:03.677
Now, what about from the
point of view of the plane?

00:04:03.677 --> 00:04:07.179
What if we're talking about
the plane's frame of reference?

00:04:07.179 --> 00:04:08.589
Why don't you pause this video

00:04:08.589 --> 00:04:11.479
and think about what
the velocities would be

00:04:11.479 --> 00:04:14.264
of the plane, the car, and the ground

00:04:14.264 --> 00:04:16.958
from the plane's point of view.

00:04:16.958 --> 00:04:18.488
All right, now let's work
through this together.

00:04:18.488 --> 00:04:20.544
So now, we're sitting in the plane

00:04:20.544 --> 00:04:22.555
and once again we shouldn't
be flying the plane,

00:04:22.555 --> 00:04:23.626
we're letting someone else do that,

00:04:23.626 --> 00:04:24.917
we have our physics instruments out

00:04:24.917 --> 00:04:27.362
and we're trying to measure the velocities

00:04:27.362 --> 00:04:30.656
of these other things from
my frame of reference.

00:04:30.656 --> 00:04:32.628
Well, the plane, first of all,

00:04:32.628 --> 00:04:35.303
is going to appear to be stationary

00:04:35.303 --> 00:04:37.959
and that might seem counterintuitive,

00:04:37.959 --> 00:04:39.099
but if you've ever sat in a plane,

00:04:39.099 --> 00:04:40.769
especially when there's no turbulence

00:04:40.769 --> 00:04:42.685
and the plane is already at altitude

00:04:42.685 --> 00:04:44.998
and it's not taking off or landing,

00:04:44.998 --> 00:04:46.523
oftentimes if you close your eyes

00:04:46.523 --> 00:04:48.627
you don't know if you are moving.

00:04:48.627 --> 00:04:50.506
In fact, if you close all of the windows,

00:04:50.506 --> 00:04:53.557
it feels like you are
in a stationary object,

00:04:53.557 --> 00:04:55.850
that you might as well be in a house.

00:04:55.850 --> 00:05:00.001
So from the plane's point
of view, you feel like,

00:05:00.001 --> 00:05:01.447
or from your point of view in the plane,

00:05:01.447 --> 00:05:03.543
it feels like the plane is stationary.

00:05:03.543 --> 00:05:05.106
Now the ground, however,

00:05:05.106 --> 00:05:07.018
looks like it's moving quite quickly.

00:05:07.018 --> 00:05:09.507
It'll look like it's moving past you

00:05:09.507 --> 00:05:11.590
at 250 meters per second.

00:05:14.411 --> 00:05:16.762
Whoops, try and draw a straight line.

00:05:16.762 --> 00:05:17.595
At 200...

00:05:20.237 --> 00:05:21.779
At 250.

00:05:21.779 --> 00:05:24.196
Sometimes my tools act funny.

00:05:25.382 --> 00:05:28.799
So, at 250 meters per second to the left.

00:05:32.135 --> 00:05:35.205
And the car, well it's moving
to the left even faster.

00:05:35.205 --> 00:05:36.416
It's going to be moving to the left

00:05:36.416 --> 00:05:39.584
50 meters per second
faster than the ground is.

00:05:39.584 --> 00:05:42.844
So the car is gonna look
not like it's just going

00:05:42.844 --> 00:05:44.476
50 meters per second,

00:05:44.476 --> 00:05:46.652
it's gonna look like it's going 50 meters

00:05:46.652 --> 00:05:50.014
plus another 250 meters per second

00:05:50.014 --> 00:05:54.097
for a total of 300 meters
per second to the left.

00:05:59.265 --> 00:06:01.864
So this gives you an appreciation

00:06:01.864 --> 00:06:03.544
for what frames of references are.

00:06:03.544 --> 00:06:05.626
You can view it for
this introductory video

00:06:05.626 --> 00:06:06.556
as a point of view

00:06:06.556 --> 00:06:08.488
from which you're making
your measurements.

00:06:08.488 --> 00:06:10.147
Now, it's tempting for
a lot of folks to say

00:06:10.147 --> 00:06:12.928
well there must be one
correct frame of reference

00:06:12.928 --> 00:06:15.064
and a lot of times in our everyday world

00:06:15.064 --> 00:06:17.077
you might say well this,
maybe this is the correct

00:06:17.077 --> 00:06:18.933
frame of reference and these are just,

00:06:18.933 --> 00:06:21.860
we're just imagining this
or this is just the mistake

00:06:21.860 --> 00:06:23.747
and the reason why we do that is because

00:06:23.747 --> 00:06:25.327
we're using the frame of reference

00:06:25.327 --> 00:06:29.020
of this big, giant thing called the earth,

00:06:29.020 --> 00:06:30.370
but it actually turns out

00:06:30.370 --> 00:06:32.608
that none of these frames of reference

00:06:32.608 --> 00:06:35.069
are more valid than the other ones,

00:06:35.069 --> 00:06:36.835
that they are all equivalent,

00:06:36.835 --> 00:06:39.556
that they are all valid
frames of reference,

00:06:39.556 --> 00:06:41.516
not, I shouldn't say they're equivalent,

00:06:41.516 --> 00:06:45.863
we're obviously getting
different measurements from them,

00:06:45.863 --> 00:06:49.019
but they're all, from a
physics point of view,

00:06:49.019 --> 00:06:51.852
equally valid frames of reference.


Strategies for dividing by tenths

WEBVTT
Kind: captions
Language: en

00:00:00.703 --> 00:00:02.184
- [Instructor] Let's
do a few more examples

00:00:02.184 --> 00:00:04.675
of thinking of strategies
for dividing decimals.

00:00:04.675 --> 00:00:06.016
In the future, we're gonna come up with

00:00:06.016 --> 00:00:07.533
a more systematic way of doing it,

00:00:07.533 --> 00:00:08.756
but it's really important

00:00:08.756 --> 00:00:10.567
to come up with some of these strategies

00:00:10.567 --> 00:00:13.189
because it gives you an
intuition for dividing decimals.

00:00:13.189 --> 00:00:14.732
And frankly, it's an easier thing to do,

00:00:14.732 --> 00:00:16.423
especially when you're trying

00:00:16.423 --> 00:00:18.630
to eventually divide
decimals in your head.

00:00:18.630 --> 00:00:22.423
So let's say we want
to figure out what six

00:00:22.423 --> 00:00:25.673
divided by 0.2, or 2/10, is.

00:00:27.745 --> 00:00:31.432
Pause the video, and see
if you can figure it out.

00:00:31.432 --> 00:00:33.775
So we've already explored
multiple strategies.

00:00:33.775 --> 00:00:36.556
One strategy is, well, let's
express this as a fraction.

00:00:36.556 --> 00:00:39.639
This is the same thing as 6 over 0.2,

00:00:42.905 --> 00:00:46.095
And maybe we can multiply the
numerator and the denominator

00:00:46.095 --> 00:00:48.936
by some value so we're not
dealing with decimals anymore.

00:00:48.936 --> 00:00:50.966
And, if you wanna get rid of
the decimal in the bottom,

00:00:50.966 --> 00:00:54.589
we have 2/10, well we can
multiply the bottom by 10.

00:00:54.589 --> 00:00:56.168
But if we multiply the bottom by 10

00:00:56.168 --> 00:00:58.534
we need to multiply the numerator by 10.

00:00:58.534 --> 00:00:59.937
So essentially we're multiplying

00:00:59.937 --> 00:01:02.167
this fraction by 10/10,
which is just multiplying

00:01:02.167 --> 00:01:05.045
it by 1, so it doesn't change its value.

00:01:05.045 --> 00:01:08.712
So this is going to be
equal to 60, so 6, 0,

00:01:11.277 --> 00:01:14.664
over 2/10 times 10, let's go and move

00:01:14.664 --> 00:01:16.792
the decimal one to the right,

00:01:16.792 --> 00:01:19.931
and that's just going to be equal to 2.

00:01:19.931 --> 00:01:22.264
Now what is 60 divided by 2?

00:01:23.970 --> 00:01:26.936
Well, 60 divided by 2 is
fairly straightforward,

00:01:26.936 --> 00:01:29.533
6 divided by 2 is 3 so 60 divided by 2

00:01:29.533 --> 00:01:31.783
is going to be equal to 30.

00:01:33.000 --> 00:01:35.658
So that's 30. And so we're done.

00:01:35.658 --> 00:01:37.541
Now another strategy is
you could've thought of

00:01:37.541 --> 00:01:39.848
all of these numbers in terms of tenths.

00:01:39.848 --> 00:01:44.309
You could've said this is
6, do that orange color,

00:01:44.309 --> 00:01:48.142
You could've said this
is 6 tenths divided by,

00:01:49.907 --> 00:01:53.324
oh, let me be careful, this is 60 tenths,

00:01:56.480 --> 00:01:57.813
6, is 60 tenths,

00:01:59.696 --> 00:02:03.196
60 tenths divided by 2 tenths is equal to,

00:02:09.257 --> 00:02:11.150
well if 60 is something
and I were to divide

00:02:11.150 --> 00:02:12.803
it into groups of two of that something

00:02:12.803 --> 00:02:14.446
I would have 30 groups.

00:02:14.446 --> 00:02:17.321
That's going to be equal to 30.

00:02:17.321 --> 00:02:19.713
Let's do another example.

00:02:19.713 --> 00:02:21.836
This will be our most involved one,

00:02:21.836 --> 00:02:23.349
before we really try to show you

00:02:23.349 --> 00:02:27.037
the standard way of dividing decimals.

00:02:27.037 --> 00:02:29.467
Let's say we wanted to compute

00:02:29.467 --> 00:02:33.300
4.2, or 4 and two tenths,
divided by 3 tenths.

00:02:37.462 --> 00:02:41.034
Pause the video and see if
you can figure that out.

00:02:41.034 --> 00:02:42.440
Well you've already seen multiple

00:02:42.440 --> 00:02:43.896
techniques for tackling this.

00:02:43.896 --> 00:02:45.960
It never hurts to try to
write this as a fraction.

00:02:45.960 --> 00:02:48.932
So this, you could write it as 4.2

00:02:48.932 --> 00:02:50.599
divided by 3 tenths.

00:02:52.959 --> 00:02:55.194
And now we could try to
get rid of the decimals.

00:02:55.194 --> 00:02:57.192
Now the best way to do
that I could imagine,

00:02:57.192 --> 00:02:59.021
I have tenths here, I have tenths here,

00:02:59.021 --> 00:03:02.067
If I multiply the numerator
and the denominator by 10,

00:03:02.067 --> 00:03:04.668
that might help out a lot,
because the numerator,

00:03:04.668 --> 00:03:07.448
this would move the
decimal one to the right,

00:03:07.448 --> 00:03:09.605
so the numerator I would get 42,

00:03:09.605 --> 00:03:13.188
4.2 times 10 is 42,
over 3 tenths times 10,

00:03:16.205 --> 00:03:19.114
well that is going to be equal to 3.

00:03:19.114 --> 00:03:21.697
And so what is 42 divided by 3?

00:03:22.785 --> 00:03:24.906
Well, there's multiple ways to do that,

00:03:24.906 --> 00:03:26.491
you can try to do it in your head,

00:03:26.491 --> 00:03:28.229
or you just try to actually do

00:03:28.229 --> 00:03:30.766
a little bit of medium long division,

00:03:30.766 --> 00:03:32.625
this shouldn't be too long.

00:03:32.625 --> 00:03:34.926
So let's see, 3 goes into 4 one time,

00:03:34.926 --> 00:03:38.176
1 times 3 is 3, subtract, 4 minus 3 is,

00:03:40.426 --> 00:03:42.366
oh why did I write 43, I knew something

00:03:42.366 --> 00:03:43.533
was fishy, 42.

00:03:44.720 --> 00:03:47.676
3 goes into 42, my
brain is malfunctioning.

00:03:47.676 --> 00:03:51.329
Alright, 3 goes into 4
one time, 1 times 3 is 3,

00:03:51.329 --> 00:03:55.144
you subtract 4 minus 3
is 1, bring down the 2,

00:03:55.144 --> 00:03:58.063
3 goes into 12 four times.

00:03:58.063 --> 00:03:59.980
So this is equal to 14.

00:04:00.895 --> 00:04:02.449
And so this one right over here

00:04:02.449 --> 00:04:04.127
is going to be equal to 14.

00:04:04.127 --> 00:04:05.925
And just like we saw in the last example,

00:04:05.925 --> 00:04:09.425
you could also think of this as 42 tenths,

00:04:11.191 --> 00:04:12.858
divided by 3 tenths,

00:04:17.190 --> 00:04:19.386
in which case 42 tenths,

00:04:19.386 --> 00:04:21.344
42 of something divided into groups of 3

00:04:21.344 --> 00:04:22.793
of that something,
you're gonna end up with

00:04:22.793 --> 00:04:26.043
14 groups, you're gonna end up with 14.

00:04:27.295 --> 00:04:29.034
So hopefully you appreciate these ideas,

00:04:29.034 --> 00:04:31.199
express it as a fraction,
see if you can multiply

00:04:31.199 --> 00:04:33.893
the numerator and
denominator by the same value

00:04:33.893 --> 00:04:36.362
so maybe the decimals get eliminated,

00:04:36.362 --> 00:04:38.436
maybe you can think of these numbers

00:04:38.436 --> 00:04:40.761
in terms of tenths, or hundredths,

00:04:40.761 --> 00:04:42.550
and then think of it that way.

00:04:42.550 --> 00:04:44.984
And any of these combinations are gonna be

00:04:44.984 --> 00:04:46.433
effective strategies, or hopefully

00:04:46.433 --> 00:04:48.176
effective strategies, for dividing

00:04:48.176 --> 00:04:49.644
decimals or dividing numbers where

00:04:49.644 --> 00:04:51.859
the quotient might be a decimal.

00:04:51.859 --> 00:04:53.142
And in future videos we're gonna learn

00:04:53.142 --> 00:04:54.171
a more standard systematic way of

00:04:54.171 --> 00:04:57.138
doing it but this is always valuable.

00:04:57.138 --> 00:04:59.724
I still, if someone walked up to me on

00:04:59.724 --> 00:05:03.017
the street and said
what's 4.2 divided by 0.3,

00:05:03.017 --> 00:05:05.105
this is how I would actually do it.

00:05:05.105 --> 00:05:06.921
I would say okay that's the same

00:05:06.921 --> 00:05:09.344
thing as 42 divided by 3, and then

00:05:09.344 --> 00:05:12.250
I would say okay, 3
goes into 42, let's see,

00:05:12.250 --> 00:05:16.018
3 times 10 is 30, and
then 3 times 4 is 12,

00:05:16.018 --> 00:05:18.035
yeah that would be 14 times.

00:05:18.035 --> 00:05:19.688
That's how my brain would do it if

00:05:19.688 --> 00:05:21.871
I was trying to do it in my head.


Multi digit division strategies for decimals

WEBVTT
Kind: captions
Language: en

00:00:00.025 --> 00:00:01.570
- [Instructor] In a previous
video we started thinking

00:00:01.570 --> 00:00:04.370
about strategies for dividing numbers

00:00:04.370 --> 00:00:06.098
where either the numbers are decimals,

00:00:06.098 --> 00:00:09.090
or their quotients are
going to be decimals.

00:00:09.090 --> 00:00:10.306
So now let's continue that,

00:00:10.306 --> 00:00:13.155
and we're gonna do a slightly
more involved examples.

00:00:13.155 --> 00:00:17.322
So let's say we wanna figure
out what 500 divided by 200 is.

00:00:19.794 --> 00:00:21.377
Divided by 200.

00:00:21.377 --> 00:00:25.545
Pause this video and see
if you can figure that out.

00:00:25.545 --> 00:00:28.313
Well, one strategy for doing this is

00:00:28.313 --> 00:00:30.425
to just re-express this as a fraction

00:00:30.425 --> 00:00:32.849
and see if you can simplify
this fraction in a way

00:00:32.849 --> 00:00:35.570
that it's straightforward
to express it as a decimal.

00:00:35.570 --> 00:00:38.457
So for example, this is
going to be the same thing.

00:00:38.457 --> 00:00:40.207
This is equal to 500,

00:00:42.045 --> 00:00:43.128
500 over 200.

00:00:46.710 --> 00:00:48.757
And now we can simplify this.

00:00:48.757 --> 00:00:53.176
We can say this is the same
thing as five times 100,

00:00:53.176 --> 00:00:54.426
five times 100,

00:00:55.584 --> 00:00:57.912
over two times 100.

00:00:57.912 --> 00:00:59.663
The reason why that is useful is you say,

00:00:59.663 --> 00:01:01.607
"Hey look, I have a
hundred in the numerator,

00:01:01.607 --> 00:01:03.079
"I have a hundred in the denominator,

00:01:03.079 --> 00:01:06.511
"a hundred divided by a hundred
is just going to be one."

00:01:06.511 --> 00:01:08.544
So, you could just do this.

00:01:08.544 --> 00:01:10.961
This is equal to five halves.

00:01:12.268 --> 00:01:13.768
Five halves times.

00:01:16.896 --> 00:01:18.479
Times 100 over 100.

00:01:20.982 --> 00:01:22.565
Times 100 over 100.

00:01:24.206 --> 00:01:26.774
Which is just going to be equal to,

00:01:26.774 --> 00:01:28.567
which is just going to be equal to one.

00:01:28.567 --> 00:01:29.575
Another way to think about it,

00:01:29.575 --> 00:01:32.481
you could divide the
numerator by a hundred.

00:01:32.481 --> 00:01:33.823
And you would have five,

00:01:33.823 --> 00:01:36.073
and as long as you divide the
denominator by the same thing,

00:01:36.073 --> 00:01:38.703
you're not changing the
value of the fraction.

00:01:38.703 --> 00:01:40.687
So if you divide the
denominator by a hundred,

00:01:40.687 --> 00:01:42.063
you're going to get two.

00:01:42.063 --> 00:01:43.391
So, any way you think about it,

00:01:43.391 --> 00:01:45.983
this could be simplified as five halves.

00:01:45.983 --> 00:01:47.223
But we're not done yet.

00:01:47.223 --> 00:01:50.199
That is what 500 divided by 200 is.

00:01:50.199 --> 00:01:52.919
But can we express this as a decimal?

00:01:52.919 --> 00:01:56.615
Well we can rewrite five
halves as a mixed number.

00:01:56.615 --> 00:01:59.865
So five halves is going to be equal to,

00:02:01.199 --> 00:02:03.415
well how many times does two go into five?

00:02:03.415 --> 00:02:05.255
Well it goes two times,

00:02:05.255 --> 00:02:08.167
and then you have one half left over.

00:02:08.167 --> 00:02:11.500
So this is going to be two and one half.

00:02:14.166 --> 00:02:17.894
And now how do we express this
right over here as a decimal?

00:02:17.894 --> 00:02:19.573
Well, you might recognize that one half

00:02:19.573 --> 00:02:21.917
is the same thing as five tenths.

00:02:21.917 --> 00:02:26.084
So this is going to be equal
to two and five over ten,

00:02:29.204 --> 00:02:32.204
which of course we can write as 2.5,

00:02:33.386 --> 00:02:35.748
or two and five tenths.

00:02:35.748 --> 00:02:38.165
So 500 divided by 200 is 2.5.

00:02:40.134 --> 00:02:42.030
Let's do another example.

00:02:42.030 --> 00:02:44.622
Let's say we wanted to figure out

00:02:44.622 --> 00:02:47.039
what 0.63 divided by 0.07 is.

00:02:56.249 --> 00:02:57.497
Pause this video and
see if you can come up

00:02:57.497 --> 00:03:00.080
with a strategy for doing this.

00:03:01.535 --> 00:03:04.056
Well there's multiple ways to tackle it.

00:03:04.056 --> 00:03:06.407
One way is to think about both of them

00:03:06.407 --> 00:03:07.865
in terms of hundredths.

00:03:07.865 --> 00:03:11.032
So for example, this is 63 hundredths.

00:03:14.399 --> 00:03:18.149
And this right over here
is seven hundredths.

00:03:20.751 --> 00:03:22.519
And so if you have 63 of something,

00:03:22.519 --> 00:03:26.269
and you're dividing that
by seven hundredths,

00:03:28.534 --> 00:03:31.126
what are you going to get?

00:03:31.126 --> 00:03:35.293
Well, you're going to, if you
took your seven hundredths,

00:03:36.200 --> 00:03:38.600
and you multiply it by nine,

00:03:38.600 --> 00:03:41.240
you're going to get 63 hundredths.

00:03:41.240 --> 00:03:44.154
And so 63 of something divided by seven

00:03:44.154 --> 00:03:47.432
of that same something is
going to be equal to nine.

00:03:47.432 --> 00:03:50.265
This is going to be equal to nine.

00:03:51.661 --> 00:03:54.141
Seven times nine is 63.

00:03:54.141 --> 00:03:58.308
So seven hundredths times nine
is going to be 63 hundredths.

00:04:00.324 --> 00:04:01.951
Another way to think about it

00:04:01.951 --> 00:04:04.732
is we can express this as a fraction.

00:04:04.732 --> 00:04:07.884
So in the numerator, you have 0.63

00:04:07.884 --> 00:04:11.051
and in the denominator, you have 0.07.

00:04:12.607 --> 00:04:14.535
And if the decimals are bothering us,

00:04:14.535 --> 00:04:16.583
we can multiply both the
numerator and the denominator

00:04:16.583 --> 00:04:19.407
by the same value to
get rid of the decimals.

00:04:19.407 --> 00:04:22.495
So let's multiply the numerator by 100,

00:04:22.495 --> 00:04:25.948
and also multiply the denominator by 100.

00:04:25.948 --> 00:04:27.532
This doesn't change the
value of the expression

00:04:27.532 --> 00:04:29.788
because multiplying by 100
over 100 is just the same thing

00:04:29.788 --> 00:04:31.636
as multiplying by one.

00:04:31.636 --> 00:04:35.803
So this would be equal to
63 hundredths times 100.

00:04:36.807 --> 00:04:39.160
Well we would move the
decimal two to the right.

00:04:39.160 --> 00:04:40.183
This is going to be equal

00:04:40.183 --> 00:04:43.350
to 63 over seven hundredths times 100.

00:04:46.385 --> 00:04:49.819
Once again we move the
decimal two to the right.

00:04:49.819 --> 00:04:51.898
This is going to be seven.

00:04:51.898 --> 00:04:54.815
So 63 divided by seven, once again,

00:04:55.963 --> 00:04:58.796
that is going to be equal to nine.


Division strategies for decimal quotients

WEBVTT
Kind: captions
Language: en

00:00:00.207 --> 00:00:01.788
- [Instructor] In this
video, we're gonna come up

00:00:01.788 --> 00:00:03.951
with some strategies for division

00:00:03.951 --> 00:00:06.908
when the quotient isn't a whole number,

00:00:06.908 --> 00:00:08.682
when it's going to be a decimal,

00:00:08.682 --> 00:00:12.432
so let's try to compute
three divided by two.

00:00:14.101 --> 00:00:16.820
Pause the video and see
if you can figure out

00:00:16.820 --> 00:00:18.930
what that is going to be,
and I'll give you a hint,

00:00:18.930 --> 00:00:22.140
it's not going to be a whole number.

00:00:22.140 --> 00:00:24.709
Alright, now let's try
to work this together.

00:00:24.709 --> 00:00:26.488
And like all things in mathematics,

00:00:26.488 --> 00:00:28.619
there's multiple
strategies that you can use

00:00:28.619 --> 00:00:33.036
to try to figure out what
three divided by two is.

00:00:33.036 --> 00:00:35.605
One strategy is, well,
let's just rewrite this

00:00:35.605 --> 00:00:38.317
as a fraction, so three divided by two,

00:00:38.317 --> 00:00:40.589
you can write that as,

00:00:40.589 --> 00:00:43.839
you could write that as three over two.

00:00:47.975 --> 00:00:51.597
Now, or you could say, this
is the same thing as 3/2.

00:00:51.597 --> 00:00:54.845
But how can we express this as a decimal?

00:00:54.845 --> 00:00:57.458
Well, you might recognize that 3/2

00:00:57.458 --> 00:01:01.625
is the same thing as 2/2 plus 1/2,

00:01:03.042 --> 00:01:04.720
let me write that down.

00:01:04.720 --> 00:01:09.550
So this is the same thing
as two plus one over two,

00:01:09.550 --> 00:01:11.064
and I'm really doing every step here

00:01:11.064 --> 00:01:13.386
to hopefully make things clear,

00:01:13.386 --> 00:01:16.719
which is the same thing as two over two,

00:01:18.506 --> 00:01:20.423
so that's two over two,

00:01:21.698 --> 00:01:24.948
plus one over two, plus 1/2.

00:01:26.567 --> 00:01:30.047
I could break this up into
two over two plus 1/2.

00:01:30.047 --> 00:01:32.216
Now, two over two is just one,

00:01:32.216 --> 00:01:36.486
and so this is going to be equal to 1 1/2.

00:01:36.486 --> 00:01:38.468
Now, you might immediately say,

00:01:38.468 --> 00:01:40.787
"Hey, 1/2, I could write that as 5/10,"

00:01:40.787 --> 00:01:42.036
and that would be exactly right.

00:01:42.036 --> 00:01:45.176
You could just, we don't
wanna spell out every step,

00:01:45.176 --> 00:01:47.196
we could say this is equal to one,

00:01:47.196 --> 00:01:49.277
and when we write it in decimal form,

00:01:49.277 --> 00:01:52.696
we express things as tenths
or hundredths or thousandths,

00:01:52.696 --> 00:01:56.542
so 1/2 is the same thing as 5/10,

00:01:56.542 --> 00:01:58.864
and if we wanna express that as a decimal,

00:01:58.864 --> 00:02:02.447
this would be equal to
one and five tenths.

00:02:05.903 --> 00:02:08.701
Now I did every step here,
as you get more practice,

00:02:08.701 --> 00:02:11.310
you say okay this is
the same thing as 3/2.

00:02:11.310 --> 00:02:14.689
3/2, two goes into three one time,

00:02:14.689 --> 00:02:16.067
and there's a 1/2 left over,

00:02:16.067 --> 00:02:19.625
so writing this as a
mixed number, it's 1 1/2,

00:02:19.625 --> 00:02:23.792
and 1/2 written as a decimal
is 0.50, so this is 1.50.

00:02:25.305 --> 00:02:28.414
Now another way that we
could've thought about this is

00:02:28.414 --> 00:02:29.940
"Okay, I'm not getting a whole number,

00:02:29.940 --> 00:02:31.850
"when I divide three divided by two.

00:02:31.850 --> 00:02:34.512
"Maybe I'll get something
in terms of tenths,

00:02:34.512 --> 00:02:38.921
"so let me express each of
these in terms of tenths."

00:02:38.921 --> 00:02:41.321
So three is how many tenths?

00:02:41.321 --> 00:02:43.404
Well, three is 30 tenths,

00:02:45.906 --> 00:02:48.239
and we'd be dividing by two,

00:02:49.129 --> 00:02:52.380
we're gonna be dividing it by two,

00:02:52.380 --> 00:02:54.560
so 30 tenths divided by two,

00:02:54.560 --> 00:02:57.430
well that's going to
be equal to 15 tenths.

00:02:57.430 --> 00:03:01.180
This is equal to 15
tenths, which is equal to

00:03:02.443 --> 00:03:05.360
10 tenths and five tenths, or 1.50.

00:03:07.329 --> 00:03:10.261
So both of these are equally
legitimate strategies

00:03:10.261 --> 00:03:13.378
for figuring out what
three divided by two is.

00:03:13.378 --> 00:03:14.699
I like the first one a little bit,

00:03:14.699 --> 00:03:17.099
it leverages what we know about fractions,

00:03:17.099 --> 00:03:20.001
but let's do another example.

00:03:20.001 --> 00:03:23.070
Let's do a few more examples, this is fun.

00:03:23.070 --> 00:03:26.737
Let's figure out what
34 divided by four is,

00:03:30.890 --> 00:03:33.101
and like before, pause this
video and try to figure it out

00:03:33.101 --> 00:03:36.151
and try to see if you can
use some of the strategies

00:03:36.151 --> 00:03:38.734
that we used in the last video.

00:03:39.852 --> 00:03:41.762
Alright, so as we just said,

00:03:41.762 --> 00:03:44.182
we can re-express this as a fraction,

00:03:44.182 --> 00:03:47.932
this is the same thing
as 34 divided by four,

00:03:49.460 --> 00:03:52.710
34 divided by four, or 34/4.

00:03:54.371 --> 00:03:56.611
Now what is this going to be equal to?

00:03:56.611 --> 00:04:00.001
Well, four goes into 34 eight times,

00:04:00.001 --> 00:04:02.172
it's gonna go eight times,

00:04:02.172 --> 00:04:04.370
and you're gonna have two left over,

00:04:04.370 --> 00:04:06.724
so this is eight and 2/4.

00:04:06.724 --> 00:04:09.165
Eight, let me write it both out.

00:04:09.165 --> 00:04:12.473
Alright, eight and 2/4,

00:04:12.473 --> 00:04:16.521
let me do that two in
that blue just for fun.

00:04:16.521 --> 00:04:18.521
Eight and two over four,

00:04:21.757 --> 00:04:24.346
8 2/4, so how do I do this again?

00:04:24.346 --> 00:04:26.807
I said four goes into 34 eight times,

00:04:26.807 --> 00:04:27.939
and then I have two left over,

00:04:27.939 --> 00:04:29.591
so I'm gonna have 2/4 left over.

00:04:29.591 --> 00:04:31.540
Another way, if you
wanna see all the steps,

00:04:31.540 --> 00:04:35.637
you can say, "Hey I can
rewrite this as 32 over four

00:04:35.637 --> 00:04:37.887
"plus two over four."

00:04:37.887 --> 00:04:42.054
The 32 over four is our eight, so 8 2/4.

00:04:43.949 --> 00:04:47.708
Well 2/4, that's the same thing as 1/2.

00:04:47.708 --> 00:04:49.578
That's the same thing as 1/2,

00:04:49.578 --> 00:04:52.500
and if we wanna express
that in terms of tenths,

00:04:52.500 --> 00:04:54.750
this is equal to eight and,

00:04:56.390 --> 00:04:59.031
1/2 is the same thing as 5/10,

00:04:59.031 --> 00:05:02.192
8 5/10, which if we wanna
express as a decimal

00:05:02.192 --> 00:05:05.192
is of course eighth and five tenths,

00:05:06.139 --> 00:05:08.222
or 8.50, and we are done.

00:05:11.418 --> 00:05:12.591
Let's do another one of these,

00:05:12.591 --> 00:05:13.880
and actually let's do one of them

00:05:13.880 --> 00:05:16.349
where we're dividing into a decimal,

00:05:16.349 --> 00:05:20.056
where a decimal itself is being divided.

00:05:20.056 --> 00:05:23.263
Let's say we wanted to calculate

00:05:23.263 --> 00:05:25.013
8.4 divided by seven.

00:05:29.970 --> 00:05:33.569
Pause this video and see
if you can figure it out.

00:05:33.569 --> 00:05:35.288
When you look at this,
you might immediately say

00:05:35.288 --> 00:05:38.338
well I know 84 is divisible by seven.

00:05:38.338 --> 00:05:40.118
We know if you know
your seven times tables,

00:05:40.118 --> 00:05:43.936
we know that seven
times 12 is equal to 84,

00:05:43.936 --> 00:05:47.269
or that 84 divided seven is equal to 12,

00:05:48.655 --> 00:05:51.704
but this isn't 84, this is 8.4,

00:05:51.704 --> 00:05:53.563
so how do we think about it?

00:05:53.563 --> 00:05:56.490
Well one way, we can think
about it in terms of tenths,

00:05:56.490 --> 00:05:59.407
8.4 is the same thing as 84 tenths,

00:06:01.432 --> 00:06:04.265
and so 84 tenths divided by seven,

00:06:06.044 --> 00:06:08.273
well 84 of anything divided by seven

00:06:08.273 --> 00:06:10.223
is going to be 12 of that thing,

00:06:10.223 --> 00:06:12.473
so it's gonna be 12 tenths,

00:06:13.555 --> 00:06:17.893
and 12 tenths we can
rewrite as one and 2/10,

00:06:17.893 --> 00:06:19.643
1.20 and we are done,

00:06:22.841 --> 00:06:25.444
so this is equal to 1.2.

00:06:25.444 --> 00:06:28.311
Another way that we could
have thought about this

00:06:28.311 --> 00:06:30.422
is we could've said, you know what,

00:06:30.422 --> 00:06:33.922
84 tenths is the same thing as 84 over 10.

00:06:35.492 --> 00:06:38.894
If fact, you would read this as 84/10,

00:06:38.894 --> 00:06:41.200
and now you wanna divided this by seven.

00:06:41.200 --> 00:06:45.701
So you wanna divide this by
seven, this is the same thing,

00:06:45.701 --> 00:06:46.669
when you divide by something,

00:06:46.669 --> 00:06:48.708
it's the same as multiplying
by the reciprocal,

00:06:48.708 --> 00:06:52.125
so it's 84 over ten times 1/7,

00:06:55.035 --> 00:06:58.733
which is equal to, we can
change, let me write it this way.

00:06:58.733 --> 00:07:02.066
This is equal to 84 over 10 times seven,

00:07:04.333 --> 00:07:09.164
over ten times seven, and
now we could simplify this.

00:07:09.164 --> 00:07:11.104
If we divide the numerator
and the denominator by seven,

00:07:11.104 --> 00:07:15.637
84 divided by seven is 12,
seven divided by seven is one.

00:07:15.637 --> 00:07:19.358
12 divided by ten, this is
gonna be equal to 12/10,

00:07:19.358 --> 00:07:23.525
which is 1.2, we could write
this as 1 2/10, or 1.2.

00:07:26.917 --> 00:07:30.739
Let's do one more example
that's kind of related.

00:07:30.739 --> 00:07:35.609
Let's say we wanna figure out
what seven divided by 70 is.

00:07:35.609 --> 00:07:38.761
Pause this video and try to figure it out.

00:07:38.761 --> 00:07:40.630
Well, we can rewrite this,

00:07:40.630 --> 00:07:43.797
as we've been doing, as seven over 70.

00:07:46.719 --> 00:07:47.691
Instead of writing it as 70,

00:07:47.691 --> 00:07:50.914
let me write that as seven times ten,

00:07:50.914 --> 00:07:52.072
and what's valuable about this is

00:07:52.072 --> 00:07:55.412
we can divide the numerator
and the denominator by seven.

00:07:55.412 --> 00:07:57.412
If we divide the numerator
by seven, we get a one,

00:07:57.412 --> 00:07:58.974
we divide the denominator
by seven, we get a one.

00:07:58.974 --> 00:08:00.001
Remember, we can do the same thing

00:08:00.001 --> 00:08:00.951
to the numerator and the denominator.

00:08:00.951 --> 00:08:02.631
If we multiply or divide
by the same value,

00:08:02.631 --> 00:08:05.750
we're not changing the value
of the actual fraction.

00:08:05.750 --> 00:08:08.311
And so you're left with 1/10,

00:08:08.311 --> 00:08:09.951
which if you express it as a decimal,

00:08:09.951 --> 00:08:10.999
you go to the tenths place,

00:08:10.999 --> 00:08:13.756
you say I have one of those
tenths, that's one tenth,

00:08:13.756 --> 00:08:16.082
so this is one tenth.

00:08:16.082 --> 00:08:18.381
Another way to think about it is

00:08:18.381 --> 00:08:21.053
this is the same thing as,

00:08:21.053 --> 00:08:23.333
and really this is what
we wrote over here,

00:08:23.333 --> 00:08:27.500
but you could write this
as seven divided by seven,

00:08:29.297 --> 00:08:33.464
divided by the blue seven,
and then you divide by 10.

00:08:34.675 --> 00:08:36.482
If you're dividing by seven times ten,

00:08:36.482 --> 00:08:38.361
actually let me write that down.

00:08:38.361 --> 00:08:41.528
If you're dividing by seven times ten,

00:08:42.695 --> 00:08:44.445
and this essentially
another way of writing

00:08:44.445 --> 00:08:47.455
what we have over here,
this is going to be equal to

00:08:47.455 --> 00:08:50.538
seven divided by seven divided by 10.

00:08:56.043 --> 00:08:58.312
Well, seven divided by seven is one,

00:08:58.312 --> 00:09:02.395
so you get one divided by
10, which is one tenth.