How to determine the domain of a modeling function

WEBVTT
Kind: captions
Language: en

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- [Voiceover] This right
over here is a screenshot

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from a Khan Academy exercise, and it says,

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"Mason stands on the 5th
step of a vertical ladder.

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"The ladder has 15 steps,
and the height difference

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"between consecutive steps is 0.5 meters.

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"He is thinking about moving
up, down, or staying put."

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Let me draw this ladder that Mason is on.

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It's a vertical ladder,

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that's one side of the ladder,

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this is the other side of the ladder,

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and it has 15 steps.

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Let me see if I can draw that.

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This is the first one, two, three, four.

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I'm gonna run out of space,

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I need to make 'em closer together.

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It's gonna be one, two, three, four,

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five, six, seven, eight, nine,

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10, 11, 12, 13,

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14, and 15.

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15 steps.

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Let me make sure it's even.

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The top and the bottom,

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and the distance between each of these,

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I guess you could say, steps,
or the rungs of the ladder,

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are half a meter.

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This distance right
over here is 0.5 meters.

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And it says that he's on the 5th step

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of this vertical ladder.

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He's on the 5th step,

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One, two, three, four, five.

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This is where he is right
now. He's on this 5th step.

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And he's thinking about moving up,

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or down, or staying put.

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"Let h of n denote the
height above the ground h

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"of Mason's feet (measured in meters)

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"after moving n steps (if Mason went

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"down the ladder, n is negative.)"

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All right, h of n.

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Denote the height above the ground

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after moving n steps.

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Make sure we understand this.

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If I were to say h of zero,

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what is that going to be?

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Well, h of zero means that
he's moved zero steps.

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He's moved zero steps,
he's still going to be

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on this 5th step of the ladder.

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And so how high is he going to be?

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If he's on the 5th step
of a vertical ladder...

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I'm assuming that there's
0.5 from the ground.

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This is the ground right over here.

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He is one, two, three, four, five steps,

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each of 'em is half a meter.

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Five times 0.5 is going

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to give us 2.5 meters,

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so h of zero is 2.5 meters.

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If I said h of one, that means he goes up.

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H of one means he goes up one step.

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Here, n would be equal to one.

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If he goes up one step, h of one,

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he's going to half a meter higher,

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so it's going to be equal to three meters.

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We could keep doing that for
a bunch of different inputs.

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Let me write that, that's going
to be equal to three meters.

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But anyway, that's not what
they're asking us about.

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They're saying, "Which number
type is more appropriate

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"for the domain of the function?"

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The domain, just as
reviewed, that's the set

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of numbers that we could
input into the function

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and get a valid output.

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And it's clear here, see, we have

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to pick between integers or real numbers.

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Well, n, which is our input,

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that's the number of
steps he goes up or down.

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It could be positive or negative,

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but we're not gonna talk about half steps.

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Then he'll put his foot
in the air, right over.

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He has to take integer
valued steps up or down.

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Or, I guess, he's taking
integer valued steps,

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if it's positive it's up,
if it's negative it's down,

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if it's zero that means he's staying put.

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If n is zero, that means he's staying put.

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It's not real numbers.

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He can't move pi steps from where he is.

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He can't move square root of
two steps from where he is.

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He can't even move 0.25 steps,

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then he'd put his foot in the air.

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This is definitely going
to be about the integers,

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not the real numbers.

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This function right over
here, the valid inputs,

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I want to be able to input an integer.

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In fact, it's not even all integers,

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because he can't go down
an arbitrary amount.

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In fact, he can't go up an
arbitrary amount either.

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The domain is going to
be a subset of integers.

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Then they say, "Define the
interval of the domain."

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And we have these little toggles here...

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to define the interval of the domain.

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And let's see, the lowest value for n,

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he can go as far as one, two, three,

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four, five steps down.

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In that case, n would be
equal to negative five.

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And then the highest value for n is

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if he takes one, two,
three, four, five, six,

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seven, eight, nine, 10 steps up.

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And so that would be n is equal to 10.

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The interval of the domain,

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and actually I just copy and pasted this

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onto my scratch pad,

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n can be as low as negative
five, and as high as 10,

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and it can include them as well,

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so I'm gonna use brackets.

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My domain would include negative five.

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If it didn't include negative five,

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I would put a parentheses,
but I could put brackets here,

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and I could put brackets there as well.

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Just for fun, let me actually input it

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into the actual exercise.

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I'm saying integers, and I'm as low,

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and I can go down five steps,

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and I can go up 10 steps,

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and 10 is also included in my interval.

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Then I can check my
answer, and I got it right.


Equations of proportional relationships

WEBVTT
Kind: captions
Language: en

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- So, let's set up a relationship

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between the variables x and y.

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So, let's say, so this is x and this is y,

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and when x is one, y is four,

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and when x is two, y is eight,

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and when x is three, y is 12.

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Now, you might immediately recognize

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that this is a proportional relationship.

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And remember, in order for it to be

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a proportional relationship, the ratio

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between the two variables
is always constant.

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So, for example, if I look
at y over x here, we see

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that y over x, here it's four
over one, which is just four.

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Eight over two is just four.

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Eight halves is the same thing as four.

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12 over three it's the same thing as four.

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Y over x is always equal to four.

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In fact, I can make another column here.

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I can make another column
here where I have y over x,

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here it's four over one,
which is equal to four.

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Here it's eight over two,
which is equal to four.

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Here it's 12 over three,
which is equal to four.

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And so, you can actually
use this information,

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the ratio, the ratio between
y and x is this constant four,

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to express the relationship
between y and x as an equation.

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In fact, in some ways this
is, or in a lot of ways,

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this is already an
equation, but I can make it

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a little bit clearer, if I
multiply both sides by x.

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If I multiply both sides by x,

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if I multiply both sides
by x, I am left with, well,

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x divided by x, you'd just
have y on the left hand side.

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Y is equal to 4x and
you see that's the case.

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X is one, four times that is four.

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X is two, four times that is eight.

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So, here you go, we're
multiplying by four.

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We are multiplying by four,
we are multiplying by four.

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And so, four, in this
case, four, in this case,

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in this situation, this is our
constant of proportionality.

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Constant, constant, sometimes people

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will say proportionality constant.

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Constant of proportionality, portionality.

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Now sometimes, it might even be described

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as a rate of change and
you're like well, Sal,

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how is this a, how would
four be a rate of change?

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And, to make that a little bit clearer,

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let me actually do another example,

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but this time, I'll actually
put some units there.

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So let's say that, let's say
that I have, let's say that x--

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Let me do this, I already
used yellow, let me use blue.

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So let's x, let's say
that's a measure of time

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and y is a measure of distance.

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Or, let me put it this way, x
is time in terms of seconds.

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Let me write it this way.

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So, x, x is going to
be in seconds and then,

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y is going to be in meters.

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So, this is meters, the units,

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and this right over here is seconds.

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So, after one second, we have traveled,

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oh, I don't know, seven meters.

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After two seconds, we've
traveled 14 meters.

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After three seconds,
we've traveled 21 meters,

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and you can verify that this
is a proportional relationship.

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The ratio between y and x is always seven.

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Seven over one, 14 over
two, 21 over three.

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But, I wanna write that
in terms of it's units.

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So, y over x is going to
be, if we look at this point

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right over here, it's seven
meters over one second.

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Seven meters over one second,

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or it's equal to seven meters per second.

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If you look at it right over
here, if you say y over x,

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it's 14 meters, 14 meters, in
two seconds, in two seconds.

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Well, 14 over two is seven,
14 over two is seven,

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and then the units are meters per second.

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So, that's why this constant, this seven,

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in all of these cases we have
y over x is equal to seven,

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that this is also sometimes
considered a rate.

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And over here it's very clear,
this is my distance per time.

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Now, if I wanted to write it generally,

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I could say that, look, if I'm dealing

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with a proportional relationship,

00:04:14.401 --> 00:04:16.624
it's going to be of the
form, I can always construct

00:04:16.624 --> 00:04:19.863
and equation of the form, of the form,

00:04:19.863 --> 00:04:23.888
either y over x is equal to k,

00:04:23.888 --> 00:04:25.494
where k is some constant.

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In this first example, k was equal to four

00:04:27.758 --> 00:04:31.035
and in this second example,
k is equal to seven.

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Or, you can just manipulate
it algebraically,

00:04:33.450 --> 00:04:38.450
multiply both sides by x and
you would have y is equal to,

00:04:38.724 --> 00:04:43.020
y is equal to kx, where
once again k is our constant

00:04:43.020 --> 00:04:46.339
of proportionality or
proportionality constant.

00:04:46.339 --> 00:04:49.581
So, this is a really, in some
ways it's a very simple idea,

00:04:49.581 --> 00:04:52.326
but in a lot of ways, you'll
see this showing up multiple,

00:04:52.326 --> 00:04:54.764
many, many times in
your mathematical career

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and it's neat to be able to recognize this

00:04:56.679 --> 00:04:59.854
as a proportional relationship.


Proportion word problem (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.225 --> 00:00:05.225
- We're told that Mika can
eat 21 hot dogs in 66 minutes.

00:00:05.623 --> 00:00:09.419
She wants to know how many
minutes, m, would it take her

00:00:09.419 --> 00:00:14.419
to eat 35 hot dogs, if she
can keep up the same pace.

00:00:14.775 --> 00:00:18.432
So, big clue is the same
pace, I have to remove a hair

00:00:18.432 --> 00:00:20.266
from my tongue, alright. (laughs)

00:00:20.266 --> 00:00:23.147
A big clue is the same pace.

00:00:23.147 --> 00:00:28.147
That means that the hot
dogs, hot dogs per minute,

00:00:29.532 --> 00:00:33.754
per minute, is going to be
constant, is always going to be

00:00:33.754 --> 00:00:37.024
the same, always the same.

00:00:37.024 --> 00:00:40.270
Always the same, because
this is essentially the pace.

00:00:40.270 --> 00:00:42.425
Her hot dogs per minute
are going to stay the same.

00:00:42.425 --> 00:00:44.545
She's gonna stay at the same pace.

00:00:44.545 --> 00:00:47.180
So, we it tells us that
she can eat 21 hot dogs

00:00:47.180 --> 00:00:49.397
in 66 minutes.

00:00:49.397 --> 00:00:52.184
So, her hot dogs per
minute, at least up here,

00:00:52.184 --> 00:00:55.016
is 21 hot dogs in 66 minutes,

00:00:55.016 --> 00:00:58.857
So, it's 21 hot dogs in 66 minutes.

00:00:58.857 --> 00:01:01.942
Well, if her pace is always
going to be the same,

00:01:01.942 --> 00:01:05.181
well it's gonna take her
this ratio over here,

00:01:05.181 --> 00:01:09.604
is going to be the ratio
between 35 hot dogs

00:01:09.604 --> 00:01:14.066
and however long it takes
her to eat 35 hot dogs.

00:01:14.066 --> 00:01:16.984
So, once again hot dogs
per minute are going,

00:01:16.984 --> 00:01:19.921
has to be a constant because
it's gonna be the same pace.

00:01:19.921 --> 00:01:21.546
Hot dogs per minute.

00:01:21.546 --> 00:01:25.037
If it takes, 21 hot dogs
takes 66 minutes, 35 hot dogs

00:01:25.037 --> 00:01:28.538
take m minutes, these two
ratios are going to be the same.

00:01:28.538 --> 00:01:31.546
We're dealing with a
proportional relationship.

00:01:31.546 --> 00:01:34.007
It's going to be happening
at the same rate.

00:01:34.007 --> 00:01:35.794
And then, we're left with a situation

00:01:35.794 --> 00:01:39.851
where we just have to solve
for m, and there is a bunch

00:01:39.851 --> 00:01:42.498
of different ways you could tackle this.

00:01:42.498 --> 00:01:44.988
The easiest way, that I
can think of doing it is,

00:01:44.988 --> 00:01:47.662
I like, I don't like this m
sitting here in the denominator,

00:01:47.662 --> 00:01:50.971
so let's multiply both sides by m.

00:01:50.971 --> 00:01:52.837
Let me do that in a different color.

00:01:52.837 --> 00:01:57.819
So, if I multiply that side
by m and then this side by m,

00:01:57.979 --> 00:01:59.337
and so what do we get?

00:01:59.337 --> 00:02:03.388
On the left hand side,
we have 21 over 66m.

00:02:03.388 --> 00:02:08.378
21 over 66 times m, times m,

00:02:08.837 --> 00:02:12.011
is equal to, is equal to,

00:02:12.011 --> 00:02:14.391
well, you divide by m and multiply by m.

00:02:14.391 --> 00:02:17.788
Those are gonna cancel out
and you're just gonna have 35.

00:02:17.788 --> 00:02:20.469
And now, you just have to
solve for m and the best way

00:02:20.469 --> 00:02:22.965
I can think of doing that,
is multiply both sides

00:02:22.965 --> 00:02:26.227
times the reciprocal on, both
sides times the reciprocal

00:02:26.227 --> 00:02:28.079
of the coefficient on the m.

00:02:28.079 --> 00:02:30.749
So, let's multiply both sides by,

00:02:30.749 --> 00:02:34.126
let's multiply both side by 66 over 21.

00:02:34.126 --> 00:02:36.447
Once again, I've just
swapped the numerator

00:02:36.447 --> 00:02:38.518
and the denominator here
to get the reciprocal,

00:02:38.518 --> 00:02:40.200
but I can't just do it to
one side of the equation,

00:02:40.200 --> 00:02:41.291
I have to do it to both sides, otherwise,

00:02:41.291 --> 00:02:42.417
it's not going to be an equa,

00:02:42.417 --> 00:02:43.903
it's not gonna be equal anymore.

00:02:43.903 --> 00:02:48.903
So, times 66 over 21, this
is just going to be one.

00:02:49.240 --> 00:02:51.215
You multiply something
times it's reciprocal,

00:02:51.215 --> 00:02:52.921
you're just going to end up with one.

00:02:52.921 --> 00:02:57.402
So, you're gonna be left
with, m is equal to.

00:02:57.402 --> 00:03:01.051
Now, 35 times 66 divided by 21.

00:03:01.051 --> 00:03:04.255
Well, 35 is the same thing as,

00:03:04.255 --> 00:03:07.842
35 is five times seven

00:03:07.842 --> 00:03:12.279
and 21 is three times seven.

00:03:12.279 --> 00:03:14.040
So, you're multiplying by seven up here

00:03:14.040 --> 00:03:16.129
and here, you have a
seven in the denominator,

00:03:16.129 --> 00:03:19.426
you're dividing by seven, so
they're going to cancel out.

00:03:19.426 --> 00:03:24.426
So, this is going to simplify
to five times 66 over 3,

00:03:27.134 --> 00:03:29.578
and then we could simplify it even more,

00:03:29.578 --> 00:03:34.198
because 66 is the same
thing as three times 22.

00:03:34.198 --> 00:03:38.041
Three times 22 and so, you
have a three in the numerator,

00:03:38.041 --> 00:03:39.643
you're multiplying by three
and three in the denominator,

00:03:39.643 --> 00:03:40.641
dividing by three.

00:03:40.641 --> 00:03:42.916
Three divided by three
is one, so you're left

00:03:42.916 --> 00:03:47.916
with five times 22, which is 110.

00:03:48.331 --> 00:03:51.724
So, it would take her m
minutes to eat 35 hot dogs

00:03:51.724 --> 00:03:53.326
at the same pace.

00:03:53.326 --> 00:03:54.736
Now, when some of you
might have tackled it,

00:03:54.736 --> 00:03:57.615
you might have had a different
equation set up here.

00:03:57.615 --> 00:03:59.716
Instead of thinking of
hot dogs per minute,

00:03:59.716 --> 00:04:01.852
you might have thought
about minutes per hot dog.

00:04:01.852 --> 00:04:04.210
And so, in that situation,
if you thought in terms

00:04:04.210 --> 00:04:06.633
of minutes per hot dog, you
might have said, ok look,

00:04:06.633 --> 00:04:11.143
it took Mika 66 minutes to eat 20,

00:04:11.143 --> 00:04:15.370
to eat 21 hot dogs,and it's
gonna take her m minutes

00:04:15.370 --> 00:04:19.777
to eat 35 hot dogs and
if it's the same pace,

00:04:19.777 --> 00:04:22.374
then these two rates
are going to be equal.

00:04:22.374 --> 00:04:24.346
They have to be the same pace.

00:04:24.346 --> 00:04:26.273
And so, then you can
solve for m and actually,

00:04:26.273 --> 00:04:27.639
this one's easier to solve for m,

00:04:27.639 --> 00:04:30.460
you just multiply both sides by 35.

00:04:30.460 --> 00:04:34.431
Multiply both sides by
35 and you're left with,

00:04:34.431 --> 00:04:36.632
on the right hand side
you're left with just an m,

00:04:36.632 --> 00:04:39.906
and on the left hand
side, same, same idea.

00:04:39.906 --> 00:04:44.433
You're taking 35, you have 35 times 66/21,

00:04:44.433 --> 00:04:46.651
which we already figured out is 110.

00:04:46.651 --> 00:04:50.223
So, 110 is equal, is equal to m.

00:04:50.223 --> 00:04:51.918
So, once again, multiple
ways to tackle it,

00:04:51.918 --> 00:04:56.080
but it's important that
we got the same answer.


Intervals and interval notation

WEBVTT
Kind: captions
Language: en

00:00:00.503 --> 00:00:02.023
- [Voiceover] What I hope
to do in this video is get

00:00:02.023 --> 00:00:04.868
familiar with the notion of an interval,

00:00:04.868 --> 00:00:07.329
and also think about ways
that we can show an interval,

00:00:07.329 --> 00:00:09.349
or interval notation.

00:00:09.349 --> 00:00:12.321
Right over here I have a number line.

00:00:12.321 --> 00:00:15.142
Let's say I wanted to talk
about the interval on the

00:00:15.142 --> 00:00:20.142
number line that goes from
negative three to two.

00:00:20.807 --> 00:00:24.331
So I care about this-- Let
me use a different color.

00:00:24.440 --> 00:00:26.925
Let's say I care about this
interval right over here.

00:00:26.925 --> 00:00:31.824
I care about all the numbers
from negative three to two.

00:00:31.824 --> 00:00:35.039
So in order to be more
precise, I have to be clear.

00:00:35.039 --> 00:00:37.907
Am I including negative three and two,

00:00:37.907 --> 00:00:40.287
or am I not including
negative three and two,

00:00:40.287 --> 00:00:42.562
or maybe I'm just including one of them.

00:00:42.562 --> 00:00:45.696
So if I'm including
negative three and two,

00:00:45.696 --> 00:00:47.519
then I would fill them in.

00:00:47.519 --> 00:00:51.037
So this right over here, I'm
filling negative three and

00:00:51.037 --> 00:00:53.625
two in, which means that
negative three and two

00:00:53.625 --> 00:00:55.378
are part of this interval.

00:00:55.378 --> 00:00:57.514
And when you include the endpoints,

00:00:57.514 --> 00:00:59.910
this is called a closed interval.

00:00:59.910 --> 00:01:02.156
Closed interval.

00:01:03.760 --> 00:01:05.594
And I just showed you how I can depict it

00:01:05.594 --> 00:01:09.896
on a number line, by actually
filling in the endpoints

00:01:09.896 --> 00:01:12.066
and there's multiple ways to talk about

00:01:12.066 --> 00:01:14.074
this interval mathematically.

00:01:14.074 --> 00:01:16.709
I could say that this is all of the...

00:01:16.709 --> 00:01:19.292
Let's say this number line

00:01:19.292 --> 00:01:21.213
is showing different values for x.

00:01:21.213 --> 00:01:24.418
I could say these are all of the x's

00:01:24.418 --> 00:01:29.418
that are between negative three and two.

00:01:31.322 --> 00:01:35.620
And notice, I have negative
three is less than or equal to x

00:01:35.620 --> 00:01:39.080
so that's telling us
that x could be equal to,

00:01:39.080 --> 00:01:41.924
that x could be equal to negative three.

00:01:41.924 --> 00:01:45.616
And then we have x is less
than or equal to positive two,

00:01:45.616 --> 00:01:47.984
so that means that x could
be equal to positive two,

00:01:47.984 --> 00:01:49.952
so it is a closed interval.

00:01:49.981 --> 00:01:52.836
Another way that we could
depict this closed interval

00:01:52.836 --> 00:01:56.296
is we could say, okay, we're
talking about the interval

00:01:56.296 --> 00:01:58.502
between, and we can use brackets

00:01:58.502 --> 00:02:00.405
because it's a closed interval,

00:02:00.405 --> 00:02:03.505
negative three and two, and once again

00:02:03.505 --> 00:02:06.500
I'm using brackets here,
these brackets tell us

00:02:06.500 --> 00:02:08.821
that we include, this
bracket on the left says that

00:02:08.821 --> 00:02:11.875
we include negative three,
and this bracket on the right

00:02:11.875 --> 00:02:16.078
says that we include
positive two in our interval.

00:02:16.078 --> 00:02:17.691
Sometimes you might see things

00:02:17.691 --> 00:02:19.105
written a little bit more math-y.

00:02:19.105 --> 00:02:24.105
You might see x is a member of
the real numbers such that...

00:02:25.705 --> 00:02:28.381
And I could put these curly
brackets around like this.

00:02:28.381 --> 00:02:29.356
These curly brackets say that

00:02:29.356 --> 00:02:31.817
we're talking about a set of values,

00:02:31.817 --> 00:02:33.698
and we're saying that the set of all x's

00:02:33.698 --> 00:02:35.973
that are a member of the real number,

00:02:35.973 --> 00:02:37.593
so this is just fancy math notation,

00:02:37.593 --> 00:02:39.694
it's a member of the real numbers.

00:02:39.694 --> 00:02:42.747
I'm using the Greek letter
epsilon right over here.

00:02:42.747 --> 00:02:45.069
It's a member of the
real numbers such that.

00:02:45.069 --> 00:02:48.812
This vertical line here means "such that,"

00:02:48.812 --> 00:02:50.870
negative three is less x is less than--

00:02:50.870 --> 00:02:53.409
negative three is less than or equal to x,

00:02:53.409 --> 00:02:55.464
is less than or equal to two.

00:02:55.464 --> 00:02:56.590
I could also write it this way.

00:02:56.590 --> 00:03:00.444
I could write x is a
member of the real numbers

00:03:00.444 --> 00:03:05.444
such that x is a member,
such that x is a member

00:03:05.471 --> 00:03:10.471
of this closed set, I'm
including the endpoints here.

00:03:10.625 --> 00:03:14.409
So these are all
different ways of denoting

00:03:14.409 --> 00:03:17.091
or depicting the same interval.

00:03:17.091 --> 00:03:18.844
Let's do some more examples here.

00:03:18.844 --> 00:03:22.147
So let's-- Let me draw
a number line again.

00:03:22.147 --> 00:03:24.444
So, a number line.

00:03:24.444 --> 00:03:27.690
And now let me do-- Let me
just do an open interval.

00:03:27.690 --> 00:03:29.317
An open interval just so that

00:03:29.317 --> 00:03:32.367
we clearly can see the difference.

00:03:32.628 --> 00:03:34.692
Let's say that I want to talk about

00:03:34.692 --> 00:03:37.774
the values between negative one and four.

00:03:38.047 --> 00:03:40.293
Let me use a different color.

00:03:40.426 --> 00:03:44.711
So the values between
negative one and four,

00:03:44.711 --> 00:03:48.252
but I don't want to include
negative one and four.

00:03:48.252 --> 00:03:49.552
So this is going to be an open interval.

00:03:49.552 --> 00:03:51.351
So I'm not going to include four,

00:03:51.351 --> 00:03:55.426
and I'm not going to include negative one.

00:03:55.426 --> 00:03:57.748
Notice I have open circles here.

00:03:57.748 --> 00:04:00.177
Over here had closed circles,
the closed circles told me

00:04:00.177 --> 00:04:02.448
that I included negative three and two.

00:04:02.448 --> 00:04:05.401
Now I have open circles here,
so that says that I'm not,

00:04:05.401 --> 00:04:07.873
it's all the values in
between negative one and four.

00:04:08.863 --> 00:04:11.856
Negative .999999 is going to be included,

00:04:11.856 --> 00:04:14.364
but negative one is not
going to be included.

00:04:14.364 --> 00:04:17.278
And 3.9999999 is going to be included,

00:04:17.278 --> 00:04:20.062
but four is not going to be included.

00:04:20.062 --> 00:04:21.217
So how would we--

00:04:21.217 --> 00:04:23.590
What would be the notation for this?

00:04:23.590 --> 00:04:27.165
Well, here we could say
x is going to be a member

00:04:27.165 --> 00:04:31.542
of the real numbers
such that negative one--

00:04:31.542 --> 00:04:32.889
I'm not going to say less than or equal to

00:04:32.889 --> 00:04:35.907
because x can't be equal to negative one,

00:04:35.907 --> 00:04:39.066
so negative one is strictly less than x,

00:04:39.066 --> 00:04:41.272
is strictly less than four.

00:04:41.272 --> 00:04:43.070
Notice not less than or
equal, because I can't

00:04:43.070 --> 00:04:45.852
be equal to four, four is not included.

00:04:45.950 --> 00:04:47.725
So that's one way to say it.

00:04:47.725 --> 00:04:49.989
Another way I could write it like this.

00:04:49.989 --> 00:04:54.355
x is a member of the
real numbers such that

00:04:54.355 --> 00:04:56.885
x is a member of...

00:04:56.885 --> 00:05:00.705
Now the interval is from
negative one to four

00:05:00.705 --> 00:05:02.307
but I'm not gonna use these brackets.

00:05:02.307 --> 00:05:04.767
These brackets say, "Hey,
let me include the endpoint,"

00:05:04.767 --> 00:05:05.906
but I'm not going to include them,

00:05:05.906 --> 00:05:08.251
so I'm going to put the
parentheses right over here.

00:05:08.251 --> 00:05:09.412
Parentheses.

00:05:09.412 --> 00:05:12.454
So this tells us that we're
dealing with an open interval.

00:05:12.454 --> 00:05:14.206
This right over here,
let me make it clear,

00:05:14.206 --> 00:05:18.415
this is an open interval.

00:05:18.431 --> 00:05:20.870
Now you're probably
wondering, okay, in this case

00:05:20.870 --> 00:05:23.250
both endpoints were included,
it's a closed interval.

00:05:23.250 --> 00:05:26.465
In this case both endpoints were excluded,

00:05:26.465 --> 00:05:27.545
it's an open interval.

00:05:27.545 --> 00:05:29.449
Can you have things that
have one endpoint included

00:05:29.449 --> 00:05:33.465
and one point excluded, and
the answer is absolutely.

00:05:33.465 --> 00:05:35.392
Let's see an example of that.

00:05:35.392 --> 00:05:37.698
I'll get another number line here.

00:05:37.924 --> 00:05:40.744
Another number line.

00:05:40.744 --> 00:05:43.171
And let's say that we want to--

00:05:43.171 --> 00:05:44.170
Actually, let me do it
the other way around.

00:05:44.170 --> 00:05:47.164
Let me write it first,
and then I'll graph it.

00:05:47.164 --> 00:05:49.463
So let's say we're thinking
about all of the x's

00:05:49.463 --> 00:05:53.560
that are a member of the
real numbers such that

00:05:57.160 --> 00:05:59.806
let's say negative four is
not included, is less than x,

00:05:59.806 --> 00:06:02.534
is less than or equal to negative one.

00:06:02.534 --> 00:06:05.146
So now negative one is included.

00:06:05.146 --> 00:06:06.888
So we're not going to
include negative four.

00:06:06.888 --> 00:06:08.189
Negative four is strictly less than,

00:06:08.189 --> 00:06:09.791
not less than or equal to,

00:06:09.791 --> 00:06:13.087
so x can't be equal to negative
four, open circle there.

00:06:13.087 --> 00:06:14.805
But x could be equal to negative one.

00:06:14.805 --> 00:06:17.452
It has to be less than
or equal to negative one.

00:06:17.452 --> 00:06:19.011
It could be equal to negative one

00:06:19.011 --> 00:06:21.437
so I'm going to fill
that in right over there.

00:06:21.437 --> 00:06:23.835
And it's everything in between.

00:06:23.944 --> 00:06:27.926
If I want to write it with
this notation I could write

00:06:27.926 --> 00:06:32.048
x is a member of the
real numbers such that

00:06:32.048 --> 00:06:36.250
x is a member of the interval,

00:06:36.250 --> 00:06:40.487
so it's going to go between
negative four and negative one,

00:06:40.487 --> 00:06:42.414
but we're not including negative four.

00:06:42.414 --> 00:06:43.401
We have an open circle here

00:06:43.401 --> 00:06:45.595
so I'm gonna put a
parentheses on that side,

00:06:45.595 --> 00:06:50.053
but we are including negative one.

00:06:50.053 --> 00:06:52.433
We are including negative one.

00:06:52.433 --> 00:06:55.504
So we put a bracket on that side.

00:06:56.453 --> 00:07:00.095
That right over there
would be the notation.

00:07:00.095 --> 00:07:01.581
Now there's other things that you could do

00:07:01.581 --> 00:07:03.496
with interval notation.

00:07:03.496 --> 00:07:06.480
You could say, well hey,
everything except for some values.

00:07:06.480 --> 00:07:08.435
Let me give another example.

00:07:08.435 --> 00:07:10.555
Let's get another example here.

00:07:10.555 --> 00:07:14.873
Let's say that we wanna talk
about all the real numbers

00:07:14.873 --> 00:07:17.201
except for one.

00:07:17.230 --> 00:07:22.230
We want to include all
of the real numbers.

00:07:22.953 --> 00:07:27.953
All of the real numbers except for one.

00:07:28.874 --> 00:07:31.892
Except for one, so we're gonna
exclude one right over here,

00:07:31.892 --> 00:07:36.628
open circle, but it can
be any other real number.

00:07:36.628 --> 00:07:40.953
So how would we denote this?

00:07:42.242 --> 00:07:47.242
Well, we could write x is a
member of the real numbers

00:07:48.295 --> 00:07:53.295
such that x does not equal one.

00:07:54.994 --> 00:07:57.060
So here I'm saying x can be
a member of the real numbers

00:07:57.060 --> 00:07:58.662
but x cannot be equal to one.

00:07:58.662 --> 00:08:02.226
It can be anything else, but
it cannot be equal to one.

00:08:02.226 --> 00:08:04.757
And there's other ways of
denoting this exact same interval.

00:08:04.757 --> 00:08:09.757
You could say x is a
member of the real numbers

00:08:10.503 --> 00:08:15.503
such that x is less than one,

00:08:17.379 --> 00:08:20.993
or x is greater than one.

00:08:21.288 --> 00:08:24.040
So you could write it just like that.

00:08:24.040 --> 00:08:25.642
Or you could do something interesting.

00:08:25.642 --> 00:08:28.045
This is the one that I would
use, this is the shortest

00:08:28.045 --> 00:08:29.020
and it makes it very clear.

00:08:29.020 --> 00:08:30.587
You say hey, everything except for one.

00:08:30.587 --> 00:08:32.781
But you could even do something
fancy, like you could say

00:08:32.781 --> 00:08:37.123
x is a member of the real
numbers such that x is a member

00:08:37.123 --> 00:08:41.767
of the set going from
negative infinity to one,

00:08:41.767 --> 00:08:46.767
not including one, or x is a
member of the set going from--

00:08:48.198 --> 00:08:50.717
or a member of the
interval going from one,

00:08:50.717 --> 00:08:54.026
not including one, all
the way to positive,

00:08:54.026 --> 00:08:56.440
all the way to positive infinity.

00:08:56.440 --> 00:08:58.820
And when we're talking
about negative infinity

00:08:58.820 --> 00:09:02.338
or positive infinity, you
always put a parentheses.

00:09:02.338 --> 00:09:05.519
And the view there is you
could never include everything

00:09:05.519 --> 00:09:06.935
all the way up to infinity.

00:09:06.935 --> 00:09:09.477
It needs to be at least
open at that endpoint

00:09:09.477 --> 00:09:11.544
because infinity just
keeps going on and on.

00:09:11.544 --> 00:09:13.517
So you always want to put
a parentheses if you're

00:09:13.517 --> 00:09:15.537
talking about infinity
or negative infinity.

00:09:15.537 --> 00:09:16.477
It's not really an endpoint,

00:09:16.477 --> 00:09:18.138
it keeps going on and on forever.

00:09:18.138 --> 00:09:19.821
So you use the notation for open interval,

00:09:19.821 --> 00:09:23.176
at least at that end, and
notice we're not including,

00:09:23.176 --> 00:09:25.532
we're not including one
either, so if x is a member

00:09:25.532 --> 00:09:27.622
of this interval or that interval,

00:09:27.622 --> 00:09:30.339
it essentially could be
anything other than one.

00:09:30.339 --> 00:09:32.359
But this would have been
the simplest notation

00:09:32.359 --> 00:09:34.431
to describe that.


How to visually identify proportional relationships using graphs

WEBVTT
Kind: captions
Language: en

00:00:00.257 --> 00:00:01.418
- [Voiceover] So I have three different

00:00:01.418 --> 00:00:03.705
relationships here between X and Y.

00:00:03.705 --> 00:00:04.866
And what I wanna think about

00:00:04.866 --> 00:00:08.276
which of these, if any, are
proportional relationships.

00:00:08.276 --> 00:00:09.229
And then I wanna graph them to see

00:00:09.229 --> 00:00:11.006
if we can see anything visually

00:00:11.006 --> 00:00:13.455
that makes them obviously proportional.

00:00:13.455 --> 00:00:14.546
And just as a reminder,

00:00:14.546 --> 00:00:16.276
a proportional relationship is one

00:00:16.276 --> 00:00:18.168
where the ratio between the two variables,

00:00:18.168 --> 00:00:20.792
and let's say we took the
ratio between Y and X,

00:00:20.792 --> 00:00:21.988
you could also go the other way around,

00:00:21.988 --> 00:00:23.450
the ratio between X and Y.

00:00:23.450 --> 00:00:25.285
But the ratio between
Y and X is always going

00:00:25.285 --> 00:00:28.315
to be some number, some constant number.

00:00:28.315 --> 00:00:29.919
Or you could rewrite it another way,

00:00:29.919 --> 00:00:31.593
If you were to multiply both sides

00:00:31.593 --> 00:00:33.276
of this equation times X,

00:00:33.276 --> 00:00:34.913
you could say in a
proportional relationship,

00:00:34.913 --> 00:00:39.913
Y is always going to be equal
to some constant times X.

00:00:40.160 --> 00:00:41.135
So with that out of the way,

00:00:41.135 --> 00:00:43.434
let's look at these three relationships.

00:00:43.434 --> 00:00:44.932
So this one over here,

00:00:44.932 --> 00:00:46.801
let me draw another column here.

00:00:46.801 --> 00:00:49.796
Another column.

00:00:49.796 --> 00:00:52.640
This is, let me call
this the Y over X column.

00:00:52.640 --> 00:00:54.358
I'm just gonna keep figuring
out what this ratio is

00:00:54.358 --> 00:00:56.018
for each of these pairs.

00:00:56.018 --> 00:00:58.038
So for this first pair,

00:00:58.038 --> 00:01:00.279
when X is one, Y is one half,

00:01:00.279 --> 00:01:02.531
so this ratio is one half over one.

00:01:02.531 --> 00:01:03.738
Well one half over one is just

00:01:03.738 --> 00:01:06.199
the same thing as one half.

00:01:06.199 --> 00:01:08.440
When X is four, Y is two,
this ratio is gonna be

00:01:08.440 --> 00:01:12.409
two over four, which is
the same thing as one half.

00:01:13.099 --> 00:01:15.398
When X is negative two
and Y is negative one,

00:01:15.398 --> 00:01:18.079
this ratio is negative
one over negative two,

00:01:18.079 --> 00:01:20.947
which is the same thing as one half.

00:01:20.947 --> 00:01:22.421
So for at least these three points

00:01:22.421 --> 00:01:24.139
that we've sampled from this relationship,

00:01:24.139 --> 00:01:28.667
it looks like the ratio between
Y and X is always one half.

00:01:28.667 --> 00:01:30.919
In this case K would be one half,

00:01:30.919 --> 00:01:35.137
we could write Y over X is
always equal to one half.

00:01:35.137 --> 00:01:36.972
Or at least for these three
points that we've sampled,

00:01:36.972 --> 00:01:38.841
and we'll say, well, maybe
it's always the case,

00:01:38.841 --> 00:01:41.441
for this relationship between X and Y,

00:01:41.441 --> 00:01:42.741
or if you wanted to write it another way,

00:01:42.741 --> 00:01:47.176
you could write that Y
is equal to one half X.

00:01:47.176 --> 00:01:49.521
Now let's graph this thing.

00:01:49.521 --> 00:01:52.621
Well, when X is one, Y is one half.

00:01:52.621 --> 00:01:55.999
When X is four, Y is two.

00:01:55.999 --> 00:01:59.621
When X is negative two, Y is negative one.

00:01:59.621 --> 00:02:00.921
I didn't put the marker for negative one,

00:02:00.921 --> 00:02:02.674
it would be right about there.

00:02:02.674 --> 00:02:05.019
And so if we say these three points

00:02:05.019 --> 00:02:07.039
are sampled on the entire relationship,

00:02:07.039 --> 00:02:09.918
and the entire relationship
is Y is equal to one half X,

00:02:09.918 --> 00:02:13.157
well the line that represents,

00:02:13.157 --> 00:02:15.270
or the set of all points
that would represent

00:02:15.270 --> 00:02:19.530
the possible X-Y pairs,
it would be a line.

00:02:19.530 --> 00:02:23.733
It would be a line that
goes through the origin.

00:02:23.733 --> 00:02:26.797
Because look, if X is
zero, one half times zero

00:02:26.797 --> 00:02:30.176
is going to be equal to Y.

00:02:30.176 --> 00:02:32.172
And so let's think about some
of the key characteristics.

00:02:32.172 --> 00:02:33.867
One, it is a line.

00:02:33.867 --> 00:02:36.828
This is a line here.

00:02:36.828 --> 00:02:38.836
It is a linear relationship.

00:02:38.836 --> 00:02:41.448
And it also goes through the origin.

00:02:41.448 --> 00:02:43.491
And it makes sense that
it goes through an origin.

00:02:43.491 --> 00:02:45.256
Because in a proportional relationship,

00:02:45.256 --> 00:02:46.251
actually when you look over here,

00:02:46.251 --> 00:02:48.851
zero over zero, that's indeterminate form,

00:02:48.851 --> 00:02:50.535
and then that gets a little bit strange,

00:02:50.535 --> 00:02:52.473
but when you look at this right over here,

00:02:52.473 --> 00:02:54.807
well if X is zero and you
multiply it by some constant,

00:02:54.807 --> 00:02:57.080
Y is going to need to be zero as well.

00:02:57.080 --> 00:02:59.076
So for any proportional relationship,

00:02:59.076 --> 00:03:01.839
if you're including when X equals zero,

00:03:01.839 --> 00:03:04.672
then Y would need to be
equal to zero as well.

00:03:04.672 --> 00:03:07.156
And so if you were to plot its graph,

00:03:07.156 --> 00:03:10.941
it would be a line that
goes through the origin.

00:03:10.941 --> 00:03:13.680
It would be a line that
goes through the origin.

00:03:13.680 --> 00:03:15.183
And so this is a proportional relationship

00:03:15.183 --> 00:03:17.713
and its graph is represented by a line

00:03:17.713 --> 00:03:19.942
that goes through the origin.

00:03:19.942 --> 00:03:21.289
Now let's look at this one over here,

00:03:21.289 --> 00:03:22.682
this one in blue.

00:03:22.682 --> 00:03:24.830
So let's think about
whether it is proportional.

00:03:24.830 --> 00:03:26.084
And we could do the same test,

00:03:26.084 --> 00:03:29.404
by calculating the ratio between Y and X.

00:03:29.404 --> 00:03:30.971
Y and X.

00:03:30.971 --> 00:03:32.747
So it's going to be, let's see,

00:03:32.747 --> 00:03:34.488
for this first one it's
going to be three over one,

00:03:34.488 --> 00:03:35.847
which is just three.

00:03:35.847 --> 00:03:38.726
Then it's gonna be five over two.

00:03:38.726 --> 00:03:39.829
Five over two,

00:03:39.829 --> 00:03:43.520
well five over two is not
the same thing as three.

00:03:43.520 --> 00:03:46.612
So already we know that
this is not proportional.

00:03:46.612 --> 00:03:49.935
Not proportional.

00:03:51.090 --> 00:03:52.926
We don't even have to look

00:03:52.926 --> 00:03:55.248
at this third point right over here,

00:03:55.248 --> 00:03:57.105
where if we took the
ratio between Y and X,

00:03:57.105 --> 00:03:58.603
it's negative one over negative one,

00:03:58.603 --> 00:04:00.344
which would just be one.

00:04:00.344 --> 00:04:01.981
Let's see, let's graph this just for fun,

00:04:01.981 --> 00:04:02.886
to see what it looks like.

00:04:02.886 --> 00:04:04.384
When X is one, Y is three.

00:04:04.384 --> 00:04:07.565
When X is one, Y is three.

00:04:07.565 --> 00:04:09.840
When X is two, Y is five.

00:04:09.840 --> 00:04:14.124
X is two, Y is five.

00:04:14.124 --> 00:04:18.408
And when X is negative
one, Y is negative one.

00:04:18.408 --> 00:04:22.622
When X is negative one, Y is negative one.

00:04:22.622 --> 00:04:24.386
And I forgot to put the
hash mark right there,

00:04:24.386 --> 00:04:25.965
it was right around there.

00:04:25.965 --> 00:04:27.184
And so if we said, okay,

00:04:27.184 --> 00:04:28.565
let's just give the benefit of the doubt

00:04:28.565 --> 00:04:32.002
that maybe these are
three points from a line,

00:04:32.002 --> 00:04:32.988
because it looks like I can actually

00:04:32.988 --> 00:04:34.521
connect them with a line.

00:04:34.521 --> 00:04:37.005
Then the line would look
something like this.

00:04:37.776 --> 00:04:42.776
The line would look something like this.

00:04:44.417 --> 00:04:46.367
So notice, this is linear.

00:04:46.367 --> 00:04:48.805
This is a line right over here.

00:04:48.805 --> 00:04:51.510
But it does not go through the origin.

00:04:51.510 --> 00:04:54.110
So if you're just looking
at a relationship visually,

00:04:54.110 --> 00:04:56.385
linear is good, but it needs to go through

00:04:56.385 --> 00:04:59.787
the origin as well for it to
be proportional relationship.

00:04:59.787 --> 00:05:00.785
And you see that right here.

00:05:00.785 --> 00:05:01.993
This is a linear relationship,

00:05:01.993 --> 00:05:04.628
or at least these three
pairs could be sampled

00:05:04.628 --> 00:05:06.543
from a linear relationship,

00:05:06.543 --> 00:05:09.887
but the graph does not
go through the origin.

00:05:09.887 --> 00:05:11.907
And we see here, when
we look at the ratio,

00:05:11.907 --> 00:05:13.985
that it was indeed not proportional.

00:05:13.985 --> 00:05:15.889
So this is not proportional.

00:05:15.889 --> 00:05:17.854
Now let's look at this one over here.

00:05:18.350 --> 00:05:21.540
Let's look at what we have here.

00:05:21.540 --> 00:05:24.325
So I'll look at the ratios.

00:05:25.491 --> 00:05:27.279
Y over X.

00:05:27.279 --> 00:05:29.492
So for this first pair, one over one,

00:05:29.492 --> 00:05:30.952
then we have four over two,

00:05:30.952 --> 00:05:33.425
well we immediately see that
we are not proportional.

00:05:33.425 --> 00:05:35.851
And then nine over
three, it would be three.

00:05:35.851 --> 00:05:38.452
So clearly this is not
a constant number here.

00:05:38.452 --> 00:05:40.123
We don't always have the same value here,

00:05:40.123 --> 00:05:44.906
and so this is also not proportional.

00:05:44.906 --> 00:05:47.344
Not proportional.

00:05:47.344 --> 00:05:49.248
But let's graph it just for fun.

00:05:49.248 --> 00:05:51.628
When X is one, Y is one.

00:05:51.628 --> 00:05:53.532
When X is two, Y is four.

00:05:53.532 --> 00:05:54.925
This actually looks like

00:05:54.925 --> 00:05:59.879
the graph of Y is equal to X squared.

00:05:59.879 --> 00:06:02.254
When X is three, Y is nine.

00:06:02.254 --> 00:06:04.437
At least these three points
are consistent with it.

00:06:04.437 --> 00:06:08.895
So one, three, four, five,
six, seven, eight, nine.

00:06:08.895 --> 00:06:11.240
So it's gonna look something...

00:06:11.240 --> 00:06:12.679
And so, if this really is,

00:06:12.679 --> 00:06:15.071
if these points are sampled
from Y equals X squared,

00:06:15.071 --> 00:06:18.739
then when X is zero, Y would be zero.

00:06:18.739 --> 00:06:20.353
So this one actually would
go through the origin,

00:06:20.353 --> 00:06:22.280
but notice, it's not a line.

00:06:22.280 --> 00:06:24.834
It's not a linear relationship.

00:06:24.834 --> 00:06:28.479
This right over here is the
graph of Y equals X squared.

00:06:28.479 --> 00:06:31.416
So this one also is not proportional.

00:06:31.416 --> 00:06:32.995
So once again, these three points

00:06:32.995 --> 00:06:36.068
could be sampled from Y equals one half X,

00:06:36.068 --> 00:06:39.249
these three points could be sampled from,

00:06:39.249 --> 00:06:41.989
let's see, Y is equal to, let's see,

00:06:41.989 --> 00:06:44.764
it looks like a line when...

00:06:44.764 --> 00:06:48.653
this looks like it could be Y = 2x + 1.

00:06:48.653 --> 00:06:50.348
So it's a linear relationship,

00:06:50.348 --> 00:06:52.031
but it does not go through the origin,

00:06:52.031 --> 00:06:53.552
so it's not proportional.

00:06:53.552 --> 00:06:56.628
And these three points look
like they could be sampled

00:06:56.628 --> 00:06:58.184
from Y equals X squared,

00:06:58.184 --> 00:06:59.542
which goes through the origin.

00:06:59.542 --> 00:07:01.388
When X is zero, Y is zero,

00:07:01.388 --> 00:07:03.582
but it's not a linear relationship.

00:07:03.582 --> 00:07:05.463
Any way you look at it,

00:07:05.463 --> 00:07:06.867
if you look at it visually,

00:07:06.867 --> 00:07:08.527
it has to be a line that
goes through the origin,

00:07:08.527 --> 00:07:11.464
or if you look at a table of
values, look at the ratios,

00:07:11.464 --> 00:07:14.139
and the ratios always
have to be the same value.

00:07:14.139 --> 00:07:16.101
And that was only the case
with this magenta one,

00:07:16.101 --> 00:07:17.465
right over here.


Constructing scatter plot exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.460
Arthur wanted to investigate how
the amount of exercise he does

00:00:04.460 --> 00:00:05.990
affects his weight.

00:00:05.990 --> 00:00:08.050
Each week he recorded
the number of hours

00:00:08.050 --> 00:00:11.420
he exercised during the
past week and the change

00:00:11.420 --> 00:00:13.440
in his weight during that week.

00:00:13.440 --> 00:00:17.370
Positive number means
he gained weight.

00:00:17.370 --> 00:00:20.590
Negative number or negative
means he lost weight.

00:00:20.590 --> 00:00:23.420
Suppose Arthur wants to create
a scatter plot according

00:00:23.420 --> 00:00:25.210
to the data presented below.

00:00:25.210 --> 00:00:27.180
Complete the sentences below.

00:00:27.180 --> 00:00:30.590
The amount of exercise
should go on the-- well,

00:00:30.590 --> 00:00:35.570
we want to put the driving
variable on the horizontal axis

00:00:35.570 --> 00:00:39.790
and the dependent variable
on the vertical axis.

00:00:39.790 --> 00:00:42.550
So the amount of
exercise is what drives

00:00:42.550 --> 00:00:47.440
or what in theory drives his
weight gain or weight loss.

00:00:47.440 --> 00:00:49.690
So we would want to put the
exercise, the thing that's

00:00:49.690 --> 00:00:52.080
driving the other
variable or that

00:00:52.080 --> 00:00:53.580
might be driving
the other variable,

00:00:53.580 --> 00:00:55.200
that we want to investigate
whether it's driving

00:00:55.200 --> 00:00:56.616
the other variable,
we want to put

00:00:56.616 --> 00:00:59.130
that on the horizontal axis.

00:00:59.130 --> 00:01:02.690
The range of that axis
should be from-- well,

00:01:02.690 --> 00:01:05.800
we just have to see what's
his range of exercise

00:01:05.800 --> 00:01:08.720
over this investigation
he's doing?

00:01:08.720 --> 00:01:09.590
So let's see.

00:01:09.590 --> 00:01:14.760
His largest amount of exercise
that he performs in hours

00:01:14.760 --> 00:01:18.390
is, let's see,
there's an 8.9 here.

00:01:18.390 --> 00:01:22.340
That looks like the largest
amount of exercise he performs.

00:01:22.340 --> 00:01:25.010
And what's the least, the
smallest amount of exercise

00:01:25.010 --> 00:01:26.260
he performs?

00:01:26.260 --> 00:01:32.290
Looks like it's 0.3 is the
smallest amount of exercise.

00:01:32.290 --> 00:01:34.666
And so we would want
it to be from 0.3,

00:01:34.666 --> 00:01:39.175
or I guess we could go from
0 all the way to-- well,

00:01:39.175 --> 00:01:41.800
8 won't be good enough, because
we have a couple of data points

00:01:41.800 --> 00:01:44.020
that are at 8-- 8.4, 8.9.

00:01:44.020 --> 00:01:46.710
We need to go all the way to 10.

00:01:46.710 --> 00:01:49.200
We definitely don't
have to go negative,

00:01:49.200 --> 00:01:52.520
because there's no negative
amount of exercise here.

00:01:52.520 --> 00:01:56.010
So 0 to 10 should be sufficient.


Constructing a scatter plot

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.130
Aubrey wanted to see if there's
a connection between the time

00:00:04.130 --> 00:00:08.380
a given exam takes place and
the average score of this exam.

00:00:08.380 --> 00:00:11.270
She collected data about
exams from the previous year.

00:00:11.270 --> 00:00:12.677
Plot the data in a scatter plot.

00:00:12.677 --> 00:00:14.760
And let's see, they give
us a couple of rows here.

00:00:14.760 --> 00:00:16.440
This is the class.

00:00:16.440 --> 00:00:18.340
Then they give us
the period of the day

00:00:18.340 --> 00:00:19.800
that the class happened.

00:00:19.800 --> 00:00:23.780
And then they give us the
average score on an exam.

00:00:23.780 --> 00:00:28.754
And we have to be a little
careful with the study-- maybe

00:00:28.754 --> 00:00:30.170
there's some
correlation depending

00:00:30.170 --> 00:00:32.630
on what subject is taught
during what period.

00:00:32.630 --> 00:00:35.300
But let's just use her data, at
least, just based on her data,

00:00:35.300 --> 00:00:38.500
see if-- well, definitely
do what they're asking us,

00:00:38.500 --> 00:00:42.320
plot a scatter plot, and then
see if there is any connection.

00:00:42.320 --> 00:00:43.530
So let's see.

00:00:43.530 --> 00:00:48.660
On the horizontal
axis, we have Period.

00:00:48.660 --> 00:00:52.520
And on this investigation,
this exploration she's doing,

00:00:52.520 --> 00:00:55.460
she's trying to see, well,
does the period of the day

00:00:55.460 --> 00:00:57.200
somehow drive average score?

00:00:57.200 --> 00:00:59.980
So that's why Period is
on the horizontal axis.

00:00:59.980 --> 00:01:03.590
And the thing that's driving
is on the horizontal, the thing

00:01:03.590 --> 00:01:05.680
that's being driven
is on the vertical.

00:01:05.680 --> 00:01:07.700
So let's plot each
of these points.

00:01:07.700 --> 00:01:14.950
Period 1, average score
93-- right over there.

00:01:14.950 --> 00:01:16.640
Period 6, 87.

00:01:22.799 --> 00:01:24.590
Oh, that's not the
right place, and then we

00:01:24.590 --> 00:01:27.960
can move it if we want--
87, right over there.

00:01:27.960 --> 00:01:29.390
Period 2, 70.

00:01:34.420 --> 00:01:41.890
Period 4, 62-- right over there.

00:01:41.890 --> 00:01:48.890
Period 4 and 86, that's
right over there.

00:01:48.890 --> 00:01:50.416
Period 1, 73.

00:01:55.350 --> 00:01:59.150
Period 3, average
score of 73 as well.

00:02:04.190 --> 00:02:07.320
Period 1, 80,
average score of 80.

00:02:10.780 --> 00:02:13.335
And then Period 3,
average score of 96.

00:02:18.560 --> 00:02:19.436
So there we go.

00:02:19.436 --> 00:02:21.060
And it doesn't really
seem like there's

00:02:21.060 --> 00:02:25.970
any obvious trend over here.

00:02:25.970 --> 00:02:29.180
So let's make sure
that we got this right.

00:02:29.180 --> 00:02:31.040
And we did.


People smoking less over time scatter plot

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.860
The percent of adults
who smoke recorded

00:00:03.860 --> 00:00:07.230
every few years
since 1967 suggests

00:00:07.230 --> 00:00:11.300
a negative linear
association with no outliers.

00:00:11.300 --> 00:00:17.430
On average, the percent drops
by 0.5 points each year.

00:00:17.430 --> 00:00:21.740
Which of the following plots
suits the above description?

00:00:21.740 --> 00:00:26.010
So let's see, this looks like
a negative linear association.

00:00:26.010 --> 00:00:29.020
As the years go by, you have a
smaller percentage of smokers.

00:00:29.020 --> 00:00:30.390
This one does too.

00:00:30.390 --> 00:00:32.540
As years go by, you have
the number of smokers

00:00:32.540 --> 00:00:34.090
go down and down.

00:00:34.090 --> 00:00:37.500
This one down here
also looks like that,

00:00:37.500 --> 00:00:38.874
although it's not as smooth.

00:00:38.874 --> 00:00:40.290
If you were to fit
a line here, it

00:00:40.290 --> 00:00:41.850
looks like you have
a few outliers.

00:00:41.850 --> 00:00:43.391
Well, this is a
positive correlation.

00:00:43.391 --> 00:00:46.320
So we can definitely
rule out Graph 4.

00:00:46.320 --> 00:00:48.300
Now, the other
thing that they told

00:00:48.300 --> 00:00:50.510
us is that there
are no outliers,

00:00:50.510 --> 00:00:53.840
suggests a negative linear
association with no outliers.

00:00:53.840 --> 00:00:56.095
If you were to try to
fit a line to Graph 3,

00:00:56.095 --> 00:00:57.720
you could fit a line
pretty reasonably.

00:00:57.720 --> 00:00:59.240
That would go
someplace like that.

00:00:59.240 --> 00:01:03.130
But it would have this
outlier right over here.

00:01:03.130 --> 00:01:08.760
It looks like it's 12
or 13 years after 1967.

00:01:08.760 --> 00:01:10.875
So that would be 1980.

00:01:10.875 --> 00:01:12.291
It looks like an outlier there.

00:01:12.291 --> 00:01:14.040
But they said it didn't
have any outliers.

00:01:14.040 --> 00:01:17.070
So we would rule out Graph 3.

00:01:17.070 --> 00:01:20.420
And so we have to pick
between Graph 1 and Graph 2.

00:01:20.420 --> 00:01:23.370
So the other hint they give
us or piece of data they

00:01:23.370 --> 00:01:30.360
give us is that the percent
drops by 0.5% each year.

00:01:30.360 --> 00:01:31.490
So here's what's happening.

00:01:37.140 --> 00:01:40.700
In 1967, it looks like
we're at about 55%.

00:01:40.700 --> 00:01:43.080
And then 10 years go by.

00:01:43.080 --> 00:01:48.940
We are roughly at around
45, a little under 45%.

00:01:48.940 --> 00:01:51.660
So we dropped 10% in 10 years.

00:01:51.660 --> 00:01:55.930
That seems to be how much
this is dropping, roughly 10%

00:01:55.930 --> 00:01:56.680
in 10 years.

00:01:56.680 --> 00:01:58.440
Another 10 years go by.

00:01:58.440 --> 00:02:02.510
We go from 45, a little
more than 10% in 10 years.

00:02:02.510 --> 00:02:04.850
And so that would mean that
we're dropping, on average,

00:02:04.850 --> 00:02:07.980
more than one percentage
point per year.

00:02:07.980 --> 00:02:09.740
That seems more than
what's going on here.

00:02:09.740 --> 00:02:11.250
Now, let's look over here.

00:02:11.250 --> 00:02:14.340
Over here, we're starting,
it looks like, at around 42%.

00:02:14.340 --> 00:02:20.540
And then after 10 years,
it looks like we're at 37%.

00:02:20.540 --> 00:02:23.156
So it looks like
we've dropped about 5%

00:02:23.156 --> 00:02:25.680
in 10 years, which is
consistent with this.

00:02:25.680 --> 00:02:27.450
If you drop 5% in
10 years, that means

00:02:27.450 --> 00:02:29.640
you drop half a
percent per year.

00:02:29.640 --> 00:02:33.030
So we'll go with Graph 2.


Studying, shoe size, and test scores scatter plots

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.960
The graphs below
show the test grades

00:00:03.960 --> 00:00:06.730
of the students
in Dexter's class.

00:00:06.730 --> 00:00:10.020
The first graph shows the
relationship between test

00:00:10.020 --> 00:00:13.240
grades and the amount of time
the students spent studying.

00:00:13.240 --> 00:00:15.510
So this is study
time on this axis

00:00:15.510 --> 00:00:17.949
and this is the test
grade on this axis.

00:00:17.949 --> 00:00:19.740
And the second graph
shows the relationship

00:00:19.740 --> 00:00:22.740
between test grades
and shoes size.

00:00:22.740 --> 00:00:26.390
So shoe size on this
axis and then test grade.

00:00:26.390 --> 00:00:28.560
Choose the best description
of the relationship

00:00:28.560 --> 00:00:30.630
between the graphs.

00:00:30.630 --> 00:00:36.420
So first, before looking
at the explanations,

00:00:36.420 --> 00:00:38.820
let's look at the actual graphs.

00:00:38.820 --> 00:00:40.570
So this one on the
left right over here,

00:00:40.570 --> 00:00:45.374
it looks like there is a
positive linear relationship

00:00:45.374 --> 00:00:46.040
right over here.

00:00:46.040 --> 00:00:48.332
I could almost fit a line
that would go just like that.

00:00:48.332 --> 00:00:49.790
And it makes sense
that there would

00:00:49.790 --> 00:00:52.550
be, that the more time
that you spend studying,

00:00:52.550 --> 00:00:54.390
the better score
that you would get.

00:00:54.390 --> 00:00:56.500
Now for a certain
amount of time studying,

00:00:56.500 --> 00:00:58.250
some people might do
better than others,

00:00:58.250 --> 00:01:01.000
but it does seem like
there's this relationship.

00:01:01.000 --> 00:01:02.570
Here it doesn't
seem like there's

00:01:02.570 --> 00:01:05.190
really much of a relationship.

00:01:05.190 --> 00:01:08.060
You see the shoe sizes,
for a given shoe size,

00:01:08.060 --> 00:01:12.480
some people do not so well
and some people do very well.

00:01:12.480 --> 00:01:16.267
Someone with a size 10
and 1/2, it looks like,

00:01:16.267 --> 00:01:18.100
someone it looks like
they flunked the exam.

00:01:18.100 --> 00:01:21.000
Someone else, looks like
they got A minus or a B plus

00:01:21.000 --> 00:01:21.950
on the exam.

00:01:21.950 --> 00:01:25.780
And it really would be hard
to somehow fit a line here.

00:01:25.780 --> 00:01:30.030
No matter how you draw
a line, these dots

00:01:30.030 --> 00:01:31.900
don't seem to form a trend.

00:01:31.900 --> 00:01:34.370
So let's see which of
these choices apply.

00:01:34.370 --> 00:01:36.000
There's a negative
linear relationship

00:01:36.000 --> 00:01:37.350
between study time and score.

00:01:37.350 --> 00:01:38.183
No, that's not true.

00:01:38.183 --> 00:01:41.250
It looks like there's a
positive linear relationship.

00:01:41.250 --> 00:01:43.550
The more you study, the
better your score would be.

00:01:43.550 --> 00:01:45.970
A negative linear relationship
would trend downwards

00:01:45.970 --> 00:01:47.190
like that.

00:01:47.190 --> 00:01:50.140
There is a non-linear
relationship between study time

00:01:50.140 --> 00:01:52.590
and score and a negative
linear relationship

00:01:52.590 --> 00:01:53.790
between shoe size and score.

00:01:53.790 --> 00:01:55.340
Well that doesn't
seem right either.

00:01:55.340 --> 00:01:58.090
A non-linear
relationship, it would not

00:01:58.090 --> 00:01:59.800
be easy to fit a line to it.

00:01:59.800 --> 00:02:02.440
And this one seems like a
line would be very reasonable.

00:02:02.440 --> 00:02:06.290
And it doesn't seem like
there's any type of relationship

00:02:06.290 --> 00:02:07.605
between shoe size and score.

00:02:07.605 --> 00:02:09.520
So I wouldn't pick
this one either.

00:02:09.520 --> 00:02:11.152
There's a positive
linear relationship

00:02:11.152 --> 00:02:12.360
between study time and score.

00:02:12.360 --> 00:02:13.150
That's right.

00:02:13.150 --> 00:02:15.530
And no relationship between
shoe size and score.

00:02:15.530 --> 00:02:17.030
Well, I'm going to
go with that one.

00:02:17.030 --> 00:02:19.060
Both graphs show
positive linear trends

00:02:19.060 --> 00:02:20.430
of approximately equal strength.

00:02:20.430 --> 00:02:21.140
No, not at all.

00:02:21.140 --> 00:02:23.440
This one doesn't show
a linear relationship

00:02:23.440 --> 00:02:25.550
of really any strength.


Reasonable samples

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.766
City Councilwomen
Kelly wants to know

00:00:02.766 --> 00:00:04.140
how the residents
of her district

00:00:04.140 --> 00:00:07.920
feel about a proposed
school redistricting plan.

00:00:07.920 --> 00:00:09.690
Which of the following
survey methods

00:00:09.690 --> 00:00:13.130
will allow councilwomen Kelly
to make a valid conclusion

00:00:13.130 --> 00:00:15.350
about how residence
of her district

00:00:15.350 --> 00:00:17.980
feel about the proposed plan?

00:00:17.980 --> 00:00:19.980
So before we even
look at these, we just

00:00:19.980 --> 00:00:20.960
have to realize that
if you're trying

00:00:20.960 --> 00:00:22.630
to make a valid
conclusion about how

00:00:22.630 --> 00:00:24.360
the residents of
her entire district

00:00:24.360 --> 00:00:26.420
feel about the
proposed plan, she

00:00:26.420 --> 00:00:28.200
has to find a
representative sample,

00:00:28.200 --> 00:00:31.770
or not kind of a skewed
sample that would just

00:00:31.770 --> 00:00:34.030
sample parts of her district.

00:00:34.030 --> 00:00:35.810
So let's look at her choices.

00:00:35.810 --> 00:00:37.970
Should she just
ask her neighbors?

00:00:37.970 --> 00:00:40.980
Well, she might live in a
part of the neighborhood that

00:00:40.980 --> 00:00:44.070
might unusually benefit
from the redistricting plan

00:00:44.070 --> 00:00:46.304
or might get hurt from
the redistricting plan.

00:00:46.304 --> 00:00:47.720
And so just her
neighbors wouldn't

00:00:47.720 --> 00:00:50.500
be representative of
the district as a whole.

00:00:50.500 --> 00:00:53.900
So just asking her neighbors
probably does not make sense.

00:00:53.900 --> 00:00:58.410
Ask the residents of Whispering
Pines Retirement Community.

00:00:58.410 --> 00:01:02.430
So once again, the first
one skews by geography.

00:01:02.430 --> 00:01:07.330
She's oversampling her neighbors
and not the entire district.

00:01:07.330 --> 00:01:10.700
Here, she's oversampling a
specific age demographic.

00:01:10.700 --> 00:01:13.710
So here she is oversampling
older residents

00:01:13.710 --> 00:01:15.690
who might have very
different opinions then

00:01:15.690 --> 00:01:17.160
middle aged or
younger residents.

00:01:17.160 --> 00:01:19.060
So that doesn't
make sense either.

00:01:19.060 --> 00:01:21.100
Ask 200 residents
of her district

00:01:21.100 --> 00:01:23.904
whose names are
chosen at random.

00:01:23.904 --> 00:01:25.070
Well, that seems reasonable.

00:01:25.070 --> 00:01:27.877
It doesn't seem like there's
some chance that you somehow

00:01:27.877 --> 00:01:29.460
over sampled one
direction or another.

00:01:29.460 --> 00:01:33.120
But it's most likely to give
a reasonably representative

00:01:33.120 --> 00:01:33.620
sample.

00:01:33.620 --> 00:01:35.620
And this is a pretty
large sample size.

00:01:35.620 --> 00:01:38.292
So it's important to say, what
is the random process, where

00:01:38.292 --> 00:01:39.500
she getting these names from?

00:01:39.500 --> 00:01:42.010
But this actually
does seem reasonable.

00:01:42.010 --> 00:01:45.420
Ask a group of parents
at the local playground.

00:01:45.420 --> 00:01:49.020
Well once again, this is just
like asking your neighbors.

00:01:49.020 --> 00:01:51.500
And it's also sampling
a specific demographic.

00:01:51.500 --> 00:01:53.070
Now, this might
be the demographic

00:01:53.070 --> 00:01:54.860
that cares most
about the schools.

00:01:54.860 --> 00:01:57.310
But she wants to know how
the whole district feels

00:01:57.310 --> 00:01:59.290
about the redistricting plan.

00:01:59.290 --> 00:02:01.470
And once ago, this is
at a local playground.

00:02:01.470 --> 00:02:03.530
This isn't at all the
playgrounds in the district

00:02:03.530 --> 00:02:04.029
somehow.

00:02:04.029 --> 00:02:06.180
So I wouldn't do
this one either.

00:02:06.180 --> 00:02:08.500
Let's do one more of these.

00:02:08.500 --> 00:02:11.650
Mimi wants to conduct a
survey of her 300 classmates

00:02:11.650 --> 00:02:14.700
to determine which candidate
for class president-- Napoleon

00:02:14.700 --> 00:02:18.610
Dynamite or Blair Waldorf--
is in the lead in the upcoming

00:02:18.610 --> 00:02:19.540
election.

00:02:19.540 --> 00:02:22.110
Mimi will ask the question,
if the election were

00:02:22.110 --> 00:02:24.380
today which candidate
would get your vote?

00:02:24.380 --> 00:02:27.100
Which of the following methods
of surveying her classmates

00:02:27.100 --> 00:02:29.710
will allow Mimi to make
valid conclusions about which

00:02:29.710 --> 00:02:32.070
candidate is in the lead?

00:02:32.070 --> 00:02:33.720
So let's see, ask
all of the students

00:02:33.720 --> 00:02:34.980
at Blair's lunch table?

00:02:34.980 --> 00:02:35.660
No.

00:02:35.660 --> 00:02:38.202
That would skew it in
Blair's favor, probably.

00:02:38.202 --> 00:02:39.660
That's not a
representative sample.

00:02:39.660 --> 00:02:41.690
Ask all the members of
Napoleon's soccer team?

00:02:41.690 --> 00:02:42.520
No, same thing.

00:02:42.520 --> 00:02:44.200
They're likely to
go Napoleon's way

00:02:44.200 --> 00:02:45.320
or maybe they don't
like Napoleon,

00:02:45.320 --> 00:02:46.736
maybe they'll go
against Napoleon.

00:02:46.736 --> 00:02:48.850
But either way this seems
like a skewed sample.

00:02:48.850 --> 00:02:50.710
Put the names of all
the students in a hat

00:02:50.710 --> 00:02:51.740
and draw 50 names.

00:02:51.740 --> 00:02:53.650
Ask those students
whose names are drawn.

00:02:53.650 --> 00:02:55.900
Well this seems like
a nice random sample

00:02:55.900 --> 00:02:58.230
that could be nicely
representative

00:02:58.230 --> 00:03:00.100
of the entire population.

00:03:00.100 --> 00:03:04.540
Ask all students whose
names begin with N or B?

00:03:04.540 --> 00:03:07.870
Well, this could be
perceived as kind of random.

00:03:07.870 --> 00:03:12.380
But notice, N is the same
starting letter as Napoleon,

00:03:12.380 --> 00:03:14.020
B is the same starter
letter as Blair.

00:03:14.020 --> 00:03:14.760
You might say,
well, that's fair.

00:03:14.760 --> 00:03:16.000
You're doing it for
each of their letters.

00:03:16.000 --> 00:03:18.860
But maybe there's like 10 people
whose names start with an N

00:03:18.860 --> 00:03:22.580
and only two people whose names
start with a B. Once again,

00:03:22.580 --> 00:03:24.237
you're not even
getting a large sample.

00:03:24.237 --> 00:03:25.820
And then on top of
that, maybe there's

00:03:25.820 --> 00:03:28.142
some type of people with
the same starting letter

00:03:28.142 --> 00:03:29.350
somehow like each other more.

00:03:29.350 --> 00:03:31.600
So I would steer
clear of this one.

00:03:31.600 --> 00:03:34.610
Ask every student in the class?

00:03:34.610 --> 00:03:36.010
Well, that would work.

00:03:36.010 --> 00:03:38.150
There's 300 classmates,
that might not

00:03:38.150 --> 00:03:40.927
be that time consuming.

00:03:40.927 --> 00:03:43.010
You can't get a better
sample than asking everyone

00:03:43.010 --> 00:03:43.750
in the population.

00:03:43.750 --> 00:03:45.645
Which of the following methods
of surveying our class which

00:03:45.645 --> 00:03:47.770
will allow Mimi to make a
valid conclusion about which

00:03:47.770 --> 00:03:48.686
candidate in the lead?

00:03:48.686 --> 00:03:50.409
Well, that's a pretty
good conclusion.

00:03:50.409 --> 00:03:51.700
People might change their mind.

00:03:51.700 --> 00:03:53.790
So it's not a done deal.

00:03:53.790 --> 00:03:55.540
You can't get a
better sample size

00:03:55.540 --> 00:03:57.260
then the entire population.

00:03:57.260 --> 00:04:01.937
Assign numbers to each
student in the class

00:04:01.937 --> 00:04:03.520
and use a computer
program to generate

00:04:03.520 --> 00:04:05.780
50 random numbers
between 1 and 300.

00:04:05.780 --> 00:04:07.697
Ask those students whose
numbers are selected.

00:04:07.697 --> 00:04:10.238
Well this is pretty close to
put the names of all the student

00:04:10.238 --> 00:04:11.400
in a hat and draw 50 names.

00:04:11.400 --> 00:04:12.740
So I would give this one.

00:04:12.740 --> 00:04:15.475
That seems reasonable as well.


Chain rule example using visual information

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.340
Given capital F of x is equal
to g of x to the third power

00:00:05.340 --> 00:00:08.940
where the graph of g and its
tangent line at x equals 4

00:00:08.940 --> 00:00:12.970
are shown, what is the
value of F prime of 4?

00:00:12.970 --> 00:00:15.710
So they give us g of x
right over here in blue.

00:00:15.710 --> 00:00:19.650
And they show us
the tangent line

00:00:19.650 --> 00:00:22.940
at x equals 4 right over here.

00:00:22.940 --> 00:00:24.860
So we need to figure
out F prime of 4.

00:00:24.860 --> 00:00:28.040
So let's just rewrite this
information they've given us.

00:00:28.040 --> 00:00:35.640
We know that F of x is equal
to g of x to the third power.

00:00:35.640 --> 00:00:41.230
So I'll write it like this,
g of x to the third power.

00:00:41.230 --> 00:00:43.560
So we want to figure
out what F prime of x

00:00:43.560 --> 00:00:44.636
is when x is equal to 4.

00:00:44.636 --> 00:00:46.010
So let's just take
the derivative

00:00:46.010 --> 00:00:48.710
here of both sides
with respect to x.

00:00:48.710 --> 00:00:50.760
So take the derivative
of the left-hand side

00:00:50.760 --> 00:00:54.190
with respect to x, and
take the derivative

00:00:54.190 --> 00:00:57.842
of the right-hand side
with respect to x.

00:00:57.842 --> 00:00:59.300
So the left-hand
side, this is just

00:00:59.300 --> 00:01:01.390
going to be capital
F prime of x.

00:01:04.044 --> 00:01:05.960
Now on the right-hand
side I have a composite.

00:01:05.960 --> 00:01:08.850
I have g of x to
the third power.

00:01:08.850 --> 00:01:12.770
So first we can view this as the
product of the derivative of g

00:01:12.770 --> 00:01:16.410
of x to the third power
with respect to g of x.

00:01:16.410 --> 00:01:18.360
So there we could
literally just apply

00:01:18.360 --> 00:01:20.990
what we know about
the power rule.

00:01:20.990 --> 00:01:26.150
The derivative of x to the
third with respect to x is 3x

00:01:26.150 --> 00:01:26.860
squared.

00:01:26.860 --> 00:01:30.840
So the derivative of gx to the
third with respect to g of x

00:01:30.840 --> 00:01:37.210
is just going to be three times
g of x to the second power.

00:01:37.210 --> 00:01:39.850
And then we're going to multiply
that times the derivative of g

00:01:39.850 --> 00:01:41.680
of x with respect to x.

00:01:41.680 --> 00:01:44.904
So times g prime of x

00:01:44.904 --> 00:01:46.820
And this comes straight
out of the chain rule.

00:01:46.820 --> 00:01:48.030
Derivative of this,
the derivative

00:01:48.030 --> 00:01:49.880
of g of x to the third
with respect to g

00:01:49.880 --> 00:01:52.690
of x, which is this,
times the derivative

00:01:52.690 --> 00:01:58.450
of g of x with respect to x,
which is that right over there.

00:01:58.450 --> 00:01:59.670
So now let's just substitute.

00:01:59.670 --> 00:02:01.940
We want to figure out
what this derivative is

00:02:01.940 --> 00:02:03.900
when x is equal to 4.

00:02:03.900 --> 00:02:08.550
So we could say
that F prime of 4

00:02:08.550 --> 00:02:19.590
is equal to 3 times g of 4
squared times g prime of 4.

00:02:19.590 --> 00:02:24.779
So what is g of 4 going to be?

00:02:24.779 --> 00:02:27.070
Well, we can just look at
our function right over here.

00:02:27.070 --> 00:02:31.785
When x equals 4, our
function is equal to 3.

00:02:34.930 --> 00:02:38.160
So g of 4 is equal to 3.

00:02:38.160 --> 00:02:40.340
And what's g prime of 4?

00:02:40.340 --> 00:02:43.320
So when x equals
4, g prime of 4 is

00:02:43.320 --> 00:02:45.020
the slope of the tangent line.

00:02:45.020 --> 00:02:49.160
And they've drawn the tangent
line when x equals 4 here.

00:02:49.160 --> 00:02:51.200
So what is the
slope of this line?

00:02:51.200 --> 00:02:53.700
So we just have to think about
change in y over change in x.

00:02:53.700 --> 00:02:57.280
And I'll look at that between
two integer-valued coordinates.

00:02:57.280 --> 00:02:59.970
So it looks like between
these two points.

00:02:59.970 --> 00:03:09.460
And when we increase x
by 2, we decrease y by 4.

00:03:09.460 --> 00:03:11.450
So as you remember,
slope is rise

00:03:11.450 --> 00:03:14.170
over run, or change
in y over change in x.

00:03:14.170 --> 00:03:17.790
So the slope of the tangent
line here, the slope

00:03:17.790 --> 00:03:20.250
is equal to our
change in y negative 4

00:03:20.250 --> 00:03:21.810
over our change in x.

00:03:21.810 --> 00:03:25.820
And this is going to
be equal to negative 2.

00:03:28.930 --> 00:03:36.141
So this simplifies to F
prime of 4 is equal to-- I'll

00:03:36.141 --> 00:03:40.590
do this in a new color--
3 squared is 9 times 3

00:03:40.590 --> 00:03:51.450
is 27 times negative 2, which
is equal to negative 54.

00:03:51.450 --> 00:03:56.160
So F prime of 4 is negative 54.


Chain rule example using visual function definitions

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.570
Consider the functions f and
g with the graphs shown below.

00:00:04.570 --> 00:00:07.500
If capital G of x is
equal to lowercase

00:00:07.500 --> 00:00:10.870
g of lowercase f of
x, what is the value

00:00:10.870 --> 00:00:14.810
of capital G prime of 2.5.?

00:00:14.810 --> 00:00:17.790
So G of x is a
composition of g and f.

00:00:17.790 --> 00:00:21.470
So it's g of f of x, or
lowercase g of f of x.

00:00:21.470 --> 00:00:23.460
And they don't graph
capital G of x here.

00:00:23.460 --> 00:00:26.860
They just give us the graphs of
lowercase g of x and lowercase

00:00:26.860 --> 00:00:27.820
f of x.

00:00:27.820 --> 00:00:29.810
This is the graph
of lowercase f of x.

00:00:29.810 --> 00:00:33.110
This is a graph of
lowercase g of x.

00:00:33.110 --> 00:00:36.060
So let's just try to think
how we could evaluate this

00:00:36.060 --> 00:00:39.120
and then see if they've given
us the right information here.

00:00:39.120 --> 00:00:42.430
So let me just rewrite a lot of
what they've already told us.

00:00:42.430 --> 00:00:45.130
They've already told
us that capital G of x

00:00:45.130 --> 00:00:53.035
is equal to lowercase
g of f of x.

00:00:56.770 --> 00:00:59.470
So if we wanted to take
the derivative of capital

00:00:59.470 --> 00:01:01.370
G of x-- and we do want
to think about what

00:01:01.370 --> 00:01:03.620
the derivative of capital G
of x is, because they want

00:01:03.620 --> 00:01:07.880
us to evaluate the derivative
at x is equal to 2.5.

00:01:07.880 --> 00:01:08.590
So let's do that.

00:01:08.590 --> 00:01:11.420
Let's take the derivative
of both sides of this.

00:01:11.420 --> 00:01:15.910
So if we take the derivative
of the left hand side,

00:01:15.910 --> 00:01:20.230
we end up with G prime of x.

00:01:20.230 --> 00:01:21.910
And on the derivative
on the right hand,

00:01:21.910 --> 00:01:23.993
since we have a composition
here of two functions,

00:01:23.993 --> 00:01:25.820
we would apply the chain rule.

00:01:25.820 --> 00:01:30.900
So this is going to be the
derivative of g with respect

00:01:30.900 --> 00:01:32.340
to f.

00:01:32.340 --> 00:01:36.310
So we could write
that as g prime

00:01:36.310 --> 00:01:44.650
of f of x times the derivative
of f with respect to x.

00:01:44.650 --> 00:01:49.420
So times f prime of x.

00:01:49.420 --> 00:01:52.310
So if we want to evaluate
what G prime of 2.5

00:01:52.310 --> 00:01:53.930
is, then every place
we see an x here,

00:01:53.930 --> 00:01:55.640
we have to start
with 2.5 in there.

00:01:55.640 --> 00:01:57.300
So let's try to do that.

00:01:57.300 --> 00:02:02.410
So G prime of-- and I'll do this
in white so it sticks out-- G

00:02:02.410 --> 00:02:09.470
prime of 2.5 is going
to be equal to lowercase

00:02:09.470 --> 00:02:30.397
g prime of f of 2.5
times f prime of 2.5.

00:02:30.397 --> 00:02:32.480
So let's think about what
these would evaluate to.

00:02:32.480 --> 00:02:35.500
What is f of 2.5?

00:02:35.500 --> 00:02:39.330
Well, when x is
equal to 2.5-- let

00:02:39.330 --> 00:02:41.840
me just get a color
you can actually see.

00:02:41.840 --> 00:02:46.290
When x is equal to 2.5, our
function here is equal to 1.

00:02:46.290 --> 00:02:51.610
So f of 2.5, so we know
that f of 2.5 is equal to 1.

00:02:51.610 --> 00:02:56.180
Let me write that down,
f of 2.5 is equal to 1.

00:02:56.180 --> 00:02:59.750
And we also need to figure
out what f prime of 2.5 is.

00:02:59.750 --> 00:03:08.350
So f-- let me write it
this way-- f prime of 2.5

00:03:08.350 --> 00:03:09.990
is equal to.

00:03:09.990 --> 00:03:11.860
Now what is f prime of 2.5?

00:03:11.860 --> 00:03:13.940
That's just
essentially the slope

00:03:13.940 --> 00:03:16.640
of the tangent line
at the function

00:03:16.640 --> 00:03:18.450
when x is equal to 2.5.

00:03:18.450 --> 00:03:21.600
So it's really just the
slope right over here.

00:03:21.600 --> 00:03:23.850
And at least right over at
this part of the function,

00:03:23.850 --> 00:03:24.725
it's actually a line.

00:03:24.725 --> 00:03:27.770
And the slope is actually
very easy to spot out.

00:03:27.770 --> 00:03:30.092
If we were to go from this
point to this point here--

00:03:30.092 --> 00:03:32.050
and I'm just picking
those points because those

00:03:32.050 --> 00:03:36.740
are on kind of
integer-valued coordinates--

00:03:36.740 --> 00:03:44.030
we see that for every three
that we run, we go up two,

00:03:44.030 --> 00:03:47.900
or that we rise two for
every three that we run.

00:03:47.900 --> 00:03:53.500
Or that our change in y
over change in x is 2/3.

00:03:53.500 --> 00:03:58.270
So the slope of the function
right over there is 2/3.

00:03:58.270 --> 00:04:00.910
So this is equal to 2/3.

00:04:00.910 --> 00:04:06.480
And so we can substitute back
in here, f of 2.5 is equal to 1.

00:04:06.480 --> 00:04:09.500
And this right over
here is equal to 2/3.

00:04:09.500 --> 00:04:10.850
Now, we're not done yet.

00:04:10.850 --> 00:04:15.140
Now we have to evaluate,
what is G prime of 1?

00:04:15.140 --> 00:04:18.760
So when x is equal to 1,
this is the function g.

00:04:18.760 --> 00:04:21.550
We're not evaluating g of 1.

00:04:21.550 --> 00:04:24.370
We're evaluating g prime of 1.

00:04:24.370 --> 00:04:26.520
So what is the slope
of the line here?

00:04:26.520 --> 00:04:30.740
Well, our change in y
over change in x is 2/1.

00:04:30.740 --> 00:04:33.250
If we go one in the
horizontal direction,

00:04:33.250 --> 00:04:35.860
we go up two in the
vertical direction.

00:04:35.860 --> 00:04:39.860
Change in y over
change in x is 2/1.

00:04:39.860 --> 00:04:48.100
So g prime-- let me write
this down-- g prime of 1

00:04:48.100 --> 00:04:50.400
is equal to 2.

00:04:53.170 --> 00:04:56.597
So this whole thing
evaluates to 2.

00:04:56.597 --> 00:04:58.430
And so this simplifies--
let me scratch that

00:04:58.430 --> 00:05:03.020
out-- the simplifies to 2 times
2/3, which is equal to 4/3.

00:05:03.020 --> 00:05:10.384
So we could write G prime
of 2.5 is equal to 4/3.

00:05:10.384 --> 00:05:11.800
And this is a
pretty neat problem,

00:05:11.800 --> 00:05:13.870
because we didn't get to
see the actual function

00:05:13.870 --> 00:05:14.996
definition from G of x.

00:05:14.996 --> 00:05:17.370
But just using the chain rule
and the information they're

00:05:17.370 --> 00:05:19.030
giving us, we were
able to figure out

00:05:19.030 --> 00:05:20.920
what the value of
this derivative

00:05:20.920 --> 00:05:23.673
is when x is equal to 2.5.


Squeeze theorem exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:06.120
The graphs of f of x, g of x,
and h of x are shown below.

00:00:06.120 --> 00:00:08.880
Select and drag cards to
create a compound inequality

00:00:08.880 --> 00:00:10.920
that orders the
values of f of x,

00:00:10.920 --> 00:00:17.860
g of x, and h of x for x-values
near 2 but not at 2 itself.

00:00:17.860 --> 00:00:20.780
So for any of the x-values that
are depicted right over here,

00:00:20.780 --> 00:00:24.810
say, x is equal to 3, we see
that h of 3 is the largest,

00:00:24.810 --> 00:00:28.870
f of 3 is the smallest,
and g of 3 is in between.

00:00:28.870 --> 00:00:30.660
And that's true for
any of the x-values

00:00:30.660 --> 00:00:31.785
that they've depicted here.

00:00:31.785 --> 00:00:35.860
If we look at when x is equal
to 1, h of 1 is the largest,

00:00:35.860 --> 00:00:39.710
f of 1 is the smallest,
g of 1 is in between.

00:00:39.710 --> 00:00:43.200
So for all of the x-values
that they're depicting,

00:00:43.200 --> 00:00:50.580
f of x is less than or equal
to g of x, which is less than

00:00:50.580 --> 00:00:53.210
or equal to h of x.

00:00:53.210 --> 00:00:55.850
And the only place where they
equal based on this graph

00:00:55.850 --> 00:01:00.290
comes into play, it looks like
as we approach x equals 2,

00:01:00.290 --> 00:01:05.530
it looks like all the
functions are approaching 1.

00:01:05.530 --> 00:01:09.590
So that's where the equal
might come into play.

00:01:09.590 --> 00:01:12.400
But let's keep seeing what
they want us to do after that.

00:01:12.400 --> 00:01:14.790
So then it says it follows that.

00:01:14.790 --> 00:01:17.262
And instead of writing f
of x, g of x, and h of x,

00:01:17.262 --> 00:01:19.220
they've written the actual
definitions of them.

00:01:19.220 --> 00:01:23.330
So let's just remind
ourselves f of x is 2x times

00:01:23.330 --> 00:01:25.060
the square root of
x minus 1 minus 1.

00:01:25.060 --> 00:01:27.410
That's this blue
one right over here.

00:01:27.410 --> 00:01:29.570
So instead writing
f of x, we can

00:01:29.570 --> 00:01:33.400
write 2 times the square
root of x minus 1.

00:01:33.400 --> 00:01:37.435
Minus 1 is less than
or equal to g of x.

00:01:37.435 --> 00:01:42.080
G of x was this rational
expression right over here.

00:01:42.080 --> 00:01:44.210
So let's go back down here.

00:01:44.210 --> 00:01:45.995
We get this rational expression.

00:01:45.995 --> 00:01:48.120
And then that's going to
be less than or equal to h

00:01:48.120 --> 00:01:51.270
of x, which was, I believe,
e to the x minus 2.

00:01:51.270 --> 00:01:51.920
Is that right?

00:01:51.920 --> 00:01:54.050
Yep, e to the x minus 2.

00:01:57.170 --> 00:01:59.530
So all we've really
done is replaced

00:01:59.530 --> 00:02:02.930
f, g, and h with
their definitions.

00:02:02.930 --> 00:02:07.300
And then this means that
the limit-- so they're

00:02:07.300 --> 00:02:08.990
looking at the limit
as x approaches

00:02:08.990 --> 00:02:11.450
2 of these different
expressions.

00:02:11.450 --> 00:02:15.290
So the limit as x approaches
2 of this expression

00:02:15.290 --> 00:02:18.810
is going to be less than
or equal to the limit

00:02:18.810 --> 00:02:22.070
as x approaches 2 of
this expression, which

00:02:22.070 --> 00:02:24.960
is this right over here,
which is going to be less than

00:02:24.960 --> 00:02:27.070
or equal to the
limit as x approaches

00:02:27.070 --> 00:02:31.880
2 of this expression, which
is that right over there.

00:02:31.880 --> 00:02:33.750
And then they say
finally the value

00:02:33.750 --> 00:02:37.060
of the limit as x approaches 2
of this thing right over here

00:02:37.060 --> 00:02:40.280
is-- well, this is where
the squeeze theorem

00:02:40.280 --> 00:02:41.494
comes into play.

00:02:41.494 --> 00:02:43.160
We just have to remind
ourselves-- well,

00:02:43.160 --> 00:02:43.450
let's just think about it.

00:02:43.450 --> 00:02:45.430
Can we figure out the
limit as x approaches

00:02:45.430 --> 00:02:47.280
2 of this right over here?

00:02:47.280 --> 00:02:49.940
Well, the limit as x approaches
2-- let's see, 2 minus 1.

00:02:49.940 --> 00:02:51.940
So we're taking the
principal root of 2 minus 1,

00:02:51.940 --> 00:02:53.315
which is the
principal root of 1.

00:02:53.315 --> 00:02:56.000
So you have 2 times 1 minus 1.

00:02:56.000 --> 00:02:57.950
So this is 1.

00:02:57.950 --> 00:03:00.940
This right over here
is e to the 2 minus 2.

00:03:00.940 --> 00:03:03.430
That's either the 0,
or that's 1 as well.

00:03:03.430 --> 00:03:05.760
So the limit of all
of this is going

00:03:05.760 --> 00:03:07.155
to be greater than
or equal to 1,

00:03:07.155 --> 00:03:09.030
and it's going to be
less than or equal to 1.

00:03:09.030 --> 00:03:10.450
Or it's right in
between 1 and 1.

00:03:10.450 --> 00:03:12.700
And the only way that it's
going to be between 1 and 1

00:03:12.700 --> 00:03:16.050
is if it is equal to 1.

00:03:16.050 --> 00:03:19.890
This is the squeeze theorem
at play right over here.

00:03:19.890 --> 00:03:23.030
g of x, over the domain
that we've been looking at,

00:03:23.030 --> 00:03:26.800
or over the x-values
that we care about-- g

00:03:26.800 --> 00:03:29.950
of x was less than or equal
to h of x, which was--

00:03:29.950 --> 00:03:32.890
or f of x was less than or equal
to g of x, which was less than

00:03:32.890 --> 00:03:34.420
or equal to h of x.

00:03:34.420 --> 00:03:37.800
And then we took the limit for
all of them as x approached 2.

00:03:37.800 --> 00:03:41.029
For the lower function, for
f of x, it approached 1.

00:03:41.029 --> 00:03:42.820
And we see it in the
graph right over here.

00:03:42.820 --> 00:03:44.610
In the lower function,
f of x approaches

00:03:44.610 --> 00:03:47.160
1, h of x approaches 1,
and therefore, g of x

00:03:47.160 --> 00:03:48.540
must also approach 1.

00:03:48.540 --> 00:03:51.530
And we actually see that in
this graph right over here.

00:03:51.530 --> 00:03:54.020
But anyway, we could
check our answer just

00:03:54.020 --> 00:03:56.410
to feel good about ourselves.

00:03:56.410 --> 00:03:58.330
We got it right.


Limit by factoring cubic expression

WEBVTT
Kind: captions
Language: en

00:00:00.599 --> 00:00:02.390
Let's try to find the
limit as x approaches

00:00:02.390 --> 00:00:06.120
1 of x to the third minus
1 over x squared minus 1.

00:00:06.120 --> 00:00:09.400
And at first when you just
try to substitute x equals 1,

00:00:09.400 --> 00:00:12.530
you get 0/0 1 minus
1 over 1 minus 1.

00:00:12.530 --> 00:00:13.870
So that doesn't help us.

00:00:13.870 --> 00:00:17.075
So let's see if we can try
to simplify this in some way.

00:00:17.075 --> 00:00:18.575
So you might
immediately recognize--

00:00:18.575 --> 00:00:21.270
so let's rewrite this
expression right over

00:00:21.270 --> 00:00:28.010
here so it's x to the third
minus 1 over x squared minus 1.

00:00:28.010 --> 00:00:30.130
This on the bottom
immediately jumps out

00:00:30.130 --> 00:00:32.009
as a difference of squares.

00:00:32.009 --> 00:00:33.460
So we know on the
bottom that this

00:00:33.460 --> 00:00:39.750
could be factored as x
minus 1 times x plus 1.

00:00:39.750 --> 00:00:42.920
And so if somehow
this thing on the top

00:00:42.920 --> 00:00:48.059
also has an x minus 1 as a
factor, then that x minus 1

00:00:48.059 --> 00:00:49.600
will cancel with
this, and then we're

00:00:49.600 --> 00:00:52.030
not going to have an
issue of dividing by 0.

00:00:52.030 --> 00:00:54.280
The reason why I care
about the x minus 1 term

00:00:54.280 --> 00:00:57.160
is that this is what's making
our denominator equal 0.

00:00:57.160 --> 00:01:01.190
When you say x equals 1, you
have 1 minus 1 times 1 plus 1.

00:01:01.190 --> 00:01:06.240
So 0 times 2, it's this 0
that's making our denominator 0.

00:01:06.240 --> 00:01:09.980
So if we can have an
x minus 1 up here,

00:01:09.980 --> 00:01:14.470
then we can cancel these out
for any x not equal to 1.

00:01:14.470 --> 00:01:17.260
And then we might have
a much simpler thing

00:01:17.260 --> 00:01:19.030
to find the limit of.

00:01:19.030 --> 00:01:21.670
So let's think about whether
x to the third minus 1

00:01:21.670 --> 00:01:24.329
is the product of x minus
1 and something else.

00:01:24.329 --> 00:01:25.870
And to do that we
can do a little bit

00:01:25.870 --> 00:01:27.130
of algebraic long division.

00:01:27.130 --> 00:01:29.940
Some of you guys might already
recognize a pattern here,

00:01:29.940 --> 00:01:32.740
but we'll try to do-- well,
let's divide x minus 1

00:01:32.740 --> 00:01:36.880
into it to see whether
it divides evenly

00:01:36.880 --> 00:01:40.620
into x to the third minus 1.

00:01:40.620 --> 00:01:43.560
So x minus 1-- we just
look at the highest degree

00:01:43.560 --> 00:01:48.690
term-- x goes into x to
the third x squared times.

00:01:48.690 --> 00:01:50.374
Goes x squared times.

00:01:50.374 --> 00:01:51.790
Actually, let me
do it this way so

00:01:51.790 --> 00:01:53.894
that way we can keep
track of the place.

00:01:53.894 --> 00:01:56.060
So this would be x-- this
would be the second degree

00:01:56.060 --> 00:01:59.120
place, first degree place, and
this would be the constant.

00:01:59.120 --> 00:02:01.050
So x to the third minus 1.

00:02:01.050 --> 00:02:04.190
x goes into x to the
third x squared times.

00:02:04.190 --> 00:02:06.780
x squared times x
is x to the third.

00:02:06.780 --> 00:02:10.710
x squared times negative
1 is minus x squared.

00:02:10.710 --> 00:02:13.490
And now we're going to
want to subtract this.

00:02:13.490 --> 00:02:16.990
So we are then left
with x squared.

00:02:16.990 --> 00:02:21.320
x goes into x squared
x times plus x.

00:02:21.320 --> 00:02:23.580
x times x is x squared.

00:02:23.580 --> 00:02:26.487
x times minus 1 is minus x.

00:02:26.487 --> 00:02:28.320
And once again we're
going to subtract this.

00:02:28.320 --> 00:02:31.880
We'll swap the signs,
negative and positive.

00:02:31.880 --> 00:02:34.530
And so these cancel out,
and we're left with x.

00:02:34.530 --> 00:02:38.010
And then we bring
down a minus 1.

00:02:38.010 --> 00:02:42.300
x minus 1 goes into x
minus 1 exactly one time.

00:02:42.300 --> 00:02:44.840
1 times x minus 1 is x minus 1.

00:02:44.840 --> 00:02:48.850
And then you subtract, and
then you have no remainder.

00:02:48.850 --> 00:02:50.320
So this numerator
right over here

00:02:50.320 --> 00:02:58.030
can be factored as x minus 1
times x squared plus x plus 1.

00:02:58.030 --> 00:03:01.140
And so we can say that this
is the same exact thing.

00:03:01.140 --> 00:03:05.210
We can have these cancel out if
we assume x does not equal 1.

00:03:05.210 --> 00:03:11.990
So that is equal to x squared
plus x plus 1 over x plus 1,

00:03:11.990 --> 00:03:16.206
for x does not equal 1.

00:03:16.206 --> 00:03:18.080
And that's completely
fine, because we're not

00:03:18.080 --> 00:03:20.050
evaluating x equals 1.

00:03:20.050 --> 00:03:22.810
We're evaluating
as x approaches 1.

00:03:22.810 --> 00:03:26.690
So this is going to be the
same thing as the limit

00:03:26.690 --> 00:03:36.780
as x approaches 1 of x squared
plus x plus 1 over x plus 1.

00:03:36.780 --> 00:03:39.872
And now this is
much easier to find.

00:03:39.872 --> 00:03:41.330
You could literally
just say, well,

00:03:41.330 --> 00:03:44.230
what happens as we get
right to x equals 1?

00:03:44.230 --> 00:03:47.520
Then you have 1 squared,
which is 1 plus 1

00:03:47.520 --> 00:03:52.420
plus 1, which is 3, over
1 plus 1, which is 2.

00:03:52.420 --> 00:03:55.390
So we get that equaling 3/2.


Two-sided limit from graph

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.070
So we have the graph of a
function right over here,

00:00:03.070 --> 00:00:04.750
and we want to think
about what does

00:00:04.750 --> 00:00:09.861
the limit of f of x as x
approaches 3 appear to be?

00:00:09.861 --> 00:00:11.860
And to do that, let's
just think about the limit

00:00:11.860 --> 00:00:14.550
as x approaches 3 from
values less than 3,

00:00:14.550 --> 00:00:16.680
and from values greater than 3.

00:00:16.680 --> 00:00:18.920
So let's first think
about the limit

00:00:18.920 --> 00:00:24.580
of f of x as x approaches
3 from values less than 3.

00:00:24.580 --> 00:00:26.640
So this little
negative superscript

00:00:26.640 --> 00:00:29.350
says we're going to
approach 3 from below 3.

00:00:29.350 --> 00:00:33.880
From 1, 2, 2.5, 2.99, 2.999.

00:00:33.880 --> 00:00:38.220
So if we approach-- so
this is 3 right over here.

00:00:38.220 --> 00:00:41.530
And we're going take the left
handed, or the left sided

00:00:41.530 --> 00:00:42.030
limit.

00:00:42.030 --> 00:00:45.270
We're going to approach 3
from this direction first.

00:00:45.270 --> 00:00:47.570
So when x is 0, f of x is there.

00:00:47.570 --> 00:00:49.680
When x is 1, f of x is there.

00:00:49.680 --> 00:00:51.900
When x is 2, f of x is there.

00:00:51.900 --> 00:00:54.980
When x is 2 and
1/2, f of x is at 5.

00:00:54.980 --> 00:01:00.780
When x is at-- looks like
roughly 2 and 3/4, we get to 4.

00:01:00.780 --> 00:01:03.740
Looks like about f
of x gets to 4.5.

00:01:03.740 --> 00:01:06.510
And so it looks like
as x approaches 3

00:01:06.510 --> 00:01:09.210
from values less
than 3, it looks

00:01:09.210 --> 00:01:13.470
like our function
is approaching 4.

00:01:13.470 --> 00:01:17.230
So I would say it looks
like the left sided

00:01:17.230 --> 00:01:20.970
limit of f of x as
x approaches 3 is 4.

00:01:20.970 --> 00:01:23.490
Now let's do the same thing
for the right hand side.

00:01:23.490 --> 00:01:32.310
So the limit of f of x
as x approaches 3 from

00:01:32.310 --> 00:01:34.570
values larger than 3.

00:01:34.570 --> 00:01:38.500
So notice when x is equal
to 5, our f of x is up here.

00:01:38.500 --> 00:01:42.340
When x is equal to
4, f of x is here.

00:01:42.340 --> 00:01:45.930
When x is 3 and 1/2, it looks
like we're a little under 2

00:01:45.930 --> 00:01:47.110
for f of x.

00:01:47.110 --> 00:01:49.370
And it looks like we're
getting closer and closer

00:01:49.370 --> 00:01:53.105
as x approaches 3 from
the positive direction,

00:01:53.105 --> 00:01:55.960
or from the right
side, it looks like f

00:01:55.960 --> 00:02:01.410
of x is getting closer
and closer to 1.

00:02:01.410 --> 00:02:04.300
So I would estimate,
based on this graph,

00:02:04.300 --> 00:02:06.430
that the limit of f
of x as x approaches 3

00:02:06.430 --> 00:02:10.820
from the positive
direction is equal to 1.

00:02:10.820 --> 00:02:12.090
Now we have an issue.

00:02:12.090 --> 00:02:15.230
In order for this
limit to exist,

00:02:15.230 --> 00:02:18.600
we have to get the same value as
we approach from the left hand

00:02:18.600 --> 00:02:21.290
side and the right hand
side, but it's clear

00:02:21.290 --> 00:02:23.340
that we are not approaching
the same value when

00:02:23.340 --> 00:02:25.360
we go from the left
hand side as we

00:02:25.360 --> 00:02:27.690
do when we go from
the right hand side.

00:02:27.690 --> 00:02:32.010
So this limit right over
here does not exist.

00:02:32.010 --> 00:02:35.882
Does not exist.

00:02:35.882 --> 00:02:37.590
The only way that this
would have existed

00:02:37.590 --> 00:02:40.430
is if we got the same
value for both of these,

00:02:40.430 --> 00:02:42.140
and then the limit
would be that value.

00:02:42.140 --> 00:02:45.280
But we're clearly not
getting the same value.