Tangent slope as limiting value of secant slope example 2

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.230
Let g of x equal cosine of x.

00:00:04.230 --> 00:00:06.017
So let's actually graph that.

00:00:06.017 --> 00:00:07.850
So that would be the
graph right over there.

00:00:07.850 --> 00:00:09.600
I drew it ahead of time.

00:00:09.600 --> 00:00:13.230
So this is a segment of cosine
of x between x is equal to 0

00:00:13.230 --> 00:00:14.120
and x equals pi.

00:00:14.120 --> 00:00:17.150
Obviously, it keeps on going
in both directions after that.

00:00:17.150 --> 00:00:19.090
And then they ask
us, what is the limit

00:00:19.090 --> 00:00:24.280
as h approaches 0 of g of pi
plus h minus g of pi over h?

00:00:24.280 --> 00:00:26.050
So let's just think
about it a little bit.

00:00:26.050 --> 00:00:28.925
So they are concerning
themselves with g of pi.

00:00:32.360 --> 00:00:35.760
So let's look at the
point pi comma g of pi.

00:00:35.760 --> 00:00:40.130
So that right there is the
point pi comma g of pi.

00:00:40.130 --> 00:00:47.040
And they're also concerning
themselves with g of pi plus h.

00:00:47.040 --> 00:00:49.490
So let's say that
this right over here

00:00:49.490 --> 00:00:53.510
is the x value pi plus h.

00:00:53.510 --> 00:00:59.112
And then this right over here
would be the point pi plus h,

00:00:59.112 --> 00:01:04.857
g of pi plus h-- would be
this point right over here.

00:01:04.857 --> 00:01:06.440
And they're essentially
trying to find

00:01:06.440 --> 00:01:08.530
the slope between
these two points.

00:01:08.530 --> 00:01:11.040
If we wanted to find the
slope between these points,

00:01:11.040 --> 00:01:14.190
it would be change in
our y value over change

00:01:14.190 --> 00:01:17.390
in x, or change in the vertical
over change in the horizontal.

00:01:17.390 --> 00:01:19.530
What is our change
in the vertical?

00:01:19.530 --> 00:01:21.406
What is the change
in the vertical?

00:01:21.406 --> 00:01:22.780
Well, the change
in the vertical,

00:01:22.780 --> 00:01:28.970
we'll take this y value
here-- so g of pi plus h.

00:01:28.970 --> 00:01:33.440
We'll subtract this y
value here-- minus g of pi.

00:01:33.440 --> 00:01:35.420
So that was our
change in the vertical

00:01:35.420 --> 00:01:36.840
over the change
in the horizontal.

00:01:39.390 --> 00:01:40.900
Well, what is that going to be?

00:01:40.900 --> 00:01:45.710
Well that's going to
be pi plus h minus pi.

00:01:48.670 --> 00:01:52.140
And this is exactly what we have
or, before we take the limit,

00:01:52.140 --> 00:01:54.230
what we have right over here.

00:01:54.230 --> 00:02:00.410
So this is going to be g
of pi plus h minus g of pi,

00:02:00.410 --> 00:02:03.640
all of that over--
these pi's cancel out--

00:02:03.640 --> 00:02:05.650
all of that over h.

00:02:05.650 --> 00:02:08.960
Now, so this is the
slope, this slope the way

00:02:08.960 --> 00:02:13.590
I've-- it's a slope of the
secant line right over here.

00:02:13.590 --> 00:02:16.040
Now let's think
about what is going

00:02:16.040 --> 00:02:20.160
to happen as h gets
closer and closer to 0.

00:02:20.160 --> 00:02:22.300
So the way I've drawn
it right over here,

00:02:22.300 --> 00:02:24.230
that means that
this point is going

00:02:24.230 --> 00:02:26.530
to go further and
further to the left.

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This point is
going to go further

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and further and
further to the left.

00:02:30.540 --> 00:02:34.280
So pi plus h, as h approaches
0, is going to approach pi.

00:02:34.280 --> 00:02:37.330
If we assumed h was a
negative value, then

00:02:37.330 --> 00:02:39.920
we would be
approaching from here.

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But what is going to
happen as h becomes

00:02:43.500 --> 00:02:45.530
smaller and smaller and smaller?

00:02:45.530 --> 00:02:52.010
Well, the point pi
plus h, g of pi plus h

00:02:52.010 --> 00:02:54.610
is going to get closer
and closer to this point.

00:02:54.610 --> 00:02:57.800
And the slope of
those secant lines--

00:02:57.800 --> 00:03:01.949
I know it's hard to see,
because it's small here--

00:03:01.949 --> 00:03:03.490
they're going to
start getting closer

00:03:03.490 --> 00:03:07.830
and closer to the slope of
the tangent line right at x

00:03:07.830 --> 00:03:09.450
equals pi.

00:03:09.450 --> 00:03:11.850
So this right over here
really is another way

00:03:11.850 --> 00:03:27.205
of saying the slope of the
tangent line at x equals pi.

00:03:27.205 --> 00:03:28.730
Well, what does that look like?

00:03:28.730 --> 00:03:30.188
Well the slope of
the tangent line,

00:03:30.188 --> 00:03:32.660
we're at a minimum point at
x equals pi right over here.

00:03:32.660 --> 00:03:34.580
Cosine of pi we know
is a negative 1.

00:03:34.580 --> 00:03:36.149
That's a minimum value.

00:03:36.149 --> 00:03:37.440
It's one of the minimum values.

00:03:37.440 --> 00:03:40.160
It keeps going to
that for cosine of x.

00:03:40.160 --> 00:03:43.680
And so its tangent line is just
going to be a horizontal line.

00:03:46.290 --> 00:03:50.650
So we know that this right
over here is going to be 0.

00:03:50.650 --> 00:03:52.876
Now, there are other ways
you could have tackled it.

00:03:52.876 --> 00:03:54.250
We don't have the
tools right now

00:03:54.250 --> 00:03:57.090
to do it purely
algebraically-- to say cosine

00:03:57.090 --> 00:03:59.190
of pi plus h minus cosine of pi.

00:03:59.190 --> 00:04:00.550
There are ways to do it.

00:04:00.550 --> 00:04:02.008
But we're not going
to do that now.

00:04:02.008 --> 00:04:05.640
The other option that you
could do is use a calculator.

00:04:05.640 --> 00:04:08.130
So for example, you
could say, well, OK,

00:04:08.130 --> 00:04:12.640
let's just take very small
h's and just evaluate them.

00:04:12.640 --> 00:04:21.339
So we'll evaluate cosine of pi
plus some h minus cosine of pi

00:04:21.339 --> 00:04:23.450
over that small h.

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And we're going to get
smaller and smaller h's here.

00:04:25.640 --> 00:04:27.223
And so actually,
let's try to do that.

00:04:27.223 --> 00:04:28.830
That could be interesting.

00:04:28.830 --> 00:04:30.490
So let me clear this.

00:04:30.490 --> 00:04:33.230
So let's just take a
really, really small h's,

00:04:33.230 --> 00:04:35.600
h's getting really,
really close to 0.

00:04:35.600 --> 00:04:39.030
So let me make sure I'm in
Radian mode, first of all.

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I'm in Degree.

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So let me fix that.

00:04:41.800 --> 00:04:43.380
All right, perfect.

00:04:43.380 --> 00:04:48.260
And now let me
take cosine of pi.

00:04:51.220 --> 00:05:01.850
I'll take a reasonably small
h, 0.1, minus cosine of pi.

00:05:01.850 --> 00:05:03.260
I always forget where the pi is.

00:05:03.260 --> 00:05:04.970
So that's the numerator.

00:05:04.970 --> 00:05:09.060
And then let me divide
that by the same h, by 0.1.

00:05:09.060 --> 00:05:12.100
So this is just my previous
answer divided by 0.1.

00:05:12.100 --> 00:05:14.440
So I get 0.04.

00:05:14.440 --> 00:05:16.299
Now let me make h even smaller.

00:05:16.299 --> 00:05:18.340
And I'll actually do in
one expression, actually,

00:05:18.340 --> 00:05:18.920
this time.

00:05:18.920 --> 00:05:24.330
So cosine of-- actually,
make it a lot smaller--

00:05:24.330 --> 00:05:34.470
of pi plus 0.0001-- so
1/10,000 more than pi,

00:05:34.470 --> 00:05:37.580
right over there--
minus cosine of pi.

00:05:41.410 --> 00:05:49.930
And now we are going to
divide by this h, 0.0001.

00:05:49.930 --> 00:05:51.610
And what do we get?

00:05:51.610 --> 00:05:54.740
5 times 10 to the
negative 5th-- so you

00:05:54.740 --> 00:05:57.580
see clearly that we're getting
to some really, really, really,

00:05:57.580 --> 00:06:02.980
really small numbers, that this
expression is approaching 0.


Tangent slope as limiting value of secant slope example 1

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:05.490
Let g of x equal
negative 4x plus 7.

00:00:05.490 --> 00:00:08.870
What is the value of the limit
as x approaches negative 1

00:00:08.870 --> 00:00:10.060
of all of this?

00:00:10.060 --> 00:00:12.900
So before we think about this,
let's just visualize the line.

00:00:12.900 --> 00:00:15.140
And then we can think about
what they're asking here.

00:00:15.140 --> 00:00:17.110
So let me draw some axes here.

00:00:17.110 --> 00:00:21.875
So this is my vertical axis
and this is my horizontal axis.

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And let's say this is my x-axis.

00:00:28.360 --> 00:00:30.250
We'll label that the x-axis.

00:00:30.250 --> 00:00:32.159
I'll graph g of
x. g of x is going

00:00:32.159 --> 00:00:35.290
to have a positive-- I guess
you would say y-intercept.

00:00:35.290 --> 00:00:36.650
or vertical axis intercept.

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It's going to have a
slope of negative 4,

00:00:38.358 --> 00:00:40.367
so it's going to look
something like this.

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Let me draw my best.

00:00:41.200 --> 00:00:44.700
So it's going to look
something like that.

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And we already know the slope
here is going to be negative 4.

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We get that right from
this slope intercept

00:00:52.340 --> 00:00:55.774
form of the equation, slope
is equal to negative 4.

00:00:55.774 --> 00:00:57.190
And they ask us,
what is the limit

00:00:57.190 --> 00:00:59.710
as x approaches negative 1
of all of this kind of stuff?

00:00:59.710 --> 00:01:02.110
So let's plot the
point negative 1.

00:01:02.110 --> 00:01:03.980
So when x is
negative 1, so that's

00:01:03.980 --> 00:01:05.610
this point right over here.

00:01:05.610 --> 00:01:10.630
And this point right over here
would be the point negative 1,

00:01:10.630 --> 00:01:12.660
g of negative 1.

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Let me label everything else.

00:01:14.620 --> 00:01:16.710
So I could call this my y-axis.

00:01:16.710 --> 00:01:18.360
I could call this graph.

00:01:18.360 --> 00:01:23.310
This is the graph of
y is equal to g of x.

00:01:23.310 --> 00:01:25.030
So what they're
doing right over here

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is they're finding the slope
between an arbitrary point x, g

00:01:29.310 --> 00:01:31.530
of x, and this point
right over here.

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So let's do that.

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So let's take another x.

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So let's say this is x.

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This would be the x, g of x.

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And this expression
right over here,

00:01:42.290 --> 00:01:47.200
notice it is your change
in the vertical axis.

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That would be your g of x.

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Let me make it this way.

00:01:49.940 --> 00:01:53.240
So this would be your
change in the vertical axis.

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That would be g of x
minus g of negative 1.

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And then that's over-- actually,
let me write it this way

00:02:02.010 --> 00:02:07.420
so you can keep track of the
colors-- minus g of negative 1,

00:02:07.420 --> 00:02:11.920
all of that over your change
in the horizontal axis.

00:02:11.920 --> 00:02:13.860
Well, your change in
the horizontal axis

00:02:13.860 --> 00:02:19.270
is this distance, which is the
same thing as this distance.

00:02:19.270 --> 00:02:22.520
Notice your change in vertical
over change in horizontal,

00:02:22.520 --> 00:02:25.670
change in vertical over
change in horizontal,

00:02:25.670 --> 00:02:28.310
reviewing the green
point as the endpoint.

00:02:28.310 --> 00:02:33.270
So it's going to be
x minus negative 1.

00:02:33.270 --> 00:02:36.240
And this is the exact
same expression.

00:02:36.240 --> 00:02:37.830
These are the exact
same expression.

00:02:37.830 --> 00:02:40.210
You can simplify them,
minus negative 1,

00:02:40.210 --> 00:02:42.640
and this becomes plus.

00:02:42.640 --> 00:02:44.480
This could become a plus 1.

00:02:44.480 --> 00:02:46.914
But these are the
exact same expression.

00:02:46.914 --> 00:02:48.330
So this is the
expression, really,

00:02:48.330 --> 00:02:52.710
for the slope between
negative 1 and g of negative 1

00:02:52.710 --> 00:02:54.840
and an arbitrary x.

00:02:54.840 --> 00:02:59.040
Well, we already know that
no matter what x you pick,

00:02:59.040 --> 00:03:03.180
the slope between x, g of , and
this point right over here is

00:03:03.180 --> 00:03:04.310
going to be constant.

00:03:04.310 --> 00:03:06.090
It's going to be the
slope of the line.

00:03:06.090 --> 00:03:08.310
It's going to be
equal to negative 4.

00:03:08.310 --> 00:03:11.620
This thing is going to
be equal to negative 4.

00:03:11.620 --> 00:03:13.640
It's going to be
equal to negative 4.

00:03:13.640 --> 00:03:16.760
Doesn't matter how close
x gets, and weather

00:03:16.760 --> 00:03:21.140
x comes from the right or
whether x comes from the left.

00:03:21.140 --> 00:03:23.750
So this thing, taking
the limit of this,

00:03:23.750 --> 00:03:25.400
this just gets
you to negative 4.

00:03:25.400 --> 00:03:27.580
It's really just the
slope of the line.

00:03:27.580 --> 00:03:30.670
So even if you were to take
the limit as x approaches

00:03:30.670 --> 00:03:33.740
negative 1, as x gets
closer and closer and closer

00:03:33.740 --> 00:03:35.400
to negative 1, well
then, these points

00:03:35.400 --> 00:03:38.170
are just going to get closer
and closer and closer.

00:03:38.170 --> 00:03:40.067
But every time you
calculate the slope,

00:03:40.067 --> 00:03:42.150
it's just going to be the
slope of the line, which

00:03:42.150 --> 00:03:42.810
is negative 4.

00:03:42.810 --> 00:03:44.560
Now, you could also
do this algebraically.

00:03:44.560 --> 00:03:46.360
And let's try to do
it algebraically.

00:03:46.360 --> 00:03:50.760
So let's actually
just take the limit

00:03:50.760 --> 00:03:54.410
as x approaches
negative 1 of g of x.

00:03:54.410 --> 00:03:56.640
Well, they already
told us what g of x is.

00:03:56.640 --> 00:04:02.980
It is negative 4x plus
7, minus g of negative 1.

00:04:02.980 --> 00:04:05.890
So that's minus, what
is g of negative 1?

00:04:05.890 --> 00:04:08.310
Negative 1 times
4 is positive 4.

00:04:08.310 --> 00:04:12.430
Positive 4 plus 7 is 11.

00:04:12.430 --> 00:04:22.210
All of that over x plus 1,
all of that over x plus 1.

00:04:22.210 --> 00:04:24.791
And that's really
x minus negative 1,

00:04:24.791 --> 00:04:26.290
is you want to think
of it that way.

00:04:26.290 --> 00:04:29.630
But I'll just write x
plus 1 this way here.

00:04:29.630 --> 00:04:33.870
So this is going to
be equal to the limit

00:04:33.870 --> 00:04:38.230
as x approaches negative 1 of,
in our numerator-- let's see.

00:04:38.230 --> 00:04:40.680
7 minus 11 is negative 4.

00:04:40.680 --> 00:04:42.120
We can factor out a negative 4.

00:04:42.120 --> 00:04:47.490
It's a negative 4 times x plus
1, all of that over x plus 1.

00:04:47.490 --> 00:04:50.130
And then since we're just
trying to find the limit as x

00:04:50.130 --> 00:04:54.350
approaches negative 1, so
we can cancel those out.

00:04:54.350 --> 00:04:59.120
And this is going to be
non-zero for any x value

00:04:59.120 --> 00:05:00.650
other than negative 1.

00:05:00.650 --> 00:05:03.800
And so this is going to
be equal to negative 4.

00:05:03.800 --> 00:05:05.346
So either way, we
get negative 4.

00:05:05.346 --> 00:05:07.220
But if you just realize,
hey, this is a line.

00:05:07.220 --> 00:05:08.719
It's going to have
a constant slope.

00:05:08.719 --> 00:05:12.190
This is just the slope
of between some arbitrary

00:05:12.190 --> 00:05:19.390
point on the line and the
point negative 1, 11, really.

00:05:19.390 --> 00:05:22.000
Negative 1, 11, you say, well
that's just going to the same

00:05:22.000 --> 00:05:22.958
as the slope of a line.

00:05:22.958 --> 00:05:24.840
It's negative 4.


Approximating instantaneous rate of change word problem

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.860
The table gives a position
s of a motorcyclist

00:00:03.860 --> 00:00:07.660
for t between 0 and
3, including 0 and 3.

00:00:07.660 --> 00:00:10.950
This is just saying that
t is part of the interval,

00:00:10.950 --> 00:00:13.390
or t in the interval
between 0 and 3.

00:00:13.390 --> 00:00:16.470
And we see that right over here,
where the distance traveled

00:00:16.470 --> 00:00:21.160
s is measured in meters,
and t is time in seconds.

00:00:21.160 --> 00:00:23.460
Assume the motorcyclist
is accelerating

00:00:23.460 --> 00:00:24.720
during a three-second period.

00:00:24.720 --> 00:00:26.094
And they give us
the information.

00:00:26.094 --> 00:00:28.680
This is time between 0
seconds and 3 seconds.

00:00:28.680 --> 00:00:31.170
And here we have the
corresponding distance

00:00:31.170 --> 00:00:33.710
that you could view
as a function of time.

00:00:33.710 --> 00:00:38.470
The average velocity
for t between 1.5 and 2,

00:00:38.470 --> 00:00:43.580
so t between 1.5 and 2
is 23 meters per second.

00:00:43.580 --> 00:00:46.020
So what they did over here
is they figured out, well,

00:00:46.020 --> 00:00:50.700
what is delta s over
delta t in this interval?

00:00:50.700 --> 00:00:55.540
And they figured out that
it was 23 meters per second.

00:00:55.540 --> 00:00:56.670
And you can verify that.

00:00:56.670 --> 00:00:59.450
Your change in s
looks like it's 12.5.

00:00:59.450 --> 00:01:01.960
Your change in time is
point-- or actually,

00:01:01.960 --> 00:01:07.230
this looks like it's 11.5.

00:01:07.230 --> 00:01:09.020
Yeah, 11.5.

00:01:09.020 --> 00:01:11.090
Your change in time is 0.5.

00:01:11.090 --> 00:01:15.040
11.5 divided by 0.5 is 23.

00:01:15.040 --> 00:01:17.340
So that makes sense.

00:01:17.340 --> 00:01:19.280
And then they tell us
the average velocity

00:01:19.280 --> 00:01:24.810
for t between 2 and 2.5.

00:01:24.810 --> 00:01:29.500
So change in our distance
over change in time,

00:01:29.500 --> 00:01:33.280
they say is 31.8
meters per second.

00:01:33.280 --> 00:01:36.440
And then they say, estimate
the instantaneous velocity at t

00:01:36.440 --> 00:01:39.380
equals 2 seconds and
use this value to write

00:01:39.380 --> 00:01:44.050
the equation of a line tangent
to s of t at the time t

00:01:44.050 --> 00:01:45.280
equals 2.

00:01:45.280 --> 00:01:47.240
So we can try to approximate.

00:01:47.240 --> 00:01:50.900
We can approximate the
slope of the tangent line

00:01:50.900 --> 00:01:53.990
right over here, right when
t equals 2 seconds, by taking

00:01:53.990 --> 00:01:58.250
the average of the slopes of
the tangent lines between 1.5

00:01:58.250 --> 00:02:00.150
and 2, and 2 and 2.5.

00:02:00.150 --> 00:02:03.100
So essentially, to approximate
the slope of the tangent line,

00:02:03.100 --> 00:02:05.740
we're going to take the average
of these two rates of change

00:02:05.740 --> 00:02:08.240
right over here, the
average of these two slopes.

00:02:08.240 --> 00:02:09.070
So let's do that.

00:02:09.070 --> 00:02:17.049
So the average is going
to be 23 plus 31.8 over 2.

00:02:17.049 --> 00:02:18.590
And let's see, what
is that equal to?

00:02:18.590 --> 00:02:25.000
That is equal to 54.8/2.

00:02:25.000 --> 00:02:26.711
And what is that equal to?

00:02:26.711 --> 00:02:27.210
Let's see.

00:02:27.210 --> 00:02:29.740
54 divided by 2 is 27.

00:02:29.740 --> 00:02:34.840
So it's 27.4.

00:02:34.840 --> 00:02:37.340
So we can use that
as our approximation

00:02:37.340 --> 00:02:38.980
for the instantaneous
rate of change

00:02:38.980 --> 00:02:40.800
for the slope of
the tangent line.

00:02:40.800 --> 00:02:43.360
And now we have to
actually figure out

00:02:43.360 --> 00:02:45.010
what that equation actually is.

00:02:45.010 --> 00:02:46.600
They don't just want the slope.

00:02:46.600 --> 00:02:48.440
So this is the slope
right over here.

00:02:48.440 --> 00:02:52.560
And they say that they want
it in point-slope form.

00:02:52.560 --> 00:02:55.592
And they remind us that t
is the independent variable.

00:02:55.592 --> 00:02:57.800
So when you're putting
something in point-slope form,

00:02:57.800 --> 00:03:00.008
it really just comes out of
the definition of a line.

00:03:00.008 --> 00:03:02.550
A line always has
a constant slope.

00:03:02.550 --> 00:03:09.910
So let's just imagine taking
a random point on that line,

00:03:09.910 --> 00:03:15.550
t-- let me write it this
way-- t comma capital S--

00:03:15.550 --> 00:03:19.620
a random point on the line,
on the tangent line here.

00:03:19.620 --> 00:03:22.600
Well, the slope between
that and this point

00:03:22.600 --> 00:03:24.800
is always going to be constant.

00:03:24.800 --> 00:03:28.060
So what's the slope between
this point and this point?

00:03:28.060 --> 00:03:33.120
Well, your change in S
is going to be S minus 2

00:03:33.120 --> 00:03:40.110
over your change in t,
which is t minus-- oh sorry,

00:03:40.110 --> 00:03:42.770
your S is S. This is
confusing sometimes.

00:03:42.770 --> 00:03:49.230
S minus 30.2 over your
change in t, t minus 2,

00:03:49.230 --> 00:03:51.366
is equal to your slope.

00:03:51.366 --> 00:03:52.740
Anywhere along
that tangent line,

00:03:52.740 --> 00:03:55.720
you're going to have
that slope, 27.4.

00:03:55.720 --> 00:03:58.250
And then you multiply
both sides by t minus 2,

00:03:58.250 --> 00:04:00.890
and you've put it
in point-slope form.

00:04:00.890 --> 00:04:05.950
So this is the same
thing as S minus 30.2

00:04:05.950 --> 00:04:13.750
is equal to 27.4
times t minus 2.


Tangent slope as limiting value of secant slope example 3

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:03.930
With the graph of the
function f as an aid,

00:00:03.930 --> 00:00:05.960
evaluate the following limits.

00:00:05.960 --> 00:00:08.109
So the first one is the
limit as x approaches

00:00:08.109 --> 00:00:13.440
3 of f of x minus f
of 3 over x minus 3.

00:00:13.440 --> 00:00:17.300
So let's think about x minus--
x equals 3 is right over here.

00:00:17.300 --> 00:00:21.920
This right over here is f
of 3, or we could say f of 3

00:00:21.920 --> 00:00:23.410
is 1 right over here.

00:00:23.410 --> 00:00:27.580
That's the point 3 comma f of 3.

00:00:27.580 --> 00:00:29.200
And they're essentially
trying to find

00:00:29.200 --> 00:00:32.790
the slope between an
arbitrary x and that

00:00:32.790 --> 00:00:36.210
point as that x gets
closer and closer to 3.

00:00:36.210 --> 00:00:39.200
So we can imagine an
x that is above 3,

00:00:39.200 --> 00:00:41.490
that is, say, right over here.

00:00:41.490 --> 00:00:45.120
Well, if we're trying to find
the slope between this x comma

00:00:45.120 --> 00:00:49.000
f of x and 3 comma
f of 3, we see

00:00:49.000 --> 00:00:51.350
that it gets this
exact same form.

00:00:51.350 --> 00:00:52.730
Your end point is f of x.

00:00:52.730 --> 00:00:56.930
So it's f of x minus
f of 3 is your change

00:00:56.930 --> 00:00:58.470
in the vertical axis.

00:00:58.470 --> 00:01:00.350
That's this distance
right over here.

00:01:00.350 --> 00:01:02.660
And we would divide
by your change

00:01:02.660 --> 00:01:04.810
in the horizontal axis,
which is your change in x.

00:01:04.810 --> 00:01:06.850
And that's going
to be x minus 3.

00:01:06.850 --> 00:01:09.380
So that's the exact expression
that we have up here when

00:01:09.380 --> 00:01:12.150
I picked this as an arbitrary x.

00:01:12.150 --> 00:01:14.680
And we see that that
slope, just by looking

00:01:14.680 --> 00:01:16.890
at the line between
those two intervals,

00:01:16.890 --> 00:01:18.420
seems to be negative 2.

00:01:18.420 --> 00:01:21.080
And the slope was the same
thing if we go the other side.

00:01:21.080 --> 00:01:24.420
If x was less than
3, then we also

00:01:24.420 --> 00:01:27.090
would have a slope
of negative 2.

00:01:27.090 --> 00:01:28.890
Either way, we have a
slope of negative 2.

00:01:28.890 --> 00:01:30.720
And that's important
because this limit

00:01:30.720 --> 00:01:32.550
is just the limit
as x approaches 3.

00:01:32.550 --> 00:01:37.540
So it can be as x approaches
3 from the positive direction

00:01:37.540 --> 00:01:40.780
or from the negative direction.

00:01:40.780 --> 00:01:45.460
But in either case, the slope,
as we get closer and closer

00:01:45.460 --> 00:01:50.027
to this point right over
here, is negative 2.

00:01:50.027 --> 00:01:52.110
Now let's think about what
they're asking us here.

00:01:52.110 --> 00:01:54.626
So we have 8, f of 8.

00:01:54.626 --> 00:01:55.280
So let's think.

00:01:55.280 --> 00:01:56.370
We have 8.

00:01:56.370 --> 00:01:58.240
This is 8 comma f of 8.

00:01:58.240 --> 00:02:02.380
So that's 8 comma f
of 8 right over there.

00:02:02.380 --> 00:02:05.480
And they have f of 8 plus h.

00:02:05.480 --> 00:02:07.165
So our temptation
might be to say, hey,

00:02:07.165 --> 00:02:09.680
8 plus h is going to
be someplace out here.

00:02:09.680 --> 00:02:11.710
It's going to be
something larger than 8.

00:02:11.710 --> 00:02:13.950
But notice, they
have the limit as h

00:02:13.950 --> 00:02:17.767
approaches 0 from the
negative direction.

00:02:17.767 --> 00:02:19.600
So approaching 0 from
the negative direction

00:02:19.600 --> 00:02:21.460
means you're coming
to 0 from below.

00:02:21.460 --> 00:02:24.380
You're at negative 1,
negative 0.5, negative 0.1,

00:02:24.380 --> 00:02:26.690
negative 0.0001.

00:02:26.690 --> 00:02:29.920
So h is actually going
to be a negative number.

00:02:29.920 --> 00:02:33.040
So 8 plus h would actually
be-- we could just

00:02:33.040 --> 00:02:34.150
pick an arbitrary point.

00:02:34.150 --> 00:02:36.630
It could be something
like this right over here.

00:02:36.630 --> 00:02:40.120
So this might be the
value of 8 plus h.

00:02:40.120 --> 00:02:46.660
And this would be the
value of f of 8 plus h.

00:02:46.660 --> 00:02:50.870
So once again, they're
finding-- or this expression

00:02:50.870 --> 00:02:54.550
is the slope between
these two points.

00:02:54.550 --> 00:02:57.680
And then we are taking the
limit as h approaches 0

00:02:57.680 --> 00:02:58.980
from the negative direction.

00:02:58.980 --> 00:03:01.760
So as h gets closer
and closer to 0,

00:03:01.760 --> 00:03:04.920
this down here moves further
and further to the right.

00:03:04.920 --> 00:03:08.950
And these points move closer
and closer and closer together.

00:03:08.950 --> 00:03:10.450
So this is really
just an expression

00:03:10.450 --> 00:03:12.930
of the slope of
the line, roughly--

00:03:12.930 --> 00:03:14.210
and we see that it's constant.

00:03:14.210 --> 00:03:16.490
So what's the slope of the
line over this interval?

00:03:16.490 --> 00:03:18.573
Well, you can just eyeball
it and see, well, look.

00:03:18.573 --> 00:03:24.390
Every time x changes by 1,
our f of x changes by 1.

00:03:24.390 --> 00:03:27.680
So the slope of the
line there is 1.

00:03:27.680 --> 00:03:29.640
It would have been a
completely different thing

00:03:29.640 --> 00:03:32.050
if this said limit
as h approaches 0

00:03:32.050 --> 00:03:33.570
from the positive direction.

00:03:33.570 --> 00:03:38.400
Then we would be looking
at points over here.

00:03:38.400 --> 00:03:41.190
And we would see that we would
slowly approach, essentially,

00:03:41.190 --> 00:03:45.058
a vertical slope, kind
of an infinite slope.


Simplifying square roots

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:09.060
Let's see if we can simplify 5
times the square root of 117.

00:00:09.060 --> 00:00:13.060
So 117 doesn't jump out at me as
some type of a perfect square.

00:00:13.060 --> 00:00:14.990
So let's actually take
its prime factorization

00:00:14.990 --> 00:00:20.130
and see if any of those prime
factors show up more than once.

00:00:20.130 --> 00:00:21.750
So clearly it's an odd number.

00:00:21.750 --> 00:00:24.140
It's clearly not divisible by 2.

00:00:24.140 --> 00:00:25.727
To test whether
it's divisible by 3,

00:00:25.727 --> 00:00:27.060
we can add up all of the digits.

00:00:27.060 --> 00:00:29.810
And we explain why this works in
another place on Khan Academy.

00:00:29.810 --> 00:00:31.860
But if you add up all
the digits, you get a 9.

00:00:31.860 --> 00:00:36.225
And 9 is divisible by 3, so 117
is going to be divisible by 3.

00:00:36.225 --> 00:00:37.600
Now, let's do a
little aside here

00:00:37.600 --> 00:00:41.340
to figure out what 117
divided by 3 actually is.

00:00:41.340 --> 00:00:43.700
So 3 doesn't go into 1.

00:00:43.700 --> 00:00:46.010
It does go into 11, three times.

00:00:46.010 --> 00:00:47.670
3 times 3 is 9.

00:00:47.670 --> 00:00:50.390
Subtract, you got
a remainder of 2.

00:00:50.390 --> 00:00:53.400
Bring down a 7.

00:00:53.400 --> 00:00:55.850
3 goes into 27 nine times.

00:00:55.850 --> 00:00:58.087
9 times 3 is 27.

00:00:58.087 --> 00:00:59.170
Subtract, and you're done.

00:00:59.170 --> 00:01:02.080
It goes in perfectly.

00:01:02.080 --> 00:01:07.550
So we can factor
117 as 3 times 39.

00:01:07.550 --> 00:01:10.935
Now 39, we can factor as--
that jumps out more at us

00:01:10.935 --> 00:01:13.010
that that's divisible by 3.

00:01:13.010 --> 00:01:15.820
That's equivalent to 3 times 13.

00:01:15.820 --> 00:01:18.320
And then all of these
are now prime numbers.

00:01:18.320 --> 00:01:23.580
So we could say that this
thing is the same as 5 times

00:01:23.580 --> 00:01:34.585
the square root of
3 times 3 times 13.

00:01:37.061 --> 00:01:39.560
And this is going to be the
same thing as-- and we know this

00:01:39.560 --> 00:01:43.210
from our exponent
properties-- 5 times

00:01:43.210 --> 00:01:54.880
the square root of 3 times 3
times the square root of 13.

00:01:54.880 --> 00:01:56.744
Now, what's the square
root of 3 times 3?

00:01:56.744 --> 00:01:58.160
Well, that's the
square root of 9.

00:01:58.160 --> 00:01:59.730
That's the square
root of 3 squared.

00:01:59.730 --> 00:02:02.120
Any of those-- well, that's
just going to give you 3.

00:02:02.120 --> 00:02:04.590
So this is just going
to simplify to 3.

00:02:04.590 --> 00:02:10.470
So this whole thing is 5 times
3 times the square root of 13.

00:02:10.470 --> 00:02:14.750
So this part right over
here would give us 15 times

00:02:14.750 --> 00:02:19.850
the square root of 13.

00:02:19.850 --> 00:02:21.750
Let's do one more example here.

00:02:21.750 --> 00:02:29.896
So let's try to simplify 3
times the square root of 26.

00:02:29.896 --> 00:02:31.770
I'm actually going
to put 26 in yellow,

00:02:31.770 --> 00:02:35.160
like I did in the
previous problem.

00:02:35.160 --> 00:02:37.442
Well, 26 is clearly
an even number,

00:02:37.442 --> 00:02:38.900
so it's going to
be divisible by 2.

00:02:38.900 --> 00:02:41.917
We can rewrite it as 2 times 13.

00:02:41.917 --> 00:02:42.750
And then we're done.

00:02:42.750 --> 00:02:43.820
13 is a prime number.

00:02:43.820 --> 00:02:45.860
We can't factor this any more.

00:02:45.860 --> 00:02:48.204
And so 26 doesn't have
any perfect squares in it.

00:02:48.204 --> 00:02:49.620
It's not like we
can factor it out

00:02:49.620 --> 00:02:50.970
as a factor of
some other numbers

00:02:50.970 --> 00:02:52.720
and some perfect squares
like we did here.

00:02:52.720 --> 00:02:55.430
117 is 13 times 9.

00:02:55.430 --> 00:02:58.740
It's the product of a
perfect square and 13.

00:02:58.740 --> 00:03:01.645
26 isn't, so we've simplified
this about as much as we can.

00:03:01.645 --> 00:03:08.138
We would just leave this as 3
times the square root of 26.


Cube root of a non-perfect cube

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:07.460
Let's see if we can find
the cube root of 3,430.

00:00:07.460 --> 00:00:10.680
And if you're like me, it
doesn't jump out of your mind

00:00:10.680 --> 00:00:13.374
what number times that same
number times that same number--

00:00:13.374 --> 00:00:14.790
if you have three
of those numbers

00:00:14.790 --> 00:00:16.456
and you were to
multiply them together--

00:00:16.456 --> 00:00:18.230
would be equal to 3,430.

00:00:18.230 --> 00:00:21.050
So what I'm going to do is
to try to prime factorize

00:00:21.050 --> 00:00:24.780
this to find all the
prime factors of 3,430

00:00:24.780 --> 00:00:26.360
and see if any of
those prime factors

00:00:26.360 --> 00:00:27.850
show up at least three times.

00:00:27.850 --> 00:00:29.600
And that'll help us with this.

00:00:29.600 --> 00:00:35.930
So 3,430-- it's clearly
divisible by 5 and 2,

00:00:35.930 --> 00:00:37.620
or it's divisible by 10.

00:00:37.620 --> 00:00:39.090
So let's do that.

00:00:39.090 --> 00:00:41.040
So first we can divide it by 2.

00:00:41.040 --> 00:00:43.560
It's 2 times-- let's see.

00:00:43.560 --> 00:00:48.430
3,430 divided by 2 is 1,715.

00:00:48.430 --> 00:00:50.820
Then we can divide
it by 5, as well.

00:00:50.820 --> 00:00:55.072
We can factor 1,715
into 5 and-- let

00:00:55.072 --> 00:00:57.280
me do a little bit of long
division on the side here.

00:00:57.280 --> 00:01:02.730
So if I have 1,715, and I'm
going to divide it by 5.

00:01:02.730 --> 00:01:04.269
5 doesn't go into 1.

00:01:04.269 --> 00:01:06.620
It goes into 17 three times.

00:01:06.620 --> 00:01:08.790
3 times 5 is 15.

00:01:08.790 --> 00:01:14.150
Subtract, you get 2, and
then you bring down a 1.

00:01:14.150 --> 00:01:16.760
5 goes into 21 four times.

00:01:16.760 --> 00:01:18.710
4 times 5 is 20.

00:01:18.710 --> 00:01:20.090
Subtract.

00:01:20.090 --> 00:01:22.770
Bring down the 5.

00:01:22.770 --> 00:01:27.320
5 goes into 15 three times,
so it goes exactly 343 times.

00:01:27.320 --> 00:01:34.470
So 1,715 can be factored
into 5 times 343.

00:01:34.470 --> 00:01:36.940
Now, 343 might not
jump out at you

00:01:36.940 --> 00:01:39.510
as a number that
is easy to factor.

00:01:39.510 --> 00:01:43.040
It's clearly an odd number,
so it won't be divisible by 2.

00:01:43.040 --> 00:01:46.870
Its digits add up to 10,
which is not divisible by 3.

00:01:46.870 --> 00:01:48.579
So this isn't going
to be divisible by 3.

00:01:48.579 --> 00:01:50.078
It's not going to
be divisible by 4,

00:01:50.078 --> 00:01:51.760
because it's not divisible by 2.

00:01:51.760 --> 00:01:53.860
It's not going to
be divisible by 5.

00:01:53.860 --> 00:01:56.570
If it wasn't
divisible by 3 or 2,

00:01:56.570 --> 00:01:58.780
it's not going to
be divisible by 6.

00:01:58.780 --> 00:02:01.070
And now we get to 7.

00:02:01.070 --> 00:02:02.700
Usually when you
see a nutty number

00:02:02.700 --> 00:02:04.075
like this that
doesn't seem to be

00:02:04.075 --> 00:02:05.500
divisible by a lot
of things, it's

00:02:05.500 --> 00:02:09.190
always a good idea to try
things like 7, 11, 13.

00:02:09.190 --> 00:02:12.020
Because those tend to construct
very interesting numbers.

00:02:12.020 --> 00:02:14.390
So let's see if this
is divisible by 7.

00:02:14.390 --> 00:02:21.730
So if I take 343 and if
I want to divide it by 7,

00:02:21.730 --> 00:02:26.370
7 goes into 30-- it doesn't
go into 3-- 7 goes into 34

00:02:26.370 --> 00:02:27.520
four times.

00:02:27.520 --> 00:02:30.800
4 times 7 is 28.

00:02:30.800 --> 00:02:33.970
Subtract, 34 minus 28 is 6.

00:02:33.970 --> 00:02:35.690
Bring down a 3.

00:02:35.690 --> 00:02:37.830
7 goes into 63 nine times.

00:02:37.830 --> 00:02:39.800
9 times 7 is 63.

00:02:39.800 --> 00:02:40.914
Subtract.

00:02:40.914 --> 00:02:42.080
We don't have any remainder.

00:02:42.080 --> 00:02:43.830
And I forgot to do
that last step up here.

00:02:43.830 --> 00:02:45.470
3 times 15 is 15.

00:02:45.470 --> 00:02:46.750
Subtract, no remainder.

00:02:46.750 --> 00:02:48.110
It went in exactly.

00:02:48.110 --> 00:02:53.480
So here, 343 can be
factored into 7 and 49.

00:02:57.550 --> 00:02:59.510
And 49 might jump out at you.

00:02:59.510 --> 00:03:04.502
It can be factored
into 7 times 7.

00:03:04.502 --> 00:03:05.460
So this is interesting.

00:03:05.460 --> 00:03:09.560
I can rewrite all of this
here-- the cube root of 3,430--

00:03:09.560 --> 00:03:15.160
now as the cube
root of-- I'm just

00:03:15.160 --> 00:03:18.170
going to write it in its
factored form-- 2 times

00:03:18.170 --> 00:03:22.280
5 times-- I could write
7 times 7 times 7,

00:03:22.280 --> 00:03:26.450
or I could write times
7 to the third power.

00:03:26.450 --> 00:03:29.320
That captures these three
7's right over here.

00:03:29.320 --> 00:03:31.700
I have three 7's, and then
I'm multiplying them together.

00:03:31.700 --> 00:03:34.660
So that's 7 to the third power.

00:03:34.660 --> 00:03:36.050
And from our
exponent properties,

00:03:36.050 --> 00:03:41.310
we know that this is the
exact same thing as the cube

00:03:41.310 --> 00:03:50.850
root of 2 times 5 times
the cube root-- so

00:03:50.850 --> 00:03:53.210
let me do that in
that same, just so we

00:03:53.210 --> 00:03:55.180
see what colors
we're dealing with.

00:03:55.180 --> 00:03:59.110
So the cube root of 2 times 5,
which is the cube root of 10,

00:03:59.110 --> 00:04:03.470
times the cube
root-- and I think

00:04:03.470 --> 00:04:09.340
you see where this is going--
of 7 to the third power.

00:04:09.340 --> 00:04:13.510
Keeping track of the
colors is the hard part.

00:04:13.510 --> 00:04:15.510
And the cube root of 10,
we just leave it as 10.

00:04:15.510 --> 00:04:17.982
We know the prime factorization
of 10 is 2 times 5,

00:04:17.982 --> 00:04:20.190
so you're not going to just
get a very simple integer

00:04:20.190 --> 00:04:20.689
answer here.

00:04:20.689 --> 00:04:22.340
You would get some
decimal answer here,

00:04:22.340 --> 00:04:24.660
but here you get a very
clear integer answer.

00:04:24.660 --> 00:04:26.850
The cube root of 7 to
the third, well, that's

00:04:26.850 --> 00:04:27.940
just going to be 7.

00:04:27.940 --> 00:04:29.780
So this is just going to be 7.

00:04:29.780 --> 00:04:31.360
So our entire thing simplifies.

00:04:31.360 --> 00:04:39.090
This is equal to 7 times
the cube root of 10.

00:04:39.090 --> 00:04:40.700
And this is about
as simplified as we

00:04:40.700 --> 00:04:42.650
can get just using
hand arithmetic.

00:04:42.650 --> 00:04:46.120
If you want to get
the exact number here,

00:04:46.120 --> 00:04:49.725
you're probably best
off using a calculator.


Dividing negative fractions

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.750
Let's do some examples
dividing fractions.

00:00:03.750 --> 00:00:15.810
Let's say that I have negative
5/6 divided by positive 3/4.

00:00:15.810 --> 00:00:18.660
Well, we've already talked about
when you divide by something,

00:00:18.660 --> 00:00:22.520
it's the exact same thing as
multiplying by its reciprocal.

00:00:22.520 --> 00:00:30.540
So this is going to be the
exact same thing as negative 5/6

00:00:30.540 --> 00:00:34.850
times the reciprocal
of 3/4, which is 4/3.

00:00:34.850 --> 00:00:37.850
I'm just swapping the
numerator and the denominator.

00:00:37.850 --> 00:00:40.460
So this is going to be 4/3.

00:00:40.460 --> 00:00:43.520
And we've already seen lots of
examples multiplying fractions.

00:00:43.520 --> 00:00:46.350
This is going to be the
numerators times each other.

00:00:46.350 --> 00:00:50.120
So we're going to multiply
negative 5 times 4.

00:00:50.120 --> 00:00:51.620
I'll give the
negative sign to the 5

00:00:51.620 --> 00:00:54.730
there, so negative 5 times 4.

00:00:54.730 --> 00:00:56.730
Let me do 4 in
that yellow color.

00:00:56.730 --> 00:00:58.930
And then the denominator
is 6 times 3.

00:01:03.520 --> 00:01:07.460
Now, in the numerator here, you
see we have a negative number.

00:01:07.460 --> 00:01:09.670
You might already know
that 5 times 4 is 20,

00:01:09.670 --> 00:01:11.336
and you just have to
remember that we're

00:01:11.336 --> 00:01:13.610
multiplying a negative
times a positive.

00:01:13.610 --> 00:01:15.930
We're essentially going to
have negative 5 four times.

00:01:15.930 --> 00:01:19.700
So negative 5 plus negative 5
plus negative 5 plus negative 5

00:01:19.700 --> 00:01:21.500
is negative 20.

00:01:21.500 --> 00:01:24.400
So the numerator
here is negative 20.

00:01:24.400 --> 00:01:26.840
And the denominator here is 18.

00:01:26.840 --> 00:01:30.450
So we get 20/18, but
we can simplify this.

00:01:30.450 --> 00:01:32.400
Both the numerator
and the denominator,

00:01:32.400 --> 00:01:33.970
they're both divisible by 2.

00:01:33.970 --> 00:01:35.450
So let's divide them both by 2.

00:01:35.450 --> 00:01:37.155
Let me give myself
a little more space.

00:01:37.155 --> 00:01:39.990
So if we divide both the
numerator and the denominator

00:01:39.990 --> 00:01:42.102
by 2, just to
simplify this-- and I

00:01:42.102 --> 00:01:43.560
picked 2 because
that's the largest

00:01:43.560 --> 00:01:45.060
number that goes
into both of these.

00:01:45.060 --> 00:01:48.530
It's the greatest common
divisor of 20 and 18.

00:01:48.530 --> 00:01:53.120
20 divided by 2 is 10,
and 18 divided by 2 is 9.

00:01:53.120 --> 00:01:57.240
So negative 5/6 divided
by 3/4 is-- oh, I

00:01:57.240 --> 00:01:58.960
have to be very careful here.

00:01:58.960 --> 00:02:02.830
It's negative 10/9, just
how we always learned.

00:02:02.830 --> 00:02:05.325
If you have a negative
divided by a positive,

00:02:05.325 --> 00:02:07.350
if the signs are
different, then you're

00:02:07.350 --> 00:02:09.270
going to get a negative value.

00:02:09.270 --> 00:02:11.320
Let's do another example.

00:02:11.320 --> 00:02:20.100
Let's say that I have negative
4 divided by negative 1/2.

00:02:24.150 --> 00:02:26.295
So using the exact
logic that we just said,

00:02:26.295 --> 00:02:27.920
we say, hey look,
dividing by something

00:02:27.920 --> 00:02:31.040
is equivalent to multiplying
by its reciprocal.

00:02:31.040 --> 00:02:34.350
So this is going to be
equal to negative 4.

00:02:34.350 --> 00:02:36.050
And instead of writing
it as negative 4,

00:02:36.050 --> 00:02:37.466
let me just write
it as a fraction

00:02:37.466 --> 00:02:39.232
so that we are clear
what its numerator is

00:02:39.232 --> 00:02:40.960
and what its denominator is.

00:02:40.960 --> 00:02:44.900
So negative 4 is the exact
same thing as negative 4/1.

00:02:44.900 --> 00:02:49.130
And we're going to multiply
that times the reciprocal

00:02:49.130 --> 00:02:51.040
of negative 1/2.

00:02:51.040 --> 00:02:53.630
The reciprocal of negative
1/2 is negative 2/1.

00:02:56.630 --> 00:02:58.420
You could view it
as negative 2/1,

00:02:58.420 --> 00:03:00.740
or you could view it as
positive 2 over negative 1,

00:03:00.740 --> 00:03:02.600
or you could view
it as negative 2.

00:03:02.600 --> 00:03:04.880
Either way, these are
all the same value.

00:03:04.880 --> 00:03:06.990
And now we're ready to multiply.

00:03:06.990 --> 00:03:09.200
Notice, all I did here,
I rewrote the negative 4

00:03:09.200 --> 00:03:10.540
just as negative 4/1.

00:03:10.540 --> 00:03:13.190
Negative 4 divided
by 1 is negative 4.

00:03:13.190 --> 00:03:17.220
And here, for the negative
1/2, since I'm multiplying now,

00:03:17.220 --> 00:03:19.150
I'm multiplying
by its reciprocal.

00:03:19.150 --> 00:03:21.710
I've swapped the denominator
and the numerator.

00:03:21.710 --> 00:03:24.440
Or I swapped the denominator
and the numerator.

00:03:24.440 --> 00:03:26.630
What was the denominator
is now the numerator.

00:03:26.630 --> 00:03:29.117
What was the numerator
is now the denominator.

00:03:29.117 --> 00:03:30.200
And I'm ready to multiply.

00:03:30.200 --> 00:03:32.991
This is going to be equal to--
I gave both the negative signs

00:03:32.991 --> 00:03:34.490
to the numerator
so it's going to be

00:03:34.490 --> 00:03:41.130
negative 4 times negative
2 in the numerator.

00:03:41.130 --> 00:03:44.542
And then in the denominator,
it's going to be 1 times 1.

00:03:44.542 --> 00:03:45.500
Let me write that down.

00:03:45.500 --> 00:03:51.150
1 times 1.

00:03:51.150 --> 00:03:54.180
And so this gives us,
so we have a negative 4

00:03:54.180 --> 00:03:56.300
times a negative 2.

00:03:56.300 --> 00:03:58.470
So it's a negative
times a negative,

00:03:58.470 --> 00:04:00.400
so we're going to get
a positive value here.

00:04:00.400 --> 00:04:01.800
And 4 times 2 is 8.

00:04:01.800 --> 00:04:05.060
So this is a positive 8 over 1.

00:04:05.060 --> 00:04:10.274
And 8 divided by 1
is just equal to 8.


Converting a fraction to a repeating decimal (example)

WEBVTT
Kind: captions
Language: en

00:00:00.551 --> 00:00:03.607
PROBLEM: "Express the rational number 19/27

00:00:03.607 --> 00:00:06.538
(or 19 27ths) as a terminating decimal

00:00:06.538 --> 00:00:08.666
or a decimal that eventually repeats.

00:00:08.666 --> 00:00:10.371
Include only the first six digits

00:00:10.371 --> 00:00:12.651
of the decimal in your answer."

00:00:12.651 --> 00:00:14.599
Let me give this a shot.

00:00:14.599 --> 00:00:18.791
So we want to express 19/27 –

00:00:18.791 --> 00:00:23.654
which is the same thing as 19 ÷ 27 – as a decimal.

00:00:23.654 --> 00:00:26.349
So let's divide 27 into 19.

00:00:26.349 --> 00:00:29.485
So 27 going into 19.

00:00:29.485 --> 00:00:31.264
And we know it's going to involve some decimals

00:00:31.264 --> 00:00:33.707
over here, because 27 is larger than 19,

00:00:33.707 --> 00:00:36.291
and it doesn't divide perfectly.

00:00:36.291 --> 00:00:38.131
So let's get into this.

00:00:38.131 --> 00:00:39.305
So 27 doesn't go into 1.

00:00:39.305 --> 00:00:40.923
It doesn't go into 19.

00:00:40.923 --> 00:00:43.994
It does go into 190.

00:00:43.994 --> 00:00:47.224
And it looks like 27 is roughly 30.

00:00:47.224 --> 00:00:48.794
It's a little less than 30.

00:00:48.794 --> 00:00:50.430
30 times 6 would be 180.

00:00:50.430 --> 00:00:53.132
So let's go with it going 6 times.

00:00:53.132 --> 00:00:54.045
Let's see if that works out.

00:00:54.045 --> 00:00:57.402
Well, 6 × 7 is 42.

00:00:57.402 --> 00:01:01.604
6 × 2 is 12, + 4 is 16.

00:01:01.604 --> 00:01:06.365
And when we subtract, 190 - 162 is going to get us –

00:01:06.365 --> 00:01:08.942
Actually, we could've had another 27 in there.

00:01:08.942 --> 00:01:10.716
Because when we subtract –

00:01:10.716 --> 00:01:12.510
So we get a 10 from the 10's place.

00:01:12.510 --> 00:01:13.966
So that becomes 8 10's.

00:01:13.966 --> 00:01:15.751
This became 28.

00:01:15.751 --> 00:01:17.522
So we could have put one more 27 in there.

00:01:17.522 --> 00:01:19.143
So let's do that.

00:01:19.143 --> 00:01:22.476
So let's put one more 27 in there.

00:01:22.476 --> 00:01:24.328
So 7 27's.

00:01:24.328 --> 00:01:26.923
7 × 7 is 49.

00:01:26.923 --> 00:01:31.190
7 × 2 is 14, + 4 is 18.

00:01:31.190 --> 00:01:33.459
And now our remainder is 1.

00:01:33.459 --> 00:01:37.859
We can bring down another 0.

00:01:37.859 --> 00:01:39.912
27 goes into 10 0 times.

00:01:39.912 --> 00:01:42.146
0 × 27 is 0. [Not "10," as Sal states by mistake.]

00:01:42.146 --> 00:01:44.102
Subtract – we have a remainder of 10.

00:01:44.102 --> 00:01:46.694
But now, we have to bring down another 0.

00:01:46.694 --> 00:01:51.014
So let's bring down this 0 right over here.

00:01:51.014 --> 00:01:56.494
So now, 27 goes into 100 3 times.

00:01:56.494 --> 00:02:05.944
3 × 27 is 60 + 21, is 81.

00:02:05.944 --> 00:02:09.154
And then we subtract: 100 - 81.

00:02:09.154 --> 00:02:10.436
Well, we could take 100 from

00:02:10.436 --> 00:02:13.331
the 100's place, and make it 10 10's.

00:02:13.331 --> 00:02:15.390
And then we could take 1 of those 10's from

00:02:15.390 --> 00:02:18.139
the 10's place and turn it into 10 1's.

00:02:18.139 --> 00:02:22.386
And so 9 10's minus 8 10's is equal to 1 10.

00:02:22.386 --> 00:02:24.719
And then 10 -1 is 9.

00:02:24.719 --> 00:02:25.616
So it's equal to 19.

00:02:25.616 --> 00:02:26.291
You probably –

00:02:26.291 --> 00:02:28.343
You might have been able to do that in your head.

00:02:28.343 --> 00:02:29.254
And then we have –

00:02:29.254 --> 00:02:30.732
And I see something interesting here –

00:02:30.732 --> 00:02:33.593
because when we bring down our next 0,

00:02:33.593 --> 00:02:34.899
we see 190 again.

00:02:34.899 --> 00:02:36.814
We saw 190 up here.

00:02:36.814 --> 00:02:38.329
But let's just keep going.

00:02:38.329 --> 00:02:40.572
So 27 goes into 190 –

00:02:40.572 --> 00:02:42.654
And we already played this game.

00:02:42.654 --> 00:02:44.551
It goes into it 7 times.

00:02:44.551 --> 00:02:48.283
7 × 27 – we already figured out – was 189.

00:02:48.283 --> 00:02:49.146
We subtracted.

00:02:49.146 --> 00:02:50.591
We had a remainder of 1.

00:02:50.591 --> 00:02:54.013
Then we brought down another 0.

00:02:54.013 --> 00:02:57.307
We said 27 goes into 10 0 times.

00:02:57.307 --> 00:02:59.068
0 × 27 is 0.

00:02:59.068 --> 00:03:00.261
Subtract.

00:03:00.261 --> 00:03:01.641
Then you have –

00:03:01.641 --> 00:03:02.945
We still have the 10,

00:03:02.945 --> 00:03:07.298
but we've got to bring down another 0.

00:03:07.298 --> 00:03:09.083
So you have 27, which goes into 100 –

00:03:09.083 --> 00:03:10.438
(We've already done this.)

00:03:10.438 --> 00:03:11.599
–3 times.

00:03:11.599 --> 00:03:13.750
So you see something happening here.

00:03:13.750 --> 00:03:16.671
It's 0.703703.

00:03:16.671 --> 00:03:18.947
And we're just going to keep repeating 703.

00:03:18.947 --> 00:03:27.218
This is going to be equal to 0.703703703703 –

00:03:27.218 --> 00:03:29.718
on and on and on forever.

00:03:29.718 --> 00:03:32.267
So the notation for representing

00:03:32.267 --> 00:03:34.419
a repeating decimal like this

00:03:34.419 --> 00:03:36.405
is to write the numbers that repeat –

00:03:36.405 --> 00:03:38.547
in this case 7, 0, and 3 –

00:03:38.547 --> 00:03:40.057
and then you put a line over all of

00:03:40.057 --> 00:03:40.999
the repeating decimal numbers

00:03:40.999 --> 00:03:41.789
to indicate that they repeat.

00:03:41.789 --> 00:03:44.984
So you put a line over the 7, the 0, and the 3,

00:03:44.984 --> 00:03:47.130
which means that the 703 will keep

00:03:47.130 --> 00:03:49.423
repeating on and on and on.

00:03:49.423 --> 00:03:52.871
So let's actually input it into the answer box now.

00:03:52.871 --> 00:04:02.107
So it's 0.703703.

00:04:02.107 --> 00:04:03.341
And they tell us to include only

00:04:03.341 --> 00:04:05.838
the first six digits of the decimal in your answer.

00:04:05.838 --> 00:04:07.636
And they don't tell us to round or approximate –

00:04:07.636 --> 00:04:10.196
because, obviously, if they said to round

00:04:10.196 --> 00:04:13.957
to that smallest, sixth decimal place,

00:04:13.957 --> 00:04:15.017
then you would round up

00:04:15.017 --> 00:04:16.107
because the next digit is a 7.

00:04:16.107 --> 00:04:17.485
But they don't ask us to round.

00:04:17.485 --> 00:04:19.337
They just say, "Include only the first six digits

00:04:19.337 --> 00:04:20.988
of the decimal in your answer."

00:04:20.988 --> 00:04:23.135
So that should do the trick.

00:04:23.135 --> 00:04:24.604
And it did.


Comparing decimals example 2

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:06.540
Let's compare 0.1 to 0.070.

00:00:06.540 --> 00:00:11.130
So this 1 right over here,
it is in the tenths place.

00:00:11.130 --> 00:00:15.990
So it literally
represents 1 times 1/10,

00:00:15.990 --> 00:00:18.709
which is obviously the
same thing as 1/10.

00:00:18.709 --> 00:00:20.750
Now, when we look at this
number right over here,

00:00:20.750 --> 00:00:22.430
it has nothing in
the tenths place.

00:00:22.430 --> 00:00:25.480
It has 7 in the
hundredths place.

00:00:25.480 --> 00:00:28.110
So this is the hundredths
place right over here.

00:00:28.110 --> 00:00:31.520
And then it also has nothing
in the thousandths place.

00:00:31.520 --> 00:00:41.339
So this number can be rewritten
as 7 times 1/100, or 7/100.

00:00:41.339 --> 00:00:43.130
And now we could compare
these two numbers.

00:00:43.130 --> 00:00:45.046
And there's two ways you
could think about it.

00:00:45.046 --> 00:00:47.797
You could try to turn
1/10 into hundredths.

00:00:47.797 --> 00:00:50.380
And the best way to do that, if
you want the denominator to be

00:00:50.380 --> 00:00:53.173
increased by a factor
of 10, you need

00:00:53.173 --> 00:00:55.030
to do the same thing
to the numerator.

00:00:55.030 --> 00:00:56.790
So all I did is I multiplied
the numerator and denominator

00:00:56.790 --> 00:00:57.580
by 10.

00:00:57.580 --> 00:01:00.810
Ten 100's is the exact
same thing as 1/10.

00:01:00.810 --> 00:01:03.220
And here it becomes
very clear, 10/100

00:01:03.220 --> 00:01:06.220
is definitely larger than 700.

00:01:06.220 --> 00:01:07.970
Another way you could
think about this is,

00:01:07.970 --> 00:01:10.070
look, if you were to
increment by hundredths here,

00:01:10.070 --> 00:01:15.150
you would start at
7/100, 8/100, 9/100,

00:01:15.150 --> 00:01:17.119
and then you would
get to 10/100.

00:01:17.119 --> 00:01:18.660
So then you would
get to that number.

00:01:18.660 --> 00:01:21.180
So this number, multiple ways
you could think about it,

00:01:21.180 --> 00:01:23.550
is definitely larger.

00:01:23.550 --> 00:01:24.780
So let me write this down.

00:01:24.780 --> 00:01:27.460
This is definitely
larger, greater than.

00:01:27.460 --> 00:01:29.260
This is greater than that.

00:01:29.260 --> 00:01:33.150
The greater than symbol
opens to the larger value.

00:01:33.150 --> 00:01:38.490
So here we have 0.093
and here we have 0.01.

00:01:38.490 --> 00:01:40.860
So let's just think
about this a little bit.

00:01:40.860 --> 00:01:44.010
So this 9-- get
a new color here.

00:01:44.010 --> 00:01:46.620
This 9 is not in the
tenths, the hundredths.

00:01:46.620 --> 00:01:49.080
It's in the thousandths place.

00:01:49.080 --> 00:01:51.570
It's in the thousands place.

00:01:51.570 --> 00:01:56.580
And this 3 is in the-- I'm
running out of colors again.

00:01:56.580 --> 00:01:59.960
This 3 is in the ten
thousandths place.

00:01:59.960 --> 00:02:02.740
So the 3 is in the
ten thousandths place.

00:02:02.740 --> 00:02:14.455
So you could literally view
this as 9/1,000 plus 3/10,000.

00:02:14.455 --> 00:02:17.080
And if you just wanted to write
it in terms of ten thousandths,

00:02:17.080 --> 00:02:20.380
you can multiply the
9 and 1,000 by 0.

00:02:20.380 --> 00:02:23.002
And so it becomes 90/10,000.

00:02:23.002 --> 00:02:25.460
And if you want to add them
together, you could, of course,

00:02:25.460 --> 00:02:30.386
write this as 93/10,000.

00:02:30.386 --> 00:02:31.280
Ten thousandths.

00:02:31.280 --> 00:02:34.700
I always have trouble with
that "-ths" at the end.

00:02:34.700 --> 00:02:39.100
Now, let's think about this
number right over here, 0.01.

00:02:39.100 --> 00:02:42.662
Well, this 1 right over here
is in the hundredths place.

00:02:42.662 --> 00:02:43.870
It's in the hundredths place.

00:02:43.870 --> 00:02:47.040
So it literally
represents 1/100.

00:02:47.040 --> 00:02:51.931
So how can we compare
1/100 to 93/10,000?

00:02:51.931 --> 00:02:53.306
So the best way
to think about it

00:02:53.306 --> 00:02:56.270
is, well, what's 1/100 in
terms of ten thousandths?

00:02:56.270 --> 00:02:58.720
Well, let's just multiply
both the numerator

00:02:58.720 --> 00:03:00.897
and the denominator
here by 10 twice.

00:03:00.897 --> 00:03:02.980
Or you could say, let's
multiply them both by 100.

00:03:02.980 --> 00:03:06.290
If you multiply by 10 once,
you get to ten thousandths.

00:03:06.290 --> 00:03:08.310
It's the same thing as 1/100.

00:03:08.310 --> 00:03:12.940
Multiply by 10 again, you get
100/10,000 is the same thing

00:03:12.940 --> 00:03:13.580
as 1/100.

00:03:13.580 --> 00:03:16.270
And we know that, 100
times 100 is 10,000.

00:03:16.270 --> 00:03:20.630
So here it becomes very
clear, 100/10,000, or 1/100,

00:03:20.630 --> 00:03:25.110
is definitely a
larger than 93/10,000.

00:03:25.110 --> 00:03:26.910
So this quantity
right over here is

00:03:26.910 --> 00:03:29.320
less than this quantity there.

00:03:29.320 --> 00:03:32.957
Less than symbol, the small end
points to the smaller number,

00:03:32.957 --> 00:03:34.290
larger end to the larger number.

00:03:34.290 --> 00:03:36.720
In fact, that's true with the
less than and greater than.

00:03:36.720 --> 00:03:42.770
So let's see, this one right
over here, 0.6 versus 0.06.

00:03:42.770 --> 00:03:45.620
So here, I have a 6
in the tenths place.

00:03:45.620 --> 00:03:49.250
So it literally represents 6/10.

00:03:49.250 --> 00:03:52.440
And in the second, I have a
6 in the hundredths place.

00:03:52.440 --> 00:03:55.830
Well, 6/100 is definitely
smaller than 6/10.

00:03:55.830 --> 00:03:57.920
A hundredth is a
tenth of a tenth.

00:03:57.920 --> 00:04:00.680
So this one is pretty
straightforward.

00:04:00.680 --> 00:04:02.450
This is going to be
the larger value.

00:04:02.450 --> 00:04:07.670
0.6 is greater than 0.06.

00:04:07.670 --> 00:04:11.640
Now, let's think
about 0.3 versus 0.06.

00:04:11.640 --> 00:04:16.010
So this 3 literally
represents 3/10

00:04:16.010 --> 00:04:24.450
while this 6 right over
here represents 6/100.

00:04:24.450 --> 00:04:26.430
And if you wanted to
compare them directly,

00:04:26.430 --> 00:04:30.030
you could multiply 3/10 times--
well, both the numerator

00:04:30.030 --> 00:04:32.700
and the denominator by 10 so
you're not changing its value.

00:04:32.700 --> 00:04:35.170
10/10 is essentially
1, or it is 1.

00:04:35.170 --> 00:04:37.100
So this becomes 30/300.

00:04:37.100 --> 00:04:39.916
3/10 is the same
thing as 30/100.

00:04:39.916 --> 00:04:45.440
And 30/100 is a lot
larger than 6/100.

00:04:45.440 --> 00:04:49.280
So this is greater than.


Comparing decimals: difference in largest place value

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:09.240
Let's compare 9.97 to 9.798.

00:00:09.240 --> 00:00:11.870
So to figure out which
one of these is greater,

00:00:11.870 --> 00:00:14.010
I like to start with
the largest place values

00:00:14.010 --> 00:00:16.550
and then keep moving to
smaller and smaller ones

00:00:16.550 --> 00:00:18.500
until we actually
see a difference.

00:00:18.500 --> 00:00:22.622
So they both have nine 1's.

00:00:22.622 --> 00:00:24.080
So at least in the
ones place, they

00:00:24.080 --> 00:00:26.590
seem comparable to each other.

00:00:26.590 --> 00:00:29.280
Now let's go to
the tenths place.

00:00:29.280 --> 00:00:32.790
So this number on the left
has a 9 in the tenths place,

00:00:32.790 --> 00:00:36.306
while the number on the right
has a 7 in the tenths place.

00:00:36.306 --> 00:00:38.180
So right now, we could
view this-- let's just

00:00:38.180 --> 00:00:39.346
write the whole numbers out.

00:00:39.346 --> 00:00:44.960
So this one is 9 plus 9/10.

00:00:44.960 --> 00:00:47.010
We haven't gone to the
hundredths place yet.

00:00:47.010 --> 00:00:50.510
So far, out of the two digits,
the two places we've looked at,

00:00:50.510 --> 00:00:55.210
this one on the
right is 9 plus 7/10.

00:01:00.320 --> 00:01:03.056
So this immediately cues to
me that the one on the left

00:01:03.056 --> 00:01:03.930
is the larger number.

00:01:03.930 --> 00:01:05.390
You're like, hey, how
do I know immediately

00:01:05.390 --> 00:01:06.490
that's the larger number?

00:01:06.490 --> 00:01:08.198
I have all this other
stuff to the right.

00:01:08.198 --> 00:01:09.770
I have this 98 to the right.

00:01:09.770 --> 00:01:11.260
I have this 7 to the right.

00:01:11.260 --> 00:01:15.050
And the way to think about it
is, no matter what you have,

00:01:15.050 --> 00:01:18.250
even if you really increase
this right-hand side

00:01:18.250 --> 00:01:22.510
here as much as possible,
you're still less than 9.8.

00:01:22.510 --> 00:01:25.110
In fact, if you keep
incrementing the thousandths

00:01:25.110 --> 00:01:33.340
here, you go from
9.798 to 9.799 to 9.8.

00:01:33.340 --> 00:01:37.490
So you would have to actually
increase to get to even 9.8.

00:01:37.490 --> 00:01:40.050
And this is at 9.9.

00:01:40.050 --> 00:01:42.460
So you can really just
look at the discrepancy

00:01:42.460 --> 00:01:44.570
in the largest place
value to recognize

00:01:44.570 --> 00:01:45.570
which number is greater.

00:01:45.570 --> 00:01:47.330
This has 9/10.

00:01:47.330 --> 00:01:48.750
This has 7/10.

00:01:48.750 --> 00:01:51.260
It doesn't matter what's
going on in the hundredths

00:01:51.260 --> 00:01:52.672
and the thousandths place.

00:01:52.672 --> 00:01:55.130
And to make that clear, let's
actually add up these numbers

00:01:55.130 --> 00:01:57.740
and compare them as fractions.

00:01:57.740 --> 00:01:59.740
So let's keep on
going with this.

00:01:59.740 --> 00:02:01.310
So you have 7/100 here.

00:02:04.310 --> 00:02:09.169
And here you have 9/100.

00:02:09.169 --> 00:02:13.952
And then finally,
here you have 0/1000.

00:02:13.952 --> 00:02:15.910
And here-- let me do that
in a different color.

00:02:15.910 --> 00:02:17.280
I already used blue.

00:02:17.280 --> 00:02:19.460
And here, you have 8/1000.

00:02:19.460 --> 00:02:23.040
So plus 8/1000.

00:02:23.040 --> 00:02:26.730
So let's put everything
in terms of thousandths

00:02:26.730 --> 00:02:30.220
so that we can add
these all up and have

00:02:30.220 --> 00:02:32.500
two fractions over
thousandths, or things

00:02:32.500 --> 00:02:34.440
in terms of thousandths.

00:02:34.440 --> 00:02:40.930
So 9 is the same
thing as 9000/1000.

00:02:40.930 --> 00:02:42.520
9/10-- well, let's see.

00:02:42.520 --> 00:02:47.230
If you multiplied it by 10,
you would get 90 over 100.

00:02:47.230 --> 00:02:51.070
Multiply by 10 again,
you get 900/1000.

00:02:51.070 --> 00:02:56.660
7/100 multiplied
by 10 is 70/1000.

00:02:56.660 --> 00:02:57.930
And let's do that over here.

00:02:57.930 --> 00:03:10.560
Once again, 9 is 9000/1000,
and then plus 700/1000

00:03:10.560 --> 00:03:17.450
plus 90/1000-- just multiply
the numerator and denominator

00:03:17.450 --> 00:03:23.200
by 10-- plus 8/1000.

00:03:23.200 --> 00:03:25.460
And so what is this
number on the left?

00:03:25.460 --> 00:03:30.450
This number on the left is--
how many thousandths is it?

00:03:30.450 --> 00:03:36.813
It's 9,970.

00:03:36.813 --> 00:03:49.730
So it's 9970/1000, while
this number on the right

00:03:49.730 --> 00:04:05.400
here is 9798/1000.

00:04:05.400 --> 00:04:07.750
So here, once again, you're
comparing two numbers.

00:04:07.750 --> 00:04:09.570
They have the same
number of thousandths.

00:04:09.570 --> 00:04:10.990
This has 900.

00:04:10.990 --> 00:04:12.560
This only has 700.

00:04:12.560 --> 00:04:14.380
So even though
this is almost 800,

00:04:14.380 --> 00:04:16.240
800 is still less than 900.

00:04:16.240 --> 00:04:20.320
So no matter how you think
about it, the number on the left

00:04:20.320 --> 00:04:23.620
is greater than the
number on the right.


Placing positive and negative decimals on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:04.430
We're asked to move the
orange dot to negative 0.6

00:00:04.430 --> 00:00:05.850
on the number line.

00:00:05.850 --> 00:00:08.760
So the dot right now is at 0.

00:00:08.760 --> 00:00:10.710
And let's see,
this is negative 2.

00:00:10.710 --> 00:00:11.760
This is positive 2.

00:00:11.760 --> 00:00:13.910
So each of these big
slashes look like it's 1.

00:00:13.910 --> 00:00:16.690
So this would be
negative 1, negative 2.

00:00:16.690 --> 00:00:19.550
This right over here
would be negative 0.5.

00:00:19.550 --> 00:00:21.810
So we're going to
go a little bit more

00:00:21.810 --> 00:00:23.240
negative than negative 0.5.

00:00:23.240 --> 00:00:24.530
That would be negative 0.6.

00:00:24.530 --> 00:00:26.230
We want to go another tenth.

00:00:26.230 --> 00:00:29.880
And so that looks right
at about negative 0.6.

00:00:29.880 --> 00:00:31.840
Let's do a couple more of these.

00:00:31.840 --> 00:00:34.876
Move the orange dot to
1.9 on the number line.

00:00:34.876 --> 00:00:36.500
So it seems like we
have the same scale

00:00:36.500 --> 00:00:38.340
as in the previous example.

00:00:38.340 --> 00:00:39.810
This gets us to 1.

00:00:39.810 --> 00:00:43.720
In fact, this is 0.5, 1, 1.5.

00:00:43.720 --> 00:00:46.830
And 1.9 is only going to
be a tenth less than 2.

00:00:46.830 --> 00:00:49.890
So it's going to be--
let's see, that's 2,

00:00:49.890 --> 00:00:52.000
so a tenth less than that.

00:00:52.000 --> 00:00:55.080
And I'm assuming that it's
locking us to the tenths.

00:00:55.080 --> 00:00:59.020
And it looks like it is, so
that looks pretty close to 1.9.

00:00:59.020 --> 00:01:01.280
Let's do one more of these.

00:01:01.280 --> 00:01:04.400
Move the orange dot to
0.5 on the number line.

00:01:04.400 --> 00:01:06.500
Well, once again, we
have the same scale.

00:01:06.500 --> 00:01:08.320
It's important to
always check the scale.

00:01:08.320 --> 00:01:09.790
And this right over here is 1.

00:01:09.790 --> 00:01:12.590
So 0.5 would be halfway, so
that little small mark right

00:01:12.590 --> 00:01:14.300
over there.


Interpreting a graph exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.389
The illustration below shows the
graph of y as a function of x.

00:00:04.389 --> 00:00:05.930
So that's this graph
right over here.

00:00:05.930 --> 00:00:07.805
And then they start to
ask us some questions.

00:00:07.805 --> 00:00:10.550
Complete the sentences based
on the graph of the function.

00:00:10.550 --> 00:00:12.870
So this axis is our
y-axis, the vertical axis.

00:00:12.870 --> 00:00:14.900
Horizontal axis is x-axis.

00:00:14.900 --> 00:00:17.710
Initially, as x increases--
so let's think about it.

00:00:17.710 --> 00:00:21.710
Initially, so when we start from
x equals 0 and x is increasing,

00:00:21.710 --> 00:00:23.160
what's happening to y?

00:00:23.160 --> 00:00:24.920
Well, y is decreasing.

00:00:24.920 --> 00:00:27.820
So y decreases.

00:00:27.820 --> 00:00:31.270
So as x increases
initially, y decreases.

00:00:31.270 --> 00:00:35.920
The slope of the graph is equal
to blank for all x between

00:00:35.920 --> 00:00:37.690
x equals 0 and x equals 3.

00:00:37.690 --> 00:00:40.950
So x equals 0 and x equals
3, what's the slope?

00:00:40.950 --> 00:00:43.050
Well, every time we move
1 in the x direction,

00:00:43.050 --> 00:00:44.900
we move down in the y direction.

00:00:44.900 --> 00:00:46.510
We go negative 1
in the y direction.

00:00:46.510 --> 00:00:49.420
Move up 1 in the x
direction, we go negative 1

00:00:49.420 --> 00:00:50.290
in the y direction.

00:00:50.290 --> 00:00:53.730
So our change in y over our
change in x-- our change in y

00:00:53.730 --> 00:00:56.440
is negative 1 whenever
our change in x is 1.

00:00:56.440 --> 00:00:58.320
So our change in
y over our change

00:00:58.320 --> 00:01:01.970
in x, which is the definition
of slope, is negative 1/1.

00:01:01.970 --> 00:01:03.573
So it's negative 1.

00:01:03.573 --> 00:01:04.239
And we see that.

00:01:04.239 --> 00:01:08.030
Every time x increases
by 1, y decreases by 1.

00:01:08.030 --> 00:01:14.990
Starting at x equals 3,
y blank as x increases.

00:01:14.990 --> 00:01:18.000
So starting at x equals 3,
y increases as x increases.

00:01:18.000 --> 00:01:20.680
As x increases, y is increasing.

00:01:20.680 --> 00:01:23.560
So y increases as x increases.

00:01:23.560 --> 00:01:29.830
The slope of the graph is equal
to blank for x between 3 and 5.

00:01:29.830 --> 00:01:34.920
So when x increases by
1, y is increasing by 3.

00:01:34.920 --> 00:01:37.560
So change in y is
3, change in x is 1.

00:01:37.560 --> 00:01:40.600
Slope is change in y over
change in x, which is 3/1.

00:01:40.600 --> 00:01:42.270
So the slope here is 3.

00:01:42.270 --> 00:01:46.880
Every time x increases
by 1, y increases by 3.

00:01:46.880 --> 00:01:52.310
For x between x equals 0 and
x equals 4, y-- let's see,

00:01:52.310 --> 00:01:56.750
we can pick less than or equal
to, greater than or equal to,

00:01:56.750 --> 00:01:58.070
or equal.

00:01:58.070 --> 00:02:02.450
So for x equals
0 and x equals 4,

00:02:02.450 --> 00:02:07.562
y is less than or equal to 0.

00:02:07.562 --> 00:02:09.990
So let's do less
than or equal to 0.

00:02:09.990 --> 00:02:12.450
And then they say,
for x between x

00:02:12.450 --> 00:02:14.630
equals 4 and x
equals 8-- well, then

00:02:14.630 --> 00:02:17.370
y is greater than or equal to 0.

00:02:17.370 --> 00:02:18.800
So let's make sure
that we didn't

00:02:18.800 --> 00:02:20.133
make any careless mistakes here.

00:02:20.133 --> 00:02:21.480
Let's check our answer.

00:02:21.480 --> 00:02:23.450
We got it right.


Constructing and interpreting graphs

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.900
Jeff works as a waiter
at a fancy restaurant.

00:00:03.900 --> 00:00:08.850
He receives a base amount of $70
each day plus approximately $10

00:00:08.850 --> 00:00:11.465
of tips for each
client he serves,

00:00:11.465 --> 00:00:14.880
or $10 in tips for
each client he serves.

00:00:14.880 --> 00:00:17.200
Assuming x represents
the number of clients

00:00:17.200 --> 00:00:20.530
Jeff will serve in a day and
y represents his daily salary,

00:00:20.530 --> 00:00:23.200
draw the graph which
represents Jeff's daily salary

00:00:23.200 --> 00:00:26.510
as a function of the number
of clients he serves.

00:00:26.510 --> 00:00:30.300
So he's going to get
$70 no matter what.

00:00:30.300 --> 00:00:36.340
So even when x-- so even if he
serves absolutely no clients,

00:00:36.340 --> 00:00:37.810
he's going to get $70.

00:00:37.810 --> 00:00:42.360
So when x is equal to 0, his
pay for that day is still $70.

00:00:42.360 --> 00:00:44.430
Then he gets $10--
they already told us

00:00:44.430 --> 00:00:47.680
he gets $10 in tips for
each client he serves.

00:00:50.210 --> 00:00:54.210
So if he goes from serving
0 clients to 1 client,

00:00:54.210 --> 00:00:56.340
he's going to get $10 in tips.

00:00:56.340 --> 00:00:59.200
If he goes from serving
1 client to 2 clients,

00:00:59.200 --> 00:01:01.280
he's going to get
another $10 in tips.

00:01:01.280 --> 00:01:03.337
If he goes from serving
2 clients to 3 clients,

00:01:03.337 --> 00:01:04.920
he's going to get
another $10 in tips.

00:01:04.920 --> 00:01:06.461
And you see that
this line represents

00:01:06.461 --> 00:01:08.620
all the different
possibilities for him.

00:01:08.620 --> 00:01:11.620
If he goes and serves
6 clients, he's

00:01:11.620 --> 00:01:14.460
going to get $60 in
tips plus the $70.

00:01:14.460 --> 00:01:18.010
So the $70 he would have made
otherwise plus $60 in tips

00:01:18.010 --> 00:01:19.510
gets him to $130.

00:01:19.510 --> 00:01:22.020
So this right here is the line.

00:01:22.020 --> 00:01:23.790
So we could check our answer.

00:01:23.790 --> 00:01:25.670
We got it right.


Finding intercepts from a table

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.850
The following table
of values represents

00:00:02.850 --> 00:00:06.510
points x comma y on the
graph of a linear function.

00:00:06.510 --> 00:00:09.770
Determine the y-intercept
of this graph.

00:00:09.770 --> 00:00:13.420
So just as a reminder of
what the y-intercept even is,

00:00:13.420 --> 00:00:17.185
if you imagine a linear function
or a line if we're graphing it,

00:00:17.185 --> 00:00:18.840
if we imagine a
line, so let's say

00:00:18.840 --> 00:00:20.950
that is our line
right over there.

00:00:20.950 --> 00:00:22.190
This is our y-axis.

00:00:22.190 --> 00:00:23.760
This is our x-axis.

00:00:23.760 --> 00:00:28.020
The y-intercept is where
we intersect the y-axis.

00:00:28.020 --> 00:00:30.890
Now, what do we know
about the y-intercept?

00:00:30.890 --> 00:00:34.930
Well, at the y-intercept x
is going to be equal to 0.

00:00:34.930 --> 00:00:38.690
So this is the point
0 comma something.

00:00:38.690 --> 00:00:40.290
And so when people
are talking about,

00:00:40.290 --> 00:00:41.520
what is your y-intercept?

00:00:41.520 --> 00:00:43.360
They're usually
saying, well, what

00:00:43.360 --> 00:00:46.409
is the y-coordinate
when x equals 0.

00:00:46.409 --> 00:00:47.950
So we're really
trying to figure out,

00:00:47.950 --> 00:00:52.060
what is the y-coordinate
when x equals 0?

00:00:52.060 --> 00:00:54.830
So we know the x-coordinate
when y is equal to 0.

00:00:54.830 --> 00:00:56.560
So this is actually
the x-intercept.

00:00:56.560 --> 00:01:00.460
So this point right over
here is the point 2 comma 0.

00:01:00.460 --> 00:01:02.830
So when people say
x-intercept, that's

00:01:02.830 --> 00:01:04.750
the x-coordinate
when y equals 0.

00:01:04.750 --> 00:01:06.970
Well, they gave us
the x-intercept.

00:01:06.970 --> 00:01:10.910
So that right over there
is the x-intercept.

00:01:10.910 --> 00:01:12.150
But what's the y-intercept?

00:01:12.150 --> 00:01:15.135
What is the y-value
when x equals 0?

00:01:15.135 --> 00:01:15.820
Well, let's see.

00:01:15.820 --> 00:01:18.620
They give us what happens to
y when x is negative 2, when

00:01:18.620 --> 00:01:21.270
it's 1, when it's
2, when it's 4.

00:01:21.270 --> 00:01:24.390
So maybe we can backtrack
from one of these

00:01:24.390 --> 00:01:27.870
to get back to what happens
when x is equal to 0.

00:01:27.870 --> 00:01:31.430
So let me rewrite this table
so I can give ourselves

00:01:31.430 --> 00:01:33.350
a little bit more
breathing room.

00:01:33.350 --> 00:01:36.695
So let's say we have
x and we have y.

00:01:36.695 --> 00:01:38.520
x and y.

00:01:38.520 --> 00:01:44.226
And they already tell us that
when x is negative 2, y is 8.

00:01:44.226 --> 00:01:45.850
And I actually want
to think about what

00:01:45.850 --> 00:01:48.550
happens when x is
negative 1, when x is 0.

00:01:48.550 --> 00:01:52.130
Then they tell us
when x is 1, y is 2.

00:01:52.130 --> 00:01:54.720
When x is 2, y is 0.

00:01:54.720 --> 00:01:56.950
This right over here
is the x-intercept.

00:01:56.950 --> 00:02:00.220
When x is 4, y is negative 4.

00:02:00.220 --> 00:02:03.410
So they skip 2 right over
here. y is negative 4.

00:02:03.410 --> 00:02:07.430
So let's just see how y changes
with respect to changes in x.

00:02:07.430 --> 00:02:17.067
So when we go here, when x
changes by 1, y goes down by 2.

00:02:17.067 --> 00:02:18.650
And it's a line, so
it's going to have

00:02:18.650 --> 00:02:22.080
a constant rate of change
of y with respect to x.

00:02:22.080 --> 00:02:25.830
So similarly, when
x increases by 1,

00:02:25.830 --> 00:02:28.650
y is going to decrease by 2.

00:02:28.650 --> 00:02:30.500
So y is going to be 6 here.

00:02:30.500 --> 00:02:35.945
When x increases by 1 again,
y is going to decrease by 2.

00:02:35.945 --> 00:02:37.070
So we're going to get to 4.

00:02:37.070 --> 00:02:38.510
And we see it works.

00:02:38.510 --> 00:02:40.790
Because if we
increase by 1 again,

00:02:40.790 --> 00:02:44.930
then it is indeed the case
that y decreased by 2.

00:02:44.930 --> 00:02:48.640
And you see here when
we increase x by 2,

00:02:48.640 --> 00:02:51.250
then y decreases
at twice the rate.

00:02:51.250 --> 00:02:52.995
Because now we didn't
just increase by 1,

00:02:52.995 --> 00:02:54.030
we increased by 2.

00:02:54.030 --> 00:02:57.160
So now y is going
to decrease by 4.

00:02:57.160 --> 00:02:59.410
And what's constant
here is your change

00:02:59.410 --> 00:03:02.320
in y over your change in x.

00:03:02.320 --> 00:03:07.590
When x increases by
1, y decreases by 2.

00:03:07.590 --> 00:03:11.840
When x increases by
2, y decreases by 4.

00:03:11.840 --> 00:03:15.130
Either way you think
about it, your change

00:03:15.130 --> 00:03:18.060
in y for a unit
change in x is going

00:03:18.060 --> 00:03:20.230
to be equal to negative 2.

00:03:20.230 --> 00:03:22.660
But anyway, we actually
answered the question

00:03:22.660 --> 00:03:24.520
before without even
realizing it when

00:03:24.520 --> 00:03:26.160
we filled in all
of these values.

00:03:26.160 --> 00:03:30.490
What is the y-value
when x equals 0?

00:03:30.490 --> 00:03:32.650
Well, the y-value is 4.

00:03:32.650 --> 00:03:35.330
So the y-intercept here is 4.

00:03:35.330 --> 00:03:37.300
We didn't really
graph this to scale.

00:03:37.300 --> 00:03:39.760
It would actually look a
little bit more like this

00:03:39.760 --> 00:03:41.970
if we were to try to
graph it properly.

00:03:41.970 --> 00:03:45.790
So this right over here is 4.

00:03:45.790 --> 00:03:47.942
This right over here is 2.

00:03:47.942 --> 00:03:51.470
And our line looks
something like this.

00:03:51.470 --> 00:03:58.943
Our line will look
something like that.


Finding the x-intercept of a line

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.660
The graph of the line
2y plus 3x equals

00:00:03.660 --> 00:00:05.560
7 is given right over here.

00:00:05.560 --> 00:00:08.109
Determine its x-intercept.

00:00:08.109 --> 00:00:13.770
The x-intercept is the x
value when y is equal to 0,

00:00:13.770 --> 00:00:16.640
or it's the x value
where our graph actually

00:00:16.640 --> 00:00:18.550
intersects the x-axis.

00:00:18.550 --> 00:00:21.740
Notice right over here
our y value is exactly 0.

00:00:21.740 --> 00:00:23.170
We're sitting on the x-axis.

00:00:23.170 --> 00:00:25.820
So let's think about what
this x value must be.

00:00:25.820 --> 00:00:29.520
Well, just trying to
eyeball a little bit,

00:00:29.520 --> 00:00:30.870
it's a little over 2.

00:00:30.870 --> 00:00:31.765
It's between 2 and 3.

00:00:31.765 --> 00:00:34.450
It looks like it's
less than 2 and 1/2.

00:00:34.450 --> 00:00:35.910
But we don't know
the exact value.

00:00:35.910 --> 00:00:39.590
So let's go turn to the equation
to figure out the exact value.

00:00:39.590 --> 00:00:41.020
We essentially
have to figure out

00:00:41.020 --> 00:00:43.890
what x value, when
y is equal to 0,

00:00:43.890 --> 00:00:45.830
will have this equation be true.

00:00:45.830 --> 00:00:51.300
So we could just say 2 times
0 plus 3x is equal to 7.

00:00:51.300 --> 00:00:53.950
Well, 2 times 0 is
just going to be 0,

00:00:53.950 --> 00:00:57.170
so we have 3x is equal to 7.

00:00:57.170 --> 00:01:01.080
Then we can divide both
sides by 3 to solve for x,

00:01:01.080 --> 00:01:04.670
and we get x is equal to 7/3.

00:01:04.670 --> 00:01:06.860
Does that look like 7/3?

00:01:06.860 --> 00:01:08.790
Well, we just have
to remind ourselves

00:01:08.790 --> 00:01:15.190
that 7/3 is the same
thing as 6/3 plus 1/3.

00:01:15.190 --> 00:01:16.640
And 6/3 is 2.

00:01:16.640 --> 00:01:19.232
So this is the same
thing as 2 and 1/3.

00:01:19.232 --> 00:01:20.940
Another way you could
think about it is 3

00:01:20.940 --> 00:01:24.680
goes into 7 two times, and
then you have a remainder of 1.

00:01:24.680 --> 00:01:26.600
So you've still got
to divide that 1 by 3.

00:01:26.600 --> 00:01:29.490
It's 2 full times
and then a 1/3,

00:01:29.490 --> 00:01:31.720
so this looks like 2 and 1/3.

00:01:31.720 --> 00:01:35.140
And so that's its
x-intercept, 7/3.

00:01:35.140 --> 00:01:37.700
If I was doing this on the
exercise on Khan Academy,

00:01:37.700 --> 00:01:40.330
it's always a little easier to
type in the improper fraction,

00:01:40.330 --> 00:01:41.991
so I would put in 7/3.


Interpreting intercepts of linear functions

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.000
After several days of camping
and long-distance bicycle

00:00:04.000 --> 00:00:07.540
riding, you decide to
ride straight home riding

00:00:07.540 --> 00:00:08.960
at a constant rate.

00:00:08.960 --> 00:00:14.360
After 2 hours of riding, you
are 10 kilometers from home.

00:00:14.360 --> 00:00:18.980
And after 4 hours of riding,
you are 5 kilometers from home.

00:00:18.980 --> 00:00:21.640
Once you begin to ride
home, in how many hours

00:00:21.640 --> 00:00:23.786
will you finish your bike
ride and arrive at home?

00:00:23.786 --> 00:00:25.660
This last sentence is
a little bit redundant.

00:00:25.660 --> 00:00:27.820
It really should say,
what was your total length

00:00:27.820 --> 00:00:28.820
of your bike ride home?

00:00:28.820 --> 00:00:31.530
But let's just think about
the data that they gave us.

00:00:31.530 --> 00:00:35.950
So let's just think
about on one column,

00:00:35.950 --> 00:00:38.650
let's put how long
you've been riding.

00:00:38.650 --> 00:00:40.260
So time.

00:00:40.260 --> 00:00:41.720
Time riding.

00:00:41.720 --> 00:00:46.570
And then in this column let's
put the distance from home.

00:00:46.570 --> 00:00:48.850
So they tell us
that after 2 hours,

00:00:48.850 --> 00:00:51.410
and the time is in
hours, after 2 hours

00:00:51.410 --> 00:00:54.320
we are 10 kilometers from home.

00:00:54.320 --> 00:01:00.080
And after 4 hours, we are
5 kilometers from home.

00:01:00.080 --> 00:01:02.800
And we are going home
at a constant rate.

00:01:02.800 --> 00:01:04.940
So let's think about
what this rate is.

00:01:04.940 --> 00:01:10.060
When our time
increased by 2 hours,

00:01:10.060 --> 00:01:13.700
our distance decreased by 5.

00:01:13.700 --> 00:01:18.720
So if our time increased
by another 2 hours,

00:01:18.720 --> 00:01:21.910
so now 6 hours have gone
by, then our distance

00:01:21.910 --> 00:01:27.610
will have decreased
by 5 again, and we

00:01:27.610 --> 00:01:30.970
are 0 kilometers from home.

00:01:30.970 --> 00:01:32.500
So we've answered our question.

00:01:32.500 --> 00:01:34.460
Once you begin to ride
home, in how many hours

00:01:34.460 --> 00:01:36.820
will you finish your bike
ride and arrive home?

00:01:36.820 --> 00:01:42.100
Well, it'll take you
a total of 6 hours.

00:01:42.100 --> 00:01:44.100
Now, I'm going to ask
ourselves another question

00:01:44.100 --> 00:01:45.480
that they didn't ask us here.

00:01:45.480 --> 00:01:51.559
Which is, how far away was I
when I started riding home?

00:01:51.559 --> 00:01:53.350
Well, we could back
things up a little bit.

00:01:53.350 --> 00:01:56.130
Actually, let me copy
and paste all of this.

00:01:56.130 --> 00:01:57.390
And move it down.

00:01:57.390 --> 00:02:01.140
So let me cut, let me paste
this all down a little bit,

00:02:01.140 --> 00:02:03.235
so we have some extra space.

00:02:03.235 --> 00:02:06.880
And let me draw a
little white line here.

00:02:06.880 --> 00:02:10.420
And then finally, let me erase
this business right over here.

00:02:10.420 --> 00:02:12.170
And so we could think
of it the other way.

00:02:12.170 --> 00:02:16.330
Where were we at time 0?

00:02:16.330 --> 00:02:22.830
So time 0, that was 2
hours before this point

00:02:22.830 --> 00:02:24.040
right over here.

00:02:24.040 --> 00:02:26.440
So here we are
subtracting 2 hours.

00:02:26.440 --> 00:02:29.360
So instead of subtracting
5, we will add 5.

00:02:29.360 --> 00:02:31.540
They're going to
have opposite signs.

00:02:31.540 --> 00:02:34.650
So we were 15 kilometers away.

00:02:34.650 --> 00:02:37.920
Another way that you
could think about this

00:02:37.920 --> 00:02:46.880
is we started 15 kilometers
away, 2 hours goes by

00:02:46.880 --> 00:02:50.026
and our distance has
decreased 5 kilometers.

00:02:50.026 --> 00:02:51.400
So let's actually
graph this just

00:02:51.400 --> 00:02:53.990
to visualize things
in graphical form.

00:02:53.990 --> 00:02:56.720
So let's say that
this is our time axis,

00:02:56.720 --> 00:02:57.867
actually our distance axis.

00:02:57.867 --> 00:02:59.700
Distance in this case
is a function of time,

00:02:59.700 --> 00:03:03.410
so the convention is to plot
that on the vertical axis.

00:03:03.410 --> 00:03:06.740
And then let's put time
on the horizontal axis.

00:03:06.740 --> 00:03:09.310
So that is time
right over there.

00:03:09.310 --> 00:03:14.910
So when time equals 0 our
distance is 15 kilometers.

00:03:14.910 --> 00:03:20.080
This is 5, 10, 15.

00:03:20.080 --> 00:03:23.980
So when time is equal to 0,
distance is 15 kilometers.

00:03:23.980 --> 00:03:26.130
That's this point
right over here.

00:03:26.130 --> 00:03:27.890
And then when time
is equal to 6,

00:03:27.890 --> 00:03:30.461
our distance is 0 kilometers.

00:03:30.461 --> 00:03:31.960
So it's not perfectly--
so let's say

00:03:31.960 --> 00:03:39.657
that this is 1 hour, 2 hours, 3
hours, 4 hours, 5 hours, and 6

00:03:39.657 --> 00:03:41.990
hours, obviously, it should
be drawn a little bit neater

00:03:41.990 --> 00:03:43.435
than that.

00:03:43.435 --> 00:03:44.310
Let's do it this way.

00:03:44.310 --> 00:03:48.030
Let's call this
2, 4, and 6 hours.

00:03:48.030 --> 00:03:49.470
Obviously, not the
same scales for

00:03:49.470 --> 00:03:51.250
the vertical and
horizontal axes.

00:03:51.250 --> 00:03:56.480
But this last point right over
here when time is 6 hours,

00:03:56.480 --> 00:03:58.640
our distance from
home is 0 kilometers.

00:03:58.640 --> 00:04:00.070
So that's that point there.

00:04:00.070 --> 00:04:01.940
And we're traveling
at a constant rate.

00:04:01.940 --> 00:04:03.475
So our line will look like this.

00:04:06.320 --> 00:04:08.320
Now one question you might
ask yourselves, well,

00:04:08.320 --> 00:04:10.110
can we quantify this
rate, or how would

00:04:10.110 --> 00:04:13.890
this rate relate to the
inclination of this line?

00:04:13.890 --> 00:04:15.830
Well remember, the
inclination of this line

00:04:15.830 --> 00:04:17.600
is defined as slope,
and that's the rate

00:04:17.600 --> 00:04:21.300
of change of our vertical
axis to our horizontal axis.

00:04:21.300 --> 00:04:25.890
So the slope of this line
is equal to the change

00:04:25.890 --> 00:04:27.500
in our vertical axis.

00:04:27.500 --> 00:04:29.330
This triangle is the
Greek letter delta,

00:04:29.330 --> 00:04:30.720
shorthand for change in.

00:04:30.720 --> 00:04:33.060
Change in our vertical
axis with respect

00:04:33.060 --> 00:04:35.829
to a change in our
horizontal axis.

00:04:35.829 --> 00:04:37.370
So in our horizontal
axis, we already

00:04:37.370 --> 00:04:41.530
see when we have a change
of 2 in our horizontal axis,

00:04:41.530 --> 00:04:43.820
and we saw that with
the first two points,

00:04:43.820 --> 00:04:48.760
when we went from
2 comma 10, that

00:04:48.760 --> 00:04:50.990
was this point right
over here, to 4 comma

00:04:50.990 --> 00:04:55.590
5, what was our change in time?

00:04:55.590 --> 00:05:00.236
Well, our change in time
was we went from 2 to 4.

00:05:00.236 --> 00:05:01.610
So I could do it
right over here.

00:05:01.610 --> 00:05:04.770
So our change in
time was equal to 2.

00:05:04.770 --> 00:05:06.600
And what was our
change in distance?

00:05:06.600 --> 00:05:10.230
Well our change in distance,
we went from 10 to 5.

00:05:10.230 --> 00:05:13.120
So our change in distance
was equal to negative 5.

00:05:13.120 --> 00:05:15.620
It went down when
our time increased.

00:05:15.620 --> 00:05:17.180
So it's negative 5.

00:05:17.180 --> 00:05:19.740
So the slope of this
line is negative 5/2.

00:05:19.740 --> 00:05:23.350
We started at time
0 at 15 kilometers,

00:05:23.350 --> 00:05:25.780
so actually we could express
the equation for this line.

00:05:25.780 --> 00:05:28.490
We could write that
distance-- and we could even

00:05:28.490 --> 00:05:31.260
write it as distance as a
function of time like that,

00:05:31.260 --> 00:05:34.650
or we could just write
distance-- is equal to,

00:05:34.650 --> 00:05:39.620
we started at 15
kilometers, and then

00:05:39.620 --> 00:05:42.900
we decreased our distance
from there as time increases.

00:05:42.900 --> 00:05:45.900
So it's 15 plus
negative 5/2 times time,

00:05:45.900 --> 00:05:54.030
or we could say
minus 5/2 times time.

00:05:54.030 --> 00:05:55.530
So we didn't have
to do all of this.

00:05:55.530 --> 00:05:57.260
We could have just reasoned
through it with the table

00:05:57.260 --> 00:05:58.770
that we did really
in the first minute,

00:05:58.770 --> 00:06:00.430
but it is nice to
just think about this

00:06:00.430 --> 00:06:02.790
in all the different ways.


Interpreting linear tables word problems example 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.510
Zane is a dangerous
fellow who likes

00:00:03.510 --> 00:00:08.180
to go rock climbing
inside an active volcano.

00:00:08.180 --> 00:00:10.000
He is a dangerous fellow.

00:00:10.000 --> 00:00:12.070
He just heard some
rumblings, so he's

00:00:12.070 --> 00:00:15.030
decided to climb out
as quickly as he can.

00:00:15.030 --> 00:00:17.890
Zane's elevation
relative to the edge

00:00:17.890 --> 00:00:20.630
of the inside of the
volcano in meters,

00:00:20.630 --> 00:00:24.560
E, as a function of time in
seconds is shown in the table

00:00:24.560 --> 00:00:25.410
below.

00:00:25.410 --> 00:00:27.140
Zane climbs at a constant rate.

00:00:27.140 --> 00:00:29.850
So this guy, I mean if we
were to draw a volcano here,

00:00:29.850 --> 00:00:32.210
this guy is just kind of silly.

00:00:32.210 --> 00:00:35.790
So this is my volcano.

00:00:35.790 --> 00:00:40.280
And he's actually
climbing on the inside

00:00:40.280 --> 00:00:41.680
of an active volcano.

00:00:41.680 --> 00:00:45.310
So there's probably smoke and
ash and all the other stuff

00:00:45.310 --> 00:00:46.870
coming out of this thing.

00:00:46.870 --> 00:00:50.240
So this really is
dangerous for him.

00:00:50.240 --> 00:00:53.320
And let's say that this
right over here is Zane.

00:00:53.320 --> 00:01:00.230
He's climbing up from
inside the active volcano.

00:01:00.230 --> 00:01:02.610
So let's think about
what they're telling us.

00:01:02.610 --> 00:01:05.069
So based on the table, which
of these statements is true?

00:01:05.069 --> 00:01:07.360
So I'm not going to even look
at these statements here.

00:01:07.360 --> 00:01:11.210
I'm just going to try
to interpret this.

00:01:11.210 --> 00:01:16.510
So his elevation as a
function of time in seconds

00:01:16.510 --> 00:01:18.240
is shown in the table below.

00:01:18.240 --> 00:01:21.930
So his elevation is negative
24 when time is equal to 0.

00:01:21.930 --> 00:01:24.230
And this table is done in a
kind of nontraditional way.

00:01:24.230 --> 00:01:26.640
Normally, we would
have the input

00:01:26.640 --> 00:01:29.270
into the function on
the left-hand side.

00:01:29.270 --> 00:01:32.172
And then we would have
the function of it

00:01:32.172 --> 00:01:33.130
on the right-hand side.

00:01:33.130 --> 00:01:34.690
And actually I like
looking at things that way,

00:01:34.690 --> 00:01:36.240
so I'm going to
make it like that.

00:01:36.240 --> 00:01:41.120
So let me copy and paste this so
I can put it on the other side.

00:01:41.120 --> 00:01:46.090
So let me cut and let me paste
it, paste it right over here.

00:01:46.090 --> 00:01:49.190
So this one, now I can think
of it a little bit clearer.

00:01:49.190 --> 00:01:53.750
So at time 0, he's going to
be at negative 24 meters.

00:01:53.750 --> 00:01:56.770
At time 4 seconds, he's going
to be at negative 21 meters.

00:01:56.770 --> 00:02:01.800
So this makes a little bit
clearer, at least in my head.

00:02:01.800 --> 00:02:04.060
So let's think about
what's happening.

00:02:04.060 --> 00:02:05.060
So where does he start?

00:02:05.060 --> 00:02:07.000
At time equals 0, where is he?

00:02:07.000 --> 00:02:09.501
Well at time equals
0, he is 24 meters

00:02:09.501 --> 00:02:10.750
below the edge of the volcano.

00:02:10.750 --> 00:02:13.200
So this distance
at time equals 0,

00:02:13.200 --> 00:02:16.235
this distance right
over here is 24 meters.

00:02:16.235 --> 00:02:17.860
And we could even
plot this in a graph.

00:02:21.710 --> 00:02:25.470
So this is his elevation
relative to the edge,

00:02:25.470 --> 00:02:27.070
and it is a function of time.

00:02:27.070 --> 00:02:28.420
I'll write it like that.

00:02:28.420 --> 00:02:30.700
And it is negative
most of this time.

00:02:30.700 --> 00:02:35.160
So I'm going to make the
t-axis a little bit higher.

00:02:35.160 --> 00:02:38.350
So it looks something like that.

00:02:38.350 --> 00:02:39.880
That's our t-axis.

00:02:39.880 --> 00:02:42.630
And when t is equal to 0,
we see that his elevation

00:02:42.630 --> 00:02:44.660
is negative 24 meters.

00:02:44.660 --> 00:02:49.310
So his elevation is
negative 24 meters.

00:02:49.310 --> 00:02:53.720
So he's going to, this is
right here at 0 seconds.

00:02:53.720 --> 00:02:58.080
And then when time increases
by 4, so our change in time

00:02:58.080 --> 00:03:01.235
is equal to 4, what's
his change in elevation?

00:03:01.235 --> 00:03:04.210
Well, his change in
elevation is, let's see,

00:03:04.210 --> 00:03:07.610
he's going from negative
24 to negative 21.

00:03:07.610 --> 00:03:08.970
He increased by 3.

00:03:08.970 --> 00:03:13.690
So his change in elevation
is equal to positive 3.

00:03:13.690 --> 00:03:15.580
He increased by 3.

00:03:15.580 --> 00:03:18.240
So at what rate is he increasing
his elevation with respect

00:03:18.240 --> 00:03:19.130
to time?

00:03:19.130 --> 00:03:24.130
Well, change in elevation
is equal to 3 per unit.

00:03:24.130 --> 00:03:28.470
And that's 3 when
his change in time.

00:03:28.470 --> 00:03:32.900
And remember this triangle just
means a Greek letter delta,

00:03:32.900 --> 00:03:34.340
shorthand for change in.

00:03:34.340 --> 00:03:38.686
So change in elevation over
change in times is 3 over 4.

00:03:38.686 --> 00:03:40.060
So one way to
think about this is

00:03:40.060 --> 00:03:43.575
that he goes 3/4 of
a meter per second.

00:03:46.800 --> 00:03:47.970
The units up here is meter.

00:03:47.970 --> 00:03:49.890
The units down here is second.

00:03:49.890 --> 00:03:51.940
So he goes 3/4 of
a meter per second.

00:03:51.940 --> 00:03:53.410
And we can verify that.

00:03:53.410 --> 00:03:58.840
The next row here, we see
our change in time is 8.

00:04:01.460 --> 00:04:03.116
So it's twice as
much time has passed,

00:04:03.116 --> 00:04:04.990
so he should have gone
twice as much distance

00:04:04.990 --> 00:04:05.990
if his rate is constant.

00:04:05.990 --> 00:04:07.920
Let's verify that
that's the case.

00:04:07.920 --> 00:04:10.980
So he went from negative
21 to negative 15.

00:04:10.980 --> 00:04:14.460
His elevation increased by 6.

00:04:14.460 --> 00:04:16.399
So change in elevation
over change in time

00:04:16.399 --> 00:04:18.640
is 6/8, which is the
same thing as 3/4.

00:04:18.640 --> 00:04:21.200
So you see that he has
this constant change.

00:04:21.200 --> 00:04:22.950
So let's plot a few
of these points.

00:04:22.950 --> 00:04:27.360
So when time is 0, his
elevation is negative 24.

00:04:27.360 --> 00:04:32.240
When time is 4,
right over there,

00:04:32.240 --> 00:04:34.410
his elevation is negative 21.

00:04:34.410 --> 00:04:38.110
Let's say this looks
something like this.

00:04:38.110 --> 00:04:42.040
And so his elevation
as a function of time

00:04:42.040 --> 00:04:43.635
is going to look
something like this.

00:04:46.240 --> 00:04:49.140
Let me actually draw it a
little bit more to scale.

00:04:49.140 --> 00:04:52.520
Because the other
thing that we do know

00:04:52.520 --> 00:04:55.420
is that when time is
32, his elevation is 0.

00:04:55.420 --> 00:04:56.920
So let me put that
right over there.

00:04:56.920 --> 00:04:59.890
When time is 32,
his elevation is 0.

00:04:59.890 --> 00:05:02.580
So his elevation as
a function of time

00:05:02.580 --> 00:05:03.795
looks something like this.

00:05:07.030 --> 00:05:10.480
And we could plot other
points there when time is 4.

00:05:10.480 --> 00:05:12.360
So 4 is going to
be at this half.

00:05:12.360 --> 00:05:12.880
That's a 4.

00:05:12.880 --> 00:05:15.080
So 4 is going to be
right over there.

00:05:15.080 --> 00:05:18.410
His elevation is negative 21.

00:05:18.410 --> 00:05:20.250
So this is a general idea.

00:05:20.250 --> 00:05:22.510
He starts at negative
24 meters and he

00:05:22.510 --> 00:05:25.160
increases at a rate of
3/4 meters per second.

00:05:25.160 --> 00:05:27.980
So which of these
choices is correct?

00:05:27.980 --> 00:05:30.970
Zane was 24 meters below
the edge of the volcano when

00:05:30.970 --> 00:05:34.160
he decided to leave,
and he climbs 3 meters

00:05:34.160 --> 00:05:36.350
every 4 seconds on the way out.

00:05:36.350 --> 00:05:37.180
That seems right.

00:05:37.180 --> 00:05:39.092
He climbs 3 meters
every 4 seconds.

00:05:39.092 --> 00:05:40.550
So we're going to
go with that one.

00:05:40.550 --> 00:05:42.270
Let's make sure that
these aren't right.

00:05:42.270 --> 00:05:44.204
Zane was 24 meters
below the volcano when

00:05:44.204 --> 00:05:45.620
he decided to
leave, and he climbs

00:05:45.620 --> 00:05:47.210
4 meters every 3 seconds.

00:05:47.210 --> 00:05:49.190
No, no, it's 3 meters
every 4 seconds.

00:05:49.190 --> 00:05:50.640
So that's not right.

00:05:50.640 --> 00:05:53.060
Zane was 32 meters below
the edge of the volcano.

00:05:53.060 --> 00:05:54.420
No, that's not right.

00:05:54.420 --> 00:05:55.620
Zane was 32 meters.

00:05:55.620 --> 00:05:57.950
That's not right either.