Writing standard equation of a circle

WEBVTT
Kind: captions
Language: en

00:00:00.213 --> 00:00:01.397
- [Voiceover] So we have a circle here

00:00:01.397 --> 00:00:03.115
and they specified some points for us.

00:00:03.115 --> 00:00:05.846
This little orangeish, or,
I guess, maroonish-red point

00:00:05.846 --> 00:00:09.027
right over here is the
center of the circle,

00:00:09.027 --> 00:00:11.720
and then this blue point
is a point that happens

00:00:11.720 --> 00:00:13.634
to sit on the circle.

00:00:13.634 --> 00:00:15.956
And so with that information,
I want you to pause the video

00:00:15.956 --> 00:00:20.274
and see if you can figure out
the equation for this circle.

00:00:20.274 --> 00:00:21.807
Alright, let's work through this together.

00:00:21.807 --> 00:00:25.725
So let's first think about
the center of the circle.

00:00:25.725 --> 00:00:29.208
And the center of the
circle is just going to be

00:00:29.208 --> 00:00:30.787
the coordinates of that point.

00:00:30.787 --> 00:00:32.923
So, the x-coordinate is negative one

00:00:32.923 --> 00:00:35.105
and then the y-coordinate is one.

00:00:35.105 --> 00:00:38.333
So center is negative one comma one.

00:00:38.333 --> 00:00:40.413
And now, let's think about what the radius

00:00:40.413 --> 00:00:41.871
of the circle is.

00:00:41.871 --> 00:00:43.542
Well, the radius is going
to be the distance between

00:00:43.542 --> 00:00:45.795
the center and any point on the circle.

00:00:45.795 --> 00:00:48.720
So, for example,

00:00:48.720 --> 00:00:50.206
for example,

00:00:50.206 --> 00:00:51.785
this distance.

00:00:51.785 --> 00:00:53.410
The distance of that line.

00:00:53.410 --> 00:00:55.036
Let's see I can do it thicker.

00:00:55.036 --> 00:00:57.311
A thicker version of that.

00:00:57.311 --> 00:01:00.748
This line,

00:01:00.748 --> 00:01:01.653
right over there.

00:01:01.653 --> 00:01:04.486
Something strange about my...

00:01:04.486 --> 00:01:07.110
Something strange about my pen tool.

00:01:07.110 --> 00:01:08.503
It's making that very thin.

00:01:08.503 --> 00:01:10.407
Let me do it one more time.

00:01:10.407 --> 00:01:11.777
Okay, that's better.

00:01:11.777 --> 00:01:14.122
(laughs) The distance of
that line right over there,

00:01:14.122 --> 00:01:16.095
that is going to be the radius.

00:01:16.095 --> 00:01:18.255
So how can we figure that out?

00:01:18.255 --> 00:01:20.948
Well, we can set up a right triangle

00:01:20.948 --> 00:01:22.481
and essentially use the distance formula

00:01:22.481 --> 00:01:24.919
which comes from the Pythagorean Theorem.

00:01:24.919 --> 00:01:26.939
To figure out the length of that line,

00:01:26.939 --> 00:01:29.214
so this is the radius,

00:01:29.214 --> 00:01:34.143
we could figure out a change in x.

00:01:34.143 --> 00:01:38.478
So, if we look at our
change in x right over here.

00:01:38.478 --> 00:01:41.404
Our change in x as we go from the center

00:01:41.404 --> 00:01:42.774
to this point.

00:01:42.774 --> 00:01:45.189
So this is our change in x.

00:01:45.189 --> 00:01:49.345
And then we could say that
this is our change in y.

00:01:49.345 --> 00:01:52.178
That right over there is our change in y.

00:01:52.178 --> 00:01:53.826
And so our change in x-squared

00:01:53.826 --> 00:01:55.173
plus our change in y-squared

00:01:55.173 --> 00:01:56.589
is going to be our radius squared.

00:01:56.589 --> 00:01:59.422
That comes straight out of
the Pythagorean Theorem.

00:01:59.422 --> 00:02:02.162
This is a right triangle.

00:02:02.162 --> 00:02:05.482
And so we can say that r-squared

00:02:05.482 --> 00:02:07.874
is going to be equal to

00:02:07.874 --> 00:02:11.960
our change in x-squared

00:02:11.960 --> 00:02:15.048
plus our change in y-squared.

00:02:15.048 --> 00:02:18.828
Plus our change in y-squared.

00:02:18.828 --> 00:02:20.848
Now, what is our change in x-squared?

00:02:20.848 --> 00:02:22.914
Or, what is our change in x going to be?

00:02:22.914 --> 00:02:27.465
Our change in x is going to be equal to,

00:02:27.465 --> 00:02:28.998
well, when we go from the radius

00:02:28.998 --> 00:02:30.600
to this point over here,

00:02:30.600 --> 00:02:34.919
our x goes from negative one to six.

00:02:34.919 --> 00:02:37.148
So you can view it as our ending x

00:02:37.148 --> 00:02:39.400
minus our starting x.

00:02:39.400 --> 00:02:41.570
So negative one minus negative,

00:02:41.570 --> 00:02:43.695
sorry, six minus negative one

00:02:43.695 --> 00:02:46.717
is equal to seven.

00:02:46.717 --> 00:02:49.611
So, let me...

00:02:49.611 --> 00:02:51.817
So, we have

00:02:51.817 --> 00:02:53.512
our change in x, this right over here,

00:02:53.512 --> 00:02:55.718
is equal to seven.

00:02:55.718 --> 00:02:57.877
If we viewed this as the start point

00:02:57.877 --> 00:02:58.546
and this as the end point,

00:02:58.546 --> 00:02:59.683
it would be negative seven,

00:02:59.683 --> 00:03:00.887
but we really care
about the absolute value

00:03:00.887 --> 00:03:03.301
in the change of x, and once you square it

00:03:03.301 --> 00:03:05.159
it all becomes a positive anyway.

00:03:05.159 --> 00:03:06.784
So our change in x right over here

00:03:06.784 --> 00:03:09.129
is going to be positive seven.

00:03:09.129 --> 00:03:11.451
And our change in y,

00:03:11.451 --> 00:03:14.748
well, we are starting at,

00:03:14.748 --> 00:03:17.790
we are starting at y is equal to one

00:03:17.790 --> 00:03:20.692
and we are going to y is
equal to negative four.

00:03:20.692 --> 00:03:24.779
So it would be negative four minus one

00:03:24.779 --> 00:03:27.867
which is equal to negative five.

00:03:27.867 --> 00:03:30.444
And so our change in y is negative five.

00:03:30.444 --> 00:03:32.070
You can view this distance right over here

00:03:32.070 --> 00:03:34.879
as the absolute value of our change in y,

00:03:34.879 --> 00:03:38.408
which of course would be
the absolute value of five.

00:03:38.408 --> 00:03:39.685
But once you square it, it doesn't matter.

00:03:39.685 --> 00:03:41.311
The negative sign goes away.

00:03:41.311 --> 00:03:44.492
And so, this is going to
simplify to seven squared,

00:03:44.492 --> 00:03:47.208
change in x-squared, is 49.

00:03:47.208 --> 00:03:52.206
Change in y-squared,
negative five squared, is 25.

00:03:52.206 --> 00:03:55.344
So we get r-squared,

00:03:55.344 --> 00:04:00.148
we get r-squared is equal to 49 plus 25.

00:04:00.958 --> 00:04:03.359
So what's 49 plus 25?

00:04:03.359 --> 00:04:06.610
Let's see, that's going
to be 54, or was it 74.

00:04:06.610 --> 00:04:11.137
r-squared is equal to 74.

00:04:11.137 --> 00:04:11.945
Did I do that right?

00:04:11.945 --> 00:04:13.130
Yep, 74.

00:04:13.130 --> 00:04:16.009
And so now we can write the
equation for the circle.

00:04:16.009 --> 00:04:18.586
The circle is going to be
all of the points that are,

00:04:18.586 --> 00:04:20.926
well, in fact, let me right all of the,

00:04:20.926 --> 00:04:22.778
so if r-squared is equal to 74,

00:04:22.778 --> 00:04:26.122
r is equal to the square-root of 74.

00:04:26.122 --> 00:04:27.840
And so the equation of
the circle is going to be

00:04:27.840 --> 00:04:29.755
all points x comma y

00:04:29.755 --> 00:04:33.047
that are this far away from the center.

00:04:33.047 --> 00:04:35.601
And so what are those points going to be?

00:04:35.601 --> 00:04:38.666
Well, the distance is going to be x minus

00:04:38.666 --> 00:04:40.779
the x-coordinate of the center.

00:04:40.779 --> 00:04:45.779
x minus negative one squared.

00:04:45.864 --> 00:04:47.907
Let me do that in a blue color.

00:04:47.907 --> 00:04:50.949
Minus negative one squared.

00:04:50.949 --> 00:04:55.949
Plus y minus,

00:04:55.956 --> 00:05:00.246
y minus the y-coordinate of the center.

00:05:00.246 --> 00:05:04.889
y minus one squared.

00:05:04.889 --> 00:05:05.957
Squared.

00:05:05.957 --> 00:05:09.765
Is equal, is going to
be equal to r-squared.

00:05:09.765 --> 00:05:12.505
Is going to be equal to the
length of the radius squared.

00:05:12.505 --> 00:05:16.151
Well, r-squared we already
know is going to be 74.

00:05:16.151 --> 00:05:17.335
74.

00:05:17.335 --> 00:05:18.937
And then if we want to
simplify it a little bit,

00:05:18.937 --> 00:05:21.584
you subtract a negative,
this becomes a positive.

00:05:21.584 --> 00:05:25.601
So it simplifies to x plus one squared

00:05:25.601 --> 00:05:30.519
plus y minus one squared

00:05:30.519 --> 00:05:33.774
is equal to 74.

00:05:33.774 --> 00:05:36.119
Is equal to 74.

00:05:36.119 --> 00:05:38.459
And, we are all done.


Features of a circle from its graph

WEBVTT
Kind: captions
Language: en

00:00:00.214 --> 00:00:01.921
- [Voiceover] So we have
a circle right over here

00:00:01.921 --> 00:00:03.639
and the first question we'll ask ourselves

00:00:03.639 --> 00:00:06.693
is what are the coordinates
of the center of that circle?

00:00:06.693 --> 00:00:09.446
Well, we can eyeball that,

00:00:09.446 --> 00:00:12.073
we can see, look, it looks like the circle

00:00:12.073 --> 00:00:14.209
is centered on that point right over there

00:00:14.209 --> 00:00:15.570
and the coordinates of that point,

00:00:15.570 --> 00:00:17.950
the x-coordinate is negative four

00:00:17.950 --> 00:00:20.377
and the y-coordinate is negative seven.

00:00:20.377 --> 00:00:24.984
So the center of that
circle would be the point

00:00:24.984 --> 00:00:27.236
negative four comma negative seven.

00:00:27.236 --> 00:00:30.700
Now let's say, on top of
that, someone were to tell us,

00:00:30.700 --> 00:00:32.533
someone were to tell us,

00:00:32.533 --> 00:00:35.447
that this point, negative
five comma negative nine

00:00:35.447 --> 00:00:37.258
is also on the circle.

00:00:37.258 --> 00:00:39.964
So negative five

00:00:40.964 --> 00:00:42.438
comma negative nine

00:00:42.438 --> 00:00:46.311
is on the circle.

00:00:46.650 --> 00:00:48.739
So based on this information,

00:00:48.739 --> 00:00:50.921
the coordinate of the center and a point

00:00:50.921 --> 00:00:54.254
that sits on the circle, can
we figure out the radius?

00:00:54.254 --> 00:00:56.356
Well the radius is just
the distance between

00:00:56.356 --> 00:00:59.363
the center of the circle
and any point on the circle.

00:00:59.363 --> 00:01:01.789
In fact, one of the most
typical definitions of a circle

00:01:01.789 --> 00:01:04.922
is all of the points that
are the same distance,

00:01:04.922 --> 00:01:09.080
or that are the radius
away from another point,

00:01:10.260 --> 00:01:12.971
and that other point would
be the center of the circle.

00:01:12.971 --> 00:01:14.887
So how do we find out the distance

00:01:14.887 --> 00:01:17.778
between these two points,

00:01:17.778 --> 00:01:20.901
between these two points?

00:01:20.901 --> 00:01:23.002
So the length of that orange line.

00:01:23.002 --> 00:01:25.057
Well we can use the distance formula,

00:01:25.057 --> 00:01:27.716
which is essentially
the Pythagorean theorem.

00:01:27.716 --> 00:01:32.035
The distance squared,
so if the length of that

00:01:32.035 --> 00:01:35.288
is the distance, so we could
say the distance squared

00:01:35.288 --> 00:01:37.006
is going to be equal to,

00:01:37.006 --> 00:01:39.363
is going to be equal to

00:01:39.363 --> 00:01:42.858
our change in x squared,
so that right there

00:01:42.858 --> 00:01:45.168
is our change in x, and I
have to write really small,

00:01:45.168 --> 00:01:46.898
but that's our change in x,

00:01:46.898 --> 00:01:51.898
our change in x squared,
plus our change in y squared.

00:01:52.204 --> 00:01:55.536
Our change in y squared.

00:01:55.536 --> 00:01:58.578
Change in y squared.

00:01:58.578 --> 00:02:00.888
Now what is our change in x?

00:02:00.888 --> 00:02:04.348
Our change in x, and you
can even eyeball it here,

00:02:04.348 --> 00:02:06.647
it looks like it's one,
but let's verify it.

00:02:08.007 --> 00:02:10.379
It doesn't matter which one,
just use the start or the end,

00:02:10.379 --> 00:02:11.819
as long as you're consistent.

00:02:11.819 --> 00:02:14.245
So, let's see, if we view this as the end,

00:02:14.245 --> 00:02:15.990
we'd say negative five

00:02:15.990 --> 00:02:17.560
it'd be negative five,

00:02:17.560 --> 00:02:18.990
minus negative four

00:02:18.990 --> 00:02:21.850
minus negative four

00:02:21.850 --> 00:02:24.381
And so this would be
equal to negative one.

00:02:24.381 --> 00:02:27.469
So when you go from the
center to this outer point,

00:02:27.469 --> 00:02:31.010
negative five comma negative nine,

00:02:31.010 --> 00:02:33.320
you go one back in the x direction.

00:02:33.320 --> 00:02:36.803
Now the actual distance would
just be the absolute value

00:02:36.803 --> 00:02:39.032
of that, but it doesn't
matter that this is a negative

00:02:39.032 --> 00:02:40.923
because we're about to square
it and so that negative sign

00:02:40.923 --> 00:02:42.202
will go away.

00:02:42.202 --> 00:02:43.726
Now what is our change in y?

00:02:44.280 --> 00:02:46.085
Our change in y.

00:02:46.706 --> 00:02:49.618
Well, this is the
finishing y, negative nine

00:02:49.618 --> 00:02:54.067
minus negative seven,

00:02:55.588 --> 00:02:57.033
minus our initial y

00:02:57.033 --> 00:02:59.006
is equal to negative two.

00:02:59.006 --> 00:03:00.249
And notice, just to go from that point,

00:03:00.249 --> 00:03:02.675
that y to that y, we go to negative two.

00:03:02.675 --> 00:03:04.568
So actually, we could call
the length of that side

00:03:04.568 --> 00:03:07.038
as the absolute value of our change in y.

00:03:07.038 --> 00:03:09.209
And we could view this
as the absolute value

00:03:09.209 --> 00:03:10.404
of our change in x.

00:03:10.404 --> 00:03:12.309
And it doesn't really matter
because once we square them

00:03:12.309 --> 00:03:13.945
the negatives go away.

00:03:13.945 --> 00:03:15.233
So our distance squared,

00:03:15.233 --> 00:03:17.123
our distance squared,

00:03:17.123 --> 00:03:19.199
I really could call
this the radius squared,

00:03:19.199 --> 00:03:21.591
is going to be equal to
our change in x squared.

00:03:21.591 --> 00:03:23.437
Well, it's negative one squared,

00:03:23.437 --> 00:03:25.782
which is just going to be one

00:03:25.782 --> 00:03:29.971
plus our change in y
squared, negative two squared

00:03:29.971 --> 00:03:32.167
is just positive four.

00:03:32.167 --> 00:03:35.383
One plus four, and so you
have your distance squared

00:03:35.383 --> 00:03:38.100
is equal to five

00:03:38.100 --> 00:03:41.478
or that the distance is equal
to the square root of five.

00:03:41.478 --> 00:03:43.452
And I could have just called
this variable the radius,

00:03:43.452 --> 00:03:45.263
so we could say the radius is equal to

00:03:45.263 --> 00:03:47.771
the square root of five.

00:03:47.771 --> 00:03:48.938
And we're done.


Graphing circles from features

WEBVTT
Kind: captions
Language: en

00:00:00.256 --> 00:00:02.078
- [Voiceover] We're
asked to graph the circle

00:00:02.078 --> 00:00:05.375
which is centered at 3, -2

00:00:05.375 --> 00:00:08.382
and has a radius of 5 units.

00:00:08.382 --> 00:00:10.646
I got this exercise
off of the Khan Academy

00:00:10.646 --> 00:00:13.978
graph a circle according
to its features exercise.

00:00:13.978 --> 00:00:15.696
It's a pretty neat little widget here,

00:00:15.696 --> 00:00:17.396
because what I can do is I can

00:00:17.396 --> 00:00:19.202
take this dot

00:00:19.202 --> 00:00:20.119
and I can move it around

00:00:20.119 --> 00:00:21.674
to redefine the center of the circle

00:00:21.674 --> 00:00:24.101
so it's centered at 3, -2.

00:00:24.101 --> 00:00:25.714
So, X is 3

00:00:25.714 --> 00:00:29.116
and Y is -2, so that's its center.

00:00:29.116 --> 00:00:30.822
It has to have a radius of 5.

00:00:30.822 --> 00:00:31.600
The way it's drawn right now

00:00:31.600 --> 00:00:32.842
it has a radius of 1.

00:00:32.842 --> 00:00:34.154
The distance between the center

00:00:34.154 --> 00:00:36.139
and the actual circle,

00:00:36.139 --> 00:00:37.904
the points that define the circle,

00:00:37.904 --> 00:00:39.401
right now it's 1.

00:00:39.401 --> 00:00:41.398
I need to make this radius equal to 5.

00:00:41.398 --> 00:00:42.954
If I take that,

00:00:42.954 --> 00:00:47.621
so now the radius is equal to 2, 3, 4

00:00:47.621 --> 00:00:49.374
and 5.

00:00:49.374 --> 00:00:50.093
There you go.

00:00:50.093 --> 00:00:51.811
Centered at 3, -2,

00:00:51.811 --> 00:00:53.402
radius of 5.

00:00:53.402 --> 00:00:55.735
Notice you go from the
center to the actual circle

00:00:55.735 --> 00:00:59.439
it's 5 no matter where you go.

00:00:59.439 --> 00:01:01.934
Let's do one more of these.

00:01:02.314 --> 00:01:05.417
Graph the circle which
is centered at -4, 1

00:01:05.417 --> 00:01:09.120
and which has the point 0, 4 on it.

00:01:09.120 --> 00:01:10.595
Once again, let's drag the center,

00:01:10.595 --> 00:01:11.988
so it's going to be -4,

00:01:11.988 --> 00:01:13.416
X is -4.

00:01:13.416 --> 00:01:15.494
Y is 1, so that's the center.

00:01:15.494 --> 00:01:18.976
It has the point 0, 4 on it.

00:01:18.976 --> 00:01:21.391
X is 0, Y is 4.

00:01:21.391 --> 00:01:22.993
I have to drag,

00:01:22.993 --> 00:01:25.060
I have to increase the
radius of the circle.

00:01:25.060 --> 00:01:25.953
Let's see,

00:01:25.953 --> 00:01:26.789
whoops,

00:01:26.789 --> 00:01:29.018
nope, I want to make sure
I don't change the center.

00:01:29.018 --> 00:01:30.655
I want to increase the
radius of the circle

00:01:30.655 --> 00:01:35.577
until it includes this
point right over here, 0,4.

00:01:35.577 --> 00:01:37.795
I'm not there quite yet,

00:01:37.795 --> 00:01:38.909
there you go.

00:01:38.909 --> 00:01:42.032
I'm now including the point 0, 4.

00:01:42.032 --> 00:01:43.855
If we're curious what the radius is,

00:01:43.855 --> 00:01:45.898
we could just go along the X axis.

00:01:45.898 --> 00:01:49.949
X = -4 is the X coordinate for the center

00:01:49.949 --> 00:01:51.958
and we see that this point ...

00:01:51.958 --> 00:01:56.958
This is -4,1 and we see that
1,1 is actually on the circle.

00:01:57.553 --> 00:02:00.432
The distance here is you go 4

00:02:00.432 --> 00:02:01.535
then another 1, it's 5.

00:02:01.535 --> 00:02:03.996
This has a radius of 5.

00:02:03.996 --> 00:02:06.991
But either way, we did
what they asked us to do.


Periodicity of algebraic models

WEBVTT
Kind: captions
Language: en

00:00:00.451 --> 00:00:02.408
- [Voiceover] We're told Divya
is seated on a Ferris Wheel

00:00:02.408 --> 00:00:04.217
at time t equals zero.

00:00:04.217 --> 00:00:06.683
The graph below shows her height h

00:00:06.683 --> 00:00:10.350
in meters, t seconds
after the ride starts.

00:00:10.350 --> 00:00:15.350
So at time equals zero, she
is, looks like about two,

00:00:15.648 --> 00:00:17.551
what is this, this
would be one and a half,

00:00:17.551 --> 00:00:22.290
so it looks like she's about
two meters off the ground

00:00:22.290 --> 00:00:25.384
and then as time
increases, she gets as high

00:00:25.384 --> 00:00:30.364
as, it looks like this is
close to 30, maybe 34 meters

00:00:30.364 --> 00:00:34.284
and then she comes back
down, looks like two meters

00:00:34.284 --> 00:00:38.434
and up to 34 meters again,
so let's read the question.

00:00:39.117 --> 00:00:43.249
So the question asks us,
approximately how long

00:00:43.249 --> 00:00:45.484
does it take Divya to complete one

00:00:45.484 --> 00:00:47.818
revolution on the Ferris Wheel?

00:00:47.818 --> 00:00:50.018
All right, so this is
interesting, so this is when

00:00:50.018 --> 00:00:52.351
she's at the bottom of the Ferris Wheel,

00:00:52.351 --> 00:00:55.151
so then she gets to the
top of the Ferris Wheel,

00:00:55.151 --> 00:00:57.617
and then she keeps rotating until she gets

00:00:57.617 --> 00:01:00.418
back to the bottom of
the Ferris Wheel again.

00:01:00.418 --> 00:01:05.405
So it took her 60 and t
is in terms of seconds.

00:01:05.405 --> 00:01:08.017
So it took her 60 seconds
to go from the bottom

00:01:08.017 --> 00:01:10.218
to the bottom again, and
in another 60 seconds,

00:01:10.218 --> 00:01:12.817
she would have completed
another revolution.

00:01:12.817 --> 00:01:15.784
And so let me fill that in, it is going

00:01:15.784 --> 00:01:19.650
to take her 60 seconds, 60 seconds,

00:01:19.650 --> 00:01:23.000
and we of course can check
our answer if we like.

00:01:24.218 --> 00:01:26.084
Let's do another one of these.

00:01:26.084 --> 00:01:29.484
So here we have,

00:01:29.484 --> 00:01:31.950
a doctor observes the electrical activity

00:01:31.950 --> 00:01:34.451
of Finn's heart over a period of time.

00:01:34.451 --> 00:01:37.885
The electrical activity of
Finn's heart is cyclical,

00:01:37.885 --> 00:01:39.718
as we hope it would be, and peaks

00:01:39.718 --> 00:01:41.985
every 0.9 seconds.

00:01:41.985 --> 00:01:43.385
Which of the following graphs could model

00:01:43.385 --> 00:01:45.850
the situation if t stands
for time in seconds,

00:01:45.850 --> 00:01:47.517
and e stands for the electrical activity

00:01:47.517 --> 00:01:49.452
of Finn's heart in volts?

00:01:49.452 --> 00:01:51.618
Over here it looks like
we peaked at zero seconds,

00:01:51.618 --> 00:01:53.518
and then here we're peaking
a little bit more than one,

00:01:53.518 --> 00:01:57.151
this looks maybe at 1.1,
maybe at 2.2 and 3.3.

00:01:57.151 --> 00:01:58.818
This looks like it's
peaking a little bit more

00:01:58.818 --> 00:02:01.584
than every one second, so
like maybe every 1.1 seconds,

00:02:01.584 --> 00:02:04.984
not every 0.9 seconds, so I'd rule out A.

00:02:04.984 --> 00:02:07.783
This one is peaking, it
looks like the interval

00:02:07.783 --> 00:02:09.884
between peaks is less than a second,

00:02:09.884 --> 00:02:11.417
but it looks like a good
bit less than a second.

00:02:11.417 --> 00:02:13.251
It looks like maybe every three quarters

00:02:13.251 --> 00:02:16.185
of a second, or maybe every
four-fifths of a second.

00:02:16.185 --> 00:02:18.118
Not quite nine-tenths,
nine-tenths this first peak

00:02:18.118 --> 00:02:20.017
would be a little bit closer to one,

00:02:20.017 --> 00:02:22.085
but this one is close.

00:02:22.085 --> 00:02:23.684
Choice C is looking good.

00:02:23.684 --> 00:02:25.508
The first we're at zero,
then the first peak,

00:02:25.508 --> 00:02:27.617
this looks pretty close to
one but it's less than one.

00:02:27.617 --> 00:02:31.351
It looks like a tenth less
than one, so I like choice C.

00:02:31.351 --> 00:02:32.784
Now choice D, it looks like we're peaking

00:02:32.784 --> 00:02:34.917
every half-second, so
it's definitely not that.

00:02:34.917 --> 00:02:38.084
So this looks like a peak
of every 0.9 seconds.

00:02:38.084 --> 00:02:41.185
This is the best representation that I...

00:02:41.185 --> 00:02:43.552
This is the best representation
that I can think of.

00:02:43.552 --> 00:02:44.950
And you can actually verify that.

00:02:44.950 --> 00:02:48.150
If you have a peak every 0.9 seconds,

00:02:48.150 --> 00:02:51.484
you're going to have four
peaks in 3.6 seconds.

00:02:51.484 --> 00:02:53.954
So one, two, three, four, this looks

00:02:53.954 --> 00:02:56.118
like it's at 3.6.

00:02:56.118 --> 00:02:59.018
Over here, you have one, two, three, four,

00:02:59.018 --> 00:03:01.851
you've had four peaks in
less than three seconds.

00:03:01.851 --> 00:03:04.879
So this one definitely isn't 0.9.

00:03:04.879 --> 00:03:06.221
So instead of just even forcing yourself

00:03:06.221 --> 00:03:09.085
to eyeball just between
this peak and that peak,

00:03:09.085 --> 00:03:10.985
you can say well, if
we're every 0.9 seconds,

00:03:10.985 --> 00:03:13.017
how long would three
peaks take or four peaks,

00:03:13.017 --> 00:03:14.850
and then you can actually
get a little bit more precise

00:03:14.850 --> 00:03:16.684
as you try to eyeball it.

00:03:16.684 --> 00:03:19.834
So we can check our answer and
verify that we got it right.


End behavior of algebraic models

WEBVTT
Kind: captions
Language: en

00:00:00.348 --> 00:00:02.600
- [Voiceover] A barista
poured a cup of coffee.

00:00:02.600 --> 00:00:04.319
The initial temperature of the coffee

00:00:04.319 --> 00:00:06.341
was 90 degrees Celsius.

00:00:06.341 --> 00:00:08.872
At time t increased (laughs).

00:00:08.872 --> 00:00:12.889
As time t increased, the
temperature C of the coffee

00:00:12.889 --> 00:00:15.489
began to decrease exponentially

00:00:15.489 --> 00:00:19.042
and approach room temperature
of 20 degrees Celsius.

00:00:19.042 --> 00:00:23.384
Which of the following graphs
could model this relationship?

00:00:23.384 --> 00:00:25.241
So we're starting at 90 degrees Celsius.

00:00:25.241 --> 00:00:28.167
It looks like all of the graphs
start at 90 degrees Celsius

00:00:28.167 --> 00:00:30.930
at t equals zero.

00:00:30.930 --> 00:00:34.506
And we're going to approach
the room temperature

00:00:34.506 --> 00:00:36.154
of 20 degrees Celsius.

00:00:36.154 --> 00:00:38.731
So this first one does
approach the room temperature

00:00:38.731 --> 00:00:41.849
of 20 degrees Celsius as t increases.

00:00:41.849 --> 00:00:45.761
Now this one, when t is 70,

00:00:45.761 --> 00:00:48.199
when t is 70.

00:00:48.199 --> 00:00:50.312
I'm assuming this is in minutes.

00:00:50.312 --> 00:00:53.725
When t is 70 it looks like
it has the temperature

00:00:53.725 --> 00:00:56.209
going to zero degrees Celsius

00:00:56.209 --> 00:00:58.708
so that cup of coffee is
going to start freezing.

00:00:58.708 --> 00:01:00.450
So I think I can rule out B.

00:01:00.450 --> 00:01:03.909
Also this looks like a linear
model not an exponential one.

00:01:03.909 --> 00:01:05.657
C, it does get us to this end state

00:01:05.657 --> 00:01:07.677
that stays at 20 degrees,

00:01:07.677 --> 00:01:09.674
but it doesn't look like
an exponential model.

00:01:09.674 --> 00:01:11.787
It looks like it's linearly decreasing

00:01:11.787 --> 00:01:14.620
and then it stops linearly
decreasing after 50 minutes

00:01:14.620 --> 00:01:16.492
and then it just stays
constant at that temperature

00:01:16.492 --> 00:01:17.792
of 20 degrees.

00:01:17.792 --> 00:01:18.999
So even though it gets
us to the right place,

00:01:18.999 --> 00:01:21.507
it does not look like
an exponential decay.

00:01:21.507 --> 00:01:23.574
So I would rule choice C out as well.

00:01:23.574 --> 00:01:25.106
So A is looking good.

00:01:25.106 --> 00:01:26.964
D, we are starting at 90,

00:01:26.964 --> 00:01:30.005
it does look like an exponential function.

00:01:30.005 --> 00:01:32.926
We have exponential decay right over here

00:01:32.926 --> 00:01:34.516
and we are approaching something

00:01:34.516 --> 00:01:35.700
but it's not the room temperature

00:01:35.700 --> 00:01:36.814
of 20 degrees Celsius.

00:01:36.814 --> 00:01:39.183
We're approaching 30 degrees Celsius here.

00:01:39.183 --> 00:01:40.715
So I'd also rule out D.

00:01:40.715 --> 00:01:42.364
So A is looking good.

00:01:42.364 --> 00:01:44.035
It's an exponential.

00:01:44.035 --> 00:01:46.195
It's decreasing exponentially.

00:01:46.195 --> 00:01:47.843
Starting at 90 degrees Celsius

00:01:47.843 --> 00:01:49.979
and it's approaching the room temperature

00:01:49.979 --> 00:01:52.533
of 20 degrees Celsius.

00:01:52.533 --> 00:01:54.609
Let's do another one of these.

00:01:56.193 --> 00:01:57.633
So it says.

00:01:57.633 --> 00:02:00.535
Let me scroll up a little bit.

00:02:00.535 --> 00:02:03.368
So it says, after the
closing of the mills,

00:02:03.368 --> 00:02:07.013
the population of the town
starts decreasing exponentially.

00:02:07.013 --> 00:02:10.032
The graph below represents
the population, P,

00:02:10.032 --> 00:02:13.143
in thousands, of the town t years

00:02:13.143 --> 00:02:15.558
after the closing of the mill.

00:02:15.558 --> 00:02:16.324
Alright.

00:02:16.324 --> 00:02:19.691
So it looks like the
population starts at 40,000.

00:02:19.691 --> 00:02:21.130
It's decreasing exponentially.

00:02:21.130 --> 00:02:22.547
It looks like, over time, the population

00:02:22.547 --> 00:02:25.542
is approaching 20,000 people.

00:02:25.542 --> 00:02:27.391
So, what are the question here?

00:02:27.391 --> 00:02:29.573
Based on the graph, with the mill closed,

00:02:29.573 --> 00:02:31.260
what does the population
of the town approach

00:02:31.260 --> 00:02:32.722
as time increases?

00:02:32.722 --> 00:02:33.541
We just said it.

00:02:33.541 --> 00:02:36.466
As time increases, it looks
like it's coming close to,

00:02:36.466 --> 00:02:38.904
it's approaching, 20,000.

00:02:38.904 --> 00:02:41.366
It's approaching 20,000.

00:02:41.366 --> 00:02:45.313
It's already gotten
below 22,000 as far as,

00:02:45.313 --> 00:02:47.719
you know, it looks like
after 20 or 22 years.

00:02:47.719 --> 00:02:49.321
We've already gotten below 22,000,

00:02:49.321 --> 00:02:51.202
so we're definitely below 30 or 40,000.

00:02:51.202 --> 00:02:53.593
But we haven't gotten below 20,000

00:02:53.593 --> 00:02:55.613
but we are approaching it.

00:02:55.613 --> 00:02:57.680
We can even check our answer if we like.


Interpreting equations graphically

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.347
- [Voiceover] Let f of x
equal three to the negative x

00:00:03.347 --> 00:00:06.575
minus five, and g of x equals
negative x to the third

00:00:06.575 --> 00:00:08.548
minus four x-squared plus x plus six.

00:00:08.548 --> 00:00:11.520
The graphs of y equals f
of x and y equals g of x

00:00:11.520 --> 00:00:13.397
are shown below, and we
see them right over here.

00:00:13.397 --> 00:00:17.042
This y equals f of x is
in that purplish color.

00:00:17.042 --> 00:00:21.756
Let me see if I can get
that same purplish color.

00:00:21.756 --> 00:00:24.607
So, that is, in that.

00:00:24.607 --> 00:00:27.045
So, let me underline it now.

00:00:27.045 --> 00:00:31.442
So, f of x is

00:00:31.442 --> 00:00:33.160
in that purplish color.

00:00:33.160 --> 00:00:37.149
So y equal f of x, that's
right over here in purple.

00:00:37.149 --> 00:00:40.586
And then y equals g of x, that is in blue.

00:00:40.586 --> 00:00:42.443
So y equals g of x.

00:00:42.443 --> 00:00:45.480
g of x is defined right over there.

00:00:45.480 --> 00:00:49.311
y equals g of x, well that
is graphed in the blue.

00:00:49.311 --> 00:00:51.029
And we see that they intersect

00:00:51.029 --> 00:00:53.978
at the point a comma b.

00:00:53.978 --> 00:00:57.879
So there's a couple of
ways to think about this.

00:00:57.879 --> 00:01:00.828
We could say that when x is equal to a,

00:01:00.828 --> 00:01:04.125
f of x and g of x equal each other.

00:01:04.125 --> 00:01:07.120
Or, we could say, f of a,

00:01:07.120 --> 00:01:09.604
and this is coming from
this point of intersection.

00:01:09.604 --> 00:01:12.344
Let me draw a little arrow here.

00:01:12.344 --> 00:01:13.737
Or a big arrow.

00:01:13.737 --> 00:01:16.586
That point of intersection lets us know

00:01:16.586 --> 00:01:18.612
that f of a,

00:01:18.612 --> 00:01:23.517
f of a is equal to g of a.

00:01:23.517 --> 00:01:27.827
Is equal to g of a, which is equal to b.

00:01:27.827 --> 00:01:32.006
g of a, which is equal to b.

00:01:32.006 --> 00:01:35.513
They both, if you input
a into the function f

00:01:35.513 --> 00:01:36.720
you're going to get b.

00:01:36.720 --> 00:01:38.438
If you input a into the function g,

00:01:38.438 --> 00:01:39.576
you're going to get b.

00:01:39.576 --> 00:01:44.071
And so the point a comma
b is on both graphs.

00:01:44.071 --> 00:01:47.717
Both y equals g of x and y equals f of x.

00:01:47.717 --> 00:01:48.635
And so from here, you could

00:01:48.635 --> 00:01:52.582
make some interesting statements.

00:01:52.582 --> 00:01:56.181
For example, you could just
say, well, what is f of a?

00:01:56.181 --> 00:02:00.329
f of a is 3 to the negative a power

00:02:00.329 --> 00:02:03.584
minus five, is going to be equal to,

00:02:03.584 --> 00:02:05.158
what's g of a?

00:02:05.158 --> 00:02:09.350
g of a is negative a to the third

00:02:09.350 --> 00:02:12.137
minus four a-squared

00:02:12.137 --> 00:02:15.782
plus a plus six, so you could say this.

00:02:15.782 --> 00:02:19.218
And that would be equal to that.

00:02:19.218 --> 00:02:21.865
Either of those would be equal to b.

00:02:21.865 --> 00:02:24.443
Alright, I think we've
analyzed that a good bit,

00:02:24.443 --> 00:02:26.811
so now let's actually
answer their questions.

00:02:26.811 --> 00:02:28.831
And normally I'd suggest
you look at the questions

00:02:28.831 --> 00:02:29.644
before you actually try to solve it,

00:02:29.644 --> 00:02:32.662
but I just wanted to do this
just to really squeeze out

00:02:32.662 --> 00:02:35.768
as much as we could out of
the information they gave us.

00:02:35.768 --> 00:02:38.020
So they tell us the value
x equals a is the solution

00:02:38.020 --> 00:02:39.445
to which of the following equations?

00:02:39.445 --> 00:02:40.908
Select all that apply.

00:02:40.908 --> 00:02:43.299
So this first one is three
to the negative x minus five

00:02:43.299 --> 00:02:44.646
is equal to b.

00:02:44.646 --> 00:02:46.332
Well, we already know that,

00:02:46.332 --> 00:02:51.170
we already know that three
to the negative a minus five,

00:02:51.170 --> 00:02:53.740
Three to the negative a minus

00:02:53.740 --> 00:02:56.065
is going to be equal to b.

00:02:56.065 --> 00:02:59.130
Is going to be equal to b.

00:02:59.130 --> 00:03:01.846
This over here, this is
equivalent to saying,

00:03:01.846 --> 00:03:05.120
this is equivalent to saying f of x,

00:03:05.120 --> 00:03:07.280
f of x is equal to b.

00:03:07.280 --> 00:03:10.832
And we know f of x
equals B when x equals a.

00:03:10.832 --> 00:03:14.501
When x equals a, f of x is equal to b.

00:03:14.501 --> 00:03:16.149
This expression is equal to b.

00:03:16.149 --> 00:03:19.539
So we know that that first one is true.

00:03:19.539 --> 00:03:21.722
Now the second one is just saying,

00:03:21.722 --> 00:03:24.592
f of x is equal to g of x.

00:03:24.592 --> 00:03:27.076
Well, we know that when x is equal to a,

00:03:27.076 --> 00:03:29.630
f of x is equal to g of x.

00:03:29.630 --> 00:03:33.183
That f of a is equal to g of a.

00:03:33.183 --> 00:03:35.925
Because, as a reminder,
this right over here

00:03:35.925 --> 00:03:38.363
is our definition for f of x,

00:03:38.363 --> 00:03:43.357
and this over here is
our definition of g of x.

00:03:43.357 --> 00:03:45.835
So this is just saying
f of x equals g of x,

00:03:45.835 --> 00:03:46.926
when does that happen?

00:03:46.926 --> 00:03:48.807
Well, that happens when x equals a.

00:03:48.807 --> 00:03:49.852
We already saw it up here.

00:03:49.852 --> 00:03:51.867
f of a is equal to g of a.

00:03:51.867 --> 00:03:54.022
They both equal b.

00:03:54.022 --> 00:03:55.461
And so both of these are going to be

00:03:55.461 --> 00:03:59.246
equal to each other when x is equal to A.

00:03:59.246 --> 00:04:02.915
So I will check that one as well.


Interpreting equations graphically (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.399 --> 00:00:02.094
- [Voiceover] Let f of t equal either

00:00:02.094 --> 00:00:04.230
the 2t minus 2t squared,

00:00:04.230 --> 00:00:08.038
and h of t equal four minus 5t squared.

00:00:08.038 --> 00:00:10.081
The graphs of y equals f of t,

00:00:10.081 --> 00:00:13.099
and y equals h of t, are shown below.

00:00:13.099 --> 00:00:16.396
So y equals f of t is here in green,

00:00:16.396 --> 00:00:18.649
so this is really y is equal to

00:00:18.649 --> 00:00:21.133
e to the 2t minus 2t squared.

00:00:21.133 --> 00:00:22.967
We see f of t right over there.

00:00:22.967 --> 00:00:27.026
And y is equal to h of
t is shown in yellow.

00:00:27.026 --> 00:00:28.117
All right.

00:00:28.117 --> 00:00:30.274
Now below that, they say,
"Which of the following

00:00:30.274 --> 00:00:35.246
"appear to be solutions of
e to the 2t minus 2t squared

00:00:35.246 --> 00:00:38.408
"equals four minus 5t squared?

00:00:38.408 --> 00:00:39.685
"Select all that apply."

00:00:39.685 --> 00:00:41.102
And I encourage you to pause the video

00:00:41.102 --> 00:00:43.191
and try to think about it.

00:00:43.191 --> 00:00:44.608
Now the key here is to realize

00:00:44.608 --> 00:00:47.278
that either the 2t minus 2t squared,

00:00:47.278 --> 00:00:49.368
that was f of t.

00:00:49.368 --> 00:00:53.268
And four minus 5t squared is h of t.

00:00:53.268 --> 00:00:55.316
So another way of thinking about it,

00:00:55.316 --> 00:01:00.313
select all of the Ts for which f of t

00:01:01.757 --> 00:01:04.423
is equal to h of t.

00:01:04.423 --> 00:01:08.533
So all of the Ts where f
of t is equal to h of t,

00:01:08.533 --> 00:01:11.203
well that's going to happen
at the points of intersection.

00:01:11.203 --> 00:01:13.842
So for example, at t1,

00:01:13.842 --> 00:01:17.116
we see at this point right here, t1 y1.

00:01:17.116 --> 00:01:19.461
So this tells us f of t1

00:01:19.461 --> 00:01:22.991
is equal to h of t1,

00:01:22.991 --> 00:01:26.218
which is equal to y1.

00:01:26.218 --> 00:01:30.490
So f of t is going to be equal to h of t

00:01:30.490 --> 00:01:33.880
at t is equal to t1.

00:01:33.880 --> 00:01:36.574
We see that there because
it's a point of intersection.

00:01:36.574 --> 00:01:38.863
Now let's keep on going.

00:01:38.863 --> 00:01:40.530
Well they have another
point of intersection

00:01:40.530 --> 00:01:44.797
right over here at t4, t4 y4.

00:01:44.797 --> 00:01:48.326
If you took f of t4,
you're going to get y4.

00:01:48.326 --> 00:01:51.832
Or if you take h of t4,
you're going to get y4.

00:01:51.832 --> 00:01:55.501
So f of t4 is equal to h of t4.

00:01:55.501 --> 00:01:57.753
f of t4 is equal to h of t4.

00:01:57.753 --> 00:02:01.129
So if you took e to the two times t4

00:02:01.129 --> 00:02:03.288
minus two t4 squared,

00:02:03.288 --> 00:02:05.216
and all the way, that's going to equal to

00:02:05.216 --> 00:02:08.254
four minus five times t4 squared.

00:02:12.564 --> 00:02:15.768
Since f of t and h of t equal each other,

00:02:15.768 --> 00:02:17.579
when t is equal to t4,

00:02:17.579 --> 00:02:19.901
these two things are
going to equal each other,

00:02:19.901 --> 00:02:22.548
when t is equal to t4.

00:02:22.548 --> 00:02:24.028
And those are the only ones

00:02:26.198 --> 00:02:29.101
that are at a point of intersection.

00:02:29.101 --> 00:02:32.955
And so I think we are done.

00:02:32.955 --> 00:02:34.279
I can check my answer,

00:02:34.279 --> 00:02:35.527
and I got it right.


Exponential model word problem: medication dissolve

WEBVTT
Kind: captions
Language: en

00:00:00.326 --> 00:00:02.427
- [Voiceover] Carlos has
taken an initial dose

00:00:02.427 --> 00:00:04.563
of a prescription medication.

00:00:04.563 --> 00:00:06.758
The relationship between the time,

00:00:06.758 --> 00:00:09.276
between the elapsed time t, in hours,

00:00:09.276 --> 00:00:12.225
since he took the first dose
and the amount of medication,

00:00:12.225 --> 00:00:14.977
M of t, in milligrams, in
his bloodstream is modeled

00:00:14.977 --> 00:00:17.078
by the following function.

00:00:17.078 --> 00:00:21.558
Alright, in how many hours
will Carlos have 1 milligram

00:00:21.558 --> 00:00:25.459
of medication remaining
in his bloodstream?

00:00:25.459 --> 00:00:29.279
So M of what t is equal to, so
we need essentially to solve

00:00:29.279 --> 00:00:32.421
for M of t is equal to 1 milligram.

00:00:32.421 --> 00:00:35.091
Because M of t outputs, whatever
value it outputs is going

00:00:35.091 --> 00:00:36.713
to be in milligram.

00:00:36.713 --> 00:00:37.987
So let's just solve that.

00:00:37.987 --> 00:00:40.601
So M of t is, they give us a definition,

00:00:40.601 --> 00:00:44.722
it's model is an exponential
function, 20 times e

00:00:44.722 --> 00:00:48.193
to the negative 0.8 t

00:00:48.193 --> 00:00:50.712
is equal to one.

00:00:50.712 --> 00:00:54.392
So let's see, we can
divide both sides by 20

00:00:54.392 --> 00:00:57.741
and so we will get e

00:00:57.741 --> 00:01:01.165
to the negative 0.8 t

00:01:01.165 --> 00:01:03.551
is equal to one over 20,

00:01:04.369 --> 00:01:06.671
one over 20.

00:01:06.671 --> 00:01:11.082
Which we could write as 0.05, 0.05.

00:01:11.082 --> 00:01:12.673
I have a feeling we're gonna
have to deal with decimals

00:01:12.673 --> 00:01:14.093
here regardless.

00:01:14.093 --> 00:01:16.971
And so how do we, how do we solve this?

00:01:16.971 --> 00:01:20.103
Well one way to think about
it, one way to think about it

00:01:20.103 --> 00:01:22.011
if we took, what happens
if we took the natural log

00:01:22.011 --> 00:01:23.149
of both sides?

00:01:23.149 --> 00:01:26.353
And just a remember, a
reminder, the natural log

00:01:26.353 --> 00:01:29.546
is the logarithm base e.

00:01:29.546 --> 00:01:30.999
So actually let me write
this, let me write this

00:01:30.999 --> 00:01:32.983
a little bit differently.

00:01:32.983 --> 00:01:35.676
So this is zero and

00:01:35.676 --> 00:01:40.676
that is 0.05.

00:01:40.683 --> 00:01:45.678
So I'm gonna take the natural
log of both sides, so ln, ln.

00:01:45.678 --> 00:01:49.232
So the natural log, this
says, what power do I have

00:01:49.232 --> 00:01:54.232
to raise e to, to get to
e to the negative 0.8 t?

00:01:54.232 --> 00:01:58.318
Well I've got to raise e
to the, this simplifies

00:01:58.318 --> 00:02:02.265
to negative 0.8 t.

00:02:02.265 --> 00:02:04.414
Once again natural log this thing...

00:02:04.414 --> 00:02:09.405
let me clarify ln of e
to the negative 0.8 t.

00:02:09.405 --> 00:02:13.071
This is equivalent to if
I were to write log base e

00:02:13.071 --> 00:02:16.973
of e to the negative 0.8 t.

00:02:16.973 --> 00:02:19.038
What power do I have to raise e to, to get

00:02:19.038 --> 00:02:21.257
to e to the negative 0.8 t?

00:02:21.257 --> 00:02:24.053
Well I have to raise it to
the negative 0.8 t power.

00:02:24.053 --> 00:02:26.712
So that's why the left-hand
side simplified to this

00:02:26.712 --> 00:02:29.452
and that's going to be
equal to the natural log,

00:02:29.452 --> 00:02:31.297
actually I'll just
leave it in those terms,

00:02:31.297 --> 00:02:34.515
the natural log of 0.05,

00:02:34.515 --> 00:02:39.303
the natural log of 0.05

00:02:39.303 --> 00:02:42.458
all of that and now we
can divide both sides

00:02:42.458 --> 00:02:46.742
by negative 0.8 to solve for t.

00:02:46.742 --> 00:02:49.172
So let's do that.

00:02:49.172 --> 00:02:54.172
So we divide by negative 0.8,

00:02:54.174 --> 00:02:58.118
divide by negative 0.8

00:02:58.118 --> 00:03:01.077
and so t is going to be equal
to all of this business.

00:03:01.077 --> 00:03:03.121
On the left-hand side now we just have a t

00:03:03.121 --> 00:03:05.362
and on the right-hand side
we have all of this business

00:03:05.362 --> 00:03:07.568
which I think a calculator
will be valuable for.

00:03:07.568 --> 00:03:09.641
So let me get a calculator out.

00:03:11.120 --> 00:03:16.120
Clear it out and let's start with 0.05.

00:03:16.123 --> 00:03:17.733
Let's take the natural log, that's

00:03:17.733 --> 00:03:20.450
that button right over
there, the natural log.

00:03:20.450 --> 00:03:23.805
We get that value and we want
to divide it by negative .8.

00:03:23.805 --> 00:03:26.147
So divided by,

00:03:26.147 --> 00:03:31.147
divided by .8 negative.

00:03:31.151 --> 00:03:35.254
So we're gonna divide by
.8 negative is equal to,

00:03:35.254 --> 00:03:38.361
let's see they wanted us to
round to the nearest hundredth

00:03:38.361 --> 00:03:41.484
so 3.74,

00:03:41.484 --> 00:03:44.921
so it'll take 3.74,

00:03:44.921 --> 00:03:47.786
seven four hours for his dosage

00:03:47.786 --> 00:03:50.491
to go down to one milligram
where it actually started

00:03:50.491 --> 00:03:51.871
at 20 milligrams.

00:03:51.871 --> 00:03:54.019
When t equals zero it's 20

00:03:54.019 --> 00:03:57.641
after 3.74 hours he's down

00:03:57.641 --> 00:03:59.521
in his bloodstream to one milligram.

00:03:59.521 --> 00:04:03.397
I guess his body has metabolized
the rest of it in some way.


Exponential model word problem: bacteria growth

WEBVTT
Kind: captions
Language: en

00:00:00.853 --> 00:00:02.730
- The bacteria in a Petri dish culture

00:00:02.730 --> 00:00:05.202
are self-duplicating at a rapid pace.

00:00:05.202 --> 00:00:07.524
The relationship between
the elapsed time T,

00:00:07.524 --> 00:00:09.927
in minutes, and the number
of bacteria, B of T,

00:00:09.927 --> 00:00:13.201
in the Petri dish is modeled
by the following function.

00:00:13.201 --> 00:00:15.140
And we see it's an exponential model here.

00:00:15.140 --> 00:00:17.044
How many bacteria will make up the culture

00:00:17.044 --> 00:00:20.085
after 120 minutes?

00:00:20.085 --> 00:00:21.710
So, really they just want to say,

00:00:21.710 --> 00:00:25.286
well what is B of 120 going to be?

00:00:25.286 --> 00:00:28.649
And so it's going to be 10 times two

00:00:28.649 --> 00:00:31.439
to the 120

00:00:31.439 --> 00:00:33.959
divided by 12th power.

00:00:33.959 --> 00:00:37.428
So, this is going to be
equal to 10 times two to the,

00:00:37.428 --> 00:00:42.025
120 divided by 12 is, 10th power.

00:00:42.025 --> 00:00:45.001
So this is going to be equal to 10 times,

00:00:45.001 --> 00:00:48.891
two to the 10th power is

00:00:48.891 --> 00:00:51.128
1,024.

00:00:51.128 --> 00:00:52.551
If you want to verify that,

00:00:52.551 --> 00:00:57.495
you can say, well two to
the 5th is equal to 32,

00:00:57.495 --> 00:00:59.493
and so two to the 10th is going to be

00:00:59.493 --> 00:01:00.851
two to the 5th times two to the 5th.

00:01:00.851 --> 00:01:04.322
And 32 times 32 is...

00:01:04.322 --> 00:01:06.489
Let's see, 64. Zero.

00:01:06.489 --> 00:01:09.659
So, Let's see we're gonna have...

00:01:09.659 --> 00:01:13.936
Sorry, three times 32 is 96.

00:01:13.936 --> 00:01:16.119
Let's see you have a four and 12,

00:01:16.119 --> 00:01:17.872
1024.

00:01:17.872 --> 00:01:19.091
So this is gonna be 1024.

00:01:19.091 --> 00:01:20.995
10 times that is going to be equal to

00:01:20.995 --> 00:01:23.932
one zero two four zero.

00:01:23.932 --> 00:01:26.264
So, 10,200...

00:01:26.264 --> 00:01:29.648
10,240 bacteria,

00:01:30.606 --> 00:01:32.376
and we're done.


Using the logarithm change of base rule

WEBVTT
Kind: captions
Language: en

00:00:00.109 --> 00:00:02.036
- [Voiceover] So we have
two different logarithmic

00:00:02.036 --> 00:00:02.778
expressions here,

00:00:02.778 --> 00:00:05.297
one in yellow and one
in this pinkish color.

00:00:05.297 --> 00:00:06.528
And what I want you to do,

00:00:06.528 --> 00:00:09.209
like always, pause the
video and see if you can

00:00:09.209 --> 00:00:10.518
re-write each of these

00:00:10.518 --> 00:00:14.534
logarithmic expressions in a simpler way.

00:00:14.534 --> 00:00:17.460
And I'll give you a hint in
case you haven't started yet.

00:00:17.460 --> 00:00:19.758
The hint is that if
you think about how you

00:00:19.758 --> 00:00:23.143
might be able to change the
base of the logarithmic,

00:00:23.143 --> 00:00:26.022
or the logarithms or the
logarithmic expressions,

00:00:26.022 --> 00:00:28.611
you might be able to
simplify this a good bit.

00:00:28.611 --> 00:00:31.026
And I'll give you an even further hint.

00:00:31.026 --> 00:00:32.883
When I'm talking about change of base,

00:00:32.883 --> 00:00:37.422
I'm saying that if I have the log base,

00:00:37.422 --> 00:00:38.852
and I'll color code it,

00:00:38.852 --> 00:00:42.763
log base A of B,

00:00:42.763 --> 00:00:46.524
log base A of B,

00:00:46.524 --> 00:00:50.808
this is going to be equal to log of B,

00:00:50.808 --> 00:00:55.280
log of B over log of A,

00:00:55.280 --> 00:00:56.783
over

00:00:56.783 --> 00:00:59.604
log of A.

00:00:59.604 --> 00:01:01.067
Now you might be saying wait, wait, wait,

00:01:01.067 --> 00:01:02.425
we wrote a logarithm here but you didn't

00:01:02.425 --> 00:01:04.201
write what the base is.

00:01:04.201 --> 00:01:05.524
Well this is going to be true regardless

00:01:05.524 --> 00:01:07.335
of which base you choose
as long as you pick

00:01:07.335 --> 00:01:08.961
the same base.

00:01:08.961 --> 00:01:13.465
This could be base nine,
base nine in either case.

00:01:13.465 --> 00:01:16.948
Now typically, people choose base 10.

00:01:16.948 --> 00:01:19.478
So 10 is the most typical one to use

00:01:19.478 --> 00:01:21.359
and that's because most
peoples calculators

00:01:21.359 --> 00:01:24.656
or they might be logarithmic
tables for base 10.

00:01:24.656 --> 00:01:28.023
So here you're saying the
exponent that I have to raise

00:01:28.023 --> 00:01:31.378
A to to get to B is equal to the exponent

00:01:31.378 --> 00:01:33.200
I have to raise 10 to to get to B,

00:01:33.200 --> 00:01:35.244
divided by the exponent
I have to raise 10 to

00:01:35.244 --> 00:01:37.078
to get to A.

00:01:37.078 --> 00:01:40.212
This is a really really
useful thing to know

00:01:40.212 --> 00:01:42.174
if you are dealing with logarithms.

00:01:42.174 --> 00:01:45.355
And we prove it in another video.

00:01:45.355 --> 00:01:48.129
But now we'll see if we can apply it.

00:01:48.129 --> 00:01:50.788
So now going back to
this yellow expression,

00:01:50.788 --> 00:01:52.463
this once again, is the same thing

00:01:52.463 --> 00:01:56.422
as one divided by this right over here.

00:01:56.422 --> 00:01:57.826
So let me write it that way actually.

00:01:57.826 --> 00:02:00.601
This is one divided by

00:02:00.601 --> 00:02:04.815
log base B of four.

00:02:04.815 --> 00:02:07.021
Well let's use what we just said over here

00:02:07.021 --> 00:02:08.135
to re-write it.

00:02:08.135 --> 00:02:10.225
So this is going to be equal to,

00:02:10.225 --> 00:02:11.896
this is going to be equal to one,

00:02:11.896 --> 00:02:13.661
divided by,

00:02:13.661 --> 00:02:15.739
instead of writing it log base B of four,

00:02:15.739 --> 00:02:19.988
we could write it as log of four,

00:02:19.988 --> 00:02:22.124
and if I just, if I don't
write the base there

00:02:22.124 --> 00:02:24.353
we can assume that it's base 10,

00:02:24.353 --> 00:02:27.285
log of four over

00:02:27.285 --> 00:02:30.280
log of B.

00:02:30.280 --> 00:02:34.262
Now if I divide by some fraction,

00:02:34.262 --> 00:02:35.562
or some rational expression,

00:02:35.562 --> 00:02:36.851
it's the same thing as multiplying by

00:02:36.851 --> 00:02:38.209
the reciprocal.

00:02:38.209 --> 00:02:40.566
So this is going to be equal one times

00:02:40.566 --> 00:02:42.284
the reciprocal of this.

00:02:42.284 --> 00:02:45.767
Log of B over

00:02:45.767 --> 00:02:48.390
log of four,

00:02:48.390 --> 00:02:49.505
which of course is just going to be

00:02:49.505 --> 00:02:52.767
log of B over log of four,

00:02:52.767 --> 00:02:54.984
I just multiplied it by one,

00:02:54.984 --> 00:02:56.969
and so we can go in the
other direction now,

00:02:56.969 --> 00:02:59.326
using this little tool we established at

00:02:59.326 --> 00:03:00.406
the beginning of the video.

00:03:00.406 --> 00:03:02.022
This is the same thing as

00:03:02.022 --> 00:03:06.468
log base four of B,

00:03:06.468 --> 00:03:10.032
log base four of B.

00:03:10.032 --> 00:03:12.210
So we have a pretty neat result

00:03:12.210 --> 00:03:13.864
that actually came out here,

00:03:13.864 --> 00:03:15.217
we didn't prove it for any values,

00:03:15.217 --> 00:03:17.492
although we have a pretty general B here.

00:03:17.492 --> 00:03:19.350
If I take the,

00:03:19.350 --> 00:03:22.856
If I take the reciprocal of

00:03:22.856 --> 00:03:25.293
a logarithmic expression,

00:03:25.293 --> 00:03:28.428
I essentially have swapped the bases.

00:03:28.428 --> 00:03:30.518
This is log base B,

00:03:30.518 --> 00:03:31.969
what exponent do I have to raise B to

00:03:31.969 --> 00:03:33.698
to get to four?

00:03:33.698 --> 00:03:35.115
And then here I have what exponent

00:03:35.115 --> 00:03:37.378
do I have to raise four to to get to B?

00:03:37.378 --> 00:03:40.292
Now it might seem a
little bit magical until

00:03:40.292 --> 00:03:43.067
you actually put some
tangible numbers here.

00:03:43.067 --> 00:03:44.437
Then it starts to make sense,

00:03:44.437 --> 00:03:46.712
especially relative to
fractional exponents.

00:03:46.712 --> 00:03:49.289
For example, we know that
four to the third power

00:03:49.289 --> 00:03:51.576
is equal to 64.

00:03:51.576 --> 00:03:55.790
So if I had log base four of 64,

00:03:55.790 --> 00:03:57.880
that's going to be equal to three.

00:03:57.880 --> 00:04:00.140
And if I were to say

00:04:00.140 --> 00:04:04.342
log base 64 of four,

00:04:04.342 --> 00:04:05.851
well now I'm going to have to raise that

00:04:05.851 --> 00:04:07.871
to the one third power.

00:04:07.871 --> 00:04:10.542
So notice, they are the
reciprocal of each other.

00:04:10.542 --> 00:04:12.411
So actually not so magical after all,

00:04:12.411 --> 00:04:15.638
but it's nice to see how
everything fits together.

00:04:15.638 --> 00:04:16.658
Now let's try to,

00:04:16.658 --> 00:04:19.537
now let's try to tackle
this one over here.

00:04:19.537 --> 00:04:21.638
So I've log base C of 16,

00:04:21.638 --> 00:04:26.305
times log base two of C, alright.

00:04:26.305 --> 00:04:28.534
So this one, once again it might be nice

00:04:28.534 --> 00:04:31.053
to re-write these, each of these,

00:04:31.053 --> 00:04:34.873
as a rational expression
dealing with log base 10.

00:04:34.873 --> 00:04:37.368
So this first one,

00:04:37.368 --> 00:04:38.889
this first one I could write this as

00:04:38.889 --> 00:04:43.272
log base 10 of 16,

00:04:43.272 --> 00:04:44.747
remember if I don't write the base

00:04:44.747 --> 00:04:46.732
you can assume it's 10,

00:04:46.732 --> 00:04:49.435
over log

00:04:49.435 --> 00:04:53.475
over log base 10 of C,

00:04:53.475 --> 00:04:56.772
and we're going to be multiplying this by,

00:04:56.772 --> 00:04:58.618
now this is going to be,

00:04:58.618 --> 00:05:01.160
we can write this as,

00:05:01.160 --> 00:05:04.144
log base 10 of C,

00:05:04.144 --> 00:05:06.061
log base 10 of C

00:05:06.061 --> 00:05:09.864
over, over log base

00:05:09.864 --> 00:05:12.128
10 of two.

00:05:12.128 --> 00:05:15.065
Log base 10 of two.

00:05:15.065 --> 00:05:16.713
Once again I could have
these little 10's here

00:05:16.713 --> 00:05:19.233
if it makes you comfortable.

00:05:19.233 --> 00:05:21.473
I could do something like
that but I don't have to.

00:05:21.473 --> 00:05:23.342
And now this is interesting,
cuz if I'm multiplying

00:05:23.342 --> 00:05:26.175
by log of C, and dividing by log of C,

00:05:26.175 --> 00:05:27.835
both of them base 10,

00:05:27.835 --> 00:05:29.716
well those are going to cancel out

00:05:29.716 --> 00:05:31.341
and I'm going to be left with

00:05:31.341 --> 00:05:34.139
log base 16,

00:05:34.139 --> 00:05:38.167
sorry log base 10 of 16 over,

00:05:38.167 --> 00:05:42.683
over log base 10 of two.

00:05:42.683 --> 00:05:44.889
And we know how to go
the other direction here,

00:05:44.889 --> 00:05:46.804
this is going to be,

00:05:46.804 --> 00:05:49.567
this is gong to be the logarithm,

00:05:49.567 --> 00:05:53.224
log base two of 16.

00:05:53.224 --> 00:05:57.542
Log base two of 16,

00:05:57.542 --> 00:05:59.446
and we're not done yet because all this is

00:05:59.446 --> 00:06:01.687
is what power do I need to raise two to

00:06:01.687 --> 00:06:04.415
to get to 16?

00:06:04.415 --> 00:06:06.725
We'll have to raise two to the,

00:06:06.725 --> 00:06:09.349
I have to raise two to the fourth power.

00:06:09.349 --> 00:06:11.810
We did it in the blue color.

00:06:11.810 --> 00:06:13.609
To raise two to the fourth power

00:06:13.609 --> 00:06:15.130
to get to 16.

00:06:15.130 --> 00:06:16.813
So that's, this is kind of a cool thing,

00:06:16.813 --> 00:06:17.672
cuz in the beginning,

00:06:17.672 --> 00:06:19.309
I started with this variable C,

00:06:19.309 --> 00:06:20.528
it looked like we were
going to have deal with

00:06:20.528 --> 00:06:21.782
a pretty abstract thing,

00:06:21.782 --> 00:06:24.673
but you can actually
evaluate this kind of crazy

00:06:24.673 --> 00:06:26.402
looking expression right over here,

00:06:26.402 --> 00:06:28.747
evaluates to the number four.

00:06:28.747 --> 00:06:29.827
In fact if I had to run some type

00:06:29.827 --> 00:06:31.290
of a math scavenger hunt or something,

00:06:31.290 --> 00:06:33.043
this could be a pretty good clue for

00:06:33.043 --> 00:06:35.272
evaluating to four.

00:06:35.272 --> 00:06:38.685
You know walk this many
steps forward or something.

00:06:38.685 --> 00:06:41.419
It'd be pretty cool.


Linear vs. exponential growth: from data

WEBVTT
Kind: captions
Language: en

00:00:00.192 --> 00:00:02.711
- [Voiceover] The number
of branches of an oak tree

00:00:02.711 --> 00:00:06.087
and a birch tree since
1950 are represented

00:00:06.087 --> 00:00:08.351
by the following tables.

00:00:08.351 --> 00:00:11.077
So for the oak tree we
see when time equals zero

00:00:11.077 --> 00:00:12.443
has 34 branches.

00:00:12.443 --> 00:00:15.131
After three years it has 46 branches,

00:00:15.131 --> 00:00:16.605
so on and so forth.

00:00:16.605 --> 00:00:18.602
And in the birch tree
they give us similar data.

00:00:18.602 --> 00:00:21.609
At the beginning has eight branches.

00:00:21.609 --> 00:00:25.521
In 10 years has 33 branches
and they give us all of that.

00:00:25.521 --> 00:00:26.751
What I want to think about in this video

00:00:26.751 --> 00:00:28.690
is how should we model these?

00:00:28.690 --> 00:00:30.295
If we want to model these with functions

00:00:30.295 --> 00:00:31.929
and the choices we'll give ourselves,

00:00:31.929 --> 00:00:33.241
there are other options,

00:00:33.241 --> 00:00:35.211
but the choices we'll give
ourselves in this video

00:00:35.211 --> 00:00:38.722
are linear, and linear

00:00:38.722 --> 00:00:41.123
versus exponential functions.

00:00:41.123 --> 00:00:43.669
Which of these are going to be better

00:00:45.359 --> 00:00:48.990
for modeling this data?

00:00:48.990 --> 00:00:50.988
So let's first look at the oak tree.

00:00:50.988 --> 00:00:53.287
A key to realization is,
whenever I have a fixed

00:00:53.287 --> 00:00:54.947
increase in time, so each of these steps,

00:00:54.947 --> 00:00:57.466
this is plus three years.

00:00:57.466 --> 00:00:59.265
So it's a fixed increase in time.

00:00:59.265 --> 00:01:01.135
What happens to my number of branches?

00:01:01.135 --> 00:01:03.455
Is it going to be a fixed change,

00:01:03.455 --> 00:01:05.597
or roughly a fixed change in which case

00:01:05.597 --> 00:01:07.250
a linear model might be good,

00:01:07.250 --> 00:01:09.409
or is it going to be a
change that's dependent

00:01:09.409 --> 00:01:10.988
on where we were?

00:01:10.988 --> 00:01:12.219
So what am I talking about?

00:01:12.219 --> 00:01:16.491
So 34 to 46, that is plus 12.

00:01:16.491 --> 00:01:20.997
46 to 59 is plus 13.

00:01:20.997 --> 00:01:25.290
59 to 70 is plus 11.

00:01:25.290 --> 00:01:29.353
70 to 82 is plus 12.

00:01:29.353 --> 00:01:31.689
So at first you mention,
well this isn't exactly

00:01:31.689 --> 00:01:32.569
a fixed change.

00:01:32.569 --> 00:01:37.066
These numbers they seem to
average right around 12,

00:01:37.066 --> 00:01:38.687
but when you're looking at real world data

00:01:38.687 --> 00:01:40.974
you're never gonna get
something that is exact.

00:01:40.974 --> 00:01:43.900
The models are just going
to give us a good fit.

00:01:43.900 --> 00:01:45.839
Are going to give us a good approximation

00:01:45.839 --> 00:01:49.376
of the behavior of the
number of branches over time.

00:01:49.376 --> 00:01:53.515
For me, this is pretty close
to a constant 12 branches

00:01:53.515 --> 00:01:54.591
a year.

00:01:54.591 --> 00:01:56.889
So I would construct a linear model here.

00:01:56.889 --> 00:01:59.871
I would say here branches

00:01:59.871 --> 00:02:01.310
as a function of time.

00:02:01.310 --> 00:02:04.389
Let me be clear, this
isn't 12 branches per year,

00:02:04.389 --> 00:02:07.361
this is 12 branches every three years.

00:02:07.361 --> 00:02:09.437
This is 13 branches over three years.

00:02:09.437 --> 00:02:11.099
This was 11 branches over three years.

00:02:11.099 --> 00:02:14.173
We're going to average 12
branches over three years.

00:02:14.173 --> 00:02:16.240
So the number of branches we have,

00:02:16.240 --> 00:02:20.036
we're gonna start at 34 branches

00:02:20.036 --> 00:02:21.754
and then minus.

00:02:21.754 --> 00:02:23.544
If we have 12 branches every three years,

00:02:23.544 --> 00:02:26.351
that's four branches every,

00:02:26.351 --> 00:02:27.500
or I should say plus.

00:02:27.500 --> 00:02:30.495
Four branches every year.

00:02:30.495 --> 00:02:31.633
And you could test this out.

00:02:31.633 --> 00:02:34.454
B of zero is gonna give us 34 branches.

00:02:34.454 --> 00:02:35.735
B of 12.

00:02:35.735 --> 00:02:38.935
Let's just really test out
the extreme part of the model.

00:02:38.935 --> 00:02:43.888
B of 12 is going to be 34 plus 48,

00:02:45.768 --> 00:02:47.872
which is equal to 82.

00:02:47.872 --> 00:02:50.217
So this model works quite well.

00:02:50.217 --> 00:02:51.076
It's gonna have a couple of places

00:02:51.076 --> 00:02:53.258
where it's not exactly fitting the data,

00:02:53.258 --> 00:02:55.313
but it fits it quite, quite well.

00:02:55.313 --> 00:02:57.019
This is the linear model.

00:02:57.019 --> 00:02:58.633
So this one is linear.

00:02:58.633 --> 00:03:00.572
We also got the birch tree.

00:03:00.572 --> 00:03:04.986
So time equals zero, so
fixed change in time.

00:03:06.556 --> 00:03:09.167
Alright, so we have a fixed change in time

00:03:09.167 --> 00:03:13.527
every time we are moving
into the future, a decade.

00:03:13.527 --> 00:03:15.501
Let's see, our change in branches.

00:03:15.501 --> 00:03:17.602
We go from eight to 33.

00:03:17.602 --> 00:03:18.577
So what is that?

00:03:18.577 --> 00:03:21.758
That is plus 25 branches.

00:03:21.758 --> 00:03:24.939
Then we go from 33 to 128.

00:03:24.939 --> 00:03:27.170
Well that's way more than 25 branches.

00:03:27.170 --> 00:03:29.445
That's going to be what?

00:03:30.965 --> 00:03:32.108
Five less than 100.

00:03:32.108 --> 00:03:35.858
So that's gonna be plus 95 branches.

00:03:35.858 --> 00:03:39.404
So this clearly is not a linear model.

00:03:39.404 --> 00:03:41.644
And so let's think in terms
of an exponential model.

00:03:41.644 --> 00:03:43.897
How much do we have to
multiply to go from...

00:03:43.897 --> 00:03:44.581
Did I do that right?

00:03:44.581 --> 00:03:47.101
128 minus, yeah, if it was 133

00:03:47.101 --> 00:03:48.123
then it would be 100

00:03:48.123 --> 00:03:49.794
and it's five less than that, okay.

00:03:49.794 --> 00:03:50.781
So now let's think about it in terms

00:03:50.781 --> 00:03:51.767
of an exponential model.

00:03:51.767 --> 00:03:53.996
In terms of an exponential model

00:03:53.996 --> 00:03:56.055
what do we have to multiply

00:03:57.285 --> 00:03:58.559
for each step?

00:03:58.559 --> 00:04:00.021
So if we have a constant step in time,

00:04:00.021 --> 00:04:02.622
what do we multiply for how much

00:04:02.622 --> 00:04:04.282
we increase our branches?

00:04:04.282 --> 00:04:06.906
So if we go to eight to 33,

00:04:08.576 --> 00:04:11.015
it's gonna be approximately four.

00:04:11.015 --> 00:04:13.058
It's gonna be a little bit more than four.

00:04:13.058 --> 00:04:16.193
33 to 128,

00:04:16.193 --> 00:04:18.921
well that's gonna be a
little bit less than four,

00:04:18.921 --> 00:04:20.999
but it's approximately four.

00:04:20.999 --> 00:04:24.876
33 times four would be
132, so we're close.

00:04:24.876 --> 00:04:29.629
128 to 512, that's exactly four, right?

00:04:29.629 --> 00:04:34.187
That's exactly 120 times four is 480

00:04:34.187 --> 00:04:37.642
plus 32, yup, that is exactly four.

00:04:37.642 --> 00:04:39.051
So times four.

00:04:39.051 --> 00:04:41.814
And so it looks like we
keep multiplying by four

00:04:41.814 --> 00:04:44.310
every 10 years that go by.

00:04:44.310 --> 00:04:47.897
So one way to think about it is

00:04:47.897 --> 00:04:51.772
we could say here, BFT, the branches of T,

00:04:51.772 --> 00:04:54.950
our initial condition,
our initial state is eight

00:04:54.950 --> 00:04:59.069
and now we could say our
common factor is four.

00:05:01.139 --> 00:05:04.098
But if we want T to be in years,

00:05:04.098 --> 00:05:08.219
well every 10 years we
multiply by a factor of four.

00:05:08.219 --> 00:05:11.122
So T has to go to 10 before we increase

00:05:11.122 --> 00:05:13.285
the exponent to one,

00:05:13.285 --> 00:05:16.220
or has to go to 20 until
this exponent becomes two.

00:05:16.220 --> 00:05:19.422
So eight times four to the T over 10 power

00:05:19.422 --> 00:05:23.149
seems like a pretty good model.

00:05:23.149 --> 00:05:25.703
And you could even verify
this for yourself if you like.

00:05:25.703 --> 00:05:28.872
Try out what B of 30 is going to be.

00:05:28.872 --> 00:05:33.409
B of 30 would be eight times four.

00:05:33.409 --> 00:05:36.571
30 divided by 10 to the third power

00:05:36.571 --> 00:05:38.333
and what is that going to be?

00:05:38.333 --> 00:05:42.396
That's going to be four
to the third is 64.

00:05:42.396 --> 00:05:43.815
64.

00:05:43.815 --> 00:05:46.623
Eight times 64 is,

00:05:46.623 --> 00:05:48.375
it's 480 plus 32.

00:05:48.375 --> 00:05:49.839
It is 512.

00:05:49.839 --> 00:05:53.189
So once again, this exponential model

00:05:53.189 --> 00:05:55.892
for this data does a pretty good job.


Linear vs. exponential growth: from data (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.218
- [Voiceover] The temperature
of a glass of warm water

00:00:02.218 --> 00:00:03.625
after it's put in a freezer

00:00:03.625 --> 00:00:06.250
is represented by the following table.

00:00:06.250 --> 00:00:08.139
So we have time in minutes

00:00:08.139 --> 00:00:09.731
and then we have the
corresponding temperature

00:00:09.731 --> 00:00:11.864
at different times in minutes.

00:00:11.864 --> 00:00:13.826
Which model for C of t,

00:00:13.826 --> 00:00:15.785
the temperature of the glass of water

00:00:15.785 --> 00:00:17.478
t minutes after it's served,

00:00:17.478 --> 00:00:19.405
best fits the data?

00:00:19.405 --> 00:00:20.754
So pause the video

00:00:20.754 --> 00:00:25.700
and see which of these
models best fit the data.

00:00:25.700 --> 00:00:29.908
Alright, now let's work
through this together.

00:00:29.908 --> 00:00:31.284
So in order for it,

00:00:31.284 --> 00:00:32.322
we see these choices.

00:00:32.322 --> 00:00:33.844
Some of these choices
are exponential models.

00:00:33.844 --> 00:00:35.636
Some of these are linear models.

00:00:35.636 --> 00:00:39.679
In order for a linear model
to be a good description,

00:00:39.679 --> 00:00:41.527
when you have a fixed change in time,

00:00:41.527 --> 00:00:44.476
you should have a fixed
change in temperature.

00:00:44.476 --> 00:00:46.769
If you're dealing with
an exponential model,

00:00:46.769 --> 00:00:48.368
then as you have a fixed change in time,

00:00:48.368 --> 00:00:51.715
you should be changing by the same factor

00:00:51.715 --> 00:00:53.395
so the amount you change

00:00:53.395 --> 00:00:55.480
from say minute one to minute two

00:00:55.480 --> 00:00:58.024
or from minute two to minute three,

00:00:58.024 --> 00:00:59.694
it's not going to be the exact same amount

00:00:59.694 --> 00:01:02.199
but it should be the same
factor of where you started.

00:01:02.199 --> 00:01:03.507
So let's think about this.

00:01:03.507 --> 00:01:05.063
So here,

00:01:05.063 --> 00:01:09.821
our change in time is two minutes.

00:01:09.821 --> 00:01:13.004
What is the absolute
change in temperature?

00:01:13.004 --> 00:01:14.795
So our absolute change in temperature

00:01:14.795 --> 00:01:18.733
is negative what, 15.7.

00:01:18.733 --> 00:01:22.661
Negative 15.7

00:01:22.661 --> 00:01:25.889
and what if we viewed
it as a multiplication?

00:01:25.889 --> 00:01:27.853
So what do multiply 80 by

00:01:27.853 --> 00:01:30.430
to get 64.3?

00:01:30.430 --> 00:01:33.675
Well, I can get a calculator out for that.

00:01:33.675 --> 00:01:38.675
So 64.3 divided by 80

00:01:39.111 --> 00:01:41.739
is equal to 0.8

00:01:41.739 --> 00:01:44.642
I'll just say approximately .8

00:01:44.642 --> 00:01:47.246
So we could multiply by

00:01:47.246 --> 00:01:49.728
.0.8

00:01:49.728 --> 00:01:51.443
This is going to be approximate.

00:01:51.443 --> 00:01:53.110
So to get from 80 to 64.3

00:01:53.110 --> 00:01:55.087
I could either subtract by 15.7

00:01:55.087 --> 00:01:57.093
if I'm dealing with a linear model

00:01:57.093 --> 00:02:00.949
or I can multiply by 0.8

00:02:00.949 --> 00:02:04.302
Now if I increase my time again by two,

00:02:04.302 --> 00:02:06.391
I'm going from minute two to minute four,

00:02:06.391 --> 00:02:08.974
so Delta t is equal to two,

00:02:08.974 --> 00:02:12.248
the absolute change here is what?

00:02:12.248 --> 00:02:15.851
This is going to be not 12.

00:02:15.851 --> 00:02:17.568
This is going to be,

00:02:17.568 --> 00:02:20.143
my brain isn't functioning optimally.

00:02:20.143 --> 00:02:23.908
This was 64.7 then this would be 12

00:02:23.908 --> 00:02:25.124
but it's four less than that

00:02:25.124 --> 00:02:26.563
so it's 11.6.

00:02:26.563 --> 00:02:29.018
Negative 11.6

00:02:29.018 --> 00:02:31.303
but if you look at it as
multiplying it by a factor,

00:02:31.303 --> 00:02:34.402
what would you have to
multiply by approximately?

00:02:34.402 --> 00:02:36.338
Let's get the calculator back out.

00:02:36.338 --> 00:02:40.761
So if I said 52.7

00:02:40.873 --> 00:02:42.884
divided by 64.3.

00:02:42.884 --> 00:02:45.627
Divided by 64.3

00:02:45.627 --> 00:02:46.560
is equal to,

00:02:46.560 --> 00:02:48.899
that's about .82.

00:02:48.899 --> 00:02:52.377
So times 0.82.

00:02:52.377 --> 00:02:53.579
So just by looking at this,

00:02:53.579 --> 00:02:54.658
I can keep going

00:02:54.658 --> 00:02:57.549
but it looks like for our
give and change in time,

00:02:57.549 --> 00:03:00.174
my absolute change in the number

00:03:00.174 --> 00:03:03.167
is not even close to being the same.

00:03:03.167 --> 00:03:04.690
If this was 15.6,

00:03:04.690 --> 00:03:06.817
I'd be like okay maybe there's
a little bit of error here.

00:03:06.817 --> 00:03:08.248
Data that you collect in the real world

00:03:08.248 --> 00:03:09.391
is never going to be perfect.

00:03:09.391 --> 00:03:11.397
These are models that try to get us close

00:03:11.397 --> 00:03:12.986
to describing the data.

00:03:12.986 --> 00:03:15.159
But over here we keep multiplying it

00:03:15.159 --> 00:03:18.271
by a factor of roughly .8,

00:03:18.271 --> 00:03:19.543
roughly .8.

00:03:19.543 --> 00:03:21.063
Now you might be tempted
to immediately say,

00:03:21.063 --> 00:03:23.336
okay, well that means that C of t

00:03:23.336 --> 00:03:26.011
is going to be equal to
our initial temperature,

00:03:26.011 --> 00:03:30.636
80 times a common ratio of 0.8

00:03:30.636 --> 00:03:33.655
to the number of minutes that passed by.

00:03:33.655 --> 00:03:34.646
Now this is very tempting

00:03:34.646 --> 00:03:37.457
and it would be the case
if this was one minute

00:03:37.457 --> 00:03:39.226
and if this was two minutes

00:03:39.226 --> 00:03:42.087
but our change in temperature each time

00:03:42.087 --> 00:03:45.698
is our change in temperature
each time is two minutes

00:03:45.698 --> 00:03:48.643
so what we really should say is this is

00:03:48.643 --> 00:03:50.987
one way to think about it is that it takes

00:03:50.987 --> 00:03:55.068
two minutes to have a 0.8 change

00:03:55.068 --> 00:03:57.077
or to be multiplied by 0.8.

00:03:57.077 --> 00:03:58.345
So the real way to describe this

00:03:58.345 --> 00:04:00.835
would be t over two.

00:04:00.835 --> 00:04:03.670
Every two minutes when t is zero,

00:04:03.670 --> 00:04:04.733
we'd be at 80.

00:04:04.733 --> 00:04:06.033
After two minutes,

00:04:06.033 --> 00:04:08.375
we would take 80 times 0.8,

00:04:08.375 --> 00:04:09.882
which is what we got over here.

00:04:09.882 --> 00:04:11.196
After four minutes,

00:04:11.196 --> 00:04:13.244
it would be 80 times 0.8 squared.

00:04:13.244 --> 00:04:14.830
In fact, let's just verify

00:04:14.830 --> 00:04:17.544
that we feel pretty good about this.

00:04:17.544 --> 00:04:21.509
So if we had something like this.

00:04:21.859 --> 00:04:24.047
So t

00:04:24.047 --> 00:04:27.235
and C of t.

00:04:27.235 --> 00:04:29.796
So when t is zero, C of t is 80.

00:04:29.796 --> 00:04:31.574
When t is,

00:04:31.574 --> 00:04:33.747
well let me just do the
same data that we have here.

00:04:33.747 --> 00:04:35.454
When t is two,

00:04:35.454 --> 00:04:38.133
we have 80 times,

00:04:38.133 --> 00:04:39.016
two over two is one

00:04:39.016 --> 00:04:40.910
so it's 80 times 0.8,

00:04:40.910 --> 00:04:43.380
which is pretty close to
what we have over here.

00:04:43.380 --> 00:04:44.729
When t is four,

00:04:44.729 --> 00:04:49.186
it would be 80 .0.8 squared

00:04:49.186 --> 00:04:52.506
which is pretty close to
what we have right over here.

00:04:52.506 --> 00:04:54.376
I can just calculate it for you.

00:04:54.376 --> 00:04:58.960
If I have .8 squared

00:04:58.960 --> 00:05:01.655
times 80,

00:05:01.965 --> 00:05:03.212
51.2

00:05:03.212 --> 00:05:04.156
Getting pretty close.

00:05:04.156 --> 00:05:06.037
This is a pretty good approximation.

00:05:06.037 --> 00:05:07.763
Pretty good model.

00:05:07.763 --> 00:05:10.872
So I'm liking this model.

00:05:10.872 --> 00:05:13.574
This isn't exactly one of the choices

00:05:13.574 --> 00:05:16.110
so how do we manipulate this a little bit?

00:05:16.110 --> 00:05:17.591
Well, we can remind ourselves

00:05:17.591 --> 00:05:19.792
that this is the same thing as 80

00:05:19.792 --> 00:05:23.889
times 0.8 to the one half

00:05:23.889 --> 00:05:26.933
and then that to the t power.

00:05:26.933 --> 00:05:29.924
And what's .8 to the one half?

00:05:29.924 --> 00:05:33.341
So .8.

00:05:33.341 --> 00:05:35.262
That's the same thing as
taking the square root of .8

00:05:35.262 --> 00:05:37.973
It's roughly .89

00:05:37.973 --> 00:05:41.379
So this is approximately 80

00:05:41.379 --> 00:05:44.545
times 0.89

00:05:44.545 --> 00:05:46.075
to the t power.

00:05:46.075 --> 00:05:47.548
And if you look at all of these choices,

00:05:47.548 --> 00:05:49.837
this one is pretty close to this.

00:05:49.837 --> 00:05:52.542
This model best fits the data,

00:05:52.542 --> 00:05:53.648
especially of the choices going,

00:05:53.648 --> 00:05:55.548
this is pretty close to the model

00:05:55.548 --> 00:05:57.289
that I just thought about.

00:05:57.289 --> 00:05:58.312
Now another way of doing it

00:05:58.312 --> 00:05:59.412
that might have been a little bit simpler.

00:05:59.412 --> 00:06:00.075
I like to do it this way

00:06:00.075 --> 00:06:01.381
because even if I didn't have choices,

00:06:01.381 --> 00:06:03.381
we would have gotten to
something reasonable.

00:06:03.381 --> 00:06:05.093
Another way to do it is to say okay,

00:06:05.093 --> 00:06:06.288
80 is our initial state.

00:06:06.288 --> 00:06:07.435
All of these whether you're talking about

00:06:07.435 --> 00:06:08.776
exponential or linear models,

00:06:08.776 --> 00:06:11.338
start with 80 when t is equal to zero

00:06:11.338 --> 00:06:12.654
but it's clearly not a linear model

00:06:12.654 --> 00:06:13.672
because we're not changing by

00:06:13.672 --> 00:06:16.412
even roughly the same amount every time

00:06:16.412 --> 00:06:17.524
but it looks like every two minutes

00:06:17.524 --> 00:06:19.107
we're changing by a factor of .8

00:06:19.107 --> 00:06:20.540
so we're going to have
an exponential model

00:06:20.540 --> 00:06:23.869
so you say okay, it will be
one of these two choices.

00:06:23.869 --> 00:06:26.464
Now this one down here you could rule out

00:06:26.464 --> 00:06:29.152
because we're not changing
by a factor of 0.8

00:06:29.152 --> 00:06:31.008
or 0.81 every minute.

00:06:31.008 --> 00:06:32.851
We're changing by a factor of 0.81

00:06:32.851 --> 00:06:34.779
every two minutes

00:06:34.779 --> 00:06:36.130
so you could have ruled that one out

00:06:36.130 --> 00:06:37.077
and then you could have deduced

00:06:37.077 --> 00:06:38.217
to this right over here

00:06:38.217 --> 00:06:38.672
and you could have said,

00:06:38.672 --> 00:06:41.575
look, if I'm changing by a
factor of .9 every minute

00:06:41.575 --> 00:06:44.892
then that would have been
.81 every two minutes

00:06:44.892 --> 00:06:46.695
which is pretty close to
what we're seeing here

00:06:46.695 --> 00:06:49.437
changing by a factor of about .8 or .81

00:06:49.437 --> 00:06:50.871
every two minutes.

00:06:50.871 --> 00:06:54.424
So once again that's why
we like that first choice.


Interpreting change in exponential models: changing units

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.523
- [Voiceover] The amount
of carbon dioxide, CO2,

00:00:02.523 --> 00:00:04.345
in the atmosphere increases rapidly

00:00:04.345 --> 00:00:06.980
as we continue to rely on fossil fuels.

00:00:06.980 --> 00:00:09.639
The relationship between
the elapsed time, t,

00:00:09.639 --> 00:00:11.624
in decades, let me highlight that,

00:00:11.624 --> 00:00:13.667
because that's not a typical unit,

00:00:13.667 --> 00:00:17.413
but, in decades, since CO2
levels were first measured,

00:00:17.413 --> 00:00:20.297
and the total amount of
CO2 in the atmosphere,

00:00:20.425 --> 00:00:24.999
so the amount of CO2,
A, of sub decade of (t),

00:00:24.999 --> 00:00:26.798
in parts per million, is modeled

00:00:26.798 --> 00:00:28.342
by the following function.

00:00:28.342 --> 00:00:30.524
So, the amount of CO2 as a function

00:00:30.524 --> 00:00:32.277
of how many decades have passed

00:00:32.277 --> 00:00:34.564
is going to be this, so t is in decades

00:00:34.564 --> 00:00:37.258
in this model right over here.

00:00:37.258 --> 00:00:38.627
Complete the following sentence

00:00:38.627 --> 00:00:41.065
about the yearly rate of change,

00:00:41.495 --> 00:00:44.989
the yearly rate of change in the amount

00:00:44.989 --> 00:00:46.461
of CO2 in the atmosphere.

00:00:46.461 --> 00:00:48.667
Round your answer to two decimal places.

00:00:48.667 --> 00:00:50.108
Every year, the amount of CO2

00:00:50.108 --> 00:00:52.874
in the atmosphere
increases by a factor of?

00:00:52.874 --> 00:00:55.043
If they said every decade, well,

00:00:55.043 --> 00:00:56.157
this would be pretty straightforward.

00:00:56.157 --> 00:00:59.503
Every decade, you increase t
by one, and so you're going

00:00:59.503 --> 00:01:01.988
to multiply by 1.06 again.

00:01:01.988 --> 00:01:05.865
So every decade, you
increase by a factor of 1.06.

00:01:05.865 --> 00:01:07.885
But what about every year?

00:01:07.885 --> 00:01:09.824
Now, I always find it
helpful to make a bit

00:01:09.824 --> 00:01:13.651
of a table, just so we can
really digest things properly.

00:01:14.061 --> 00:01:17.265
So I'll say t, and I'll say A of t.

00:01:17.725 --> 00:01:19.668
So, when t is equal to zero,

00:01:19.668 --> 00:01:21.503
so at the beginning of our study,

00:01:21.503 --> 00:01:24.521
well, 1.06 to the zeroth
power is just going to be one,

00:01:24.521 --> 00:01:27.922
you have 315 parts per million.

00:01:28.742 --> 00:01:30.767
So, what's a year later?

00:01:30.767 --> 00:01:33.181
So, a year later is going
to be a tenth of a decade.

00:01:33.181 --> 00:01:35.039
Remember, t is in decades.

00:01:35.039 --> 00:01:37.397
So a year later is

00:01:37.397 --> 00:01:40.577
0.1 of a decade.

00:01:40.577 --> 00:01:42.817
So, 0.1 of a decade later,

00:01:42.817 --> 00:01:46.845
what is going to be the
amount of carbon we have?

00:01:46.845 --> 00:01:49.102
Well, it's going to be 315

00:01:49.802 --> 00:01:53.962
times 1.06

00:01:53.962 --> 00:01:58.946
to the 0.1 power.

00:01:59.556 --> 00:02:01.600
And what is that going to be?

00:02:01.600 --> 00:02:03.666
Let's see, if we...

00:02:03.666 --> 00:02:08.034
so 1.06 to the,

00:02:08.774 --> 00:02:10.494
so to the

00:02:11.824 --> 00:02:15.634
0.1 power,

00:02:15.634 --> 00:02:17.737
I didn't have to actually
use the parentheses there,

00:02:17.737 --> 00:02:22.191
is equal to 1.0058,

00:02:22.191 --> 00:02:23.681
I'll just take it with that.

00:02:23.681 --> 00:02:26.525
1.0058.

00:02:27.485 --> 00:02:30.809
So this is the same thing as 3.5

00:02:30.809 --> 00:02:34.839
times 1.0058.

00:02:34.839 --> 00:02:36.713
And I should say approximately equal to,

00:02:36.713 --> 00:02:38.343
I did a little bit of rounding there.

00:02:38.343 --> 00:02:40.455
So, after another year,
so now now we're at

00:02:40.455 --> 00:02:43.903
t equals 0.2, we're at
two-tenths of a decade.

00:02:43.903 --> 00:02:44.983
Where're we going to be?

00:02:44.983 --> 00:02:49.696
We're going to be at 3.5 times 1.06

00:02:50.286 --> 00:02:53.562
to the 0.2, which is the same thing

00:02:53.562 --> 00:02:56.569
as 3.5

00:02:56.569 --> 00:02:59.649
times 1.06

00:02:59.649 --> 00:03:01.969
to the 0.1,

00:03:02.619 --> 00:03:05.275
and then that raised to the second power.

00:03:05.275 --> 00:03:07.946
So we're going to multiply by this 1.06

00:03:07.946 --> 00:03:10.720
to the one-tenth power again,

00:03:10.720 --> 00:03:15.062
or we're going to multiply
by 1.0058 a second time.

00:03:15.062 --> 00:03:16.141
Another way to think about it,

00:03:16.141 --> 00:03:18.672
if we wanted to reformulate this model

00:03:18.672 --> 00:03:23.281
in terms of years, so per year of t,

00:03:23.281 --> 00:03:26.938
it's going to be 315
and now our common ratio

00:03:26.938 --> 00:03:31.175
wouldn't be 1.06, it would
be 1.06 to the .1 power,

00:03:31.175 --> 00:03:35.841
or 1.0058,

00:03:36.271 --> 00:03:37.601
and then we would raise that.

00:03:37.601 --> 00:03:39.580
Now t would be in years now.

00:03:39.580 --> 00:03:41.379
Here, it is in decades.

00:03:41.379 --> 00:03:42.687
And I can say approximately,

00:03:42.687 --> 00:03:44.978
since this is rounded a little bit.

00:03:44.978 --> 00:03:46.696
And so, every year, the amount of CO2

00:03:46.696 --> 00:03:48.914
in the atmosphere
increases by a factor of,

00:03:48.914 --> 00:03:53.418
I could say 1.06 to the 0.1
power, but if I'm rounding

00:03:53.418 --> 00:03:56.111
my answer to two decimal places,

00:03:56.111 --> 00:03:58.713
the, well, we're going to increase

00:03:58.713 --> 00:04:03.693
by 1.0058,

00:04:03.693 --> 00:04:07.093
in fact, they should, the
increase is by a factor of,

00:04:07.093 --> 00:04:09.496
they should, I'm guessing they want more

00:04:09.496 --> 00:04:11.830
than two decimal places.

00:04:11.830 --> 00:04:15.452
Well, anyway, this is arguably,
this right over here is

00:04:15.452 --> 00:04:17.170
five significant digits.

00:04:17.170 --> 00:04:19.075
But anyway, I'll leave it there.


Interpreting change in exponential models: with manipulation

WEBVTT
Kind: captions
Language: en

00:00:00.654 --> 00:00:02.465
- [Voiceover] Ocean
sunfishes are well-known

00:00:02.465 --> 00:00:04.485
for rapidly gaining a lot of weight

00:00:04.485 --> 00:00:06.772
on a diet based on jellyfish.

00:00:06.772 --> 00:00:09.826
The relationship between the
elapsed time, t, in days,

00:00:09.826 --> 00:00:14.678
since an ocean sunfish
is born, and its mass,

00:00:14.678 --> 00:00:18.858
M of t, in milligrams, is modeled
by the following function.

00:00:18.858 --> 00:00:19.903
All right.

00:00:19.903 --> 00:00:21.133
Complete the following sentence

00:00:21.133 --> 00:00:23.223
about the daily percent change

00:00:23.223 --> 00:00:25.150
in the mass of the sunfish.

00:00:25.150 --> 00:00:27.658
Every day, there is a blank percent

00:00:27.658 --> 00:00:31.419
addition or removal from
the mass of the sunfish.

00:00:31.419 --> 00:00:34.809
So one thing that we know
almost from the get-go,

00:00:34.809 --> 00:00:38.594
we know that the sunfish gains weight,

00:00:38.594 --> 00:00:42.541
and we also see that
as t grows, as t grows,

00:00:42.541 --> 00:00:44.654
the exponent here is going to grow.

00:00:44.654 --> 00:00:46.136
And if you grow an exponent on something

00:00:46.136 --> 00:00:49.317
that is larger than one,
M of t is going to grow.

00:00:49.317 --> 00:00:52.010
So I already know it's going
to be about addition to

00:00:52.010 --> 00:00:53.705
the mass of the sunfish.

00:00:53.705 --> 00:00:56.654
But let's think about how
much is added every day.

00:00:56.654 --> 00:00:58.233
Well, let's think about it.

00:00:58.233 --> 00:00:59.928
Well let's see if we can rewrite this.

00:00:59.928 --> 00:01:02.161
I'm going to just focus
on the right-hand side

00:01:02.161 --> 00:01:03.531
of this expression.

00:01:03.531 --> 00:01:06.758
So 1.35

00:01:06.758 --> 00:01:09.963
to the t/6 plus five.

00:01:09.963 --> 00:01:11.425
That's the same thing as

00:01:11.425 --> 00:01:15.972
1.35 to the fifth power,

00:01:15.972 --> 00:01:20.082
times 1.35

00:01:20.082 --> 00:01:23.077
to the t/6 power.

00:01:23.077 --> 00:01:25.492
And that's going to be equal to

00:01:25.492 --> 00:01:29.722
1.35 to the fifth power,

00:01:29.722 --> 00:01:34.683
times 1.35,

00:01:34.683 --> 00:01:37.961
and I can separate this
t/6 as 1/6 times t.

00:01:37.961 --> 00:01:40.817
So 1.35 to the 1/6 power,

00:01:40.817 --> 00:01:45.530
and then that being
raised to the t-th power.

00:01:45.530 --> 00:01:46.343
So let's think about it.

00:01:46.343 --> 00:01:49.681
Every day, as t increases by one,

00:01:49.681 --> 00:01:52.050
now we can say that we're gonna take

00:01:52.050 --> 00:01:55.579
the previous day's mass, and multiply it

00:01:55.579 --> 00:01:57.088
by this common ratio.

00:01:57.088 --> 00:01:59.201
The common ratio here isn't
the way I've written it.

00:01:59.201 --> 00:02:00.594
Isn't 1.35.

00:02:00.594 --> 00:02:04.263
It's 1.35 to the 1/6 power.

00:02:04.263 --> 00:02:05.749
Let me draw a little table here

00:02:05.749 --> 00:02:08.339
to make that really, really clear.

00:02:08.339 --> 00:02:10.823
And all of that algebraic
manipulation I just did

00:02:10.823 --> 00:02:12.286
is just so I could simplify this,

00:02:12.286 --> 00:02:15.954
so I have some common
ratio to the t-th power.

00:02:15.954 --> 00:02:18.712
So t and M of t.

00:02:19.730 --> 00:02:21.183
So based on how I've just written it,

00:02:21.183 --> 00:02:23.017
when t is zero,

00:02:23.017 --> 00:02:24.759
well if t is zero, this is one,

00:02:24.759 --> 00:02:29.008
so then we just have our initial mass,

00:02:29.008 --> 00:02:31.125
it's going to be 1.35

00:02:31.125 --> 00:02:33.540
to the fifth power.

00:02:33.540 --> 00:02:36.466
And then when t is equal to one,

00:02:36.466 --> 00:02:38.904
it's going to be our initial mass,

00:02:38.904 --> 00:02:43.037
1.35 to the fifth power

00:02:43.037 --> 00:02:44.662
times our common ratio,

00:02:44.662 --> 00:02:48.103
times 1.35

00:02:48.103 --> 00:02:50.147
to the 1/6 power.

00:02:50.147 --> 00:02:52.494
When t equals two, we're
just gonna multiply

00:02:52.494 --> 00:02:53.956
what we had at t equals one,

00:02:53.956 --> 00:02:55.252
and we're just gonna multiply that

00:02:55.252 --> 00:02:58.211
times 1.35 to the 1/6 again.

00:02:58.211 --> 00:03:02.549
And so, every day, we are growing

00:03:04.054 --> 00:03:08.466
by our common ratio,
1.35 to the 1/6 power.

00:03:08.466 --> 00:03:09.975
Actually, let me get a calculator out.

00:03:09.975 --> 00:03:12.808
We're allowed to use
calculators in this exercise.

00:03:12.808 --> 00:03:15.849
So 1.35

00:03:15.849 --> 00:03:19.866
to the, open parentheses,

00:03:19.866 --> 00:03:23.837
one divided by six,

00:03:23.837 --> 00:03:25.578
close parentheses, power,

00:03:25.578 --> 00:03:27.224
is equal to,

00:03:32.068 --> 00:03:35.553
I'll say 1.051, approximately.

00:03:35.553 --> 00:03:38.339
So this

00:03:38.339 --> 00:03:41.938
is approximately 1.051.

00:03:41.938 --> 00:03:43.935
So we can say this is approximately

00:03:43.935 --> 00:03:47.186
1.35 times

00:03:47.186 --> 00:03:52.062
1.051 to the t-th power.

00:03:52.062 --> 00:03:56.311
So every day, we are growing
by a factor of 1.051.

00:03:56.311 --> 00:03:59.144
Well growing by a factor of 1.051

00:03:59.144 --> 00:04:02.580
means that you are adding
a little bit more than 5%.

00:04:02.580 --> 00:04:07.526
You're adding 0.51 every day of your mass,

00:04:07.526 --> 00:04:10.358
so you're adding 5.1%.

00:04:10.358 --> 00:04:12.718
And if you're rounding
to the nearest percent,

00:04:12.718 --> 00:04:15.592
we would say there's a 5% addition

00:04:15.592 --> 00:04:18.446
to the mass of the sunfish every day.


Interpreting time in exponential models

WEBVTT
Kind: captions
Language: en

00:00:00.306 --> 00:00:02.349
- After a special medicine is introduced

00:00:02.349 --> 00:00:04.346
into a petri dish full of bacteria,

00:00:04.346 --> 00:00:08.200
the number of bacteria
remaining in the dish

00:00:08.200 --> 00:00:09.856
decreases rapidly.

00:00:10.046 --> 00:00:13.123
The relationship between the
elapsed time t, in seconds,

00:00:13.123 --> 00:00:16.002
and the number of bacteria,
N of (t), in the petri dish

00:00:16.002 --> 00:00:19.374
is modeled by the following
function, all right.

00:00:19.728 --> 00:00:21.006
Complete the following sentence

00:00:21.006 --> 00:00:23.751
about the half-life of
the bacteria culture.

00:00:23.785 --> 00:00:27.971
The number of bacteria is
halved every blank seconds.

00:00:28.707 --> 00:00:32.057
So t is being given to us in seconds,

00:00:32.283 --> 00:00:34.628
so let's think about this a little bit.

00:00:35.287 --> 00:00:37.410
I'll draw a little table here.

00:00:38.265 --> 00:00:40.323
I'll draw a little table here,

00:00:41.009 --> 00:00:45.153
so if we have, this is
t and this is N of (t)

00:00:45.153 --> 00:00:47.150
and I'll start with a straightforward t

00:00:47.150 --> 00:00:48.950
at time equals zero right when we start,

00:00:48.950 --> 00:00:51.155
this whole thing, if this t is zero,

00:00:51.155 --> 00:00:53.957
then 1/2 to the zero over 5.5 power,

00:00:53.957 --> 00:00:56.368
this is 1/2 to the zero,
that's all gonna be one

00:00:56.368 --> 00:01:00.843
and you're just gonna be
left with 1000 bacteria

00:01:00.907 --> 00:01:02.399
in the petri dish.

00:01:02.416 --> 00:01:05.638
Now at what point do we
get to multiply by 1/2?

00:01:06.189 --> 00:01:11.164
At what point do we get
to say 1000 times 1/2?

00:01:12.005 --> 00:01:14.025
Well in order to say 1000 times 1/2,

00:01:14.025 --> 00:01:16.144
the exponent here has to be one,

00:01:16.730 --> 00:01:19.685
so at what time is the
exponent here going to be one?

00:01:20.108 --> 00:01:21.756
Well the exponent here's going to be one,

00:01:21.756 --> 00:01:23.070
this whole exponent's going to be one

00:01:23.070 --> 00:01:25.362
when t is equal to 5.5 seconds.

00:01:26.110 --> 00:01:29.877
So t is 5.5 seconds,

00:01:30.428 --> 00:01:32.797
and likewise we wait another 5.5 seconds,

00:01:32.797 --> 00:01:35.786
so if we go to 11 seconds,

00:01:36.035 --> 00:01:40.910
then this is going to be 1000 times 11

00:01:40.910 --> 00:01:42.686
divided by 5.5 is two,

00:01:42.686 --> 00:01:45.008
so times 1/2 to the second power,

00:01:45.008 --> 00:01:48.102
so times 1/2, times 1/2.

00:01:48.665 --> 00:01:51.288
So every five-and-a-half seconds we will

00:01:51.288 --> 00:01:53.227
essentially have half of the bacteria

00:01:53.227 --> 00:01:55.497
that we had five-and-a-half seconds ago.

00:01:55.688 --> 00:02:00.688
So the number of bacteria's
halved every 5.5 seconds.

00:02:01.319 --> 00:02:05.336
And you see it in the formula
or in the function definition

00:02:05.336 --> 00:02:07.402
right over there but it's
nice to reason it through

00:02:07.402 --> 00:02:10.903
and really, really digest
why it makes sense.

00:02:11.500 --> 00:02:13.329
Let's do a few more of these.

00:02:14.333 --> 00:02:18.953
The chemical element einsteinium-253

00:02:18.953 --> 00:02:21.223
naturally loses its mass over time.

00:02:21.333 --> 00:02:26.081
A sample of einsteinium-253 had
an initial mass of 320 grams

00:02:26.081 --> 00:02:27.632
when we measured it.

00:02:27.753 --> 00:02:30.411
The relationship between
the elapsed time t, in days,

00:02:30.411 --> 00:02:33.012
and the mass, M of (t) in
grams, left in the sample

00:02:33.012 --> 00:02:34.910
is modeled by the following function.

00:02:35.252 --> 00:02:37.435
Complete the following sentence
about the rate of change

00:02:37.435 --> 00:02:38.985
in the mass sample.

00:02:39.199 --> 00:02:42.953
The sample loses 87.5% of its mass

00:02:43.485 --> 00:02:45.455
every blank days.

00:02:45.885 --> 00:02:49.356
So this one, instead of
saying how much we grew

00:02:49.356 --> 00:02:52.029
or shrunk by, we're
saying a percent change.

00:02:54.499 --> 00:02:59.268
So if you lose 87.5%,

00:02:59.642 --> 00:03:02.109
that means that you are left with,

00:03:02.427 --> 00:03:06.590
left with 12.5%.

00:03:09.021 --> 00:03:13.026
Which is the same thing,
is the same way of saying

00:03:13.026 --> 00:03:17.410
that you have 0.125% of your mass.

00:03:17.937 --> 00:03:20.395
So another way of thinking about it is

00:03:21.505 --> 00:03:26.505
the sample is 0.125 of its mass,

00:03:27.288 --> 00:03:28.228
of its original mass

00:03:28.228 --> 00:03:29.505
or how long does it take

00:03:29.505 --> 00:03:33.180
for the sample to be 0.125 of its mass?

00:03:33.790 --> 00:03:36.669
And there we could do a similar idea,

00:03:36.669 --> 00:03:39.483
you see the 0.125 right over here

00:03:39.792 --> 00:03:41.567
and so I could draw a table if you like,

00:03:41.567 --> 00:03:44.301
although I think you might
guess where this is going.

00:03:44.806 --> 00:03:46.716
But let me draw a little table here.

00:03:47.673 --> 00:03:52.673
So, t and M of (t), when t is a zero,

00:03:52.961 --> 00:03:54.785
M of (t) is 320

00:03:57.323 --> 00:03:59.884
and so at what time

00:04:00.233 --> 00:04:05.138
is M of t going to be 320 times 0.125?

00:04:05.631 --> 00:04:07.732
Because this, going from this to this,

00:04:07.732 --> 00:04:10.722
that is losing 87.5% of your mass.

00:04:13.915 --> 00:04:18.014
Losing 87-- so let me try this way.

00:04:18.066 --> 00:04:22.472
So this is minus 87.5% of the mass,

00:04:23.152 --> 00:04:26.559
you've lost 0.875 to get to 0.125.

00:04:27.284 --> 00:04:30.796
So this, you could just use
0.125 to the first power,

00:04:30.987 --> 00:04:35.034
so what t do you have to
make this exponent equal one?

00:04:35.050 --> 00:04:37.030
Well t has to be 61.4.

00:04:37.303 --> 00:04:41.018
61.4 and where t is in days,

00:04:41.290 --> 00:04:43.734
61.4 days.

00:04:44.193 --> 00:04:46.387
Now you might be tempted to
always just pattern match,

00:04:46.387 --> 00:04:48.488
so, oh what ever's in the denominator,

00:04:48.488 --> 00:04:50.310
but I really encourage you to think about,

00:04:50.310 --> 00:04:51.692
cause that's the whole
point of these problems.

00:04:51.692 --> 00:04:53.190
If you just are pattern matching these,

00:04:53.190 --> 00:04:55.494
well I don't know how helpful
that's going to be for you.

00:04:55.825 --> 00:04:57.596
Let's do one more of these.

00:04:57.763 --> 00:04:58.924
Howard started studying

00:04:58.924 --> 00:05:01.792
how the number of branches
on his tree grows over time.

00:05:01.792 --> 00:05:05.089
The relationship between the
elapsed time t, in years,

00:05:05.089 --> 00:05:06.784
since Howard studying the tree,

00:05:06.784 --> 00:05:08.467
and the number of its branches, N of (t),

00:05:08.467 --> 00:05:10.104
is modeled by the following function.

00:05:10.104 --> 00:05:11.427
Complete the following sentence

00:05:11.427 --> 00:05:13.424
about the rate of change
in the number of branches.

00:05:13.424 --> 00:05:18.424
Howard's tree gains 4/5 more branches

00:05:18.662 --> 00:05:20.410
every blank years.

00:05:21.864 --> 00:05:25.275
So gaining 4/5

00:05:26.175 --> 00:05:28.944
is equivalent to multiplying,

00:05:30.122 --> 00:05:35.122
multiplying by,

00:05:36.387 --> 00:05:37.467
now remember

00:05:37.467 --> 00:05:39.417
you're gaining 4/5 of
what you already are,

00:05:39.417 --> 00:05:41.530
you're not just gaining the number 4/5,

00:05:41.530 --> 00:05:43.545
you're getting 4/5 of
what you already are.

00:05:43.735 --> 00:05:46.344
So that's equivalent of multiplying

00:05:46.344 --> 00:05:51.073
by one plus 4/5, or 9/5.

00:05:51.646 --> 00:05:55.276
So gaining 4/5 is the same
thing as multiplying by 9/5.

00:05:55.361 --> 00:05:59.529
If I'm 5 years old and if
I gained 4/5 of my age,

00:05:59.557 --> 00:06:03.673
I've gained 4 years to
get to be 9 years old,

00:06:03.783 --> 00:06:06.192
which means I multiplied my age by 9/5.

00:06:06.801 --> 00:06:09.518
So Howard's tree, you could say,

00:06:09.518 --> 00:06:12.426
grows by a factor of 9/5
every how many years?

00:06:12.722 --> 00:06:15.468
Well you can see over here
the common ratio is 9/5

00:06:15.717 --> 00:06:18.283
and so you're going to grow by 9/5

00:06:18.283 --> 00:06:23.283
every time t is a multiple of 7.3.

00:06:23.878 --> 00:06:26.757
Or I guess you could say
every time t increases by 7.3

00:06:26.757 --> 00:06:31.488
then this exponent is going
to increase by a whole

00:06:31.679 --> 00:06:35.470
and so you could do that as
multiplying again by 9/5.

00:06:35.754 --> 00:06:40.754
So Howard's tree gains 4/5
more branches every 7.3 years.