Calculus BC 2008 2 a

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Kind: captions
Language: en

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Let's keep going with our
2008 Calculus BC free

00:00:04.980 --> 00:00:05.780
answer questions.

00:00:05.780 --> 00:00:07.250
So we're on problem number 2.

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And I cut and pasted the table
that they gave for the problem.

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And I'll read the
rest of the problem.

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And let's see what we can do.

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So it says concert tickets went
on sale at noon, t equals 0,

00:00:17.250 --> 00:00:19.630
and were sold out
within 9 hours.

00:00:19.630 --> 00:00:23.130
The number of people waiting in
line to purchase tickets at

00:00:23.130 --> 00:00:26.780
time t, is modeled by a
twice differentiable

00:00:26.780 --> 00:00:28.560
function L, L of t.

00:00:28.560 --> 00:00:33.350
So this is the number of people
waiting in line at anytime.

00:00:33.350 --> 00:00:38.180
And this is noon, 12:00
noon, 1:00 pm, 9:00 pm.

00:00:38.180 --> 00:00:40.070
All the tickets are
sold by 9:00 pm.

00:00:40.070 --> 00:00:41.710
And they tell us it's
twice differentiable.

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So that means that whatever
function we're modeling this,

00:00:44.900 --> 00:00:49.020
this L of t, that it's
continuous because

00:00:49.020 --> 00:00:50.960
it's differentiable.

00:00:50.960 --> 00:00:53.300
And since it's twice
differentiable, we also know

00:00:53.300 --> 00:00:56.350
that its derivative is
continuous because the second

00:00:56.350 --> 00:00:59.330
derivative exists
at all points.

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So it says part a-- let me make
sure I'm not writing too

00:01:04.140 --> 00:01:11.440
thickly-- use the data in the
table to estimate the rate--

00:01:11.440 --> 00:01:13.740
let me actually just copy
and paste this, I've just

00:01:13.740 --> 00:01:15.890
figured out how to do this.

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So I figure it doesn't hurt.

00:01:24.310 --> 00:01:24.700
There you go.

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You probably can't read it.

00:01:28.350 --> 00:01:32.690
But it says use the data in the
table to estimate the rate at

00:01:32.690 --> 00:01:34.970
which the number of people
waiting in line was

00:01:34.970 --> 00:01:37.030
changing at 5:30 pm.

00:01:37.030 --> 00:01:39.370
So they don't give us
any data on 5:30 pm.

00:01:39.370 --> 00:01:43.980
They give us 4:00 pm and
7:00 pm, or t equals 5.5.

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Show the computations that
lead to your answer.

00:01:45.710 --> 00:01:47.430
Indicate the units of measure.

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So what do they want to know?

00:01:48.280 --> 00:01:52.520
They want to know an estimate
of the rate at which the

00:01:52.520 --> 00:01:56.220
number of people waiting
in line was changing.

00:01:56.220 --> 00:01:59.830
So they don't give us a
continuous function definition.

00:01:59.830 --> 00:02:02.200
They just give us a bunch
of sample points of

00:02:02.200 --> 00:02:03.620
this function L of t.

00:02:03.620 --> 00:02:08.700
So the best estimate I can do,
of the rate at which this L of

00:02:08.700 --> 00:02:13.520
t is changing at time 5.5-- and
5.5 is in between these two--

00:02:13.520 --> 00:02:15.840
is to just figure out the
average rate of change

00:02:15.840 --> 00:02:18.770
between time 4 and time 7.

00:02:18.770 --> 00:02:19.860
So how do we figure that out?

00:02:19.860 --> 00:02:23.760
Well the average rate of
change is just the slope.

00:02:23.760 --> 00:02:24.790
So let's write it.

00:02:27.810 --> 00:02:35.760
So we could write average
rate of change or delta

00:02:35.760 --> 00:02:40.770
L over delta t at 5.5.

00:02:40.770 --> 00:02:42.790
You can write it however you
want, however you think that

00:02:42.790 --> 00:02:44.770
the exam graders would
best like to see it.

00:02:44.770 --> 00:02:48.130
We could say at time 5.5.

00:02:48.130 --> 00:02:49.835
We could say approximately
equals, or whatever.

00:02:49.835 --> 00:02:52.140
But it's just going to be the
slope between these two points.

00:02:52.140 --> 00:03:00.400
So it's L of 7 minus L of 4,
all of that over 7 over 4.

00:03:00.400 --> 00:03:03.210
Rise over run or change in
the value of the function

00:03:03.210 --> 00:03:06.930
divided by change in the
independent variable.

00:03:06.930 --> 00:03:09.520
L of 7 is 154.

00:03:09.520 --> 00:03:11.250
They give us that.

00:03:11.250 --> 00:03:16.930
L of 4 is 126.

00:03:16.930 --> 00:03:20.790
And we divide it by 7 minus 4.

00:03:20.790 --> 00:03:31.880
So that equals 54 minus
126 is 24 plus 26 is 50.

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So it's equal to 28.

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Right, if this was 2
less, it would be 30.

00:03:39.910 --> 00:03:42.660
And 7 minus 4 is 3.

00:03:42.660 --> 00:03:44.010
So you could say that
the average rate of

00:03:44.010 --> 00:03:45.950
change is 28 over 3.

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Or you could write
that as 9 1/3.

00:03:50.550 --> 00:03:52.530
And they want us to use
the units of measure.

00:03:52.530 --> 00:03:54.450
Indicate units of measure.

00:03:54.450 --> 00:03:56.905
So the numerator,
this is people.

00:04:01.850 --> 00:04:02.570
And what's the denominator?

00:04:02.570 --> 00:04:06.150
It's hours, people per hour.

00:04:06.150 --> 00:04:10.290
So my best estimate, or our
best estimate of the rate at

00:04:10.290 --> 00:04:12.580
which the number of people
waiting in line was changing at

00:04:12.580 --> 00:04:15.410
5:30 pm, which is between these
two points, is the average

00:04:15.410 --> 00:04:16.640
slope between these two points.

00:04:16.640 --> 00:04:20.890
Which is 9 1/3 people per hour.

00:04:20.890 --> 00:04:21.790
That's it.

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Let's do part b.

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Let me clear all of this so we
have enough space for part b.

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And I'll copy and don't
know if you can read it.

00:04:31.370 --> 00:04:33.770
But maybe you can.

00:04:33.770 --> 00:04:37.735
So doesn't hurt to
just copy and paste.

00:04:42.510 --> 00:04:48.060
OK part b says, use a
trapezoidal sum with

00:04:48.060 --> 00:04:49.550
three subintervals.

00:04:49.550 --> 00:04:51.780
Maybe I can make this
a little bit bigger.

00:04:51.780 --> 00:04:54.930
Let me see if I can grow
that a little bit.

00:04:54.930 --> 00:04:57.170
I don't want to take up
all the space though.

00:04:57.170 --> 00:04:58.850
No that doesn't look good.

00:05:04.570 --> 00:05:07.590
OK well I'll read it out loud
in case you can't see it.

00:05:07.590 --> 00:05:11.670
Use a trapezoidal sum with
three subintervals to estimate

00:05:11.670 --> 00:05:13.810
the average number of people
waiting in line during the

00:05:13.810 --> 00:05:16.420
first 4 hours that the
tickets were on sale.

00:05:16.420 --> 00:05:18.550
If you want to do this
really fast, you wouldn't

00:05:18.550 --> 00:05:19.480
have to graph it.

00:05:19.480 --> 00:05:21.500
But I'm about to graph it
because I want you to

00:05:21.500 --> 00:05:23.250
understand how to
do this problem.

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If you understand it, you
don't have to graph it.

00:05:24.770 --> 00:05:26.930
So let's graph some
of these points.

00:05:26.930 --> 00:05:29.770
And we really just have
to do the first 4 hours.

00:05:29.770 --> 00:05:31.580
Actually let's graph all
of them, just because it

00:05:31.580 --> 00:05:37.710
might be useful in future
parts of the problem.

00:05:37.710 --> 00:05:40.023
If you weren't graphing it
while recording a video, it

00:05:40.023 --> 00:05:41.080
probably would be a
little bit faster.

00:05:44.470 --> 00:05:45.305
That should be good enough.

00:05:48.760 --> 00:05:49.920
What are our data points?

00:05:49.920 --> 00:06:03.800
We have 1, 2, 3,
4, 5, 6, 7, 8, 9.

00:06:03.800 --> 00:06:04.930
You don't have do it so neatly.

00:06:04.930 --> 00:06:06.690
Well it's good to do it neatly
when you're doing the exam,

00:06:06.690 --> 00:06:08.600
because otherwise you'll
confuse yourself.

00:06:08.600 --> 00:06:12.640
And let's see, how high
should my L values?

00:06:12.640 --> 00:06:14.272
This is time right here.

00:06:14.272 --> 00:06:18.250
This is my L of t axis.

00:06:18.250 --> 00:06:21.180
It goes up to about, 176
is a high point, at least

00:06:21.180 --> 00:06:24.040
on the data we have.

00:06:24.040 --> 00:06:28.270
Let's say this is 200.

00:06:28.270 --> 00:06:33.410
So halfway up would be 100.

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This would be 150.

00:06:37.320 --> 00:06:39.770
This would be 50.

00:06:39.770 --> 00:06:41.080
I don't know what
that sound was from.

00:06:41.080 --> 00:06:42.370
But let's keep going forward.

00:06:42.370 --> 00:06:44.570
Let's plot the points.

00:06:44.570 --> 00:06:47.260
At t equals 0 there are
120 people in line.

00:06:47.260 --> 00:06:48.910
That's about right there.

00:06:48.910 --> 00:06:51.520
At t equals 1, 156.

00:06:51.520 --> 00:06:52.630
I'm just approximating.

00:06:52.630 --> 00:06:54.280
It's about right there.

00:06:54.280 --> 00:06:57.180
At t equals 3-- they skip
2-- at t equals 3 it's

00:06:57.180 --> 00:06:58.730
176 people in line.

00:06:58.730 --> 00:07:01.640
So that's going to be
right about there,

00:07:01.640 --> 00:07:03.520
just approximating it.

00:07:03.520 --> 00:07:08.520
At t equals 4, we have
126 people in line.

00:07:08.520 --> 00:07:12.870
It's a little bit more
than at t equals 0.

00:07:12.870 --> 00:07:17.242
At t equals 7, we have
150 people in line.

00:07:17.242 --> 00:07:18.840
It's right about there.

00:07:18.840 --> 00:07:21.480
At t equals 8 we have 80.

00:07:21.480 --> 00:07:25.050
That's about right there.

00:07:25.050 --> 00:07:27.500
And t equal 9, the
line is gone.

00:07:27.500 --> 00:07:29.000
Everyone has gotten
their tickets, or maybe

00:07:29.000 --> 00:07:30.220
they got sold out.

00:07:30.220 --> 00:07:31.660
Let's connect those points.

00:07:31.660 --> 00:07:33.681
Connect the dots.

00:07:33.681 --> 00:07:36.850
So from here to there.

00:07:36.850 --> 00:07:39.490
And from there to there.

00:07:39.490 --> 00:07:41.890
There to there.

00:07:41.890 --> 00:07:45.240
There to there, almost done.

00:07:45.240 --> 00:07:49.570
OK so we've at least plotted
the sample points and

00:07:49.570 --> 00:07:51.160
connected them with lines.

00:07:51.160 --> 00:07:53.830
We know that the real L of t,
whatever we used to approximate

00:07:53.830 --> 00:07:56.590
it, it's not going to have
these sharp edges because

00:07:56.590 --> 00:07:57.660
it's differentiable.

00:07:57.660 --> 00:07:58.660
It's actually twice
differentiable.

00:07:58.660 --> 00:08:01.460
So it's actually going to be
a smoother curve, right?

00:08:01.460 --> 00:08:02.790
Because we can take the
derivative of any point.

00:08:02.790 --> 00:08:04.200
If this was the actual
function, you wouldn't be

00:08:04.200 --> 00:08:05.950
able to take a derivative
at this point.

00:08:05.950 --> 00:08:10.640
Because there's a positive
slope here and then it

00:08:10.640 --> 00:08:12.190
immediately switches to a
negative slope like the

00:08:12.190 --> 00:08:13.160
absolute value functions.

00:08:13.160 --> 00:08:14.220
You actually wouldn't
be able to take the

00:08:14.220 --> 00:08:15.660
derivative at that point.

00:08:15.660 --> 00:08:17.090
But anyway back to the problem.

00:08:17.090 --> 00:08:20.650
Use a trapezoidal sum with
three subintervals to estimate

00:08:20.650 --> 00:08:22.390
the average number of people
waiting in line during

00:08:22.390 --> 00:08:23.020
the first 4 hours.

00:08:23.020 --> 00:08:26.600
So the first 4 hours,
that's right here.

00:08:26.600 --> 00:08:29.210
Now this might seem
kind of daunting.

00:08:29.210 --> 00:08:32.080
The trapezoidal sum
for the fourth.

00:08:32.080 --> 00:08:33.860
Actually I just realized.

00:08:33.860 --> 00:08:37.270
For some reason, YouTube used
to let me do longer videos

00:08:37.270 --> 00:08:38.270
because I thought
I was a partner.

00:08:38.270 --> 00:08:40.270
But now for some reason it's
been limiting me again.

00:08:40.270 --> 00:08:43.380
So I will continue part
b in the next video.

00:08:43.380 --> 00:08:44.882
See you soon.


Calculus BC 2008 2d

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Kind: captions
Language: en

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Welcome

00:00:01.130 --> 00:00:01.400
back.

00:00:01.400 --> 00:00:04.350
We're ready to do part
D, and let me copy and

00:00:04.350 --> 00:00:05.990
paste that in as well.

00:00:10.210 --> 00:00:12.330
See, I don't think that's going
to need this graph, so let

00:00:12.330 --> 00:00:15.290
me just remove that with a
color other than yellow.

00:00:23.750 --> 00:00:25.150
It's copy and pasted.

00:00:25.150 --> 00:00:27.610
I don't know if you can read
it, but it's helpful for me

00:00:27.610 --> 00:00:32.060
to review the problem
on our clipboard.

00:00:32.060 --> 00:00:34.430
OK.

00:00:34.430 --> 00:00:42.010
The rate at which tickets were
sold for t -- for over this

00:00:42.010 --> 00:00:45.120
range is modeled by r of t--
let me write that in case you

00:00:45.120 --> 00:00:48.270
can't see it-- is the rate at
which tickets were sold. r of t

00:00:48.270 --> 00:00:56.320
is equal to 550 te to the minus
t over 2 tickets per hour.

00:00:56.320 --> 00:01:00.590
Based on the model, how many
tickets were sold by 3:00 PM?

00:01:00.590 --> 00:01:04.480
So my t equals 3-- to the
nearest whole number?

00:01:04.480 --> 00:01:05.650
So that's sometimes important.

00:01:05.650 --> 00:01:09.980
You don't want to give
a decimal answer.

00:01:09.980 --> 00:01:12.790
So this is the rate at
which tickets are sold.

00:01:12.790 --> 00:01:17.020
So this is the derivative
of the total tickets

00:01:17.020 --> 00:01:18.940
sold function.

00:01:18.940 --> 00:01:21.870
Or another way that we could
write it is the total tickets

00:01:21.870 --> 00:01:26.600
sold-- so let's call that, I
don't know, capital T sub--

00:01:26.600 --> 00:01:29.740
well let me-- I don't want to
do T of t, that's

00:01:29.740 --> 00:01:30.490
[UNINTELLIGIBLE]

00:01:30.490 --> 00:01:37.150
So let's say the tickets sold
as a function of time is going

00:01:37.150 --> 00:01:41.547
to be equal to the definite
integral-- well, we could say

00:01:41.547 --> 00:01:45.260
is at any time t, the tickets
sold-- and this is the

00:01:45.260 --> 00:01:47.680
fundamental theorems calculus,
I think it might be one of its

00:01:47.680 --> 00:01:49.500
correlaries or actually
sometimes it is the fundamental

00:01:49.500 --> 00:01:53.810
theorem of calculus, I always
forget my definitions.

00:01:53.810 --> 00:01:58.340
Between time equals 0 and t--
or if we want to know the

00:01:58.340 --> 00:02:01.260
tickets sold, between time
equals 0 and t is equal to the

00:02:01.260 --> 00:02:05.580
integral of the rate at which
the tickets sold was changing.

00:02:05.580 --> 00:02:13.970
So that's equal to 550te
to the minus t over 2dt.

00:02:13.970 --> 00:02:14.210
Right?

00:02:14.210 --> 00:02:15.020
That's it.

00:02:15.020 --> 00:02:18.180
And so if we want to know how
many tickets were sold at time

00:02:18.180 --> 00:02:23.120
equals 3, that's just equal to
the definite integral from 0 to

00:02:23.120 --> 00:02:26.520
3, or we could also view it as
the area under this curve, from

00:02:26.520 --> 00:02:31.390
time equals 0 to time equal to
3 of 550te to the

00:02:31.390 --> 00:02:34.640
minus t over 2dt.

00:02:34.640 --> 00:02:36.990
Now this integral right here,
you can solve it analytically

00:02:36.990 --> 00:02:40.120
using integration of parts,
which I just called the reverse

00:02:40.120 --> 00:02:43.840
product rule, but you only have
45 minutes to do all three of

00:02:43.840 --> 00:02:46.980
these problems, and they'll let
you use your graphing

00:02:46.980 --> 00:02:50.050
calculator, and your graphic
calculator is excellent at

00:02:50.050 --> 00:02:52.210
doing definite integrals, and
they just want the

00:02:52.210 --> 00:02:53.180
number, right?

00:02:53.180 --> 00:02:59.210
So let's use our graphic
calculators to get that number.

00:02:59.210 --> 00:03:00.350
Let's see.

00:03:00.350 --> 00:03:06.090
I don't want to copy,
so how do we do that?

00:03:06.090 --> 00:03:12.320
We just do second the division
button but that's calc--

00:03:12.320 --> 00:03:16.620
definite, let me use the
definite integral, and like

00:03:16.620 --> 00:03:18.790
what was, let's see, let me
make sure I have that--

00:03:18.790 --> 00:03:22.560
550 let's just use x.

00:03:22.560 --> 00:03:37.180
550 times x times second e to
the minus x divided by 2.

00:03:37.180 --> 00:03:40.220
I think that's the
whole function.

00:03:40.220 --> 00:03:40.730
And let's see.

00:03:40.730 --> 00:03:46.610
My independent variable is x,
I've just swapped t for x

00:03:46.610 --> 00:03:53.060
there, and I'm taking the
integral from 0 to 3.

00:03:55.590 --> 00:03:57.520
Click enter, let the
calculator do the work.

00:03:57.520 --> 00:03:59.720
This would have taken you quite
a while if you had to actually

00:03:59.720 --> 00:04:01.140
do the integral yourself.

00:04:01.140 --> 00:04:04.810
972.78, and they want
us to round to the

00:04:04.810 --> 00:04:06.030
nearest whole number.

00:04:06.030 --> 00:04:09.770
So the nearest whole
number is 973.

00:04:09.770 --> 00:04:18.840
So we say 973 tickets
sold by 3:00 PM.

00:04:18.840 --> 00:04:19.490
And we're done.

00:04:19.490 --> 00:04:21.970
That only took us four minutes.

00:04:21.970 --> 00:04:23.230
And it would have taken us
even less if we didn't

00:04:23.230 --> 00:04:24.130
have to explain it.

00:04:24.130 --> 00:04:25.020
Anyway.

00:04:25.020 --> 00:04:27.840
I will see you in
problem number three.


AP Calculus BC exams: 2008 1 d

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Kind: captions
Language: en

00:00:00.630 --> 00:00:01.380
Welcome back.

00:00:01.380 --> 00:00:04.460
We're doing the last part
of Problem 1 of the

00:00:04.460 --> 00:00:07.390
2008 Calculus BC exam.

00:00:07.390 --> 00:00:08.670
And I'll repeat the problem.

00:00:08.670 --> 00:00:12.800
It says, the region R models
the surface of a small pond.

00:00:12.800 --> 00:00:20.550
At all points in R at a
distance x from the y-axis, the

00:00:20.550 --> 00:00:24.300
depth of the water is given by
h of x is equal to 3 minus x.

00:00:24.300 --> 00:00:26.400
Find the volume of the
water in the pond.

00:00:26.400 --> 00:00:28.900
So we thought about, this is,
you know, a perspective drawing

00:00:28.900 --> 00:00:31.950
of this, this is the surface
of the pond, right here.

00:00:31.950 --> 00:00:33.560
And I was trying to
draw the depth.

00:00:33.560 --> 00:00:39.480
So when x is equal to 0, the
pond is exactly 3, I guess, you

00:00:39.480 --> 00:00:42.060
know, we could say feet, or
whatever the units are, deep,

00:00:42.060 --> 00:00:48.100
and then when x is equal to
2, the pond is 1 unit deep.

00:00:48.100 --> 00:00:50.340
And so to find the volume, we
think about it the same way.

00:00:50.340 --> 00:00:52.730
What does a cross section
at any point of the

00:00:52.730 --> 00:00:55.040
pond look like?

00:00:55.040 --> 00:00:56.430
Well if we, let's see.

00:00:56.430 --> 00:00:59.300
So if we were to take a cross
section, so let's say that

00:00:59.300 --> 00:01:01.970
this, let's take this cross
section of the pond.

00:01:01.970 --> 00:01:04.230
Let me do it in a different
color, I know it can

00:01:04.230 --> 00:01:05.560
be kind of confusing.

00:01:05.560 --> 00:01:06.610
That's not too different.

00:01:06.610 --> 00:01:09.110
So that's green.

00:01:09.110 --> 00:01:12.480
So the height along the surface
of the pond is going to be the

00:01:12.480 --> 00:01:14.730
difference between
the two functions.

00:01:14.730 --> 00:01:20.220
And we know that the top
function is sine of pi x, and

00:01:20.220 --> 00:01:22.990
we know that this bottom
function is x to the

00:01:22.990 --> 00:01:26.280
third minus 4x.

00:01:26.280 --> 00:01:29.530
So if I were to draw, let
me see if I can draw

00:01:29.530 --> 00:01:33.030
this cross section.

00:01:36.530 --> 00:01:39.730
So we know this the height,
sorry, the width along the

00:01:39.730 --> 00:01:42.410
surface of this cross section
is going to be the difference

00:01:42.410 --> 00:01:46.460
between this function and
this function, right?

00:01:46.460 --> 00:01:50.850
So if I were to draw, let me
see if I can, so this is that

00:01:50.850 --> 00:01:58.790
same cross section, it's the
width along the top, I know I'm

00:01:58.790 --> 00:02:02.120
confusing you, is going to be
this function minus

00:02:02.120 --> 00:02:02.870
this function.

00:02:02.870 --> 00:02:15.580
It's going to be sine of pi x
minus x to the third minus 4x.

00:02:15.580 --> 00:02:18.200
And what's going to be
the height, right here?

00:02:18.200 --> 00:02:19.020
Well, they told us.

00:02:19.020 --> 00:02:21.400
The height of the pond, or
the depth of the pond, at

00:02:21.400 --> 00:02:24.630
any point, is 3 minus x,
whatever x value we're at.

00:02:24.630 --> 00:02:28.720
So this is going
to be 3 minus x.

00:02:28.720 --> 00:02:32.450
So the area of this cross
section of the pond is going to

00:02:32.450 --> 00:02:45.010
be sine of pi x minus x to the
third minus four x times the

00:02:45.010 --> 00:02:46.110
depth of the pond, right?

00:02:46.110 --> 00:02:48.410
That's the surface
width times the depth.

00:02:48.410 --> 00:02:52.470
So that times 3 minus x.

00:02:52.470 --> 00:02:54.610
So that's the area of
each cross section.

00:02:54.610 --> 00:02:57.630
So we want the volume for
the entire pond, we take

00:02:57.630 --> 00:03:00.150
essentially the volume of
each of these kind of

00:03:00.150 --> 00:03:01.950
slivers of the pond.

00:03:01.950 --> 00:03:04.730
So we take the area of each
cross section, and we multiply

00:03:04.730 --> 00:03:10.060
it by a very small width to
get a very small sliver.

00:03:10.060 --> 00:03:16.240
So we take that and multiply
it by dx, and we integrate.

00:03:16.240 --> 00:03:20.720
We sum up all of these slivers
of the pond from x is equal

00:03:20.720 --> 00:03:23.260
to 0 to x is equal to 2.

00:03:23.260 --> 00:03:24.680
So let's do that.

00:03:24.680 --> 00:03:28.660
Let's integrate from x is equal
to 0 from x is equal to 2.

00:03:28.660 --> 00:03:31.410
Once again, this is a
really hard integral.

00:03:31.410 --> 00:03:33.930
Or it's something that you can
do, especially if you use

00:03:33.930 --> 00:03:36.440
integration by parts, but it's
a little bit messy, and you

00:03:36.440 --> 00:03:40.170
only have 45 minutes to do all
three problems on the BC exam,

00:03:40.170 --> 00:03:42.890
so I'm assuming that they want
you to use the calculator here.

00:03:42.890 --> 00:03:46.970
So let's use the calculator
to evaluate it.

00:03:46.970 --> 00:03:48.080
So let's see.

00:03:50.580 --> 00:03:53.480
So this was part c that we
already have in our calculator.

00:03:53.480 --> 00:03:57.660
We could do second enter, and
we'll get the previous entry.

00:03:57.660 --> 00:04:00.236
Second enter.

00:04:00.236 --> 00:04:02.020
Because we already typed
in a bunch of stuff.

00:04:02.020 --> 00:04:05.350
The only difference between
what we did in Part C and now

00:04:05.350 --> 00:04:09.590
is, now, instead of having in
Part C, we had this

00:04:09.590 --> 00:04:10.960
expression squared.

00:04:10.960 --> 00:04:12.575
Now, we don't have it squared,
but we're multiplying

00:04:12.575 --> 00:04:14.355
it by times 3 minus x.

00:04:14.355 --> 00:04:16.660
So let's do that.

00:04:16.660 --> 00:04:20.970
So if we go here, we can
delete this squared.

00:04:20.970 --> 00:04:22.710
We're not squaring it anymore.

00:04:22.710 --> 00:04:23.660
Deleted.

00:04:23.660 --> 00:04:26.420
And now let's do, I want
you to see my key strokes.

00:04:26.420 --> 00:04:28.590
So let's do second insert.

00:04:28.590 --> 00:04:40.180
So let's insert times 3
minus x close parentheses.

00:04:40.180 --> 00:04:41.000
And so, let's see.

00:04:41.000 --> 00:04:45.450
We have sine of pi x minus x
to the third plus 4x, right,

00:04:45.450 --> 00:04:46.840
that's the same thing as this.

00:04:46.840 --> 00:04:49.160
Times 3 minus x.

00:04:49.160 --> 00:04:51.810
And our variable of integration
is x, and we're integrating

00:04:51.810 --> 00:04:54.520
from x is equal to zero
to x is equal to 2.

00:04:54.520 --> 00:04:55.550
Let's hit enter.

00:04:58.900 --> 00:05:00.980
And it noticed, it
is calculating.

00:05:00.980 --> 00:05:05.760
And our answer is 8.37, I
think, is a fair answer.

00:05:05.760 --> 00:05:09.470
So the volume this time,
the volume of the pond,

00:05:09.470 --> 00:05:13.090
is equal to 8.37.

00:05:13.090 --> 00:05:15.270
Anyway, hopefully you
found that useful.

00:05:15.270 --> 00:05:19.250
I will, every day, try to do
a couple of these problems.

00:05:19.250 --> 00:05:21.340
See you in the next video.


AP Calculus BC exams: 2008 1 c&d

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.370
So we were doing part c of the
first problem on the Calculus

00:00:04.370 --> 00:00:09.210
BC exam, the free response
part, and so what, I'll re-read

00:00:09.210 --> 00:00:12.420
it, it says the region
r, this is the region r,

00:00:12.420 --> 00:00:13.610
is the base of a solid.

00:00:13.610 --> 00:00:16.190
I redrew the region r here, but
with a little perspective, so

00:00:16.190 --> 00:00:19.930
we can hopefully visualize
it in three dimensions.

00:00:19.930 --> 00:00:22.590
So it says for this solid, each
cross section perpendicular

00:00:22.590 --> 00:00:24.000
to the x-axis, right?

00:00:24.000 --> 00:00:25.750
There's two ways you can do a
cross-section, you could do

00:00:25.750 --> 00:00:27.870
them, you could cut this
way, but that would be

00:00:27.870 --> 00:00:30.040
parallel to the x-axis.

00:00:30.040 --> 00:00:33.210
We wanted to cut perpendicular
to the x-axis, or parallel

00:00:33.210 --> 00:00:34.420
to the y-axis, right?

00:00:34.420 --> 00:00:38.740
So if we cut it along
the line like that.

00:00:38.740 --> 00:00:42.500
So they say, each cross-section
perpendicular to the

00:00:42.500 --> 00:00:43.810
x-axis is a square.

00:00:43.810 --> 00:00:45.120
So I drew a couple
of squares here.

00:00:45.120 --> 00:00:46.250
That's one.

00:00:46.250 --> 00:00:50.270
So here, this would be the
base, and since we know the

00:00:50.270 --> 00:00:52.290
cross-section is the square,
the height has to be the

00:00:52.290 --> 00:00:53.710
same length as the base.

00:00:53.710 --> 00:00:54.660
Same thing here.

00:00:54.660 --> 00:00:56.790
So here, the height will be
really high, because this is

00:00:56.790 --> 00:01:01.150
kind of maybe our maximum point
in terms of the base width.

00:01:01.150 --> 00:01:04.090
But it's still pretty wide,
then it gets narrow again.

00:01:04.090 --> 00:01:06.320
So how do we figure out
the volume of this solid?

00:01:06.320 --> 00:01:08.140
Which is kind of
hard to visualize.

00:01:08.140 --> 00:01:10.816
And it's in some ways you
easier to visualize it than

00:01:10.816 --> 00:01:12.680
it is to draw, so you have
to give me some credit.

00:01:12.680 --> 00:01:15.830
But anyway, what we do is, we
take the area of each of these

00:01:15.830 --> 00:01:17.960
squares, and I drew
one of them here.

00:01:17.960 --> 00:01:20.520
You take you each of the area
of each of these squares,

00:01:20.520 --> 00:01:24.600
multiply them by a super small
change in x, and that we know,

00:01:24.600 --> 00:01:27.770
from everything we've learned
in calculus hopefully, that

00:01:27.770 --> 00:01:30.180
that super small change in x,
I'm trying to draw a little

00:01:30.180 --> 00:01:32.830
perspective, is dx.

00:01:32.830 --> 00:01:36.960
So if we multiply dx times the
area of this square, that is

00:01:36.960 --> 00:01:42.190
the volume of this kind of
part the entire solid.

00:01:42.190 --> 00:01:45.900
And if we were to sum up all of
these infinitely thin solids,

00:01:45.900 --> 00:01:47.830
we would get the volume
for the whole thing.

00:01:47.830 --> 00:01:49.540
So how do we do that?

00:01:49.540 --> 00:01:53.850
Well let's write our
integral expression.

00:01:53.850 --> 00:01:59.350
So what is the area of
each of these squares?

00:01:59.350 --> 00:02:02.020
Each of these cross-sections,
what is the area?

00:02:02.020 --> 00:02:05.090
Well the base is going to
be the difference between

00:02:05.090 --> 00:02:06.520
our two functions, right?

00:02:06.520 --> 00:02:13.210
This top function right
here, that was sin of pi x.

00:02:13.210 --> 00:02:18.560
And this bottom function, right
here, that is y is equal to

00:02:18.560 --> 00:02:22.010
x to the third minus 4x.

00:02:22.010 --> 00:02:27.040
So the base of the functions,
the base of this distance right

00:02:27.040 --> 00:02:29.560
here is going to be the
difference between the top

00:02:29.560 --> 00:02:31.820
function and the
bottom function.

00:02:31.820 --> 00:02:42.540
So each base is going to be sin
of pi x, minus this function,

00:02:42.540 --> 00:02:46.580
so minus x to the third, plus
4x, right switch the sign.

00:02:46.580 --> 00:02:50.590
Minus x to the third, plus 4x.

00:02:50.590 --> 00:02:54.110
So far this might look pretty
similar to part a, but

00:02:54.110 --> 00:02:55.280
what's the twist here?

00:02:55.280 --> 00:02:56.980
We want the area.

00:02:56.980 --> 00:02:58.350
The area of each of
the squares, not

00:02:58.350 --> 00:02:59.200
just this distance.

00:02:59.200 --> 00:02:59.940
So what's the area?

00:02:59.940 --> 00:03:03.260
It's going to be this distance
squared, so we have to

00:03:03.260 --> 00:03:06.360
square this entire thing.

00:03:06.360 --> 00:03:09.090
So that's the area of each
of these squares, and then

00:03:09.090 --> 00:03:12.740
we have to multiply them
times a little bit of dx.

00:03:12.740 --> 00:03:16.950
And that gives us the volume
for each of these, I guess,

00:03:16.950 --> 00:03:18.690
parts of the entire solid.

00:03:18.690 --> 00:03:20.360
And then what are our
boundaries of integration?

00:03:20.360 --> 00:03:22.340
Well it's the same
thing as in part a.

00:03:22.340 --> 00:03:24.240
This is 0, this is 2.

00:03:24.240 --> 00:03:28.090
So the boundaries are
pretty straightforward.

00:03:28.090 --> 00:03:31.970
So we essentially just have
to evaluate this now.

00:03:31.970 --> 00:03:36.730
And just like in part b, I've
first tried to evaluate this

00:03:36.730 --> 00:03:39.620
analytically, and you end up
getting a very, very,

00:03:39.620 --> 00:03:41.000
very hairy integral.

00:03:41.000 --> 00:03:43.220
Which you can do analytically,
you but you have to know some

00:03:43.220 --> 00:03:46.010
power reduction formulas in
trigonometry, you have to do

00:03:46.010 --> 00:03:49.160
some integration by parts, it
would probably use up all of

00:03:49.160 --> 00:03:51.220
the time on the AP Exam.

00:03:51.220 --> 00:03:54.080
So since they said that a
graphing calculator is required

00:03:54.080 --> 00:03:56.700
for some parts of the problem,
I say why not use our

00:03:56.700 --> 00:03:58.110
graphing calculator?

00:03:58.110 --> 00:04:01.360
Because the graphing calculator
is very good at numerically

00:04:01.360 --> 00:04:03.450
evaluating definite
integrals like this.

00:04:03.450 --> 00:04:09.100
So let's get out my
TI-85 emulator again.

00:04:09.100 --> 00:04:11.950
Here we go.

00:04:11.950 --> 00:04:15.210
I want you see the key strokes
so that you, ok turned

00:04:15.210 --> 00:04:18.640
on, exit out of this.

00:04:18.640 --> 00:04:23.920
So we're going to use the
calculus function here,

00:04:23.920 --> 00:04:27.070
so second, calculus.

00:04:27.070 --> 00:04:29.580
And this function right here,
this is definite integral, a

00:04:29.580 --> 00:04:31.490
very useful function to use.

00:04:31.490 --> 00:04:34.490
Press F5, definite integral.

00:04:34.490 --> 00:04:36.450
And then we just type
in the expression.

00:04:36.450 --> 00:04:41.050
So the expression, let me
move it down a little bit.

00:04:41.050 --> 00:04:50.040
So it's open parentheses
sin of where is pi?

00:04:50.040 --> 00:04:54.470
Pi is -- I haven't used one of
these calculators in a long,

00:04:54.470 --> 00:04:57.160
long time -- oh there it is.

00:04:57.160 --> 00:05:13.210
Second pi x sin of pi x minus x
to the third power, plus 4x.

00:05:16.380 --> 00:05:21.010
All of that squared.

00:05:21.010 --> 00:05:24.180
And then this definite integral
function, you have to tell it

00:05:24.180 --> 00:05:26.750
which is the independent
variable, or kind of, you

00:05:26.750 --> 00:05:29.930
know, what variable are we
integrating across, and

00:05:29.930 --> 00:05:31.360
that's the variable x.

00:05:31.360 --> 00:05:33.510
And then you just tell it the
boundaries of integration,

00:05:33.510 --> 00:05:34.450
and we're done.

00:05:34.450 --> 00:05:40.420
So integrate from 0, from x is
equal to 0, to x is equal to 2.

00:05:40.420 --> 00:05:43.740
If I haven't made a mistake,
I can hit enter, and let

00:05:43.740 --> 00:05:46.100
the calculator do the
rest of the work.

00:05:46.100 --> 00:05:49.780
Let's see what it ends up with.

00:05:49.780 --> 00:05:54.460
OK, 9. -- that's the answer,
that is the volume of this

00:05:54.460 --> 00:06:00.110
solid -- it's 9.9783 So you
could write that this

00:06:00.110 --> 00:06:05.130
is equal to 9.9783.

00:06:05.130 --> 00:06:07.630
And I'm pretty sure they want
you to use a calculator,

00:06:07.630 --> 00:06:11.580
because frankly, computing the
integral, that's kind of just

00:06:11.580 --> 00:06:14.870
chug math, you know, very
mechanical math, although it's

00:06:14.870 --> 00:06:17.220
pretty sophisticated, but
it would take you forever.

00:06:17.220 --> 00:06:19.860
But this is kind of, I think,
what they wanted you to do, set

00:06:19.860 --> 00:06:23.810
up the integral, recognize that
each of the squares, the area

00:06:23.810 --> 00:06:26.790
of each of the squares is just
going to be this distance, the

00:06:26.790 --> 00:06:29.060
distance between the
functions squared.

00:06:29.060 --> 00:06:32.470
And then you integrate
that from 0 to 2.

00:06:32.470 --> 00:06:34.830
Let's see how much
time I have left.

00:06:34.830 --> 00:06:35.760
I have a couple of minutes.

00:06:35.760 --> 00:06:37.920
So let's do part d.

00:06:45.360 --> 00:06:52.620
The region r models, let me,
paste, oh, I didn't want to

00:06:52.620 --> 00:07:02.030
do that, edit, undo, edit,
paste, there you go.

00:07:02.030 --> 00:07:05.510
OK, so I wanted to make
this a little smaller.

00:07:05.510 --> 00:07:06.610
So what does part d say?

00:07:06.610 --> 00:07:09.760
The region r models the
surface of a small pond,

00:07:09.760 --> 00:07:11.420
so that's the surface now.

00:07:11.420 --> 00:07:17.880
At all points r at a distance x
from the y-axis, the depth of

00:07:17.880 --> 00:07:22.460
the water is given by h
of x is equal to 3-x.

00:07:22.460 --> 00:07:27.710
So essentially 3-x is
the depth, right?

00:07:27.710 --> 00:07:33.890
So at this point of this pond,
the depth is just 3, right 3-0.

00:07:33.890 --> 00:07:39.260
And at this point, the
depth is 3-2 which is 1.

00:07:39.260 --> 00:07:41.650
So essentially the pond is
going to get shallower and

00:07:41.650 --> 00:07:45.750
shallower or as we go
further to the right.

00:07:45.750 --> 00:07:47.640
You could almost imagine
it, let me see if I

00:07:47.640 --> 00:07:50.900
can draw it again.

00:07:50.900 --> 00:07:56.980
So this is the sin function
with some perspective.

00:07:56.980 --> 00:08:01.700
This is the polynomial
function below it.

00:08:01.700 --> 00:08:06.610
This is, it's probably drawn
in, that's the x-axis.

00:08:10.110 --> 00:08:12.480
This is the y-axis.

00:08:18.980 --> 00:08:24.460
And so here, the depth of the
pond is given by the function

00:08:24.460 --> 00:08:28.580
h of x is equal to 3-x.

00:08:28.580 --> 00:08:34.826
So over here, the depth is 3,
so if I were to draw, so if I

00:08:34.826 --> 00:08:36.790
were to go straight down the
depth is, you know

00:08:36.790 --> 00:08:38.110
maybe it's 3.

00:08:38.110 --> 00:08:41.140
And the pond essentially gets
shallower and shallower

00:08:41.140 --> 00:08:44.070
as we go to the right.

00:08:44.070 --> 00:08:48.110
So how do we figure out
the volume of this?

00:08:48.110 --> 00:08:49.880
Over here, what is the depth?

00:08:49.880 --> 00:08:52.100
It's going to be 1, right, 3-2.

00:08:52.100 --> 00:08:54.370
This is x is equal to 2.

00:08:54.370 --> 00:08:56.580
So here the depth
is going to be 1.

00:08:56.580 --> 00:09:00.890
So if we take the cross-section
just along the x-axis, the

00:09:00.890 --> 00:09:03.290
depth is going to look
something like this, but then

00:09:03.290 --> 00:09:04.190
this is the top of it.

00:09:04.190 --> 00:09:06.100
I know that's kind of
hard to visualize.

00:09:06.100 --> 00:09:11.680
But anyway, how do we
figure out what the

00:09:11.680 --> 00:09:13.500
volume of this lake is?

00:09:13.500 --> 00:09:15.840
But actually, I realize I'm
pushing nine minutes, so I

00:09:15.840 --> 00:09:17.750
will continue this
in the next video.

00:09:17.750 --> 00:09:19.252
See you soon.


AP Calculus BC exams: 2008 1 b&c

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:04.560
Let's keep doing the first
problem from the 2008

00:00:04.560 --> 00:00:06.030
Calculus BC exam.

00:00:06.030 --> 00:00:07.400
We're on Part b.

00:00:11.790 --> 00:00:13.640
I think that's too thick.

00:00:13.640 --> 00:00:18.970
OK it says, the horizontal line
y equals negative 2 splits

00:00:18.970 --> 00:00:22.460
the region r-- this is
r-- into two parts.

00:00:22.460 --> 00:00:26.340
Write, but do not evaluate, an
integral expression for the

00:00:26.340 --> 00:00:30.780
area of the part of r that is
below the horizontal lines.

00:00:30.780 --> 00:00:32.980
Let's draw y is equal
to negative 2.

00:00:32.980 --> 00:00:37.500
So y equals negative 2
would look like this.

00:00:37.500 --> 00:00:39.460
y is just a constant.

00:00:39.460 --> 00:00:40.350
That's not thick enough.

00:00:40.350 --> 00:00:41.300
I don't know if
you can see that.

00:00:41.300 --> 00:00:46.190
Let me do it as a thick line
and in a darker color maybe.

00:00:46.190 --> 00:00:51.300
y is equal to negative 2 will
look something like that.

00:00:51.300 --> 00:00:53.740
And what they're saying is if
it splits this region r into

00:00:53.740 --> 00:00:56.550
two parts, this part
and this part.

00:00:56.550 --> 00:01:00.260
And what they want to know is
an integral expression for the

00:01:00.260 --> 00:01:04.250
area of the part of r that is
below the horizontal line.

00:01:04.250 --> 00:01:09.670
So they care about the
area of this part of r.

00:01:09.670 --> 00:01:12.570
And remember they just want
us to write the expression,

00:01:12.570 --> 00:01:13.600
not evaluate it.

00:01:13.600 --> 00:01:15.470
So that'll hopefully
save us time.

00:01:15.470 --> 00:01:17.880
So how do we figure this out?

00:01:17.880 --> 00:01:20.480
Well the easy part is actually
to figure out what the

00:01:20.480 --> 00:01:22.600
expression we're going to take
the definite integral of.

00:01:22.600 --> 00:01:25.250
And it's going to be a little
bit harder to figure out

00:01:25.250 --> 00:01:26.570
the boundary points.

00:01:26.570 --> 00:01:30.510
So what is the expression
within the definite

00:01:30.510 --> 00:01:31.510
integral that we will use?

00:01:31.510 --> 00:01:34.290
Well, just like we did in Part
a, think about we're going to

00:01:34.290 --> 00:01:37.600
take a sum of a bunch
of rectangles.

00:01:37.600 --> 00:01:39.530
And the height of the
rectangles is going to be

00:01:39.530 --> 00:01:41.795
the difference between
the two functions.

00:01:44.410 --> 00:01:47.330
And this is y is equal to
negative 2, and then this

00:01:47.330 --> 00:01:50.680
function right here is y is
equal to-- and we had written

00:01:50.680 --> 00:01:54.680
it down in Part 3-- but
that's x cubed minus 4x.

00:01:54.680 --> 00:01:57.310
That's this curve right here.

00:01:57.310 --> 00:02:00.780
So the height of each of these
little rectangles is going to

00:02:00.780 --> 00:02:10.480
be minus 2 minus x to
the third minus 4x.

00:02:10.480 --> 00:02:12.150
That's the height of each
of these rectangles.

00:02:12.150 --> 00:02:15.630
And then the width of each of
those rectangles we know from--

00:02:15.630 --> 00:02:18.760
well, just learning calculus or
learning integrals--

00:02:18.760 --> 00:02:19.420
the width is dx.

00:02:19.420 --> 00:02:21.600
So we're going to
multiply that times dx.

00:02:21.600 --> 00:02:24.300
And then we're going to take
all of the sums from x is equal

00:02:24.300 --> 00:02:27.460
to whatever this point is
to whatever this point is.

00:02:27.460 --> 00:02:31.550
So we need to figure out this
point, this value of f, which

00:02:31.550 --> 00:02:34.600
is going to be here, and then
this value of x, which

00:02:34.600 --> 00:02:36.350
is going to be there.

00:02:36.350 --> 00:02:41.210
And so these are really just
two of the points where

00:02:41.210 --> 00:02:43.390
these two functions
intersect each other.

00:02:43.390 --> 00:02:44.860
So how do we figure
out those points?

00:02:44.860 --> 00:02:49.790
Well what we could do is we can
set them equal to each other,

00:02:49.790 --> 00:02:54.640
so we could say at what x
values does x to the third

00:02:54.640 --> 00:03:00.260
minus 4x equal minus 2.

00:03:00.260 --> 00:03:03.880
At what x values are
the y values the same?

00:03:03.880 --> 00:03:07.120
So we just set them
equal to each other.

00:03:07.120 --> 00:03:11.220
If we wanted to write this is a
proper polynomial expression,

00:03:11.220 --> 00:03:18.650
we would get x to the third
minus 4x plus 2 is equal to 0.

00:03:18.650 --> 00:03:22.940
And I actually just tried to
record a video where I was

00:03:22.940 --> 00:03:25.120
doing this on the fly, and I
kept staring at this and I was

00:03:25.120 --> 00:03:28.080
like, boy, this is a hard
polynomial to factor.

00:03:28.080 --> 00:03:31.220
I kept trying to guess numbers,
or figuring out-- I even tried

00:03:31.220 --> 00:03:33.670
to do Newton's method-- and I
kept getting weird numbers, and

00:03:33.670 --> 00:03:35.830
I became suspicious of myself.

00:03:35.830 --> 00:03:38.410
And then I looked at the
actual test-- actually

00:03:38.410 --> 00:03:39.350
I can show it to you.

00:03:42.840 --> 00:03:46.940
It says right there, a graphing
calculator is required for some

00:03:46.940 --> 00:03:49.390
problems or parts of problems.

00:03:49.390 --> 00:03:51.830
And I realized that they
probably want us to use a

00:03:51.830 --> 00:03:53.920
graphing calculator to
figure out the roots

00:03:53.920 --> 00:03:55.290
of this polynomial.

00:03:55.290 --> 00:03:56.550
So let's do that.

00:03:56.550 --> 00:04:00.150
It's been a long time since I
actually took AP Calculus, and

00:04:00.150 --> 00:04:05.250
now I remember that a graphing
calculator was a big deal.

00:04:05.250 --> 00:04:08.425
I just actually downloaded
this TI-85 emulator.

00:04:08.425 --> 00:04:11.940
So let's use this to figure out
the roots of this polynomial.

00:04:11.940 --> 00:04:13.130
Let's turn it on.

00:04:16.230 --> 00:04:18.040
If we want to figure out
the roots we use the poly

00:04:18.040 --> 00:04:21.900
function, so second poly.

00:04:21.900 --> 00:04:23.540
What's the order of
this polynomial?

00:04:23.540 --> 00:04:25.550
What's the third-degree
polynomial?

00:04:25.550 --> 00:04:26.670
f of x to the third.

00:04:26.670 --> 00:04:30.110
So order is 3, enter.

00:04:30.110 --> 00:04:31.980
And what are the coefficients?

00:04:31.980 --> 00:04:36.302
Well the coefficient on x
to the third term is 1.

00:04:36.302 --> 00:04:37.880
Go down.

00:04:37.880 --> 00:04:40.960
What's the coefficient
on the x squared term?

00:04:40.960 --> 00:04:43.490
Well there is no x
squared term, right?

00:04:43.490 --> 00:04:45.910
So that coefficient is 0.

00:04:45.910 --> 00:04:47.260
Go down.

00:04:47.260 --> 00:04:49.610
What's the coefficient
on the x term?

00:04:49.610 --> 00:04:52.690
It's minus 4.

00:04:52.690 --> 00:04:55.420
So minus 4.

00:04:55.420 --> 00:04:56.980
Go down again.

00:04:56.980 --> 00:04:59.720
And then the coefficient
or the constant term.

00:04:59.720 --> 00:05:02.390
Well that's just going to be 2.

00:05:02.390 --> 00:05:04.403
And now we can just hit solve.

00:05:08.440 --> 00:05:11.820
And we get three crazy numbers,
and this shows you that this

00:05:11.820 --> 00:05:16.440
would have been very hard to
solve analytically if you

00:05:16.440 --> 00:05:17.490
have a normal brain.

00:05:17.490 --> 00:05:19.260
So let's see.

00:05:19.260 --> 00:05:25.220
There are three places where y
equals negative 2 intersects y

00:05:25.220 --> 00:05:27.645
is equal to x to the
third minus 4x.

00:05:27.645 --> 00:05:31.090
It intersects at minus 2.21.

00:05:31.090 --> 00:05:33.650
Well that's off of this
graph, that's not even here.

00:05:33.650 --> 00:05:35.290
That's somewhere
off to the left.

00:05:35.290 --> 00:05:37.520
This curve probably comes
back down and intersects

00:05:37.520 --> 00:05:39.040
over there at minus 2.

00:05:39.040 --> 00:05:45.680
But it also intersects at 1.675
which is probably right here.

00:05:45.680 --> 00:05:47.980
Right, that looks like 1.675.

00:05:47.980 --> 00:05:51.660
It also intersects at 0.539,
which is right there.

00:05:51.660 --> 00:05:57.100
So we can use those values that
our graphing calculator gave us

00:05:57.100 --> 00:05:59.740
and put it into our
definite integral.

00:05:59.740 --> 00:06:06.350
So this point right here our
polynomial solver told us

00:06:06.350 --> 00:06:10.526
is x is equal to 0.539.

00:06:10.526 --> 00:06:16.560
So we'll put here 0.539.

00:06:16.560 --> 00:06:19.360
And then this point right
here-- so this is our limits

00:06:19.360 --> 00:06:21.152
of integration for our
definite integral, right?

00:06:21.152 --> 00:06:25.193
We're going to sum up these
little rectangles from x is

00:06:25.193 --> 00:06:35.170
equal to 0.539 to x
is equal to 1.675.

00:06:35.170 --> 00:06:37.585
And they told us that they do
not want us to evaluate it,

00:06:37.585 --> 00:06:39.140
so we are done with Part b.

00:06:39.140 --> 00:06:41.500
We could just write this
and we should hopefully

00:06:41.500 --> 00:06:43.410
get full credit.

00:06:43.410 --> 00:06:45.250
Maybe they'd want you to
simplify this a little bit,

00:06:45.250 --> 00:06:48.040
but I'd be surprised if they
were mark off for that.

00:06:48.040 --> 00:06:52.240
Anyway, let's do Part c.

00:06:52.240 --> 00:06:54.510
If we have time.

00:06:54.510 --> 00:06:56.350
Part c.

00:06:56.350 --> 00:06:59.490
So they say the region r
is the base of a solid.

00:06:59.490 --> 00:07:02.570
For this solid each
cross-section perpendicular

00:07:02.570 --> 00:07:04.650
to the x-axis is a square.

00:07:04.650 --> 00:07:05.780
Find the volume of the solid.

00:07:05.780 --> 00:07:08.310
OK, so this is interesting.

00:07:08.310 --> 00:07:09.700
Let me see if I can draw it.

00:07:09.700 --> 00:07:11.610
So that curve, I'm going to
draw it kind of with a

00:07:11.610 --> 00:07:14.650
perspective so you can see the
solid they're talking about.

00:07:14.650 --> 00:07:16.620
So it's the same region r.

00:07:16.620 --> 00:07:19.320
So we had a sin
function on the top.

00:07:19.320 --> 00:07:20.720
Looks something like that.

00:07:20.720 --> 00:07:24.150
And then we had that polynomial
function on the bottom that

00:07:24.150 --> 00:07:26.310
looks something like that.

00:07:26.310 --> 00:07:29.260
I'm trying to draw
it at an angle.

00:07:29.260 --> 00:07:31.270
So just to show you the x-axis.

00:07:34.760 --> 00:07:36.220
This is going to be the x-axis.

00:07:44.460 --> 00:07:47.660
Let me draw the y-axis.

00:07:47.660 --> 00:07:51.110
The way I drew it now it
looks something like this.

00:07:51.110 --> 00:07:53.710
Trying to do a little bit of
perspective so that we can

00:07:53.710 --> 00:07:55.910
visualize what they're
talking about.

00:07:58.650 --> 00:08:00.300
So that's the y-axis.

00:08:00.300 --> 00:08:02.310
So that's x y.

00:08:02.310 --> 00:08:04.720
And what they're saying, this
is the region r again, just

00:08:04.720 --> 00:08:08.330
like in the previous two
parts of the problem.

00:08:08.330 --> 00:08:10.330
They say the region r is
the base of a solid.

00:08:10.330 --> 00:08:11.890
So this is the base of a solid.

00:08:11.890 --> 00:08:14.630
For this solid each
cross-section perpendicular

00:08:14.630 --> 00:08:15.490
to the x-axis.

00:08:15.490 --> 00:08:18.470
So a cross-section
perpendicular to the x-axis.

00:08:18.470 --> 00:08:20.210
Let's see if we can draw that.

00:08:20.210 --> 00:08:22.310
So this would be
a cross-section.

00:08:22.310 --> 00:08:25.100
So it's like we took a knife
and we cut like this, we cut

00:08:25.100 --> 00:08:26.140
parallel to the y-axis.

00:08:30.270 --> 00:08:34.410
Say we took this cross-section
of this solid right now.

00:08:34.410 --> 00:08:36.160
They say that it is a square.

00:08:36.160 --> 00:08:39.350
So that means that the base has
to be the same height, has

00:08:39.350 --> 00:08:42.290
to be the same disk
size as the height.

00:08:42.290 --> 00:08:44.460
So if we took the cross-section
of the solid here it

00:08:44.460 --> 00:08:46.100
would be like that.

00:08:46.100 --> 00:08:49.720
Here it would be a smaller
square like that.

00:08:49.720 --> 00:08:51.810
If we took the cross-section
there, it would be a

00:08:51.810 --> 00:08:53.080
small square as well.

00:08:59.140 --> 00:09:00.645
So what they want us to
do is figure out the

00:09:00.645 --> 00:09:01.170
volume of the solid.

00:09:01.170 --> 00:09:02.500
You can kind of imagine
what it looks like.

00:09:02.500 --> 00:09:04.920
It's small squares and the
squares get really big and

00:09:04.920 --> 00:09:05.840
then they get small again.

00:09:05.840 --> 00:09:07.470
So how do we do that?

00:09:07.470 --> 00:09:09.110
Well we do the same thing.

00:09:09.110 --> 00:09:13.920
We take the area of each of
these squares-- we know they're

00:09:13.920 --> 00:09:20.920
squares-- times each of the
dx's-- the small differential--

00:09:20.920 --> 00:09:26.200
and we sum them up
over from 0 to 2.

00:09:26.200 --> 00:09:28.440
On the first diagram,
I think that was 2.

00:09:28.440 --> 00:09:29.860
Oh, and I'm about
to be out of time.

00:09:29.860 --> 00:09:32.150
So I will continue this
problem in the next video.

00:09:32.150 --> 00:09:32.480
See you soon.


Basic subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:05.700
Welcome to the video
on basic subtraction.

00:00:05.700 --> 00:00:10.340
Let's do a little bit a review
of basic addition first.

00:00:10.340 --> 00:00:15.570
If I said 4 plus 3,
what did this mean?

00:00:19.050 --> 00:00:20.700
What did that equal?

00:00:20.700 --> 00:00:22.940
Well, there were a couple of
ways we could have viewed this.

00:00:22.940 --> 00:00:24.640
We could have said I
had 4 of something.

00:00:24.640 --> 00:00:28.100
Let's say I had 4 circles
or I don't know, I had

00:00:28.100 --> 00:00:30.160
4 lemons for breakfast.

00:00:30.160 --> 00:00:36.180
So 1, 2, 3, 4 lemons
for breakfast.

00:00:36.180 --> 00:00:40.760
And let's say, I had another
3 lemons for lunch.

00:00:40.760 --> 00:00:46.000
1, 2, 3, and so you could
view 4 plus 3 as how many

00:00:46.000 --> 00:00:47.750
total lemons did I have?

00:00:47.750 --> 00:00:49.640
I'm adding 3 to 4.

00:00:49.640 --> 00:00:50.900
So how many total did I have?

00:00:50.900 --> 00:00:55.360
Well, it's 1, 2, 3, 4, 5, 6, 7.

00:00:55.360 --> 00:00:59.190
So I had a total of 7 lemons.

00:00:59.190 --> 00:01:01.440
Another way we could have
viewed that is we could have

00:01:01.440 --> 00:01:04.060
drawn our number line.

00:01:04.060 --> 00:01:07.240
And I'll draw it in yellow
because-- now that's

00:01:07.240 --> 00:01:09.720
not wide enough.

00:01:09.720 --> 00:01:13.710
I'll draw it in yellow because
we're talking about lemons.

00:01:13.710 --> 00:01:15.355
So let's say that's
our number line.

00:01:18.070 --> 00:01:21.350
And if I start at the number--
let me draw all of the numbers.

00:01:21.350 --> 00:01:32.020
0, 1, 2, 3, 4, 5, 6, 7.

00:01:32.020 --> 00:01:34.470
So you could think about
this as saying, well,

00:01:34.470 --> 00:01:35.380
we're on the number line.

00:01:35.380 --> 00:01:37.950
We start at the number 4.

00:01:37.950 --> 00:01:39.130
That's this number 4.

00:01:39.130 --> 00:01:40.700
And we're adding 3 to it.

00:01:40.700 --> 00:01:44.170
So we'll increase, along
the number line by 3.

00:01:44.170 --> 00:01:48.650
So we'll go 1, 2, 3
and you end up at 7.

00:01:48.650 --> 00:01:51.920
So you could say if I have 4
and I get 3 more, I get 7.

00:01:51.920 --> 00:01:57.130
Or if I increase 4
by 3 I also get 7.

00:01:57.130 --> 00:01:59.350
So what's subtraction now?

00:01:59.350 --> 00:02:00.450
Because that's what
this video is about.

00:02:00.450 --> 00:02:03.030
I shouldn't waste all of our
time talking about addition.

00:02:03.030 --> 00:02:05.920
So let's take the
example of 4 minus 3.

00:02:05.920 --> 00:02:08.500
What is that equal to?

00:02:08.500 --> 00:02:11.540
Let me switch colors just to
keep things interesting.

00:02:11.540 --> 00:02:18.860
So what is 4 minus 3 equal to?

00:02:18.860 --> 00:02:23.050
Subtraction or minus is
the opposite of addition.

00:02:23.050 --> 00:02:26.990
So in addition you're doing
something more, you're adding.

00:02:26.990 --> 00:02:29.850
I don't want to use the word
adding to define addition, but

00:02:29.850 --> 00:02:31.040
that's what you're doing.

00:02:31.040 --> 00:02:33.890
I had 4 lemons and
then I had 3 more.

00:02:33.890 --> 00:02:36.280
In subtraction
you're taking away.

00:02:36.280 --> 00:02:42.916
So this example, if I started
with 4 lemons, let's say I

00:02:42.916 --> 00:02:45.210
had 4 lemons on a plate.

00:02:45.210 --> 00:02:50.070
If I'm subtracting 3, if I'm
saying minus 3, instead of

00:02:50.070 --> 00:02:53.720
adding these 3 and getting 7,
I'm going to take 3 away.

00:02:53.720 --> 00:02:55.910
So maybe I'm eating them
or maybe I'm giving them

00:02:55.910 --> 00:02:59.760
to you in exchange for
watching these videos.

00:02:59.760 --> 00:03:04.260
So to take away 3 from this 4,
let's say this one goes away,

00:03:04.260 --> 00:03:06.910
this one goes away, and
this one goes away.

00:03:06.910 --> 00:03:09.900
How many lemons
would we have left?

00:03:09.900 --> 00:03:11.820
Well, this is the only one
that I haven't crossed out.

00:03:11.820 --> 00:03:13.985
So we would have 1 lemon left.

00:03:16.570 --> 00:03:18.120
And this would be the
lemon that's left.

00:03:18.120 --> 00:03:19.440
Didn't have to be that
one, I could have crossed

00:03:19.440 --> 00:03:21.210
out any of the 3.

00:03:21.210 --> 00:03:24.520
Another way to view that,
let's draw the same lemon

00:03:24.520 --> 00:03:28.045
colored number line.

00:03:28.045 --> 00:03:30.810
Let's say that this is the
number line right here.

00:03:34.400 --> 00:03:36.180
And I'll draw all
the same numbers.

00:03:36.180 --> 00:03:47.240
So that's 0, 1, 2,
3, 4, 5, 6, 7.

00:03:47.240 --> 00:03:49.530
Of course, the number
line keeps going.

00:03:49.530 --> 00:03:51.145
There's no largest number.

00:03:51.145 --> 00:03:55.190
And you can imagine, any number
that you can think of, I can

00:03:55.190 --> 00:03:56.580
think of one higher than that.

00:03:56.580 --> 00:03:57.670
So there is no largest number.

00:03:57.670 --> 00:03:59.040
So that's why we draw
that arrow there.

00:03:59.040 --> 00:04:00.730
I could never draw the
entire number line.

00:04:00.730 --> 00:04:03.020
But anyway, back
to subtraction.

00:04:03.020 --> 00:04:06.710
So we're starting at
4 lemons, right?

00:04:06.710 --> 00:04:12.130
When we added 3-- plus 3-- we
went to the right 4 spaces

00:04:12.130 --> 00:04:12.840
on the number line.

00:04:12.840 --> 00:04:15.090
And that's because the
right is increasing value.

00:04:15.090 --> 00:04:16.270
So we went from 4 to 5.

00:04:16.270 --> 00:04:17.450
That was one more.

00:04:17.450 --> 00:04:20.990
5 to 6 was two more,
and 7 was 3 more.

00:04:20.990 --> 00:04:23.510
Now we're taking away from 4.

00:04:23.510 --> 00:04:24.860
So what do we do?

00:04:24.860 --> 00:04:26.690
What would you think we do?

00:04:26.690 --> 00:04:29.580
Well, since we're taking away
we're going to decrease the

00:04:29.580 --> 00:04:31.790
total number of lemons we have.

00:04:31.790 --> 00:04:33.680
So we take away
one, we get to 3.

00:04:33.680 --> 00:04:35.550
Take away two, get to 2.

00:04:35.550 --> 00:04:38.910
Take away three, we took
away three, right?

00:04:38.910 --> 00:04:41.980
So we'll go back 1, 2, 3
along the number line

00:04:41.980 --> 00:04:44.210
and we'll end up at 1.

00:04:44.210 --> 00:04:47.340
And that's this 1 right here.

00:04:47.340 --> 00:04:52.340
So just to review, addition is
you're doing more or something.

00:04:52.340 --> 00:04:55.120
Subtraction is you
are taking away.

00:04:55.120 --> 00:04:58.180
If you think about it on the
number line, addition is

00:04:58.180 --> 00:05:00.740
increasing along the number
line by that amount.

00:05:00.740 --> 00:05:04.200
So in this case we increased
along the number line by 3.

00:05:04.200 --> 00:05:06.340
And so we went from 4 to 7.

00:05:06.340 --> 00:05:10.170
In the subtraction case
we decrease back on

00:05:10.170 --> 00:05:11.350
the number line.

00:05:11.350 --> 00:05:13.550
So we decrease by the amount
that you're subtracting.

00:05:13.550 --> 00:05:17.480
So in this case we
decreased by 3.

00:05:17.480 --> 00:05:21.180
We went back 1, 2,
3 and we had 1.

00:05:21.180 --> 00:05:22.740
And the other way to view it,
if I have 4 of something.

00:05:22.740 --> 00:05:26.580
If I give 3 away or if I ate
3 of them or I don't know

00:05:26.580 --> 00:05:27.680
what I did to 3 of them.

00:05:27.680 --> 00:05:31.050
If I lost 3 of them I
would have 1 left.

00:05:31.050 --> 00:05:33.260
Now let me show you some
interesting things

00:05:33.260 --> 00:05:34.840
about subtraction.

00:05:34.840 --> 00:05:38.510
So we know that 4 minus
3 is equal to 1.

00:05:38.510 --> 00:05:40.900
Let me show you something
else that's interesting.

00:05:40.900 --> 00:05:46.050
What is 4 minus 1?

00:05:46.050 --> 00:05:49.510
Well, we could use
either example.

00:05:49.510 --> 00:05:51.440
Let's do the lemon example.

00:05:51.440 --> 00:05:54.130
Let's do apples now; I'm
bored of the lemons.

00:05:54.130 --> 00:06:01.160
Let's say I 1, 2, 3, 4.

00:06:01.160 --> 00:06:03.620
I have a new pen, sometimes
it doesn't draw right.

00:06:03.620 --> 00:06:05.390
Let's say I had 4 apples.

00:06:05.390 --> 00:06:07.140
This is the example
we're dealing with.

00:06:07.140 --> 00:06:09.450
And I were to eat one of them.

00:06:09.450 --> 00:06:10.830
So one of them were to go away.

00:06:10.830 --> 00:06:12.640
How many apples
would I have left?

00:06:12.640 --> 00:06:15.560
Well, 3-- 1, 2, 3.

00:06:15.560 --> 00:06:18.600
So 4 minus 1 is equal to 3.

00:06:18.600 --> 00:06:21.640
And if we did it on the number
line, if we started at 4 and we

00:06:21.640 --> 00:06:23.780
subtracted 1-- we took 1 away.

00:06:23.780 --> 00:06:26.020
So we're going to
become one smaller.

00:06:26.020 --> 00:06:28.380
We go back one, we get 3.

00:06:28.380 --> 00:06:29.570
Either way works.

00:06:29.570 --> 00:06:30.820
But isn't this interesting?

00:06:30.820 --> 00:06:37.520
4 minus 3 is equal to 1 and
4 minus 1 is equal to 3.

00:06:37.520 --> 00:06:41.070
You might say, well, did I pick
the numbers just so it worked?

00:06:41.070 --> 00:06:42.790
Well, it turns out that
it's always true.

00:06:45.360 --> 00:06:47.440
Well, I don't want to get too
technical, but we're already

00:06:47.440 --> 00:06:50.860
touching on something that
you'll learn later, maybe

00:06:50.860 --> 00:06:52.210
in algebra one day.

00:06:52.210 --> 00:06:54.840
But actually I don't want to
go into all that right now.

00:06:54.840 --> 00:06:56.460
So where does this come from?

00:06:56.460 --> 00:07:02.170
Well, this is also based on
the fact that 3 plus 1-- I

00:07:02.170 --> 00:07:03.660
didn't want to confuse you.

00:07:03.660 --> 00:07:05.700
So I apologize if I did.

00:07:05.700 --> 00:07:07.440
But I'll show you another
interesting thing.

00:07:07.440 --> 00:07:10.610
What is 3 plus 1?

00:07:10.610 --> 00:07:12.710
3 plus 1 is equal to what?

00:07:12.710 --> 00:07:13.370
Well, that's easy.

00:07:13.370 --> 00:07:15.400
You know that from
basic addition.

00:07:15.400 --> 00:07:20.720
You can start on the number
line at 3 and add 1 do it.

00:07:20.720 --> 00:07:22.000
And where do you end up?

00:07:22.000 --> 00:07:23.310
You end up at 4.

00:07:23.310 --> 00:07:25.060
3 plus 1 is equal to 4.

00:07:25.060 --> 00:07:29.420
Or you could have started at 1
on the number line and added 3.

00:07:29.420 --> 00:07:33.160
1, 2, 3 and you would
have also ended up at 4.

00:07:33.160 --> 00:07:36.690
So we also know that you could
have switch this either way.

00:07:36.690 --> 00:07:38.880
Both of those are equal 4.

00:07:38.880 --> 00:07:40.140
What do you see here?

00:07:40.140 --> 00:07:42.630
Well, there's a bunch of things
I've written here and they all

00:07:42.630 --> 00:07:43.930
kind of relate to each other.

00:07:43.930 --> 00:07:45.660
1 plus 3 is equal to 4.

00:07:45.660 --> 00:07:47.780
3 plus 1 is equal to 4.

00:07:47.780 --> 00:07:49.740
4 minus 1 is 3.

00:07:49.740 --> 00:07:53.920
Essentially, 4 minus 1 and
getting 3 is the exact same--

00:07:53.920 --> 00:07:59.200
you're saying the same thing
as 3 plus 1 is equal to 4.

00:07:59.200 --> 00:08:03.170
This statement says if
I add 1 to 3 I get 4.

00:08:03.170 --> 00:08:08.120
This is saying if I take
away 1 from 4 I get 3.

00:08:08.120 --> 00:08:10.600
So if I start at 4,
move back, I get 3.

00:08:10.600 --> 00:08:15.280
This is saying if I start at
3, and I move up 1, I get 4.

00:08:15.280 --> 00:08:17.600
Hopefully that gives you a
little bit of intuition

00:08:17.600 --> 00:08:19.670
about what subtraction is.

00:08:19.670 --> 00:08:24.870
In the next video I'll just do
as many basic subtraction

00:08:24.870 --> 00:08:27.800
problems I can do
in 10 minutes.

00:08:27.800 --> 00:08:30.030
And then you'll be ready
to do the exercises.

00:08:30.030 --> 00:08:31.532
See you soon.


Definite integral with substitution

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.430
So I've been sent this definite
integral problem and it seemed

00:00:03.430 --> 00:00:06.270
as good as any, and I think the
key with this is just to

00:00:06.270 --> 00:00:07.390
see a lot of examples.

00:00:07.390 --> 00:00:08.820
So let's do it.

00:00:08.820 --> 00:00:19.290
This definite integral is from
pi over 2 to pi of minus cosine

00:00:19.290 --> 00:00:27.610
squared of x times sin of x dx.

00:00:27.610 --> 00:00:29.770
So before we just chug
through the math and do the

00:00:29.770 --> 00:00:33.510
antiderivatives and use the
fundamental theorem of calculus

00:00:33.510 --> 00:00:35.970
to evaluate the definite
integral, let's think about

00:00:35.970 --> 00:00:37.170
what we're even doing.

00:00:37.170 --> 00:00:39.770
So I've graphed this function
right here, minus cosine

00:00:39.770 --> 00:00:43.940
squared of x times sin of x.

00:00:43.940 --> 00:00:45.980
And what we care about,
we're defining the definite

00:00:45.980 --> 00:00:49.670
integral between pi over 2,
which is roughly here.

00:00:49.670 --> 00:00:53.490
Let me see if I can make
this a little bit bigger.

00:00:53.490 --> 00:00:57.670
So between pi over 2 which is
right there, and between pi.

00:00:57.670 --> 00:01:00.190
So the definite integral of
this function between here and

00:01:00.190 --> 00:01:03.970
here is essentially the area of
the curve between the

00:01:03.970 --> 00:01:04.670
curve and the x-axis.

00:01:04.670 --> 00:01:08.370
And since the a curve is below
the x-axis here, this area is

00:01:08.370 --> 00:01:09.870
going to be a negative number.

00:01:09.870 --> 00:01:11.400
So that gives us
immediately an intuition.

00:01:11.400 --> 00:01:13.520
We should be getting a negative
number when we evaluate this

00:01:13.520 --> 00:01:17.160
and just to prove this I
actually typed it in up here.

00:01:17.160 --> 00:01:20.540
So let's now evaluate
this definite integral.

00:01:23.320 --> 00:01:26.400
Now I'll rearrange some of the
terms here just to make it a

00:01:26.400 --> 00:01:27.940
little bit easier to read.

00:01:27.940 --> 00:01:29.860
But the way I always think
about it is, well I have

00:01:29.860 --> 00:01:31.760
a cosine and a sin.

00:01:31.760 --> 00:01:34.060
The cosine is squared, so all
these crazy things are

00:01:34.060 --> 00:01:39.770
happening to it, so it seems
like I could use substitution

00:01:39.770 --> 00:01:42.320
or the reverse chain
rules some out here.

00:01:42.320 --> 00:01:43.820
And what was the chain rule?

00:01:43.820 --> 00:01:52.060
The chain rule said if I took
the derivative of f of g of x

00:01:52.060 --> 00:02:01.750
that this is equal to f prime
of g of x times g prime of x.

00:02:01.750 --> 00:02:04.610
That might completely confuse
you, but I just wrote that here

00:02:04.610 --> 00:02:09.160
because we could say, well what
if g of x is cosine of x. f

00:02:09.160 --> 00:02:12.540
prime of g of x is the cosine
of x squared, and then the

00:02:12.540 --> 00:02:14.875
derivative of g of x or the
derivative of cosine

00:02:14.875 --> 00:02:16.080
of x is sin of x.

00:02:16.080 --> 00:02:18.270
Well it's actually minus sin of
x, and we have a minus sin

00:02:18.270 --> 00:02:20.030
here, so that works
out well too.

00:02:20.030 --> 00:02:21.760
If this confuse you, ignore it.

00:02:21.760 --> 00:02:23.260
Well essentially we're just
going to do the same thing,

00:02:23.260 --> 00:02:24.470
but we're going to do
it with substitution.

00:02:24.470 --> 00:02:26.950
So let me do it
with substitution.

00:02:26.950 --> 00:02:29.985
Let me a erase this if
this confuses people.

00:02:33.360 --> 00:02:38.580
I want to do it however it
is least confusing to you.

00:02:38.580 --> 00:02:40.260
OK let me erase that.

00:02:40.260 --> 00:02:43.190
Actually, let me do it with
substitution, just because the

00:02:43.190 --> 00:02:46.540
way I was just doing is kind
of my shortcut back in the

00:02:46.540 --> 00:02:48.530
day when I was a mathlete.

00:02:48.530 --> 00:02:51.530
But it's good to be able to do
it with substitution, helps you

00:02:51.530 --> 00:02:53.000
from making careless mistakes.

00:02:53.000 --> 00:02:54.280
So let me rewrite
this first of all.

00:02:54.280 --> 00:02:58.700
This is the same thing as the
integral from pi over 2 to

00:02:58.700 --> 00:03:04.670
pi of cosine squared of x.

00:03:04.670 --> 00:03:09.340
Actually let me write that
as cosine of x squared.

00:03:09.340 --> 00:03:11.810
Same thing, right.

00:03:11.810 --> 00:03:18.730
Times minus sin of x dx.

00:03:18.730 --> 00:03:20.350
And now it should
be clearer to you.

00:03:20.350 --> 00:03:22.330
What's the derivative
of cosine of x?

00:03:22.330 --> 00:03:23.710
It's minus sin of x.

00:03:23.710 --> 00:03:26.020
So I have a function and it's
being squared, and I have its

00:03:26.020 --> 00:03:30.310
derivative, so I can figure out
its antiderivative by using

00:03:30.310 --> 00:03:31.855
substitution or the
reverse chain rule.

00:03:31.855 --> 00:03:34.360
So let's make a substitution.

00:03:34.360 --> 00:03:37.800
u is equal to cosine of x.

00:03:37.800 --> 00:03:40.385
How did I know to substitute
u is equal to cosine of x?

00:03:40.385 --> 00:03:42.960
Well because I say, well
the derivative of this

00:03:42.960 --> 00:03:43.850
function is here.

00:03:43.850 --> 00:03:46.750
So when I find du, this whole
thing is going to end up

00:03:46.750 --> 00:03:48.460
being du, and let me
show that to you.

00:03:48.460 --> 00:03:50.502
So what is du/dx?

00:03:50.502 --> 00:03:57.980
du/dx is equal to
minus sin of x.

00:03:57.980 --> 00:03:59.620
That hopefully we've
learned already.

00:03:59.620 --> 00:04:01.480
So what is du?

00:04:01.480 --> 00:04:04.750
If we multiply both sides by
the differential d of x get du

00:04:04.750 --> 00:04:10.250
is equal to minus sin of x dx.

00:04:10.250 --> 00:04:15.160
So if we look at the original
equation, this right here we

00:04:15.160 --> 00:04:20.900
just showed is equal to du,
and this right here is what?

00:04:20.900 --> 00:04:22.790
Cosine of x is u.

00:04:22.790 --> 00:04:24.440
That was our original
substitution.

00:04:24.440 --> 00:04:25.685
So we have u squared.

00:04:29.650 --> 00:04:32.100
So now let's take the integral.

00:04:32.100 --> 00:04:34.450
And I will arbitrarily
switch colors.

00:04:37.900 --> 00:04:40.160
And now this is a very
important thing.

00:04:40.160 --> 00:04:42.790
If you're going to do
substitution, if we're going to

00:04:42.790 --> 00:04:45.020
say u is equal to cosine of x,
we're going to have to actually

00:04:45.020 --> 00:04:46.450
make this substitution
on the boundaries.

00:04:46.450 --> 00:04:48.640
Or we could do the substitution
and reverse the substitution

00:04:48.640 --> 00:04:50.710
and then evaluate the
boundaries, but let's do that.

00:04:50.710 --> 00:04:55.240
So if this is going from x is
equal to pi over 2 to pi,

00:04:55.240 --> 00:04:57.160
what is u going from?

00:04:57.160 --> 00:05:00.070
Well when x is equal to
pi over 2, u is equal

00:05:00.070 --> 00:05:01.870
to cosine of pi over 2.

00:05:04.640 --> 00:05:06.390
Because u is just cosine of x.

00:05:06.390 --> 00:05:10.253
And then when x is pi, i is
going to be cosine of pi.

00:05:13.070 --> 00:05:14.720
And now the fun part.

00:05:14.720 --> 00:05:15.840
Cosine of x squared.

00:05:15.840 --> 00:05:19.400
Well that's just the same
thing is u squared.

00:05:19.400 --> 00:05:22.620
And minus sin of x dx,
that's the same thing

00:05:22.620 --> 00:05:24.560
is du-- did that here.

00:05:28.170 --> 00:05:29.800
This is pretty straightforward
and I'm just going

00:05:29.800 --> 00:05:30.370
to rewrite it.

00:05:30.370 --> 00:05:31.450
What's cosine of pi?

00:05:31.450 --> 00:05:34.120
Cosine of pi is minus 1.

00:05:34.120 --> 00:05:36.860
Cosine of pi over 2,
well that's a 0.

00:05:36.860 --> 00:05:40.420
So we have the integral from u
is equal to 0 to u is equal to

00:05:40.420 --> 00:05:44.280
negative 1 of u squared du.

00:05:44.280 --> 00:05:50.050
And now this seems like
a simple problem.

00:05:50.050 --> 00:05:53.710
So this is equal to the
antiderivative of u

00:05:53.710 --> 00:05:56.460
squared, which is
fairly straightforward.

00:05:56.460 --> 00:05:59.350
u cubed over 3.

00:05:59.350 --> 00:06:00.960
You could just take the
derivative of this,

00:06:00.960 --> 00:06:01.980
you get this.

00:06:01.980 --> 00:06:04.900
All I did is I increased this
exponent to get the third, and

00:06:04.900 --> 00:06:07.491
I divided by that exponent.

00:06:07.491 --> 00:06:10.790
Now we're going to have to
evaluate it at minus 1 and

00:06:10.790 --> 00:06:13.840
subtract from that,
it evaluated at 0.

00:06:13.840 --> 00:06:19.090
So this is equal to minus 1
to the third over 3 minus

00:06:19.090 --> 00:06:25.110
0 over 3, and so this
is equal to minus 1/3.

00:06:25.110 --> 00:06:27.310
And we are done.

00:06:27.310 --> 00:06:31.260
And if we look at this area
from our original graph, what

00:06:31.260 --> 00:06:33.400
we just solved, as we said the
area of the curve between

00:06:33.400 --> 00:06:36.405
here and here is minus 1/3.

00:06:36.405 --> 00:06:39.050
Or if we wanted the absolute
area because you can't really

00:06:39.050 --> 00:06:41.670
have a negative area, it's 1/3
but we know it's negative

00:06:41.670 --> 00:06:45.490
because this curve is
below the x-axis here.

00:06:45.490 --> 00:06:47.610
And that looks about right,
that looks about 1/3.

00:06:47.610 --> 00:06:50.580
I mean if this cube right
here is 1, then that

00:06:50.580 --> 00:06:52.600
looks about 1/3.

00:06:52.600 --> 00:06:54.520
The intuition all
works out at least.

00:06:54.520 --> 00:06:58.800
So hopefully you found
that vaguely useful.

00:06:58.800 --> 00:07:01.670
Actually let me-- since we
have a little bit of time.

00:07:01.670 --> 00:07:03.810
Hopefully you understood this,
and if you did, don't worry

00:07:03.810 --> 00:07:05.920
about what I'm going to do now,
but I want to show you how I

00:07:05.920 --> 00:07:07.330
tend to do it where I
just think of it is the

00:07:07.330 --> 00:07:09.040
reverse chain rule.

00:07:09.040 --> 00:07:12.840
It's a little bit
quicker sometimes.

00:07:12.840 --> 00:07:14.570
But it's really the same
thing as what we just

00:07:14.570 --> 00:07:16.250
did with substitution.

00:07:16.250 --> 00:07:18.450
So if we erase all of this.

00:07:21.380 --> 00:07:27.130
And so if we have this integral
right here, what I do is I say,

00:07:27.130 --> 00:07:28.690
well I have cosine
of x squared.

00:07:31.660 --> 00:07:34.980
I have cosine of x squared, and
then I have minus sin of x.

00:07:34.980 --> 00:07:38.240
This is the derivative of this.

00:07:38.240 --> 00:07:41.040
Since the derivative is here,
I can just treat this whole

00:07:41.040 --> 00:07:43.260
thing like an x term.

00:07:43.260 --> 00:07:46.130
So this is the same thing.

00:07:46.130 --> 00:07:55.330
So the antiderivative is cosine
of x to the third over 3, and I

00:07:55.330 --> 00:07:58.140
evaluate it at pi
and pi over 2.

00:07:58.140 --> 00:07:59.870
And remember, how
did I do this?

00:07:59.870 --> 00:08:03.030
What allowed me to treat this
cosine of x just like an x or

00:08:03.030 --> 00:08:05.200
like a u when I did it
with the substitution?

00:08:05.200 --> 00:08:06.960
Well I had its derivative
sitting right here,

00:08:06.960 --> 00:08:07.860
minus sin of x.

00:08:07.860 --> 00:08:10.750
So that's what gave me the
license to just take the

00:08:10.750 --> 00:08:14.920
antiderivative, pretend like
this cosine of x is just an x,

00:08:14.920 --> 00:08:18.650
is just a u, you could even
say, and just take it's

00:08:18.650 --> 00:08:21.110
exponent, raise it by 1 and
divide it by 3 and then

00:08:21.110 --> 00:08:23.740
evaluate it from
pi to pi over 2.

00:08:23.740 --> 00:08:31.540
So this is equal to cosine of
pi cubed over 3 minus cosine

00:08:31.540 --> 00:08:36.470
of pi over 2 cubed over 3.

00:08:36.470 --> 00:08:39.630
This is minus 1 to the third,
so this is equal to minus

00:08:39.630 --> 00:08:41.990
1/3 minus this is 0.

00:08:41.990 --> 00:08:43.330
So we get the same answer.

00:08:43.330 --> 00:08:44.550
I just wanted to show you that.

00:08:44.550 --> 00:08:46.450
It's exactly the same
with substitution.

00:08:46.450 --> 00:08:49.440
It's just I didn't formally do
the substitution, but it's

00:08:49.440 --> 00:08:50.320
the exact same thing.

00:08:50.320 --> 00:08:51.950
Anyway, hope you
found that helpful.


Equation of a tangent line

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:04.650
I've told you multiple times
that the derivative of a curve

00:00:04.650 --> 00:00:08.170
at a point is the slope of the
tangent line, but our

00:00:08.170 --> 00:00:09.670
friend [? Akosh ?]

00:00:09.670 --> 00:00:12.880
sent me a problem where it
actually wants you to find the

00:00:12.880 --> 00:00:13.940
equation of the tangent line.

00:00:13.940 --> 00:00:15.550
And I realize, I've never
actually done that.

00:00:15.550 --> 00:00:16.840
So it's worthwhile.

00:00:16.840 --> 00:00:17.540
So let's do that.

00:00:17.540 --> 00:00:21.370
So it says, find the equation
of the tangent line to the

00:00:21.370 --> 00:00:34.950
function f of x is equal to x e
to the x at x is equal to 1.

00:00:34.950 --> 00:00:37.130
So let's just get an intuition
of what we're even looking for.

00:00:37.130 --> 00:00:40.550
So this function is going to
look something like, I actually

00:00:40.550 --> 00:00:44.150
graphed it, because it's not
a trivial function to graph.

00:00:44.150 --> 00:00:47.340
So this is x e to the x,
this is what it looks like.

00:00:47.340 --> 00:00:49.120
I'm just using a graphing
calculator, and you can

00:00:49.120 --> 00:00:50.910
see, I just typed it in.

00:00:50.910 --> 00:00:52.680
And what this is
asking us, is ok.

00:00:52.680 --> 00:00:54.310
At the point, x is equal to 1.

00:00:54.310 --> 00:00:56.440
So this is the point
x is equal to one.

00:00:56.440 --> 00:00:59.130
So f of x is going to be
someplace up here, and

00:00:59.130 --> 00:01:02.540
actually, f of x is going
to be equal to e, right?

00:01:02.540 --> 00:01:08.220
Because f of 1 is
equal to what?

00:01:08.220 --> 00:01:09.630
1 times e to the 1.

00:01:09.630 --> 00:01:10.950
So it equals e.

00:01:10.950 --> 00:01:15.170
So we're saying at the point,
at the point 1 comma e, so at

00:01:15.170 --> 00:01:18.220
the point 1 comma 2.71,
whatever, whatever.

00:01:18.220 --> 00:01:19.420
So that's what point?

00:01:19.420 --> 00:01:20.740
That's this point.

00:01:20.740 --> 00:01:21.780
So it's right here.

00:01:21.780 --> 00:01:26.370
2 point, this is e right
here, the point 1 comma e.

00:01:26.370 --> 00:01:29.320
So we want to do is figure
out the equation of the

00:01:29.320 --> 00:01:31.690
line tangent to this point.

00:01:31.690 --> 00:01:33.810
So what we're going to do, is
we're going to solve it by

00:01:33.810 --> 00:01:35.540
figuring out its slope, which
is just the derivative

00:01:35.540 --> 00:01:36.310
at that point.

00:01:36.310 --> 00:01:37.860
So we have to figure
out the derivative at

00:01:37.860 --> 00:01:39.140
exactly this point.

00:01:39.140 --> 00:01:41.410
And then we use what we learned
from algebra 1 to figure out

00:01:41.410 --> 00:01:43.970
its equation, and we'll graph
it here, just to confirm that

00:01:43.970 --> 00:01:48.010
we actually figured out the
equation of the tangent line.

00:01:48.010 --> 00:01:50.730
So the first thing we want to
know is the slope of the

00:01:50.730 --> 00:01:54.850
tangent line, and that's just
the derivative at this point.

00:01:54.850 --> 00:01:57.840
When x is equal to 1, or
at the point 1 comma e.

00:01:57.840 --> 00:01:59.900
So what's the
derivative of this?

00:01:59.900 --> 00:02:03.030
So f prime of x.

00:02:03.030 --> 00:02:07.370
f prime of x is equal to,
well, this looks like a

00:02:07.370 --> 00:02:09.100
job for the product rule.

00:02:09.100 --> 00:02:11.080
Because we know how to figure
out the derivative of x, we

00:02:11.080 --> 00:02:12.940
know how to figure out the
derivative of e to the x, and

00:02:12.940 --> 00:02:14.370
they're just multiplying
by each other.

00:02:14.370 --> 00:02:16.140
So the product rules help us.

00:02:16.140 --> 00:02:18.105
The derivative of this thing
is going to be equal to the

00:02:18.105 --> 00:02:20.100
derivative of the first
expression of the

00:02:20.100 --> 00:02:20.820
first function.

00:02:20.820 --> 00:02:25.810
So the derivative of x is just
1, times the second function,

00:02:25.810 --> 00:02:32.220
times e to the x, plus the
first function, x, times the

00:02:32.220 --> 00:02:34.050
derivative of the
second function.

00:02:34.050 --> 00:02:36.430
So what's the derivative
of e to the x?

00:02:36.430 --> 00:02:40.280
And that's what I find so
amazing about the number e, or

00:02:40.280 --> 00:02:42.015
the function e to the x, is
that the derivative of e

00:02:42.015 --> 00:02:43.710
to the x is e to the x.

00:02:43.710 --> 00:02:45.840
The slope at any point of
this curve is equal to the

00:02:45.840 --> 00:02:47.760
value of the function.

00:02:47.760 --> 00:02:49.570
So this is the derivative.

00:02:49.570 --> 00:02:53.385
So what is the derivative of
this function at the point x

00:02:53.385 --> 00:02:55.560
is equal to 1, or at
the point 1 comma e?

00:02:55.560 --> 00:02:56.610
So we just evaluate it.

00:02:56.610 --> 00:03:06.080
We say f prime of 1 is equal to
1 time e to the 1 plus 1 times

00:03:06.080 --> 00:03:11.150
e to the 1, well, that's
just equal e plus e.

00:03:11.150 --> 00:03:14.585
And that's just equal to 2 e.

00:03:14.585 --> 00:03:17.220
And you know, we could figure
out what that number, e is just

00:03:17.220 --> 00:03:19.880
a constant number, but we write
e because it's easier to write

00:03:19.880 --> 00:03:24.060
e than 2.7 et cetera, and an
infinite number of digits,

00:03:24.060 --> 00:03:25.170
so we just write 2e.

00:03:25.170 --> 00:03:29.240
So this is the slope of the
equation, or this is the slope

00:03:29.240 --> 00:03:32.090
of the curve when x is equal to
one, or at the point

00:03:32.090 --> 00:03:35.440
1e, or 1 f of 1.

00:03:35.440 --> 00:03:38.760
So what is the equation
of the tangent line?

00:03:38.760 --> 00:03:41.350
So let's go ahead and take this
form, the equation's going to

00:03:41.350 --> 00:03:45.460
be y is equal to, I'm just
writing it in the, you know,

00:03:45.460 --> 00:03:49.090
not the point slope, the mx
plus b form that you

00:03:49.090 --> 00:03:49.840
learned in algebra.

00:03:49.840 --> 00:03:52.570
So the slope is going to be 2e.

00:03:52.570 --> 00:03:53.430
We just learned that here.

00:03:53.430 --> 00:03:56.300
That's the derivative
when x is equal to 1.

00:03:56.300 --> 00:04:02.220
So 2e times x plus
the y-intercept.

00:04:02.220 --> 00:04:04.120
So if we can figure out
the y-intercept of this

00:04:04.120 --> 00:04:04.790
line, we are done.

00:04:04.790 --> 00:04:09.230
We have figured out the
equation of the tangent line.

00:04:09.230 --> 00:04:10.690
So how do we do that?

00:04:10.690 --> 00:04:13.960
Well, if we knew a y or
an x where this equation

00:04:13.960 --> 00:04:16.080
goes through, we could
then solve for b.

00:04:16.080 --> 00:04:20.230
And we know a y and x that
satisfies this equation.

00:04:20.230 --> 00:04:22.410
The point 1 comma e.

00:04:22.410 --> 00:04:25.770
The point where we're trying to
find the tangent line, right?

00:04:25.770 --> 00:04:28.360
So this point, 1 comma e,
this is where we want to

00:04:28.360 --> 00:04:29.510
find the tangent line.

00:04:29.510 --> 00:04:30.870
And by definition, the
tangent line is going to

00:04:30.870 --> 00:04:33.080
go through that point.

00:04:33.080 --> 00:04:37.002
So let's substitute those
points back in here, or this

00:04:37.002 --> 00:04:41.050
point back into this equation,
and then solve for b.

00:04:41.050 --> 00:04:48.170
So y is equal to e, is equal to
2 e, that's just the slope at

00:04:48.170 --> 00:04:52.040
that point, times x,
times 1, plus b.

00:04:52.040 --> 00:04:55.620
It might confuse you, because
e, you'll say, oh, e,

00:04:55.620 --> 00:04:56.030
is that a variable?

00:04:56.030 --> 00:04:57.800
No, it's a number,
remember, it's like pi.

00:04:57.800 --> 00:04:58.220
It's a number.

00:04:58.220 --> 00:05:00.900
You can substitute 2.7 whatever
there, but we're not doing

00:05:00.900 --> 00:05:02.340
that, because this is cleaner.

00:05:02.340 --> 00:05:03.100
And let's solve.

00:05:03.100 --> 00:05:08.000
So you get e is
equal to 2e plus b.

00:05:08.000 --> 00:05:09.880
Let's subtract 2e
from both sides.

00:05:09.880 --> 00:05:13.030
You get b is equal
to e minus 2e.

00:05:13.030 --> 00:05:16.430
b is equal to minus e.

00:05:16.430 --> 00:05:17.010
Now we're done.

00:05:17.010 --> 00:05:21.940
What's the equation
of the tangent line?

00:05:21.940 --> 00:05:29.670
It is y is equal to
2 times e x plus b.

00:05:29.670 --> 00:05:33.000
But b is minus e,
so it's minus e.

00:05:33.000 --> 00:05:35.950
So this is the equation
of the tangent line.

00:05:35.950 --> 00:05:38.270
If you don't like these e's
there, you could replace that

00:05:38.270 --> 00:05:40.920
with the number 2.7 et cetera,
and this would become 5 point

00:05:40.920 --> 00:05:43.780
something, and this would
just be minus 2.7 something.

00:05:43.780 --> 00:05:44.920
But this looks neater.

00:05:44.920 --> 00:05:45.830
And let's confirm.

00:05:45.830 --> 00:05:49.550
Let's use this little graphing
calculator to confirm that that

00:05:49.550 --> 00:05:53.140
really is the equation
of the tangent line.

00:05:53.140 --> 00:05:54.910
So let me type it in here.

00:05:54.910 --> 00:06:13.330
So it's 2, 2 times e times x,
right, that's 2ex minus e.

00:06:13.330 --> 00:06:17.240
And let us graph this line.

00:06:17.240 --> 00:06:18.200
There we go.

00:06:18.200 --> 00:06:19.100
It graphed it.

00:06:19.100 --> 00:06:22.860
And notice that that line, that
green line, I don't know if you

00:06:22.860 --> 00:06:24.700
can, maybe I need to make this
bigger for it to

00:06:24.700 --> 00:06:26.920
show up, bolder.

00:06:26.920 --> 00:06:27.830
I don't know if that helps.

00:06:30.620 --> 00:06:33.800
But if you look here, so this
red, this is our original

00:06:33.800 --> 00:06:37.390
equation, x e to the
x, that's this curve.

00:06:37.390 --> 00:06:40.810
We want to know equation
of the tangent line

00:06:40.810 --> 00:06:42.530
at x is equal to 1.

00:06:42.530 --> 00:06:44.035
So it's the point
x is equal to 1.

00:06:44.035 --> 00:06:47.990
And when x is equal to 1, f of
x is e, right, you can just

00:06:47.990 --> 00:06:49.900
substitute back into the
original equation to get that.

00:06:49.900 --> 00:06:52.710
So this is the
point, 1 comma e.

00:06:52.710 --> 00:06:55.470
So the equation of the tangent
line, its slope is going to be

00:06:55.470 --> 00:06:56.800
the derivative at this point.

00:06:56.800 --> 00:07:00.410
So we solved the derivative of
this function, and evaluated

00:07:00.410 --> 00:07:03.310
it at x is equal to 1.

00:07:03.310 --> 00:07:03.910
That's what we did here.

00:07:03.910 --> 00:07:06.970
We figured out the derivative,
evaluated x equals 1.

00:07:06.970 --> 00:07:10.200
And so we said, OK, the slope.

00:07:10.200 --> 00:07:15.970
The slope at when x is equal to
1 and y is equal to e, the

00:07:15.970 --> 00:07:18.640
slope at that point
is equal to 2e.

00:07:18.640 --> 00:07:20.630
And we figured that out
from the derivative.

00:07:20.630 --> 00:07:23.840
And then we just used our
algebra 1 skills to figure out

00:07:23.840 --> 00:07:25.230
the equation of that line.

00:07:25.230 --> 00:07:26.160
And how did we do that?

00:07:26.160 --> 00:07:28.450
We knew the slope, because
that's just the derivative

00:07:28.450 --> 00:07:29.250
at that point.

00:07:29.250 --> 00:07:32.590
And then we just have to
solve for the y-intercept.

00:07:32.590 --> 00:07:36.100
And the way we did that is we
said, well, the point 1 comma e

00:07:36.100 --> 00:07:38.250
is on this green line as well.

00:07:38.250 --> 00:07:41.420
So we substituted that in, and
solve for our y-intercept,

00:07:41.420 --> 00:07:43.930
which we got as minus e, and
notice that this line

00:07:43.930 --> 00:07:47.200
intersects the y-axis at minus
e, that's about minus

00:07:47.200 --> 00:07:48.420
2.7 something.

00:07:48.420 --> 00:07:49.230
And there we have it.

00:07:49.230 --> 00:07:52.700
We have shown that, and
visually, it shows that

00:07:52.700 --> 00:07:54.890
this is the tangent line.

00:07:54.890 --> 00:07:59.380
Anyway, hope you found
that vaguely useful.

00:07:59.380 --> 00:08:01.400
If you did, you should
thank [? Akosh ?]

00:08:01.400 --> 00:08:05.110
for being unusually persistent,
and having me do this problem.

00:08:05.110 --> 00:08:07.000
See you in the next video.