[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.03,0:00:05.98,Default,,0000,0000,0000,,I received a suggestion that I\Ndo actual old AP exam problems,
Dialogue: 0,0:00:05.98,0:00:08.55,Default,,0000,0000,0000,,and I looked on the internet\Nand lo and behold, on the
Dialogue: 0,0:00:08.55,0:00:11.64,Default,,0000,0000,0000,,college board site, if you go\Nto collegeboard.com, you can
Dialogue: 0,0:00:11.64,0:00:14.38,Default,,0000,0000,0000,,actually get-- I couldn't find\Nthe actual multiple choice
Dialogue: 0,0:00:14.38,0:00:16.76,Default,,0000,0000,0000,,questions, but you can find the\Nfree response questions, and so
Dialogue: 0,0:00:16.76,0:00:19.79,Default,,0000,0000,0000,,this question is actually the\Nfirst free response question
Dialogue: 0,0:00:19.79,0:00:23.06,Default,,0000,0000,0000,,that they have on the calculus\NBC that was administered
Dialogue: 0,0:00:23.06,0:00:24.62,Default,,0000,0000,0000,,just recently in 2008.
Dialogue: 0,0:00:24.62,0:00:25.99,Default,,0000,0000,0000,,So let's do this problem.
Dialogue: 0,0:00:25.99,0:00:28.14,Default,,0000,0000,0000,,And frankly, if you understand\Nhow to do all of the free
Dialogue: 0,0:00:28.14,0:00:32.20,Default,,0000,0000,0000,,response questions, you\Nprobably will do fairly well on
Dialogue: 0,0:00:32.20,0:00:34.84,Default,,0000,0000,0000,,the multiple choice, because\Nthe free response tend to be a
Dialogue: 0,0:00:34.84,0:00:36.98,Default,,0000,0000,0000,,little bit more challenging,\Nespecially the last parts
Dialogue: 0,0:00:36.98,0:00:38.26,Default,,0000,0000,0000,,of the free response.
Dialogue: 0,0:00:38.26,0:00:40.11,Default,,0000,0000,0000,,Well anyway, let's do this one.
Dialogue: 0,0:00:40.11,0:00:42.20,Default,,0000,0000,0000,,So I'll just read it out,\Nbecause I don't want to write
Dialogue: 0,0:00:42.20,0:00:44.30,Default,,0000,0000,0000,,it out all here, but this\Nis the actual diagram.
Dialogue: 0,0:00:44.30,0:00:48.25,Default,,0000,0000,0000,,I actually copied and pasted\Nthis from the PDF that they
Dialogue: 0,0:00:48.25,0:00:50.36,Default,,0000,0000,0000,,provide on collegeboard.com.
Dialogue: 0,0:00:50.36,0:00:54.63,Default,,0000,0000,0000,,So it says, let r-- this is r--\Nbe the region bounded by the
Dialogue: 0,0:00:54.63,0:00:57.39,Default,,0000,0000,0000,,graphs of y equals\Nsine pi of x.
Dialogue: 0,0:00:57.39,0:00:58.85,Default,,0000,0000,0000,,So let me write that down.
Dialogue: 0,0:00:58.85,0:01:09.12,Default,,0000,0000,0000,,So this top graph is y\Nis equal to sine pi x.
Dialogue: 0,0:01:22.86,0:01:28.06,Default,,0000,0000,0000,,and then the bottom graph is y\Nis equal to x cubed minus 4x.
Dialogue: 0,0:01:37.41,0:01:39.32,Default,,0000,0000,0000,,And how did I know that\Nthis was the bottom one?
Dialogue: 0,0:01:39.32,0:01:41.78,Default,,0000,0000,0000,,Well I knew that this one\Nwas sine of pi x, right?
Dialogue: 0,0:01:41.78,0:01:42.84,Default,,0000,0000,0000,,Because sine looks like this.
Dialogue: 0,0:01:42.84,0:01:44.91,Default,,0000,0000,0000,,It doesn't look\Nlike that, right?
Dialogue: 0,0:01:44.91,0:01:48.28,Default,,0000,0000,0000,,When you go sine of pi\Nis 0, sine of 0 is
Dialogue: 0,0:01:48.28,0:01:50.38,Default,,0000,0000,0000,,0, sine of 2pi is 0.
Dialogue: 0,0:01:50.38,0:01:51.76,Default,,0000,0000,0000,,So we do this as sine of pi x.
Dialogue: 0,0:01:51.76,0:01:55.60,Default,,0000,0000,0000,,Well anyway, they want-- so\Nthis is the region between
Dialogue: 0,0:01:55.60,0:01:59.11,Default,,0000,0000,0000,,these two functions and part A\Nof this-- and this is kind of
Dialogue: 0,0:01:59.11,0:02:01.89,Default,,0000,0000,0000,,the softball question, just to\Nmake sure that you know how to
Dialogue: 0,0:02:01.89,0:02:07.04,Default,,0000,0000,0000,,do definite integrals-- and\Nit says, find the area of r.
Dialogue: 0,0:02:07.04,0:02:08.89,Default,,0000,0000,0000,,So how do we do that?
Dialogue: 0,0:02:08.89,0:02:11.80,Default,,0000,0000,0000,,I think you know that we're\Ngoing to do a little definite
Dialogue: 0,0:02:11.80,0:02:13.29,Default,,0000,0000,0000,,integration, so let's do that.
Dialogue: 0,0:02:13.29,0:02:15.78,Default,,0000,0000,0000,,So then we're going to take the\Ndefinite integral, so let's
Dialogue: 0,0:02:15.78,0:02:23.28,Default,,0000,0000,0000,,just say the area is equal to--\NI don't know if that's-- I hope
Dialogue: 0,0:02:23.28,0:02:26.14,Default,,0000,0000,0000,,I'm writing big enough for\Nyou-- the area is going to be
Dialogue: 0,0:02:26.14,0:02:28.96,Default,,0000,0000,0000,,equal to the definite\Nintegral from.
Dialogue: 0,0:02:28.96,0:02:30.15,Default,,0000,0000,0000,,So what are the x values?
Dialogue: 0,0:02:30.15,0:02:32.27,Default,,0000,0000,0000,,We're going to be going\Nfrom x is equal to 0
Dialogue: 0,0:02:32.27,0:02:34.54,Default,,0000,0000,0000,,to x is equal to 2.
Dialogue: 0,0:02:38.89,0:02:40.33,Default,,0000,0000,0000,,And what's this?
Dialogue: 0,0:02:40.33,0:02:44.51,Default,,0000,0000,0000,,At any given point value of x,\Nwhat is kind of going to be the
Dialogue: 0,0:02:44.51,0:02:46.99,Default,,0000,0000,0000,,high-- when we're taking the\Narea, we're taking a bunch of
Dialogue: 0,0:02:46.99,0:02:50.85,Default,,0000,0000,0000,,rectangles that are\Nof dx width, right?
Dialogue: 0,0:02:50.85,0:02:52.90,Default,,0000,0000,0000,,So that's-- that's not dark\Nenough, I don't think that
Dialogue: 0,0:02:52.90,0:02:55.75,Default,,0000,0000,0000,,you can see that-- so that's\None of my rectangles.
Dialogue: 0,0:02:55.75,0:02:56.89,Default,,0000,0000,0000,,Whoops.
Dialogue: 0,0:02:56.89,0:03:00.73,Default,,0000,0000,0000,,Let's say that's one of my\Nrectangles right here that
Dialogue: 0,0:03:00.73,0:03:02.07,Default,,0000,0000,0000,,I'm going to be summing up.
Dialogue: 0,0:03:02.07,0:03:04.11,Default,,0000,0000,0000,,Its width is dx.
Dialogue: 0,0:03:04.11,0:03:06.22,Default,,0000,0000,0000,,What's its height?
Dialogue: 0,0:03:06.22,0:03:09.44,Default,,0000,0000,0000,,Its height is going to be\Nthis top function minus
Dialogue: 0,0:03:09.44,0:03:12.34,Default,,0000,0000,0000,,this bottom function.
Dialogue: 0,0:03:12.34,0:03:15.24,Default,,0000,0000,0000,,So, essentially, we're going to\Ntake the sum of all of these
Dialogue: 0,0:03:15.24,0:03:18.71,Default,,0000,0000,0000,,rectangles, so its height is\Ngoing to be-- let me switch
Dialogue: 0,0:03:18.71,0:03:22.67,Default,,0000,0000,0000,,colors arbitrarily-- the height\Nis going to be the top function
Dialogue: 0,0:03:22.67,0:03:24.50,Default,,0000,0000,0000,,minus the bottom function.
Dialogue: 0,0:03:24.50,0:03:35.06,Default,,0000,0000,0000,,So sine of pi x-- parentheses\Nhere-- minus the
Dialogue: 0,0:03:35.06,0:03:35.72,Default,,0000,0000,0000,,bottom function.
Dialogue: 0,0:03:35.72,0:03:40.25,Default,,0000,0000,0000,,So minus x cubed plus 4x.
Dialogue: 0,0:03:42.81,0:03:47.27,Default,,0000,0000,0000,,Since I'm subtracting, I\Nswitched both of these signs.
Dialogue: 0,0:03:47.27,0:03:51.01,Default,,0000,0000,0000,,And all of that times the width\Nof each of these little
Dialogue: 0,0:03:51.01,0:03:54.67,Default,,0000,0000,0000,,rectangles-- which is\Ninfinitely small-- dx.
Dialogue: 0,0:03:54.67,0:03:56.81,Default,,0000,0000,0000,,And we're going to sum them\Nall up from x is equal
Dialogue: 0,0:03:56.81,0:03:59.51,Default,,0000,0000,0000,,to 0 to x is equal to 2.
Dialogue: 0,0:03:59.51,0:04:01.61,Default,,0000,0000,0000,,This should be fairly\Nstraightforward for you.
Dialogue: 0,0:04:01.61,0:04:02.85,Default,,0000,0000,0000,,So how do we evaluate this?
Dialogue: 0,0:04:02.85,0:04:06.08,Default,,0000,0000,0000,,Well, we essentially take the\Nantiderivative of this and
Dialogue: 0,0:04:06.08,0:04:08.87,Default,,0000,0000,0000,,then evaluate that at 2\Nand then evaluate at 0.
Dialogue: 0,0:04:08.87,0:04:12.59,Default,,0000,0000,0000,,What's the antiderivative\Nof sine of pi x?
Dialogue: 0,0:04:12.59,0:04:17.90,Default,,0000,0000,0000,,Well, what functions\Nderivative is sine of x.
Dialogue: 0,0:04:17.90,0:04:19.10,Default,,0000,0000,0000,,Cosine of x-- let's see.
Dialogue: 0,0:04:19.10,0:04:21.42,Default,,0000,0000,0000,,If I were to take the\Nderivative of cosine-- let's
Dialogue: 0,0:04:21.42,0:04:24.96,Default,,0000,0000,0000,,say I took the derivative\Nof cosine pi x.
Dialogue: 0,0:04:24.96,0:04:27.09,Default,,0000,0000,0000,,This should be reasonably\Nfamiliar to you.
Dialogue: 0,0:04:27.09,0:04:30.59,Default,,0000,0000,0000,,Cosine of pi x, if I were\Nto take the derivative
Dialogue: 0,0:04:30.59,0:04:34.20,Default,,0000,0000,0000,,of it, what do I get?
Dialogue: 0,0:04:34.20,0:04:36.32,Default,,0000,0000,0000,,That equals pi.
Dialogue: 0,0:04:36.32,0:04:37.98,Default,,0000,0000,0000,,You take the derivative\Nof the inside, right?
Dialogue: 0,0:04:37.98,0:04:39.12,Default,,0000,0000,0000,,By the chain rule.
Dialogue: 0,0:04:39.12,0:04:43.13,Default,,0000,0000,0000,,So it's pi times the derivative\Nof the whole thing.
Dialogue: 0,0:04:43.13,0:04:46.23,Default,,0000,0000,0000,,The derivative of cosine of x\Nis minus sine of x, so the
Dialogue: 0,0:04:46.23,0:04:54.44,Default,,0000,0000,0000,,derivative to this is going to\Nbe times minus sine of pi x, or
Dialogue: 0,0:04:54.44,0:05:02.08,Default,,0000,0000,0000,,you could say that equals\Nminus pi sine of pi x.
Dialogue: 0,0:05:02.08,0:05:06.81,Default,,0000,0000,0000,,So the derivative of cosine of\Npi x is almost this, it just
Dialogue: 0,0:05:06.81,0:05:09.27,Default,,0000,0000,0000,,has that minus pi there, right?
Dialogue: 0,0:05:09.27,0:05:12.15,Default,,0000,0000,0000,,So let's see if we can rewrite\Nthis so it looks just like the
Dialogue: 0,0:05:12.15,0:05:16.44,Default,,0000,0000,0000,,derivative of cosine pi x.
Dialogue: 0,0:05:16.44,0:05:17.69,Default,,0000,0000,0000,,And I'll switch to magenta.
Dialogue: 0,0:05:20.73,0:05:22.40,Default,,0000,0000,0000,,I want to make sure I\Nhave enough space to do
Dialogue: 0,0:05:22.40,0:05:23.22,Default,,0000,0000,0000,,this entire problem.
Dialogue: 0,0:05:27.18,0:05:36.88,Default,,0000,0000,0000,,So let's write a minus 1\Nover pi times a minus pi.
Dialogue: 0,0:05:36.88,0:05:40.02,Default,,0000,0000,0000,,All I did, when you evaluate\Nthis, this equals 1, so I can
Dialogue: 0,0:05:40.02,0:05:48.10,Default,,0000,0000,0000,,do this times sine pi x, and\Nthen that's minus x to the
Dialogue: 0,0:05:48.10,0:05:54.37,Default,,0000,0000,0000,,third plus 4x, and then all\Nof that times the width dx.
Dialogue: 0,0:05:54.37,0:05:55.20,Default,,0000,0000,0000,,Well now we have it.
Dialogue: 0,0:05:55.20,0:05:59.81,Default,,0000,0000,0000,,We know that the antiderivative\Nof this is cosine pi x, right?
Dialogue: 0,0:05:59.81,0:06:00.91,Default,,0000,0000,0000,,And this is just\Na constant term.
Dialogue: 0,0:06:00.91,0:06:03.37,Default,,0000,0000,0000,,So what's the antiderivative\Nof this whole thing?
Dialogue: 0,0:06:03.37,0:06:05.78,Default,,0000,0000,0000,,And I'll arbitrarily\Nswitch colors again.
Dialogue: 0,0:06:05.78,0:06:10.07,Default,,0000,0000,0000,,The antiderivative\Nis cosine pi x.
Dialogue: 0,0:06:10.07,0:06:18.62,Default,,0000,0000,0000,,So we have minus 1 over pi\Ncosine pi x-- remember, I could
Dialogue: 0,0:06:18.62,0:06:21.32,Default,,0000,0000,0000,,just carry this over, this is\Njust a constant term-- this
Dialogue: 0,0:06:21.32,0:06:25.59,Default,,0000,0000,0000,,antiderivative is\Nthis right here.
Dialogue: 0,0:06:25.59,0:06:28.33,Default,,0000,0000,0000,,And then these are a little\Nbit more straightforward.
Dialogue: 0,0:06:28.33,0:06:31.77,Default,,0000,0000,0000,,So minus the antiderivative of\Nx to the third is x to the
Dialogue: 0,0:06:31.77,0:06:41.30,Default,,0000,0000,0000,,fourth over 4 plus the\Nantiderivative of this is 4x
Dialogue: 0,0:06:41.30,0:06:47.25,Default,,0000,0000,0000,,squared over 2, or you could\Njust view that as 2x squared,
Dialogue: 0,0:06:47.25,0:06:52.62,Default,,0000,0000,0000,,and then we're going to\Nevaluate that at 2 and at
Dialogue: 0,0:06:52.62,0:06:55.26,Default,,0000,0000,0000,,0, and let's do that.
Dialogue: 0,0:06:55.26,0:07:03.51,Default,,0000,0000,0000,,So this is equal to cosine of\N2pi, and we'll have a minus
Dialogue: 0,0:07:03.51,0:07:09.93,Default,,0000,0000,0000,,sign out here, so minus cosine\Nof 2pi over pi, minus-- what's
Dialogue: 0,0:07:09.93,0:07:11.68,Default,,0000,0000,0000,,2 to the fourth power?
Dialogue: 0,0:07:11.68,0:07:11.96,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:07:11.96,0:07:18.17,Default,,0000,0000,0000,,2 to the third is 8, 2 the\Nfourth is 16, 16 over 4 is 4,
Dialogue: 0,0:07:18.17,0:07:26.75,Default,,0000,0000,0000,,so it's minus 4, 2 squared is\N4 times 2 is 8, so plus 8, so
Dialogue: 0,0:07:26.75,0:07:31.02,Default,,0000,0000,0000,,that's the antiderivative\Nevaluated at 2, and now let's
Dialogue: 0,0:07:31.02,0:07:35.46,Default,,0000,0000,0000,,subtract it evaluated at 0.
Dialogue: 0,0:07:35.46,0:07:46.47,Default,,0000,0000,0000,,So this will be minus cosine of\N0 over pi-- all right, that's
Dialogue: 0,0:07:46.47,0:07:50.63,Default,,0000,0000,0000,,that evaluated at 0--\Nminus 0, plus 0.
Dialogue: 0,0:07:50.63,0:07:52.54,Default,,0000,0000,0000,,So these terms don't\Ncontribute anything when
Dialogue: 0,0:07:52.54,0:07:54.88,Default,,0000,0000,0000,,you evaluate them at 0.
Dialogue: 0,0:07:54.88,0:07:56.25,Default,,0000,0000,0000,,And so what do we get?
Dialogue: 0,0:07:56.25,0:07:58.62,Default,,0000,0000,0000,,What's cosine of 2pi?
Dialogue: 0,0:07:58.62,0:08:01.11,Default,,0000,0000,0000,,Cosine of 2pi is the\Nsame thing as cosine
Dialogue: 0,0:08:01.11,0:08:03.09,Default,,0000,0000,0000,,of 0, and it equals 1.
Dialogue: 0,0:08:03.09,0:08:06.49,Default,,0000,0000,0000,,What is the x value of the\Nunit circle at 2pi, or at 0?
Dialogue: 0,0:08:06.49,0:08:07.07,Default,,0000,0000,0000,,It's equal to 1.
Dialogue: 0,0:08:07.07,0:08:15.67,Default,,0000,0000,0000,,So this equals minus 1 over pi\Nminus 4 plus 8, and so this
Dialogue: 0,0:08:15.67,0:08:19.90,Default,,0000,0000,0000,,minus minus, those both become\Npluses, cosine of 0 is also 1,
Dialogue: 0,0:08:19.90,0:08:25.84,Default,,0000,0000,0000,,so plus 1 over pi, and so this\Nminus 1 over pi and this plus 1
Dialogue: 0,0:08:25.84,0:08:30.57,Default,,0000,0000,0000,,over pi will cancel out, and\Nall we're left with is minus 4
Dialogue: 0,0:08:30.57,0:08:34.21,Default,,0000,0000,0000,,plus 8 and that is equal to 4.
Dialogue: 0,0:08:34.21,0:08:42.83,Default,,0000,0000,0000,,So that is part one, part A of\Nnumber one, on the 2008 DC
Dialogue: 0,0:08:42.83,0:08:43.92,Default,,0000,0000,0000,,free response questions.
Dialogue: 0,0:08:43.92,0:08:46.09,Default,,0000,0000,0000,,It actually took me a whole\Nvideo just to do that part.
Dialogue: 0,0:08:46.09,0:08:48.55,Default,,0000,0000,0000,,In the next video, I'll do part\NB, and we'll just keep doing
Dialogue: 0,0:08:48.55,0:08:51.12,Default,,0000,0000,0000,,this, and I'll try to do a\Ncouple of these every day.
Dialogue: 0,0:08:51.12,0:08:52.69,Default,,0000,0000,0000,,See you soon.