[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:36.25,0:00:37.33,Default,,0000,0000,0000,,Here we go.
Dialogue: 0,0:00:41.24,0:00:44.17,Default,,0000,0000,0000,,- Cool; I was gettin' worried there.\N- Me, too.
Dialogue: 0,0:01:01.03,0:01:04.27,Default,,0000,0000,0000,,Okay. They're comin' in now.\NI've got... 11.
Dialogue: 0,0:01:14.50,0:01:18.26,Default,,0000,0000,0000,,I'll wait until noon, because that's\Nwhen I set this one to start.
Dialogue: 0,0:01:18.26,0:01:21.25,Default,,0000,0000,0000,,And, um... then I'll begin.
Dialogue: 0,0:01:40.61,0:01:43.12,Default,,0000,0000,0000,,This time, I didn't need\Npassword to log in.
Dialogue: 0,0:01:43.12,0:01:44.07,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:01:51.95,0:01:57.01,Default,,0000,0000,0000,,All right, I've got Manon; I've got...\NRichard; I've got Causey.
Dialogue: 0,0:02:01.99,0:02:03.33,Default,,0000,0000,0000,,Ross. Yeah.
Dialogue: 0,0:02:06.39,0:02:07.63,Default,,0000,0000,0000,,I have Jordan...
Dialogue: 0,0:02:12.25,0:02:13.35,Default,,0000,0000,0000,,Don't have Jordan.
Dialogue: 0,0:02:16.38,0:02:17.61,Default,,0000,0000,0000,,[student] There you go. 15.
Dialogue: 0,0:02:21.01,0:02:23.50,Default,,0000,0000,0000,,[student] Oh my gosh; I was\Npanicking for the last 30 minutes.
Dialogue: 0,0:02:23.50,0:02:26.50,Default,,0000,0000,0000,,- Oh, well—never panic, because—\N- [laughs]
Dialogue: 0,0:02:26.50,0:02:30.74,Default,,0000,0000,0000,,I promise, I—even if there's\Nnobody here, I can record it...
Dialogue: 0,0:02:30.74,0:02:34.62,Default,,0000,0000,0000,,You know, just by myself.\N- Oh, sweet. Okay.
Dialogue: 0,0:02:36.03,0:02:40.14,Default,,0000,0000,0000,,[instructor] I'm kind of waiting on Jordan,\Nbecause he was having trouble.
Dialogue: 0,0:02:45.50,0:02:47.23,Default,,0000,0000,0000,,Is that Luke, Luke, Luke...?
Dialogue: 0,0:02:49.40,0:02:50.16,Default,,0000,0000,0000,,Nope.
Dialogue: 0,0:02:58.36,0:02:59.38,Default,,0000,0000,0000,,There's Jordan.
Dialogue: 0,0:03:03.68,0:03:07.19,Default,,0000,0000,0000,,- [student] Finally.\N- Heh. Do I have Audrey?
Dialogue: 0,0:03:42.07,0:03:43.65,Default,,0000,0000,0000,,[student] So, what happened there?
Dialogue: 0,0:03:45.51,0:03:47.26,Default,,0000,0000,0000,,[instructor] Your guess\Nis as good as mine.
Dialogue: 0,0:04:40.97,0:04:41.76,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:04:42.75,0:04:47.26,Default,,0000,0000,0000,,Um, so, it is 12:01, so I'm\Ngonna go ahead and start.
Dialogue: 0,0:04:47.26,0:04:50.76,Default,,0000,0000,0000,,And hopefully you printed\Nout, or, you know,
Dialogue: 0,0:04:50.76,0:04:54.07,Default,,0000,0000,0000,,copied the handouts that\Nhave been on Blackboard.
Dialogue: 0,0:04:54.07,0:04:56.50,Default,,0000,0000,0000,,Since we lost so much time today,
Dialogue: 0,0:04:56.50,0:04:59.76,Default,,0000,0000,0000,,I really think we'll probably\Njust get through 4.1,
Dialogue: 0,0:04:59.76,0:05:03.62,Default,,0000,0000,0000,,and then try to save the\N2.8 until next time,
Dialogue: 0,0:05:03.62,0:05:08.64,Default,,0000,0000,0000,,and maybe I can find a way to\Ncombine 2.8 with 4.2. We'll see.
Dialogue: 0,0:05:09.00,0:05:10.76,Default,,0000,0000,0000,,But 4.1, I'm glad you're here for me,
Dialogue: 0,0:05:10.76,0:05:15.27,Default,,0000,0000,0000,,because this one is a section on\Napplications. It's called related rates.
Dialogue: 0,0:05:15.27,0:05:19.38,Default,,0000,0000,0000,,And it... can be a little bit challenging;\Nbut it's also really fun.
Dialogue: 0,0:05:19.38,0:05:24.26,Default,,0000,0000,0000,,So let me get my screen-sharing\Ngoing for ya, here.
Dialogue: 0,0:06:03.13,0:06:04.09,Default,,0000,0000,0000,,Wow.
Dialogue: 0,0:06:04.09,0:06:06.59,Default,,0000,0000,0000,,Okay. That was a very long delay.
Dialogue: 0,0:06:07.24,0:06:11.22,Default,,0000,0000,0000,,I don't know if it's just that there's\Nso many people on Zoom, and it's slow?
Dialogue: 0,0:06:11.22,0:06:12.62,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:06:12.62,0:06:13.95,Default,,0000,0000,0000,,Hope this works.
Dialogue: 0,0:06:13.95,0:06:15.24,Default,,0000,0000,0000,,So, here's where we are.
Dialogue: 0,0:06:15.24,0:06:17.76,Default,,0000,0000,0000,,Today, I'm gonna definitely\Nget through 4.1, and like I said,
Dialogue: 0,0:06:17.76,0:06:22.51,Default,,0000,0000,0000,,if I need to try to squeeze 2.8\Ninto Wednesday, I'll do that.
Dialogue: 0,0:06:22.51,0:06:25.01,Default,,0000,0000,0000,,All right. And here's our\Nhandout for today.
Dialogue: 0,0:06:25.01,0:06:27.26,Default,,0000,0000,0000,,So this one is related rates.
Dialogue: 0,0:06:27.86,0:06:31.03,Default,,0000,0000,0000,,Try to work on my... focus.
Dialogue: 0,0:06:31.03,0:06:33.62,Default,,0000,0000,0000,,And this is the famous balloon problem.
Dialogue: 0,0:06:33.62,0:06:39.35,Default,,0000,0000,0000,,If I had a balloon, I would blow one\Nup for you; but I don't have a balloon.
Dialogue: 0,0:06:39.96,0:06:42.76,Default,,0000,0000,0000,,And I didn't want to go\Nto the store to get one.
Dialogue: 0,0:06:42.76,0:06:44.94,Default,,0000,0000,0000,,Hashtag: coronavirus.
Dialogue: 0,0:06:45.43,0:06:49.18,Default,,0000,0000,0000,,So this is the famous balloon problem\Nwhich we shall just try to imagine.
Dialogue: 0,0:06:49.63,0:06:54.51,Default,,0000,0000,0000,,It says, "Suppose I can blow up a balloon\Nat a rate of 3 cubic inches per second."
Dialogue: 0,0:06:54.51,0:06:57.49,Default,,0000,0000,0000,,So, [puffing noises] blowing it up;\Nit's getting bigger.
Dialogue: 0,0:06:58.03,0:07:03.25,Default,,0000,0000,0000,,This is a unit of volume.\NSo, ({\i1}dv{\i0}/{\i1}dt{\i0}) equals 3.
Dialogue: 0,0:07:03.25,0:07:06.00,Default,,0000,0000,0000,,It's the derivative of volume\Nwith respect to {\i1}t{\i0},
Dialogue: 0,0:07:06.00,0:07:08.51,Default,,0000,0000,0000,,because it's a rate of change in volume.
Dialogue: 0,0:07:09.56,0:07:12.51,Default,,0000,0000,0000,,Both volume and radius are changing\Nas you blow that thing up;
Dialogue: 0,0:07:12.51,0:07:15.63,Default,,0000,0000,0000,,the radius is increasing; well,\Neverything is increasing.
Dialogue: 0,0:07:15.63,0:07:18.75,Default,,0000,0000,0000,,The radius; the diameter; the\Ncircumference; the surface area;
Dialogue: 0,0:07:18.75,0:07:20.65,Default,,0000,0000,0000,,the volume; everything.
Dialogue: 0,0:07:20.65,0:07:25.35,Default,,0000,0000,0000,,The rate at which the radius is changing,\Nwe're gonna call that ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:07:25.35,0:07:28.26,Default,,0000,0000,0000,,Remember, a derivative\N{\i1}is{\i0} a rate of change.
Dialogue: 0,0:07:28.75,0:07:31.36,Default,,0000,0000,0000,,Now, at first, the radius grows quickly.
Dialogue: 0,0:07:31.36,0:07:34.69,Default,,0000,0000,0000,,So imagine, you know, when you've got\Nthe balloon, it's about this long.
Dialogue: 0,0:07:34.69,0:07:38.36,Default,,0000,0000,0000,,And you put in that first couple\Nof [puff-puff-puffff].
Dialogue: 0,0:07:38.36,0:07:43.20,Default,,0000,0000,0000,,And then it grows fast. Like, all of\Na sudden, it's a round shape.
Dialogue: 0,0:07:44.38,0:07:48.75,Default,,0000,0000,0000,,But as the balloon gets larger,\Nthe radius grows more slowly.
Dialogue: 0,0:07:48.75,0:07:52.76,Default,,0000,0000,0000,,So imagine you got a big balloon here,\Nand I put in a couple more puffs of air.
Dialogue: 0,0:07:52.76,0:07:53.49,Default,,0000,0000,0000,,[{\i1}puff{\i0}-{\i1}puff{\i0}]
Dialogue: 0,0:07:53.49,0:07:56.86,Default,,0000,0000,0000,,You hardly notice any change\Nin the shape of the balloon.
Dialogue: 0,0:07:57.66,0:08:01.52,Default,,0000,0000,0000,,So, as the balloon gets larger,\Nthe radius grows more slowly,
Dialogue: 0,0:08:01.52,0:08:06.99,Default,,0000,0000,0000,,even though the rate at which the\Nvolume is changing remains constant.
Dialogue: 0,0:08:06.99,0:08:10.93,Default,,0000,0000,0000,,So, it's always 3 cubic inches per second.
Dialogue: 0,0:08:10.93,0:08:14.75,Default,,0000,0000,0000,,It's just that that's more noticeable\Nwhen the balloon is this big;
Dialogue: 0,0:08:14.75,0:08:18.02,Default,,0000,0000,0000,,{\i1}less{\i0} noticeable when\Nthe balloon is {\i1}this{\i0} big.
Dialogue: 0,0:08:19.24,0:08:25.02,Default,,0000,0000,0000,,Volume and radius are related\Nby the formula V = (4/3) π r³.
Dialogue: 0,0:08:25.02,0:08:27.77,Default,,0000,0000,0000,,That's just the formula\Nfor volume of a sphere.
Dialogue: 0,0:08:28.48,0:08:33.02,Default,,0000,0000,0000,,And A says, "At what rate is the radius\Nincreasing with respect to time,
Dialogue: 0,0:08:33.02,0:08:36.90,Default,,0000,0000,0000,,when the radius is 2 inches?"
Dialogue: 0,0:08:39.75,0:08:42.27,Default,,0000,0000,0000,,Okay. So here's my little handout here.
Dialogue: 0,0:08:42.27,0:08:45.63,Default,,0000,0000,0000,,"At what rate is the radius increasing\Nwith respect to time
Dialogue: 0,0:08:45.63,0:08:48.02,Default,,0000,0000,0000,,when the radius is 2 inches?'
Dialogue: 0,0:08:48.61,0:08:49.35,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:08:49.35,0:08:52.03,Default,,0000,0000,0000,,So what we need to find, is this.
Dialogue: 0,0:08:52.51,0:08:55.52,Default,,0000,0000,0000,,We need ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:08:55.52,0:09:00.81,Default,,0000,0000,0000,,Because this is the rate\Nat which the {\i1}radius{\i0} is increasing.
Dialogue: 0,0:09:00.81,0:09:02.39,Default,,0000,0000,0000,,We need ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:09:02.95,0:09:04.70,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:09:04.70,0:09:11.90,Default,,0000,0000,0000,,So. I'm gonna take the formula\Nthat I have, which is V = (4/3) π r³;
Dialogue: 0,0:09:11.90,0:09:17.32,Default,,0000,0000,0000,,and we're gonna differentiate\Nthat with respect to {\i1}t{\i0}.
Dialogue: 0,0:09:17.32,0:09:18.68,Default,,0000,0000,0000,,Not {\i1}r{\i0}, but {\i1}t{\i0}.
Dialogue: 0,0:09:18.68,0:09:21.40,Default,,0000,0000,0000,,So, this is an implicit differentiation.
Dialogue: 0,0:09:21.40,0:09:23.61,Default,,0000,0000,0000,,I want it with respect to {\i1}t{\i0}.
Dialogue: 0,0:09:24.25,0:09:28.26,Default,,0000,0000,0000,,So I'm gonna find ({\i1}dv{\i0}/{\i1}dt{\i0}),
Dialogue: 0,0:09:28.26,0:09:33.02,Default,,0000,0000,0000,,and that will be the derivative with\Nrespect to {\i1}t{\i0} of the right-hand side,
Dialogue: 0,0:09:33.02,0:09:36.24,Default,,0000,0000,0000,,which is (4/3) π r³.
Dialogue: 0,0:09:36.24,0:09:40.02,Default,,0000,0000,0000,,So what I want you to notice here\Nis that {\i1}r{\i0} is the wrong letter.
Dialogue: 0,0:09:40.02,0:09:44.25,Default,,0000,0000,0000,,{\i1}t{\i0} is the right letter;\N{\i1}r{\i0} is the wrong letter.
Dialogue: 0,0:09:44.25,0:09:46.76,Default,,0000,0000,0000,,So, with this implicit differentiation,
Dialogue: 0,0:09:46.76,0:09:48.76,Default,,0000,0000,0000,,when I differentiate\Nthat right-hand side,
Dialogue: 0,0:09:48.76,0:09:53.50,Default,,0000,0000,0000,,I've got to follow it with an {\i1}r{\i0} prime,\N'cause it's the wrong letter.
Dialogue: 0,0:09:53.50,0:09:57.98,Default,,0000,0000,0000,,Now on the left-hand side, you know,\Nwe just leave that, (dv/dt),
Dialogue: 0,0:09:57.98,0:10:00.60,Default,,0000,0000,0000,,and here we go on the\Nright-hand side differentiating.
Dialogue: 0,0:10:00.60,0:10:03.75,Default,,0000,0000,0000,,So we'll do 3 times (4/3) is 4.
Dialogue: 0,0:10:03.75,0:10:06.75,Default,,0000,0000,0000,,And that's π r².
Dialogue: 0,0:10:06.75,0:10:10.49,Default,,0000,0000,0000,,Now, the chain rule says, multiply\Nby the derivative of the inside.
Dialogue: 0,0:10:10.49,0:10:12.32,Default,,0000,0000,0000,,And that inside is the {\i1}r{\i0}.
Dialogue: 0,0:10:12.32,0:10:16.41,Default,,0000,0000,0000,,So since it was the wrong letter,\Nthis is what we'd follow it with; {\i1}r{\i0} prime.
Dialogue: 0,0:10:16.41,0:10:18.04,Default,,0000,0000,0000,,Now here's the deal.
Dialogue: 0,0:10:18.04,0:10:23.27,Default,,0000,0000,0000,,So, instead of using {\i1}r{\i0} prime,\Nwhich is a perfectly fine symbol;
Dialogue: 0,0:10:23.27,0:10:24.36,Default,,0000,0000,0000,,but instead of using that one—
Dialogue: 0,0:10:24.36,0:10:26.38,Default,,0000,0000,0000,,- Professor?\N- Uh-huh.
Dialogue: 0,0:10:26.38,0:10:31.77,Default,,0000,0000,0000,,[student] Um, how did you get the r²?\NThe derivative function?
Dialogue: 0,0:10:31.77,0:10:34.88,Default,,0000,0000,0000,,Minus one?\N- [instructor] So, the 3 ti—mhmm.
Dialogue: 0,0:10:34.88,0:10:40.68,Default,,0000,0000,0000,,3 times (4/3) is 4,\Nπ, and then r ⁿ - 1. So r².
Dialogue: 0,0:10:40.68,0:10:42.36,Default,,0000,0000,0000,,[student] Okay. Gotcha.
Dialogue: 0,0:10:42.36,0:10:44.38,Default,,0000,0000,0000,,[instructor] Instead of\Nusing this {\i1}r{\i0} prime,
Dialogue: 0,0:10:44.38,0:10:47.51,Default,,0000,0000,0000,,we're gonna use {\i1}this{\i0}\Nnotation for the derivative.
Dialogue: 0,0:10:47.51,0:10:51.93,Default,,0000,0000,0000,,The only reason is because with\N({\i1}dr{\i0}/{\i1}dt{\i0}), it really makes it noticeable
Dialogue: 0,0:10:51.93,0:10:56.50,Default,,0000,0000,0000,,that we're talking about a rate of change\Nof radius with respect to time here.
Dialogue: 0,0:10:56.50,0:11:00.25,Default,,0000,0000,0000,,And with {\i1}r{\i0} prime, that's not\Nas obvious what the intent is.
Dialogue: 0,0:11:00.25,0:11:04.04,Default,,0000,0000,0000,,So we'll follow this with ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:11:04.51,0:11:09.11,Default,,0000,0000,0000,,So that's implicit differentiation\Ninvolving a chain rule.
Dialogue: 0,0:11:09.11,0:11:11.75,Default,,0000,0000,0000,,Now, what we need is ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:11:11.75,0:11:16.02,Default,,0000,0000,0000,,Well, we've got an equation here;\Nwe're just going to isolate ({\i1}dr{\i0}/{\i1}dt{\i0})
Dialogue: 0,0:11:16.02,0:11:17.75,Default,,0000,0000,0000,,in this equation.
Dialogue: 0,0:11:17.75,0:11:20.08,Default,,0000,0000,0000,,So if I isolate this equation...
Dialogue: 0,0:11:20.08,0:11:24.35,Default,,0000,0000,0000,,Let's see. How can I do this?\NI can... ummm... hmmm.
Dialogue: 0,0:11:24.35,0:11:29.93,Default,,0000,0000,0000,,I can first substitute in\Nplace of ({\i1}dv{\i0}/{\i1}dt{\i0}) the 3,
Dialogue: 0,0:11:29.93,0:11:35.60,Default,,0000,0000,0000,,because I was told that\N({\i1}dv{\i0}/{\i1}dt{\i0}) equals 3.
Dialogue: 0,0:11:35.60,0:11:41.75,Default,,0000,0000,0000,,So now, 3 equals 4π r² ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:11:41.94,0:11:48.52,Default,,0000,0000,0000,,And I say we divide everything by the\N4πr², and then we've got ({\i1}dr{\i0}/{\i1}dt{\i0}) all alone!
Dialogue: 0,0:11:49.37,0:11:56.40,Default,,0000,0000,0000,,So we'll say (dr/dt) = (3/4 π r²).
Dialogue: 0,0:11:56.40,0:11:57.75,Default,,0000,0000,0000,,Hmmmm.
Dialogue: 0,0:11:57.75,0:12:00.68,Default,,0000,0000,0000,,What was the {\i1}r{\i0} in this problem?
Dialogue: 0,0:12:00.68,0:12:02.26,Default,,0000,0000,0000,,What was the {\i1}r{\i0}.
Dialogue: 0,0:12:03.03,0:12:04.08,Default,,0000,0000,0000,,See right here?
Dialogue: 0,0:12:04.08,0:12:08.00,Default,,0000,0000,0000,,- [student] Two?\N- When the radius is 2 inches.
Dialogue: 0,0:12:08.00,0:12:11.79,Default,,0000,0000,0000,,So now we'll just substitute in\Nthe 2; we'll have a number.
Dialogue: 0,0:12:11.79,0:12:15.95,Default,,0000,0000,0000,,3 over 4 times π times r².
Dialogue: 0,0:12:15.95,0:12:18.39,Default,,0000,0000,0000,,So, 2 squared is another 4.
Dialogue: 0,0:12:18.39,0:12:21.77,Default,,0000,0000,0000,,This point, I'm gonna go to\Nmy calculator; see what that is.
Dialogue: 0,0:12:25.37,0:12:26.55,Default,,0000,0000,0000,,And we'll clear.
Dialogue: 0,0:12:26.55,0:12:29.52,Default,,0000,0000,0000,,Okay. So I want 3 divided by...
Dialogue: 0,0:12:30.51,0:12:32.63,Default,,0000,0000,0000,,And I'm gonna put that\Ndenominator in parentheses,
Dialogue: 0,0:12:32.63,0:12:36.25,Default,,0000,0000,0000,,so the calculator understands\NI'm dividing by all of this stuff.
Dialogue: 0,0:12:36.76,0:12:40.11,Default,,0000,0000,0000,,So that would be a 16π down there.
Dialogue: 0,0:12:40.11,0:12:48.77,Default,,0000,0000,0000,,And I get... ({\i1}dr{\i0}/{\i1}dt{\i0}) is\Napproximately 0.0597.
Dialogue: 0,0:12:48.77,0:12:52.90,Default,,0000,0000,0000,,I just chose to round that\Nto four decimal places.
Dialogue: 0,0:12:55.23,0:12:56.89,Default,,0000,0000,0000,,I need a unit, though.
Dialogue: 0,0:12:56.89,0:13:00.99,Default,,0000,0000,0000,,So this was a rate of change of\Nthe radius with respect to time.
Dialogue: 0,0:13:01.53,0:13:04.74,Default,,0000,0000,0000,,So the unit for radius was...
Dialogue: 0,0:13:04.74,0:13:07.19,Default,,0000,0000,0000,,-Inches per second.\N[Prof] -Inches.
Dialogue: 0,0:13:07.19,0:13:10.15,Default,,0000,0000,0000,,And for time, it was seconds.
Dialogue: 0,0:13:10.15,0:13:12.50,Default,,0000,0000,0000,,So, I'll circle this by itself.
Dialogue: 0,0:13:12.50,0:13:18.18,Default,,0000,0000,0000,,So, ({\i1}dr{\i0}/{\i1}dt{\i0}) equals\N0.0597 inches per second.
Dialogue: 0,0:13:18.18,0:13:21.61,Default,,0000,0000,0000,,The rate of change of the radius\Nwith respect to time.
Dialogue: 0,0:13:22.55,0:13:24.24,Default,,0000,0000,0000,,Okay, so that was the first example.
Dialogue: 0,0:13:24.24,0:13:26.51,Default,,0000,0000,0000,,And that's the way all of\Nthese are gonna work.
Dialogue: 0,0:13:26.51,0:13:28.26,Default,,0000,0000,0000,,You're gonna have a formula.
Dialogue: 0,0:13:29.00,0:13:30.76,Default,,0000,0000,0000,,Sometimes you're given the formula,
Dialogue: 0,0:13:30.76,0:13:34.65,Default,,0000,0000,0000,,and sometimes you have to figure\Nthe formula out; that's comin' up.
Dialogue: 0,0:13:35.28,0:13:38.11,Default,,0000,0000,0000,,And then once you get that formula,\Nyou're gonna be doing an, um—
Dialogue: 0,0:13:38.11,0:13:42.64,Default,,0000,0000,0000,,implicit differentiation with\Nrespect to time on all of these.
Dialogue: 0,0:13:42.64,0:13:47.18,Default,,0000,0000,0000,,With respect to time. And then,\Nsolving for an unknown.
Dialogue: 0,0:13:47.74,0:13:51.31,Default,,0000,0000,0000,,Okay, so let's look at part B.\NStill about the same problem.
Dialogue: 0,0:13:51.31,0:13:53.28,Default,,0000,0000,0000,,It s-says—"It s-ssays."
Dialogue: 0,0:13:53.28,0:13:57.75,Default,,0000,0000,0000,,It says, "Suppose I increase my\Neffort when {\i1}r{\i0} equals 2 inches,
Dialogue: 0,0:13:57.75,0:14:01.50,Default,,0000,0000,0000,,and begin to blow air into\Nthe balloon at a faster rate.
Dialogue: 0,0:14:01.50,0:14:03.96,Default,,0000,0000,0000,,A rate of 4 cubic inches per second.
Dialogue: 0,0:14:03.96,0:14:06.66,Default,,0000,0000,0000,,Well, how fast is the\Nradius changing now?"
Dialogue: 0,0:14:07.25,0:14:13.76,Default,,0000,0000,0000,,Okay. So in this case, the ({\i1}dv{\i0}/{\i1}dt{\i0})\Nis now equal to 4.
Dialogue: 0,0:14:14.76,0:14:20.27,Default,,0000,0000,0000,,So, I had ({\i1}dv{\i0}/{\i1}dt{\i0}) from above,\Nso I can just write that down again.
Dialogue: 0,0:14:20.27,0:14:22.17,Default,,0000,0000,0000,,Instead of reinventing the wheel.
Dialogue: 0,0:14:22.17,0:14:28.34,Default,,0000,0000,0000,,And it was 4 times π\Ntimes r² times ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:14:28.34,0:14:33.13,Default,,0000,0000,0000,,And now I'm going to substitute\Nthe 4 in place of the ({\i1}dv{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:14:33.13,0:14:35.74,Default,,0000,0000,0000,,So we've got that faster rate\Nof change of volume.
Dialogue: 0,0:14:36.12,0:14:39.74,Default,,0000,0000,0000,,So 4 equals 4 times π.
Dialogue: 0,0:14:39.74,0:14:45.61,Default,,0000,0000,0000,,That radius is still 2, as it says;\Nwhen {\i1}r{\i0} equals 2 inches.
Dialogue: 0,0:14:45.61,0:14:51.85,Default,,0000,0000,0000,,So that's a 2² ({\i1}dr{\i0}/{\i1}dt{\i0}), and we just\Nneed to solve for ({\i1}dr{\i0}/{\i1}dt{\i0}) again.
Dialogue: 0,0:14:53.25,0:14:59.52,Default,,0000,0000,0000,,So, to do that, I'll divide both\Nsides by the 4π times 2²?
Dialogue: 0,0:15:00.73,0:15:02.26,Default,,0000,0000,0000,,2² is 4.
Dialogue: 0,0:15:02.26,0:15:04.43,Default,,0000,0000,0000,,4 times 4 is 16.
Dialogue: 0,0:15:04.43,0:15:07.71,Default,,0000,0000,0000,,So, there's that 16π again.
Dialogue: 0,0:15:07.71,0:15:09.75,Default,,0000,0000,0000,,So I'll bring out my calculator.
Dialogue: 0,0:15:10.57,0:15:13.11,Default,,0000,0000,0000,,And... turn it on.
Dialogue: 0,0:15:13.11,0:15:15.37,Default,,0000,0000,0000,,Ooh, I got a bad glare on that; sorry.
Dialogue: 0,0:15:15.76,0:15:18.75,Default,,0000,0000,0000,,Okay, so this one'll be 4 divided by.
Dialogue: 0,0:15:18.75,0:15:21.60,Default,,0000,0000,0000,,And then in parentheses, (16π).
Dialogue: 0,0:15:22.26,0:15:25.56,Default,,0000,0000,0000,,That would also reduce to ¼π.
Dialogue: 0,0:15:25.56,0:15:26.49,Default,,0000,0000,0000,,Dun't matter.
Dialogue: 0,0:15:26.49,0:15:35.88,Default,,0000,0000,0000,,At any rate, this ({\i1}dr{\i0}/{\i1}dt{\i0}) is\Napproximately 0.0796.
Dialogue: 0,0:15:35.88,0:15:39.38,Default,,0000,0000,0000,,And again, that would be\Ninches per second.
Dialogue: 0,0:15:39.38,0:15:42.26,Default,,0000,0000,0000,,So you notice, when you\Ncompare the two,
Dialogue: 0,0:15:42.26,0:15:47.33,Default,,0000,0000,0000,,that your radius here is changing\Nat a faster rate than it did here.
Dialogue: 0,0:15:47.33,0:15:50.85,Default,,0000,0000,0000,,Obviously; because you\Nwere increasing your effort,
Dialogue: 0,0:15:50.85,0:15:54.27,Default,,0000,0000,0000,,and the rate of change in\Nyour volume was higher.
Dialogue: 0,0:15:55.76,0:15:57.98,Default,,0000,0000,0000,,Okay. So, um.
Dialogue: 0,0:15:58.81,0:16:00.75,Default,,0000,0000,0000,,Hold on one second, y'all.
Dialogue: 0,0:16:02.36,0:16:06.91,Default,,0000,0000,0000,,I've got to mute you for a second\Nbecause my dog needs to go out.
Dialogue: 0,0:16:06.91,0:16:07.44,Default,,0000,0000,0000,,[exaggerated whisper]\N{\i1}I'm so sorry{\i0}.
Dialogue: 0,0:16:12.49,0:16:16.72,Default,,0000,0000,0000,,[Student] Hey, I just came back.\NMy house just had a rolling blackout.
Dialogue: 0,0:16:17.94,0:16:19.75,Default,,0000,0000,0000,,[Instructor] A rolling blackout?
Dialogue: 0,0:16:20.68,0:16:23.55,Default,,0000,0000,0000,,[Student] Yeah, I've been gone\Nfor, like, five minutes. [Chuckles]
Dialogue: 0,0:16:23.55,0:16:26.50,Default,,0000,0000,0000,,[Instructor] Whoaaa.\NSo, where do you live?
Dialogue: 0,0:16:27.97,0:16:32.51,Default,,0000,0000,0000,,[Student] Uh, right next to the\NCrow Bar. On South Congress.
Dialogue: 0,0:16:33.34,0:16:35.39,Default,,0000,0000,0000,,[Instructor] Oh, maaaan...
Dialogue: 0,0:16:35.39,0:16:38.25,Default,,0000,0000,0000,,Okay. That's all we need,\Nare blackouts.
Dialogue: 0,0:16:38.25,0:16:39.03,Default,,0000,0000,0000,,[Student] Yep.
Dialogue: 0,0:16:39.83,0:16:40.75,Default,,0000,0000,0000,,[Instructor] How fun.
Dialogue: 0,0:16:42.26,0:16:43.29,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,0:16:43.29,0:16:47.25,Default,,0000,0000,0000,,So, now, for part C it says, "At what\Nrate is the volume increasing
Dialogue: 0,0:16:47.25,0:16:52.34,Default,,0000,0000,0000,,with respect to the radius, when\Nthe radius is 1 inch or 3 inches?"
Dialogue: 0,0:16:52.34,0:16:56.42,Default,,0000,0000,0000,,"At what rate is the {\i1}volume{\i0} increasing."
Dialogue: 0,0:16:56.42,0:17:01.50,Default,,0000,0000,0000,,Okay. So, I'm gonna write a little note\Nhere to be careful with this one.
Dialogue: 0,0:17:01.50,0:17:03.43,Default,,0000,0000,0000,,You've got to read it carefully.
Dialogue: 0,0:17:03.43,0:17:07.53,Default,,0000,0000,0000,,"At what rate is the {\i1}volume{\i0}."\NSo, now, underline that.
Dialogue: 0,0:17:07.53,0:17:11.26,Default,,0000,0000,0000,,—"increasing with respect\Nto the {\i1}radius{\i0}."
Dialogue: 0,0:17:11.26,0:17:16.74,Default,,0000,0000,0000,,So, what we want here, is ({\i1}dv{\i0}/{\i1}dr{\i0}).
Dialogue: 0,0:17:17.50,0:17:21.40,Default,,0000,0000,0000,,This is the rate of change of the\Nvolume with respect to the radius.
Dialogue: 0,0:17:21.40,0:17:24.69,Default,,0000,0000,0000,,We need ({\i1}dv{\i0}/{\i1}dr{\i0}).
Dialogue: 0,0:17:24.69,0:17:28.38,Default,,0000,0000,0000,,So, we're gonna start with the\Nformula we were given again.
Dialogue: 0,0:17:28.38,0:17:33.63,Default,,0000,0000,0000,,And that was V = (4/3) π r³.
Dialogue: 0,0:17:33.63,0:17:36.25,Default,,0000,0000,0000,,That was our formula for\Nvolume of that sphere.
Dialogue: 0,0:17:36.25,0:17:37.96,Default,,0000,0000,0000,,We need ({\i1}dv{\i0}/{\i1}dr{\i0}).
Dialogue: 0,0:17:37.96,0:17:44.74,Default,,0000,0000,0000,,So, if you'll notice. When we get\N({\i1}dv{\i0}/{\i1}dr{\i0}), uh... {\i1}r{\i0} is the right letter.
Dialogue: 0,0:17:44.74,0:17:48.18,Default,,0000,0000,0000,,So this one's not gonna require\Nan implicit differentiation.
Dialogue: 0,0:17:48.18,0:17:50.19,Default,,0000,0000,0000,,This one's pretty straightforward.
Dialogue: 0,0:17:50.19,0:17:58.25,Default,,0000,0000,0000,,({\i1}dv{\i0}/{\i1}dr{\i0}). 3 times (4/3) is 4. Times π.\NTimes {\i1}r{\i0}², and there you have it.
Dialogue: 0,0:17:58.25,0:18:00.38,Default,,0000,0000,0000,,That's the rate of change of volume.
Dialogue: 0,0:18:00.38,0:18:04.77,Default,,0000,0000,0000,,Don't need to do—\Nfollowing it by a ({\i1}dr{\i0}/{\i1}dt{\i0}),
Dialogue: 0,0:18:04.77,0:18:07.79,Default,,0000,0000,0000,,because {\i1}r{\i0} was the correct\Nletter in the first place.
Dialogue: 0,0:18:08.57,0:18:14.09,Default,,0000,0000,0000,,So, now we just need to evaluate this when\N{\i1}r{\i0} is one inch, and when {\i1}r{\i0} is 3 inches.
Dialogue: 0,0:18:14.74,0:18:17.13,Default,,0000,0000,0000,,So, let's see if I can get\Na little more room here.
Dialogue: 0,0:18:20.71,0:18:21.51,Default,,0000,0000,0000,,There we go.
Dialogue: 0,0:18:21.51,0:18:29.18,Default,,0000,0000,0000,,So, at {\i1}r{\i0} = 1, ({\i1}dv{\i0}/{\i1}dr{\i0}) is equal to 4π.
Dialogue: 0,0:18:30.38,0:18:34.99,Default,,0000,0000,0000,,Okay? If I'm looking for my units here,\Nthis is a rate of change of {\i1}volume{\i0}
Dialogue: 0,0:18:34.99,0:18:37.25,Default,,0000,0000,0000,,with respect to the radius.
Dialogue: 0,0:18:37.25,0:18:41.03,Default,,0000,0000,0000,,The units of volume were inches cubed.
Dialogue: 0,0:18:41.74,0:18:45.76,Default,,0000,0000,0000,,The units for the radius were inches.
Dialogue: 0,0:18:45.76,0:18:49.50,Default,,0000,0000,0000,,Now. I don't want you to reduce\Nthat to inches squared. [Chuckles]
Dialogue: 0,0:18:49.50,0:18:50.90,Default,,0000,0000,0000,,Don't do that.
Dialogue: 0,0:18:50.90,0:18:53.28,Default,,0000,0000,0000,,So, this is a rate of change of volume.
Dialogue: 0,0:18:53.28,0:18:57.13,Default,,0000,0000,0000,,So what it says is that\Nwhen the radius is 1 inch,
Dialogue: 0,0:18:57.13,0:19:01.22,Default,,0000,0000,0000,,that your volume is changing\Nat a rate of 4π cubic inches
Dialogue: 0,0:19:01.22,0:19:04.63,Default,,0000,0000,0000,,for every {\i1}1-inch change in radius{\i0}.
Dialogue: 0,0:19:04.63,0:19:07.51,Default,,0000,0000,0000,,So, leave this be; it means something.
Dialogue: 0,0:19:07.51,0:19:10.52,Default,,0000,0000,0000,,It's describing how the\Nvolume is changing
Dialogue: 0,0:19:10.52,0:19:13.25,Default,,0000,0000,0000,,with respect to how\Nthe radius is changing.
Dialogue: 0,0:19:13.25,0:19:14.77,Default,,0000,0000,0000,,Does that make sense to y'all?
Dialogue: 0,0:19:18.100,0:19:20.14,Default,,0000,0000,0000,,[Student] Yeah.
Dialogue: 0,0:19:21.76,0:19:23.62,Default,,0000,0000,0000,,All right. Let's try {\i1}r{\i0} = 3.
Dialogue: 0,0:19:23.62,0:19:29.24,Default,,0000,0000,0000,,So, ({\i1}dv{\i0}/{\i1}dr{\i0}) in this case\Nwould be 4π times 3².
Dialogue: 0,0:19:29.24,0:19:34.78,Default,,0000,0000,0000,,3² is 9. 9 times 4.\NThis'll be 36π.
Dialogue: 0,0:19:35.65,0:19:37.38,Default,,0000,0000,0000,,And I'm gonna leave it like that.
Dialogue: 0,0:19:38.14,0:19:42.75,Default,,0000,0000,0000,,And my units, again,\Nare inches cubed, per inch.
Dialogue: 0,0:19:43.75,0:19:46.77,Default,,0000,0000,0000,,Sounds better if I say "cubic\Ninches per inch," I think.
Dialogue: 0,0:19:47.51,0:19:48.20,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:19:48.75,0:19:50.62,Default,,0000,0000,0000,,So, there's example one.
Dialogue: 0,0:19:50.62,0:19:51.25,Default,,0000,0000,0000,,And that was—
Dialogue: 0,0:19:51.25,0:19:56.51,Default,,0000,0000,0000,,[Student] Is there a place that we can\Nget our... or find our graded tests?
Dialogue: 0,0:19:56.51,0:19:58.24,Default,,0000,0000,0000,,[Student] -Like, you have—okay.\N[Instructor] -Yes. Yeah.
Dialogue: 0,0:19:58.24,0:20:02.27,Default,,0000,0000,0000,,[Instructor] So, um, when you\Ngo to your gradebook,
Dialogue: 0,0:20:02.27,0:20:06.69,Default,,0000,0000,0000,,and go down to, like, the row\Nthat the test is on...
Dialogue: 0,0:20:08.28,0:20:13.76,Default,,0000,0000,0000,,There should be a place where\Nyou can see my feedback,
Dialogue: 0,0:20:13.76,0:20:16.40,Default,,0000,0000,0000,,and that's where I uploaded\Nyour graded test.
Dialogue: 0,0:20:17.66,0:20:19.91,Default,,0000,0000,0000,,Can anybody else jump in\Nhere; if you found it,
Dialogue: 0,0:20:19.91,0:20:22.05,Default,,0000,0000,0000,,can you explain that\Nbetter than I just did?
Dialogue: 0,0:20:25.02,0:20:28.88,Default,,0000,0000,0000,,[Student #2] Just next to the grade,\Nthere's like, a little cloud thing in blue,
Dialogue: 0,0:20:28.88,0:20:30.70,Default,,0000,0000,0000,,which has the comment.
Dialogue: 0,0:20:30.70,0:20:34.27,Default,,0000,0000,0000,,[Student #1] -The little speech bubble.\N-Yeah. And you can find there.
Dialogue: 0,0:20:36.26,0:20:38.60,Default,,0000,0000,0000,,[Instructor] -Great. Thank you.\N[Student #1] -Yes, uh, thanks.
Dialogue: 0,0:20:38.60,0:20:39.54,Default,,0000,0000,0000,,[Instructor] Sure.
Dialogue: 0,0:20:41.51,0:20:44.76,Default,,0000,0000,0000,,All right; so now, related rates procedure.
Dialogue: 0,0:20:44.76,0:20:47.82,Default,,0000,0000,0000,,So we went through that first\Nexample pretty slowly.
Dialogue: 0,0:20:47.82,0:20:49.08,Default,,0000,0000,0000,,And so now I'm gonna show you;
Dialogue: 0,0:20:49.08,0:20:52.91,Default,,0000,0000,0000,,this is just the general way\Nwe're gonna handle all of these.
Dialogue: 0,0:20:52.91,0:20:56.50,Default,,0000,0000,0000,,So the first thing is, we're gonna\Ndraw a picture if we can.
Dialogue: 0,0:20:57.10,0:20:59.75,Default,,0000,0000,0000,,Uh, I didn't really need to draw\Na picture of the balloon problem.
Dialogue: 0,0:20:59.75,0:21:02.80,Default,,0000,0000,0000,,I could have drawn a sphere,\NI guess, if I wanted, but.
Dialogue: 0,0:21:02.80,0:21:04.42,Default,,0000,0000,0000,,For some of these, you {\i1}need{\i0}\Na diagram.
Dialogue: 0,0:21:04.42,0:21:07.16,Default,,0000,0000,0000,,You're gonna need a picture,\Nand you'll need to label things.
Dialogue: 0,0:21:07.75,0:21:11.05,Default,,0000,0000,0000,,And that's the second point,\Nis "label and assign variables."
Dialogue: 0,0:21:11.05,0:21:12.11,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:21:12.11,0:21:16.88,Default,,0000,0000,0000,,The third thing is, write down what\Nyou {\i1}know{\i0}, and what you {\i1}need{\i0} to know.
Dialogue: 0,0:21:17.38,0:21:20.90,Default,,0000,0000,0000,,So whatever the question's asking,\Nthat's what you {\i1}need{\i0} to know.
Dialogue: 0,0:21:20.90,0:21:24.75,Default,,0000,0000,0000,,And then what you {\i1}know{\i0}\Nis usually gonna be a formula
Dialogue: 0,0:21:24.75,0:21:27.90,Default,,0000,0000,0000,,associated with the shape\Nthat you're drawing.
Dialogue: 0,0:21:29.24,0:21:33.89,Default,,0000,0000,0000,,Then you wanna find an equation or\Na formula that relates the variables.
Dialogue: 0,0:21:34.51,0:21:39.76,Default,,0000,0000,0000,,So, oftentimes, this is gonna be\Na formula for volume, or for area.
Dialogue: 0,0:21:39.76,0:21:42.07,Default,,0000,0000,0000,,It could be the Pythagorean theorem.
Dialogue: 0,0:21:42.07,0:21:45.19,Default,,0000,0000,0000,,Just depends on the picture\Nthat we end up drawing.
Dialogue: 0,0:21:46.02,0:21:48.26,Default,,0000,0000,0000,,And then we're gonna use\Nimplicit differentiation
Dialogue: 0,0:21:48.26,0:21:51.88,Default,,0000,0000,0000,,to differentiate with respect to time.
Dialogue: 0,0:21:51.88,0:21:55.02,Default,,0000,0000,0000,,And then the last thing is just\Nsubstitute in your known values,
Dialogue: 0,0:21:55.02,0:21:57.25,Default,,0000,0000,0000,,and then solve for the unknown values.
Dialogue: 0,0:21:57.25,0:22:01.60,Default,,0000,0000,0000,,So, we're gonna follow that pattern\Non all of the rest of the problems.
Dialogue: 0,0:22:01.60,0:22:04.53,Default,,0000,0000,0000,,'Kay, next up is the famous\Nsliding-ladder problem.
Dialogue: 0,0:22:05.02,0:22:07.20,Default,,0000,0000,0000,,And I wish we were in a classroom,\Nbecause in a classroom,
Dialogue: 0,0:22:07.20,0:22:11.27,Default,,0000,0000,0000,,I bring in my meter stick,\Nand pretend it's a ladder,
Dialogue: 0,0:22:11.27,0:22:16.50,Default,,0000,0000,0000,,and then I prop it up against the wall,\Nand I pull it out slowly from the bottom,
Dialogue: 0,0:22:16.50,0:22:19.00,Default,,0000,0000,0000,,and watch it slam down on the floor.
Dialogue: 0,0:22:19.00,0:22:22.77,Default,,0000,0000,0000,,So, you know, the ladder\Nis sliding down the wall.
Dialogue: 0,0:22:22.77,0:22:26.91,Default,,0000,0000,0000,,And when you see it in class, I just\Nthink this makes a little more sense, but.
Dialogue: 0,0:22:26.91,0:22:27.82,Default,,0000,0000,0000,,Darn it!
Dialogue: 0,0:22:28.62,0:22:30.90,Default,,0000,0000,0000,,So this is the famous sliding-ladder problem.
Dialogue: 0,0:22:30.90,0:22:33.62,Default,,0000,0000,0000,,Says, "A 10-foot ladder\Nrests against a wall."
Dialogue: 0,0:22:33.62,0:22:36.94,Default,,0000,0000,0000,,So I'm just imagining a ladder\Npropped up against a wall.
Dialogue: 0,0:22:37.63,0:22:40.63,Default,,0000,0000,0000,,If the bottom of the ladder\Nslides away from the wall
Dialogue: 0,0:22:40.63,0:22:43.37,Default,,0000,0000,0000,,at a rate of 1 foot per second—
Dialogue: 0,0:22:43.37,0:22:48.10,Default,,0000,0000,0000,,so that's steady, constant pulling the\Nbottom of that ladder away from that wall—
Dialogue: 0,0:22:48.10,0:22:51.76,Default,,0000,0000,0000,,—"how fast is the top of the\Nladder sliding {\i1}down{\i0} the wall,
Dialogue: 0,0:22:51.76,0:22:55.50,Default,,0000,0000,0000,,when the bottom of the ladder\Nis 6 feet {\i1}from{\i0} the wall?"
Dialogue: 0,0:22:56.10,0:22:58.25,Default,,0000,0000,0000,,So the first thing that we talk about is,
Dialogue: 0,0:22:58.25,0:23:00.100,Default,,0000,0000,0000,,when that ladder is\Npropped up against the wall,
Dialogue: 0,0:23:00.100,0:23:05.52,Default,,0000,0000,0000,,and you're pulling the bottom\Nof the ladder, pulling it out slowly,
Dialogue: 0,0:23:05.52,0:23:09.64,Default,,0000,0000,0000,,the top of that ladder\Nis also falling down.
Dialogue: 0,0:23:09.64,0:23:13.20,Default,,0000,0000,0000,,But would it fall at the exact same rate
Dialogue: 0,0:23:13.20,0:23:16.62,Default,,0000,0000,0000,,at which you're pulling the\Nbottom of the ladder away?
Dialogue: 0,0:23:18.13,0:23:21.02,Default,,0000,0000,0000,,I mean, it's all one ladder.
Dialogue: 0,0:23:21.62,0:23:25.26,Default,,0000,0000,0000,,So, this is my ladder.\NAnd this is my wall.
Dialogue: 0,0:23:25.26,0:23:27.26,Default,,0000,0000,0000,,And I'm pulling the bottom away.
Dialogue: 0,0:23:27.26,0:23:30.26,Default,,0000,0000,0000,,It seems like whatever rate\NI'm pulling it away,
Dialogue: 0,0:23:30.26,0:23:34.50,Default,,0000,0000,0000,,that the top should slide\Ndown at that same rate.
Dialogue: 0,0:23:35.51,0:23:38.62,Default,,0000,0000,0000,,But if you think about it...\NI mean, really think about it.
Dialogue: 0,0:23:38.62,0:23:42.91,Default,,0000,0000,0000,,If there really were a ladder there, and\Nyou had a string tied around the bottom,
Dialogue: 0,0:23:42.91,0:23:44.62,Default,,0000,0000,0000,,and you're pulling it out,
Dialogue: 0,0:23:44.62,0:23:47.75,Default,,0000,0000,0000,,it's gonna slide down\Nthe wall slowly at first,
Dialogue: 0,0:23:47.75,0:23:51.10,Default,,0000,0000,0000,,but what happens when\Nit gets close to the floor?
Dialogue: 0,0:23:53.87,0:23:54.86,Default,,0000,0000,0000,,[Student] Speeds up.
Dialogue: 0,0:23:54.86,0:23:58.02,Default,,0000,0000,0000,,[Instructor] Yeah man, that thing\Nis gonna smack the floor so hard,
Dialogue: 0,0:23:58.02,0:24:01.63,Default,,0000,0000,0000,,it's gonna {\i1}damage{\i0} the floor!\NUnless it's on a carpet.
Dialogue: 0,0:24:02.19,0:24:03.61,Default,,0000,0000,0000,,So, what really happens is,
Dialogue: 0,0:24:03.61,0:24:06.90,Default,,0000,0000,0000,,even though we're pulling the\Nbottom out at a constant rate,
Dialogue: 0,0:24:06.90,0:24:10.51,Default,,0000,0000,0000,,the rate at which the top is\Nsliding down is increasing.
Dialogue: 0,0:24:11.50,0:24:13.40,Default,,0000,0000,0000,,Craziest thing about physics.
Dialogue: 0,0:24:14.37,0:24:15.92,Default,,0000,0000,0000,,So let's try to draw this.
Dialogue: 0,0:24:15.92,0:24:19.26,Default,,0000,0000,0000,,Says, "Draw a picture if you can."\NThat's the first bullet.
Dialogue: 0,0:24:19.26,0:24:24.06,Default,,0000,0000,0000,,So when I draw my picture,\NI'm just gonna draw a wall.
Dialogue: 0,0:24:24.43,0:24:28.30,Default,,0000,0000,0000,,And then, here's my ladder\Npropped against the wall.
Dialogue: 0,0:24:28.30,0:24:30.03,Default,,0000,0000,0000,,This is the floor.
Dialogue: 0,0:24:31.13,0:24:32.16,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:24:32.16,0:24:36.75,Default,,0000,0000,0000,,So, that ladder. One thing\NI know about it is that it's 10 feet.
Dialogue: 0,0:24:36.75,0:24:38.75,Default,,0000,0000,0000,,So I can label that "10."
Dialogue: 0,0:24:38.75,0:24:41.25,Default,,0000,0000,0000,,And of course, you notice,\NI just drew a triangle.
Dialogue: 0,0:24:41.25,0:24:44.28,Default,,0000,0000,0000,,So, the hypotenuse of that\Ntriangle is definitely 10.
Dialogue: 0,0:24:45.06,0:24:49.75,Default,,0000,0000,0000,,Now, I can think of this as\Nbeing in a coordinate system.
Dialogue: 0,0:24:49.75,0:24:54.23,Default,,0000,0000,0000,,And when I think of it that way,\Nthen the base of this triangle is {\i1}x{\i0},
Dialogue: 0,0:24:54.23,0:24:58.02,Default,,0000,0000,0000,,and so what {\i1}x{\i0} really represents\Nis the distance from the wall.
Dialogue: 0,0:24:58.75,0:25:01.74,Default,,0000,0000,0000,,And then {\i1}y{\i0} can be... Oh.
Dialogue: 0,0:25:01.74,0:25:04.51,Default,,0000,0000,0000,,The height of the wall\Nwhere the ladder meets.
Dialogue: 0,0:25:05.02,0:25:07.93,Default,,0000,0000,0000,,So, {\i1}x{\i0} is the distance from the base,
Dialogue: 0,0:25:07.93,0:25:12.74,Default,,0000,0000,0000,,and then {\i1}y{\i0} is the height of the\Nladder propped against the wall.
Dialogue: 0,0:25:12.74,0:25:15.50,Default,,0000,0000,0000,,So, I've labeled it with what I know.
Dialogue: 0,0:25:15.50,0:25:18.76,Default,,0000,0000,0000,,And I've assigned variables\Nto what I don't know.
Dialogue: 0,0:25:19.63,0:25:22.01,Default,,0000,0000,0000,,Um. There is something else I knew.
Dialogue: 0,0:25:22.01,0:25:25.100,Default,,0000,0000,0000,,It says, "If the bottom of the ladder\Nslides away from the wall
Dialogue: 0,0:25:25.100,0:25:28.64,Default,,0000,0000,0000,,at a rate of 1 foot per second."
Dialogue: 0,0:25:28.64,0:25:30.49,Default,,0000,0000,0000,,So, here's the bottom of the ladder.
Dialogue: 0,0:25:30.49,0:25:36.25,Default,,0000,0000,0000,,It's being pulled away from the wall\Nat a rate of 1 foot per second.
Dialogue: 0,0:25:36.25,0:25:42.62,Default,,0000,0000,0000,,That's one of our rates. And {\i1}x{\i0} is\Nthe thing that's changing there.
Dialogue: 0,0:25:42.62,0:25:49.57,Default,,0000,0000,0000,,So the distance from the base\Nis changing. That's actually ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:25:49.57,0:25:55.00,Default,,0000,0000,0000,,So I'm gonna write down what\NI know; that ({\i1}dx{\i0}/{\i1}dt{\i0}) equals 1.
Dialogue: 0,0:25:55.52,0:25:57.07,Default,,0000,0000,0000,,So I really know two things about this:
Dialogue: 0,0:25:57.07,0:26:01.05,Default,,0000,0000,0000,,I know the length of the ladder.\NAnd I know ({\i1}dx{\i0}/{\i1}dt{\i0}) is 1.
Dialogue: 0,0:26:02.85,0:26:05.02,Default,,0000,0000,0000,,So I wrote down what I know\Nand what I need to know—
Dialogue: 0,0:26:05.02,0:26:08.93,Default,,0000,0000,0000,,Oh no, I didn't. What do I {\i1}need{\i0}\Nto know? What's it asking for?
Dialogue: 0,0:26:09.40,0:26:13.39,Default,,0000,0000,0000,,How fast is the {\i1}top{\i0} of the\Nladder sliding {\i1}down{\i0} the wall.
Dialogue: 0,0:26:13.39,0:26:15.39,Default,,0000,0000,0000,,Well, as the top of that\Nladder slides down—
Dialogue: 0,0:26:15.39,0:26:16.88,Default,,0000,0000,0000,,[Student] ({\i1}dr{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:26:16.88,0:26:19.70,Default,,0000,0000,0000,,[Instructor] True. It's {\i1}y{\i0} that's changing.
Dialogue: 0,0:26:19.70,0:26:25.27,Default,,0000,0000,0000,,So what I want to know, what I\N{\i1}need{\i0} to know, is ({\i1}dy{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:26:26.34,0:26:28.38,Default,,0000,0000,0000,,Which is why it's called "related rates."
Dialogue: 0,0:26:28.38,0:26:31.75,Default,,0000,0000,0000,,You're gonna have multiple\Nrates in the same problem.
Dialogue: 0,0:26:31.75,0:26:37.13,Default,,0000,0000,0000,,({\i1}dx{\i0}/{\i1}dt{\i0}) is given to us as 1.\NWe wanna find ({\i1}dy{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:26:37.95,0:26:41.51,Default,,0000,0000,0000,,So now, the next thing says,\N"Find an equation or a formula
Dialogue: 0,0:26:41.51,0:26:43.96,Default,,0000,0000,0000,,that relates all the variables."
Dialogue: 0,0:26:43.96,0:26:46.51,Default,,0000,0000,0000,,So, back to our draw-ring.
Dialogue: 0,0:26:46.51,0:26:49.64,Default,,0000,0000,0000,,We have a 10, an {\i1}x{\i0}, and a {\i1}y{\i0}.
Dialogue: 0,0:26:49.64,0:26:53.50,Default,,0000,0000,0000,,What's a formula you know\Nthat relates these three numbers?
Dialogue: 0,0:26:54.26,0:26:55.75,Default,,0000,0000,0000,,[Student] Pythagoras theorem.
Dialogue: 0,0:26:56.67,0:26:58.75,Default,,0000,0000,0000,,[Instructor] Thank you, Pythagoras.
Dialogue: 0,0:26:58.75,0:27:01.44,Default,,0000,0000,0000,,Pythagoras makes our lives easy.
Dialogue: 0,0:27:01.44,0:27:04.13,Default,,0000,0000,0000,,"Py-tha-gor...as."
Dialogue: 0,0:27:04.13,0:27:06.62,Default,,0000,0000,0000,,So thank you, Pythagoras.\NAnd here's your theorem.
Dialogue: 0,0:27:06.62,0:27:12.36,Default,,0000,0000,0000,,It says that {\i1}x{\i0}² plus {\i1}y{\i0}²\Nis equal to 10².
Dialogue: 0,0:27:13.46,0:27:14.36,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:27:14.36,0:27:15.92,Default,,0000,0000,0000,,So that's our equation.
Dialogue: 0,0:27:15.92,0:27:18.87,Default,,0000,0000,0000,,And {\i1}that's{\i0} the thing\Nwe need to differentiate.
Dialogue: 0,0:27:18.87,0:27:20.62,Default,,0000,0000,0000,,So once we have that equation,
Dialogue: 0,0:27:20.62,0:27:26.03,Default,,0000,0000,0000,,we use implicit differentiation to\Ndifferentiate with respect to time.
Dialogue: 0,0:27:26.97,0:27:29.26,Default,,0000,0000,0000,,So, that means on the left-hand side,
Dialogue: 0,0:27:29.26,0:27:35.51,Default,,0000,0000,0000,,we want the derivative with\Nrespect to {\i1}t{\i0}, of {\i1}x{\i0}² plus {\i1}y{\i0}².
Dialogue: 0,0:27:35.51,0:27:40.75,Default,,0000,0000,0000,,On the right-hand side, the derivative\Nwith respect to {\i1}t{\i0} of 100.
Dialogue: 0,0:27:42.24,0:27:44.76,Default,,0000,0000,0000,,So now on the left,\Nwe're gonna split it up.
Dialogue: 0,0:27:44.76,0:27:47.25,Default,,0000,0000,0000,,We want the derivative of that sum.
Dialogue: 0,0:27:47.25,0:27:53.82,Default,,0000,0000,0000,,So I'm gonna write it as: the\Nderivative with respect to {\i1}t{\i0} of {\i1}x{\i0}²
Dialogue: 0,0:27:53.82,0:28:00.36,Default,,0000,0000,0000,,plus the derivative with\Nrespect to {\i1}t{\i0} of {\i1}y{\i0}², equals.
Dialogue: 0,0:28:00.36,0:28:04.17,Default,,0000,0000,0000,,And then, what is the derivative\Nwith respect to {\i1}t{\i0} of 100?
Dialogue: 0,0:28:04.85,0:28:05.93,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:28:06.74,0:28:07.76,Default,,0000,0000,0000,,[Student #1] -Zero.\N[Student #2] -Zero.
Dialogue: 0,0:28:07.76,0:28:10.100,Default,,0000,0000,0000,,It's the constant, so we're gonna\Nget a zero on the right-hand side.
Dialogue: 0,0:28:10.100,0:28:15.02,Default,,0000,0000,0000,,Now, on the ladder problems, when\Nyou know the length of the ladder,
Dialogue: 0,0:28:15.02,0:28:18.32,Default,,0000,0000,0000,,you'll have the constant on that\Nside of the Pythagorean theorem,
Dialogue: 0,0:28:18.32,0:28:20.76,Default,,0000,0000,0000,,and that derivative is\Nalways going to be zero.
Dialogue: 0,0:28:21.50,0:28:23.51,Default,,0000,0000,0000,,Now, on the left,\Nwe need to differentiate.
Dialogue: 0,0:28:23.51,0:28:27.95,Default,,0000,0000,0000,,So getting the derivative\Nwith respect to {\i1}t{\i0} of {\i1}x{\i0}²?
Dialogue: 0,0:28:28.75,0:28:30.30,Default,,0000,0000,0000,,{\i1}x{\i0} is the wrong letter.
Dialogue: 0,0:28:30.94,0:28:38.23,Default,,0000,0000,0000,,So, we'll do our 2{\i1}x{\i0}, all right, but then\Nwe've got to follow it by... ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:28:38.23,0:28:39.62,Default,,0000,0000,0000,,And that's the chain rule.
Dialogue: 0,0:28:39.62,0:28:44.50,Default,,0000,0000,0000,,Since {\i1}x{\i0} is the wrong letter, 2 times\N{\i1}x{\i0} is the derivative of the outside.
Dialogue: 0,0:28:44.50,0:28:46.65,Default,,0000,0000,0000,,This is the derivative of the inside.
Dialogue: 0,0:28:47.32,0:28:49.76,Default,,0000,0000,0000,,Now for the {\i1}y{\i0}² term?\NSame thing.
Dialogue: 0,0:28:49.76,0:28:51.38,Default,,0000,0000,0000,,{\i1}y{\i0} is the wrong letter.
Dialogue: 0,0:28:51.38,0:28:58.29,Default,,0000,0000,0000,,So we'll do 2{\i1}y{\i0}, followed by\N({\i1}dy{\i0}/{\i1}dt{\i0}) is equal to 0.
Dialogue: 0,0:28:58.29,0:28:59.15,Default,,0000,0000,0000,,Great.
Dialogue: 0,0:28:59.15,0:29:03.39,Default,,0000,0000,0000,,So now that we've differentiated, we're\Ngonna sub in the things that we know.
Dialogue: 0,0:29:04.03,0:29:06.53,Default,,0000,0000,0000,,So, what do we know here?
Dialogue: 0,0:29:07.12,0:29:11.24,Default,,0000,0000,0000,,Um, let's see. It says,\N"10-foot ladder"...
Dialogue: 0,0:29:11.24,0:29:16.01,Default,,0000,0000,0000,,A rate of 1 foot per second;\Nthat was ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:29:16.01,0:29:19.52,Default,,0000,0000,0000,,And we also are stopping this;\Nwe're looking at this
Dialogue: 0,0:29:19.52,0:29:23.39,Default,,0000,0000,0000,,when the bottom of the ladder\Nis 6 feet from the wall?
Dialogue: 0,0:29:24.13,0:29:28.43,Default,,0000,0000,0000,,Okay. So it's 6 feet from\Nthe wall right now. That's {\i1}x{\i0}.
Dialogue: 0,0:29:29.24,0:29:37.12,Default,,0000,0000,0000,,So I'll say 2 times 6 times ({\i1}dx{\i0}/{\i1}dt{\i0}),\Nwhich is... what, now?
Dialogue: 0,0:29:38.04,0:29:40.04,Default,,0000,0000,0000,,[Students] -One.\N[Instructor] -One. Yeah.
Dialogue: 0,0:29:40.04,0:29:46.25,Default,,0000,0000,0000,,And then plus 2 times {\i1}yyyy{\i0}.
Dialogue: 0,0:29:47.48,0:29:49.98,Default,,0000,0000,0000,,[Student] You can find that\Nusing Pythagoras' theorem.
Dialogue: 0,0:29:49.98,0:29:51.75,Default,,0000,0000,0000,,[Instructor] Exactly right.
Dialogue: 0,0:29:51.75,0:29:55.75,Default,,0000,0000,0000,,So, 2 times {\i1}y{\i0}, and then\Ntimes ({\i1}dy{\i0}/{\i1}dt{\i0}) equals 0.
Dialogue: 0,0:29:55.75,0:29:59.75,Default,,0000,0000,0000,,We don't know what {\i1}y{\i0} is,\Nbut we can find it.
Dialogue: 0,0:30:00.27,0:30:04.95,Default,,0000,0000,0000,,So I'm gonna go back over here,\Nand rewrite my Pythagorean theorem,
Dialogue: 0,0:30:04.95,0:30:08.99,Default,,0000,0000,0000,,which is {\i1}x{\i0}² + {\i1}y{\i0}² = 10².
Dialogue: 0,0:30:08.99,0:30:12.06,Default,,0000,0000,0000,,I know what {\i1}x{\i0} is; {\i1}x{\i0} is 6.\NSo this is—
Dialogue: 0,0:30:12.06,0:30:13.97,Default,,0000,0000,0000,,[Student] -Professor?\N[Instructor] -Yes.
Dialogue: 0,0:30:13.97,0:30:19.39,Default,,0000,0000,0000,,[Student] So the 2 times 6 times 1.\NIs the 1 a derivative of the {\i1}x{\i0}?
Dialogue: 0,0:30:20.18,0:30:25.20,Default,,0000,0000,0000,,[Ins.] Yes. That was the rate of change\Nof {\i1}x{\i0} with respect to time. It was a 1.
Dialogue: 0,0:30:25.20,0:30:26.09,Default,,0000,0000,0000,,[Student] Okay.
Dialogue: 0,0:30:27.04,0:30:31.75,Default,,0000,0000,0000,,So, down here, 6² + {\i1}y{\i0}² = 100.
Dialogue: 0,0:30:31.75,0:30:36.39,Default,,0000,0000,0000,,That's {\i1}y{\i0}² = 100-36.
Dialogue: 0,0:30:36.39,0:30:39.64,Default,,0000,0000,0000,,{\i1}y{\i0}² is equal to... uh...
Dialogue: 0,0:30:40.91,0:30:42.62,Default,,0000,0000,0000,,Make that 64.
Dialogue: 0,0:30:43.75,0:30:46.51,Default,,0000,0000,0000,,And {\i1}y{\i0} must be 8.
Dialogue: 0,0:30:46.51,0:30:48.08,Default,,0000,0000,0000,,Negative-8 wouldn't make sense,
Dialogue: 0,0:30:48.08,0:30:51.51,Default,,0000,0000,0000,,so we're going with the\Npositive square root of 8.
Dialogue: 0,0:30:51.51,0:30:54.25,Default,,0000,0000,0000,,So now, I can plug in that unknown.
Dialogue: 0,0:30:54.25,0:30:56.59,Default,,0000,0000,0000,,And this is somethin' that\Ncommonly happens.
Dialogue: 0,0:30:56.59,0:31:00.75,Default,,0000,0000,0000,,So, once you've got your equation,\Nyou do your implicit differentiation;
Dialogue: 0,0:31:00.75,0:31:03.01,Default,,0000,0000,0000,,you fill in the stuff you know.
Dialogue: 0,0:31:03.01,0:31:08.16,Default,,0000,0000,0000,,A lot of times, there's another unknown\Nvariable that you've gotta go find,
Dialogue: 0,0:31:08.16,0:31:11.25,Default,,0000,0000,0000,,but you will be given the\Ninformation to find it,
Dialogue: 0,0:31:11.25,0:31:13.100,Default,,0000,0000,0000,,and it's usually from your formula.
Dialogue: 0,0:31:13.100,0:31:17.26,Default,,0000,0000,0000,,So you'll plug something in\Nto find something else;
Dialogue: 0,0:31:17.26,0:31:22.25,Default,,0000,0000,0000,,then you can sub it all in, and finally,\Njust be left with that one unknown,
Dialogue: 0,0:31:22.25,0:31:25.49,Default,,0000,0000,0000,,which is ({\i1}dy{\i0}/{\i1}dt{\i0}),\Nand that's what we want.
Dialogue: 0,0:31:25.91,0:31:28.16,Default,,0000,0000,0000,,Well, 2 times 6 is 12.
Dialogue: 0,0:31:28.16,0:31:34.76,Default,,0000,0000,0000,,12 plus 2 times 8 is 16;\Ntimes ({\i1}dy{\i0}/{\i1}dt{\i0}) equals 0.
Dialogue: 0,0:31:34.76,0:31:42.96,Default,,0000,0000,0000,,Let's try to isolate ({\i1}dy{\i0}/{\i1}dt{\i0}),\Nso I have 16 ({\i1}dy{\i0}/{\i1}dt{\i0}) = -12;
Dialogue: 0,0:31:42.96,0:31:46.89,Default,,0000,0000,0000,,and the last step is just\Ndividing both sides by 16.
Dialogue: 0,0:31:46.89,0:31:54.11,Default,,0000,0000,0000,,({\i1}dy{\i0}/{\i1}dt{\i0}) is equal to -12 over 16,\Nand that is negative...
Dialogue: 0,0:31:54.78,0:31:57.25,Default,,0000,0000,0000,,-It's like, three-fourths?\N[Student] - Yep.
Dialogue: 0,0:31:58.06,0:32:03.33,Default,,0000,0000,0000,,So negative 0.75.\NAnd then our units for this,
Dialogue: 0,0:32:03.33,0:32:07.26,Default,,0000,0000,0000,,since this is a change in {\i1}y{\i0}\Nwith respect to {\i1}t{\i0},
Dialogue: 0,0:32:07.26,0:32:10.39,Default,,0000,0000,0000,,is gonna be feet per second.
Dialogue: 0,0:32:11.07,0:32:14.61,Default,,0000,0000,0000,,The units of {\i1}y{\i0} were feet;\Nthe units of time were seconds,
Dialogue: 0,0:32:14.61,0:32:19.50,Default,,0000,0000,0000,,so in ({\i1}dy{\i0}/{\i1}dt{\i0}), units are feet per second.\NMan, I've almost run out of—
Dialogue: 0,0:32:19.50,0:32:22.50,Default,,0000,0000,0000,,[Student] Would you have a\Npreference on fraction or decimal?
Dialogue: 0,0:32:22.50,0:32:24.74,Default,,0000,0000,0000,,[Instructor] Oh no, I don't. Nah.
Dialogue: 0,0:32:25.92,0:32:28.96,Default,,0000,0000,0000,,To me, on these kind of problems,\Nthough, the decimals...
Dialogue: 0,0:32:30.07,0:32:32.97,Default,,0000,0000,0000,,I guess I like 'em better because\NI can imagine that better.
Dialogue: 0,0:32:33.74,0:32:38.02,Default,,0000,0000,0000,,Like, I have an idea of -0.75 feet\Nper second, but -3/4—
Dialogue: 0,0:32:38.02,0:32:39.76,Default,,0000,0000,0000,,Well, I guess it wouldn't {\i1}matter{\i0}.
Dialogue: 0,0:32:40.25,0:32:41.38,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,0:32:42.77,0:32:44.36,Default,,0000,0000,0000,,Whatever makes you happy.
Dialogue: 0,0:32:49.30,0:32:51.86,Default,,0000,0000,0000,,Okay. Now, that was the famous\Nladder problem, and—
Dialogue: 0,0:32:51.86,0:32:55.41,Default,,0000,0000,0000,,[chuckles] because, in every calculus\Nbook since the history of calculus,
Dialogue: 0,0:32:55.41,0:32:58.16,Default,,0000,0000,0000,,there has been a ladder problem.
Dialogue: 0,0:32:58.16,0:33:01.10,Default,,0000,0000,0000,,And you will have more ladder\Nproblems in your homework.
Dialogue: 0,0:33:01.10,0:33:04.83,Default,,0000,0000,0000,,And you will most likely have\Na ladder problem on your next test.
Dialogue: 0,0:33:04.83,0:33:05.98,Default,,0000,0000,0000,,[Whispers] It's famous.
Dialogue: 0,0:33:06.94,0:33:10.02,Default,,0000,0000,0000,,Okay, the last question says,\N"How fast is the top moving down
Dialogue: 0,0:33:10.02,0:33:12.92,Default,,0000,0000,0000,,when the ladder is\N9 feet from the wall."
Dialogue: 0,0:33:12.92,0:33:15.09,Default,,0000,0000,0000,,How about 9.9 feet.
Dialogue: 0,0:33:15.09,0:33:18.63,Default,,0000,0000,0000,,How about 9.99999999999 feet?
Dialogue: 0,0:33:19.50,0:33:21.100,Default,,0000,0000,0000,,So in other words:\Nthe ladder's only 10 feet.
Dialogue: 0,0:33:21.100,0:33:25.12,Default,,0000,0000,0000,,So, when you're pulling it out.\NWhen it's 9 feet—
Dialogue: 0,0:33:25.12,0:33:29.18,Default,,0000,0000,0000,,I mean, most of the ladder is down.\NIt only has another foot to fall;
Dialogue: 0,0:33:29.18,0:33:33.08,Default,,0000,0000,0000,,so we're looking at the speed\Nat which it's falling at that point.
Dialogue: 0,0:33:33.71,0:33:34.47,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:33:35.51,0:33:39.100,Default,,0000,0000,0000,,So, let's go back to when {\i1}x{\i0} equals 9.
Dialogue: 0,0:33:39.100,0:33:43.27,Default,,0000,0000,0000,,Because we need to figure out\Nwhat {\i1}y{\i0} is at that point.
Dialogue: 0,0:33:43.27,0:33:45.75,Default,,0000,0000,0000,,'Cause, you know, if I'm\Ndrawing a picture of it...
Dialogue: 0,0:33:46.64,0:33:49.74,Default,,0000,0000,0000,,It now looks like that, right?
Dialogue: 0,0:33:49.74,0:33:52.76,Default,,0000,0000,0000,,So it's almost all the way on the ground.
Dialogue: 0,0:33:52.76,0:33:56.13,Default,,0000,0000,0000,,So when {\i1}x{\i0} is 9, let's\Nfigure out what {\i1}y{\i0} is.
Dialogue: 0,0:33:56.13,0:34:01.26,Default,,0000,0000,0000,,So using our Pythagorean\Ntheorem, {\i1}x{\i0}² + {\i1}y{\i0}² = 10².
Dialogue: 0,0:34:02.00,0:34:04.78,Default,,0000,0000,0000,,That is, 9² is 81.
Dialogue: 0,0:34:05.37,0:34:07.75,Default,,0000,0000,0000,,Plus {\i1}y{\i0}² equals 100.
Dialogue: 0,0:34:07.75,0:34:13.25,Default,,0000,0000,0000,,So then y² is 100 minus 81,\Nwhich would be 19,
Dialogue: 0,0:34:13.25,0:34:17.01,Default,,0000,0000,0000,,and {\i1}y{\i0} will be the square root of 19.
Dialogue: 0,0:34:17.63,0:34:23.26,Default,,0000,0000,0000,,So then, we'll go back to\Nour "({\i1}dy{\i0}/{\i1}dt{\i0}) equals."
Dialogue: 0,0:34:23.63,0:34:26.77,Default,,0000,0000,0000,,And our ({\i1}dy{\i0}/{\i1}dt{\i0}) was...
Dialogue: 0,0:34:27.16,0:34:30.62,Default,,0000,0000,0000,,Oh, man. Do I have to reinvent that wheel?
Dialogue: 0,0:34:33.100,0:34:35.50,Default,,0000,0000,0000,,Shoot. I do.
Dialogue: 0,0:34:35.99,0:34:41.50,Default,,0000,0000,0000,,So, ({\i1}dy{\i0}/{\i1}dt{\i0}) would equal. I'm gonna go\Nback to this step so I can isolate ({\i1}dy{\i0}/{\i1}dt{\i0})
Dialogue: 0,0:34:41.50,0:34:45.08,Default,,0000,0000,0000,,before I've substituted in\Nnumbers for {\i1}x{\i0} and for {\i1}y{\i0}.
Dialogue: 0,0:34:45.08,0:34:47.24,Default,,0000,0000,0000,,({\i1}dy{\i0}/{\i1}dt{\i0}) would be...
Dialogue: 0,0:34:48.18,0:34:58.97,Default,,0000,0000,0000,,Will be -2{\i1}x{\i0} times ({\i1}dx{\i0}/{\i1}dt{\i0}),\Nand then that would be divided by 2{\i1}y{\i0}.
Dialogue: 0,0:34:59.85,0:35:00.82,Default,,0000,0000,0000,,Think I got it.
Dialogue: 0,0:35:00.82,0:35:05.94,Default,,0000,0000,0000,,({\i1}dy{\i0}/{\i1}dt{\i0}) would be -2{\i1}x{\i0}({\i1}dx{\i0}/{\i1}dt{\i0}) when\Nyou subtract this from both sides,
Dialogue: 0,0:35:05.94,0:35:10.53,Default,,0000,0000,0000,,and then to isolate the ({\i1}dy{\i0}/{\i1}dt{\i0}),\Nyou're dividing both sides by 2{\i1}y{\i0}.
Dialogue: 0,0:35:10.53,0:35:12.79,Default,,0000,0000,0000,,So. It looks ugly, but\Nthis is what it looks like.
Dialogue: 0,0:35:13.21,0:35:15.31,Default,,0000,0000,0000,,Now we'll substitute in\Nour new information.
Dialogue: 0,0:35:15.31,0:35:20.79,Default,,0000,0000,0000,,So, our new {\i1}x{\i0} is a 9.\NSo this is -2 times 9.
Dialogue: 0,0:35:20.79,0:35:23.14,Default,,0000,0000,0000,,({\i1}dx{\i0}/{\i1}dt{\i0}) is still 1.
Dialogue: 0,0:35:23.63,0:35:27.99,Default,,0000,0000,0000,,And then 2 times {\i1}y{\i0} would be\N2 times the square root of 19.
Dialogue: 0,0:35:28.59,0:35:31.54,Default,,0000,0000,0000,,Now, I plugged all that into my\Ncalculator already,
Dialogue: 0,0:35:31.54,0:35:38.40,Default,,0000,0000,0000,,and that was approximately\N-2.06 feet per second.
Dialogue: 0,0:35:38.40,0:35:42.15,Default,,0000,0000,0000,,So it sped up. Remember when it was\N6 feet away,
Dialogue: 0,0:35:42.15,0:35:46.40,Default,,0000,0000,0000,,the speed at which the top was falling\Nwas -0.75 feet per second.
Dialogue: 0,0:35:46.40,0:35:51.13,Default,,0000,0000,0000,,Now, it sped up to -2.06\Nfeet per second.
Dialogue: 0,0:35:51.13,0:35:53.19,Default,,0000,0000,0000,,[Student] -Professor?\N[Instructor] -Yes, go ahead.
Dialogue: 0,0:35:53.19,0:35:55.96,Default,,0000,0000,0000,,[Student] The 9.9, did you round it up?
Dialogue: 0,0:35:57.76,0:35:59.26,Default,,0000,0000,0000,,[Student] The 10? The 10²?
Dialogue: 0,0:36:01.76,0:36:05.50,Default,,0000,0000,0000,,[Student] Know when it says\N{\i1}x{\i0} + {\i1}y{\i0}² = the 10². Is it from the—
Dialogue: 0,0:36:05.50,0:36:06.25,Default,,0000,0000,0000,,[Instructor] Yes.
Dialogue: 0,0:36:06.25,0:36:09.51,Default,,0000,0000,0000,,[Student] But it's a question,\Nor you just rounded it up?
Dialogue: 0,0:36:10.50,0:36:13.26,Default,,0000,0000,0000,,[Instructor] So this is still going back\Nto my Pythagorean theorem.
Dialogue: 0,0:36:13.26,0:36:14.14,Default,,0000,0000,0000,,[Student] Oh, okay.
Dialogue: 0,0:36:14.14,0:36:17.78,Default,,0000,0000,0000,,[Instructor] I still have a hypotenuse\Nof 10 there; the base is 9.
Dialogue: 0,0:36:18.75,0:36:20.27,Default,,0000,0000,0000,,And we were looking for {\i1}y{\i0}.
Dialogue: 0,0:36:20.72,0:36:21.97,Default,,0000,0000,0000,,-Gotcha.\N-Yeah.
Dialogue: 0,0:36:22.77,0:36:25.76,Default,,0000,0000,0000,,Turned out to be...\Nthe square root of 19.
Dialogue: 0,0:36:25.76,0:36:26.91,Default,,0000,0000,0000,,That fits in there.
Dialogue: 0,0:36:26.91,0:36:32.16,Default,,0000,0000,0000,,So then I would do it again for\N9.9, and then for 9.9999999...
Dialogue: 0,0:36:32.16,0:36:34.76,Default,,0000,0000,0000,,I don't have room, so I'm\Ngonna talk you through it.
Dialogue: 0,0:36:35.37,0:36:40.28,Default,,0000,0000,0000,,So, when you get to 9.9. That\Nladder's almost all the way down.
Dialogue: 0,0:36:40.80,0:36:45.11,Default,,0000,0000,0000,,When you go through and calculate\Nthe rate of change of {\i1}y{\i0} with respect to {\i1}t{\i0},
Dialogue: 0,0:36:45.11,0:36:49.97,Default,,0000,0000,0000,,when {\i1}x{\i0} is 9.9, your rate is then...
Dialogue: 0,0:36:50.50,0:36:54.21,Default,,0000,0000,0000,,Uh, -7 feet per second.
Dialogue: 0,0:36:54.21,0:36:58.94,Default,,0000,0000,0000,,When you go to 9.9999999,\Nit's approaching infinity.
Dialogue: 0,0:36:59.66,0:37:03.76,Default,,0000,0000,0000,,It is negative, but so large,\Nit's incredible.
Dialogue: 0,0:37:03.76,0:37:08.76,Default,,0000,0000,0000,,So, as it's slamming the floor, the rate\Nat which it's slamming the floor?
Dialogue: 0,0:37:08.76,0:37:10.76,Default,,0000,0000,0000,,That rate is approaching infinity.
Dialogue: 0,0:37:11.76,0:37:14.61,Default,,0000,0000,0000,,Can't make this stuff up.\NIt's really true.
Dialogue: 0,0:37:15.73,0:37:17.50,Default,,0000,0000,0000,,That's why it damages the floor.
Dialogue: 0,0:37:18.45,0:37:19.74,Default,,0000,0000,0000,,It's pretty darn fast.
Dialogue: 0,0:37:21.50,0:37:25.49,Default,,0000,0000,0000,,All right, and that is another\Nfamous sliding-ladder problem.
Dialogue: 0,0:37:26.51,0:37:27.78,Default,,0000,0000,0000,,We'll take that one away.
Dialogue: 0,0:37:28.26,0:37:32.21,Default,,0000,0000,0000,,And now I'm lookin' at\Nnumber 10 from the exercises.
Dialogue: 0,0:37:32.21,0:37:33.86,Default,,0000,0000,0000,,This one's comin' up next.
Dialogue: 0,0:37:35.38,0:37:39.01,Default,,0000,0000,0000,,Probably better also check what\Ntime it is. 12:34? We're good.
Dialogue: 0,0:37:40.24,0:37:46.79,Default,,0000,0000,0000,,So exercise 10 says: "A particle\Nmoves along the curve; {\i1}y{\i0} = √(1+{\i1}x{\i0}³)."
Dialogue: 0,0:37:47.76,0:37:53.06,Default,,0000,0000,0000,,"As it reaches the 0.23, the\N{\i1}y{\i0} coordinate is increasing at a rate
Dialogue: 0,0:37:53.06,0:37:55.37,Default,,0000,0000,0000,,of 4 centimeters per second."
Dialogue: 0,0:37:56.02,0:37:57.86,Default,,0000,0000,0000,,That's ({\i1}dy{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:37:58.52,0:38:02.62,Default,,0000,0000,0000,,"How fast is the {\i1}x{\i0} coordinate of\Nthe point changing at that instant?"
Dialogue: 0,0:38:02.62,0:38:07.10,Default,,0000,0000,0000,,Okay. So here, the graph that we draw\Nis the graph of the function.
Dialogue: 0,0:38:07.10,0:38:11.51,Default,,0000,0000,0000,,So the curve is {\i1}y{\i0} = √(1+{\i1}x{\i0}³).
Dialogue: 0,0:38:11.51,0:38:13.43,Default,,0000,0000,0000,,That's the graph we want to draw.
Dialogue: 0,0:38:13.98,0:38:16.62,Default,,0000,0000,0000,,So I'm gonna draw my\Ncoordinate system here.
Dialogue: 0,0:38:17.10,0:38:18.36,Default,,0000,0000,0000,,Like so.
Dialogue: 0,0:38:18.94,0:38:22.51,Default,,0000,0000,0000,,And I graphed this on a graphing\Ncalculator earlier to see what it looks like;
Dialogue: 0,0:38:22.51,0:38:28.66,Default,,0000,0000,0000,,and you don't have to be exactly right,\Nbut it looks something like that.
Dialogue: 0,0:38:29.36,0:38:34.50,Default,,0000,0000,0000,,And then this point, I'm gonna\Nlabel this point right here at (2,3),
Dialogue: 0,0:38:34.50,0:38:37.76,Default,,0000,0000,0000,,because the particle is moving along,
Dialogue: 0,0:38:37.76,0:38:40.52,Default,,0000,0000,0000,,and at some point, it's\Ngonna reach that point.
Dialogue: 0,0:38:41.25,0:38:45.75,Default,,0000,0000,0000,,Particle's moving along the curve.\NAs it reaches the point (2,3),
Dialogue: 0,0:38:45.75,0:38:50.91,Default,,0000,0000,0000,,the {\i1}y{\i0} coordinate is increasing at\Na rate of 4 centimeters per second.
Dialogue: 0,0:38:51.49,0:38:56.16,Default,,0000,0000,0000,,So we know that ({\i1}dy{\i0}/{\i1}dt{\i0}) equals 4.
Dialogue: 0,0:38:56.55,0:39:00.03,Default,,0000,0000,0000,,The question is, how fast is\Nthe {\i1}x{\i0} coordinate of the point
Dialogue: 0,0:39:00.03,0:39:01.77,Default,,0000,0000,0000,,changing at that instant?
Dialogue: 0,0:39:02.25,0:39:07.25,Default,,0000,0000,0000,,So, what we want is ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:39:07.74,0:39:10.65,Default,,0000,0000,0000,,We know ({\i1}dy{\i0}/{\i1}dt{\i0});\Nwe want ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:39:11.50,0:39:15.51,Default,,0000,0000,0000,,So if I look at, you know, my little\Nbullets, and see where I'm at.
Dialogue: 0,0:39:15.51,0:39:16.90,Default,,0000,0000,0000,,I drew a picture.
Dialogue: 0,0:39:17.51,0:39:20.07,Default,,0000,0000,0000,,It says, "Label and assign variables."
Dialogue: 0,0:39:20.50,0:39:23.37,Default,,0000,0000,0000,,Well, I guess I kind of did.\NI've got the point labeled,
Dialogue: 0,0:39:23.37,0:39:26.100,Default,,0000,0000,0000,,and I wrote down\Nwhat ({\i1}dy{\i0}/{\i1}dt{\i0}) is, and...
Dialogue: 0,0:39:26.100,0:39:30.27,Default,,0000,0000,0000,,I wrote down what I don't\Nknow, which is ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:39:30.87,0:39:34.76,Default,,0000,0000,0000,,So then I find an equation or formula\Nthat relates all of these variables.
Dialogue: 0,0:39:34.76,0:39:41.49,Default,,0000,0000,0000,,Well, that equation or formula\Nis the {\i1}y{\i0} = √(1+{\i1}x{\i0}³).
Dialogue: 0,0:39:41.49,0:39:43.76,Default,,0000,0000,0000,,That's relating {\i1}x{\i0} and {\i1}y{\i0}.
Dialogue: 0,0:39:44.50,0:39:49.30,Default,,0000,0000,0000,,We wanna use implicit differentiation\Nnow to differentiate with respect to time.
Dialogue: 0,0:39:49.30,0:39:54.75,Default,,0000,0000,0000,,And then, we'll substitute in what we\Nknow; solve for what we don't know.
Dialogue: 0,0:39:55.90,0:40:00.62,Default,,0000,0000,0000,,So now I need to find the\Nderivative with respect to {\i1}t{\i0}.
Dialogue: 0,0:40:00.62,0:40:05.02,Default,,0000,0000,0000,,So I want derivative with respect\Nto {\i1}t{\i0} of the left-hand side.
Dialogue: 0,0:40:05.02,0:40:09.10,Default,,0000,0000,0000,,I want derivative with respect\Nto {\i1}t{\i0} of the right-hand side,
Dialogue: 0,0:40:09.10,0:40:14.51,Default,,0000,0000,0000,,which I'm going to rewrite\Nas (1+{\i1}x{\i0}³) to the ½ power.
Dialogue: 0,0:40:14.51,0:40:16.72,Default,,0000,0000,0000,,Just makes it easier\Nfor me to differentiate.
Dialogue: 0,0:40:17.50,0:40:21.51,Default,,0000,0000,0000,,So now on the left, it is just ({\i1}dy{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:40:22.24,0:40:26.96,Default,,0000,0000,0000,,And then on the right, derivative\Nof that (1+{\i1}x{\i0}³) to the ½.
Dialogue: 0,0:40:26.96,0:40:29.38,Default,,0000,0000,0000,,So bring my ½ down in front.
Dialogue: 0,0:40:29.38,0:40:33.74,Default,,0000,0000,0000,,(1+{\i1}x{\i0}³) to the -½ power.
Dialogue: 0,0:40:33.74,0:40:36.76,Default,,0000,0000,0000,,Now multiply by the\Nderivative of the inside.
Dialogue: 0,0:40:36.76,0:40:40.68,Default,,0000,0000,0000,,Okay, now. Your inside is this (1+{\i1}x{\i0}³).
Dialogue: 0,0:40:40.68,0:40:44.10,Default,,0000,0000,0000,,Derivative of (1+{\i1}x{\i0}³) is...
Dialogue: 0,0:40:45.18,0:40:46.26,Default,,0000,0000,0000,,3{\i1}x{\i0}².
Dialogue: 0,0:40:46.89,0:40:50.67,Default,,0000,0000,0000,,But now, chain rule says,\N"Do it again"; it's a double chain.
Dialogue: 0,0:40:51.28,0:40:55.52,Default,,0000,0000,0000,,Now we need to multiply by the\Nderivative of {\i1}x{\i0} with respect to {\i1}t{\i0}.
Dialogue: 0,0:40:55.52,0:40:58.76,Default,,0000,0000,0000,,Because {\i1}x{\i0} was the wrong letter.
Dialogue: 0,0:40:58.76,0:41:00.57,Default,,0000,0000,0000,,{\i1}t{\i0}'s the right letter;\N{\i1}x{\i0} is the wrong letter,
Dialogue: 0,0:41:00.57,0:41:03.46,Default,,0000,0000,0000,,so I've gotta follow it\Nwith that ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:41:04.26,0:41:10.28,Default,,0000,0000,0000,,Now I've differentiated implicitly;\Nnow it's time to sub in what I know.
Dialogue: 0,0:41:10.75,0:41:14.39,Default,,0000,0000,0000,,So, I do know that ({\i1}dy{\i0}/{\i1}dt{\i0}) is 4.
Dialogue: 0,0:41:15.50,0:41:21.39,Default,,0000,0000,0000,,That's 4 equals ½ times 1 plus...\NWhat's {\i1}x{\i0} at this point?
Dialogue: 0,0:41:23.06,0:41:23.79,Default,,0000,0000,0000,,[Student] -2.\N[Instructor] -2.
Dialogue: 0,0:41:24.33,0:41:31.77,Default,,0000,0000,0000,,So that's a 2³, to the -½.\NAnd that's times 3 times a 2².
Dialogue: 0,0:41:31.77,0:41:34.72,Default,,0000,0000,0000,,And then that's times ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:41:34.72,0:41:38.40,Default,,0000,0000,0000,,({\i1}dx{\i0}/{\i1}dt{\i0}) is the unknown;\Nthat's what I need to solve for.
Dialogue: 0,0:41:38.95,0:41:41.08,Default,,0000,0000,0000,,All right, so this is 4 equals.
Dialogue: 0,0:41:41.65,0:41:46.50,Default,,0000,0000,0000,,Ummm, 2³ is 8.\N8 plus 1 is 9.
Dialogue: 0,0:41:46.50,0:41:54.27,Default,,0000,0000,0000,,9 to the -½, so it's like 1 over\Nthe square root of 9, is... 3, I think.
Dialogue: 0,0:41:54.72,0:41:58.56,Default,,0000,0000,0000,,So this would be 1 over, 2 times...
Dialogue: 0,0:41:59.11,0:42:02.27,Default,,0000,0000,0000,,8+1 is 9; square root of that is 3.
Dialogue: 0,0:42:02.27,0:42:04.36,Default,,0000,0000,0000,,So that's 1 over 6.
Dialogue: 0,0:42:04.36,0:42:08.52,Default,,0000,0000,0000,,And then 3 times 2²;\Nthat's 4 times 3; that's 12.
Dialogue: 0,0:42:09.03,0:42:10.64,Default,,0000,0000,0000,,({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:42:11.26,0:42:16.51,Default,,0000,0000,0000,,So, this is 4 equals. 2 goes into 12\Nsix times; 6 over 3—
Dialogue: 0,0:42:16.51,0:42:20.15,Default,,0000,0000,0000,,that's just a 2 times ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:42:20.15,0:42:25.50,Default,,0000,0000,0000,,I think I'm ready to isolate my ({\i1}dx{\i0}/{\i1}dt{\i0})\Nby dividing both sides by 2.
Dialogue: 0,0:42:26.14,0:42:30.62,Default,,0000,0000,0000,,And ({\i1}dx{\i0}/{\i1}dt{\i0}) is 4 over 2, which is 2.
Dialogue: 0,0:42:31.21,0:42:34.89,Default,,0000,0000,0000,,And the rate is in centimeters per second.
Dialogue: 0,0:42:36.99,0:42:40.50,Default,,0000,0000,0000,,Okay. So sometimes, I guess, solving\Nthe equation after you substitute in
Dialogue: 0,0:42:40.50,0:42:43.50,Default,,0000,0000,0000,,your known values can get\Na little tricky, but you know;
Dialogue: 0,0:42:43.50,0:42:46.76,Default,,0000,0000,0000,,just take it one step at a time,\Nand you'll get there.
Dialogue: 0,0:42:47.40,0:42:49.27,Default,,0000,0000,0000,,So let me know how that one went.
Dialogue: 0,0:42:52.38,0:42:55.04,Default,,0000,0000,0000,,[Student] Can you just go over\Nwhat happened to, uh, 12?
Dialogue: 0,0:42:56.10,0:42:57.25,Default,,0000,0000,0000,,[Instructor] Yeah, sure.
Dialogue: 0,0:42:57.25,0:43:01.50,Default,,0000,0000,0000,,So, the ½, times the 12? Is 6.
Dialogue: 0,0:43:02.75,0:43:06.26,Default,,0000,0000,0000,,So I just canceled the 2 with\Nthe 12, leaving me a 6 on top;
Dialogue: 0,0:43:06.26,0:43:09.27,Default,,0000,0000,0000,,but 6 over 3 is 2.
Dialogue: 0,0:43:14.91,0:43:15.80,Default,,0000,0000,0000,,Good?
Dialogue: 0,0:43:17.76,0:43:18.52,Default,,0000,0000,0000,,[Student] Yeah.
Dialogue: 0,0:43:20.67,0:43:25.85,Default,,0000,0000,0000,,All right. So, you guys are so quiet.\NI don't—I don't like that about Zoom;
Dialogue: 0,0:43:25.85,0:43:28.77,Default,,0000,0000,0000,,it's different than being in a classroom;\Nin a classroom, you know...
Dialogue: 0,0:43:28.77,0:43:31.95,Default,,0000,0000,0000,,We can see each other's eyeballs,\Nand you can just ask a question;
Dialogue: 0,0:43:31.95,0:43:34.33,Default,,0000,0000,0000,,or sometimes I'll look at you,\Nand I know you have a question,
Dialogue: 0,0:43:34.33,0:43:35.67,Default,,0000,0000,0000,,and I'll say "What's up."
Dialogue: 0,0:43:36.01,0:43:39.27,Default,,0000,0000,0000,,Um, jump in there; really. Stop me\Nany time you wanna stop me.
Dialogue: 0,0:43:39.27,0:43:43.26,Default,,0000,0000,0000,,Don't be shy or embarrassed about it.\NStop me, and ask your question.
Dialogue: 0,0:43:43.26,0:43:46.28,Default,,0000,0000,0000,,Because the most important thing\Nis that you guys continue to learn.
Dialogue: 0,0:43:47.50,0:43:51.24,Default,,0000,0000,0000,,Exercise 4 says, "The length of a\Nrectangle is increasing at a rate
Dialogue: 0,0:43:51.24,0:43:54.65,Default,,0000,0000,0000,,of 8 centimeters per second."\NGot a rectangle.
Dialogue: 0,0:43:54.65,0:43:58.98,Default,,0000,0000,0000,,"And its width is increasing at a rate\Nof 3 centimeters per second."
Dialogue: 0,0:43:58.98,0:44:02.50,Default,,0000,0000,0000,,"When the length is 20,\Nand the width is 10,
Dialogue: 0,0:44:02.50,0:44:06.74,Default,,0000,0000,0000,,how fast is the {\i1}area{\i0} of\Nthe rectangle increasing?"
Dialogue: 0,0:44:06.74,0:44:10.76,Default,,0000,0000,0000,,Okay. So the first thing we're\Ngonna do? Draw a picture.
Dialogue: 0,0:44:11.07,0:44:13.26,Default,,0000,0000,0000,,So, I've got a rectangle here.
Dialogue: 0,0:44:14.50,0:44:17.40,Default,,0000,0000,0000,,Here we go. And I'm gonna\Nlabel this thing.
Dialogue: 0,0:44:17.89,0:44:20.51,Default,,0000,0000,0000,,So it says, "The length of the rectangle\Nis increasing at a rate
Dialogue: 0,0:44:20.51,0:44:22.50,Default,,0000,0000,0000,,of 8 centimeters per second;
Dialogue: 0,0:44:22.50,0:44:26.04,Default,,0000,0000,0000,,width is increasing at a rate of\N3 centimeters per second."
Dialogue: 0,0:44:26.04,0:44:30.21,Default,,0000,0000,0000,,"When the length is 20,\Nand the width is 10,
Dialogue: 0,0:44:30.38,0:44:34.39,Default,,0000,0000,0000,,how fast is the area of\Nthe rectangle increasing."
Dialogue: 0,0:44:34.68,0:44:41.11,Default,,0000,0000,0000,,So, like, right now, the area is\N20 times 10, or 200, but.
Dialogue: 0,0:44:41.11,0:44:43.50,Default,,0000,0000,0000,,We're gonna be increasing the length\Nand the width,
Dialogue: 0,0:44:43.50,0:44:46.77,Default,,0000,0000,0000,,and looking at how fast\Nthat area is changing.
Dialogue: 0,0:44:46.77,0:44:50.26,Default,,0000,0000,0000,,So I'm gonna write down the things\Nthat I know. I've given a lot in this problem.
Dialogue: 0,0:44:50.26,0:44:56.10,Default,,0000,0000,0000,,It says the length is increasing at a rate\Nof 8 centimeters per second.
Dialogue: 0,0:44:56.10,0:45:03.31,Default,,0000,0000,0000,,So, that would be the derivative\Nof {\i1}l{\i0}, with respect to time.
Dialogue: 0,0:45:03.92,0:45:05.45,Default,,0000,0000,0000,,That is 8.
Dialogue: 0,0:45:05.75,0:45:09.98,Default,,0000,0000,0000,,It says, the width is increasing at a\Nrate of 3 centimeters, so.
Dialogue: 0,0:45:10.39,0:45:16.03,Default,,0000,0000,0000,,{\i1}dw{\i0}, the change in width,\Nwith respect to time. That one is 3.
Dialogue: 0,0:45:16.40,0:45:24.95,Default,,0000,0000,0000,,We know that we're kind of stopping this\Nwhen {\i1}l{\i0} is 20, and when {\i1}w{\i0} is 10.
Dialogue: 0,0:45:25.22,0:45:27.26,Default,,0000,0000,0000,,So, there are four things that I know.
Dialogue: 0,0:45:27.26,0:45:30.09,Default,,0000,0000,0000,,What do I not know? What do I need.
Dialogue: 0,0:45:30.61,0:45:34.37,Default,,0000,0000,0000,,I need, or want to know...\Nhow fast the area—
Dialogue: 0,0:45:34.37,0:45:37.47,Default,,0000,0000,0000,,[Student] -({\i1}da{\i0}/{\i1}dt{\i0})?\N[Instructor] Yeah. How fast the area.
Dialogue: 0,0:45:37.90,0:45:40.62,Default,,0000,0000,0000,,Derivative of area with respect to time.
Dialogue: 0,0:45:40.62,0:45:44.52,Default,,0000,0000,0000,,I need the rate of change of\Nthe area with respect to time.
Dialogue: 0,0:45:45.10,0:45:48.37,Default,,0000,0000,0000,,So, if I'm looking for the\Nrate of change of area,
Dialogue: 0,0:45:48.37,0:45:50.92,Default,,0000,0000,0000,,then I want to use\Nthe area formula here.
Dialogue: 0,0:45:51.29,0:45:53.00,Default,,0000,0000,0000,,Area of a rectangle?
Dialogue: 0,0:45:58.65,0:46:00.01,Default,,0000,0000,0000,,Length times width.
Dialogue: 0,0:46:00.01,0:46:03.02,Default,,0000,0000,0000,,So there's my formula\Nrelating all of my variables;
Dialogue: 0,0:46:03.02,0:46:05.63,Default,,0000,0000,0000,,it's time to differentiate implicitly.
Dialogue: 0,0:46:06.14,0:46:12.08,Default,,0000,0000,0000,,So now we'll get the derivative\Nwith respect to {\i1}t{\i0} of the left-hand side.
Dialogue: 0,0:46:12.42,0:46:16.81,Default,,0000,0000,0000,,And the derivative with respect\Nto {\i1}t{\i0} of the right-hand side.
Dialogue: 0,0:46:17.25,0:46:21.02,Default,,0000,0000,0000,,Now on the left, there's your ({\i1}da{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:46:21.02,0:46:23.13,Default,,0000,0000,0000,,This is the very thing we're lookin' for.
Dialogue: 0,0:46:23.13,0:46:26.99,Default,,0000,0000,0000,,So then on the right, we need to get\Nthe derivative of length times width.
Dialogue: 0,0:46:26.99,0:46:31.90,Default,,0000,0000,0000,,So I said it: length times width.\NThis is a...?
Dialogue: 0,0:46:32.45,0:46:34.33,Default,,0000,0000,0000,,[Student] -Product rule.\N[Instructor] -Product rule.
Dialogue: 0,0:46:34.33,0:46:40.08,Default,,0000,0000,0000,,So we want the first function, {\i1}l{\i0}. Times\Nthe derivative of the second function.
Dialogue: 0,0:46:40.08,0:46:44.61,Default,,0000,0000,0000,,Okay now, remember: {\i1}t{\i0}'s the right letter.\NEverything else is the wrong letter.
Dialogue: 0,0:46:44.61,0:46:47.51,Default,,0000,0000,0000,,So when I do first times\Nthe derivative of the second,
Dialogue: 0,0:46:47.51,0:46:50.26,Default,,0000,0000,0000,,I don't know what the\Nderivative of the second is,
Dialogue: 0,0:46:50.26,0:46:53.37,Default,,0000,0000,0000,,so I have to write ({\i1}dw{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:46:54.13,0:46:56.97,Default,,0000,0000,0000,,So, {\i1}l{\i0} times ({\i1}dw{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:46:57.50,0:47:03.37,Default,,0000,0000,0000,,And then plus the second, which is {\i1}w{\i0},\Ntimes the derivative of the first,
Dialogue: 0,0:47:03.37,0:47:06.82,Default,,0000,0000,0000,,which has to be ({\i1}dl{\i0} /{\i1}dt{\i0}).
Dialogue: 0,0:47:07.62,0:47:08.95,Default,,0000,0000,0000,,So now I'm gonna go back up here,
Dialogue: 0,0:47:08.95,0:47:11.76,Default,,0000,0000,0000,,where I was given these\Nfour pieces of information.
Dialogue: 0,0:47:11.76,0:47:13.92,Default,,0000,0000,0000,,I'm gonna substitute them in.
Dialogue: 0,0:47:14.61,0:47:19.76,Default,,0000,0000,0000,,({\i1}da{\i0}/{\i1}dt{\i0}) = {\i1}l{\i0}, at this moment\Nin time, is 20.
Dialogue: 0,0:47:20.42,0:47:23.27,Default,,0000,0000,0000,,({\i1}dw{\i0}/{\i1}dt{\i0}) is 3.
Dialogue: 0,0:47:24.26,0:47:27.61,Default,,0000,0000,0000,,Plus {\i1}w{\i0} at this moment is 10.
Dialogue: 0,0:47:27.61,0:47:30.42,Default,,0000,0000,0000,,And ({\i1}dl{\i0}/{\i1}dt{\i0}) is 8.
Dialogue: 0,0:47:30.42,0:47:32.09,Default,,0000,0000,0000,,Okay, this one's gonna be easy.
Dialogue: 0,0:47:32.98,0:47:36.51,Default,,0000,0000,0000,,Don't have to isolate anything;\Njust multiply and add.
Dialogue: 0,0:47:36.51,0:47:39.76,Default,,0000,0000,0000,,So, that's gonna be 60 plus 80,
Dialogue: 0,0:47:39.76,0:47:43.53,Default,,0000,0000,0000,,and 80 plus 60 would be 140.
Dialogue: 0,0:47:43.95,0:47:46.94,Default,,0000,0000,0000,,Now, units. What are the\Nunits for the area?
Dialogue: 0,0:47:49.39,0:47:51.48,Default,,0000,0000,0000,,[Student] Centimeters\Nsquared per second?
Dialogue: 0,0:47:51.84,0:47:52.66,Default,,0000,0000,0000,,[Instructor] Mhm.
Dialogue: 0,0:47:53.25,0:47:56.51,Default,,0000,0000,0000,,So, units for area are\Ncentimeters squared.
Dialogue: 0,0:47:56.51,0:48:00.28,Default,,0000,0000,0000,,The units for time are second.\NSo it says that,
Dialogue: 0,0:48:00.28,0:48:04.90,Default,,0000,0000,0000,,at this point in time, when our\Nrectangle is this big, and it's increasing?
Dialogue: 0,0:48:04.90,0:48:10.87,Default,,0000,0000,0000,,That the rate of change in the area is\N140 square centimeters for every second.
Dialogue: 0,0:48:11.66,0:48:13.96,Default,,0000,0000,0000,,Okay. Almost done.\NThat was an easy one.
Dialogue: 0,0:48:15.02,0:48:16.80,Default,,0000,0000,0000,,Maybe we should've done that one first.
Dialogue: 0,0:48:21.52,0:48:28.62,Default,,0000,0000,0000,,Okay. The last one on this handout,\NI believe. Exercise 32. Exercise 32.
Dialogue: 0,0:48:29.63,0:48:31.24,Default,,0000,0000,0000,,Oh, but this is a good one.
Dialogue: 0,0:48:32.03,0:48:38.06,Default,,0000,0000,0000,,So, exercise 32 says, "Two sides of\Na triangle have lengths 12 meters
Dialogue: 0,0:48:38.06,0:48:41.64,Default,,0000,0000,0000,,and 15 meters."\NTwo sides of a triangle.
Dialogue: 0,0:48:41.64,0:48:46.00,Default,,0000,0000,0000,,It didn't say a right triangle.\NJust said "a triangle."
Dialogue: 0,0:48:46.00,0:48:52.75,Default,,0000,0000,0000,,"The angle between them is increasing\Nat a rate of 2 degrees per minute."
Dialogue: 0,0:48:52.75,0:48:55.76,Default,,0000,0000,0000,,"How fast is the length of\Nthe third side increasing
Dialogue: 0,0:48:55.76,0:49:00.12,Default,,0000,0000,0000,,when the angle between\Nthe sides of fixed length is 60?"
Dialogue: 0,0:49:00.62,0:49:01.95,Default,,0000,0000,0000,,[Exaggerated shriek]
Dialogue: 0,0:49:01.95,0:49:05.76,Default,,0000,0000,0000,,If it were a right triangle, this\Nwould be so much easier to draw!
Dialogue: 0,0:49:05.76,0:49:09.52,Default,,0000,0000,0000,,But it didn't say that; and it's not;\Nand it's changing; so, man!
Dialogue: 0,0:49:09.94,0:49:11.26,Default,,0000,0000,0000,,Let me just go for it.
Dialogue: 0,0:49:11.26,0:49:13.27,Default,,0000,0000,0000,,So I'm gonna draw a triangle.
Dialogue: 0,0:49:14.08,0:49:17.26,Default,,0000,0000,0000,,Maybe something like—I'm gonna\Nmake this pretty big. [Chuckle]
Dialogue: 0,0:49:17.26,0:49:18.50,Default,,0000,0000,0000,,Something like that.
Dialogue: 0,0:49:19.26,0:49:22.28,Default,,0000,0000,0000,,And your triangle doesn't have to\Nlook exactly like mine, but.
Dialogue: 0,0:49:22.99,0:49:25.08,Default,,0000,0000,0000,,I'll be danged if that doesn't\Nlook like a right triangle.
Dialogue: 0,0:49:25.08,0:49:27.77,Default,,0000,0000,0000,,That looks like a right angle right\Nthere. I just couldn't help myself.
Dialogue: 0,0:49:27.77,0:49:29.100,Default,,0000,0000,0000,,It's not. Not a right triangle.
Dialogue: 0,0:49:29.100,0:49:31.49,Default,,0000,0000,0000,,So I'm gonna label my sides.
Dialogue: 0,0:49:31.49,0:49:35.78,Default,,0000,0000,0000,,I'm gonna call that one 12, and\Nthis one 15, because it looks longer.
Dialogue: 0,0:49:36.51,0:49:42.26,Default,,0000,0000,0000,,And then there's an angle between them,\Nand that angle between them we'll call θ.
Dialogue: 0,0:49:42.26,0:49:46.37,Default,,0000,0000,0000,,So, if that's θ, and here\Nare the two sides.
Dialogue: 0,0:49:46.37,0:49:49.16,Default,,0000,0000,0000,,What's happening is,\Nthat is opening up.
Dialogue: 0,0:49:49.16,0:49:54.26,Default,,0000,0000,0000,,So as that opens up, we're looking to\Nsee how that third side is changing.
Dialogue: 0,0:49:54.26,0:49:57.64,Default,,0000,0000,0000,,It's obviously growing; it's getting\Nlonger. We're looking for that.
Dialogue: 0,0:49:58.22,0:50:01.27,Default,,0000,0000,0000,,"How fast is the length of\Nthe third side increasing
Dialogue: 0,0:50:01.27,0:50:06.09,Default,,0000,0000,0000,,when the angle between the sides\Nof fixed length is 60 degrees."
Dialogue: 0,0:50:06.56,0:50:09.62,Default,,0000,0000,0000,,So guess let's start writing down\Nthe things that we know here.
Dialogue: 0,0:50:10.50,0:50:17.10,Default,,0000,0000,0000,,So, we know that two sides of the\Ntriangle are 12 and 15... OK, got that.
Dialogue: 0,0:50:17.10,0:50:21.51,Default,,0000,0000,0000,,The angle between them is increasing\Nat a rate. Ah. This is a rate that we know.
Dialogue: 0,0:50:21.51,0:50:25.90,Default,,0000,0000,0000,,And it's the rate of change of that\Nangle with respect to time.
Dialogue: 0,0:50:25.90,0:50:32.50,Default,,0000,0000,0000,,So, we know ({\i1}dθ{\i0}/{\i1}dt{\i0}). ({\i1}dθ{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:50:32.50,0:50:35.37,Default,,0000,0000,0000,,And that is a rate of 2 degrees\Nper minute.
Dialogue: 0,0:50:35.37,0:50:38.03,Default,,0000,0000,0000,,So ({\i1}dθ{\i0}/{\i1}dt{\i0}) is 2.
Dialogue: 0,0:50:38.61,0:50:41.51,Default,,0000,0000,0000,,"How fast is the length of\Nthe third side increasing
Dialogue: 0,0:50:41.51,0:50:45.26,Default,,0000,0000,0000,,when the angle between the\Nsides of fixed length is 60?
Dialogue: 0,0:50:45.92,0:50:48.75,Default,,0000,0000,0000,,So, it's telling us that we're kind of\Nstopping this,
Dialogue: 0,0:50:48.75,0:50:52.62,Default,,0000,0000,0000,,looking at when that angle\Nis 60 degrees right then,
Dialogue: 0,0:50:52.62,0:50:54.81,Default,,0000,0000,0000,,how fast is the third side changing.
Dialogue: 0,0:50:55.25,0:50:57.99,Default,,0000,0000,0000,,Well, we need to give a name\Nto that third side.
Dialogue: 0,0:50:58.63,0:51:02.25,Default,,0000,0000,0000,,Hmm, I don't—what do you wanna\Ncall that third side? Anybody?
Dialogue: 0,0:51:03.75,0:51:04.94,Default,,0000,0000,0000,,Any variable?
Dialogue: 0,0:51:06.38,0:51:08.66,Default,,0000,0000,0000,,[Student] -{\i1}x{\i0}.\N[Instructor] -Why not?
Dialogue: 0,0:51:08.66,0:51:12.18,Default,,0000,0000,0000,,So we'll call that third side {\i1}x{\i0}.\NWorks for me.
Dialogue: 0,0:51:12.82,0:51:14.51,Default,,0000,0000,0000,,Now we need a formula.
Dialogue: 0,0:51:14.51,0:51:17.26,Default,,0000,0000,0000,,We need a formula that relates\Nwhat's going on here.
Dialogue: 0,0:51:17.26,0:51:21.50,Default,,0000,0000,0000,,So, look at your picture.\NYour knowns; your unknowns.
Dialogue: 0,0:51:21.50,0:51:23.95,Default,,0000,0000,0000,,Does a formula come to mind—
Dialogue: 0,0:51:23.95,0:51:27.80,Default,,0000,0000,0000,,and it cannot be Pythagoras,\Nbecause this is not a right triangle.
Dialogue: 0,0:51:28.77,0:51:31.88,Default,,0000,0000,0000,,[Student] This is the\Ndouble-angle thing? I mean...
Dialogue: 0,0:51:31.88,0:51:36.40,Default,,0000,0000,0000,,It's sine over hypotenuse\Nequals sine over hypotenuse?
Dialogue: 0,0:51:36.40,0:51:37.65,Default,,0000,0000,0000,,[Instructor] Not that one.
Dialogue: 0,0:51:38.63,0:51:42.41,Default,,0000,0000,0000,,[Student #2] Is this a sine-angle-sine\Nproblem? Or a side-angle-side problem?
Dialogue: 0,0:51:42.41,0:51:45.51,Default,,0000,0000,0000,,[Student #3] -Is it the law of sines?\N[Instructor] -Yes. Yes, it's SAS.
Dialogue: 0,0:51:48.99,0:51:51.15,Default,,0000,0000,0000,,So, think trigonometry.
Dialogue: 0,0:51:53.38,0:51:55.13,Default,,0000,0000,0000,,[Student #3] It's not the\Nlaw of sines or anything, is it?
Dialogue: 0,0:51:56.04,0:51:58.14,Default,,0000,0000,0000,,[Instructor] Keep thinkin'. You're close.
Dialogue: 0,0:52:05.51,0:52:06.50,Default,,0000,0000,0000,,[Student] Law of cosine?
Dialogue: 0,0:52:06.50,0:52:08.95,Default,,0000,0000,0000,,[Instructor] Yeah, that might\Nhelp if I write that in there.
Dialogue: 0,0:52:08.95,0:52:14.20,Default,,0000,0000,0000,,So when you know two sides and the\Nincluded angle, that's a law of cosines.
Dialogue: 0,0:52:14.20,0:52:19.62,Default,,0000,0000,0000,,And we know two sides. And we know\Nthe included angle at this moment is 60°.
Dialogue: 0,0:52:19.62,0:52:22.61,Default,,0000,0000,0000,,So definitely a law of cosines.
Dialogue: 0,0:52:23.22,0:52:25.75,Default,,0000,0000,0000,,Yayyy, I love the law of cosines!
Dialogue: 0,0:52:25.75,0:52:30.25,Default,,0000,0000,0000,,That part of trig was so fun; solving\Nfor the triangles using the law of sines
Dialogue: 0,0:52:30.25,0:52:32.50,Default,,0000,0000,0000,,and the law of cosines.\NI loved doing those problems.
Dialogue: 0,0:52:32.50,0:52:35.100,Default,,0000,0000,0000,,Remember the vector problems?\NThey were great.
Dialogue: 0,0:52:36.62,0:52:39.49,Default,,0000,0000,0000,,Okay, now, what does the\Nlaw of cosines say?
Dialogue: 0,0:52:39.49,0:52:42.34,Default,,0000,0000,0000,,Well, the law of cosines says this:
Dialogue: 0,0:52:42.34,0:52:46.25,Default,,0000,0000,0000,,That your side opposite,\Nwhich we're calling {\i1}x{\i0}.
Dialogue: 0,0:52:46.25,0:52:49.63,Default,,0000,0000,0000,,We're gonna square it.\N{\i1}x{\i0}² is equal to.
Dialogue: 0,0:52:49.63,0:52:53.51,Default,,0000,0000,0000,,And it's the sum of the squares\Nof the other two sides;
Dialogue: 0,0:52:53.51,0:52:56.91,Default,,0000,0000,0000,,so it starts out kind of looking\Nlike the Pythagorean theorem.
Dialogue: 0,0:52:56.91,0:53:04.62,Default,,0000,0000,0000,,But then it's minus 2 times {\i1}a{\i0}\Ntimes {\i1}b{\i0} times the cosine of θ.
Dialogue: 0,0:53:04.62,0:53:06.78,Default,,0000,0000,0000,,This is the law of cosines.
Dialogue: 0,0:53:07.28,0:53:10.04,Default,,0000,0000,0000,,So that's our formula relating everything.
Dialogue: 0,0:53:10.51,0:53:12.78,Default,,0000,0000,0000,,Ummmm, what do we next?
Dialogue: 0,0:53:14.33,0:53:16.03,Default,,0000,0000,0000,,Implicit differentiation.
Dialogue: 0,0:53:16.88,0:53:20.29,Default,,0000,0000,0000,,So, we want derivative\Nwith respect to {\i1}t{\i0}.
Dialogue: 0,0:53:21.36,0:53:24.26,Default,,0000,0000,0000,,Of the left-hand side, which is {\i1}x{\i0}².
Dialogue: 0,0:53:24.26,0:53:28.24,Default,,0000,0000,0000,,And the derivative with respect\Nto {\i1}t{\i0} of the right-hand side,
Dialogue: 0,0:53:28.24,0:53:34.31,Default,,0000,0000,0000,,which is ({\i1}a{\i0}² + {\i1}b{\i0}² − 2{\i1}ab{\i0}cosθ).
Dialogue: 0,0:53:34.76,0:53:39.78,Default,,0000,0000,0000,,Well, I say before we get this derivative,\Nmaybe we substitute in what we know,
Dialogue: 0,0:53:39.78,0:53:42.59,Default,,0000,0000,0000,,with the sides 12 and 15?
Dialogue: 0,0:53:42.59,0:53:44.93,Default,,0000,0000,0000,,Ummm, I can do that.
Dialogue: 0,0:53:44.93,0:53:49.02,Default,,0000,0000,0000,,Or not; I don't have to.\NI can live with it like this.
Dialogue: 0,0:53:49.02,0:53:50.15,Default,,0000,0000,0000,,Y'all, give me a preference.
Dialogue: 0,0:53:50.15,0:53:54.09,Default,,0000,0000,0000,,Do you want me to substitute in those\Nnumbers now, or get the derivative first?
Dialogue: 0,0:53:54.09,0:53:57.64,Default,,0000,0000,0000,,If I get the derivative first, you know,\Nthese will just be zeros,
Dialogue: 0,0:53:57.64,0:53:59.75,Default,,0000,0000,0000,,because there are only constants.
Dialogue: 0,0:53:59.75,0:54:01.100,Default,,0000,0000,0000,,Weigh in with your preference here.
Dialogue: 0,0:54:05.42,0:54:06.58,Default,,0000,0000,0000,,[Student] Put the numbers?
Dialogue: 0,0:54:07.09,0:54:09.74,Default,,0000,0000,0000,,[Instructor] -Put the numbers in?\N[Student] -Yes.
Dialogue: 0,0:54:11.50,0:54:17.37,Default,,0000,0000,0000,,So, derivative with respect to {\i1}t{\i0} of {\i1}x{\i0}²\Nequals the derivative with respect to {\i1}t{\i0};
Dialogue: 0,0:54:17.37,0:54:19.41,Default,,0000,0000,0000,,and we'll put those numbers in.
Dialogue: 0,0:54:19.41,0:54:24.26,Default,,0000,0000,0000,,So, the {\i1}a{\i0}? I guess I'll just\Ncall {\i1}a{\i0} the base; 15.
Dialogue: 0,0:54:24.26,0:54:29.83,Default,,0000,0000,0000,,That would be 15². Plus the other\Nside squared; so that's 12².
Dialogue: 0,0:54:29.83,0:54:36.50,Default,,0000,0000,0000,,Minus 2 times 15 times 12.\NTimes the cosine of θ.
Dialogue: 0,0:54:36.50,0:54:38.64,Default,,0000,0000,0000,,And then we can clean that up a bit.
Dialogue: 0,0:54:38.64,0:54:41.94,Default,,0000,0000,0000,,This is the derivative\Nwith respect to {\i1}t{\i0} of {\i1}x{\i0}².
Dialogue: 0,0:54:41.94,0:54:45.61,Default,,0000,0000,0000,,Notice I'm not taking the derivative\Nyet; I'm just cleaning this up a bit.
Dialogue: 0,0:54:45.61,0:54:48.97,Default,,0000,0000,0000,,Equals derivative with respect to {\i1}t{\i0} of.
Dialogue: 0,0:54:48.97,0:54:52.67,Default,,0000,0000,0000,,If I do 15² + 12².
Dialogue: 0,0:54:52.67,0:54:55.00,Default,,0000,0000,0000,,Go into my calculator here.
Dialogue: 0,0:54:56.11,0:54:59.99,Default,,0000,0000,0000,,15² plus 12².
Dialogue: 0,0:54:59.99,0:55:02.65,Default,,0000,0000,0000,,Okay. That is 369.
Dialogue: 0,0:55:03.24,0:55:06.76,Default,,0000,0000,0000,,So that would be 369 minus.
Dialogue: 0,0:55:06.76,0:55:10.16,Default,,0000,0000,0000,,Now, the 2 times 15 times 12?
Dialogue: 0,0:55:12.03,0:55:14.52,Default,,0000,0000,0000,,That is 360.
Dialogue: 0,0:55:16.00,0:55:18.37,Default,,0000,0000,0000,,Sitting in front of the cosθ.
Dialogue: 0,0:55:18.37,0:55:21.97,Default,,0000,0000,0000,,Okay. {\i1}Now{\i0}, let's differentiate.\NLet's do it now.
Dialogue: 0,0:55:21.97,0:55:26.26,Default,,0000,0000,0000,,So then on the left-hand side,\Nremember that {\i1}x{\i0} is the wrong letter.
Dialogue: 0,0:55:26.26,0:55:32.17,Default,,0000,0000,0000,,So when I get the derivative of {\i1}x{\i0}²,\Nit's 2{\i1}x{\i0}, but follow it by...?
Dialogue: 0,0:55:36.63,0:55:37.90,Default,,0000,0000,0000,,({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:55:39.05,0:55:40.24,Default,,0000,0000,0000,,On the right-hand side.
Dialogue: 0,0:55:40.24,0:55:45.61,Default,,0000,0000,0000,,The derivative of 369 is just 0,\Nso we won't worry about that.
Dialogue: 0,0:55:45.61,0:55:50.42,Default,,0000,0000,0000,,So now let's look at the\Nderivative of -360cosθ.
Dialogue: 0,0:55:51.17,0:55:54.18,Default,,0000,0000,0000,,Well, that constant in front\Njust hangs out.
Dialogue: 0,0:55:54.53,0:55:56.26,Default,,0000,0000,0000,,What's the derivative of cosine?
Dialogue: 0,0:55:57.51,0:55:58.75,Default,,0000,0000,0000,,[Student] Negative-sine.
Dialogue: 0,0:55:58.75,0:56:01.77,Default,,0000,0000,0000,,[Instructor] So since it's\Nnegative-sine, then we can do...
Dialogue: 0,0:56:02.32,0:56:03.16,Default,,0000,0000,0000,,That.
Dialogue: 0,0:56:03.51,0:56:06.26,Default,,0000,0000,0000,,So 360sinθ.
Dialogue: 0,0:56:06.26,0:56:11.02,Default,,0000,0000,0000,,Now, θ is the wrong variable,\Nso what do we follow this by?
Dialogue: 0,0:56:15.88,0:56:17.08,Default,,0000,0000,0000,,({\i1}dθ{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:56:17.08,0:56:18.65,Default,,0000,0000,0000,,And that's the chain rule.
Dialogue: 0,0:56:18.96,0:56:24.76,Default,,0000,0000,0000,,So if you have a cosθ,\Nderivative is -sinθ ({\i1}dθ{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:56:24.76,0:56:26.51,Default,,0000,0000,0000,,That's the derivative of the inside.
Dialogue: 0,0:56:27.25,0:56:28.27,Default,,0000,0000,0000,,Okay, great.
Dialogue: 0,0:56:28.27,0:56:31.25,Default,,0000,0000,0000,,So now we're ready to substitute in\Nthings that we know;
Dialogue: 0,0:56:31.25,0:56:33.67,Default,,0000,0000,0000,,and we're solving for...
Dialogue: 0,0:56:33.67,0:56:36.82,Default,,0000,0000,0000,,What are we solving for? I didn't\Nwrite down what we needed to know.
Dialogue: 0,0:56:38.71,0:56:39.76,Default,,0000,0000,0000,,We need...
Dialogue: 0,0:56:40.35,0:56:44.21,Default,,0000,0000,0000,,And it says, "How fast is the length\Nof the third side increasing?"
Dialogue: 0,0:56:44.78,0:56:50.24,Default,,0000,0000,0000,,We need ({\i1}dx{\i0}/{\i1}dt{\i0}), the rate of\Nchange of {\i1}x{\i0} with respect to {\i1}t{\i0}.
Dialogue: 0,0:56:50.24,0:56:52.95,Default,,0000,0000,0000,,Okay, got it. So I'm solving for ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,0:56:53.38,0:56:56.75,Default,,0000,0000,0000,,Well, then on the left-hand side,\NI'll have 2 times {\i1}x{\i0}.
Dialogue: 0,0:56:57.26,0:57:00.17,Default,,0000,0000,0000,,Ummm. How are we gonna find {\i1}x{\i0} here?
Dialogue: 0,0:57:06.35,0:57:08.25,Default,,0000,0000,0000,,How we gonna find {\i1}x{\i0}.
Dialogue: 0,0:57:09.51,0:57:12.54,Default,,0000,0000,0000,,I'm bringing my picture right down\Nin front of your face, there.
Dialogue: 0,0:57:16.00,0:57:17.83,Default,,0000,0000,0000,,If I'm looking for this side...
Dialogue: 0,0:57:18.48,0:57:20.26,Default,,0000,0000,0000,,[Student] Is it a [inaudible] equation?
Dialogue: 0,0:57:20.26,0:57:23.01,Default,,0000,0000,0000,,[Instructor] We're gonna plug it\Ninto the law of cosines
Dialogue: 0,0:57:23.01,0:57:25.38,Default,,0000,0000,0000,,to find out what this\Nthird side would be
Dialogue: 0,0:57:25.38,0:57:31.82,Default,,0000,0000,0000,,when the two sides are 12 and 15,\Nand, {\i1}at this moment{\i0}, that angle is 60°.
Dialogue: 0,0:57:31.82,0:57:36.35,Default,,0000,0000,0000,,So we're going to go back to the law of\Ncosines just to determine this unknown.
Dialogue: 0,0:57:36.73,0:57:40.26,Default,,0000,0000,0000,,Remember, we had to do this\Nbefore on one of the ladder problems.
Dialogue: 0,0:57:41.28,0:57:44.68,Default,,0000,0000,0000,,Okay. So then, using a law\Nof cosines, it would say...
Dialogue: 0,0:57:44.68,0:57:46.93,Default,,0000,0000,0000,,I'll try to do this over here on the side.
Dialogue: 0,0:57:46.93,0:57:51.85,Default,,0000,0000,0000,,It would say that {\i1}x{\i0}²\Nis equal to {\i1}a{\i0}² + {\i1}b{\i0}².
Dialogue: 0,0:57:51.85,0:57:55.37,Default,,0000,0000,0000,,So that's 15² + 12² again.
Dialogue: 0,0:57:55.37,0:58:03.74,Default,,0000,0000,0000,,Minus 15 times 12 times the 2;\Ntimes the cosine of 60°.
Dialogue: 0,0:58:04.63,0:58:09.17,Default,,0000,0000,0000,,So our {\i1}x{\i0}² equals.\NThat 15² +12² ?
Dialogue: 0,0:58:09.17,0:58:12.65,Default,,0000,0000,0000,,That was the 369.
Dialogue: 0,0:58:12.65,0:58:20.95,Default,,0000,0000,0000,,And then 15 times 12 times 2,\Nthat was the -360cos60°.
Dialogue: 0,0:58:20.95,0:58:26.31,Default,,0000,0000,0000,,So {\i1}x{\i0}² is 369 minus 360 times.
Dialogue: 0,0:58:26.31,0:58:29.76,Default,,0000,0000,0000,,And the cosine of 60°\Nis one that we know.
Dialogue: 0,0:58:29.76,0:58:31.63,Default,,0000,0000,0000,,[Student] -One-half?\N[Instructor] -Is what?
Dialogue: 0,0:58:31.63,0:58:33.01,Default,,0000,0000,0000,,[Student] -One-half, I think?\N[Instructor] -One over two.
Dialogue: 0,0:58:33.01,0:58:34.35,Default,,0000,0000,0000,,One-half is right.
Dialogue: 0,0:58:34.35,0:58:36.06,Default,,0000,0000,0000,,So this is ½.
Dialogue: 0,0:58:36.06,0:58:41.28,Default,,0000,0000,0000,,{\i1}x{\i0}² is 369 minus...\NI guess that'd be 180?
Dialogue: 0,0:58:41.69,0:58:45.88,Default,,0000,0000,0000,,And then 369 minus 180 is—\N[goofy voice] {\i1}I unno{\i0}.
Dialogue: 0,0:58:51.84,0:58:53.26,Default,,0000,0000,0000,,[Instructor] -I got—\N[Student] -189.
Dialogue: 0,0:58:53.26,0:58:57.99,Default,,0000,0000,0000,,[Instructor] 189. So {\i1}x{\i0} would be\Nthe square root of that.
Dialogue: 0,0:58:57.99,0:59:03.15,Default,,0000,0000,0000,,Which is not real pretty;\Nit's 13.7-{\i1}ish{\i0}.
Dialogue: 0,0:59:03.15,0:59:06.50,Default,,0000,0000,0000,,So I'm just gonna leave it at 13.7.
Dialogue: 0,0:59:09.26,0:59:11.78,Default,,0000,0000,0000,,More decimal places would be better;
Dialogue: 0,0:59:11.78,0:59:15.61,Default,,0000,0000,0000,,but I kinda messed myself up by not\Ngiving myself very much room to write
Dialogue: 0,0:59:15.61,0:59:19.52,Default,,0000,0000,0000,,any number in here at all, sooo,\NI'm gonna have to just round it off.
Dialogue: 0,0:59:19.52,0:59:20.94,Default,,0000,0000,0000,,So 13.7.
Dialogue: 0,0:59:20.94,0:59:25.49,Default,,0000,0000,0000,,And that equals the 360\Ntimes the sine of θ.
Dialogue: 0,0:59:25.49,0:59:34.61,Default,,0000,0000,0000,,Oh, but the sine of θ is the sine of...\N60°, times ({\i1}dθ{\i0}/{\i1}dt{\i0}), which was 2.
Dialogue: 0,0:59:36.04,0:59:37.74,Default,,0000,0000,0000,,Hold on. I can fix this.
Dialogue: 0,0:59:43.100,0:59:49.87,Default,,0000,0000,0000,,My Calc 2 student told me on Wednesday,\N"So why don't you just use a pencil?"
Dialogue: 0,0:59:49.87,0:59:52.51,Default,,0000,0000,0000,,"Then if you mess up, it's no big deal!"
Dialogue: 0,0:59:53.23,0:59:54.04,Default,,0000,0000,0000,,Well.
Dialogue: 0,0:59:55.02,0:59:57.87,Default,,0000,0000,0000,,I've got a... Wite-Out tape here.
Dialogue: 0,1:00:02.09,1:00:03.64,Default,,0000,0000,0000,,So let's fix all that.
Dialogue: 0,1:00:04.85,1:00:07.30,Default,,0000,0000,0000,,Like, really? You're not gonna work?
Dialogue: 0,1:00:10.25,1:00:12.78,Default,,0000,0000,0000,,[Cries] Why is my life so hard?!
Dialogue: 0,1:00:13.50,1:00:15.06,Default,,0000,0000,0000,,All right. So I'll just rewrite it.
Dialogue: 0,1:00:15.76,1:00:20.38,Default,,0000,0000,0000,,2 times 13.7, times ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:00:20.38,1:00:25.01,Default,,0000,0000,0000,,Equals 360 times the sine of 60°;
Dialogue: 0,1:00:25.01,1:00:28.77,Default,,0000,0000,0000,,times ({\i1}dθ{\i0}/{\i1}dt{\i0}), which was 2.
Dialogue: 0,1:00:31.49,1:00:32.75,Default,,0000,0000,0000,,There we go.
Dialogue: 0,1:00:32.75,1:00:36.68,Default,,0000,0000,0000,,Now, 2 times 13.7—aw, heck.\NYou know what I'm gonna do?
Dialogue: 0,1:00:37.12,1:00:43.75,Default,,0000,0000,0000,,Say ({\i1}dx{\i0}/{\i1}dt{\i0}) is equal to.\N2 times 360 would be...
Dialogue: 0,1:00:44.52,1:00:50.06,Default,,0000,0000,0000,,720. Sine of 60°.\NI know that one, too.
Dialogue: 0,1:00:51.12,1:00:52.15,Default,,0000,0000,0000,,That would be...
Dialogue: 0,1:00:52.15,1:00:55.51,Default,,0000,0000,0000,,[Student] -Square root of 3 over 2.\N[Instructor] -Square root of 3 over 2.
Dialogue: 0,1:00:55.51,1:01:00.43,Default,,0000,0000,0000,,And then, let's divide that\Nby 2 times 13.7.
Dialogue: 0,1:01:01.77,1:01:04.76,Default,,0000,0000,0000,,All right, I'm going to my\Ncalculator to figure this one out.
Dialogue: 0,1:01:05.07,1:01:09.34,Default,,0000,0000,0000,,720. Times the square root of 3.
Dialogue: 0,1:01:18.26,1:01:19.78,Default,,0000,0000,0000,,Divided by 2.
Dialogue: 0,1:01:19.78,1:01:24.95,Default,,0000,0000,0000,,And then that divided by\N2 times 13.7.
Dialogue: 0,1:01:24.95,1:01:32.28,Default,,0000,0000,0000,,Hey y'all; if I didn't fat-finger this,\NI got approximately 22.7.
Dialogue: 0,1:01:32.95,1:01:34.44,Default,,0000,0000,0000,,And now I need a unit for that.
Dialogue: 0,1:01:34.44,1:01:38.60,Default,,0000,0000,0000,,This was a rate of change\Nof {\i1}x{\i0} with respect to time.
Dialogue: 0,1:01:38.60,1:01:44.91,Default,,0000,0000,0000,,And {\i1}x{\i0} was measured in meters,\Nand time was measured in minutes.
Dialogue: 0,1:01:44.91,1:01:50.27,Default,,0000,0000,0000,,So, 22.7 meters per minute.
Dialogue: 0,1:01:53.04,1:01:54.04,Default,,0000,0000,0000,,Okay.
Dialogue: 0,1:01:54.04,1:01:55.26,Default,,0000,0000,0000,,[Student] Umm.
Dialogue: 0,1:01:56.22,1:01:56.99,Default,,0000,0000,0000,,[Instructor] Yes?
Dialogue: 0,1:01:58.42,1:02:01.89,Default,,0000,0000,0000,,[Student] -Oh, nothing; I just said "Wow."\N[Instructor] -Oh, okay. {\i1}Wow!{\i0}
Dialogue: 0,1:02:03.23,1:02:07.64,Default,,0000,0000,0000,,So, yeah. This was a pretty\Nchallenging section; but also doable.
Dialogue: 0,1:02:07.64,1:02:11.50,Default,,0000,0000,0000,,So if you look at those bullets,\Ndraw the picture; label;
Dialogue: 0,1:02:11.50,1:02:14.96,Default,,0000,0000,0000,,write down what you know and what you\Ndon't know; what you {\i1}need{\i0} to know.
Dialogue: 0,1:02:14.96,1:02:17.26,Default,,0000,0000,0000,,Find a formula that relates everything.
Dialogue: 0,1:02:17.26,1:02:20.76,Default,,0000,0000,0000,,If you try to go through that\Nstep-by-step, I think you'll be just fine.
Dialogue: 0,1:02:20.76,1:02:24.02,Default,,0000,0000,0000,,And I tried to pick problems—\Nmost of them—
Dialogue: 0,1:02:24.02,1:02:29.17,Default,,0000,0000,0000,,are like the ones that we did...\Nin class today? On Zoom today?
Dialogue: 0,1:02:29.17,1:02:33.53,Default,,0000,0000,0000,,And so, hopefully you'll have an example\Nfor almost everything in the homework.
Dialogue: 0,1:02:34.01,1:02:38.00,Default,,0000,0000,0000,,But definitely stop by\Nduring office hours; um...
Dialogue: 0,1:02:38.00,1:02:43.05,Default,,0000,0000,0000,,I've got that all figured out now,\Nand I'm in Blackboard from 10 to 11;
Dialogue: 0,1:02:43.05,1:02:46.99,Default,,0000,0000,0000,,so, you know, before your\Nclass for an hour, stop in.
Dialogue: 0,1:02:46.99,1:02:52.04,Default,,0000,0000,0000,,Or this afternoon from 3:45 to 4:45. I'm\Non Blackboard again, and Collaborate.
Dialogue: 0,1:02:52.04,1:02:56.42,Default,,0000,0000,0000,,So, there's a link in your Blackboard.\NJust go to it; and I'll be there,
Dialogue: 0,1:02:56.42,1:02:58.100,Default,,0000,0000,0000,,and I can help you with\Nhomework problems, so.
Dialogue: 0,1:02:58.100,1:03:03.15,Default,,0000,0000,0000,,Especially some of you who were used to\Ncoming by when we were still at Hays.
Dialogue: 0,1:03:03.15,1:03:06.77,Default,,0000,0000,0000,,Come on by! I wanna still be able\Nto help you, even though it's not...
Dialogue: 0,1:03:06.77,1:03:10.52,Default,,0000,0000,0000,,{\i1}quite{\i0} as effective this way?\NIt's still better than nothing.
Dialogue: 0,1:03:11.08,1:03:14.32,Default,,0000,0000,0000,,Umm. And then the tutoring labs,\Nthe learning labs,
Dialogue: 0,1:03:14.32,1:03:17.37,Default,,0000,0000,0000,,have gone online using Brainfuse.
Dialogue: 0,1:03:18.08,1:03:20.99,Default,,0000,0000,0000,,They were supposed to send out\Nan email about that.
Dialogue: 0,1:03:20.99,1:03:24.51,Default,,0000,0000,0000,,I never got one. I'm hoping\Nthat the students {\i1}did{\i0}.
Dialogue: 0,1:03:24.51,1:03:27.39,Default,,0000,0000,0000,,Somebody let me know if\Nyou've got anything about that?
Dialogue: 0,1:03:33.10,1:03:35.29,Default,,0000,0000,0000,,[Student #1] -I didn't.\N[Instructor] -Ahh.
Dialogue: 0,1:03:35.29,1:03:38.25,Default,,0000,0000,0000,,[Student #2] I think I remember seeing\Nsomething about [inaudible]...
Dialogue: 0,1:03:40.98,1:03:43.74,Default,,0000,0000,0000,,[Student #3] So, I've been\Ntalking to a tutor. Um.
Dialogue: 0,1:03:43.74,1:03:48.62,Default,,0000,0000,0000,,And we've been meeting on Zoom.\NBut the way that {\i1}she{\i0} sees it
Dialogue: 0,1:03:48.62,1:03:52.71,Default,,0000,0000,0000,,is that it's almost like a ticketing system,\Nkinda like how Highland works now;
Dialogue: 0,1:03:52.71,1:03:55.40,Default,,0000,0000,0000,,where you send in a ticket\Nfor a singular question,
Dialogue: 0,1:03:55.40,1:03:57.58,Default,,0000,0000,0000,,and then they can reach\Nout to you and help?
Dialogue: 0,1:03:57.58,1:04:00.08,Default,,0000,0000,0000,,[Instructor] Do you do it\Nfrom the website?
Dialogue: 0,1:04:01.24,1:04:03.00,Default,,0000,0000,0000,,[Student #3] Yeah, I believe so.
Dialogue: 0,1:04:03.00,1:04:06.75,Default,,0000,0000,0000,,There {\i1}was{\i0} a post on it on the front \Npage; I don't know if it's still there.
Dialogue: 0,1:04:06.75,1:04:10.68,Default,,0000,0000,0000,,[Instructor] Oh, okay. So, maybe\Njust go to austincc.edu.
Dialogue: 0,1:04:11.21,1:04:13.27,Default,,0000,0000,0000,,And if there's not anything\Non the front page,
Dialogue: 0,1:04:13.27,1:04:15.27,Default,,0000,0000,0000,,maybe do a search for "learning lab,"
Dialogue: 0,1:04:15.27,1:04:18.24,Default,,0000,0000,0000,,and then hopefully their page\Nwill come up with information.
Dialogue: 0,1:04:18.24,1:04:20.19,Default,,0000,0000,0000,,I did not recieve anything about it.
Dialogue: 0,1:04:20.19,1:04:24.76,Default,,0000,0000,0000,,I just—I {\i1}heard{\i0} from someone who works\Nthere that they were gonna do Brainfuse.
Dialogue: 0,1:04:25.73,1:04:28.33,Default,,0000,0000,0000,,So yeah, the ticketing system,\Nthat would be okay, I guess;
Dialogue: 0,1:04:28.33,1:04:30.88,Default,,0000,0000,0000,,and just kinda wait until it's your turn.
Dialogue: 0,1:04:33.68,1:04:35.96,Default,,0000,0000,0000,,[Student #3] What problem set\Nare we working on?
Dialogue: 0,1:04:35.96,1:04:37.06,Default,,0000,0000,0000,,[Instructor] Sorry?
Dialogue: 0,1:04:37.06,1:04:40.62,Default,,0000,0000,0000,,[Student #3] What problem set are we\Ngoing to be working on, for next class?
Dialogue: 0,1:04:40.62,1:04:45.63,Default,,0000,0000,0000,,[Instructor] Oh, um. So this is \NSection... uhh, what is this? 4.1.
Dialogue: 0,1:04:47.22,1:04:49.65,Default,,0000,0000,0000,,So you'll have homework from 4.1.
Dialogue: 0,1:04:50.63,1:04:53.74,Default,,0000,0000,0000,,And then on next class,\Nwe're gonna try to do...
Dialogue: 0,1:04:53.74,1:04:56.23,Default,,0000,0000,0000,,2.8, which will go fast.
Dialogue: 0,1:04:56.23,1:04:58.50,Default,,0000,0000,0000,,And then, 4.2.
Dialogue: 0,1:04:58.50,1:05:00.76,Default,,0000,0000,0000,,We'll try to do two sections. We'll see.
Dialogue: 0,1:05:01.91,1:05:04.66,Default,,0000,0000,0000,,[Student] So the homework\Nis 4.1, right? And 4.2?
Dialogue: 0,1:05:05.39,1:05:07.71,Default,,0000,0000,0000,,[Instructor] -Yeah.\N[Student] -Okay.
Dialogue: 0,1:05:09.38,1:05:10.88,Default,,0000,0000,0000,,[Student #3] Wait, so um...
Dialogue: 0,1:05:10.88,1:05:14.63,Default,,0000,0000,0000,,We're doing the homework\Nfor 4.1 and 4.2 for next class?
Dialogue: 0,1:05:14.63,1:05:16.65,Default,,0000,0000,0000,,-We're not doing the—\N-[Instructor] No-no-no-no-no, no-no.
Dialogue: 0,1:05:16.65,1:05:19.76,Default,,0000,0000,0000,,So, all you need to be working\Non right now is 4.1.
Dialogue: 0,1:05:21.14,1:05:22.83,Default,,0000,0000,0000,,[Student] Okay. So we'll just\Ndo the normal homework.
Dialogue: 0,1:05:22.83,1:05:25.09,Default,,0000,0000,0000,,So we're not doing problem\Nsets... between Mondays
Dialogue: 0,1:05:25.09,1:05:26.87,Default,,0000,0000,0000,,-and Wednesdays anymore?\N-Oh, oh; I see what you're saying.
Dialogue: 0,1:05:26.87,1:05:31.08,Default,,0000,0000,0000,,So I'm not giving you a problem set\Nthis week, because you just had a test.
Dialogue: 0,1:05:31.08,1:05:34.12,Default,,0000,0000,0000,,So I don't really have anything\Nto problem-set you {\i1}over{\i0}.
Dialogue: 0,1:05:35.24,1:05:37.40,Default,,0000,0000,0000,,-I appreciate that.\N-[laughs] You're welcome.
Dialogue: 0,1:05:37.40,1:05:40.28,Default,,0000,0000,0000,,But we will next Monday. Next\NMonday, you'll get a problem set.
Dialogue: 0,1:05:41.72,1:05:45.63,Default,,0000,0000,0000,,Um, I was gonna say that in my\NCalc 2 class, some of those students
Dialogue: 0,1:05:45.63,1:05:49.04,Default,,0000,0000,0000,,are also meeting on Zoom,\Nto work on homework together.
Dialogue: 0,1:05:49.04,1:05:53.10,Default,,0000,0000,0000,,So, they had study groups going, and\Nthey're just keeping those going on Zoom.
Dialogue: 0,1:05:53.62,1:05:57.41,Default,,0000,0000,0000,,So, I'm gonna throw that out there.\NIf any of you guys had study groups.
Dialogue: 0,1:05:57.41,1:05:58.98,Default,,0000,0000,0000,,You know, continue to do that.
Dialogue: 0,1:06:03.20,1:06:04.24,Default,,0000,0000,0000,,[Student] Um, before we go.
Dialogue: 0,1:06:04.24,1:06:08.27,Default,,0000,0000,0000,,Is this meeting going to be\Nposted in Recorded Meetings?
Dialogue: 0,1:06:08.27,1:06:10.88,Default,,0000,0000,0000,,[Instructor] Yes, it is.\NIt just takes a while.
Dialogue: 0,1:06:10.88,1:06:15.52,Default,,0000,0000,0000,,So, once I'm finished, it has to convert\Nit, or something? I don't know.
Dialogue: 0,1:06:15.52,1:06:17.37,Default,,0000,0000,0000,,And that can take hours.
Dialogue: 0,1:06:17.37,1:06:21.36,Default,,0000,0000,0000,,So, hopefully... hopefully by\Ntonight I'll have it posted?
Dialogue: 0,1:06:21.36,1:06:24.24,Default,,0000,0000,0000,,But in the morning, as a last resort.
Dialogue: 0,1:06:25.66,1:06:26.74,Default,,0000,0000,0000,,[Student] Okay. Heard.
Dialogue: 0,1:06:30.88,1:06:32.66,Default,,0000,0000,0000,,Okay. Well, so, we are early.
Dialogue: 0,1:06:32.66,1:06:38.01,Default,,0000,0000,0000,,And I'm gonna let you go; so if you wanna\Ngo, just go ahead and exit the meeting.
Dialogue: 0,1:06:38.01,1:06:39.61,Default,,0000,0000,0000,,I'm gonna just stay here for a minute,
Dialogue: 0,1:06:39.61,1:06:42.53,Default,,0000,0000,0000,,in case anybody wants to\Ntalk or ask me a question.
Dialogue: 0,1:06:45.89,1:06:46.61,Default,,0000,0000,0000,,And if you're leaving, bye—
Dialogue: 0,1:06:46.61,1:06:50.52,Default,,0000,0000,0000,,[Student] -I actually have a question.\N[Instructor] -Sure. You can hang out.
Dialogue: 0,1:06:50.52,1:06:56.75,Default,,0000,0000,0000,,So, I actually was struggling with the\Nparticle moves along a curve equation.
Dialogue: 0,1:06:56.75,1:06:57.75,Default,,0000,0000,0000,,[Instructor] Okay.
Dialogue: 0,1:06:57.75,1:07:01.67,Default,,0000,0000,0000,,I would just like you to break down\Na little bit more what you did,
Dialogue: 0,1:07:01.67,1:07:04.15,Default,,0000,0000,0000,,um, a couple steps through it.
Dialogue: 0,1:07:04.15,1:07:06.62,Default,,0000,0000,0000,,[Instructor] Sure. Let me turn\Nmy screen-sharing back on.
Dialogue: 0,1:07:28.50,1:07:30.22,Default,,0000,0000,0000,,It's really taking forever.
Dialogue: 0,1:07:45.34,1:07:48.03,Default,,0000,0000,0000,,[Student] Uh, you said this is\Nalready recorded already, right?
Dialogue: 0,1:07:48.03,1:07:49.19,Default,,0000,0000,0000,,[Instructor] Mhmm.
Dialogue: 0,1:07:49.19,1:07:52.34,Default,,0000,0000,0000,,[Student] How do we go to view it?\NDo we just go on Blackboard, then...
Dialogue: 0,1:07:52.34,1:07:58.51,Default,,0000,0000,0000,,[Instructor] Yeah; so. Once it's ready\Nto post, then you'll see a link.
Dialogue: 0,1:07:58.51,1:08:01.52,Default,,0000,0000,0000,,I think the link's already\Nthere, right? That says...
Dialogue: 0,1:08:02.26,1:08:05.64,Default,,0000,0000,0000,,Zoom Recordings, or Recorded\NMeetings. I forgot what I called it,
Dialogue: 0,1:08:05.64,1:08:07.61,Default,,0000,0000,0000,,but. "Recorded" is in it.
Dialogue: 0,1:08:07.61,1:08:10.02,Default,,0000,0000,0000,,So you'll just click there,\Nand then you'll see them.
Dialogue: 0,1:08:10.76,1:08:12.50,Default,,0000,0000,0000,,[Student] -All right. Thank you.\N[Instructor] -Sure.
Dialogue: 0,1:08:13.89,1:08:16.64,Default,,0000,0000,0000,,Okay. So here's my particle problem, again.
Dialogue: 0,1:08:17.49,1:08:19.63,Default,,0000,0000,0000,,So the deal was, um.
Dialogue: 0,1:08:19.63,1:08:23.26,Default,,0000,0000,0000,,Little dude is moving.\NLittle particle is moving.
Dialogue: 0,1:08:23.98,1:08:28.24,Default,,0000,0000,0000,,And the curve, the {\i1}y{\i0} = √(1+{\i1}x{\i0}³)?
Dialogue: 0,1:08:28.24,1:08:32.86,Default,,0000,0000,0000,,That is the formula that relates\Nyour variables, {\i1}x{\i0} and {\i1}y{\i0}.
Dialogue: 0,1:08:33.63,1:08:35.27,Default,,0000,0000,0000,,And then, let's see.
Dialogue: 0,1:08:36.01,1:08:40.51,Default,,0000,0000,0000,,The {\i1}y{\i0} coordinate was increasing\Nat a rate of 4; so as it's moving,
Dialogue: 0,1:08:40.51,1:08:43.56,Default,,0000,0000,0000,,the rate of change of\Nthe {\i1}y{\i0} coordinate is 4.
Dialogue: 0,1:08:43.56,1:08:46.75,Default,,0000,0000,0000,,And what we wanted here was the\Nrate of change of {\i1}dx{\i0} coordinates.
Dialogue: 0,1:08:46.75,1:08:48.61,Default,,0000,0000,0000,,So we want ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:08:49.100,1:08:54.50,Default,,0000,0000,0000,,Okay, so my formula was {\i1}y{\i0} = √(1+{\i1}x{\i0}³)?
Dialogue: 0,1:08:55.36,1:08:57.39,Default,,0000,0000,0000,,And one thing that's nice\Nabout these problems is
Dialogue: 0,1:08:57.39,1:09:00.01,Default,,0000,0000,0000,,you don't have to find the formula,\Nor think about it,
Dialogue: 0,1:09:00.01,1:09:02.63,Default,,0000,0000,0000,,or figure out what it is, because\Nit's just handed to you.
Dialogue: 0,1:09:02.63,1:09:05.86,Default,,0000,0000,0000,,It's whatever the equation is.\NIt's kinda nice, really.
Dialogue: 0,1:09:06.64,1:09:09.25,Default,,0000,0000,0000,,So we're always differentiating\Nwith respect to {\i1}t{\i0};
Dialogue: 0,1:09:09.25,1:09:12.52,Default,,0000,0000,0000,,so I wrote ({\i1}d{\i0}/{\i1}dt{\i0}) of both sides.
Dialogue: 0,1:09:12.52,1:09:15.50,Default,,0000,0000,0000,,But what I did over here was,\NI just rewrote it in the form of
Dialogue: 0,1:09:15.50,1:09:19.51,Default,,0000,0000,0000,,a rational exponent, because it makes\Nit easier for me to differentiate.
Dialogue: 0,1:09:20.76,1:09:22.76,Default,,0000,0000,0000,,So my left is ({\i1}dy{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:09:22.76,1:09:25.63,Default,,0000,0000,0000,,And... here's differentiating\Non the right-hand side.
Dialogue: 0,1:09:25.63,1:09:29.50,Default,,0000,0000,0000,,½ down in front. Rewrite 1 +{\i1}x{\i0}³.
Dialogue: 0,1:09:29.50,1:09:31.100,Default,,0000,0000,0000,,Decrease by 1, so -½.
Dialogue: 0,1:09:32.50,1:09:35.27,Default,,0000,0000,0000,,This is the derivative of the inside.
Dialogue: 0,1:09:35.27,1:09:38.27,Default,,0000,0000,0000,,The inside is the (1 +{\i1}x{\i0}³).
Dialogue: 0,1:09:38.27,1:09:45.21,Default,,0000,0000,0000,,But the derivative of (1 +{\i1}x{\i0}³)\Nis 3{\i1}x{\i0}²({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:09:45.72,1:09:50.88,Default,,0000,0000,0000,,Any time it's the wrong letter,\Ngotta follow it by that ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:09:51.75,1:09:54.50,Default,,0000,0000,0000,,And then I just substituted\Nall of the stuff in.
Dialogue: 0,1:09:54.50,1:09:57.24,Default,,0000,0000,0000,,({\i1}dy{\i0}/{\i1}dt{\i0}) was 4.
Dialogue: 0,1:09:57.24,1:09:59.51,Default,,0000,0000,0000,,½ times 1 plus.
Dialogue: 0,1:09:59.51,1:10:03.76,Default,,0000,0000,0000,,{\i1}x{\i0} was 2, because that's the\Npoint we're kind of lookin' at here.
Dialogue: 0,1:10:05.66,1:10:10.75,Default,,0000,0000,0000,,Here's 3 times 2²;\Nand then ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:10:10.75,1:10:12.76,Default,,0000,0000,0000,,And then I did a bunch\Nof stuff in my head.
Dialogue: 0,1:10:12.76,1:10:14.14,Default,,0000,0000,0000,,Maybe that's where...
Dialogue: 0,1:10:14.14,1:10:16.06,Default,,0000,0000,0000,,[Student] That's the part\NI had a problem with. Yeah.
Dialogue: 0,1:10:16.06,1:10:16.91,Default,,0000,0000,0000,,[Instructor] Sorry about that.
Dialogue: 0,1:10:16.91,1:10:21.04,Default,,0000,0000,0000,,I could see that I didn't have much\Nroom left; and so, that's why I did that.
Dialogue: 0,1:10:21.80,1:10:26.33,Default,,0000,0000,0000,,So the 2³ plus 1 is 9.\NBut it's 9 to the -½.
Dialogue: 0,1:10:26.33,1:10:30.10,Default,,0000,0000,0000,,That's 1 over the square root of 9.
Dialogue: 0,1:10:31.61,1:10:32.90,Default,,0000,0000,0000,,So that's ⅓.
Dialogue: 0,1:10:34.39,1:10:37.62,Default,,0000,0000,0000,,And that's why I wrote this 3—\Noh, it paused on me—
Dialogue: 0,1:10:37.62,1:10:39.64,Default,,0000,0000,0000,,That's why I wrote this 3 down here.
Dialogue: 0,1:10:40.64,1:10:43.63,Default,,0000,0000,0000,,So my ½ is one over two.
Dialogue: 0,1:10:43.63,1:10:46.53,Default,,0000,0000,0000,,This, when I bring it downstairs, is 3.
Dialogue: 0,1:10:47.26,1:10:51.07,Default,,0000,0000,0000,,And then, 2² is 4. Times 3;\Nthere's the 12.
Dialogue: 0,1:10:51.99,1:10:53.41,Default,,0000,0000,0000,,And then ({\i1}dx{\i0}/{\i1}dt{\i0}).
Dialogue: 0,1:10:53.41,1:10:54.46,Default,,0000,0000,0000,,Did that help?
Dialogue: 0,1:10:56.76,1:10:59.02,Default,,0000,0000,0000,,[Student] -Uh, just give me a moment.\N[Instructor] -Sure.
Dialogue: 0,1:11:05.24,1:11:09.75,Default,,0000,0000,0000,,So you're multiplying ½ by -⅑ ?
Dialogue: 0,1:11:09.75,1:11:13.25,Default,,0000,0000,0000,,So, it's more like this. I'm gonna show\Ny'all the steps out here to the side.
Dialogue: 0,1:11:13.25,1:11:15.95,Default,,0000,0000,0000,,The 1 + 2³, that's a 9.
Dialogue: 0,1:11:15.95,1:11:18.62,Default,,0000,0000,0000,,But it's a 9 to the -½.
Dialogue: 0,1:11:18.62,1:11:23.01,Default,,0000,0000,0000,,Well, that's the same as\N⅑ to the positive ½.
Dialogue: 0,1:11:25.96,1:11:28.75,Default,,0000,0000,0000,,And then times 3 times 4.
Dialogue: 0,1:11:29.95,1:11:36.96,Default,,0000,0000,0000,,So that's ½ times ⅓; 'cause 9 to the\N½ is square root of 9, and that's 3.
Dialogue: 0,1:11:37.41,1:11:39.19,Default,,0000,0000,0000,,And that's times 12.
Dialogue: 0,1:11:42.38,1:11:46.04,Default,,0000,0000,0000,,So, this is 12 over 6, which is 2.
Dialogue: 0,1:11:48.08,1:11:49.76,Default,,0000,0000,0000,,And that's where that comes from.
Dialogue: 0,1:11:51.91,1:11:52.87,Default,,0000,0000,0000,,-Thank you.\N-Sure.
Dialogue: 0,1:11:52.87,1:11:54.99,Default,,0000,0000,0000,,-I really needed to see that.\N-Good.
Dialogue: 0,1:11:56.12,1:11:57.23,Default,,0000,0000,0000,,Glad to help.
Dialogue: 0,1:11:58.29,1:12:01.20,Default,,0000,0000,0000,,Anybody else still in the room,\Nyou have a question; go ahead.
Dialogue: 0,1:12:25.01,1:12:28.50,Default,,0000,0000,0000,,[Student] So, is this considered as\Na multivariable calculus, or...
Dialogue: 0,1:12:28.50,1:12:30.04,Default,,0000,0000,0000,,This is just single-variable?
Dialogue: 0,1:12:30.04,1:12:33.40,Default,,0000,0000,0000,,[Instructor] Uh, yeah; actually,\Nthat's a good question.
Dialogue: 0,1:12:34.14,1:12:39.50,Default,,0000,0000,0000,,Uh, no. It's not considered\Nmulti-variable calculus. It's not.
Dialogue: 0,1:12:41.14,1:12:41.100,Default,,0000,0000,0000,,[Student, softly] Okay.
Dialogue: 0,1:12:46.76,1:12:49.75,Default,,0000,0000,0000,,There you'll get into, you know...
Dialogue: 0,1:12:49.75,1:12:55.62,Default,,0000,0000,0000,,{\i1}xyz{\i0}, and all this cool stuff\Ncalled partial differentiation,
Dialogue: 0,1:12:55.62,1:12:59.87,Default,,0000,0000,0000,,and you'll have double and\Ntriple integrals. It's good.
Dialogue: 0,1:12:59.87,1:13:02.14,Default,,0000,0000,0000,,So, no; this is not that.
Dialogue: 0,1:13:06.49,1:13:08.61,Default,,0000,0000,0000,,-Can't wait to learn that.\N-Yeah, thank you for today; I'm leaving.
Dialogue: 0,1:13:08.61,1:13:09.87,Default,,0000,0000,0000,,Oh, you're welcome. Bye-bye.
Dialogue: 0,1:13:11.49,1:13:14.18,Default,,0000,0000,0000,,-Manon, you said you can't wait?\N-Yeah, I can't wait to learn that.
Dialogue: 0,1:13:14.18,1:13:17.07,Default,,0000,0000,0000,,Yeah, it's beautiful stuff.\NYou'll love it.
Dialogue: 0,1:13:20.12,1:13:23.50,Default,,0000,0000,0000,,I was looking at one of the\Nhardest challenging problems
Dialogue: 0,1:13:23.50,1:13:27.96,Default,,0000,0000,0000,,in the mathematics right now,\Nwhich is Riemann's data function,
Dialogue: 0,1:13:27.96,1:13:30.21,Default,,0000,0000,0000,,-that they can't prove it.\N-They did? Oh wow.
Dialogue: 0,1:13:31.56,1:13:35.14,Default,,0000,0000,0000,,Yeah; they can't, um.\NThere's no proof of it.
Dialogue: 0,1:13:35.14,1:13:38.49,Default,,0000,0000,0000,,We know the answer, like...
Dialogue: 0,1:13:39.25,1:13:42.26,Default,,0000,0000,0000,,The numbers, but. We can't\Nprove it, basically, right now.
Dialogue: 0,1:13:43.52,1:13:46.78,Default,,0000,0000,0000,,Yeah. So, if you can't prove it,\Nthen you don't know it.
Dialogue: 0,1:13:48.39,1:13:52.92,Default,,0000,0000,0000,,We can graph it, and look at the values,\Nand we know where it's approaching.
Dialogue: 0,1:13:53.50,1:13:54.84,Default,,0000,0000,0000,,The answer, but.
Dialogue: 0,1:13:55.99,1:13:59.00,Default,,0000,0000,0000,,Yeah, there's a $1,000,000 prize\Non it, if someone solves it.
Dialogue: 0,1:13:59.00,1:14:01.55,Default,,0000,0000,0000,,-Oh ho-ho, nice.\N-Yeah.
Dialogue: 0,1:14:07.50,1:14:08.96,Default,,0000,0000,0000,,Anybody else?
Dialogue: 0,1:14:08.96,1:14:10.90,Default,,0000,0000,0000,,[Student] All right, I'm gonna leave.\NThank you for the lecture.
Dialogue: 0,1:14:10.90,1:14:12.24,Default,,0000,0000,0000,,Okay. Bye, Manon!
Dialogue: 0,1:14:12.24,1:14:15.49,Default,,0000,0000,0000,,-Yeah. Stay safe.\N-You, too.
Dialogue: 0,1:14:20.74,1:14:24.21,Default,,0000,0000,0000,,All right, everybody;\NI'll end the meeting, and um...
Dialogue: 0,1:14:25.50,1:14:28.12,Default,,0000,0000,0000,,I'll see ya Wednesday. Bye!