WEBVTT 00:00:05.250 --> 00:00:09.970 Okay, let's pretend this balloon is a perfect sphere. 00:00:10.500 --> 00:00:20.804 Okay, that was exciting. So, over those, what was that, about eight seconds? Over 00:00:20.804 --> 00:00:21.440 those 00:00:21.440 --> 00:00:30.415 eight seconds while I was blowing up the balloon. What was changing? Hold on, hold 00:00:30.415 --> 00:00:31.400 on, let's 00:00:31.400 --> 00:00:38.171 go one at a time. Somebody from that row. Volume change. What else? Sorry, let's go 00:00:38.171 --> 00:00:38.660 second 00:00:38.660 --> 00:00:56.711 row. Surface area is changing. What else? Sorry, this row. Anybody from this row? 00:00:56.711 --> 00:00:57.380 The 00:00:57.380 --> 00:01:11.020 diameter change. Somebody from this row. The radius change. Somebody from this row. 00:01:11.450 --> 00:01:23.074 The size. Yes. What size? The size of the... Sure, let's go with circumference. 00:01:23.074 --> 00:01:25.870 That's another one. 00:01:26.400 --> 00:01:46.328 What else changed? Somebody from this row. Oh, is that it? Did the color change at 00:01:46.328 --> 00:01:47.328 all? 00:01:47.980 --> 00:02:01.269 The color changed a little bit. How about the thickness of the balloon, the latex, 00:02:01.269 --> 00:02:03.700 whatever it is. 00:02:09.220 --> 00:02:18.005 The air in my lungs changed. Volume of lungs. Anything else? I'm sure we can come up 00:02:18.005 --> 00:02:22.920 with more. But lots of things change over time. 00:02:22.920 --> 00:02:35.761 As I release air, all these same things change. So, lots of things change. They're 00:02:35.761 --> 00:02:45.000 all related in some way. So, we'll get to that in a second. 00:02:48.000 --> 00:03:02.848 So, if it's a sphere, what's the formula for volume of a sphere? R cubed. Somebody 00:03:02.848 --> 00:03:14.980 discovered it. We're actually going to prove this formula later on. 00:03:14.980 --> 00:03:25.604 So, this equation is how volume is related to radius. They're related by exactly 00:03:25.604 --> 00:03:36.360 this formula. At any given time, if we knew the radius, we could find the volume. 00:03:36.360 --> 00:03:42.428 At any given time, we knew the volume, we could find the radius. So, that's how 00:03:42.428 --> 00:03:48.803 these variables are related. These are also functions. So, V is a function and R is 00:03:48.803 --> 00:03:55.025 a function. It's not just a variable. What is the volume a function of over those 00:03:55.025 --> 00:03:56.100 eight seconds? 00:04:00.060 --> 00:04:08.202 Is it a function of time? We could write it like that. Because the volume changed 00:04:08.202 --> 00:04:16.646 over the time period, it's a function of time. What is radius a function of? Time as 00:04:16.646 --> 00:04:23.080 well. So, even though we just see variables, they are functions. 00:04:24.420 --> 00:04:30.915 And they're functions of time as time passed. The volume changed, the radius 00:04:30.915 --> 00:04:33.820 changed, and so they're functions. 00:04:37.440 --> 00:04:44.507 Now, if we want to know how the rates have changed, how fast the volume is changing, 00:04:44.507 --> 00:04:51.154 or how fast the radius is changing, we simply differentiate. Using what kind of 00:04:51.154 --> 00:04:52.500 differentiation? 00:04:53.700 --> 00:05:02.660 Implicit. Implicit, because we've got functions on both sides. So, we differentiate 00:05:02.660 --> 00:05:11.512 V. What is that? With respect to time, that's simply dV/dt. We differentiate this, 00:05:11.512 --> 00:05:19.500 we get 4 3rd pi times 3 R squared. And because of the chain rule, times... 00:05:23.000 --> 00:05:28.880 The derivative of R, which is dr/dt. 00:05:31.080 --> 00:05:41.702 dV/dt is the rate of change of volume with respect to time. It's how fast the volume 00:05:41.702 --> 00:05:43.220 is changing. 00:05:45.720 --> 00:05:53.056 And this is how fast the radius is changing. And how are the rates related? They're 00:05:53.056 --> 00:06:00.215 related by exactly this equation. It's that simple. So, this is how the functions 00:06:00.215 --> 00:06:07.640 are related, or the variables. This is how the rates of those functions are related. 00:06:08.640 --> 00:06:16.740 Okay, so related rates is simply an equation that involves d derivatives. 00:06:17.120 --> 00:06:30.825 Okay. As I was blowing it up, let's say I was blowing it up at 1 gallon per... Let's 00:06:30.825 --> 00:06:37.840 not 2 gallons. Let's go inches cubed per... 00:06:41.940 --> 00:06:50.885 20 seconds. Let's assume it's a constant rate. Let's assume it didn't change for a 00:06:50.885 --> 00:06:59.722 minute. That would be our function dV/dt. dV/dt is a function. It's a function of 00:06:59.722 --> 00:07:01.140 time as well. 00:07:01.510 --> 00:07:08.557 And the units for any rate of change is always the units of this variable or 00:07:08.557 --> 00:07:16.067 function divided by the units of this variable or function. Okay. That's why it's 00:07:16.067 --> 00:07:19.590 inches cubed per seconds or something. 00:07:22.150 --> 00:07:27.491 Is this a positive rate or a negative rate? I mean, obviously I've wrote it as 00:07:27.491 --> 00:07:30.710 positive. But would it be positive or negative? 00:07:35.710 --> 00:07:44.973 It is positive. Why is it positive? Yeah, so whenever this function... If V is 00:07:44.973 --> 00:07:54.591 increasing, it's a positive rate. If volume... Like when I let out the air at the 00:07:54.591 --> 00:08:00.410 end, volume was decreasing, it would not be this. 00:08:00.710 --> 00:08:10.398 It would be a negative rate. Okay. If I let out the volume at the exact same rate I 00:08:10.398 --> 00:08:20.203 blew it in, then it would be exactly dV/dt. It would simply be negative. 1 inch. Oh, 00:08:20.203 --> 00:08:27.090 that's not much. 1 inch cubed, but that's okay. 20 seconds. 00:08:27.090 --> 00:08:34.606 It works better with gallons. But that would be a very slow rate, right? One cubic 00:08:34.606 --> 00:08:41.938 inch per twenty seconds. Anyways, whenever things are changing over time, if the 00:08:41.938 --> 00:08:49.270 function V or r is increasing, the derivative is positive. The rate is positive. 00:08:49.270 --> 00:08:54.587 If the function is decreasing, like the rate is decreasing, it would be a dr, it 00:08:54.587 --> 00:09:00.169 would be a negative value. Okay, so sometimes they're just going to give you numbers 00:09:00.169 --> 00:09:05.752 for a rate. And we have to decide... They'll give you a positive number. And they'll 00:09:05.752 --> 00:09:10.485 say whether it's increasing or decreasing or you have to figure it out. 00:09:11.030 --> 00:09:20.494 And that's going to be very important, whether it's positive or negative. Surface 00:09:20.494 --> 00:09:30.075 area. What's the formula for surface area of a cube? I actually don't know what it 00:09:30.075 --> 00:09:35.450 is. Sorry, did I say a cube? I meant a sphere. 00:09:37.190 --> 00:09:48.266 Is it 4 pi R square? Yeah. Okay, I'm trusting you. Again, this is how surface area 00:09:48.266 --> 00:09:59.478 is related to radius at any point in time. They're both functions of time. We could 00:09:59.478 --> 00:10:03.530 write it as surface area of t. 00:10:04.270 --> 00:10:11.308 R of t. Okay, because they are functions of time, even though we just write them as 00:10:11.308 --> 00:10:14.530 a variable. And we differentiate this. 00:10:14.530 --> 00:10:30.828 This dS/dt is four—so that is eight pi r to the first, times the derivative of r, 00:10:30.828 --> 00:10:44.510 dr/dt. This is how the rate of change, how fast the surface area is— 00:10:44.530 --> 00:10:52.590 How fast the surface area is changing is related to how fast the radius is changing. 00:10:52.590 --> 00:11:00.267 If we were given one of these, like how fast the radius is changing, the rate of 00:11:00.267 --> 00:11:08.039 change of surface area changes at every point in time based on this and the given 00:11:08.039 --> 00:11:10.150 radius at that moment. 00:11:14.750 --> 00:11:26.346 What other equations do we have? We probably could not come up with a solution here. 00:11:26.346 --> 00:11:34.490 Again, let's say, what would be the units for surface area? 00:11:37.290 --> 00:11:45.301 Yeah, whatever the units, let's call it inches squared. And what would be the units 00:11:45.301 --> 00:11:53.120 for radius? Just be inches. Okay. The units for this are going to be what? Inches 00:11:53.120 --> 00:11:59.490 squared per whatever the units of time is. Let's call it a second. 00:12:01.030 --> 00:12:09.579 What would be the units of this? Inches, just inches is radius in seconds. And what 00:12:09.579 --> 00:12:18.127 are the units of the R? Inches. Okay. And you can see an inch times an inch divided 00:12:18.127 --> 00:12:22.350 by a second is inches squared per second. 00:12:23.750 --> 00:12:30.426 It always works out that way. And we're typically not going to put in units until 00:12:30.426 --> 00:12:37.103 the end. It just makes it way too complicated. I'll show you how we do that later 00:12:37.103 --> 00:12:38.103 on. 00:12:43.030 --> 00:12:44.050 Okay. 00:12:46.710 --> 00:12:54.330 Can we come up with a formula that relates surface area and volume? 00:12:54.750 --> 00:13:06.737 We could. We would need to take this equation, volume is four thirds pi r cubed, and 00:13:06.737 --> 00:13:10.590 replace this r with that r. 00:13:11.670 --> 00:13:24.216 Okay. So what would be three-fourths V? Let's go three over pi equals R cubed. And 00:13:24.216 --> 00:13:30.030 then take the cube root of both sides. 00:13:32.030 --> 00:13:49.788 So we get r is the cube root of 3V over 4 pi. Now we replace this with this r right 00:13:49.788 --> 00:13:59.630 here. So volume equals four thirds pi r cubed. 00:14:02.810 --> 00:14:15.577 Wait, sorry. Sorry, I meant to replace it here. Let's delete that. So surface area 00:14:15.577 --> 00:14:25.230 equals 4 pi R squared, which is the cube root of 3V over 4 pi. 00:14:26.010 --> 00:14:33.074 Okay. So that's, we can often come up, no, often come up with relationships that we 00:14:33.074 --> 00:14:40.138 rarely ever deal with. But this is exactly how surface area is related to volume at 00:14:40.138 --> 00:14:41.670 any point in time. 00:14:41.670 --> 00:14:46.531 If we differentiate this, we're not going to, what, this is kind of messy. Actually, 00:14:46.531 --> 00:14:51.392 it's not that bad. But anyways. That would give us the rate of change of the surface 00:14:51.392 --> 00:14:54.930 area. How is related to the rate of change of the volume? 00:15:00.990 --> 00:15:02.390 Okay. 00:15:02.390 --> 00:15:08.921 Times the height. Okay. Again, this is how the volume is related to the both the 00:15:08.921 --> 00:15:15.533 radius and the height. Or this is how the height is related to the volume by this 00:15:15.533 --> 00:15:22.390 equation. At any point in time, if we knew two of them, we could find the third one. 00:15:23.130 --> 00:15:31.059 Again, these are functions. They're all functions of time. So if we differentiate, 00:15:31.059 --> 00:15:37.150 we get dV/dt is, what rule do we got to use here? Product rule. 00:15:39.330 --> 00:15:55.390 Two pi r times dr/dt times h, plus pi r squared times the derivative of height, 00:15:55.390 --> 00:15:56.610 dh/dt. 00:15:58.390 --> 00:16:05.260 So again, this is how all those rates are related. We're filling this up with water. 00:16:05.260 --> 00:16:12.048 How fast the volume is changing this. This is how fast the radius is changing. This 00:16:12.048 --> 00:16:14.910 is how fast the height is changing. 00:16:14.910 --> 00:16:20.240 And again, over time, they all depend on all of them, including the radius at that 00:16:20.240 --> 00:16:24.107 point in time, including the height at that point in time. 00:16:30.390 --> 00:16:33.510 Um, what's the formula for volume of a cone? 00:16:39.610 --> 00:16:47.462 It is kind of weird, but it is exactly one third the volume of a cylinder: pi r 00:16:47.462 --> 00:16:55.514 squared h. We are going to do a lot of these, because we are filling this up with 00:16:55.514 --> 00:16:56.514 water. 00:16:58.110 --> 00:17:05.963 Now the radius at any point in time, whatever the height of the water is, the radius 00:17:05.963 --> 00:17:13.443 is changing. The height is changing and the volume is changing. So that makes it 00:17:13.443 --> 00:17:14.443 fine. 00:17:17.770 --> 00:17:24.570 The derivative looks exactly like this, except we have a one third out here. Okay. 00:17:29.450 --> 00:17:37.561 Okay. The other thing with cones, right, let's say this is the R. Let's call this 00:17:37.561 --> 00:17:44.470 big R for a second. Let's call little r the radius at any given time. 00:17:49.710 --> 00:17:56.419 Let's call this big H, the total height of the cone. The other equation we could 00:17:56.419 --> 00:18:03.297 always come up with is how H and R are related. How are H and R any, sorry, let me 00:18:03.297 --> 00:18:04.890 do a little h here. 00:18:07.430 --> 00:18:13.890 How are they related? Sorry, if capital R. 00:18:18.030 --> 00:18:21.050 So we have two triangles here, right? 00:18:23.610 --> 00:18:33.917 We got the triangle of the volume and we got the triangle of the big of the cone. 00:18:33.917 --> 00:18:42.570 Okay. R and H are fixed. R equals some number. H equals some number. 00:18:44.910 --> 00:18:55.293 And they're proportional triangles. Very important. So R over H is equal to what? 00:18:55.293 --> 00:19:04.010 Little r over h. Okay. And let's say it was, the height was 10 feet. 00:19:05.570 --> 00:19:19.962 Let’s pick R, and H is six feet. We know ten over six is equal to little r over 00:19:19.962 --> 00:19:25.610 little h, and we multiply by h. 00:19:25.610 --> 00:19:29.590 What is it? Five thirds. 00:19:33.230 --> 00:19:34.230 Equals R. 00:19:35.750 --> 00:19:46.961 So almost all cone problems. The radius and the height of the cone are some fixed 00:19:46.961 --> 00:19:47.961 number. 00:19:49.810 --> 00:19:55.285 We come up with that, and then we differentiate this, which is simply five thirds 00:19:55.285 --> 00:19:58.530 times the derivative of the height of the water. 00:20:00.190 --> 00:20:03.550 And this is the R. Okay. 00:20:08.250 --> 00:20:15.750 And then once we get our first real problem, we'll show you what we do with that. 00:20:16.210 --> 00:20:20.410 Anything that was triangles, we come up with lots of equations. 00:20:24.810 --> 00:20:30.888 The triangle has to be changing in some way. Let's say this is some person and he's 00:20:30.888 --> 00:20:33.890 walking away. And this is another person. 00:20:40.310 --> 00:20:48.830 The triangle has to be changing. The green will be my new triangle. 00:20:49.250 --> 00:20:55.746 Let me call this theta for a second. So what equations can we come up with with the 00:20:55.746 --> 00:20:56.746 triangle? 00:21:01.130 --> 00:21:07.310 I don't see A's or B's. I like that though. There we go. 00:21:08.310 --> 00:21:17.610 And why is that true? Good old Pythagorean, our favorite geometer. 00:21:21.530 --> 00:21:27.642 Okay. This is how those four functions are related. Or sorry, three functions, three 00:21:27.642 --> 00:21:28.642 variables. 00:21:29.370 --> 00:21:39.898 If we differentiate, this one is easy: two x times dx/dt plus two y times dy/dt 00:21:39.898 --> 00:21:43.230 equals two z times dz/dt. 00:21:51.530 --> 00:21:57.390 And this is how their rates of change are related. 00:21:59.710 --> 00:22:07.072 Okay. If we wanted to figure out how fast the distance between the two guys are, 00:22:07.072 --> 00:22:11.950 that would be this. We divide everything by 2z. Oops. 00:22:15.090 --> 00:22:26.750 What is that? Two x dx/dt plus two y dy/dt, over two z. 00:22:56.950 --> 00:22:58.350 Okay. 00:22:59.430 --> 00:23:04.189 This is exactly the units of this divided by the units of that. Okay. So if we were 00:23:04.189 --> 00:23:08.947 to plug everything in, we don't plug in any units. We just know at the end, we look 00:23:08.947 --> 00:23:11.630 at the units of z divided by the units of t. 00:23:11.630 --> 00:23:15.510 And that's the units of this number. Okay. 00:23:16.030 --> 00:23:23.510 What other equation can we come up with this triangle? 00:23:31.110 --> 00:23:32.370 How about area? 00:23:35.210 --> 00:23:41.749 One half times base times height. Let me just call it base times height for a 00:23:41.749 --> 00:23:48.627 second. Or let's call this the base. This is the height. You could use x and y or 00:23:48.627 --> 00:23:50.410 change the variables. 00:23:50.990 --> 00:23:57.330 Okay. So this guy's walking. This area is changing. It's not that, it's kind of a 00:23:57.330 --> 00:24:03.826 weird thing, but this is how the area is related to the base, related to the height 00:24:03.826 --> 00:24:05.470 at any point in time. 00:24:06.230 --> 00:24:14.870 We differentiate dA/dt. And what rule we got to use here? Product rule again. 00:24:15.870 --> 00:24:26.617 So that is one half db/dt times the height, plus one half the base times the 00:24:26.617 --> 00:24:29.870 derivative of h, dh/dt. 00:24:29.870 --> 00:24:34.764 This is how the rate, how fast the area is changing. It depends on how fast B is 00:24:34.764 --> 00:24:39.596 changing. Depends on the height of that given time. Depends on the base at that 00:24:39.596 --> 00:24:44.730 point in time. It depends on the rate of change of the height at any given time. 00:24:44.960 --> 00:24:51.471 We got an angle. In fact, we got three angles, right? We can come up with lots of 00:24:51.471 --> 00:24:53.480 equations involving this. 00:24:54.480 --> 00:25:05.600 We can come up with sine of theta. What would that be? X over Z. 00:25:07.080 --> 00:25:14.540 What is cosine of theta? Y over Z. 00:25:17.140 --> 00:25:21.840 What is tangent of theta? X over Y. 00:25:22.960 --> 00:25:26.765 We could obviously do secant, cosecant, and use the other derivatives. Usually we 00:25:26.765 --> 00:25:28.631 use only one of these three. 00:25:29.320 --> 00:25:34.820 This is how the angle is related to the other two sides of the triangle. 00:25:40.760 --> 00:25:46.775 If we differentiate, let's start with this one. We get, what's the derivative of 00:25:46.775 --> 00:25:48.580 theta, or sine of theta? 00:25:50.620 --> 00:25:57.000 Cosine of theta times the derivative of theta, dθ/dt. That is the left side. What 00:25:57.000 --> 00:26:00.780 rule do we use on the right side? Quotient rule. 00:26:04.600 --> 00:26:16.100 dx/dt times z, plus x times dz/dt, over z squared. 00:26:23.880 --> 00:26:33.411 Let's say we wanted to know how fast the angle is changing. We just divide both 00:26:33.411 --> 00:26:38.840 sides by cosine. That would be that equation. 00:26:39.700 --> 00:26:43.860 Did I screw up? Yes, this is definitely a minus. 00:26:48.140 --> 00:26:56.265 Let's say we do all the variables here. We plug them in. What are the units of this 00:26:56.265 --> 00:26:57.440 going to be? 00:27:05.060 --> 00:27:12.327 This is very important. In calculus, this is always in radians. If they ever give 00:27:12.327 --> 00:27:17.440 you a value in degrees, we need to convert it to radians. 00:27:19.060 --> 00:27:26.223 Except the only thing you wouldn't need to do is you can just find it there. It's 00:27:26.223 --> 00:27:31.440 guaranteed to be something radians per whatever time it is. 00:27:32.800 --> 00:27:47.060 And really radians don't have units. Why do radians don't have units? 00:27:54.560 --> 00:28:04.083 So that's some theta. Anybody remember how we defined the angle measurement in 00:28:04.083 --> 00:28:05.083 radians? 00:28:14.950 --> 00:28:24.808 Theta is simply the ratio of the arc length over the radius. And both of these are 00:28:24.808 --> 00:28:26.010 distances. 00:28:26.010 --> 00:28:33.647 Let's say it was inches. The inches cancel out and there are no units. So radians 00:28:33.647 --> 00:28:41.473 don't really have units. Sometimes we don't even write radians, we just put nothing 00:28:41.473 --> 00:28:42.510 per second. 00:28:42.510 --> 00:28:54.210 And that's okay. But we know it's radians. It's going to be very important. We 00:28:54.210 --> 00:29:00.510 always need to know the things in radians. 00:29:00.800 --> 00:29:10.499 What is the first graph going to look like? It is going to be some linear function. 00:29:10.499 --> 00:29:14.940 The rate of change of volume is dV/dt. 00:29:14.940 --> 00:29:18.900 Okay. What about the one on the right? 00:29:24.440 --> 00:29:27.620 So what happens once it gets here? 00:29:30.560 --> 00:29:39.640 Yes, the height grows a rate much faster. So it's going to be something like this. 00:29:39.640 --> 00:29:46.140 [Brief speech unclear.] 00:30:01.380 --> 00:30:04.980 It would be the height. 00:30:07.480 --> 00:30:13.400 Something like that. Third one. 00:30:17.980 --> 00:30:19.960 So slow. 00:30:22.980 --> 00:30:25.380 Faster, slower, something like that. 00:30:26.440 --> 00:30:28.173 Okay. What about this one? 00:30:31.260 --> 00:30:35.000 So faster and slower, something like that. 00:30:35.370 --> 00:30:46.990 Right. Three inches a year. Three inches a year. 00:30:47.450 --> 00:30:55.971 That's a lot of it. It depends on their point in time. Any time you have a rate, 00:30:55.971 --> 00:31:01.190 it's a d something dt. Let's change it over time. 00:31:02.150 --> 00:31:06.030 So this would be d what over dt? 00:31:09.610 --> 00:31:11.543 Inches is what kind of thing? 00:31:13.930 --> 00:31:23.463 It's a distance, but when you're growing, we call that your height. So it would be d 00:31:23.463 --> 00:31:24.030 h dt. 00:31:24.030 --> 00:31:27.963 And is it positive three or negative three inches per year? 00:31:29.350 --> 00:31:35.930 Because we're growing, it's positive three inches per year. 00:31:40.850 --> 00:31:46.383 My mutual fund is shrinking. What does that tell us since it's shrinking? Negative. 00:31:48.110 --> 00:31:50.190 Four cents per day. 00:31:51.790 --> 00:31:55.010 And this would be d what? Dt. 00:31:57.090 --> 00:32:03.708 Okay. It helps if you look at the units. What is the units? What is cents a 00:32:03.708 --> 00:32:08.650 measurement of? Money. So you could call it d money, dt. 00:32:10.550 --> 00:32:19.958 D fund, dt. D value, dt, whatever the value of the fund is. So this would be, it's 00:32:19.958 --> 00:32:21.450 called dm dt. 00:32:23.170 --> 00:32:25.850 The radius of a circle is increasing. 00:32:28.230 --> 00:32:37.187 So that would be what? D radius dt. Is it positive or negative four feet per hour? 00:32:37.187 --> 00:32:38.187 Positive. 00:32:46.490 --> 00:32:55.170 The volume of the cone is decreasing. So that would be d v dt. 00:32:57.630 --> 00:32:59.697 And is it positive or negative? 00:33:00.790 --> 00:33:07.548 When you tell if it's decreasing, negative two inches cubed per second. Again, we 00:33:07.548 --> 00:33:09.550 could look at the units. 00:33:10.290 --> 00:33:12.957 Inches cubed is a measurement of volume. 00:33:13.950 --> 00:33:16.550 So in seconds is a measurement of time. 00:33:19.090 --> 00:33:20.157 Okay, next page. 00:33:20.620 --> 00:33:24.880 We got a rectangle. Start by just drawing a rectangle. 00:33:26.520 --> 00:33:29.453 I don't know if there's much room here, but. 00:33:31.820 --> 00:33:32.820 Okay. 00:33:33.880 --> 00:33:41.176 Got a width. We got a, what else we got? So just call the width and the length. 00:33:41.176 --> 00:33:46.440 Whatever you want to call it. Let's call this the length. 00:33:47.100 --> 00:33:48.300 This is the width. 00:33:50.160 --> 00:33:51.160 Okay. 00:33:52.800 --> 00:33:59.322 And then we come up with some, some, how those two variables are related. And what 00:33:59.322 --> 00:34:02.980 are we talking about the, what's the question? 00:34:06.120 --> 00:34:12.474 Perimeter. Okay, so we come up with an equation involving a perimeter. What's the 00:34:12.474 --> 00:34:13.474 equation? 00:34:16.000 --> 00:34:21.580 Perimeter equals. 2L. 2L plus 2W. 00:34:24.840 --> 00:34:28.107 And then I automatically just differentiate that. 00:34:29.580 --> 00:34:38.231 So derivative of perimeter with respect to time. Is 2 times the derivative of L with 00:34:38.231 --> 00:34:41.320 respect to time. Plus 2 times. 00:34:43.480 --> 00:34:52.059 Okay. That's how their three rates are related. Okay, then I, I go back through the 00:34:52.059 --> 00:34:54.540 problem and label stuff. 00:34:55.960 --> 00:35:02.360 So it's the length is increasing at a rate. So what is that going to be? 00:35:05.020 --> 00:35:13.529 L. DT. Anytime you have a, they say the word rate. It's a D something DT. And since 00:35:13.529 --> 00:35:18.860 it's increasing, it is positive 2 inches per second. 00:35:22.320 --> 00:35:24.680 And the width is increasing. 00:35:26.560 --> 00:35:29.160 At a rate, that's going to be a D what? 00:35:31.960 --> 00:35:33.960 D. W. 00:35:35.860 --> 00:35:36.860 DT. 00:35:42.300 --> 00:35:46.060 Equals 3 inches per second. 00:35:50.040 --> 00:35:54.712 Okay, then how fast? Same thing the rest of the facts. They're asking for a rate. 00:35:54.712 --> 00:35:55.712 Okay. 00:35:57.060 --> 00:36:01.660 Perimeter. So we want to know. D. P. DT. 00:36:03.860 --> 00:36:04.860 Okay. 00:36:06.620 --> 00:36:12.798 And then they give us exact moments of time. We never plug these in. Until after we 00:36:12.798 --> 00:36:13.840 differentiate. 00:36:15.400 --> 00:36:21.334 Okay. Never write this formula with any numbers in there. Unless we're told one of 00:36:21.334 --> 00:36:23.360 those sides is not changing. 00:36:23.360 --> 00:36:26.980 Or you know it's not changing. Okay. So. 00:36:30.900 --> 00:36:32.167 What would this be? 00:36:34.460 --> 00:36:35.460 This is L. 00:36:37.060 --> 00:36:42.060 Of the length. At some time, unknown time. 00:36:45.360 --> 00:36:51.300 Equals 10 inches. And the width at some point in time, the same point in time. 00:36:53.140 --> 00:36:54.900 Is 6 inches. 00:36:55.350 --> 00:37:05.090 So we want to know D. B. D. So we solve for that. So we solve. So it's simply D. D. 00:37:05.090 --> 00:37:09.550 P. D. T. Is 2 times. What is D. L. DT. 00:37:12.590 --> 00:37:13.723 We wrote up here. 00:37:14.810 --> 00:37:21.630 Plus 2. Times. D. W. DT. Which is right here. 00:37:24.610 --> 00:37:31.117 And again, don't bother putting in units. It gets super messy if you do. We got 4 00:37:31.117 --> 00:37:34.330 plus 6 is 10. Okay. Now we put in units. 00:37:34.770 --> 00:37:38.946 And the way we figure out is you put in the units of P divided by the units of time. 00:37:38.946 --> 00:37:40.212 The units of P was. 00:37:41.630 --> 00:37:45.850 Inches. To distance and units of time was seconds. 00:37:47.970 --> 00:37:49.970 So at that one moment in time. 00:37:51.490 --> 00:37:57.545 When the length was exactly this and the width was exactly this. The rate of change 00:37:57.545 --> 00:38:00.390 of the perimeter is exactly this. Okay. 00:38:00.890 --> 00:38:06.150 Actually, did we even plug in those numbers at all? No. Do we need you? 00:38:07.790 --> 00:38:11.230 No. Okay. So in fact, in every moment in time. 00:38:13.670 --> 00:38:18.890 Not just a specific moment. It happens to not change. And the reason is that the 00:38:18.890 --> 00:38:21.424 derivative doesn't involve L's or W's. 00:38:22.350 --> 00:38:28.623 If this were like W squared, we would. Definitely have a W in it. And then it would 00:38:28.623 --> 00:38:30.890 depend on that one time. Okay. 00:38:31.470 --> 00:38:34.537 So in fact, this is a constant rate of change. 00:38:35.550 --> 00:38:37.750 Because these two were constants. 00:38:38.870 --> 00:38:39.870 Interesting. 00:38:41.350 --> 00:38:44.883 Now the rectangle's length is increasing at 3 inches. 00:38:49.790 --> 00:38:53.650 So the rectangle's length. What would that be? 00:38:55.630 --> 00:38:56.630 D L. 00:38:58.230 --> 00:39:03.910 D T. Is increasing. So it's 3 inches per second. 00:39:06.210 --> 00:39:09.050 The width is decreasing. So D. 00:39:11.070 --> 00:39:20.550 W. D T. H. Is very important. Negative 2 inches per second. 00:39:23.410 --> 00:39:25.277 Area. We need new equations. 00:39:26.910 --> 00:39:32.830 What is area of a rectangle? Length times width. Okay. Now we got to differentiate. 00:39:35.830 --> 00:39:37.970 D area. D T. 00:39:40.970 --> 00:39:43.130 Is D L. D T. 00:39:45.150 --> 00:39:46.510 Plus. 00:39:48.590 --> 00:39:49.590 L. 00:39:51.210 --> 00:39:53.870 Times. D W. D T. 00:40:01.890 --> 00:40:04.430 So D. 00:40:08.130 --> 00:40:10.050 D A. D T. 00:40:14.030 --> 00:40:16.730 Is. What was this rate? That's 3. 00:40:18.790 --> 00:40:23.690 Okay. Times W at that one moment in time, which is 6. 00:40:26.250 --> 00:40:27.250 Plus. 00:40:28.510 --> 00:40:37.090 L. Which is 10 at that one moment in time. Times D W D T. Which is negative 2. 00:40:48.230 --> 00:40:52.370 What is that negative 2. 00:40:57.230 --> 00:41:02.406 And the units are. The units of A divided by units of time. What is the units of 00:41:02.406 --> 00:41:03.406 area? 00:41:05.490 --> 00:41:10.390 Everything was in inches. So it's inches squared. So. 00:41:12.650 --> 00:41:14.850 Time was measured in seconds. 00:41:16.670 --> 00:41:17.670 Okay. 00:41:20.250 --> 00:41:22.917 What does it mean that this is negative? 00:41:24.090 --> 00:41:29.393 It is getting smaller. It means the area. The area is decreasing at a rate of two 00:41:29.393 --> 00:41:31.060 square inches per second. 00:41:31.670 --> 00:41:40.782 Okay, spherical snowball melts. At the instant—one moment in time—at which the 00:41:40.782 --> 00:41:44.170 radius is twenty centimeters. 00:41:44.170 --> 00:41:52.971 Its radius is decreasing at three centimeters per minute. At what rate is the volume 00:41:52.971 --> 00:41:55.590 of the snowball changing? 00:41:57.250 --> 00:42:00.510 So what formula do you come up with? 00:42:02.630 --> 00:42:08.780 We went over volume or the rate of change of volume. So volume equals volume of a 00:42:08.780 --> 00:42:11.210 sphere, which we talked earlier. 00:42:15.550 --> 00:42:26.510 We might as well label we got. What is the 20 centimeters? 00:42:28.350 --> 00:42:35.910 That's the radius at one moment in time. So that's our of some time. Let's call it a 00:42:35.910 --> 00:42:36.910 is 20. 00:42:40.550 --> 00:42:44.190 Its radius is decreasing. What is this? 00:42:46.230 --> 00:42:47.230 Dr. 00:42:48.790 --> 00:42:55.790 Dt is a positive or negative negative three centimeters per minute. 00:42:55.790 --> 00:43:05.450 And we want to know the rate of the volume that is dv dt. 00:43:09.130 --> 00:43:12.790 We differentiate this, which we did earlier, right? 00:43:13.250 --> 00:43:24.250 dV/dt is four thirds pi times three r squared times the derivative of r, dr/dt. 00:43:34.110 --> 00:43:41.076 So dv dt at that one moment in time. I called it a earlier. We don't know what it 00:43:41.076 --> 00:43:42.710 is. Doesn't matter. 00:43:50.350 --> 00:43:52.990 What was the radius of that time? 00:43:56.090 --> 00:43:57.090 Twenty. 00:43:58.670 --> 00:44:01.510 And dr/dt was negative three. 00:44:03.910 --> 00:44:10.250 And we do some math. What's this going to be? 00:44:23.730 --> 00:44:26.210 Nothing negative. 00:44:30.350 --> 00:44:42.570 Four hundred times four is sixteen hundred; times three is forty-eight hundred. 00:44:48.170 --> 00:44:57.490 And we put in pi. What did you say? Negative what? 00:45:02.570 --> 00:45:06.630 About three times that seems about right. Point what? 00:45:09.790 --> 00:45:18.699 So most of the time when it's a calculator problem, we need to leave three digits. 00:45:18.699 --> 00:45:22.610 Give me three digits: six four five. 00:45:24.090 --> 00:45:29.610 So this is the exact rate of change. This is an approximate rate of change. 00:45:32.530 --> 00:45:35.197 The units, what are the units of volume? 00:45:39.330 --> 00:45:43.797 Yeah, it's whatever the units of r is cubed. So it was centimeters. 00:45:46.890 --> 00:45:51.030 Centimeters cubed per what time was it? 00:45:53.850 --> 00:46:04.434 The volume is decreasing at forty-eight hundred pi cubic centimeters per minute. How 00:46:04.434 --> 00:46:07.710 do I know it's decreasing? 00:46:07.710 --> 00:46:21.138 Because it is negative. When you write that out, you say the volume is decreasing at 00:46:21.138 --> 00:46:29.450 forty-eight hundred pi cubic centimeters per minute. 00:46:30.450 --> 00:46:34.490 You leave off the negative when you say it is decreasing. 00:46:34.780 --> 00:46:39.513 It is leaning up against the wall, so we have to draw that. Okay. Okay. 00:47:01.280 --> 00:47:07.128 Okay. The base of the ladder is being pulled away from the wall. Later on, it is 00:47:07.128 --> 00:47:09.540 going to look like that triangle. 00:47:12.660 --> 00:47:15.720 And eventually it looks like that and so on. 00:47:18.420 --> 00:47:23.660 A rate of three feet per second. All right. Let's put it to variables. 00:47:28.420 --> 00:47:30.540 What's changing here? 00:47:34.480 --> 00:47:37.480 Right. The height of the ladder is changing. 00:47:39.560 --> 00:47:40.960 Which we call that H. 00:47:47.200 --> 00:47:51.740 Let's go right now. 00:47:55.620 --> 00:47:57.287 Let's call that the base. 00:48:01.860 --> 00:48:04.327 Is the length of the ladder changing? 00:48:05.400 --> 00:48:12.784 No. So let's not put a Z here or whatever variable you want. Let's just plug in the 00:48:12.784 --> 00:48:15.720 number. This is twenty five feet. 00:48:23.200 --> 00:48:30.391 How fast is the well, let's go here. Being pulled away at three feet at a rate of 00:48:30.391 --> 00:48:34.120 three feet per second. What would that be? 00:48:36.740 --> 00:48:37.740 Sorry, d what? 00:48:41.300 --> 00:48:48.356 That's this distance right here is changing. That's db/dt. And is it getting larger? 00:48:48.356 --> 00:48:50.960 Is it increasing or decreasing? 00:48:55.000 --> 00:48:59.533 What's that? Increasing? Yes, increasing just three feet per second. 00:49:03.820 --> 00:49:10.745 So how fast is the top moving down the wall when the base of the ladder is twenty 00:49:10.745 --> 00:49:15.020 feet from the wall? So this is one moment in time. 00:49:18.340 --> 00:49:20.140 So what are we looking for? 00:49:22.580 --> 00:49:25.020 So we're looking for dh/dt. 00:49:31.360 --> 00:49:38.020 Except our answer is not going to be that value. I'll talk about that in a second. 00:49:43.020 --> 00:49:50.455 Okay, so what equation we come up with? How are h, how are b and h related at any 00:49:50.455 --> 00:49:51.740 point in time? 00:49:53.520 --> 00:49:55.253 We always have a triangle. 00:49:57.020 --> 00:49:58.287 What should we use? 00:50:00.660 --> 00:50:04.140 Area. Do we care about this area? No. 00:50:06.720 --> 00:50:10.389 Let's go with, this would be a what kind of triangle? Right triangle. Let's go with 00:50:10.389 --> 00:50:11.389 Pythagorean. 00:50:12.400 --> 00:50:16.380 h squared plus b squared equals twenty five squared. 00:50:18.800 --> 00:50:24.520 That's how b and h are related at any moment in time. Now we differentiate. 00:50:29.980 --> 00:50:40.520 Two h times dh/dt plus two b times db/dt. 00:50:42.560 --> 00:50:45.520 And derivative of twenty five squared is? 00:50:47.900 --> 00:50:50.240 Zero. Zero. It is just a constant. 00:50:52.520 --> 00:50:59.493 You could go twenty five, two times twenty five times the derivative of twenty five, 00:50:59.493 --> 00:51:01.900 which is zero. Don't do that. 00:51:04.180 --> 00:51:05.540 Okay, we want dh/dt. 00:51:08.200 --> 00:51:09.800 So let's solve for that. 00:51:30.380 --> 00:51:36.400 dh/dt is? 00:51:47.440 --> 00:51:49.920 We can simplify this a little bit. 00:52:06.740 --> 00:52:07.740 Okay. 00:52:09.940 --> 00:52:15.420 What was the length of the base? What were we given? 00:52:19.600 --> 00:52:24.832 So how when the base of the ladder is exactly twenty feet. So the one moment in 00:52:24.832 --> 00:52:28.565 time. So this is the rate only when this is twenty feet. 00:52:31.600 --> 00:52:33.440 db/dt is three. 00:52:39.580 --> 00:52:41.900 And how do we figure out height? 00:52:46.160 --> 00:52:49.500 We use this at that one moment in time. 00:52:51.320 --> 00:52:53.840 H squared is? 00:52:55.900 --> 00:52:58.160 Square minus b squared. 00:53:07.700 --> 00:53:10.033 B squared. What was the value of b? 00:53:11.560 --> 00:53:12.560 Twenty. Twenty. 00:53:17.340 --> 00:53:19.880 And h is? What is that? 00:53:22.940 --> 00:53:23.940 Twenty five. 00:53:29.640 --> 00:53:35.547 It is fifteen. It would have been easier if we just drew this as twenty, twenty 00:53:35.547 --> 00:53:39.660 five. What kind of triangle is this? Three, four, five. 00:53:39.660 --> 00:53:41.860 So it would be fifteen. 00:53:44.980 --> 00:53:45.980 Fifteen. 00:53:51.540 --> 00:53:53.760 And negative four. 00:53:55.860 --> 00:54:04.260 And it's the units of h divided by the units of time, which is? Feet per second. 00:54:12.800 --> 00:54:16.260 So how fast is the ladder moving down? 00:54:18.580 --> 00:54:26.180 Fast is like a speed, not a velocity. What we got here is a velocity. 00:54:28.880 --> 00:54:32.813 We've talked about difference in speed and velocity, right? 00:54:36.060 --> 00:54:43.314 So h, this distance is decreasing at four feet per second. That means exactly this 00:54:43.314 --> 00:54:48.180 point is moving that direction at four feet per second. 00:54:49.740 --> 00:54:54.231 Okay. dh/dt is not really how we want to phrase the answer. We want to know how fast 00:54:54.231 --> 00:54:55.564 this is moving down. 00:55:00.920 --> 00:55:05.760 Because it's moving down, this distance is actually. 00:55:09.400 --> 00:55:13.067 It's the rate we wanted, but we don't bother with that. 00:55:16.720 --> 00:55:17.720 Okay. 00:55:19.760 --> 00:55:25.810 Next, how fast is the angle between the ladder and the wall changing when the base 00:55:25.810 --> 00:55:27.360 is fifteen feet away? 00:55:27.860 --> 00:55:29.793 And where do I put the angle? 00:55:32.900 --> 00:55:34.240 We got two options. 00:55:38.020 --> 00:55:43.380 How fast the angle between the ladder and the wall. So that would be this angle. 00:55:46.960 --> 00:55:48.600 Before we do it. 00:55:50.520 --> 00:55:54.653 Is that angle growing or is it increasing or is it decreasing? 00:55:57.700 --> 00:56:03.066 As it goes down, this is getting a larger angle. So we're definitely better get a 00:56:03.066 --> 00:56:04.066 positive value. 00:56:05.600 --> 00:56:14.886 Okay. So anything involving theta, we use sine, cosine, or tangent. Which one should 00:56:14.886 --> 00:56:15.886 we use? 00:56:25.340 --> 00:56:33.760 Well, we're given db/dt. Okay. So use the one that involved B. 00:56:36.540 --> 00:56:41.680 Okay. So we want ideally use this. 00:56:44.900 --> 00:56:50.900 Okay. We could use tangent. 00:56:54.760 --> 00:56:57.500 But we'd have to know DHDT. 00:56:59.680 --> 00:57:03.760 Which we don't know at the moment. So let's use sine. 00:57:06.720 --> 00:57:09.920 Sine of theta is b over— 00:57:11.480 --> 00:57:14.780 And this is fixed. So put in the twenty-five. 00:57:22.480 --> 00:57:29.680 And we differentiate. We get cosine of theta times D theta, DT. 00:57:33.280 --> 00:57:40.160 This is simply one over twenty-five times the derivative of b, db/dt. 00:57:50.180 --> 00:57:51.180 Okay. 00:58:22.120 --> 00:58:26.100 When the base is fifteen feet away. 00:58:27.840 --> 00:58:32.240 First off, we know DB, DT, right? So that's, what was it? 00:58:34.820 --> 00:58:35.820 Three. 00:58:37.400 --> 00:58:38.400 Okay. 00:58:48.800 --> 00:58:55.948 So we need to know cosine of theta. At the one moment, the base is fifteen feet 00:58:55.948 --> 00:58:56.948 away. 00:58:59.100 --> 00:59:04.300 Cosine would be this length divided by this length. 00:59:06.680 --> 00:59:08.347 Adjacent over hypotenuse. 00:59:09.920 --> 00:59:13.053 So we need to figure out this length, which is, 00:59:17.280 --> 00:59:20.947 the Pythagorean theorem. This is what type of triangle? 00:59:22.080 --> 00:59:27.580 Three, four, five triangles. So we could do the math or know that that's twenty. 00:59:32.580 --> 00:59:37.580 So cosine of theta is twenty over twenty-five. 00:59:39.560 --> 00:59:42.580 Which is what? Four-fifths. 00:59:50.440 --> 00:59:58.220 Three over twenty-five times five over four. 01:00:00.200 --> 01:00:04.360 Which is three over twenty. 01:00:11.520 --> 01:00:16.560 Units of theta, always in this classes. Radians. 01:00:18.780 --> 01:00:22.920 Per time, which was, was it seconds? 01:00:27.020 --> 01:00:31.087 We get a positive value because that angle is getting larger. 01:00:42.920 --> 01:00:44.660 Okay. We're going to stop there.