1 00:00:05,250 --> 00:00:09,970 Okay, let's pretend this balloon is a perfect sphere. 2 00:00:10,500 --> 00:00:20,804 Okay, that was exciting. So, over those, what was that, about eight seconds? Over 3 00:00:20,804 --> 00:00:21,440 those 4 00:00:21,440 --> 00:00:30,415 eight seconds while I was blowing up the balloon. What was changing? Hold on, hold 5 00:00:30,415 --> 00:00:31,400 on, let's 6 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 7 00:00:38,171 --> 00:00:38,660 second 8 00:00:38,660 --> 00:00:56,711 row. Surface area is changing. What else? Sorry, this row. Anybody from this row? 9 00:00:56,711 --> 00:00:57,380 The 10 00:00:57,380 --> 00:01:11,020 diameter change. Somebody from this row. The radius change. Somebody from this row. 11 00:01:11,450 --> 00:01:23,074 The size. Yes. What size? The size of the... Sure, let's go with circumference. 12 00:01:23,074 --> 00:01:25,870 That's another one. 13 00:01:26,400 --> 00:01:46,328 What else changed? Somebody from this row. Oh, is that it? Did the color change at 14 00:01:46,328 --> 00:01:47,328 all? 15 00:01:47,980 --> 00:02:01,269 The color changed a little bit. How about the thickness of the balloon, the latex, 16 00:02:01,269 --> 00:02:03,700 whatever it is. 17 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 18 00:02:18,005 --> 00:02:22,920 with more. But lots of things change over time. 19 00:02:22,920 --> 00:02:35,761 As I release air, all these same things change. So, lots of things change. They're 20 00:02:35,761 --> 00:02:45,000 all related in some way. So, we'll get to that in a second. 21 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 22 00:03:02,848 --> 00:03:14,980 discovered it. We're actually going to prove this formula later on. 23 00:03:14,980 --> 00:03:25,604 So, this equation is how volume is related to radius. They're related by exactly 24 00:03:25,604 --> 00:03:36,360 this formula. At any given time, if we knew the radius, we could find the volume. 25 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 26 00:03:42,428 --> 00:03:48,803 these variables are related. These are also functions. So, V is a function and R is 27 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 28 00:03:55,025 --> 00:03:56,100 eight seconds? 29 00:04:00,060 --> 00:04:08,202 Is it a function of time? We could write it like that. Because the volume changed 30 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 31 00:04:16,646 --> 00:04:23,080 well. So, even though we just see variables, they are functions. 32 00:04:24,420 --> 00:04:30,915 And they're functions of time as time passed. The volume changed, the radius 33 00:04:30,915 --> 00:04:33,820 changed, and so they're functions. 34 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, 35 00:04:44,507 --> 00:04:51,154 or how fast the radius is changing, we simply differentiate. Using what kind of 36 00:04:51,154 --> 00:04:52,500 differentiation? 37 00:04:53,700 --> 00:05:02,660 Implicit. Implicit, because we've got functions on both sides. So, we differentiate 38 00:05:02,660 --> 00:05:11,512 V. What is that? With respect to time, that's simply dV/dt. We differentiate this, 39 00:05:11,512 --> 00:05:19,500 we get 4 3rd pi times 3 R squared. And because of the chain rule, times... 40 00:05:23,000 --> 00:05:28,880 The derivative of R, which is dr/dt. 41 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 42 00:05:41,702 --> 00:05:43,220 is changing. 43 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 44 00:05:53,056 --> 00:06:00,215 related by exactly this equation. It's that simple. So, this is how the functions 45 00:06:00,215 --> 00:06:07,640 are related, or the variables. This is how the rates of those functions are related. 46 00:06:08,640 --> 00:06:16,740 Okay, so related rates is simply an equation that involves d derivatives. 47 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 48 00:06:30,825 --> 00:06:37,840 not 2 gallons. Let's go inches cubed per... 49 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 50 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 51 00:06:59,722 --> 00:07:01,140 time as well. 52 00:07:01,510 --> 00:07:08,557 And the units for any rate of change is always the units of this variable or 53 00:07:08,557 --> 00:07:16,067 function divided by the units of this variable or function. Okay. That's why it's 54 00:07:16,067 --> 00:07:19,590 inches cubed per seconds or something. 55 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 56 00:07:27,491 --> 00:07:30,710 positive. But would it be positive or negative? 57 00:07:35,710 --> 00:07:44,973 It is positive. Why is it positive? Yeah, so whenever this function... If V is 58 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 59 00:07:54,591 --> 00:08:00,410 end, volume was decreasing, it would not be this. 60 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 61 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, 62 00:08:20,203 --> 00:08:27,090 that's not much. 1 inch cubed, but that's okay. 20 seconds. 63 00:08:27,090 --> 00:08:34,606 It works better with gallons. But that would be a very slow rate, right? One cubic 64 00:08:34,606 --> 00:08:41,938 inch per twenty seconds. Anyways, whenever things are changing over time, if the 65 00:08:41,938 --> 00:08:49,270 function V or r is increasing, the derivative is positive. The rate is positive. 66 00:08:49,270 --> 00:08:54,587 If the function is decreasing, like the rate is decreasing, it would be a dr, it 67 00:08:54,587 --> 00:09:00,169 would be a negative value. Okay, so sometimes they're just going to give you numbers 68 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 69 00:09:05,752 --> 00:09:10,485 say whether it's increasing or decreasing or you have to figure it out. 70 00:09:11,030 --> 00:09:20,494 And that's going to be very important, whether it's positive or negative. Surface 71 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 72 00:09:30,075 --> 00:09:35,450 is. Sorry, did I say a cube? I meant a sphere. 73 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 74 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 75 00:09:59,478 --> 00:10:03,530 write it as surface area of t. 76 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 77 00:10:11,308 --> 00:10:14,530 a variable. And we differentiate this. 78 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, 79 00:10:30,828 --> 00:10:44,510 dr/dt. This is how the rate of change, how fast the surface area is— 80 00:10:44,530 --> 00:10:52,590 How fast the surface area is changing is related to how fast the radius is changing. 81 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 82 00:11:00,267 --> 00:11:08,039 change of surface area changes at every point in time based on this and the given 83 00:11:08,039 --> 00:11:10,150 radius at that moment. 84 00:11:14,750 --> 00:11:26,346 What other equations do we have? We probably could not come up with a solution here. 85 00:11:26,346 --> 00:11:34,490 Again, let's say, what would be the units for surface area? 86 00:11:37,290 --> 00:11:45,301 Yeah, whatever the units, let's call it inches squared. And what would be the units 87 00:11:45,301 --> 00:11:53,120 for radius? Just be inches. Okay. The units for this are going to be what? Inches 88 00:11:53,120 --> 00:11:59,490 squared per whatever the units of time is. Let's call it a second. 89 00:12:01,030 --> 00:12:09,579 What would be the units of this? Inches, just inches is radius in seconds. And what 90 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 91 00:12:18,127 --> 00:12:22,350 by a second is inches squared per second. 92 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 93 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 94 00:12:37,103 --> 00:12:38,103 on. 95 00:12:43,030 --> 00:12:44,050 Okay. 96 00:12:46,710 --> 00:12:54,330 Can we come up with a formula that relates surface area and volume? 97 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 98 00:13:06,737 --> 00:13:10,590 replace this r with that r. 99 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 100 00:13:24,216 --> 00:13:30,030 then take the cube root of both sides. 101 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 102 00:13:49,788 --> 00:13:59,630 here. So volume equals four thirds pi r cubed. 103 00:14:02,810 --> 00:14:15,577 Wait, sorry. Sorry, I meant to replace it here. Let's delete that. So surface area 104 00:14:15,577 --> 00:14:25,230 equals 4 pi R squared, which is the cube root of 3V over 4 pi. 105 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 106 00:14:33,074 --> 00:14:40,138 rarely ever deal with. But this is exactly how surface area is related to volume at 107 00:14:40,138 --> 00:14:41,670 any point in time. 108 00:14:41,670 --> 00:14:46,531 If we differentiate this, we're not going to, what, this is kind of messy. Actually, 109 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 110 00:14:51,392 --> 00:14:54,930 area. How is related to the rate of change of the volume? 111 00:15:00,990 --> 00:15:02,390 Okay. 112 00:15:02,390 --> 00:15:08,921 Times the height. Okay. Again, this is how the volume is related to the both the 113 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 114 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. 115 00:15:23,130 --> 00:15:31,059 Again, these are functions. They're all functions of time. So if we differentiate, 116 00:15:31,059 --> 00:15:37,150 we get dV/dt is, what rule do we got to use here? Product rule. 117 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, 118 00:15:55,390 --> 00:15:56,610 dh/dt. 119 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. 120 00:16:05,260 --> 00:16:12,048 How fast the volume is changing this. This is how fast the radius is changing. This 121 00:16:12,048 --> 00:16:14,910 is how fast the height is changing. 122 00:16:14,910 --> 00:16:20,240 And again, over time, they all depend on all of them, including the radius at that 123 00:16:20,240 --> 00:16:24,107 point in time, including the height at that point in time. 124 00:16:30,390 --> 00:16:33,510 Um, what's the formula for volume of a cone? 125 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 126 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 127 00:16:55,514 --> 00:16:56,514 water. 128 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 129 00:17:05,963 --> 00:17:13,443 is changing. The height is changing and the volume is changing. So that makes it 130 00:17:13,443 --> 00:17:14,443 fine. 131 00:17:17,770 --> 00:17:24,570 The derivative looks exactly like this, except we have a one third out here. Okay. 132 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 133 00:17:37,561 --> 00:17:44,470 big R for a second. Let's call little r the radius at any given time. 134 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 135 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 136 00:18:03,297 --> 00:18:04,890 do a little h here. 137 00:18:07,430 --> 00:18:13,890 How are they related? Sorry, if capital R. 138 00:18:18,030 --> 00:18:21,050 So we have two triangles here, right? 139 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. 140 00:18:33,917 --> 00:18:42,570 Okay. R and H are fixed. R equals some number. H equals some number. 141 00:18:44,910 --> 00:18:55,293 And they're proportional triangles. Very important. So R over H is equal to what? 142 00:18:55,293 --> 00:19:04,010 Little r over h. Okay. And let's say it was, the height was 10 feet. 143 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 144 00:19:19,962 --> 00:19:25,610 little h, and we multiply by h. 145 00:19:25,610 --> 00:19:29,590 What is it? Five thirds. 146 00:19:33,230 --> 00:19:34,230 Equals R. 147 00:19:35,750 --> 00:19:46,961 So almost all cone problems. The radius and the height of the cone are some fixed 148 00:19:46,961 --> 00:19:47,961 number. 149 00:19:49,810 --> 00:19:55,285 We come up with that, and then we differentiate this, which is simply five thirds 150 00:19:55,285 --> 00:19:58,530 times the derivative of the height of the water. 151 00:20:00,190 --> 00:20:03,550 And this is the R. Okay. 152 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. 153 00:20:16,210 --> 00:20:20,410 Anything that was triangles, we come up with lots of equations. 154 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 155 00:20:30,888 --> 00:20:33,890 walking away. And this is another person. 156 00:20:40,310 --> 00:20:48,830 The triangle has to be changing. The green will be my new triangle. 157 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 158 00:20:55,746 --> 00:20:56,746 triangle? 159 00:21:01,130 --> 00:21:07,310 I don't see A's or B's. I like that though. There we go. 160 00:21:08,310 --> 00:21:17,610 And why is that true? Good old Pythagorean, our favorite geometer. 161 00:21:21,530 --> 00:21:27,642 Okay. This is how those four functions are related. Or sorry, three functions, three 162 00:21:27,642 --> 00:21:28,642 variables. 163 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 164 00:21:39,898 --> 00:21:43,230 equals two z times dz/dt. 165 00:21:51,530 --> 00:21:57,390 And this is how their rates of change are related. 166 00:21:59,710 --> 00:22:07,072 Okay. If we wanted to figure out how fast the distance between the two guys are, 167 00:22:07,072 --> 00:22:11,950 that would be this. We divide everything by 2z. Oops. 168 00:22:15,090 --> 00:22:26,750 What is that? Two x dx/dt plus two y dy/dt, over two z. 169 00:22:56,950 --> 00:22:58,350 Okay. 170 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 171 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 172 00:23:08,947 --> 00:23:11,630 at the units of z divided by the units of t. 173 00:23:11,630 --> 00:23:15,510 And that's the units of this number. Okay. 174 00:23:16,030 --> 00:23:23,510 What other equation can we come up with this triangle? 175 00:23:31,110 --> 00:23:32,370 How about area? 176 00:23:35,210 --> 00:23:41,749 One half times base times height. Let me just call it base times height for a 177 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 178 00:23:48,627 --> 00:23:50,410 change the variables. 179 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 180 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 181 00:24:03,826 --> 00:24:05,470 at any point in time. 182 00:24:06,230 --> 00:24:14,870 We differentiate dA/dt. And what rule we got to use here? Product rule again. 183 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 184 00:24:26,617 --> 00:24:29,870 derivative of h, dh/dt. 185 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 186 00:24:34,764 --> 00:24:39,596 changing. Depends on the height of that given time. Depends on the base at that 187 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. 188 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 189 00:24:51,471 --> 00:24:53,480 equations involving this. 190 00:24:54,480 --> 00:25:05,600 We can come up with sine of theta. What would that be? X over Z. 191 00:25:07,080 --> 00:25:14,540 What is cosine of theta? Y over Z. 192 00:25:17,140 --> 00:25:21,840 What is tangent of theta? X over Y. 193 00:25:22,960 --> 00:25:26,765 We could obviously do secant, cosecant, and use the other derivatives. Usually we 194 00:25:26,765 --> 00:25:28,631 use only one of these three. 195 00:25:29,320 --> 00:25:34,820 This is how the angle is related to the other two sides of the triangle. 196 00:25:40,760 --> 00:25:46,775 If we differentiate, let's start with this one. We get, what's the derivative of 197 00:25:46,775 --> 00:25:48,580 theta, or sine of theta? 198 00:25:50,620 --> 00:25:57,000 Cosine of theta times the derivative of theta, dθ/dt. That is the left side. What 199 00:25:57,000 --> 00:26:00,780 rule do we use on the right side? Quotient rule. 200 00:26:04,600 --> 00:26:16,100 dx/dt times z, plus x times dz/dt, over z squared. 201 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 202 00:26:33,411 --> 00:26:38,840 sides by cosine. That would be that equation. 203 00:26:39,700 --> 00:26:43,860 Did I screw up? Yes, this is definitely a minus. 204 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 205 00:26:56,265 --> 00:26:57,440 going to be? 206 00:27:05,060 --> 00:27:12,327 This is very important. In calculus, this is always in radians. If they ever give 207 00:27:12,327 --> 00:27:17,440 you a value in degrees, we need to convert it to radians. 208 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 209 00:27:26,223 --> 00:27:31,440 guaranteed to be something radians per whatever time it is. 210 00:27:32,800 --> 00:27:47,060 And really radians don't have units. Why do radians don't have units? 211 00:27:54,560 --> 00:28:04,083 So that's some theta. Anybody remember how we defined the angle measurement in 212 00:28:04,083 --> 00:28:05,083 radians? 213 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 214 00:28:24,808 --> 00:28:26,010 distances. 215 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 216 00:28:33,647 --> 00:28:41,473 don't really have units. Sometimes we don't even write radians, we just put nothing 217 00:28:41,473 --> 00:28:42,510 per second. 218 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 219 00:28:54,210 --> 00:29:00,510 always need to know the things in radians. 220 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. 221 00:29:10,499 --> 00:29:14,940 The rate of change of volume is dV/dt. 222 00:29:14,940 --> 00:29:18,900 Okay. What about the one on the right? 223 00:29:24,440 --> 00:29:27,620 So what happens once it gets here? 224 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. 225 00:29:39,640 --> 00:29:46,140 [Brief speech unclear.] 226 00:30:01,380 --> 00:30:04,980 It would be the height. 227 00:30:07,480 --> 00:30:13,400 Something like that. Third one. 228 00:30:17,980 --> 00:30:19,960 So slow. 229 00:30:22,980 --> 00:30:25,380 Faster, slower, something like that. 230 00:30:26,440 --> 00:30:28,173 Okay. What about this one? 231 00:30:31,260 --> 00:30:35,000 So faster and slower, something like that. 232 00:30:35,370 --> 00:30:46,990 Right. Three inches a year. Three inches a year. 233 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, 234 00:30:55,971 --> 00:31:01,190 it's a d something dt. Let's change it over time. 235 00:31:02,150 --> 00:31:06,030 So this would be d what over dt? 236 00:31:09,610 --> 00:31:11,543 Inches is what kind of thing? 237 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 238 00:31:23,463 --> 00:31:24,030 h dt. 239 00:31:24,030 --> 00:31:27,963 And is it positive three or negative three inches per year? 240 00:31:29,350 --> 00:31:35,930 Because we're growing, it's positive three inches per year. 241 00:31:40,850 --> 00:31:46,383 My mutual fund is shrinking. What does that tell us since it's shrinking? Negative. 242 00:31:48,110 --> 00:31:50,190 Four cents per day. 243 00:31:51,790 --> 00:31:55,010 And this would be d what? Dt. 244 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 245 00:32:03,708 --> 00:32:08,650 measurement of? Money. So you could call it d money, dt. 246 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 247 00:32:19,958 --> 00:32:21,450 called dm dt. 248 00:32:23,170 --> 00:32:25,850 The radius of a circle is increasing. 249 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? 250 00:32:37,187 --> 00:32:38,187 Positive. 251 00:32:46,490 --> 00:32:55,170 The volume of the cone is decreasing. So that would be d v dt. 252 00:32:57,630 --> 00:32:59,697 And is it positive or negative? 253 00:33:00,790 --> 00:33:07,548 When you tell if it's decreasing, negative two inches cubed per second. Again, we 254 00:33:07,548 --> 00:33:09,550 could look at the units. 255 00:33:10,290 --> 00:33:12,957 Inches cubed is a measurement of volume. 256 00:33:13,950 --> 00:33:16,550 So in seconds is a measurement of time. 257 00:33:19,090 --> 00:33:20,157 Okay, next page. 258 00:33:20,620 --> 00:33:24,880 We got a rectangle. Start by just drawing a rectangle. 259 00:33:26,520 --> 00:33:29,453 I don't know if there's much room here, but. 260 00:33:31,820 --> 00:33:32,820 Okay. 261 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. 262 00:33:41,176 --> 00:33:46,440 Whatever you want to call it. Let's call this the length. 263 00:33:47,100 --> 00:33:48,300 This is the width. 264 00:33:50,160 --> 00:33:51,160 Okay. 265 00:33:52,800 --> 00:33:59,322 And then we come up with some, some, how those two variables are related. And what 266 00:33:59,322 --> 00:34:02,980 are we talking about the, what's the question? 267 00:34:06,120 --> 00:34:12,474 Perimeter. Okay, so we come up with an equation involving a perimeter. What's the 268 00:34:12,474 --> 00:34:13,474 equation? 269 00:34:16,000 --> 00:34:21,580 Perimeter equals. 2L. 2L plus 2W. 270 00:34:24,840 --> 00:34:28,107 And then I automatically just differentiate that. 271 00:34:29,580 --> 00:34:38,231 So derivative of perimeter with respect to time. Is 2 times the derivative of L with 272 00:34:38,231 --> 00:34:41,320 respect to time. Plus 2 times. 273 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 274 00:34:52,059 --> 00:34:54,540 problem and label stuff. 275 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? 276 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 277 00:35:13,529 --> 00:35:18,860 it's increasing, it is positive 2 inches per second. 278 00:35:22,320 --> 00:35:24,680 And the width is increasing. 279 00:35:26,560 --> 00:35:29,160 At a rate, that's going to be a D what? 280 00:35:31,960 --> 00:35:33,960 D. W. 281 00:35:35,860 --> 00:35:36,860 DT. 282 00:35:42,300 --> 00:35:46,060 Equals 3 inches per second. 283 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. 284 00:35:54,712 --> 00:35:55,712 Okay. 285 00:35:57,060 --> 00:36:01,660 Perimeter. So we want to know. D. P. DT. 286 00:36:03,860 --> 00:36:04,860 Okay. 287 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 288 00:36:12,798 --> 00:36:13,840 differentiate. 289 00:36:15,400 --> 00:36:21,334 Okay. Never write this formula with any numbers in there. Unless we're told one of 290 00:36:21,334 --> 00:36:23,360 those sides is not changing. 291 00:36:23,360 --> 00:36:26,980 Or you know it's not changing. Okay. So. 292 00:36:30,900 --> 00:36:32,167 What would this be? 293 00:36:34,460 --> 00:36:35,460 This is L. 294 00:36:37,060 --> 00:36:42,060 Of the length. At some time, unknown time. 295 00:36:45,360 --> 00:36:51,300 Equals 10 inches. And the width at some point in time, the same point in time. 296 00:36:53,140 --> 00:36:54,900 Is 6 inches. 297 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. 298 00:37:05,090 --> 00:37:09,550 P. D. T. Is 2 times. What is D. L. DT. 299 00:37:12,590 --> 00:37:13,723 We wrote up here. 300 00:37:14,810 --> 00:37:21,630 Plus 2. Times. D. W. DT. Which is right here. 301 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 4 302 00:37:31,117 --> 00:37:34,330 plus 6 is 10. Okay. Now we put in units. 303 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. 304 00:37:38,946 --> 00:37:40,212 The units of P was. 305 00:37:41,630 --> 00:37:45,850 Inches. To distance and units of time was seconds. 306 00:37:47,970 --> 00:37:49,970 So at that one moment in time. 307 00:37:51,490 --> 00:37:57,545 When the length was exactly this and the width was exactly this. The rate of change 308 00:37:57,545 --> 00:38:00,390 of the perimeter is exactly this. Okay. 309 00:38:00,890 --> 00:38:06,150 Actually, did we even plug in those numbers at all? No. Do we need you? 310 00:38:07,790 --> 00:38:11,230 No. Okay. So in fact, in every moment in time. 311 00:38:13,670 --> 00:38:18,890 Not just a specific moment. It happens to not change. And the reason is that the 312 00:38:18,890 --> 00:38:21,424 derivative doesn't involve L's or W's. 313 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 314 00:38:28,623 --> 00:38:30,890 depend on that one time. Okay. 315 00:38:31,470 --> 00:38:34,537 So in fact, this is a constant rate of change. 316 00:38:35,550 --> 00:38:37,750 Because these two were constants. 317 00:38:38,870 --> 00:38:39,870 Interesting. 318 00:38:41,350 --> 00:38:44,883 Now the rectangle's length is increasing at 3 inches. 319 00:38:49,790 --> 00:38:53,650 So the rectangle's length. What would that be? 320 00:38:55,630 --> 00:38:56,630 D L. 321 00:38:58,230 --> 00:39:03,910 D T. Is increasing. So it's 3 inches per second. 322 00:39:06,210 --> 00:39:09,050 The width is decreasing. So D. 323 00:39:11,070 --> 00:39:20,550 W. D T. H. Is very important. Negative 2 inches per second. 324 00:39:23,410 --> 00:39:25,277 Area. We need new equations. 325 00:39:26,910 --> 00:39:32,830 What is area of a rectangle? Length times width. Okay. Now we got to differentiate. 326 00:39:35,830 --> 00:39:37,970 D area. D T. 327 00:39:40,970 --> 00:39:43,130 Is D L. D T. 328 00:39:45,150 --> 00:39:46,510 Plus. 329 00:39:48,590 --> 00:39:49,590 L. 330 00:39:51,210 --> 00:39:53,870 Times. D W. D T. 331 00:40:01,890 --> 00:40:04,430 So D. 332 00:40:08,130 --> 00:40:10,050 D A. D T. 333 00:40:14,030 --> 00:40:16,730 Is. What was this rate? That's 3. 334 00:40:18,790 --> 00:40:23,690 Okay. Times W at that one moment in time, which is 6. 335 00:40:26,250 --> 00:40:27,250 Plus. 336 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. 337 00:40:48,230 --> 00:40:52,370 What is that negative 2. 338 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 339 00:41:02,406 --> 00:41:03,406 area? 340 00:41:05,490 --> 00:41:10,390 Everything was in inches. So it's inches squared. So. 341 00:41:12,650 --> 00:41:14,850 Time was measured in seconds. 342 00:41:16,670 --> 00:41:17,670 Okay. 343 00:41:20,250 --> 00:41:22,917 What does it mean that this is negative? 344 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 345 00:41:29,393 --> 00:41:31,060 square inches per second. 346 00:41:31,670 --> 00:41:40,782 Okay, spherical snowball melts. At the instant—one moment in time—at which the 347 00:41:40,782 --> 00:41:44,170 radius is twenty centimeters. 348 00:41:44,170 --> 00:41:52,971 Its radius is decreasing at three centimeters per minute. At what rate is the volume 349 00:41:52,971 --> 00:41:55,590 of the snowball changing? 350 00:41:57,250 --> 00:42:00,510 So what formula do you come up with? 351 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 352 00:42:08,780 --> 00:42:11,210 sphere, which we talked earlier. 353 00:42:15,550 --> 00:42:26,510 We might as well label we got. What is the 20 centimeters? 354 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 355 00:42:35,910 --> 00:42:36,910 is 20. 356 00:42:40,550 --> 00:42:44,190 Its radius is decreasing. What is this? 357 00:42:46,230 --> 00:42:47,230 Dr. 358 00:42:48,790 --> 00:42:55,790 Dt is a positive or negative negative three centimeters per minute. 359 00:42:55,790 --> 00:43:05,450 And we want to know the rate of the volume that is dv dt. 360 00:43:09,130 --> 00:43:12,790 We differentiate this, which we did earlier, right? 361 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. 362 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 363 00:43:41,076 --> 00:43:42,710 is. Doesn't matter. 364 00:43:50,350 --> 00:43:52,990 What was the radius of that time? 365 00:43:56,090 --> 00:43:57,090 Twenty. 366 00:43:58,670 --> 00:44:01,510 And dr/dt was negative three. 367 00:44:03,910 --> 00:44:10,250 And we do some math. What's this going to be? 368 00:44:23,730 --> 00:44:26,210 Nothing negative. 369 00:44:30,350 --> 00:44:42,570 Four hundred times four is sixteen hundred; times three is forty-eight hundred. 370 00:44:48,170 --> 00:44:57,490 And we put in pi. What did you say? Negative what? 371 00:45:02,570 --> 00:45:06,630 About three times that seems about right. Point what? 372 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. 373 00:45:18,699 --> 00:45:22,610 Give me three digits: six four five. 374 00:45:24,090 --> 00:45:29,610 So this is the exact rate of change. This is an approximate rate of change. 375 00:45:32,530 --> 00:45:35,197 The units, what are the units of volume? 376 00:45:39,330 --> 00:45:43,797 Yeah, it's whatever the units of r is cubed. So it was centimeters. 377 00:45:46,890 --> 00:45:51,030 Centimeters cubed per what time was it? 378 00:45:53,850 --> 00:46:04,434 The volume is decreasing at forty-eight hundred pi cubic centimeters per minute. How 379 00:46:04,434 --> 00:46:07,710 do I know it's decreasing? 380 00:46:07,710 --> 00:46:21,138 Because it is negative. When you write that out, you say the volume is decreasing at 381 00:46:21,138 --> 00:46:29,450 forty-eight hundred pi cubic centimeters per minute. 382 00:46:30,450 --> 00:46:34,490 You leave off the negative when you say it is decreasing. 383 00:46:34,780 --> 00:46:39,513 It is leaning up against the wall, so we have to draw that. Okay. Okay. 384 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 385 00:47:07,128 --> 00:47:09,540 going to look like that triangle. 386 00:47:12,660 --> 00:47:15,720 And eventually it looks like that and so on. 387 00:47:18,420 --> 00:47:23,660 A rate of three feet per second. All right. Let's put it to variables. 388 00:47:28,420 --> 00:47:30,540 What's changing here? 389 00:47:34,480 --> 00:47:37,480 Right. The height of the ladder is changing. 390 00:47:39,560 --> 00:47:40,960 Which we call that H. 391 00:47:47,200 --> 00:47:51,740 Let's go right now. 392 00:47:55,620 --> 00:47:57,287 Let's call that the base. 393 00:48:01,860 --> 00:48:04,327 Is the length of the ladder changing? 394 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 395 00:48:12,784 --> 00:48:15,720 number. This is twenty five feet. 396 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 397 00:48:30,391 --> 00:48:34,120 three feet per second. What would that be? 398 00:48:36,740 --> 00:48:37,740 Sorry, d what? 399 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? 400 00:48:48,356 --> 00:48:50,960 Is it increasing or decreasing? 401 00:48:55,000 --> 00:48:59,533 What's that? Increasing? Yes, increasing just three feet per second. 402 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 403 00:49:10,745 --> 00:49:15,020 feet from the wall? So this is one moment in time. 404 00:49:18,340 --> 00:49:20,140 So what are we looking for? 405 00:49:22,580 --> 00:49:25,020 So we're looking for dh/dt. 406 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. 407 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 408 00:49:50,455 --> 00:49:51,740 point in time? 409 00:49:53,520 --> 00:49:55,253 We always have a triangle. 410 00:49:57,020 --> 00:49:58,287 What should we use? 411 00:50:00,660 --> 00:50:04,140 Area. Do we care about this area? No. 412 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 413 00:50:10,389 --> 00:50:11,389 Pythagorean. 414 00:50:12,400 --> 00:50:16,380 h squared plus b squared equals twenty five squared. 415 00:50:18,800 --> 00:50:24,520 That's how b and h are related at any moment in time. Now we differentiate. 416 00:50:29,980 --> 00:50:40,520 Two h times dh/dt plus two b times db/dt. 417 00:50:42,560 --> 00:50:45,520 And derivative of twenty five squared is? 418 00:50:47,900 --> 00:50:50,240 Zero. Zero. It is just a constant. 419 00:50:52,520 --> 00:50:59,493 You could go twenty five, two times twenty five times the derivative of twenty five, 420 00:50:59,493 --> 00:51:01,900 which is zero. Don't do that. 421 00:51:04,180 --> 00:51:05,540 Okay, we want dh/dt. 422 00:51:08,200 --> 00:51:09,800 So let's solve for that. 423 00:51:30,380 --> 00:51:36,400 dh/dt is? 424 00:51:47,440 --> 00:51:49,920 We can simplify this a little bit. 425 00:52:06,740 --> 00:52:07,740 Okay. 426 00:52:09,940 --> 00:52:15,420 What was the length of the base? What were we given? 427 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 428 00:52:24,832 --> 00:52:28,565 time. So this is the rate only when this is twenty feet. 429 00:52:31,600 --> 00:52:33,440 db/dt is three. 430 00:52:39,580 --> 00:52:41,900 And how do we figure out height? 431 00:52:46,160 --> 00:52:49,500 We use this at that one moment in time. 432 00:52:51,320 --> 00:52:53,840 H squared is? 433 00:52:55,900 --> 00:52:58,160 Square minus b squared. 434 00:53:07,700 --> 00:53:10,033 B squared. What was the value of b? 435 00:53:11,560 --> 00:53:12,560 Twenty. Twenty. 436 00:53:17,340 --> 00:53:19,880 And h is? What is that? 437 00:53:22,940 --> 00:53:23,940 Twenty five. 438 00:53:29,640 --> 00:53:35,547 It is fifteen. It would have been easier if we just drew this as twenty, twenty 439 00:53:35,547 --> 00:53:39,660 five. What kind of triangle is this? Three, four, five. 440 00:53:39,660 --> 00:53:41,860 So it would be fifteen. 441 00:53:44,980 --> 00:53:45,980 Fifteen. 442 00:53:51,540 --> 00:53:53,760 And negative four. 443 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. 444 00:54:12,800 --> 00:54:16,260 So how fast is the ladder moving down? 445 00:54:18,580 --> 00:54:26,180 Fast is like a speed, not a velocity. What we got here is a velocity. 446 00:54:28,880 --> 00:54:32,813 We've talked about difference in speed and velocity, right? 447 00:54:36,060 --> 00:54:43,314 So h, this distance is decreasing at four feet per second. That means exactly this 448 00:54:43,314 --> 00:54:48,180 point is moving that direction at four feet per second. 449 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 450 00:54:54,231 --> 00:54:55,564 this is moving down. 451 00:55:00,920 --> 00:55:05,760 Because it's moving down, this distance is actually. 452 00:55:09,400 --> 00:55:13,067 It's the rate we wanted, but we don't bother with that. 453 00:55:16,720 --> 00:55:17,720 Okay. 454 00:55:19,760 --> 00:55:25,810 Next, how fast is the angle between the ladder and the wall changing when the base 455 00:55:25,810 --> 00:55:27,360 is fifteen feet away? 456 00:55:27,860 --> 00:55:29,793 And where do I put the angle? 457 00:55:32,900 --> 00:55:34,240 We got two options. 458 00:55:38,020 --> 00:55:43,380 How fast the angle between the ladder and the wall. So that would be this angle. 459 00:55:46,960 --> 00:55:48,600 Before we do it. 460 00:55:50,520 --> 00:55:54,653 Is that angle growing or is it increasing or is it decreasing? 461 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 462 00:56:03,066 --> 00:56:04,066 positive value. 463 00:56:05,600 --> 00:56:14,886 Okay. So anything involving theta, we use sine, cosine, or tangent. Which one should 464 00:56:14,886 --> 00:56:15,886 we use? 465 00:56:25,340 --> 00:56:33,760 Well, we're given db/dt. Okay. So use the one that involved B. 466 00:56:36,540 --> 00:56:41,680 Okay. So we want ideally use this. 467 00:56:44,900 --> 00:56:50,900 Okay. We could use tangent. 468 00:56:54,760 --> 00:56:57,500 But we'd have to know DHDT. 469 00:56:59,680 --> 00:57:03,760 Which we don't know at the moment. So let's use sine. 470 00:57:06,720 --> 00:57:09,920 Sine of theta is b over— 471 00:57:11,480 --> 00:57:14,780 And this is fixed. So put in the twenty-five. 472 00:57:22,480 --> 00:57:29,680 And we differentiate. We get cosine of theta times D theta, DT. 473 00:57:33,280 --> 00:57:40,160 This is simply one over twenty-five times the derivative of b, db/dt. 474 00:57:50,180 --> 00:57:51,180 Okay. 475 00:58:22,120 --> 00:58:26,100 When the base is fifteen feet away. 476 00:58:27,840 --> 00:58:32,240 First off, we know DB, DT, right? So that's, what was it? 477 00:58:34,820 --> 00:58:35,820 Three. 478 00:58:37,400 --> 00:58:38,400 Okay. 479 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 480 00:58:55,948 --> 00:58:56,948 away. 481 00:58:59,100 --> 00:59:04,300 Cosine would be this length divided by this length. 482 00:59:06,680 --> 00:59:08,347 Adjacent over hypotenuse. 483 00:59:09,920 --> 00:59:13,053 So we need to figure out this length, which is, 484 00:59:17,280 --> 00:59:20,947 the Pythagorean theorem. This is what type of triangle? 485 00:59:22,080 --> 00:59:27,580 Three, four, five triangles. So we could do the math or know that that's twenty. 486 00:59:32,580 --> 00:59:37,580 So cosine of theta is twenty over twenty-five. 487 00:59:39,560 --> 00:59:42,580 Which is what? Four-fifths. 488 00:59:50,440 --> 00:59:58,220 Three over twenty-five times five over four. 489 01:00:00,200 --> 01:00:04,360 Which is three over twenty. 490 01:00:11,520 --> 01:00:16,560 Units of theta, always in this classes. Radians. 491 01:00:18,780 --> 01:00:22,920 Per time, which was, was it seconds? 492 01:00:27,020 --> 01:00:31,087 We get a positive value because that angle is getting larger. 493 01:00:42,920 --> 01:00:44,660 Okay. We're going to stop there.