1 00:00:00,012 --> 00:00:06,492 [music] Here, I've got an orange. I'd like to measure say, the radius of the 2 00:00:06,492 --> 00:00:11,275 orange, thinking of this orange as a perfect sphere. 3 00:00:11,275 --> 00:00:17,885 One way to do this would be to measure, say, the circumference of the orange 4 00:00:17,885 --> 00:00:23,282 around its equator. So here if I measure that it looks like 5 00:00:23,282 --> 00:00:29,798 it's about 24 centimeters. So the circumference of the orange is 24 6 00:00:29,798 --> 00:00:34,460 centimeters. What does that tell me about, say the 7 00:00:34,460 --> 00:00:39,727 radius of the orange? Well that circumference Will be 2 pi times 8 00:00:39,727 --> 00:00:44,019 the radius of the orange. So if I divide both sides by 2 pi. 9 00:00:44,019 --> 00:00:48,684 I get that the radius of the orange is about 12 over pi centimeters. 10 00:00:48,684 --> 00:00:53,793 How much rind is there on this orange? That could be formulated as a linear 11 00:00:53,793 --> 00:00:57,997 approximation problem. Well let's fit this up./g So, I just 12 00:00:57,997 --> 00:01:02,554 calculated the orange radius to be about 12 over pi centimeters. 13 00:01:02,554 --> 00:01:05,378 I want to figure out how thick the rind is. 14 00:01:05,378 --> 00:01:08,901 Let me. Let me peel off a bit of the orange here. 15 00:01:08,901 --> 00:01:14,787 Just want to measure how thick this is with, say my little ruler here and yeah, 16 00:01:14,787 --> 00:01:19,512 maybe three millimeters. So, point three centimeters will be the 17 00:01:19,512 --> 00:01:23,547 rind thickness. Now how does knowing the rind thickness 18 00:01:23,547 --> 00:01:28,817 going to help me to calculate, at least approximate, the volume of the rind in 19 00:01:28,817 --> 00:01:33,302 this whole orange. Well I know the volume of a sphere, which 20 00:01:33,302 --> 00:01:36,584 by modeling the orange to be, is 4 3rds pi r q. 21 00:01:36,584 --> 00:01:42,046 And the rind volume, is at least approximately how much the volume changes 22 00:01:42,046 --> 00:01:45,976 when I take this formula, and replace r with r minus h. 23 00:01:45,976 --> 00:01:49,461 Right? When I take that sphere and I make it just 24 00:01:49,461 --> 00:01:54,073 a little bit smaller. That change in volume is the volume of the 25 00:01:54,073 --> 00:01:57,436 rind. And to understand how wiggling the input 26 00:01:57,436 --> 00:02:01,054 effects the output, I'm going to use the derivative. 27 00:02:01,054 --> 00:02:05,734 The formula that I want to be differentiating here is this volume of a 28 00:02:05,734 --> 00:02:10,041 sphere of radius r. Which is 4 thirds pi r cubed [NOISE]And if 29 00:02:10,041 --> 00:02:16,016 I differentiate this with respect to r. The r cubed differentiates to 3r squared. 30 00:02:16,016 --> 00:02:21,090 And the 3 in the denominator here will cancel this 3 by the power rule. 31 00:02:21,090 --> 00:02:27,254 So the derivative will be 4 pi r squared. Now what I'm trying to approximate is the 32 00:02:27,254 --> 00:02:33,113 rind volume which is really the difference between a sphere of radius r and a 33 00:02:33,113 --> 00:02:39,003 slightly smaller sphere where I've shrunk down just past the rind, right. 34 00:02:39,003 --> 00:02:42,685 That difference will be the volume of the rind. 35 00:02:42,685 --> 00:02:47,114 And approximately this h times the derivitive of v at r. 36 00:02:47,115 --> 00:02:52,130 Right, because this is asking how does the output change when the input goes from R 37 00:02:52,130 --> 00:02:55,448 minus H to R. And it's approximately the input change 38 00:02:55,448 --> 00:02:58,861 times the derivative. Now in this case what do I know? 39 00:02:58,861 --> 00:03:02,456 The orange's radius is 12 over Pi, so I'll use that for R. 40 00:03:02,456 --> 00:03:07,262 And the rind's thickness is point three centimeters so I'm going to use that for 41 00:03:07,262 --> 00:03:11,227 H. So this is point three centimeters Times 4 42 00:03:11,227 --> 00:03:17,266 pi, times this radius, which is 12 over pi centimeters squared. 43 00:03:17,266 --> 00:03:23,657 So this is cubic centimeters. Well that's good because I'm trying to 44 00:03:23,657 --> 00:03:29,481 calculate a volume. And I'll just keep calculating here. 45 00:03:29,481 --> 00:03:34,656 So it's point 3 times 4 pi times 144 over pi squared. 46 00:03:34,656 --> 00:03:41,388 Cubic centimeters. And 4 times 144 is 576 times .3, the pi 47 00:03:41,388 --> 00:03:47,945 divided the pi squared is going to be a pi in the denominator. 48 00:03:47,945 --> 00:03:52,913 So I've got .3 times 576 over pi Cubic centimeters. 49 00:03:52,913 --> 00:03:58,834 That's about 55 cubic centimeters. So the volume of the rind is approximately 50 00:03:58,834 --> 00:04:03,869 55 cubic centimeters. Now admittedly I could have calculated the 51 00:04:03,869 --> 00:04:09,133 volume of the rind just by calculating the volume of the whole thing. 52 00:04:09,133 --> 00:04:13,716 Calculating the volume of the whole thing with the radius reduced by h, and taking a 53 00:04:13,716 --> 00:04:16,563 difference. But the neat thing here is that you can 54 00:04:16,563 --> 00:04:19,597 approximate that quantity by using calculus, right? 55 00:04:19,597 --> 00:04:22,680 By thinking about this linear approximation business. 56 00:04:22,680 --> 00:04:26,796 I also want to see now how close the linear approximation was. 57 00:04:26,796 --> 00:04:31,886 Alright, so I've taken my orange and peeled off all the rind, and I've taken 58 00:04:31,886 --> 00:04:37,380 the rind and smooched it down into this ball, and I'll try to calculate the volume 59 00:04:37,380 --> 00:04:41,919 now of this rind ball. I want to compute the volume of a rind, 60 00:04:41,919 --> 00:04:48,569 and I made that rind ball here, let me Let me compute the circumference of this rind 61 00:04:48,569 --> 00:04:52,214 ball. That's about 15 centimeters. 62 00:04:52,214 --> 00:04:59,632 So the circumference of my ball of rind is about 15 centimeters, that means my rind 63 00:04:59,632 --> 00:05:03,868 ball has a radius. 15 over 2 pi centimeters. 64 00:05:03,868 --> 00:05:08,167 And here, assuming my Rind ball is a perfect sphere. 65 00:05:08,167 --> 00:05:14,918 This would be the volume of the Rind ball. But in here I gotta put the radius, which 66 00:05:14,918 --> 00:05:16,837 is 15 over. 2 pie. 67 00:05:16,837 --> 00:05:22,665 So, 4 3rds pie the radius cubed, and that's about 57 cubic centimeters. 68 00:05:22,665 --> 00:05:29,205 So, yeah, I mean, I actually took all the rind off the orange and the value that we 69 00:05:29,205 --> 00:05:35,937 got is pretty close to what the linear approximation told us the volume of rind 70 00:05:35,937 --> 00:05:39,407 should be. This perspective sheds some light on what 71 00:05:39,407 --> 00:05:42,998 might otherwise be seen as just some random coincidence. 72 00:05:42,998 --> 00:05:47,792 Now what you notice is that the volume of a sphere which is 4 3rds pi r cubed and 73 00:05:47,792 --> 00:05:51,043 the surface area of a sphere which is 4 pi r squared. 74 00:05:51,043 --> 00:05:55,907 These two formulas are related right. If I differentiate this volume formula 75 00:05:55,907 --> 00:05:59,142 with respect to r, I get the surface area. Formula. 76 00:05:59,142 --> 00:06:03,658 Again we're seeing that calculus isn't really about solving specific problems. 77 00:06:03,658 --> 00:06:06,835 I don't care about the volume of the rind of this orange. 78 00:06:06,835 --> 00:06:09,536 Right? If that was what calculus was good for, 79 00:06:09,536 --> 00:06:10,918 who would care? Right. 80 00:06:10,918 --> 00:06:16,414 What calculus is really good at Is providing a general framework and with 81 00:06:16,414 --> 00:06:22,835 that general framework we can begin to see the patterns between different concepts. 82 00:06:22,835 --> 00:06:28,921 Like this fact, the surface area of a sphere is the derivative of the volume of 83 00:06:28,921 --> 00:06:35,052 a sphere with respect to the radius. That's not really a coincidence, right? 84 00:06:35,052 --> 00:06:40,547 Calculus is providing an explanation for that supposed coincidence.