1 00:00:00,350 --> 00:00:05,100 What is the first derivative of that mysterious function s? 2 00:00:06,050 --> 00:00:09,000 And here it comes the amazing part ... 3 00:00:09,950 --> 00:00:16,250 ... the function s is so mysterious. It's hard to even say something, well anything about it. 4 00:00:16,350 --> 00:00:17,350 But ... 5 00:00:22,500 --> 00:00:27,000 We are going to be able to compute what the derivative is ... 6 00:00:27,500 --> 00:00:30,700 ... without even knowing the formula for s. 7 00:00:31,400 --> 00:00:38,200 Well the way we'll do it is by saying by definition the derivative is a limit ... 8 00:00:39,400 --> 00:00:43,200 ... limit as delta t goes to zero ... 9 00:00:43,650 --> 00:00:45,400 ... of the ratio ... 10 00:00:45,700 --> 00:00:48,100 ... the change of the function s ... 11 00:00:48,200 --> 00:00:53,700 ... s of t plus delta t minus s of t ... 12 00:00:54,300 --> 00:00:56,400 ... divided by the change of ... 13 00:00:57,000 --> 00:01:00,100 ... the function t [...] delta t. 14 00:01:00,250 --> 00:01:02,200 That's the definition. 15 00:01:04,000 --> 00:01:06,500 And now I'll do a little trick ... 16 00:01:07,400 --> 00:01:11,000 ... rewriting that fraction in a little bit different way. 17 00:01:12,750 --> 00:01:15,400 And what I [...] is ... 18 00:01:17,900 --> 00:01:18,900 ... this ... 19 00:01:20,600 --> 00:01:25,200 ... this picture of how the cars are moving. 20 00:01:25,300 --> 00:01:27,500 So what I have there is ... 21 00:01:27,600 --> 00:01:30,100 ... I have some time t. 22 00:01:30,250 --> 00:01:38,200 And at that time the first car is in the position f of t right here. 23 00:01:39,100 --> 00:01:40,600 And ... 24 00:01:41,700 --> 00:01:44,200 That position corresponds ... 25 00:01:45,800 --> 00:01:46,800 ... to ... 26 00:01:50,100 --> 00:01:52,150 ... the time s of t for the second car ... 27 00:01:52,250 --> 00:01:57,600 ... when the second car was exactly at the same position ... 28 00:01:59,000 --> 00:02:00,200 ... g of s of t. 29 00:02:00,300 --> 00:02:01,400 Right? 30 00:02:01,500 --> 00:02:03,300 So that was ... 31 00:02:03,950 --> 00:02:06,800 ... a part of the definition of the function s of t. 32 00:02:07,400 --> 00:02:11,500 Now what happens when the time delta t passes? 33 00:02:11,750 --> 00:02:13,600 t becomes t plus delta t. 34 00:02:13,700 --> 00:02:17,500 And the first car shows up at ... 35 00:02:18,700 --> 00:02:22,000 ... a different point. Somewhere there. 36 00:02:23,000 --> 00:02:28,800 What about that point? Well this is going exactly the point where the second car is ... 37 00:02:30,000 --> 00:02:37,300 ... when t is equal to t plus delta t. 38 00:02:42,150 --> 00:02:46,800 So at that time the second car is exactly in the same point. 39 00:02:49,300 --> 00:02:53,000 So what is it that I see there in the fraction? 40 00:02:54,100 --> 00:02:59,500 What is going to be s of t plus delta t minus s of t? 41 00:02:59,600 --> 00:03:01,600 What's the meaning of the difference? 42 00:03:01,700 --> 00:03:04,500 Well that is the time ... 43 00:03:05,100 --> 00:03:07,800 ... spent by the second car ... 44 00:03:07,900 --> 00:03:11,200 ... as it moved from this point to that point. 45 00:03:14,000 --> 00:03:17,900 So this is delta s -- the time spent by the second car. 46 00:03:19,300 --> 00:03:23,100 And what I want to relate it to is I want to relate it ... 47 00:03:23,200 --> 00:03:26,500 ... to the time spent by the first car ... 48 00:03:27,750 --> 00:03:32,500 ... as it moved from the same first point to the same ending point. 49 00:03:33,500 --> 00:03:39,900 So two cars were travelling along the same segment of the road ... 50 00:03:40,000 --> 00:03:42,400 ... and spent different times. 51 00:03:43,200 --> 00:03:45,400 Now if I relate ... 52 00:03:48,750 --> 00:03:50,800 ... that time spent ... 53 00:03:50,900 --> 00:03:56,700 ... to the length of the segment -- let's call it delta L -- the length of the segment of the road. 54 00:03:57,700 --> 00:04:03,200 And I multiply it by delta L back so that the fraction is kept. 55 00:04:06,700 --> 00:04:09,000 Then this fraction ... 56 00:04:11,750 --> 00:04:13,900 ... has the perfect meaning. 57 00:04:14,000 --> 00:04:18,000 The distance travelled by the first car divided by the time spent. 58 00:04:19,350 --> 00:04:23,200 That's the speed of the first car. 59 00:04:23,800 --> 00:04:26,800 That speed of the first car is the magnitude ... 60 00:04:28,500 --> 00:04:30,700 ... of the velocity. 61 00:04:31,400 --> 00:04:33,300 Now what is that fraction? 62 00:04:35,300 --> 00:04:37,500 Well that fraction is ... 63 00:04:39,750 --> 00:04:44,600 ... the reciprocal of the speed of the second car ... 64 00:04:47,250 --> 00:04:49,900 ... which is [...] to the magnitude of velocity. 65 00:04:50,300 --> 00:04:55,200 So the whole thing, the whole limit is going to be the fraction ... 66 00:04:55,900 --> 00:05:00,000 ... f prime divided by g prime. 67 00:05:01,500 --> 00:05:04,800 It is simply the ratio of the speeds. 68 00:05:05,100 --> 00:05:07,700 Well it should be the speed, not the vector. 69 00:05:08,500 --> 00:05:09,500 Right? 70 00:05:11,300 --> 00:05:12,900 So that's the remaining thing. 71 00:05:13,000 --> 00:05:17,500 We found the formula for the derivative of s. 72 00:05:18,300 --> 00:05:21,400 An exact explicit formula in terms of f and g. 73 00:05:21,500 --> 00:05:24,900 Although we had no idea what the formula for s is. 74 00:05:25,800 --> 00:05:28,000 That's what I like about this. Er ... 75 00:05:29,250 --> 00:05:34,100 Well the use ... highly non-trivial use of the concept of a function. 76 00:05:34,500 --> 00:05:36,300 I didn't use any formula for the function. 77 00:05:36,400 --> 00:05:39,400 Just the idea of the existence of a function. 78 00:05:39,500 --> 00:05:43,000 And some definition of derivative helps me to differentiate. |

Video Lectures > Calculus III (2012 Spring) > Lecture 2012.02.21 Curvature Formula > Part 06 Finding the First Derivative >