1 00:00:01,060 --> 00:00:02,630 So now moving on. So 2 00:00:04,850 --> 00:00:06,880 the first six lessons I was just trying to 3 00:00:06,880 --> 00:00:10,350 talk about what an integral is, the problem I'm trying 4 00:00:10,350 --> 00:00:13,240 to answer and then some techniques that we're going to 5 00:00:13,240 --> 00:00:18,570 use in 461 and in later courses for evaluating integrals. 6 00:00:18,570 --> 00:00:22,570 Now I need some, some special results, eh, in 7 00:00:22,570 --> 00:00:26,750 particular I'm driving at is, is differentiating improper integrals. 8 00:00:26,750 --> 00:00:30,400 Because this is going to be something you need to be able to do 9 00:00:30,400 --> 00:00:31,890 when you want to work with the black 10 00:00:31,890 --> 00:00:31,990 [UNKNOWN], 11 00:00:31,990 --> 00:00:32,810 spacing formula. 12 00:00:32,810 --> 00:00:35,950 Eh, and so I'm going to first 13 00:00:35,950 --> 00:00:39,310 get started by just differentiating definite integrals. 14 00:00:39,310 --> 00:00:42,120 Then I'll introduce what an improper integral is. 15 00:00:42,120 --> 00:00:44,080 And then we will learn how to take their derivatives. 16 00:00:45,120 --> 00:00:47,300 So this is actually not too difficult of a topic. 17 00:00:47,300 --> 00:00:49,400 So hopefully this set will go by quickly. 18 00:00:51,930 --> 00:00:55,780 So first start out by pointing out that a definite integral, so if I 19 00:00:55,780 --> 00:01:00,920 integrate a function f of x over an interval ab, that's a real number. 20 00:01:00,920 --> 00:01:03,290 So it's constant, it's not changing. 21 00:01:03,290 --> 00:01:05,310 So if I try and take the derivitive of 22 00:01:05,310 --> 00:01:08,720 this function, it's constant so it's derivative is zero. 23 00:01:09,960 --> 00:01:13,530 So the only way this is going to get exciting is 24 00:01:13,530 --> 00:01:17,360 if the limits so that the A and the B are 25 00:01:17,360 --> 00:01:21,310 now functions of a, of another variable. So I, I chose T. 26 00:01:23,810 --> 00:01:27,370 Then when I evaluate this definite integral 27 00:01:27,370 --> 00:01:28,820 it still going to be a function of 28 00:01:28,820 --> 00:01:33,440 T and so I might want to consider taking its derivative with respect to T. 29 00:01:36,950 --> 00:01:42,460 And then more generally the integrand could also have T in it. 30 00:01:42,460 --> 00:01:46,150 But since we haven't talked about functions of more than 31 00:01:46,150 --> 00:01:48,860 one variable yet, I'm going to leave that for lesson three. 32 00:01:51,880 --> 00:01:54,280 And so the question we want to consider is 33 00:01:54,280 --> 00:01:57,920 how can I calculate the derivative, with respect to T, 34 00:01:57,920 --> 00:02:01,950 of the integral of, of function f of x 35 00:02:01,950 --> 00:02:05,310 where the limits depend our functions of the value t. 36 00:02:09,390 --> 00:02:11,250 So it's actually pretty straightforward. 37 00:02:11,250 --> 00:02:15,100 If I wanted to just evaluate this integral. 38 00:02:15,100 --> 00:02:19,220 What I would do is find an antiderivative of my function. 39 00:02:23,880 --> 00:02:25,859 And then by the fundamental theorem of calculus. 40 00:02:27,290 --> 00:02:28,780 I could evaluate this integral. 41 00:02:28,780 --> 00:02:32,200 It's just going to be the anti-derivative evaluated at the top 42 00:02:32,200 --> 00:02:35,900 end point, and the anti-derivative evaluated at the bottom end point. 43 00:02:35,900 --> 00:02:35,900 [BLANK_AUDIO] 44 00:02:35,900 --> 00:02:36,400 And 45 00:02:41,050 --> 00:02:44,360 now, if I take the derivative of this well, 46 00:02:44,360 --> 00:02:47,360 all I have here is a function of a function. 47 00:02:47,360 --> 00:02:49,780 So I can take the derivative of that, using the chain rule. 48 00:02:52,830 --> 00:02:57,240 And so what I'm going to get is G prime of T, so G 49 00:02:57,240 --> 00:03:00,520 is a function of T, so I'm taking it's derivative with respect to T. 50 00:03:02,190 --> 00:03:06,070 That's just going to be equal to the derivative 51 00:03:06,070 --> 00:03:09,800 of the outside function, evaluated at the inside function. 52 00:03:09,800 --> 00:03:11,860 And then by the chain rule, I have to 53 00:03:11,860 --> 00:03:16,350 also multiply that by the derivative of the inside function. 54 00:03:16,350 --> 00:03:17,740 And it looks a little bit complicated, 55 00:03:17,740 --> 00:03:20,030 but all I'm doing is just I have to put once 56 00:03:20,030 --> 00:03:23,690 for the upper end point, and once for the lower end point. 57 00:03:25,160 --> 00:03:29,120 And then by the way we've defined this capital F, I know its derivative is 58 00:03:29,120 --> 00:03:34,450 lowercase f, so the formula for differentiating a definite integral is 59 00:03:34,450 --> 00:03:40,955 just going to be f(b(t)) times b'(t) minus f(a(t)) times a'(t). 60 00:03:44,300 --> 00:03:48,790 And so just to state it with all of its actual conditions. 61 00:03:48,790 --> 00:03:53,050 So this is the, this is the definition that came out of the course textbook. 62 00:03:53,050 --> 00:03:56,450 So F has to be a continuous function. 63 00:03:56,450 --> 00:03:59,699 And a of t and b of t have to be differentiable functions. 64 00:04:01,330 --> 00:04:05,070 Then, the derivative with respect to T, of the integral from A of 65 00:04:05,070 --> 00:04:09,320 T to B of T of F of X DX is just F of 66 00:04:09,320 --> 00:04:13,980 B of T times B prime of T minus F of A of T times A prime of T. 67 00:04:16,670 --> 00:04:18,820 A so just a, a quick example of this. 68 00:04:18,820 --> 00:04:23,320 I think this is my simplified version of exercise 11 from, from chapter one. 69 00:04:25,140 --> 00:04:30,750 So, let's let g of x be defined as 1 over the square root of 70 00:04:30,750 --> 00:04:36,330 2 pi times the integral from 0 to b of x of e to the minus t squared over 2. 71 00:04:36,330 --> 00:04:41,830 And then, b of x is just going to be defined by this 72 00:04:41,830 --> 00:04:42,510 function here. 73 00:04:43,870 --> 00:04:48,620 Which can be simplified the way I've done it in the light grey on the right. 74 00:04:50,820 --> 00:04:54,080 And so this is actually a calculation you'll, you'll come across 75 00:04:56,420 --> 00:04:58,440 These are, these are sort of pieces of the 76 00:04:58,440 --> 00:05:01,540 black sholes pricing formula, so this is the interest rate. 77 00:05:01,540 --> 00:05:05,160 This is the, this is the volatility but again since 78 00:05:05,160 --> 00:05:08,100 I haven't talked about functions of multiple variables yet I 79 00:05:08,100 --> 00:05:10,700 just wanted to plug in specific numbers so it doesn't 80 00:05:10,700 --> 00:05:13,520 get too confusing about what I'm actually trying to do. 81 00:05:14,610 --> 00:05:17,680 And so the, the goal then is to compute g prime of x, 82 00:05:20,230 --> 00:05:24,180 and so is this the, the formula from the previous slide. 83 00:05:24,180 --> 00:05:29,620 But now since my lower limit is zero, so it's constant with respect to x, if I 84 00:05:29,620 --> 00:05:33,540 take its derivative, this, this a prime of t is just going to be equal to 0. 85 00:05:33,540 --> 00:05:35,000 So derivative of a constant. 86 00:05:35,000 --> 00:05:38,840 So this formula will simplify a bit, because of a being equal to 0. 87 00:05:38,840 --> 00:05:42,710 So all I need to do is just evaluate my function. 88 00:05:44,210 --> 00:05:45,240 And then multiply by the 89 00:05:45,240 --> 00:05:46,390 derivative of the inside. 90 00:05:51,490 --> 00:05:56,680 So the first step I will do is just evaluate my function g of x, 91 00:05:56,680 --> 00:06:03,250 so this bit here. I'm just going to evaluate that at b. 92 00:06:03,250 --> 00:06:08,092 And so I get, I had 5 times log of x plus 0.06. 93 00:06:08,092 --> 00:06:10,740 So when I square that, I'm going to end up with 94 00:06:10,740 --> 00:06:15,010 25 times the log of x plus 0.06, that quantity squared. 95 00:06:15,010 --> 00:06:15,010 [BLANK_AUDIO] 96 00:06:15,010 --> 00:06:15,510 And 97 00:06:17,990 --> 00:06:25,070 then I still need to multiply it by the derivative, then of b of x. 98 00:06:26,620 --> 00:06:30,930 So, b of x was equal to 5 times the log of x plus 0.06. 99 00:06:30,930 --> 00:06:37,530 And so that's pretty easy to see the derivative of, that's 5 times log of x. 100 00:06:37,530 --> 00:06:43,290 That's derivative is 5 divided by x and 5 times 0.06, that's just going to be 101 00:06:43,290 --> 00:06:44,940 a constant, so we don't even need to bother 102 00:06:44,940 --> 00:06:47,880 doing the math, that's going to have a derivative of 0. 103 00:06:47,880 --> 00:06:51,480 So b'(x) is just going to be 5 divided by x, 104 00:06:52,870 --> 00:06:54,580 and so I just need to put all of the pieces 105 00:06:54,580 --> 00:06:59,463 together, which just means substituting in 5 over x for b'(x), 106 00:07:00,770 --> 00:07:05,009 And that gives me the derivative of my function g(x).