1 00:00:07,620 --> 00:00:09,570 PROFESSOR: Welcome back to recitation. 2 00:00:09,570 --> 00:00:12,720 In this video I want us to work on the following problem. 3 00:00:12,720 --> 00:00:16,400 So we're going to assume little f is a continuous function. 4 00:00:16,400 --> 00:00:20,380 And if we know that the integral from 0 to x of f of t dt 5 00:00:20,380 --> 00:00:24,470 is equal to x squared sine pi x, I 6 00:00:24,470 --> 00:00:27,305 want us to find little f of 2. 7 00:00:27,305 --> 00:00:28,930 So I'm going to give you a little while 8 00:00:28,930 --> 00:00:30,080 to work on the problem. 9 00:00:30,080 --> 00:00:32,990 When you're ready, come back and we'll take a look 10 00:00:32,990 --> 00:00:34,130 at how I do the problem. 11 00:00:43,070 --> 00:00:44,010 Welcome back. 12 00:00:44,010 --> 00:00:45,510 So again, we're working on a problem 13 00:00:45,510 --> 00:00:48,800 where we know that f is a continuous function. 14 00:00:48,800 --> 00:00:51,450 We know that the integral from 0 to x of f of t dt 15 00:00:51,450 --> 00:00:52,820 is equal to a certain function. 16 00:00:52,820 --> 00:00:55,477 And we want to find f of 2. 17 00:00:55,477 --> 00:00:57,060 So what we're going to use, because we 18 00:00:57,060 --> 00:00:59,050 know little f is continuous, we can actually 19 00:00:59,050 --> 00:01:01,520 use the fundamental theorem of calculus. 20 00:01:01,520 --> 00:01:06,650 And we know that d/dx of this whole expression 21 00:01:06,650 --> 00:01:09,950 here is actually little f of x. 22 00:01:09,950 --> 00:01:12,925 So let's just remind us-- I'll write it down formally. 23 00:01:16,380 --> 00:01:26,395 FTC implies that d/dx, the interval from 0 to x f of t dt 24 00:01:26,395 --> 00:01:27,186 is equal to f of x. 25 00:01:27,186 --> 00:01:27,686 Right? 26 00:01:27,686 --> 00:01:29,960 We know that. 27 00:01:29,960 --> 00:01:35,680 So, we know that d/dx of the right-hand side 28 00:01:35,680 --> 00:01:39,190 and d/dx of the left-hand side are the same. 29 00:01:39,190 --> 00:01:41,910 And so we know that if we take d/dx of the right-hand side, 30 00:01:41,910 --> 00:01:43,040 we'll get little f of x. 31 00:01:43,040 --> 00:01:47,200 Because d/dx of the left-hand side of this equation 32 00:01:47,200 --> 00:01:48,680 is little f of x. 33 00:01:48,680 --> 00:01:52,430 So d/dx of the right-hand side is also little f of x. 34 00:01:52,430 --> 00:01:54,360 So now what we have to do is apply 35 00:01:54,360 --> 00:01:57,900 what we know about taking derivatives to find little 36 00:01:57,900 --> 00:01:59,090 f of x. 37 00:01:59,090 --> 00:02:00,630 Then once we find little f of x we 38 00:02:00,630 --> 00:02:04,880 can evaluate it at f equal 2, at 2-- sorry-- at x equal 2. 39 00:02:04,880 --> 00:02:07,210 Sorry about that. 40 00:02:07,210 --> 00:02:10,370 So let's come over to the right part of the board 41 00:02:10,370 --> 00:02:13,530 and we'll find d/dx of this function here. 42 00:02:21,747 --> 00:02:22,830 So what do we have to use? 43 00:02:22,830 --> 00:02:24,496 We have to use the product rule and then 44 00:02:24,496 --> 00:02:27,420 we'll have to use a little chain rule, a really simple chain 45 00:02:27,420 --> 00:02:29,286 rule in the second term. 46 00:02:29,286 --> 00:02:30,660 I'm going to write it down below, 47 00:02:30,660 --> 00:02:31,770 so we have some room here. 48 00:02:31,770 --> 00:02:36,550 So the first term, we get 2x sine pi*x. 49 00:02:36,550 --> 00:02:38,690 The second term we get x squared. 50 00:02:38,690 --> 00:02:41,010 Derivative of sine is cosine. 51 00:02:41,010 --> 00:02:41,690 pi*x. 52 00:02:41,690 --> 00:02:44,210 And then we have to pull out a pi. 53 00:02:44,210 --> 00:02:48,940 Which I'll just put, well I'll put it here for the moment. 54 00:02:48,940 --> 00:02:51,980 The pi comes from taking the derivative of pi*x. 55 00:02:51,980 --> 00:02:56,670 And now all we need, this is then f of x, right? 56 00:02:56,670 --> 00:02:59,610 So I just will remind us this is f of x. 57 00:02:59,610 --> 00:03:02,920 Let me put it right here, a little f of x. 58 00:03:02,920 --> 00:03:05,570 So if I want to find f of 2, all I have to do 59 00:03:05,570 --> 00:03:09,100 is evaluate this at x equal 2. 60 00:03:09,100 --> 00:03:13,450 So f of 2 is going to equal-- 2 times 2 is 4-- 61 00:03:13,450 --> 00:03:20,050 sine 2*pi plus 4-- I'll bring the pi in front-- 62 00:03:20,050 --> 00:03:23,040 4*pi cosine 2*pi. 63 00:03:23,040 --> 00:03:24,610 So what do I get? 64 00:03:24,610 --> 00:03:27,010 Sine 2*pi is 0. 65 00:03:27,010 --> 00:03:28,970 Cosine 2*pi is 1. 66 00:03:28,970 --> 00:03:32,669 So ultimately, I just get 4*pi. 67 00:03:32,669 --> 00:03:34,710 So, hopefully you were able to solve this problem 68 00:03:34,710 --> 00:03:36,950 and get the same answer. 69 00:03:36,950 --> 00:03:40,220 Now let me remind you, one more time what we did. 70 00:03:40,220 --> 00:03:41,440 Let's come over here. 71 00:03:41,440 --> 00:03:44,410 We started with a function that we knew was continuous. 72 00:03:44,410 --> 00:03:47,760 We knew the integral from 0 to x of that function 73 00:03:47,760 --> 00:03:50,610 was equal to a certain function of x. 74 00:03:50,610 --> 00:03:53,447 And we wanted to evaluate this function at a certain, 75 00:03:53,447 --> 00:03:55,430 at a certain point. 76 00:03:55,430 --> 00:03:57,430 So what we exploited was the fundamental theorem 77 00:03:57,430 --> 00:03:58,986 of calculus. 78 00:03:58,986 --> 00:04:00,610 And the fundamental theorem of calculus 79 00:04:00,610 --> 00:04:06,010 tells us that d/dx of the left-hand side is f of x. 80 00:04:06,010 --> 00:04:09,260 And so that we can take d/dx of the right-hand side 81 00:04:09,260 --> 00:04:11,070 and figure out what f of x is. 82 00:04:11,070 --> 00:04:13,290 And then we just evaluate. 83 00:04:13,290 --> 00:04:15,240 So I think I'll stop there.