WEBVTT 00:00:07.620 --> 00:00:09.570 PROFESSOR: Welcome back to recitation. 00:00:09.570 --> 00:00:12.720 In this video I want us to work on the following problem. 00:00:12.720 --> 00:00:16.400 So we're going to assume little f is a continuous function. 00:00:16.400 --> 00:00:20.380 And if we know that the integral from 0 to x of f of t dt 00:00:20.380 --> 00:00:24.470 is equal to x squared sine pi x, I 00:00:24.470 --> 00:00:27.305 want us to find little f of 2. 00:00:27.305 --> 00:00:28.930 So I'm going to give you a little while 00:00:28.930 --> 00:00:30.080 to work on the problem. 00:00:30.080 --> 00:00:32.990 When you're ready, come back and we'll take a look 00:00:32.990 --> 00:00:34.130 at how I do the problem. 00:00:43.070 --> 00:00:44.010 Welcome back. 00:00:44.010 --> 00:00:45.510 So again, we're working on a problem 00:00:45.510 --> 00:00:48.800 where we know that f is a continuous function. 00:00:48.800 --> 00:00:51.450 We know that the integral from 0 to x of f of t dt 00:00:51.450 --> 00:00:52.820 is equal to a certain function. 00:00:52.820 --> 00:00:55.477 And we want to find f of 2. 00:00:55.477 --> 00:00:57.060 So what we're going to use, because we 00:00:57.060 --> 00:00:59.050 know little f is continuous, we can actually 00:00:59.050 --> 00:01:01.520 use the fundamental theorem of calculus. 00:01:01.520 --> 00:01:06.650 And we know that d/dx of this whole expression 00:01:06.650 --> 00:01:09.950 here is actually little f of x. 00:01:09.950 --> 00:01:12.925 So let's just remind us-- I'll write it down formally. 00:01:16.380 --> 00:01:26.395 FTC implies that d/dx, the interval from 0 to x f of t dt 00:01:26.395 --> 00:01:27.186 is equal to f of x. 00:01:27.186 --> 00:01:27.686 Right? 00:01:27.686 --> 00:01:29.960 We know that. 00:01:29.960 --> 00:01:35.680 So, we know that d/dx of the right-hand side 00:01:35.680 --> 00:01:39.190 and d/dx of the left-hand side are the same. 00:01:39.190 --> 00:01:41.910 And so we know that if we take d/dx of the right-hand side, 00:01:41.910 --> 00:01:43.040 we'll get little f of x. 00:01:43.040 --> 00:01:47.200 Because d/dx of the left-hand side of this equation 00:01:47.200 --> 00:01:48.680 is little f of x. 00:01:48.680 --> 00:01:52.430 So d/dx of the right-hand side is also little f of x. 00:01:52.430 --> 00:01:54.360 So now what we have to do is apply 00:01:54.360 --> 00:01:57.900 what we know about taking derivatives to find little 00:01:57.900 --> 00:01:59.090 f of x. 00:01:59.090 --> 00:02:00.630 Then once we find little f of x we 00:02:00.630 --> 00:02:04.880 can evaluate it at f equal 2, at 2-- sorry-- at x equal 2. 00:02:04.880 --> 00:02:07.210 Sorry about that. 00:02:07.210 --> 00:02:10.370 So let's come over to the right part of the board 00:02:10.370 --> 00:02:13.530 and we'll find d/dx of this function here. 00:02:21.747 --> 00:02:22.830 So what do we have to use? 00:02:22.830 --> 00:02:24.496 We have to use the product rule and then 00:02:24.496 --> 00:02:27.420 we'll have to use a little chain rule, a really simple chain 00:02:27.420 --> 00:02:29.286 rule in the second term. 00:02:29.286 --> 00:02:30.660 I'm going to write it down below, 00:02:30.660 --> 00:02:31.770 so we have some room here. 00:02:31.770 --> 00:02:36.550 So the first term, we get 2x sine pi*x. 00:02:36.550 --> 00:02:38.690 The second term we get x squared. 00:02:38.690 --> 00:02:41.010 Derivative of sine is cosine. 00:02:41.010 --> 00:02:41.690 pi*x. 00:02:41.690 --> 00:02:44.210 And then we have to pull out a pi. 00:02:44.210 --> 00:02:48.940 Which I'll just put, well I'll put it here for the moment. 00:02:48.940 --> 00:02:51.980 The pi comes from taking the derivative of pi*x. 00:02:51.980 --> 00:02:56.670 And now all we need, this is then f of x, right? 00:02:56.670 --> 00:02:59.610 So I just will remind us this is f of x. 00:02:59.610 --> 00:03:02.920 Let me put it right here, a little f of x. 00:03:02.920 --> 00:03:05.570 So if I want to find f of 2, all I have to do 00:03:05.570 --> 00:03:09.100 is evaluate this at x equal 2. 00:03:09.100 --> 00:03:13.450 So f of 2 is going to equal-- 2 times 2 is 4-- 00:03:13.450 --> 00:03:20.050 sine 2*pi plus 4-- I'll bring the pi in front-- 00:03:20.050 --> 00:03:23.040 4*pi cosine 2*pi. 00:03:23.040 --> 00:03:24.610 So what do I get? 00:03:24.610 --> 00:03:27.010 Sine 2*pi is 0. 00:03:27.010 --> 00:03:28.970 Cosine 2*pi is 1. 00:03:28.970 --> 00:03:32.669 So ultimately, I just get 4*pi. 00:03:32.669 --> 00:03:34.710 So, hopefully you were able to solve this problem 00:03:34.710 --> 00:03:36.950 and get the same answer. 00:03:36.950 --> 00:03:40.220 Now let me remind you, one more time what we did. 00:03:40.220 --> 00:03:41.440 Let's come over here. 00:03:41.440 --> 00:03:44.410 We started with a function that we knew was continuous. 00:03:44.410 --> 00:03:47.760 We knew the integral from 0 to x of that function 00:03:47.760 --> 00:03:50.610 was equal to a certain function of x. 00:03:50.610 --> 00:03:53.447 And we wanted to evaluate this function at a certain, 00:03:53.447 --> 00:03:55.430 at a certain point. 00:03:55.430 --> 00:03:57.430 So what we exploited was the fundamental theorem 00:03:57.430 --> 00:03:58.986 of calculus. 00:03:58.986 --> 00:04:00.610 And the fundamental theorem of calculus 00:04:00.610 --> 00:04:06.010 tells us that d/dx of the left-hand side is f of x. 00:04:06.010 --> 00:04:09.260 And so that we can take d/dx of the right-hand side 00:04:09.260 --> 00:04:11.070 and figure out what f of x is. 00:04:11.070 --> 00:04:13.290 And then we just evaluate. 00:04:13.290 --> 00:04:15.240 So I think I'll stop there.