1 00:00:00,012 --> 00:00:09,952 [music] There's a visual way to gain some insight into these anti-differentiation 2 00:00:09,952 --> 00:00:14,019 problems. A visual method goes by the name of slope 3 00:00:14,019 --> 00:00:17,174 fields. So, what's a slope field? 4 00:00:17,175 --> 00:00:20,376 The idea is that I start with some function. 5 00:00:20,376 --> 00:00:23,954 Like, here's an example of function x squared minus x. 6 00:00:23,954 --> 00:00:29,414 But instead of just graphing the function, like I normally do here, I'm graphing the 7 00:00:29,414 --> 00:00:31,848 function's values. I make a graph. 8 00:00:31,848 --> 00:00:37,007 But the thing that I'm graphing is little tiny line segments of that given slope, 9 00:00:37,007 --> 00:00:40,594 right? So, instead of plotting a value at some 10 00:00:40,594 --> 00:00:45,264 height, I draw little tiny line segments with that slope. 11 00:00:45,264 --> 00:00:51,445 Now, you can see in this example that if you plug in 0 or 1 into this function, the 12 00:00:51,445 --> 00:00:55,812 output 0. And in this slope field, if you look at a 13 00:00:55,812 --> 00:01:03,138 point whose x value is 0 or whose x value is 1, there I'm plotting just horizontal 14 00:01:03,138 --> 00:01:08,317 lines, lines of slope zero. Over here, I've got very high slope, and 15 00:01:08,317 --> 00:01:13,336 over here I've got very high slope. And that just reflects the fact that this 16 00:01:13,336 --> 00:01:18,059 function's value is very high when x is negative and when x is positive. 17 00:01:18,060 --> 00:01:22,879 And in between here, I've got just a little tiny region where the lines are 18 00:01:22,879 --> 00:01:26,780 pointing down, right? Where the slope of these little line 19 00:01:26,780 --> 00:01:30,420 segments is negative. And of course, that's because there's a 20 00:01:30,420 --> 00:01:33,805 little tiny region here where this function's value is negative. 21 00:01:33,806 --> 00:01:40,320 Well, let's look at a specific example. With this specific example, function F of 22 00:01:40,320 --> 00:01:45,141 x equals x times cosine x. Now, I can draw the slope field for this 23 00:01:45,141 --> 00:01:48,957 example. Well, in that case, here's the slope field 24 00:01:48,957 --> 00:01:54,780 that I get. Imagine that I have an anti-derivative, a 25 00:01:54,780 --> 00:01:59,621 function f, whose derivative was F. That is, suppose I had some function, f, 26 00:01:59,621 --> 00:02:03,018 whose derivative was F. The slope field is drawn so that all these 27 00:02:03,018 --> 00:02:08,760 little line segments have slope equal to the value of F, so I could use this slope 28 00:02:08,760 --> 00:02:13,624 field to try to draw a picture, the graph of my function f, right? 29 00:02:13,624 --> 00:02:16,150 Let's suppose that my function maybe starts here. 30 00:02:16,150 --> 00:02:23,222 And then, I could try to follow the slope field along and try to use the information 31 00:02:23,222 --> 00:02:29,774 in the, in the slope field to try to draw a picture of what my function, f, must 32 00:02:29,774 --> 00:02:32,910 look like. So, these slope fields let you see the 33 00:02:32,910 --> 00:02:36,721 anti-derivative, right? If you follow the slope field, you're 34 00:02:36,721 --> 00:02:40,847 tracing out a function whose derivative is given by the slope field. 35 00:02:40,848 --> 00:02:46,356 Now, we can find this anti-derivative, but we're trying to anti-differentiate x 36 00:02:46,356 --> 00:02:49,480 cosine x. Well, our first guess might be x sine x. 37 00:02:49,480 --> 00:02:53,995 I mean, that's a reasonable guess because differentiating sine will give me the 38 00:02:53,995 --> 00:02:58,445 cosine and it, but what happens if I actually differentiate this product? 39 00:02:58,445 --> 00:03:04,127 Well, that's the sort of thing that I do with the product rule which is x times the 40 00:03:04,127 --> 00:03:07,626 derivative, which is cosine. So, that's looking good. 41 00:03:07,627 --> 00:03:14,042 But it's plus the derivative of the first, which is 1 times the second, which is sine 42 00:03:14,042 --> 00:03:16,585 x. So, the anti-derivative of x cosine x 43 00:03:16,585 --> 00:03:21,012 isn't just x sine x because I've got this pesky additional sine x term. 44 00:03:21,012 --> 00:03:25,141 But I also know that the derivative of cosine is minus sine. 45 00:03:25,141 --> 00:03:34,408 So, if I add these things together, right? If I add them together, what's the 46 00:03:34,408 --> 00:03:39,636 derivative of x sine x plus cosine x? Well, it's these two things added 47 00:03:39,636 --> 00:03:44,040 together. But that's x cosine x, which is good, plus 48 00:03:44,040 --> 00:03:47,892 sine x minus sine x. So, it's just x cosine x. 49 00:03:47,893 --> 00:03:53,750 So that means the general anti-derivative of x cosine x is x sine x plus cosine x 50 00:03:53,750 --> 00:03:57,695 plus some constant c. And yeah, we can see this in the graph. 51 00:03:57,696 --> 00:04:01,296 So here, I've graphed x sine x plus cosine x. 52 00:04:01,296 --> 00:04:05,387 And you can see that it's really moving in the direction of the slope field, which 53 00:04:05,387 --> 00:04:08,284 makes sense. Because the slope field is coming from x 54 00:04:08,284 --> 00:04:11,272 cosine x, which is the derivative of this thing. 55 00:04:11,272 --> 00:04:16,226 And, you know, I drew the slope field by drawing little tiny line segments whose 56 00:04:16,226 --> 00:04:20,282 slope is x cosine x. So, since this thing differentiates to x 57 00:04:20,282 --> 00:04:24,480 cosine x, this thing is really moving along the slope field. 58 00:04:24,480 --> 00:04:29,088 The slope field picture also gives you some insight into how the plus c really 59 00:04:29,088 --> 00:04:31,366 works. What is the plus c really accomplishing, 60 00:04:31,366 --> 00:04:34,607 right? It's moving the graph up and down. 61 00:04:34,608 --> 00:04:39,828 And if you start with, you know, say x sine x plus cosine x, and then you slide 62 00:04:39,828 --> 00:04:44,702 that graph up and down, it's still following the slope field, right? 63 00:04:44,703 --> 00:04:50,194 It's still an anti-derivative for x cosine x. 64 00:04:50,195 --> 00:04:56,016 .