1 00:00:00,000 --> 00:00:06,272 [music] Sometimes people ask for the anti derivative. 2 00:00:06,272 --> 00:00:14,722 But instead of talking about the anti derivative it's probably better to talk 3 00:00:14,722 --> 00:00:20,942 about an anti deriviative. So why is this the and an distinction 4 00:00:20,942 --> 00:00:25,984 really important. Because a function can have more than one 5 00:00:25,984 --> 00:00:29,125 anti-derivative. Let's take a look at an example. 6 00:00:29,126 --> 00:00:36,256 Let's think about the function f of x equals 2 x, to find on the whole real 7 00:00:36,256 --> 00:00:39,347 line. Can you think of a function whose 8 00:00:39,347 --> 00:00:42,898 derivative is 2 x? I'm looking for a function which 9 00:00:42,898 --> 00:00:46,820 differentiates to give me this function. So here's an example. 10 00:00:46,820 --> 00:00:53,882 A big f of x equals x squared, and yes, if I differentiate that function I get back 11 00:00:53,882 --> 00:00:59,067 2x, which is in fact little f. So the derivative of big f is little f, or 12 00:00:59,067 --> 00:01:02,544 in other words an anti-derivative for little f. 13 00:01:02,545 --> 00:01:04,368 Is big F. Okay. 14 00:01:04,369 --> 00:01:06,421 Good. Now, can we think of another function 15 00:01:06,421 --> 00:01:08,600 whose derivative is 2x? Well, yeah. 16 00:01:08,600 --> 00:01:14,785 Think about this function big G of x which I'll define to be x squared plus 17 and 17 00:01:14,785 --> 00:01:20,770 big G's derivative is the derivative of this sum which is the derivative of x 18 00:01:20,770 --> 00:01:25,168 squared which is 2x plus the derivative of 17 which is 0. 19 00:01:25,168 --> 00:01:32,425 So big G differentiates to little f. And that means another anti-derivative for 20 00:01:32,425 --> 00:01:34,280 little f is big G. Okay great. 21 00:01:34,280 --> 00:01:37,545 Now let's think of yet another function whose derivative is 2x. 22 00:01:37,545 --> 00:01:44,724 Yeah, I can play this game all day. Big H of x is equal to say x squared plus 23 00:01:44,724 --> 00:01:49,120 123. Big h differentiates to 2 x again. 24 00:01:49,120 --> 00:01:52,847 So big h is another anti derivative of little f. 25 00:01:52,848 --> 00:01:56,527 There's clearly a pattern here, right. What's the general story? 26 00:01:56,528 --> 00:02:01,154 Well here's the general story. So I got this function f of x equals 2 x. 27 00:02:01,155 --> 00:02:05,380 Any anti derivative of little f has the form big f. 28 00:02:05,381 --> 00:02:09,656 Equals x squared plus c for some constant c. 29 00:02:09,656 --> 00:02:12,912 All of out previous examples had this form. 30 00:02:12,912 --> 00:02:19,248 Like here the constant is 0, here the constant is 17, here the constant is 123. 31 00:02:19,248 --> 00:02:24,624 And what I'm claiming here is that any other anti derivative is just x squared 32 00:02:24,624 --> 00:02:27,910 plus some fixed number. How do we know this? 33 00:02:27,910 --> 00:02:30,778 Well, it goes back to the mean value theorem. 34 00:02:30,778 --> 00:02:38,327 So, suppose I've got 2 anti-derivatives, big F and big G for little f. 35 00:02:38,328 --> 00:02:44,476 In other words, I mean that if I differentiate big F I get little f, and if 36 00:02:44,476 --> 00:02:50,713 I differentiate big G I get little f. So, suppose I've got Two anti derivatives 37 00:02:50,713 --> 00:02:55,410 for little F. Now what can I do, like I define a new 38 00:02:55,410 --> 00:03:01,730 function called big H and big H will just be big F minus big G. 39 00:03:01,730 --> 00:03:05,955 Now what do I know about the derivative of big H, the derivative of big H is the 40 00:03:05,955 --> 00:03:10,440 derivative of F minus the derivative of G because the derivative of the difference 41 00:03:10,440 --> 00:03:16,286 is the difference of derivatives. So this is the derivative of f minus the 42 00:03:16,286 --> 00:03:19,200 derivative of g. But what is the derivative of f? 43 00:03:19,200 --> 00:03:21,230 It's little f. What's the derivative of g? 44 00:03:21,230 --> 00:03:25,927 It's little f? This is little f minus little f. 45 00:03:25,928 --> 00:03:28,883 Oh noes. This is 0, right? 46 00:03:28,883 --> 00:03:32,050 The derivative of big H is 0. Now what does that mean? 47 00:03:32,050 --> 00:03:38,782 Well if you think back to the mean value theorem From earlier in the course, if a 48 00:03:38,782 --> 00:03:46,009 functions derivative is equal to zero, and say that function is defined on the whole 49 00:03:46,009 --> 00:03:52,741 real line, like our example with x squared and two x, then, that means that this 50 00:03:52,741 --> 00:03:55,459 function is a constant. Right? 51 00:03:55,459 --> 00:04:03,360 So that means that h of x Is a constant. Now, what that means is that the 52 00:04:03,360 --> 00:04:09,591 difference between F and G, these two antiderivatives for little f, is just some 53 00:04:09,591 --> 00:04:16,851 fixed constant, right? In other words, F of x, is equal to G of x 54 00:04:16,851 --> 00:04:23,692 plus some constant. And this is exactly what we're trying to 55 00:04:23,692 --> 00:04:27,728 get at, right? This idea that if a function's got two 56 00:04:27,728 --> 00:04:32,040 different anti-derivatives. And the function's defined say on the 57 00:04:32,040 --> 00:04:36,572 whole real line. Then those two different anti-derivatives 58 00:04:36,572 --> 00:04:39,557 just differ by a constant or said differently. 59 00:04:39,558 --> 00:04:44,883 If you've got a specific anti-derivative, than any other anti-derivative is just 60 00:04:44,883 --> 00:04:48,340 that specific anti-derivative plus some constant. 61 00:04:48,340 --> 00:04:50,734 Let me say that again, and, and actually write it down. 62 00:04:50,735 --> 00:04:54,870 Suppose that little f is a function that's defined on an interval I. 63 00:04:54,870 --> 00:04:57,837 That's really crucial that it's defined on an interval. 64 00:04:57,838 --> 00:05:01,747 And suppose that big F is an anti-derivative of little f. 65 00:05:01,748 --> 00:05:05,558 In other words if I differentiate big F, I get little f. 66 00:05:05,558 --> 00:05:14,308 Well then any antiderivative of little f, not just this one, any antiderivative has 67 00:05:14,308 --> 00:05:18,820 the form big F of x plus c for some constant c.