1 00:00:00,000 --> 00:00:07,690 JOEL LEWIS: Hi. 2 00:00:07,690 --> 00:00:09,150 Welcome to recitation. 3 00:00:09,150 --> 00:00:12,840 In lecture you've learned how to compute derivatives 4 00:00:12,840 --> 00:00:14,900 of polynomials, and you've learned 5 00:00:14,900 --> 00:00:18,050 the relationship between derivatives and tangents lines. 6 00:00:18,050 --> 00:00:22,340 So let's do a quick example that puts those two ideas together. 7 00:00:22,340 --> 00:00:24,450 So here I have a question on the board: 8 00:00:24,450 --> 00:00:26,840 compute the tangent line to the curve y 9 00:00:26,840 --> 00:00:30,710 equals x cubed minus x at the point (2, 6). 10 00:00:30,710 --> 00:00:32,210 So why don't you take a minute, work 11 00:00:32,210 --> 00:00:34,559 on that yourself, pause the video, we'll come back 12 00:00:34,559 --> 00:00:35,600 and we'll do it together. 13 00:00:38,691 --> 00:00:39,190 All right. 14 00:00:39,190 --> 00:00:40,170 Welcome back. 15 00:00:40,170 --> 00:00:44,510 So we have this function, y equals x cubed minus x. 16 00:00:44,510 --> 00:00:49,270 Let's just draw a quick sketch of it. 17 00:00:49,270 --> 00:00:57,760 So looks to me like it has zeros at 0, 1, minus 1. 18 00:00:57,760 --> 00:01:02,860 And it sort of does something like this in between. 19 00:01:02,860 --> 00:01:05,070 Very rough sketch there. 20 00:01:05,070 --> 00:01:07,820 And way over-- well, OK. 21 00:01:07,820 --> 00:01:10,060 We'll call that the point (2, 6). 22 00:01:10,060 --> 00:01:13,560 Feel a little sketchy, but all right. 23 00:01:13,560 --> 00:01:14,060 OK. 24 00:01:14,060 --> 00:01:16,220 So we want to know what the tangent line 25 00:01:16,220 --> 00:01:18,270 to the curve at that point is. 26 00:01:18,270 --> 00:01:23,440 So in order to do that, we need to know what its derivative is, 27 00:01:23,440 --> 00:01:25,050 and then that'll give us the slope. 28 00:01:25,050 --> 00:01:26,799 And then with the slope, we have the slope 29 00:01:26,799 --> 00:01:28,930 and we have a point, so we can slap 30 00:01:28,930 --> 00:01:32,320 that into, say, your point-slope formula for a line. 31 00:01:35,120 --> 00:01:41,150 So, all right, so the derivative of this function is y prime. 32 00:01:41,150 --> 00:01:43,940 So, OK, so here we have a sum of two things, 33 00:01:43,940 --> 00:01:45,810 and they're both powers of x. 34 00:01:45,810 --> 00:01:48,060 And so we learned our rules for a power 35 00:01:48,060 --> 00:01:50,440 of x that the derivative of x to the n 36 00:01:50,440 --> 00:01:52,510 is n times x to the n minus 1. 37 00:01:52,510 --> 00:01:54,640 And so we also learned that the derivative 38 00:01:54,640 --> 00:01:58,470 of a sum of two things is the sum of the derivatives. 39 00:01:58,470 --> 00:02:01,230 So in this case, so the derivative of x cubed 40 00:02:01,230 --> 00:02:08,076 minus x is 3 x squared minus 1. 41 00:02:08,076 --> 00:02:08,576 OK. 42 00:02:08,576 --> 00:02:13,480 So this is the slope of the function in terms of x. 43 00:02:13,480 --> 00:02:15,840 But in order to compute the tangent line, 44 00:02:15,840 --> 00:02:20,035 we need the slope at the particular point in question. 45 00:02:20,035 --> 00:02:20,535 Right? 46 00:02:20,535 --> 00:02:22,120 This is really important. 47 00:02:22,120 --> 00:02:25,870 So we aren't going to use 3 x squared plus 1 as our slope. 48 00:02:25,870 --> 00:02:26,780 Right? 49 00:02:26,780 --> 00:02:29,750 We want the slope at the point x equals 2. 50 00:02:29,750 --> 00:02:30,250 Right? 51 00:02:33,530 --> 00:02:34,030 OK. 52 00:02:34,030 --> 00:02:37,100 So what we want for the slope of the tangent line 53 00:02:37,100 --> 00:02:38,640 is y prime of 2. 54 00:02:38,640 --> 00:02:39,410 Right? 55 00:02:39,410 --> 00:02:42,050 We want it at this point when x is equal to 2. 56 00:02:42,050 --> 00:02:47,155 So that's equal to, well, 3 times 2 squared is 12, minus 1 57 00:02:47,155 --> 00:02:47,655 is 11. 58 00:02:58,681 --> 00:03:01,180 So this is the slope, this is the slope of the tangent line. 59 00:03:01,180 --> 00:03:03,310 I just want to say this one more time for emphasis. 60 00:03:03,310 --> 00:03:05,690 This is a really common mistake that we 61 00:03:05,690 --> 00:03:10,500 see on lots of homework and exams when teaching calculus. 62 00:03:10,500 --> 00:03:12,460 You have to remember that when you compute 63 00:03:12,460 --> 00:03:15,390 the slope of the tangent line, you compute the derivative 64 00:03:15,390 --> 00:03:18,980 and then you need to plug in the value of x 65 00:03:18,980 --> 00:03:20,160 at the point in question. 66 00:03:20,160 --> 00:03:22,620 Or the value of x and y. 67 00:03:22,620 --> 00:03:24,970 You know, you need to plug in the values of the point 68 00:03:24,970 --> 00:03:27,380 that you have. 69 00:03:27,380 --> 00:03:31,410 So here, the derivative is 3 x squared minus 1, 70 00:03:31,410 --> 00:03:34,760 so the slope at the point (2, 6) is 11. 71 00:03:34,760 --> 00:03:36,240 It's just a number. 72 00:03:36,240 --> 00:03:39,940 The slope at that point is that particular number. 73 00:03:39,940 --> 00:03:41,910 OK, so now, to compute the tangent line, 74 00:03:41,910 --> 00:03:45,950 we have a point, (2, 6), and we have a slope, 11. 75 00:03:45,950 --> 00:03:48,400 So we can plug into point-slope form. 76 00:03:57,060 --> 00:04:11,572 So the equation of the-- on the-- tangent line is y minus 6 77 00:04:11,572 --> 00:04:16,107 is equal to 11 times x minus 2. 78 00:04:16,107 --> 00:04:16,870 Right? 79 00:04:16,870 --> 00:04:22,340 So it's y minus y_0 is equal to the slope times x minus x_0. 80 00:04:22,340 --> 00:04:24,530 If you like, some people prefer to write 81 00:04:24,530 --> 00:04:28,209 their equations of their lines in slope-intercept form. 82 00:04:28,209 --> 00:04:30,000 So if you wanted to do that, you could just 83 00:04:30,000 --> 00:04:34,500 multiply through by 11 and then bring the constants together. 84 00:04:34,500 --> 00:04:42,680 So we could say, or y equals 11x-- well, 85 00:04:42,680 --> 00:04:48,050 we get minus 22 plus 6 is minus 16. 86 00:04:48,050 --> 00:04:50,540 So either of those is a perfectly good answer 87 00:04:50,540 --> 00:04:51,720 to the question. 88 00:04:51,720 --> 00:04:53,410 So that's that.