WEBVTT 00:00:00.000 --> 00:00:07.690 JOEL LEWIS: Hi. 00:00:07.690 --> 00:00:09.150 Welcome to recitation. 00:00:09.150 --> 00:00:13.010 In lecture you've learned how to compute derivatives of 00:00:13.010 --> 00:00:15.650 polynomials, and you've learned the relationship 00:00:15.650 --> 00:00:18.050 between derivatives and tangents lines. 00:00:18.050 --> 00:00:19.880 So let's do a quick example that puts 00:00:19.880 --> 00:00:22.340 those two ideas together. 00:00:22.340 --> 00:00:24.450 So here I have a question on the board-- 00:00:24.450 --> 00:00:27.840 compute the tangent line to the curve y equals x cubed 00:00:27.840 --> 00:00:30.970 minus x at the point 2, 6. 00:00:30.970 --> 00:00:33.080 So why don't you take a minute, work on that yourself, 00:00:33.080 --> 00:00:35.600 pause the video, we'll come back and we'll do it together. 00:00:38.730 --> 00:00:39.190 All right. 00:00:39.190 --> 00:00:40.170 Welcome back. 00:00:40.170 --> 00:00:44.510 So we have this function, y equals x cubed minus x. 00:00:44.510 --> 00:00:49.270 Let's just draw a quick sketch of it. 00:00:49.270 --> 00:00:57.760 So looks to me like it has 0's at 0, 1, minus 1. 00:00:57.760 --> 00:01:02.860 And it sort of does something like this in between. 00:01:02.860 --> 00:01:05.070 Very rough sketch there. 00:01:05.070 --> 00:01:07.820 And way over-- well, OK. 00:01:07.820 --> 00:01:10.060 We'll call that the point, 2, 6. 00:01:10.060 --> 00:01:13.810 Feel a little sketchy, but all right. 00:01:13.810 --> 00:01:14.060 OK. 00:01:14.060 --> 00:01:17.560 So we want to know what the tangent line to the curve at 00:01:17.560 --> 00:01:18.270 that point is. 00:01:18.270 --> 00:01:21.320 So in order to do that, we need to know what its 00:01:21.320 --> 00:01:25.050 derivative is, and then that'll give us the slope. 00:01:25.050 --> 00:01:26.950 And then with the slope, we have the slope and we have a 00:01:26.950 --> 00:01:30.480 point, so we can slap that into, say, your point-slope 00:01:30.480 --> 00:01:32.320 formula for a line. 00:01:35.120 --> 00:01:37.270 So, all right, so the derivative of this 00:01:37.270 --> 00:01:41.150 function is y prime. 00:01:41.150 --> 00:01:44.270 So, OK, so here we have a sum of two things, and they're 00:01:44.270 --> 00:01:45.810 both powers of x. 00:01:45.810 --> 00:01:48.600 And so we learned our rules for a power of x that the 00:01:48.600 --> 00:01:52.510 derivative of x to the n is n times x to the n minus 1. 00:01:52.510 --> 00:01:55.600 And so we also learned that the derivative of a sum of two 00:01:55.600 --> 00:01:58.470 things is the sum of the derivatives. 00:01:58.470 --> 00:02:03.410 So in this case, so the derivative of x cubed minus x 00:02:03.410 --> 00:02:08.470 is 3x squared minus 1. 00:02:08.470 --> 00:02:08.576 OK. 00:02:08.576 --> 00:02:13.480 So this is the slope of the function in terms of x. 00:02:13.480 --> 00:02:16.670 But in order to compute the tangent line, we need the 00:02:16.670 --> 00:02:20.460 slope at the particular point in question. 00:02:20.460 --> 00:02:20.535 Right? 00:02:20.535 --> 00:02:22.120 This is really important. 00:02:22.120 --> 00:02:25.870 So we aren't going to use 3x squared plus 1 as our slope. 00:02:25.870 --> 00:02:26.780 Right? 00:02:26.780 --> 00:02:30.040 We want the slope at the point, x equals 2. 00:02:30.040 --> 00:02:31.290 Right? 00:02:33.940 --> 00:02:34.030 OK. 00:02:34.030 --> 00:02:37.100 So what we want for the slope of the tangent line 00:02:37.100 --> 00:02:38.640 is y prime of 2. 00:02:38.640 --> 00:02:39.410 Right? 00:02:39.410 --> 00:02:42.050 We want it at this point when x is equal to 2. 00:02:42.050 --> 00:02:45.300 So that's equal to, well, 3 times 2 squared is 00:02:45.300 --> 00:02:59.330 12 minus 1 is 11. 00:02:59.330 --> 00:03:01.180 So this is the slope, this is the slope of the tangent line. 00:03:01.180 --> 00:03:03.310 I just want to say this one more time for emphasis. 00:03:03.310 --> 00:03:07.200 This is a really common mistake that we see on lots of 00:03:07.200 --> 00:03:10.500 homework and exams when teaching calculus. 00:03:10.500 --> 00:03:13.110 You have to remember that when you compute the slope of the 00:03:13.110 --> 00:03:16.020 tangent line, you compute the derivative and then you need 00:03:16.020 --> 00:03:20.160 to plug in the value of x at the point in question. 00:03:20.160 --> 00:03:22.620 Or the value of x and y. 00:03:22.620 --> 00:03:25.120 You know, you need to plug in the values of the point that 00:03:25.120 --> 00:03:30.990 you have. So here, the derivative is 3x squared minus 00:03:30.990 --> 00:03:34.760 1, so the slope at the point, 2, 6 is 11. 00:03:34.760 --> 00:03:36.240 It's just a number. 00:03:36.240 --> 00:03:39.940 The slope at that point is that particular number. 00:03:39.940 --> 00:03:42.830 OK, so now, to compute the tangent line, we have a point, 00:03:42.830 --> 00:03:45.950 2, 6 and we have a slope, 11. 00:03:45.950 --> 00:03:48.400 So we can plug into point-slope form. 00:03:57.060 --> 00:04:00.930 So the equation of the-- one the-- 00:04:04.140 --> 00:04:16.107 tangent line is y minus 6 is equal to 11 times x minus 2. 00:04:16.107 --> 00:04:16.870 Right? 00:04:16.870 --> 00:04:22.340 So it's y minus y0 is equal to the slope times x minus x0. 00:04:22.340 --> 00:04:25.210 If you like, some people prefer to write their 00:04:25.210 --> 00:04:28.440 equations of their lines in slope-intercept form. 00:04:28.440 --> 00:04:30.440 So if you wanted to do that, you could just multiply 00:04:30.440 --> 00:04:34.500 through by 11 and then bring the constants together. 00:04:34.500 --> 00:04:42.370 So we could say, or, y equals 11x-- 00:04:42.370 --> 00:04:45.480 well, we get minus 22 plus 6-- 00:04:45.480 --> 00:04:48.050 is minus 16. 00:04:48.050 --> 00:04:50.230 So either of those is a perfectly good 00:04:50.230 --> 00:04:51.720 answer to the question. 00:04:51.720 --> 00:04:53.410 So that's that.