WEBVTT 00:00:07.230 --> 00:00:07.690 Hi. 00:00:07.690 --> 00:00:09.200 Welcome to recitation. 00:00:09.200 --> 00:00:13.240 Last time in lecture we graphed some trigonometric 00:00:13.240 --> 00:00:15.560 functions and some inverse trigonometric functions. 00:00:15.560 --> 00:00:18.570 And there was a slight error in one of the graphs that 00:00:18.570 --> 00:00:19.650 Professor Jerison did. 00:00:19.650 --> 00:00:22.280 So I just wanted to talk a little bit about it and about 00:00:22.280 --> 00:00:25.590 what the problem was and with the correction is. 00:00:25.590 --> 00:00:28.960 So the function in question is the arctangent 00:00:28.960 --> 00:00:29.950 or the inverse tangent. 00:00:29.950 --> 00:00:33.610 And so I like to write arctan, where Professor Jerison 00:00:33.610 --> 00:00:35.870 usually writes tan to the minus 1. 00:00:35.870 --> 00:00:38.070 But they just meet, they're just two different names for 00:00:38.070 --> 00:00:39.140 the same function. 00:00:39.140 --> 00:00:41.010 So the inverse function of tangent. 00:00:41.010 --> 00:00:43.400 Now I've got a graph set up here. 00:00:43.400 --> 00:00:47.230 And what I've graphed are the lines, y equals x-- 00:00:47.230 --> 00:00:50.900 that's this diagonal line-- and the graph y 00:00:50.900 --> 00:00:53.490 equals tangent of x-- 00:00:53.490 --> 00:00:54.830 so that's this curve-- 00:00:54.830 --> 00:01:00.240 and here I've got one of the asymptotes of y equals tangent 00:01:00.240 --> 00:01:01.700 x at pi over 2. 00:01:01.700 --> 00:01:02.120 Right? 00:01:02.120 --> 00:01:06.300 So as x approaches pi over 2 from the right, tangent of x 00:01:06.300 --> 00:01:08.680 shoots off to infinity getting closer and 00:01:08.680 --> 00:01:10.170 closer to this line. 00:01:10.170 --> 00:01:12.880 And, you know, it does something similar down here. 00:01:12.880 --> 00:01:15.370 And then of course, it's a periodic function so there are 00:01:15.370 --> 00:01:16.890 many copies of this. 00:01:16.890 --> 00:01:22.180 So one thing to notice about this is that the 00:01:22.180 --> 00:01:24.680 tangent comes in here. 00:01:24.680 --> 00:01:29.820 The graph y equals tan x comes in and it is tangent to the 00:01:29.820 --> 00:01:32.250 line y equals x at the origin. 00:01:32.250 --> 00:01:36.990 So the slope of tan x is just its derivative. 00:01:36.990 --> 00:01:43.350 So we saw in an earlier recitation that d over dx of 00:01:43.350 --> 00:01:48.650 tan x is equal to secant squared of x. 00:01:48.650 --> 00:01:53.800 And so the derivative at 0 is secant squared of 0, which is 00:01:53.800 --> 00:01:57.260 1/1 squared, which is just 1. 00:01:57.260 --> 00:01:58.400 So the slope is 1. 00:01:58.400 --> 00:02:02.300 And in fact, another, a stronger thing is true, which 00:02:02.300 --> 00:02:08.070 is that for positive x, tangent of x is larger than x. 00:02:08.070 --> 00:02:09.390 So this falls away. 00:02:09.390 --> 00:02:12.110 So you can figure that out, for example by looking at the 00:02:12.110 --> 00:02:16.330 difference and higher derivatives if you wanted to. 00:02:16.330 --> 00:02:20.260 So the result of this, is that the graph of the arctangent, 00:02:20.260 --> 00:02:22.890 that is what you get when you reflect this graph across the 00:02:22.890 --> 00:02:27.680 line y equals x, and because of the way these graphs, 00:02:27.680 --> 00:02:29.940 because of this property that this graph has, that it lies 00:02:29.940 --> 00:02:33.620 above the line y equals x for positive x, when you reflect 00:02:33.620 --> 00:02:37.500 it what you get is something that lies just below the line 00:02:37.500 --> 00:02:38.120 y equals x. 00:02:38.120 --> 00:02:42.470 When you reflect this whole picture, that the piece, this 00:02:42.470 --> 00:02:46.450 piece gets reflected and comes entirely on the other 00:02:46.450 --> 00:02:49.560 side of that line. 00:02:49.560 --> 00:02:56.470 So the height here will be pi over 2. 00:02:56.470 --> 00:03:00.086 That'll be the horizontal asymptote. 00:03:00.086 --> 00:03:01.336 And it'll come below-- 00:03:06.350 --> 00:03:12.470 so this is y equals arctan x. 00:03:12.470 --> 00:03:14.110 So it will come below that line. 00:03:14.110 --> 00:03:18.840 And similarly, over here it'll come, it'll be the reflection, 00:03:18.840 --> 00:03:21.380 so it'll come above that line. 00:03:21.380 --> 00:03:30.400 And again is has an asymptote, horizontal asymptote at 00:03:30.400 --> 00:03:32.000 minus pi over 2. 00:03:32.000 --> 00:03:35.050 So the one feature I want to point out is specifically 00:03:35.050 --> 00:03:37.770 these two curves only intersect at the origin. 00:03:37.770 --> 00:03:41.840 So in the graph Professor Jerison showed you, they 00:03:41.840 --> 00:03:44.370 looked more like square root of x and x squared, which have 00:03:44.370 --> 00:03:45.920 a later intersection point. 00:03:45.920 --> 00:03:50.420 But here, for x bigger than 0, y equals tan x is always 00:03:50.420 --> 00:03:53.950 bigger than x, which is always bigger than y equals arctan x. 00:03:53.950 --> 00:03:56.870 And then they come in and right at the origin, their 00:03:56.870 --> 00:04:00.900 tangent to each other, they both have derivative 1 here. 00:04:00.900 --> 00:04:04.760 And then for negative x, and then they cross. 00:04:04.760 --> 00:04:09.255 And so arctan x is larger than x is larger than tan x when x 00:04:09.255 --> 00:04:11.190 is less than 0. 00:04:11.190 --> 00:04:14.740 So that was all I wanted to share with you, just this 00:04:14.740 --> 00:04:19.280 slightly cleaner picture of the arctan of x that I get by 00:04:19.280 --> 00:04:21.600 being able to put it up on the board ahead of time. 00:04:21.600 --> 00:04:23.360 So that's that.