WEBVTT 00:00:05.100 --> 00:00:11.260 Okay, so today we're talking about, given a function, how do we graph a derivative? 00:00:13.940 --> 00:00:17.200 This is what we did last class, on Wednesday. 00:00:18.400 --> 00:00:22.820 We're given a function, and we found the derivative. 00:00:23.700 --> 00:00:24.860 And how do we find the derivative? 00:00:27.220 --> 00:00:32.080 by the definition, right, which is the limit as h goes to zero. 00:00:34.320 --> 00:00:44.592 I worked this one out for us and it ends up with this, okay, which is another 00:00:44.592 --> 00:00:45.800 function, okay. 00:00:45.920 --> 00:00:50.840 So this is a quadratic function. The derivative turned out to be a linear function. 00:00:51.660 --> 00:00:54.280 The derivative is the slope of the tangent line. 00:00:56.320 --> 00:00:57.280 We talked about this. 00:00:57.840 --> 00:01:01.120 A slope is a number, not a function. 00:01:02.120 --> 00:01:04.860 So how is this still the slope of the tangent line? 00:01:10.100 --> 00:01:11.700 It depends on the value of x. 00:01:11.880 --> 00:01:15.560 Every single x value, the function has a different slope. 00:01:16.040 --> 00:01:17.920 So you want any slope at any particular point. 00:01:17.920 --> 00:01:24.980 you plug in an x value okay so the derivative is a whole nother function 00:01:26.760 --> 00:01:31.474 This was a quadratic, and the derivative happens to be linear. Given a graph, we 00:01:31.474 --> 00:01:33.360 will attempt to graph its derivative. 00:01:33.360 --> 00:01:37.980 keep giving a graph of some function and we're gonna attempt to graph the 00:01:37.980 --> 00:01:45.840 derivative so here are those two functions the quadratic and then the 00:01:45.840 --> 00:01:56.191 derivatives in red. How are we gonna be given this and come up with this? Why don't 00:01:56.191 --> 00:01:56.941 you 00:01:57.030 --> 00:02:00.090 We always start with the easiest value we know. 00:02:00.090 --> 00:02:00.990 Zero. 00:02:01.210 --> 00:02:03.310 And that's when the slope of the tangent line is zero. 00:02:04.310 --> 00:02:05.210 Right here. 00:02:06.370 --> 00:02:09.810 Anytime you have a minimum or a maximum, when it's a smooth, 00:02:10.810 --> 00:02:14.530 OK, the tangent line is horizontal, 00:02:15.530 --> 00:02:17.250 and the derivative is zero. 00:02:18.790 --> 00:02:19.910 So we're going to start with that. 00:02:21.370 --> 00:02:25.930 And then we find the zeros of the derivative. 00:02:26.990 --> 00:02:29.130 And then you go to the right and the left. 00:02:29.130 --> 00:02:32.170 Soon as we go to the right is it a positive slope or negative slope? 00:02:33.530 --> 00:02:34.430 Positive 00:02:35.330 --> 00:02:38.650 It's just gonna keep getting larger and larger and larger and larger 00:02:39.570 --> 00:02:44.383 So I know this is to the right of whatever number. This is one and a half or 00:02:44.383 --> 00:02:45.133 something 00:02:45.450 --> 00:02:46.750 It's gonna go up 00:02:48.090 --> 00:02:50.010 It's not obvious. It's a straight line 00:02:51.470 --> 00:02:54.845 And for right now it really doesn't matter we know it's positive just keep getting 00:02:54.845 --> 00:02:55.595 larger 00:02:55.610 --> 00:02:59.771 And then we go to the left of that zero or where the slope of the tangent line is 00:02:59.771 --> 00:03:00.521 zero. 00:03:01.090 --> 00:03:03.010 Look, is that a positive slope or negative? 00:03:03.630 --> 00:03:05.810 It's negative, it just keeps getting steeper and steeper. 00:03:07.050 --> 00:03:10.970 And so I know our derivative is a negative number down here, just keep it steeper. 00:03:11.620 --> 00:03:17.816 If we wanted to we could estimate the values of the slope plug them in here and then 00:03:17.816 --> 00:03:18.160 plot 00:03:18.160 --> 00:03:19.960 it but we almost never do that. 00:03:20.820 --> 00:03:26.411 Okay so we just first figure out where the derivative is zero how many times is the 00:03:26.411 --> 00:03:26.740 derivative 00:03:26.740 --> 00:03:27.640 zero? 00:03:30.700 --> 00:03:37.764 So when the has a horizontal tangent line we got one right here one right here and 00:03:37.764 --> 00:03:38.180 one 00:03:38.180 --> 00:03:44.160 There are three horizontal tangents. Wherever we have a horizontal tangent line, the 00:03:44.160 --> 00:03:45.540 derivative is zero. 00:03:45.540 --> 00:03:50.880 zero, the derivative is zero, and the derivative is zero. 00:04:00.900 --> 00:04:06.408 And then between these two zeros of the derivative, just you can put a pencil or you 00:04:06.408 --> 00:04:07.380 can do whatever, 00:04:08.180 --> 00:04:11.380 The tangent line negative or positive 00:04:12.140 --> 00:04:16.660 The slope of the tangent line is negative throughout this interval. 00:04:17.480 --> 00:04:22.501 Here the slope is most negative. Then it is still negative, but becomes less steep 00:04:22.501 --> 00:04:23.840 as it approaches zero. 00:04:24.620 --> 00:04:26.480 so wherever the 00:04:27.740 --> 00:04:29.820 Slope is the steepest 00:04:29.820 --> 00:04:31.800 that's going to be our 00:04:31.800 --> 00:04:34.060 least slope here and 00:04:35.340 --> 00:04:37.720 Just put it some it doesn't matter how down 00:04:38.180 --> 00:04:42.540 If you want to estimate that, I don't know, it's about a slope of negative 2. 00:04:42.780 --> 00:04:46.260 Maybe I should have put it down here at negative 2. 00:04:46.860 --> 00:04:47.760 It doesn't really matter. 00:04:48.600 --> 00:04:50.180 And then we just make a nice smooth curve. 00:05:00.380 --> 00:05:03.220 And then between this 0 and this 0, 00:05:05.140 --> 00:05:08.720 Is the slope of the tangent positive or negative? 00:05:09.740 --> 00:05:10.640 Positive. 00:05:10.840 --> 00:05:12.020 All the way to the next one. 00:05:13.400 --> 00:05:17.700 And somewhere right here is the steepest. 00:05:19.860 --> 00:05:22.140 So I'm going to put some point about here. 00:05:37.020 --> 00:05:42.196 And notice I didn't draw it very high because this is not a very steep value, 00:05:42.196 --> 00:05:42.520 especially 00:05:42.520 --> 00:05:43.580 compared to this. 00:05:45.560 --> 00:05:51.220 And now to the right of that zero the derivative that that local maximum 00:05:51.740 --> 00:05:54.040 Slope is positive negative 00:05:54.860 --> 00:05:57.860 Negative and just keep getting more and more and more and more negative 00:05:59.420 --> 00:06:00.860 So we just 00:06:04.300 --> 00:06:08.060 Put some arrow and then to the left of this 00:06:09.260 --> 00:06:10.160 zero 00:06:10.600 --> 00:06:12.380 So for the tangent line positive, 00:06:12.420 --> 00:06:14.700 The slope of the tangent line is positive, and it keeps getting larger. 00:06:15.480 --> 00:06:19.060 So it just keeps going up and up and up and up and up. 00:06:27.700 --> 00:06:28.780 And that's it. 00:06:28.860 --> 00:06:32.300 That's our best guess as to the graph of the derivative. 00:06:32.710 --> 00:06:40.677 Assuming this original function is a polynomial, what's the least degree this could 00:06:40.677 --> 00:06:41.427 be? 00:06:44.710 --> 00:06:45.850 First how about this? 00:06:46.490 --> 00:06:50.830 Is the degree of this polynomial is an even degree or an odd degree? 00:06:51.250 --> 00:06:52.150 Even on okay now we most of us agree. Why is he even? 00:06:53.590 --> 00:06:55.090 The end behavior 00:06:56.210 --> 00:06:58.490 Both going down to negative infinity. 00:06:59.630 --> 00:07:03.310 Okay. It's some even degree. 00:07:03.590 --> 00:07:06.330 Either x squared, x to the fourth, x to the sixth, x to the eighth. 00:07:07.010 --> 00:07:09.650 So we look at the end behavior. Okay. 00:07:10.330 --> 00:07:15.090 And then we have one, two, three extrema or turning points. 00:07:16.910 --> 00:07:19.290 Where it changes from increasing to decreasing. 00:07:19.990 --> 00:07:22.050 So what does it have to at least be? 00:07:22.050 --> 00:07:24.690 Oh, could it be quadratic? 00:07:26.350 --> 00:07:30.990 No, that's what we know a quadratic look like yeah only has one turning point 00:07:32.290 --> 00:07:34.110 This has three 00:07:36.010 --> 00:07:40.150 Extrema a local min or max so it has to be at least a fourth degrees 00:07:41.210 --> 00:07:45.990 We're good that it could be a sixth degree. We can't tell could be an eighth degree 00:07:45.990 --> 00:07:49.710 but actually at least a fourth degree polynomial. 00:07:50.710 --> 00:07:53.150 And what about the derivative, the Green function? 00:07:54.290 --> 00:07:56.390 Is it an odd degree or even degree? 00:07:57.410 --> 00:07:58.310 Odd. 00:08:00.150 --> 00:08:01.050 It's definitely not linear. 00:08:02.490 --> 00:08:03.590 Could it be cubic? 00:08:06.070 --> 00:08:07.850 It could be cubic. 00:08:08.840 --> 00:08:11.920 It has one, two turning points or two little, 00:08:12.340 --> 00:08:14.320 it could be a cubic, okay? 00:08:15.340 --> 00:08:17.440 And this is always gonna happen with polynomials. 00:08:18.700 --> 00:08:20.320 Whatever the degree of the polynomials, 00:08:20.600 --> 00:08:23.160 the derivative is gonna be one degree less. 00:08:24.180 --> 00:08:26.040 And we'll learn exactly why later. 00:08:26.690 --> 00:08:29.610 So first we figure out where the slope of the tangent line is zero. 00:08:30.130 --> 00:08:36.910 horizontal tangent lines which obviously should be here we put a zero 00:08:37.630 --> 00:08:45.250 we got one and to the right is it negative or positive negative okay it 00:08:45.250 --> 00:08:52.170 just keeps getting more and more and more and more negative and to the left 00:08:52.790 --> 00:08:55.830 positive, steep is getting steeper and steeper. 00:08:56.670 --> 00:08:58.070 It's going to look like this. 00:08:59.490 --> 00:09:05.770 This is a quadratic, 00:09:06.950 --> 00:09:08.470 or could it be a fourth degree problem? 00:09:08.950 --> 00:09:12.841 The derivative of a quadratic is a linear function, so that is why I drew a straight 00:09:12.841 --> 00:09:13.591 line. 00:09:14.710 --> 00:09:15.550 Okay. 00:09:15.550 --> 00:09:17.070 Should have drawn it something like that? 00:09:17.990 --> 00:09:18.790 Yes. 00:09:18.790 --> 00:09:20.630 What type of function is this? 00:09:23.010 --> 00:09:24.370 Definitely not. 00:09:24.770 --> 00:09:25.670 Why not? 00:09:28.810 --> 00:09:31.230 Do polynomials have asymptotes? 00:09:32.450 --> 00:09:33.350 No. 00:09:33.430 --> 00:09:34.330 Okay. 00:09:34.670 --> 00:09:35.890 Polynomials are continuous. 00:09:36.230 --> 00:09:38.390 It's definitely not a polynomial. 00:09:38.610 --> 00:09:39.930 What type of function is this? 00:09:40.930 --> 00:09:45.330 Rational, which is the ratio of two polynomials. 00:09:47.950 --> 00:09:52.410 Are there any parts where the derivative is zero here? 00:09:54.630 --> 00:09:55.530 No. 00:09:56.250 --> 00:09:57.710 So we can't start by doing that. 00:09:58.850 --> 00:10:01.590 So now we just start some other slide. 00:10:02.090 --> 00:10:02.150 Just pick a value. 00:10:02.150 --> 00:10:03.140 This would be the tangent line. 00:10:05.040 --> 00:10:07.620 What's approximately the slope of that line? 00:10:09.260 --> 00:10:10.740 It's about one. 00:10:11.380 --> 00:10:18.440 Right at 45 degree angle has a slope of one. 00:10:19.680 --> 00:10:20.580 Okay. 00:10:20.980 --> 00:10:22.520 So I'm putting it right there at one. 00:10:24.840 --> 00:10:30.120 And then when we go to the right, does it have a positive slope or a negative slope? 00:10:31.900 --> 00:10:36.940 The closer we get to that asymptote, the steeper the slope becomes. 00:10:38.720 --> 00:10:43.160 So in fact that's going to be the derivative. 00:10:43.160 --> 00:10:51.700 and then the farther we go to negative infinity that closer the derivative gets 00:10:51.700 --> 00:11:09.567 The derivative approaches zero farther to the left. It is not the same function as 00:11:09.567 --> 00:11:13.140 the original graph. 00:11:13.140 --> 00:11:16.800 And then right here, what's approximately the slope of that line? 00:11:18.300 --> 00:11:19.720 One, positive one. 00:11:21.200 --> 00:11:23.240 So we need a dot up here at one. 00:11:28.240 --> 00:11:35.694 And then when we go to the right, it's still positive but just keeps getting flatter 00:11:35.694 --> 00:11:36.160 and 00:11:36.160 --> 00:11:37.440 flatter and flatter. 00:11:39.880 --> 00:11:43.280 So positive, but gets very, very, very close to zero. 00:11:49.040 --> 00:11:54.700 And when we go to the left of this x value of two or so, it's positive, but it keeps 00:11:54.700 --> 00:11:58.540 getting steeper and steeper and steeper and steeper and steeper and steeper. 00:12:00.220 --> 00:12:03.220 So this goes up, up, up, up, up, up. 00:12:05.000 --> 00:12:05.900 OK. 00:12:10.160 --> 00:12:12.780 This is also a rational function. 00:12:13.920 --> 00:12:15.620 The derivative of rational functions 00:12:15.620 --> 00:12:17.800 are other rational functions. 00:12:25.120 --> 00:12:27.180 And we'll see why that is later. 00:12:28.320 --> 00:12:29.220 We do the derivative. 00:12:29.980 --> 00:12:31.160 OK, questions on this? 00:12:31.740 --> 00:12:38.100 We go to the right of the zero it's the steepest right, I don't know about here 00:12:39.920 --> 00:12:41.820 Okay right here is the steepest 00:12:43.200 --> 00:12:46.240 So that's gonna be our it's gonna go up to here 00:12:47.840 --> 00:12:53.045 And then the closer we get whatever this is x equals 5 it starts to level off get 00:12:53.045 --> 00:12:54.780 close to zero it's never zero 00:13:00.660 --> 00:13:01.100 And 00:13:01.100 --> 00:13:06.360 And then what's the slope of the tangent line right here at the end point? 00:13:09.060 --> 00:13:10.020 Very important. 00:13:11.060 --> 00:13:12.260 It does not exist. 00:13:14.580 --> 00:13:16.640 Derivatives do not exist at end points. 00:13:16.840 --> 00:13:17.740 Why is that? 00:13:19.060 --> 00:13:21.220 Let's call this number 5. 00:13:23.620 --> 00:13:26.220 The limit as h approaches... 00:13:29.400 --> 00:13:30.940 Let's go to the other definition. 00:13:32.580 --> 00:13:39.940 Approaches 5, f of x minus f of 5, over x minus 5. 00:13:44.900 --> 00:13:47.220 Why does this limit not exist? 00:13:56.260 --> 00:14:00.880 The limit as x approaches five from the left does exist. 00:14:03.600 --> 00:14:08.020 Okay but the limit as x approaches five from the right. 00:14:19.260 --> 00:14:23.860 Okay, this does not exist there are no values to the right of five 00:14:26.380 --> 00:14:32.912 Okay, so that's why the limit as X approaches five does not exist endpoints are not 00:14:32.912 --> 00:14:33.662 differentiable 00:14:35.560 --> 00:14:40.260 Very important so it's same thing on the left here. This is positive gets steeper 00:14:42.140 --> 00:14:45.440 And then we got to put an open circle on the end point 00:14:46.040 --> 00:14:55.758 Problem 42 has one horizontal tangent. To the right, the slope is always negative 00:14:55.758 --> 00:14:57.840 and becomes steeper. 00:14:57.840 --> 00:15:10.571 is always negative and just keeps getting steeper. And again if the function ends, 00:15:10.571 --> 00:15:11.480 there's 00:15:11.860 --> 00:15:24.100 there it's not differentiable somewhere right here is the steepest 00:15:28.540 --> 00:15:36.400 so it's gonna go up and then get closer and closer and closer to zero okay do 43 00:15:38.630 --> 00:15:40.990 Shouldn't get something like that. 00:15:46.590 --> 00:15:54.710 Okay, if you're not understanding where I, or how I do this, please ask. 00:15:56.650 --> 00:15:59.310 And again, where I put the steepest point doesn't really matter. 00:15:59.870 --> 00:16:01.850 I could have drawn it way up here 00:16:03.430 --> 00:16:07.510 We can't really tell because we don't really have numbers to deal with 00:16:07.980 --> 00:16:12.760 Okay, we'll talk about this x point in a second. 00:16:13.640 --> 00:16:18.140 We definitely have a zero here, a zero here, and a zero here. 00:16:18.680 --> 00:16:22.840 When we go to the right of this point, where's the steepest point? 00:16:25.500 --> 00:16:29.280 It just keeps getting steeper and steeper and steeper and steeper and steeper until 00:16:29.280 --> 00:16:30.660 we hit that sharp point. 00:16:34.080 --> 00:16:36.000 So it's just going to get steeper. 00:16:38.400 --> 00:16:40.560 Okay I could have drawn it there or I could have drawn it here. 00:16:40.740 --> 00:16:41.640 It doesn't really matter. 00:16:41.980 --> 00:16:46.000 Just to the right of that sharp point is the slope positive or negative? 00:16:47.320 --> 00:16:48.180 Negative. 00:16:48.180 --> 00:16:52.400 And in fact every single point on that line has the same slope. 00:16:53.080 --> 00:16:56.380 So we are going to draw a negative, some negative value here. 00:16:57.500 --> 00:16:59.240 And it is a horizontal line. 00:16:59.240 --> 00:17:12.100 it's always the same value let's say this is at 7 at x equals 7 that sharp 00:17:12.100 --> 00:17:18.740 point what's the derivative okay definitely does not exist and why is 00:17:18.740 --> 00:17:29.160 that okay it's definitely a cusp but what makes it not differentiable that we 00:17:29.160 --> 00:17:36.380 We need an open circle on each side. Why is the derivative undefined? 00:17:38.180 --> 00:17:45.780 yes this is exactly right the derp the definition is the limit as let's do the 00:17:45.780 --> 00:17:53.040 other definition X approaches 7 of the difference formula or the slope of the 00:17:53.040 --> 00:17:53.940 secant line. 00:18:00.260 --> 00:18:05.941 And the reason it's not differentiable, the limit from the left is going to equal 00:18:05.941 --> 00:18:06.320 whatever 00:18:06.320 --> 00:18:07.280 this number is. 00:18:08.180 --> 00:18:10.060 Let us say the left-hand slope is 8; these are illustrative values. 00:18:10.060 --> 00:18:10.960 Let's make that. 00:18:12.880 --> 00:18:17.600 The limit as X approaches seven from the right 00:18:23.220 --> 00:18:25.380 is gonna be whatever this number is. 00:18:26.200 --> 00:18:27.960 I'm gonna just call it negative two. 00:18:31.100 --> 00:18:34.500 Therefore the limit as X approaches seven. 00:18:40.800 --> 00:18:45.220 It does not exist because the limit from the left and the limit from the right not 00:18:45.220 --> 00:18:46.780 of the function but of the 00:18:50.380 --> 00:18:53.240 We call this the difference quotient 00:18:53.240 --> 00:18:55.380 We call this the difference quotient, or the slope of the secant line. 00:18:55.880 --> 00:19:01.499 Are different so it does not exist. So the reason sharp points are not 00:19:01.499 --> 00:19:02.249 differentiable 00:19:03.400 --> 00:19:06.687 It's by the definition of the derivative the derivative or the limit for the left 00:19:06.687 --> 00:19:08.440 and the right are not the same therefore 00:19:08.440 --> 00:19:18.820 the limit doesn't exist therefore there is no derivative there and it will be 00:19:18.820 --> 00:19:19.430 important to state that later on. Okay question on that problem is that the 00:19:19.430 --> 00:19:21.730 What function is this? 00:19:21.730 --> 00:19:23.350 Is it a cosine? 00:19:24.130 --> 00:19:25.550 Looks just like cosine. 00:19:27.310 --> 00:19:30.050 In fact I believe it is cosine. 00:19:32.470 --> 00:19:33.590 That's right at 2 pi. 00:19:35.310 --> 00:19:37.210 Does it have any horizontal tangent lines? 00:19:39.070 --> 00:19:40.310 Yeah we got lots of them. 00:19:44.410 --> 00:19:46.370 Right here, right here, right here. 00:19:47.290 --> 00:19:49.210 So the derivatives are going to have lots of zeros. 00:19:49.210 --> 00:19:59.450 one here, one here at pi, at 2 pi, 3 pi, negative pi, negative 2 pi. 00:20:04.210 --> 00:20:08.070 Between zero and pi, where is it the steepest? 00:20:11.710 --> 00:20:14.650 Exactly here, at pi over 2. 00:20:16.170 --> 00:20:21.850 In fact, I gave you grid marks. What is the slope of that line? 00:20:27.230 --> 00:20:29.510 Plus one. Negative one. 00:20:30.230 --> 00:20:35.410 It goes down one over one. It's exactly negative one. 00:20:36.570 --> 00:20:39.630 So we go to negative one, which is right here. 00:20:45.070 --> 00:20:56.350 okay and a nice smooth point and then between pi and 2 pi the tangent line the 00:20:56.640 --> 00:21:00.200 The derivative looks a lot like a trig function as well. 00:21:01.020 --> 00:21:02.520 If it were, what would it be? 00:21:03.160 --> 00:21:04.340 It's definitely not sine. 00:21:06.220 --> 00:21:10.180 Sine starts at zero and goes up to one at pi over two. 00:21:11.460 --> 00:21:18.580 But this is the reflection of sine, so it would be negative sine. 00:21:30.040 --> 00:21:34.313 In fact, it turns out, we're going to show this later on, the derivative of cosine 00:21:34.313 --> 00:21:34.740 is 00:21:34.740 --> 00:21:36.240 Exactly negative sine. 00:21:36.870 --> 00:21:39.730 What type of function is this? 00:21:41.370 --> 00:21:42.690 Looks like an exponential function. 00:21:46.230 --> 00:21:48.890 What is the base of the exponential function? 00:21:50.950 --> 00:21:55.010 How do we look at a graph and figure out what the base is? 00:21:56.030 --> 00:21:58.490 X equals 1, what is the value? 00:22:04.150 --> 00:22:08.590 X equals 1, it's right here. 00:22:11.310 --> 00:22:15.950 It's not obvious, but this turns out to be E. What is E? 00:22:16.230 --> 00:22:21.170 This is our function, e to the x. 00:22:29.670 --> 00:22:31.990 It has no minimums or maximums. 00:22:33.210 --> 00:22:35.210 This tangent line will never be zero. 00:22:35.210 --> 00:22:37.430 So we just got to go to some other point. 00:22:39.890 --> 00:22:43.610 Let's pick a point, let's say, right here. 00:22:44.670 --> 00:22:47.450 What is the slope of this line? 00:22:50.770 --> 00:22:52.450 OK, it looks like it's one. 00:22:52.550 --> 00:22:54.110 In fact, it is exactly one. 00:22:55.270 --> 00:22:57.710 I'll put a dot right there at 0 comma 1. 00:23:00.290 --> 00:23:00.810 OK. 00:23:00.810 --> 00:23:04.810 And then to the right here, it's definitely positive. 00:23:06.270 --> 00:23:09.690 And it just keeps getting steeper and steeper and steeper. 00:23:11.350 --> 00:23:22.170 Okay, and this is not exactly obvious here, but that is the derivative to the right. 00:23:23.950 --> 00:23:28.169 And to the left is still positive but keeps getting flatter and flatter and flatter 00:23:28.169 --> 00:23:28.919 and 00:23:29.330 --> 00:23:30.230 flatter. 00:23:33.790 --> 00:23:34.690 Ok. 00:23:36.910 --> 00:23:44.830 And for this function it's not obvious but it turns out the derivative of e to the x 00:23:45.590 --> 00:23:47.750 is exactly the same function. 00:23:47.950 --> 00:23:49.110 It is e to the x. 00:23:49.830 --> 00:23:54.470 Which is kind of crazy and we'll prove that later for you 00:23:59.950 --> 00:24:02.210 Super cool a 00:24:02.210 --> 00:24:06.110 Derivatives of exponential functions are exponential functions as well. 00:24:06.710 --> 00:24:08.930 Sometimes it's exactly the same 00:24:08.930 --> 00:24:13.480 The other times it'll be just a slight coefficient different have a different 00:24:13.480 --> 00:24:15.930 coefficient. It'll still be an exponential function 00:24:16.490 --> 00:24:18.390 Let's assume this goes up forever. 00:24:21.610 --> 00:24:23.330 What type of function is this? 00:24:27.050 --> 00:24:30.230 Or if I gave you an equation for this... 00:24:31.610 --> 00:24:34.330 We have to be some piecewise function. 00:24:35.530 --> 00:24:37.330 OK, that's three different pieces. 00:24:41.910 --> 00:24:45.110 And we'll talk about what the actual function is in a minute. 00:24:47.450 --> 00:24:50.030 It does have a horizontal tangent line right here. 00:24:51.590 --> 00:24:54.590 So the derivative has a 0 right there. 00:24:56.130 --> 00:24:57.410 The derivative is 0 there. 00:24:58.850 --> 00:25:00.250 And then to the right here 00:25:00.990 --> 00:25:03.130 Does have a positive negative slope? 00:25:04.130 --> 00:25:08.668 negative just keeps getting more and more and more negative and then what happens at 00:25:08.668 --> 00:25:09.418 the 00:25:11.610 --> 00:25:17.530 What's the derivative of that sharp point does not exist so 00:25:20.710 --> 00:25:25.330 It's more and more negative we definitely need to put a hole right there 00:25:28.990 --> 00:25:33.510 And instantly to the right, does it have a positive slope or negative slope? 00:25:35.110 --> 00:25:39.067 It's not obvious, but this is getting larger, just barely larger and larger and 00:25:39.067 --> 00:25:39.817 larger. 00:25:40.070 --> 00:25:44.430 So from here, it doesn't exist. 00:25:50.330 --> 00:25:52.410 It looks something like that. 00:25:56.150 --> 00:26:02.470 To the left here, the positive slope, it keeps me steeper and steeper and steeper. 00:26:06.970 --> 00:26:12.350 And then right here, whatever value that is, doesn't exist. 00:26:16.570 --> 00:26:19.570 And then instantly has negative slope 00:26:33.220 --> 00:26:38.100 Derivatives of piecewise functions are also described piecewise. 00:26:41.420 --> 00:26:50.100 In fact, this function, the equation is the absolute value of some quadratic. 00:26:50.360 --> 00:26:52.620 I don't know exactly what quadratic it is. 00:26:52.620 --> 00:26:54.840 I'm just let me make something up. 00:26:55.340 --> 00:26:57.080 x squared minus 00:27:00.300 --> 00:27:01.200 plus 00:27:03.940 --> 00:27:05.940 three x minus 00:27:11.960 --> 00:27:14.220 16 or something like that. 00:27:18.020 --> 00:27:21.600 Anytime you see an absolute value, it's a piecewise function. 00:27:24.180 --> 00:27:26.480 This actually was some quadratic. 00:27:28.440 --> 00:27:34.020 If it was without the absolute value, it'd be exactly this function. 00:27:35.920 --> 00:27:40.500 But because of the absolute value, it flipped this to this. 00:27:41.740 --> 00:27:46.240 It's kind of interesting. 00:27:50.960 --> 00:27:53.980 And again, why is it not differentiable there? 00:27:54.960 --> 00:27:57.520 Let's say this is at 4.2 or something. 00:27:57.520 --> 00:28:02.400 The left- and right-hand limits of the difference quotient are not equal. 00:28:02.670 --> 00:28:04.610 Okay, this is some piecewise function. 00:28:05.470 --> 00:28:09.595 It does have extreme, there's a maximum here and a maximum there and a minimum 00:28:09.595 --> 00:28:10.345 there. 00:28:12.750 --> 00:28:14.330 Is the derivative zero there? 00:28:15.830 --> 00:28:17.070 Definitely not, it's sharp. 00:28:18.070 --> 00:28:20.030 Okay, sharp points are not differentiable. 00:28:20.750 --> 00:28:23.710 The limit from the left of the difference quotient limit from the right are not the 00:28:23.710 --> 00:28:24.550 same. 00:28:24.550 --> 00:28:25.810 So let's just choose some other point. 00:28:26.710 --> 00:28:30.350 Let's say, I don't know, right here. 00:28:31.870 --> 00:28:34.450 What is the slope of that tangent line there? 00:28:37.270 --> 00:28:39.190 It's whatever the slope of that line is. 00:28:39.270 --> 00:28:40.370 What is the slope of that line? 00:28:41.870 --> 00:28:42.770 One. 00:28:43.570 --> 00:28:45.090 Okay, in fact it's right there. 00:28:45.610 --> 00:28:49.070 Everywhere on this point has the same exact slope of the tangent line. 00:28:49.570 --> 00:28:52.670 The slope of a tangent line of a line is always the slope of the line. 00:28:52.670 --> 00:28:57.710 So everywhere here, it's going to have a slope of one 00:28:59.670 --> 00:29:03.550 Sharp points are not differentiable open circle 00:29:08.210 --> 00:29:11.850 So that's this piece here the slope of this line is one 00:29:15.490 --> 00:29:19.410 Between zero and five the slope of that line is negative one 00:29:19.520 --> 00:29:23.660 So it's gonna be like this. 00:29:27.280 --> 00:29:29.100 And one again. 00:29:31.790 --> 00:29:34.950 This is y equals the cube root of x. 00:29:40.910 --> 00:29:44.010 It's the continuous function everywhere. 00:29:45.610 --> 00:29:46.890 Domain's all real numbers. 00:29:53.150 --> 00:29:55.270 It has no minimum or maximum, right? 00:29:56.090 --> 00:29:58.150 So the derivative is going to have no zeros. 00:29:58.610 --> 00:30:03.550 I don't know, somewhere right about here, the slope of the tangent line is one. 00:30:07.450 --> 00:30:08.850 So I'm going to put a dot here. 00:30:09.370 --> 00:30:13.477 Ok, we have a positive slope but it just keeps decreasing and decreasing and 00:30:13.477 --> 00:30:14.227 decreasing. 00:30:15.290 --> 00:30:17.350 So it's going to look something like this. 00:30:19.470 --> 00:30:21.230 And have a horizontal asymptote. 00:30:24.590 --> 00:30:26.290 What happens when I get closer to zero? 00:30:28.230 --> 00:30:30.190 It's steeper and steeper and steeper. 00:30:30.190 --> 00:30:32.530 And right here at zero 00:30:33.830 --> 00:30:36.590 The tangent line is vertical 00:30:38.170 --> 00:30:41.210 What's the slope of vertical lines 00:30:51.030 --> 00:30:53.690 Vertical lines don't have slopes 00:30:54.990 --> 00:30:56.090 Okay like 00:30:57.330 --> 00:31:02.030 x equals 2 is a vertical line. It does not have a slope. 00:31:02.630 --> 00:31:04.730 Vertical lines do not have slopes. 00:31:06.070 --> 00:31:09.210 So right here the derivative does not exist 00:31:09.870 --> 00:31:11.970 because we have a vertical tangent line. 00:31:12.990 --> 00:31:14.410 And the closer you get to 0, 00:31:16.010 --> 00:31:18.510 This has a vertical asymptote, the derivative. 00:31:21.970 --> 00:31:28.370 And then just to the left of zero, it has a positive, very very very steep slope. 00:31:29.270 --> 00:31:31.830 It's going to look something like this. 00:31:31.830 --> 00:31:33.830 It has... 00:31:39.370 --> 00:31:45.455 At this vertical tangent, the derivative does not exist; its graph has a vertical 00:31:45.455 --> 00:31:46.205 asymptote. 00:31:46.710 --> 00:31:47.750 vertical asymptote 00:31:48.450 --> 00:31:51.390 Okay, it might go to positive, it might go to negative, they both... 00:31:51.930 --> 00:31:53.090 which is a semi-circle. 00:32:02.990 --> 00:32:04.930 Let's say this is the semi-circle. 00:32:06.310 --> 00:32:07.310 We definitely have a zero. 00:32:11.190 --> 00:32:12.870 And the slope here is negative. 00:32:14.710 --> 00:32:16.650 We're going to have a vertical asymptote here. 00:32:17.650 --> 00:32:18.610 It's going to look like this. 00:32:19.270 --> 00:32:23.936 Approaching the endpoint, the derivative does not exist at the endpoint, and the 00:32:23.936 --> 00:32:25.730 slopes become steeper and steeper. 00:32:25.730 --> 00:32:26.650 and steeper and steeper. 00:32:26.870 --> 00:32:28.930 It goes all the way to negative infinity. 00:32:30.210 --> 00:32:33.310 And this will go all the way to positive infinity. 00:32:33.850 --> 00:32:38.801 But, okay, yeah, so that makes sure when you draw the graph of a semicircle, the 00:32:38.801 --> 00:32:39.110 closer 00:32:39.110 --> 00:32:41.370 we get to this value, the steeper it gets. 00:32:41.470 --> 00:32:44.230 You're gonna have vertical asymptotes at those two points.