WEBVTT 00:00:05.500 --> 00:00:13.200 In topic seven we established a need to analyze further a function in order to be able to sketch the graph. 00:00:13.202 --> 00:00:19.100 If you have not already reviewed topic seven, you should do so before you review this topic. 00:00:19.102 --> 00:00:24.803 Let's take a look at the graph of the function f that we have here on the board. We're going to draw some 00:00:24.805 --> 00:00:31.501 generalizations from this graph and then hope this will help us sketch the graph of a function when all 00:00:31.503 --> 00:00:38.704 we have is the equation. We notice that there are some particular points here where something happens. 00:00:38.706 --> 00:00:47.603 So let's take a look at these points. First of all, you can see right about here something changes. And then 00:00:47.605 --> 00:00:56.305 right about here something changes, and then here we know that this, where the slope of the 00:00:56.307 --> 00:01:06.407 tangent line is 0, where the first derivative is 0, is going to be the point where the relative max occurs. 00:01:06.409 --> 00:01:12.207 Now let's label these points. I'm going to call this one x-sub-one, 00:01:12.209 --> 00:01:22.205 this one right in here, that one, x-sub-two, and this one x-sub-three. 00:01:22.205 --> 00:01:34.606 So I'm going to label this point (x-sub-one, f of x-sub-one). This is a lot of notation here. I'm going to 00:01:34.608 --> 00:01:50.903 label this point (x-sub-two, f of x-sub-two), and this one is (x-sub-three, f of x-sub-three), but we already 00:01:50.905 --> 00:01:55.700 know that it's a relative max so I'm not going to write a label for that one at this time. Alright, 00:01:55.702 --> 00:02:04.203 let's see if we can answer the questions. State the intervals where the function f of x is concave down. 00:02:04.205 --> 00:02:13.105 Let's talk about what concave down means. If you look at the shapes on this blouse, you can see 00:02:13.107 --> 00:02:21.004 that it looks like a piece of a parabola that opens downward. Anywhere on this shape, the graph 00:02:21.006 --> 00:02:27.405 of a function would be concave down. You don't have to have the whole parabola, just any piece. 00:02:27.407 --> 00:02:35.103 So let's look. From negative infinity to here, x-sub-one, this function is concave down. So we'll put that 00:02:35.105 --> 00:02:45.704 in here. Negative infinity to x-sub-one, and then the concavity changes here and it looks like a piece, 00:02:45.706 --> 00:02:53.302 just a piece, not the whole thing, but a piece of a parabola that opens upward. So this is called concave up. 00:02:53.304 --> 00:03:07.504 So from x-sub-1 to x-sub-2 the function is concave up. Ok and then again right here at this point the 00:03:07.506 --> 00:03:14.005 concavity changes from up, and then it changes to down again and here we get the entire shape. This is 00:03:14.007 --> 00:03:27.903 down from here all the way to here, so this would be concave down from x-sub-two on out to infinity. 00:03:27.905 --> 00:03:35.203 Now an inflection point is a point where the concavity changes. So let's go back and identify these points. 00:03:35.205 --> 00:03:47.201 The concavity changes here at (x-sub-one, f of x-sub-one), and we list this as a point, not just the y-value 00:03:47.203 --> 00:03:54.806 as we did with the relative max, because this is called an inflection point. There's another inflection point 00:03:54.808 --> 00:04:08.404 here at (x-sub-two, f of x-sub-two). So one more time, what did we learn from this graph? Well first of all, 00:04:08.406 --> 00:04:17.000 we learned how to read concavity, and second of all, we learned how to identify inflection points. Ok, it's 00:04:17.002 --> 00:04:26.703 calculus time. So what does that have to do with calculus? Well if you look, let's just take this interval here. 00:04:26.705 --> 00:04:38.103 If you look at the tangent line here, you can see that its [slope is] positive. Now if you look at the tangent line at 00:04:38.105 --> 00:04:45.900 another point, say further to the right, the tangent line's slope is still positive. Ok, that's fine because we know 00:04:45.902 --> 00:04:53.405 the function is increasing over this interval. But what else is happening is that the tangent line is... 00:04:53.407 --> 00:05:04.902 Here it maybe has a slope of oh let's say something like rise over run might be slope of 2 or 3, whereas 00:05:04.904 --> 00:05:15.000 here the slope of the tangent line, which is the first derivative, is smaller. So over this interval where 00:05:15.002 --> 00:05:24.601 the function is concave down, the change in the slope of the tangent line is a decreasing change. So what 00:05:24.603 --> 00:05:32.701 we realize here is that this represents the derivative of the tangent line, or the derivative of the derivative, 00:05:32.703 --> 00:05:39.604 which we call the second derivative or f double prime. Now one more time, let's look at why this 00:05:39.606 --> 00:05:45.206 is happening. Because the tangent line, the slope of the tangent line, which is the derivative, is changing. 00:05:45.208 --> 00:05:55.403 So here that means the second derivative is negative. It's getting smaller, negative, changing in a negative 00:05:55.405 --> 00:06:01.304 direction. Now let's look here where the function is concave up. What are the tangent lines looking like? 00:06:01.306 --> 00:06:09.403 Well, this is positive. Fine, because the slope is positive, the function is increasing. If you pick a tangent line 00:06:09.405 --> 00:06:17.902 further to the right, you can see that the tangent line here has a slope that is greater than the previous one. 00:06:17.904 --> 00:06:25.106 So when the function is concave up, the rate of change of the derivative is positive. So what does that mean? 00:06:25.108 --> 00:06:31.604 That means if the function is concave down over an interval then the second derivative is negative because 00:06:31.606 --> 00:06:37.405 the rate of change of the first derivative is the second derivative and then here where the function is 00:06:37.407 --> 00:06:45.906 concave up, the slopes of the tangent lines are increasing, and so the second derivative would be positive. 00:06:45.908 --> 00:06:54.304 Now let's look at this interval. It's even clearer here. Look at the tangent line. Look. This function is 00:06:54.306 --> 00:07:05.009 concave down and if you look at, it's a little harder to draw but you can see that the tangent lines start out 00:07:05.011 --> 00:07:14.606 here and the rate of change which is the, the rate of change of the derivative [which is the slope of] the 00:07:14.608 --> 00:07:20.802 tangent line and you can see that it's decreasing. It actually goes here from positive values to negative 00:07:20.804 --> 00:07:29.503 and that's certainly decreasing. So let's see if we can generalize this into some form that we can use. 00:07:29.505 --> 00:07:34.603 Alright, we're going to find partition numbers and critical values of the first derivative. 00:07:34.605 --> 00:07:40.201 That's going to help us discover where to partition the axis and look for possible inflection points. 00:07:40.203 --> 00:07:46.200 Inflection points will occur where the concavity changes. Where do we look? Well we actually look at 00:07:46.202 --> 00:07:51.204 the same places we looked when we did the first derivative test. We identify partition numbers which 00:07:51.206 --> 00:07:58.703 are going to be values of x where the function is discontinuous or the second derivative is 0, or the 00:07:58.705 --> 00:08:04.708 second derivative does not exist, because we want the rate of change of the first derivative which is the 00:08:04.710 --> 00:08:16.304 second derivative. So what happens at that point? Well, if we have a function f of x defined on an interval 00:08:16.306 --> 00:08:21.500 where x is on the interval, and the function of course has to be differentiable over this interval, then 00:08:21.502 --> 00:08:29.300 what happens? If the second derivative is positive, the function is concave up. So that's what happens 00:08:29.302 --> 00:08:36.408 in this interval here. If the second derivative is negative, the function is concave down. 00:08:36.410 --> 00:08:46.702 That's what happens here and here. So where do we identify that we have inflection points? Well, an 00:08:46.704 --> 00:08:55.000 inflection point is a point on the graph of a function where the concavity changes either from up to down, 00:08:55.002 --> 00:09:04.701 which is what this one is up to down, or down to up, which is this one. Those are called inflection points. 00:09:04.703 --> 00:09:13.405 It's time to look at an example of how we can use this to help us analyze and sketch the graph of a function. 00:09:13.407 --> 00:09:29.904 Let's use the same function we used in topic seven. Let's examine the function f of x equals negative x to the 00:09:29.906 --> 00:09:43.403 fourth plus 4 x-cubed plus 5. Alright, now we already know a little bit about this function, but let's. 00:09:43.405 --> 00:09:48.607 Wait, the function we used in the first one was minus 5, and if we're going to get the same results it would 00:09:48.609 --> 00:09:55.003 be helpful to use exactly the same function. Ok, first we're going to do the first derivative test for relative 00:09:55.005 --> 00:10:01.208 extrema. Get the extrema just like we did before. Then we're going to do the second derivative test for 00:10:01.210 --> 00:10:08.404 inflection points, and then we're going to put it all together on one graph. Ok, first derivative test. 00:10:08.406 --> 00:10:16.104 We take the derivative, let's back up a second. One more time, polynomial function. Everywhere defined, 00:10:16.106 --> 00:10:22.103 the domain is all reals. We don't have to worry about any points of discontinuity. We take the first derivative. 00:10:22.105 --> 00:10:32.801 We get negative 4 x-cubed plus 12 x-squared, the derivative of 5 is 0. We want to find the values of x 00:10:32.803 --> 00:10:40.300 where this, the derivative, is either 0 or not defined. Well once again we have a polynomial function so 00:10:40.302 --> 00:10:45.404 we're not going to have to worry about the not defined. We're going to set this equal to 0, and we're going 00:10:45.406 --> 00:10:55.200 to solve for x. We factor out negative 4 x-squared. Be careful with the signs here. We set each factor equal 00:10:55.202 --> 00:11:03.600 to 0. We get critical values at x equals 0 and x equals 3. These are critical values of the function. 00:11:03.602 --> 00:11:11.003 If there are any relative extrema, this is where they're going to be. We do a sign analysis chart of the 00:11:11.005 --> 00:11:23.704 first derivative. We partition the axis at 0 and at 3, giving us three intervals, and we test the sign of the 00:11:23.706 --> 00:11:30.204 first derivative. Now I can remember from topic seven. You can always go back and check this. The sign 00:11:30.206 --> 00:11:35.503 here was positive. The sign here was positive, and this was negative. Let's just review how we got that. 00:11:35.505 --> 00:11:42.901 We picked a test value in each interval and we plugged that value into what? Into the first derivative. 00:11:42.903 --> 00:11:50.603 What that did was that told us that the function was increasing, increasing, and then decreasing over 00:11:50.605 --> 00:12:01.302 these three intervals. So we have no extrema here, and we have here a relative max of f of 3, and 00:12:01.304 --> 00:12:08.704 f of 3 plugged back into here was 22. So we're right back to where we were. But what we want this time is 00:12:08.706 --> 00:12:17.503 we want to be able to sketch the entire function. So what we're going to do now is we're going to use the 00:12:17.505 --> 00:12:23.204 second derivative test. Now we're going to leave the sign chart here because this is going to tell us where 00:12:23.206 --> 00:12:30.204 the function is increasing and decreasing. We're going to go right in here, and we're going to work on 00:12:30.206 --> 00:12:37.901 the second derivative. Now you have to keep this straight so we can get it all on the board at the same time. 00:12:37.903 --> 00:12:46.701 We're going to do f double prime this time. Ok, it would help if I still had the derivative here, so let's 00:12:46.703 --> 00:12:57.900 go back and write it back in. It was negative 4 x-cubed plus 12 x-squared. So the second derivative would 00:12:57.902 --> 00:13:09.105 be negative 12 x-squared plus 24x. Again, critical values of the first derivative occur where the second 00:13:09.107 --> 00:13:14.200 derivative is 0 or not defined. This is not going to have any places where the function is not defined. 00:13:14.202 --> 00:13:21.603 It's a polynomial. So all we have to do is set it equal to 0. We factor out the negative 12x. Watch the signs 00:13:21.605 --> 00:13:33.099 again. We get x minus 2, and so we get two critical values of the first derivative. We get x equals 0 00:13:33.101 --> 00:13:34.000 and x equals 2. 00:13:34.100 --> 00:13:39.500 Alright, now we need a sign chart to evaluate the concavity and then the 00:13:39.600 --> 00:13:45.300 inflection points of the function. We're going to just leave this here so we can see it when we 00:13:45.400 --> 00:13:51.800 get ready to put this all together. We want it lined up together. Right below this we're going to put 00:13:51.900 --> 00:14:03.700 f double prime sign chart, and now we have partitions at 0 and 2. Now do yourself a favor here. 00:14:03.800 --> 00:14:12.200 Line up the 0 and when you mark the 2 put it where you think the 2 might be. It would just roughly be 00:14:12.300 --> 00:14:18.000 about say here. We're going to put this all together at the end. We want the 3 to be further over. Here's 00:14:18.100 --> 00:14:25.200 negative infinity and here's infinity. Ok, we're going to have to take test values in each of these three 00:14:25.300 --> 00:14:32.200 intervals to get the sign of the second derivative. Now how does this go again? A plus here means the 00:14:32.300 --> 00:14:40.000 function is concave up. Minus here means the function is concave down. Ok, we're going to the 00:14:40.100 --> 00:14:49.300 second derivative which is going to be this one. Usually it's faster though to use the factored form because 00:14:49.400 --> 00:14:54.200 all we want is the plus or minus. We don't really care about the actual value. So we're going to pick 00:14:54.200 --> 00:14:59.500 something in here. Negative 1 is probably the best number to pick. When I put negative 1 into here, 00:14:59.600 --> 00:15:05.500 negative times negative is positive, so we can forget about that part. Then we have negative 1, negative 2, 00:15:05.600 --> 00:15:12.300 so we have positive times negative is negative. Be very careful when you do this. 00:15:12.400 --> 00:15:23.400 Let's pick 1 here. That would make this factor negative and this factor negative, so we get positive. If you 00:15:23.500 --> 00:15:28.700 make a sign error here you'll know, because when you try to put this all together it won't make any sense. 00:15:28.800 --> 00:15:34.700 Then we pick something out here. Let's pick 3. When we put 3 into here, we get a negative number here 00:15:34.800 --> 00:15:42.300 and a positive number here, and so that's negative. So what did we determine? Well we determined that 00:15:42.400 --> 00:15:53.000 the function itself over the first interval here is concave down, and that's some part of this shape, but not 00:15:53.600 --> 00:16:01.400 necessarily all of it. Here the function is concave up. And then what happens here is concave down. 00:16:01.500 --> 00:16:12.700 So did we get any inflection points? We got two. We got one here at 0, and one here at 2. We're going to 00:16:12.800 --> 00:16:20.500 have to find the rest of the ordered pair here. We need 0, something. And here we need 2, something, 00:16:20.600 --> 00:16:30.700 because these are going to be our inflection points, points where the concavity changes. So where do we 00:16:30.800 --> 00:16:36.700 go to find this? Do we go to the second derivative? No, we're not graphing the second derivative. We are 00:16:36.800 --> 00:16:44.300 graphing f of x, way back here. So we're going to have to go back all the way to here to find f of 0. 00:16:44.400 --> 00:16:58.000 F of 0 would be 0, 0, negative 5, get that negative in there, and then 2 is going to take a little bit more work. 00:16:58.100 --> 00:17:12.375 F of 2. Alright, well 2 to the fourth is 16, so that would be negative 16, 2 cubed is 8, times 4 is 32, and then 00:17:12.385 --> 00:17:28.500 we've got the negative 5. So what do I have here? I have 32 minus 21 is 11. So those are our inflection points. 00:17:28.600 --> 00:17:37.000 Now we need to put this together and get the picture. We leave the sign charts here. We have the key points. 00:17:37.100 --> 00:17:44.700 We have the relative max, and we have the two inflection points. Now what I'm going to show you here 00:17:44.800 --> 00:17:51.250 feels a little silly when you first do it. But trust me, it works, and you won't be the first person that ever 00:17:51.350 --> 00:17:58.000 sat in a test and did this with your hands. So here's what we're going to do. We're going to realize that 00:17:58.100 --> 00:18:08.000 there are actually four intervals here. There's negative infinity to 0, 0 to 2, 2 to 3, and then 3 to infinity. 00:18:08.100 --> 00:18:17.000 Over each one we have to determine increasing or decreasing, and then we have to determine concavity 00:18:17.100 --> 00:18:30.700 up or down. First thing we're going to do is plot these key points that we have. So let's get the sketch started 00:18:30.800 --> 00:18:41.000 and you notice right away when you start to do this that your x-values, 0, 2, and 3, are small, but your 00:18:41.100 --> 00:18:52.500 y-values go from negative 5 to 11 to 22. So we're going to need to make the scale on the y-axis larger than 1 00:18:52.600 --> 00:18:59.000 so we can get a reasonably good picture here. I'm going to make my scale here, aou're going to have to 00:18:59.100 --> 00:19:05.875 make this fit each function that you sketch the graph of. So we're going to make our scale on this axis 1. 00:19:05.975 --> 00:19:12.000 So if I put nothing that just means 1. But on the y-axis we're going to make our scale, we've got to go all the 00:19:12.100 --> 00:19:19.750 way up to 22, so we're going to make it 5, 10, 15, 20, and 22 will have to be a little bit above that. So this 00:19:19.850 --> 00:19:35.000 scale is 5. I don't want to overcrowd the graph so there's a 5, and then down 5, 10, 15. What do we have here? 00:19:35.100 --> 00:19:47.000 Well at f of 3 we have a max. So at (3, 22) which is somewhere about here we have a relative max. 00:19:47.100 --> 00:20:03.700 At (0, negative 5) I have the first inflection point, and at (2, 11), which is about here, I have another 00:20:03.800 --> 00:20:15.900 inflection point. Ok, get ready with your hands. First interval, from negative infinity to 0, the function is 00:20:16.000 --> 00:20:23.000 increasing, so that looks like this, and it's concave down, so it's shaped like this. Now you have to make 00:20:23.100 --> 00:20:30.300 that fit so that it actually hits right here at the inflection point. Now you're not going to get the whole thing 00:20:30.400 --> 00:20:36.300 of the concave down because you just want the part that's increasing. So this is increasing and concave down. 00:20:36.400 --> 00:20:44.200 So you piece this together and it's going to look something like that. That's concave down and increasing, 00:20:44.300 --> 00:20:50.000 We'll put an arrow on here just to show that we're going to sort of run out of paper. Alright, second interval. 00:20:50.100 --> 00:21:03.000 We're only going to go here to 2. So from 0 to 2 the function is still increasing but the concavity is down, 00:21:03.100 --> 00:21:11.300 so it looks like that. Up. Sorry. See how you could tell. If you do it wrong it just doesn't make sense. 00:21:11.400 --> 00:21:16.300 I looked at this and I saw concave down and you can see there isn't any way that's going to fit here. 00:21:16.400 --> 00:21:26.300 So it's increasing and concave up. It looks like this and that little piece fits right here. So we go there, 00:21:26.400 --> 00:21:35.000 increasing and concave up. Alright, from 2 to 3, use your hand. The function is still increasing, 00:21:35.100 --> 00:21:48.900 but from 2 to 3, which is about here, we have increasing and concave down. So that fits right there. 00:21:49.000 --> 00:21:54.700 And I know it feels kind of stupid, but it does work really well. And then from 3 to infinity the function 00:21:54.800 --> 00:22:07.500 is decreasing and it's still concave down. So that looks like this, and that piece is right there. So what 00:22:07.600 --> 00:22:13.500 we just did was, we were able to sketch the graph of a fourth degree polynomial function, leading 00:22:13.400 --> 00:22:19.200 coefficient negative. We were able to get things that you cannot get with algebra, because we found 00:22:19.300 --> 00:22:25.000 inflection points. We found a relative max. We found concavity, and we found increasing and decreasing. 00:22:25.100 --> 00:22:34.300 We put it all together into a sketch. We are now well on our way to be able to do some real things with 00:22:34.400 --> 00:22:39.000 some complicated functions. That will be in the topics to come.