 Welcome back everyone to our lecture series, Math 1210, Calculus I at Southern Utah University. As usual, I am your professor today, Dr. Andrew Missildine. This part of our series is lecture 36, which we entitled Curve Sketching. The examples from this lecture are derived from examples from Calculus by James Stewart. In particular, these examples are coming from section 4.5, Summary of Curve Sketching. So what do we mean by curve sketchy? It sounds a little sketchy to me, right? I'm sorry, it's a horrible pun, but I have to try it anyways. So based upon what we've learned over the last couple lectures here in our chapter 4 of Stewart's Calculus here, we've been learning about applications of the derivative. In particular, as we've been talking about Lobie-Tall's rule, extrema, max and men's, and how the first and second derivative affect the shape of the graph, we want to put all of these together and see exactly why does a functions graph look the way it does. Turns out the algebraic formula that gives us the function says so much about it, it's not just some random string of points, the behavior of the function indicates how this graph is going to be shaped here. So if we follow along with the same suggestions that Stewart offers in his Calculus textbook, we have a list of things A through E that go into a good curve sketch. So how does one sketch the graph of a function f? So the first on our list is the idea of domain. We should know what the domain of the function is. And so the domain, remember, these are going to be all of the possible x-coordinates, and you'll apologize for my horrible handwriting, but it's what happens here. So we want to figure out all the possible x-coordinates that go onto the graph. And by the domain convention that we've talked about previously, we're going to assume the domain is as large as possible, meaning that we accept all possible x's such that the formula makes sense for those real numbers x. Now, there might be some situations like we have a piecewise function or we might specify the domain otherwise, but unless otherwise stated, we assume it to be as big as possible. So some things we have to watch out for, we don't want to divide by zero, right? When we have a rational expression of some kind, if you divide by zero, this typically gives us some type of vertical asymptote. We'll talk about these a little bit more in a little bit, but vertical asymptotes often happen from division by zero. We have to look out for square roots of a negative. This will give us imaginary numbers. Likewise, if we take the log of a negative, same basic problem, but I should also mention that the natural log of zero or any log of zero is going to be undefined. So these are the three main problems we have to look out for. Division by zero, square roots of a negative, and of course, when we talk about a square root, it's really just eat any even root, right? If you take a fourth root of negative one, that's equally a problem. So we want to make sure we avoid even radicals of negatives and any logarithm of a negative or zero is also problematic. Now, admittedly, the trigonometric functions have some restrictions on the domains like tangent, secant, but if you view those as quotients, they apply by the divide division by zero rule applies there. So I'm not going to make much effort about that. The inverse trigonometric functions also have some restrictions, but we'll be honest, we're not going to be graphing those functions. So I'm not going to pay much more attention to them. So look out for the domain of these things. The range on the other hand, like all the set of all possible Y coordinates, that's a much harder question to answer. And in fact, by curve sketch, you will actually learn very well how to identify the range of a function. But I would recommend start off with identifying the domain of these things. The x or the intercepts, excuse me, part B, intercepts are an important part of any graph because the intercepts tell us where does the graph touch the x-axis or the y-axis if it touches at all. We're going to use the x and y-axis to give us frame of reference as we're graphing these things. So it's nice to know they're intercepts. A y-intercept is pretty easy to come by. If you want to find the y-intercept, you just have to look at the point f of zero and then compute the corresponding y-intercept or the y-intercept will be just f of zero. That's how you find it. So those are you'll be pretty straightforward. Your graph will have one y-intercept, except for if f of zero is undefined. It could be that x equals zero is outside the domain. X-intercepts are a little bit more challenging because one there could be multiple x-intercepts. And to find x-intercepts we have to we actually have to solve the equation f of x equals zero. And that could be more challenging depending on the nature of the function. Like if f is a polynomial function, like we'll see in a little bit we might have to factor it. We might have to use other algebraic tools to help us there. And so finding the x-intercepts will be important for us. But this is one of the harder parts of the problem, finding the x-intercepts. There could be lots of x-intercepts. There could be one. There could be none. It depends on the function. And we'll see some examples to show you how that works. The notion of symmetry. Well, what do we mean by symmetry here? Well, there's a couple of different types of symmetry we've talked about this semester already. So one of the types of symmetry we've talked about is the idea of even symmetry when your function is will say symmetric with respect to the y-axis. That is if we reflect the graph across the y-axis we get the exact same thing again. This is the geometric interpretation of symmetry along the y-axis. I mean what an even function is. Algebraically we can find out this symmetry by looking at f of negative x. Because algebraically if we replace x with negative x we get a reflection across the y-axis. If the function's even this should then look just like f of x. That is it's unaffected by that change. Going hand in hand with even symmetry there's the notion of odd symmetry. Odd symmetry is when you are symmetric with respect to the origin. Origin symmetry meaning that if you were to go through the origin you would come out the exact same distance of their side or another way I like to think of it is if you were to rotate the function around the origin 180 degrees a half spin this gives you the same graph again. Algebraically we can test for symmetry by looking again at f of negative x and we get negative f of x like so. And so the idea here is that if you were to reflect the graph across the y-axis that's the same thing as reflected across the x-axis. Now when it comes to these two types of symmetries there's a simple test that we do to do for symmetry. You look at f of f of negative x and so you look at f of negative x. If it turns out f of negative x simplifies just to be f of x you're even. If it simplifies to be negative f of x then you're odd. And if neither of those things happens well then neither of these two symmetries are exhibited on the function. Now I should mention there is one other type of symmetry that is mentioned here the idea of your periodic. So it could be that you have a function which if you take the function f of x and you add some period p to it it might repeat itself after a while. The idea is like with trig functions if you take sine of x plus two pi then this thing just repeats itself. In fact adding two pi to the angle doesn't make any bit of difference to it whatsoever. So trig functions are periodic and those are the types of periodic behavior that will be really interested with the trigometric functions. There's really not going to be other type of periodic functions we have here but that is a type of symmetry I would like to mention because after all if you know how to graph one period of a trigometric function you can just replicate that over and over and over again and that gives you the whole picture. So knowing the period of a function if it has one can be helpful in simplifying the graph here and that's kind of the reason why we care about symmetry in general is that if we know the function is symmetric of any kind if it's an even function then I can just take the right hand side of the graph and copy it and that gives me the whole graph. So I basically only have to graph half of it and if the function's odd same basic thing you have to rotate it but if you can graph the right side of the graph you get the left side for free. So this can be useful when you're graphing but honestly symmetry in my opinion doesn't play too much of a benefit as we curve sketch here. Again it's useful it is don't get me wrong but if you don't put a lot of emphasis on symmetry I'll be okay with that. I'm just kind of following James Stewart's list here but again I do recommend symmetry it's good but it's not the most important thing. Discontinuities is going to be a pretty important thing for us that I can't emphasize here that if your graph has any type of discontinuities of some kind to it I want to see that as you draw your your graph of the function there and we've learned in the past a couple different types of discontinuities and so there's three discontinuities that I want you to be looking for as you're drawing your functions here. So the first kind if you remember was what we called the removable discontinuities and so visually this off this was the case that there was like a point missing on the graph. Some hobbits stole it from us right and so it could be that the point's just missing or maybe it got moved to be somewhere else. There's a couple of ways this this basically these are one of two pictures you get either the point was removed or it's just missing in general. This happens when the limit of our function it exists but for whatever reason it doesn't equal the function at that value. I guess I should probably add a limit here as x approaches a the limit exists but it doesn't equal f of a and that could be because either f of a is a different number or it could be that f of a is just missing entirely. Someone might again have moved it so there's no point there at all and so these this is a removable discontinuity that's that those are sort of the most benign type of discontinuities but if your graph has any of those we want to see them all right. What was the second type of discontinuities the second type is what we call a jump discontinuity jump of some kind and so maybe your function was coming along and then it jumped maybe because it's defined as a piecewise function and so there actually might be this gap that lives between the function. This you're gonna get a jump discontinuity because with your function as you're approaching say x equals a if you come from the left hand side this might disagree with the limit from the right hand side and so this is what we have this is when we get a jump discontinuity if our graph has one I want to see it as you're graphing it all right and so I'm trying to kind of erase things to keep everything organized here by all means if I'm going too fast please pause the video and copy things down if you need it for your notes here. The last type of discontinuity that I kind of mentioned earlier but the idea of a vertical asymptote this is a long word isn't it vertical asymptote the vertical asymptote the idea behind that is we might have some vertical line that our function is avoiding for whatever reason maybe you get something like this going on something like this probably because this line is outside the domain although that's not always required and so our function is going to be avoiding it as it goes it's going to be going off towards infinity or negative infinity so a vertical asymptote occurs whenever x approaches a from the left or right f of x here we get plus or minus infinity now it doesn't have to be that both the left limits in the right limit are infinite but we just we need something like this for a vertical asymptote occur and so these are the three types of discontinuities that we are looking out for jump discontinuities removal discontinuities and these vertical asymptotes all right so in behavior what do we mean by this in behavior we're trying to figure out what does the function do as it goes to the extremes as it goes to the extreme of its domain which oftentimes will oftentimes that will be negative infinity or positive infinity and so like with a polynomial is it going up on the right hand side is it going up on the left hand side maybe it goes up on the right hand side but down on the left hand side these are the type of questions we're trying to ask ourselves so what's the limit as x approaches plus or minus infinity right what is that that's what we mean by in behavior here now oftentimes this might be plus or minus infinity itself or it might be a finite number if this limit turns out to be finite that's because we have a horizontal asymptote and so if that asymptote exists include that on the graph as well now another issue that does come up here is if what if your domain's not uh negative infinity to positive infinity like if you took for example the natural log y equals the natural log here its domain is actually zero to infinity in which case we do care what happens as x approaches infinity but on the other hand we the left hand side we actually care what happens as you approach zero from the right and so in behavior depends on the function as you go to the extremes what's happening to our function we want to make sure we put that on our graph as well all right what does the first derivative have to do with with graphing a function well as we've seen before whenever your first derivative is positive this is exactly when your function is increasing and when your second or when your first derivative is negative this is exactly when your function is decreasing and so this type of monotonic behavior is essential to know as we graph our functions and we also want to keep our eye out for extrema right do we have a local maximum a local minimum we should put those on the graph if they're there extrema will be something I absolutely look for on graphs make sure you have the appropriate extrema there um we can say similar things with the second derivative right because with the second derivative if it's positive this means your graph is concave upward if your second derivative is negative that means your graph is concave downward and lastly I'll be interested in those points of inflection and we'll see some examples where we go through all of these in just a second points of inflection we're going to care about those and so we'll see some sign charts to handle all of those and so in the end once you have all this information I want you to plot this stuff on on some grid some grid lines here so if there's intercepts plot them if there's critical numbers inflections put those on there in behavior draw it what else do we have on our list discount who's put them there so plot all the points and then the very last step is one of the most critical steps because the function otherwise except for those points that you looked at is going to be smooth and continuous you can smoothly connect the dots uh I know it sounds like a kindergarten skill but turns out kindergarten is actually it's called out college level mathematics right we have to appropriately connect the dots here and so these steps will help us as we try to sketch a curve here