In today's video, we bring together a range of ideas and techniques that we've been discussing over several videos, that lead to an ordered and systematic checklist for sketching a curve in the plane. Of central importance are the sign diagram for the derivative, which tells us where the curve is increasing or decreasing, and the sign diagram for the second derivative, which tells us where the curve is concave up or down. We begin by making some general remarks about curve sketching, by which we mean more specifically, sketching or drawing the graph of the function y equals f of x in the xy-plane. The scales of the x and y axes don't need to be the same, and the axes are positioned on the page depending on where the important features of the graph will appear. Always aim for simplicity and clarity, and focus on the main qualitative features. Extra details, such as coordinates of points can be added depending on the requirements of any particular problem. It's a fairly natural list of typical features to look for. Firstly, look for the y-intercept, where the curve crosses the vertical axis. In the case of x equals 0 is in the domain of the function, which is simply found by evaluating f of 0. Then, look for the x-intercepts, where the curve crosses the horizontal axis. If they're likely to be straightforward to find or simple to describe. But they may be difficult, as they're the solutions to the equation, f of x equals 0. Indeed, a given problem might be to attempt to solve the equation, f of x equals 0, typically by using some approximation method from advanced calculus. The main purpose of the sketch of the curve may be just to get started with the final aim of knowing roughly where the x-intercepts might be. Then look for asymptotic behaviour which roughly speaking means, seeing what happens when things get large. Such as when x gets arbitrarily large and positive or arbitrarily large and negative, leading to possible horizontal or oblique asymptotes. Or whether there are any vertical asymptotes near which y can get arbitrarily large and positive or negative. Then, we look for where the curve might be increasing, where the derivative could be positive or decreasing, where the derivative could be negative, which helps us find the turning points, where the derivative might be zero. This could be local or global maxima or minima. This information typically is stored in the sign diagram for the derivative. Then, we look for where the curve might be concave up, where the second derivative might be positive. Or concave down, where the second derivative might be negative. Which helps us find possible points of inflection, where the second derivative might be zero. This information typically is stored in the sign diagram for the second derivative. Let's work through this list to sketch the following cubic. The y-intercept is minus 1, the constant term which results when you evaluate the polynomial at zero. The x-intercepts look tricky. We can say more about them later and roughly where they might exist. In fact, there's only one x-intercept in this case, but that's not obvious from the rule itself. We now consider asymptotic behavior. As x gets large and positive, the cubic is dominated by the leading term two x cubed, and also gets arbitrarily large and positive indicated by the use of the infinity symbol. As x gets large and negative, now the cubic also gets arbitrarily large and negative, indicated by the use of the negative infinity symbol. There are certainly no horizontal asymptotes. Nor there any oblique asymptotes, though it's not obvious why there are none and some careful reasoning is given in the notes. The derivative is this quadratic, which quickly factorizes. We can see it's 0 when x equals 1 or 2, and then build its sign diagram. With this pattern, plus, minus, plus, corresponding to increasing, decreasing, increasing with a maximum occurring at x equals 1, and a minimum at x equals 2. With y values four and three, respectively. Giving a turning point one, four with local maximum four and a turning point two, three with local minimum three. We get the second derivative y double dash by taking the derivative of y dash and get 12x minus 18, which factorizes six times 2x minus three. So, the y double dash equals 0, when x equals three on two. We can build its sign diagram with a pattern of negative, positive. So, concave down followed by concave up with changing concavity and inflection, when x equals three on two. We can evaluate y for this input yielding seven on two. So, the point of inflection has coordinates three on two, seven on two. We can now put all of this information together. Noting important points on the axes and the position of the y-intercept, the two extrema and the inflection, and then, drawing a smooth curve that passes through these points, consistent with the information of sign diagrams. This completes the sketch of the cubic. One may, if it's important, highlight the point of inflection and the points where the local maximum and minimum occur. Notice, that a sketch tells us that there's only one x-intercept, and the curve crosses the x-axis somewhere between zero and one, and quite close to zero. If one were trying to solve the equation, for x where this cubic is set equal to 0, then this sketch would tell you firstly, that there's only one solution, and secondly, where to start looking to find an approximation to this unique solution. Now, we'll sketch the graph of a by now familiar, rational function g with the rule g of x, equals x squared minus 2, over x minus 1. Many of its main features have been explored in earlier videos. Let's go through the checklist systematically. Firstly, the y-intercept is two, the result of evaluating the rule when x equals zero. The x-intercepts turned out to be straightforward, simply when the numerator x squared minus 2 is 0, which occurs when x is plus or minus the square root of two. The asymptotic behavior has been explored thoroughly in an earlier video. The limit as x approaches one from above of g of x is minus infinity, which means g of x gets arbitrarily large and negative. The limit is x approaches one from below becomes positive infinity, which means g of x gets arbitrarily large and positive. X equals 1 is a vertical asymptote. The interesting asymptotic behavior occurs as x gets large and positive or large and negative. By rewriting the rule for g as x plus 1 minus 1 on x minus 1, we saw that g of x is approximately x plus 1 for large positive or negative x so that the line y equals x plus 1 becomes an oblique asymptote. The derivative of g we looked at in detail in an earlier video, which is 1 plus 1 on x minus 1 squared and positive for all x not equal to 1. The sign diagram, you might recall, is particularly simple, with plus plus on either side of x equals 1, where y dash is undefined so that the curve is increasing for x less than 1 and increasing again for x greater than 1, so that there are no turning points. We can rewrite the derivative as 1 plus x minus 2 to the negative 2. So, the second derivative y double dash becomes 0 plus negative 2 times x minus 1 to the negative 3. I'll just explain briefly where the last part comes from. The derivative of x to the n is n times X to the n minus 1 for any exponent n so that the derivative of x minus 1 to the n must similarly be n times x minus 1 to the n minus 1, because the curve y equals x minus 1 to the n is obtained from the curve x equals x to the n by shifting the curve to the right horizontally by 1 unit and slopes of tangent lines are not altered at all by horizontal shifting. This result was then applied with n equal to negative 2. The second derivative then becomes negative 2 divided by x minus 1 cubed, and its sign, positive or negative, must be the opposite of the sign of x minus 1. This produces the following sign diagram, with y double dash undefined at x equals 1 and the pattern plus followed by minus so that the curve is concave up for x less than 1, followed by concave down for x greater than 1. Now, we can gather all of this information and set up the axes for the sketch of the curve, noting the important points two on the y-axis and plus and minus root two on the x-axis and the oblique and vertical asymptotes. Then, we see this familiar curve in two branches consistent with all of this information. The feature that we didn't discuss previously was concavity. We see the concave up branch to the left of the vertical asymptote and the concave down branch to the right. Notice that these are not full smiley or sad faces, so to speak. The slope is increasing in the left-hand branch of the graph and decreasing in the right-hand branch, which is consistently the pattern of plus and minus in a sign diagram for the second derivative. The reason for the oblique asymptote is so the rule for the function splits up into a linear phase, x plus 1. Take away this extra piece, 1 over x minus 1. This means that to the right, the curve is approaching the oblique asymptote from below because we are taking a small positive phase away from the linear piece. But to the left, the curve is approaching the oblique asymptote from above because we are taking away a small negative piece , which is the same as adding a small positive piece. It's interesting to ask, what happens if we add 1 over x minus 1 in the rule for the function instead of taking it away? So, let's work through the checklist for sketching the curve with this variation of the previous example. We recall the function f with rule y equals f of x equals x plus 1 plus 1 over x minus 1. Observe after a little bit of algebraic manipulation that the rule becomes the rational function x squared over x minus 1. The y-intercept is now just 0, and we quickly see there's exactly 1 x-intercept, which is also 0. As in the previous example, x equals 1 becomes the vertical asymptote, but notice that the behavior is different. The limit as x approaches 1 from above now turns out to be plus instead of minus infinity, and the limit as x approaches 1 from below now turns out to be minus instead of plus infinity. The oblique asymptote, again, is the line y equals x plus 1. The derivative y dash now becomes 1 minus instead of plus 1 over x minus 1 squared, which, after a few steps, you can see becomes the rational function x squared minus 2x over x minus 1 squared, and the numerator factorizes as x times x minus 2 so that y dash equals 0 when x equals 0 and 2. Notice that y dash is undefined at x equals 1. We can now build the sign diagram for y dash, which produces the pattern of signs plus, minus, minus again, plus, as we move from left to right past the important points for x, producing the pattern increasing, decreasing, decreasing again, and increasing, with a maximum at x equals 0 and a minimum at x equals 2. Evaluating f at x equals 0 and 2 produces a turning point 0, 0, the origin, in fact, with local maximum of 0 and 2, 4, with a local minimum of 4. To find the second derivative y double dash , first notice that y dash can be rewritten as 1 minus x minus 1 to the minus 2. So, the differentiating produces minus negative 2 times x minus 1 to the minus 3 by bringing the exponent down and making a new exponent by subtracting 1 to produce simply 2 over x minus 1 cubed. Then, y double dash is undefined at x equals 1 and otherwise, non 0, in which case, it's clear that the sign of y double dash is the same as the sign of x minus 1. So, we can quickly build the sign diagram with the pattern minus, plus, sad face, followed by smiley face so that the curve is concave down for x less than 1 and concave up for x greater than 1. Putting this information together, drawing the axes into asymptotes, we get the curve in two branches, where the first branch concave down, achieving a local maximum corresponding to the origin, and the second branch concave up, achieving a local minimum corresponding to the point 2, 4. Notice how by contrast with the previous example, the curve gets closer and closer to the oblique asymptote from above as we move to the right and from below as we move to the left. In today's video, we produced a checklist for sketching curves in the plane and worked through the details in three contrasting examples. The first involved a cubic function, where there are no asymptotes, but there were two turning points and an inflection. Interestingly, the sketch shows there's exactly 1 x-intercept, in this case, roughly where to find it, which will be useful if one wanted to go further and solve an associated cubic equation. The second and third examples involved rational functions with similar rules for contrasting behavior. Both of them had two branches, an identical vertical and oblique asymptotes but approached differently. One curve had no turning points, whilst the other curve had two. They both had shifts in concavity as one passes across the vertical asymptotes, so to speak, which we may think of as creating some kind of inflection of plus and minus infinity. Please read the notes. When you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon.
