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=f(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 that x=0 is in the domain of the function, which is simply found by evaluating f(0). And 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(x)=0. Indeed, a given problem might be to attempt to solve the equation f(x)=0, typically by using some approximation method from advanced calculus, and 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 turning points, where the derivative might be zero, and these 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 and 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 are 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. And 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 4 and 3 respectively, giving a turning point (1,4) with local maximum 4, and a turning point (2,3) with local minimum 3. We get the second derivative y double dash by taking the derivative of y dash and get 12x-18, which factorizes as 6 times (2x-3), so that y double dash equals 0 when x equals 3/2. And 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 3/2. We can evaluate y for this input yielding 7/2, so the point of inflection has coordinates (3/2,7/2). 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 the sign diagrams. This completes the sketch of the cubic. And 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(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 turn 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(x) is minus infinity, which means g(x) gets arbitrarily large and negative, and the limit as x approaches one from below becomes positive infinity, which means g(x) gets arbitrarily large and positive, and x=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(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. And 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 consistent with the pattern of plus and minus in the sign diagram for the second derivative. The reason for the oblique asymptote is that the rule for the function splits up into a linear piece, 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 piece 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, where we call the function f, with rule y equals f(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 one x-intercept, which is also 0. As in the previous example, x=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 dashed equals 0 when x equals 0 and 2. Notice that y dashed is undefined at x equals 1. We can now build the sign diagram for y dashed, 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 dashed, first notice that y dashed 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 dashed is undefined at x equals 1 and otherwise nonzero, in which case, it's clear that the sign of y double dashed is the same as the sign of x minus 1. And 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 and two 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 one x-intercept, and 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 but contrasting behavior. Both of them had two branches and 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 asymptote, so to speak, which we may think of as creating some kind of inflection of plus and minus infinity. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon.
