In today's video, we introduce general sign diagrams, for rules of functions, and provide contrasting examples, illustrating how they clarify and inform our understanding of graphs of functions. A sign diagram is a special type of table with two rows. The first row for indicating values of a variable x, say, and the second row for some expression involving the variable. For example, x squared minus x, which we factorize if possible. The line separating the rows is a copy of the real number line. On that line, we indicate important points relevant to the behavior of the expression, such as when it becomes zero or undefined. For example, x squared minus x, becomes zero if x equals zero and one. So, we mark off two points on the line and label them with zero and one. Below this line, we indicate the way the expression behaves with a zero or undefined at the indicated points, and whether positive or negative between these points and on neither side to the right and left. So, for this particular expression, the value is positive for x less than zero or greater than one, and negative only for x between zero and one. Later, I will give an example with expression is undefined at a particular point. This completes the sign diagram. This particular expression is a quadratic, so the associated curve is a parabola. Bowl-shaped up because the coefficient of x squared is positive and passing through the x-axis at zero and one. The parabola sits above the x-axis for x less than zero and x greater than one, corresponding to the plus signs in the sign diagram, and below the x-axis for x between zero and one, corresponding to minus sign. The sign of the derivative gives us information about where a function might be increasing or decreasing. So, let's investigate the derivative y, denoted by y dash, which in this case is 2x minus 1. Observe, that this derivative is zero precisely when x equals one half. So, we can draw a sign diagram marking off the point half x, zero directly beneath, and then plus to the right, where 2x minus 1 will be positive, and minus to the left where it will be negative. The signs tells us that the curve is increasing for x greater than a half and decreasing for x less than a half. Both of these, exactly match what we see on the curve. This also confirms that the point with coordinates of half negative a quarter is the lowest point on the parabola. We informally draw a small line sloping downwards beneath any minus sign to indicate decreasing, and informally, a line sloping upwards beneath any plus sign to indicate increasing. Now, what about the derivative of y dash called the second derivative denoted by y double dash? The derivative 2x minus 1 is just the constant two and the sign diagram is particularly simple. There are no x values where the second derivative is zero or undefined. The second derivative is positive everywhere indicated by a plus sign. We have an informal notation which is a bowl-shaped up symbol, which has a technical term, concave up. When we see a plus sign for the second derivative, then we know the curve has some kind of bowl-shaped up behavior over the relevant interval. We'll say more about the technical meaning of concavity in a later video. In the case of this parabola, you can imagine the tangent launch to the curve as you pass from left to right having increasing slope in the sense of becoming more and more positive as you move from left to right. This is exactly what the sign diagram for the second derivative is telling us, causing the curve to have the shape of a ball facing upwards. This upright parabola is concave up. But if we tipped it over, it will become what we call concave down. Then, there will be a negative sign instead of a plus sign and an inverted bowl-shaped down or sad face symbol in the sign diagram. We'll illustrate that in a moment. We revisit an example discussed towards the end of the previous video. Consider y equal to x cubed minus x, which factorizes as x times x plus 1 times x minus 1. So, that y equals zero when x equals minus one, zero and one. We can draw a sign diagram for y, noting these important values x that cause y to be zero. Then noting an alternating pattern of plus and minus symbols as you move right to left past the points that we've marked on the x-axis. This tells us when the curve sits above and below the x-axis. To find out where the curve is increasing or decreasing, we form a sign diagram for the derivative, which in this case, is three x squared minus 1. So, that y dash to zero, when x equals plus or minus 1 over root 3. We get the sign diagram that we mentioned at the end of the last video. We now add our up or down notation to indicate visually when the curve is increasing or decreasing according to whether the sign of the derivative is positive or negative. Now, we can be sure how the curve should undulate. In this case, up, down, and then up again as we move from left to right. What about the second derivative? Can this give us any useful information? The first derivative is 3x squared minus 1. So, the second derivative, y double dash is 6x. This is zero when x is zero. We can draw the sign diagram for y double dash, indicating that it's zero when x equals zero, and then it has the same sign as x, positive, when x is positive, and negative when x is negative. You might remember in the last example, we used a bowl-shaped apple happy face symbol when the second derivative was positive. So, we'll use that again here for x greater than zero. But when the second derivative is negative, we use a bowl-shaped down or sad face symbol. We now have three sign diagrams that gives a fairly complete description of how the curve should behave. Where it sits above or below the x-axis, where it's increasing or decreasing, and where it's bowl-shaped up or down. The important points marked off on the sign diagrams, then correspond to important features of the graph so we note them on the x-axis. Then, fitting the curve to match the information about the signs of y, y dash, and y double dash. Of note, other points where the curve is turning around. A peak at minus 1 over root 3 and a trough at plus 1 over root 3, then this sad face shape for negative x and this happy face shape for positive x. The point where the curve changes from sad to happy is called an inflection. The tangent line actually passes through the curve at a point of inflection. It's worth noticing and relates to something we call the second derivative test, that we'll talk about in a future video, that the point on the curve for x equals negative 1 over root 3, that looks like the top of a hill, correlates with the imagery suggested by the sign diagrams for both the first, and second derivatives. Similarly, for the point on the curve, x equals 1 on root 3, that looks like the bottom of the valley. The next example also was discussed in an earlier video and it's quite difficult. Let g be the rational function with rule x squared minus 2 divided by x minus 1, which can be re-written as x plus 1 minus 1 over x minus 1. Here's the graph of g with vertical and oblique asymptotes. We now explore the use of sign diagrams to gain a better understanding of the graph. First of all, g of x is zero precisely when the numerator of the rational expression is zero, which occurs when x equals plus or minus root two. We can form the sign diagram for g of x by first noting plus or minus root two, and that g of x is zero at those points. We also leave room to insert the point 1 because this is the input for which the rule for g does not make sense, and we write u beneath, an abbreviation for undefined. Then, we note an alternating pattern of signs for g of x as we move from right to left, past these important points on the x-axis. Each part of the sign diagram corresponds to some feature of the graph, sitting above or below the x-axis, being zero or being undefined. What about a sign diagram for the derivative? As we've noted, the row for g can be rewritten as x plus 1 takeaway 1 over x minus 1. Put h of x equal to 1 over x minus 1, and then the derivative g dashed of x becomes 1 minus h dashed of x. But, what's h dashed of x? The answer turns out to be negative 1 over x minus 1 squared. I'll try to give an explanation, which also serves as good revision of one of the limit definitions of the derivative. We have the rule for h. If any a, h dashed of a is the limit as b approaches a of h of b minus h of a over b minus a, which becomes a limit of this fraction after subbing in the rule for h. We can rewrite this, simplify the numerator, and do some cancellation, so that the fraction simplifies further. As b approaches a, this fraction approaches negative one over a minus 1 squared. Replacing a by x, gives the formula that we mentioned earlier. So, we can now complete the calculation for the derivative of g, which becomes 1 plus 1 over x minus 1 squared. But all parts of this expression are positive. So, g dashed of x is positive always provided x is not equal to one. Now, we can provide a sign diagram for the derivative of g. We note that g dashed of x is undefined for x equals one, and positive everywhere to the left and right of one. So, we add an upward sloping symbol in each case to indicate that the function is increasing on both intervals. This behavior predicted by the sign diagram is exactly matched on the curve, but for the branch of the curve to the left to the vertical asymptote and to the branch to the right. There's more that we can say by considering the sign diagram for the second derivative. But we'll save that up for later video. Today, we discussed sign diagrams in general for rules of functions, and interpreted a positive or a negative sign for the derivative as indicating that the function is increasing or decreasing respectively. A positive sign for the second derivative as indicating that the graph of the function has a bowl-shaped up or smiley face appearance. A negative sign for the second derivative as indicating that the graph has a bowl-shaped down or sad face appearance. 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.