 Another way we can use the derivative is to sketch the graph of a function. There are two things you want to remember about the derivative. First, the derivative is the slope of the line tangent to the curve at the point of tangency. And second, the tangent line is a good approximation to the curve through the point of tangency. So suppose we know that the derivative is positive. Then the slope of the line tangent to the graph of y equals f of x at x equals a will be positive. So wherever our point is, the tangent line will look like this. And because a tangent line is a good approximation to the curve through the point of tangency, we know what the curve looks like. Not this, not this, but maybe something like this. And so we say that the graph is rising as it passes through x equals a, or we say that the function is increasing through x equals a. We could say the graph is increasing at x equals a, or the function is rising, but that would be nonsense. The graph isn't getting bigger, the function isn't going anywhere. By a similar argument, if f prime of a is negative, then the slope of the line tangent to the graph of y equals f of x at x equals a will be negative and will look like this. And because the tangent line is a good approximation to the curve, then we know that the curve looks like not this, not this, but maybe this. And the graph will also look like this, and so we can say the graph is falling as it passes through x equals a, or we could say that the function is decreasing through x equals a. This means that the derivative gives us a lot of information about what the graph of a function looks like. So suppose f of x is a continuous function over the interval between 0 and 10, and suppose I know the values of f of x and f prime of x. Just based on this information, we can sketch a graph of y equals f of x. Now it's important to remember, sketch means just that. We're not trying to get a perfect graph because we don't have enough information, but we're trying to get something that conveys the essential features of this function. In particular, we are given a bunch of information about f of x and f prime of x. Whatever the sketch of our graph looks like should take into account all of the information given. So where can we start? Since we're graphing y equals f of x, the f of x values give the y coordinates of points on the graph, and so we know that the graph passes through the points 0, 3, 5, 8, and 10, 2. So let's start by plotting these points. Next, we know the values of f prime of x give the slope of the tangent lines through these points. So since we know the derivative of x equals 0 is 2, then the tangent line through 0, 3 has slope 2. Likewise, the tangent line through 5, 8 has slope negative 3. And finally, the tangent line through 10, 2 has slope 0. And so we might try to draw these tangent lines. Now one important thing, we don't want to draw too long a segment because that will clutter up our graph. Remember that the tangent line is an approximation to the curve as long as we don't go too far from the point of tangency. So we'll draw short segments of these tangent lines. And because f of x is continuous, we know that somehow these segments will join up. And since we've used up all of the information that we have available, at this point we just want to make sure that the sections of the curve join up, and so we might join them up something like this. What if we have different information? For example, suppose I have this information about the function and its derivatives. So again, we might want to start with a point that the graph passes through, and we know what the tangent line through that point looks like. Now because the graph is continuous over the interval, we know that there has to be a point where x equals 0, but we don't know where. If we don't have the f of x value, we don't have the y value. One of the nice things about mathematics is we can put something down and fix it later. So we'll put down a point as a placeholder, and we do know that our derivative is negative 3, so we know that the slope of the tangent line will be negative 3. Likewise, we know that the graph has a point with x equal to 10, but we don't know where. But we do know the slope through this point will be 5. So we can put the tangent line and fix the location of the point later. Well, it's later now, so let's go ahead and fix that. Because we know our function is continuous, we need to adjust the vertical locations of these points to give us a continuous graph. Because we've already incorporated all of the available information, we can adjust the vertical position any way that we want to, provided that we don't contradict any of the given information. So we might randomly move some pieces up and end up with a graph looking something like this. We'll take a look at another example. Suppose f of x is continuous over the interval between 0 and 5, and between 5 and 10, but not including x equals 5. And suppose we have the following information. Again, it's convenient to start with a point if we have that information. And we know the tangent line through this point has slope negative 1. How about these limit statements? Well, since we're graphing y equals f of x, then any statement about the limit of f of x is a statement about the limit of y. We have this information about the limit, and so we know that as x gets close to 5 from below, y gets close to 4. But again, we have to make no commitment as to what happens when x is actually equal to 5. And in fact, we have this other bit of information about the limit as x approaches 5. And so from this information, we know that as x gets close to 5, but stays a little bit to the right of it, y goes to minus infinity. Notice this also means we have a vertical asymptote at x equal to 5. And finally, this last bit of information tells us that there is a point where x is equal to 10, and the tangent line through that point has slope negative 3. So again, we don't know exactly where that point is except that its x value is equal to 10. So we'll put a placeholder point and run a line with slope negative 3. And since the only place where the curve is not continuous is at x equals 5, we need to make sure the pieces connect. And so we'll adjust our pieces, keeping in mind that there are two points that are fixed. This point at 0, 3, and this point at 5, 4, which is a limit point.