 Concavity is a geometric concept, but it's hard to compute with geometric concepts. So let's see if we can tie this to our algebraic concept of the derivative. So one of the things to remember is that the derivative of is the rate of change of whatever happens to be. The geometry is that the derivative gives the slope of the tangent line to the graph of y equals f of x. So if is the derivative, then if the derivative of the derivative is greater than zero, then the derivative itself is increasing. Now there's two possibilities to consider. If the derivative is negative, then my tangent line looks something like this. But if that slope is increasing, that means my slope is getting less and less negative. My slope is getting shallower. And so my tangent lines as I move to the right might look like this. And that suggests a graph that looks something like this. If my derivative is positive and increasing, then my slope is getting steeper. And so maybe my first tangent line looks like this, but then later tangent lines look steeper. And my graph is going to look something like this. And based on this, it appears that my graph of y equals f of x will be concave up. On the other hand, suppose my derivative is negative, then f prime of x is going to be decreasing. And if f prime of x is positive, my tangent line might look something like this. But if f prime of x is decreasing, then the slope is getting shallower. And so later tangent lines will look like this. And putting it together, my graph might look something like this. And if f prime is negative, my tangent line has a negative slope, so it'll look something like this. And if it's decreasing, then the slope is getting steeper. And my graph will look something like this. And overall, y equals f of x will be concave down. And so we have to consider the derivative of the derivative. And in a fit of mathematical creativity, we'll call this the second derivative. And we'll write it in a couple of different ways. If we are using prime notation, the second derivative is going to be indicated by using two primes. And sometimes we call this f double prime. If we're using differential notation, then our derivative of the derivative is going to be expressed this way. And note that this superscript 2, which looks a lot like an exponent, is going to be above the d and above the x. And it's vitally important for later use that this 2 is not attached to the f in the top and not attached to the d in the bottom. And finally, again, mostly for future reference, we can indicate this second derivative in subscript notation, f subscript x, x. For example, suppose we have the following information. Again, it's convenient to start with an actual point on the graph. And we know the graph goes through the point 6, negative 2. We also know the slope of the tangent line at x equals 4, x equals 8, and x equals 10, but we don't know the height at the point of tangency. So I'll put down a couple of placeholder points and run some tangent lines with the appropriate slopes, but we know we will have to adjust their heights later. At this point, we could adjust the height and get a stick figure graph. However, we'd be ignoring the information we have about the second derivative, and so we want to incorporate that. Since f double prime of 4 is positive, we know that the graph is concave up, which means that the tangent line is below the curve. And so our graph is going to curl upwards away from the tangent line. Since f double prime of 8 is negative, we know the graph is concave down and the tangent line is above the curve. And so this means the graph will curl downwards away from the tangent line. Since f double prime of 10 is negative, then we know, and now we'll want to adjust the height so our graph looks something like a continuous curve. And one thing you'll notice is because of our concavity, there's no way of avoiding some weirdness that occurs between x equals 8 and x equals 10. But a little bit of weirdness is normal, and what we have is consistent with the given information, and that's the important thing. What about linear approximations? Say we want to find a linear approximation to square root of 9.1, and then we want to determine whether our approximation is too large or too small. So we'll use the function f of x equals square root of x, and we want to approximate f of 9.1. So we want an x0 value that is close to 9.1 and has computably exact values for f of x and f prime of x. So we might use x0 equals 9, and we'll find both the function value and the derivative at 9. And so we find that the tangent line through x equals 9 passes through the point 9.3 with slope 1.6, so its equation is going to be... Now to determine whether our approximation is too large or too small, we can take a look at the second derivative. So finding the second derivative and then evaluating that at x equals 9. Now we don't need the actual value, we just need to know the sign, and in this case the second derivative is negative, and consequently the graph is concave down, and the tangent line at x equals 9 is going to be above the curve. This means that the y values of the line are going to be greater than the function values, or the function values are going to be less than the y values. And so we can write that using this less than approximately symbol. And so the square root of 9.1 is less than but approximately equal to 1.6 times 0.1 plus 3.