 One extremely important idea is that we can define a function as an integral. For example, let's define the function n of x be the definite integral for minus x to x of e to power minus t squared over 2. Now, fair warning, you can try to find an antiderivative for this function, but you won't be able to, at least not among the familiar algebraic or transcendental functions. There is no simple antiderivative of e to minus t squared over 2. Nevertheless, even though n of x is defined as a definite integral of this function, we can still find quite a bit about n of t. So let's start off by showing that n of x is 2 times the definite integral from 0 to x. So let's do this problem the hardest way possible and not draw a graph. Let's not. Let's actually draw a graph. So the geometric content of the definite integral says that n of x is going to be the area of the region under the graph of y equals e to the minus t squared over 2 above the x-axis from t equals minus x to t equals positive x. However, it's useful to note that y equals e to minus t squared over 2 is actually symmetric about the y-axis. And the beautiful thing about symmetry is that it tells you some things are equal. In this particular case, it tells us that the integral from minus x to 0, the area of this region, must be the same as the integral from 0 to x, the area of this region. Is that useful? I don't know. Let's try it and find out. If you don't play, you can't win. We know that n of x is defined as this definite integral. We also have this additivity property of the definite integral. We can split the integral at 0 until our first integral runs from minus x to 0 and our second runs from 0 to x. But wait, we know that this integral from minus x to 0 is the same as the integral from 0 to x. And so now I have two copies of this definite integral from 0 to x, which gives us the result that we want. Well, let's see if we can find some actual function values. How about approximating n of 0.1? Again, it's worth emphasizing that you can try all you want. You will not find an antiderivative of e to the minus t squared over 2 among the familiar functions. Nevertheless, we can still approximate n of 0.1. We'll try a linear approximation. But we'll need a value of n of x, where x is close to 0.1, where we know both the function and its derivative. Since n of x is defined by a definite integral, then we know n of 0 is 0. What about n prime of 0? The fundamental theorem of calculus tells us something about the derivative of a function defined as an integral from 0. Unfortunately, this function is defined from minus x. Fortunately, we showed that this could be written as a function defined from 0. So we can apply the fundamental theorem of calculus to find the derivative. And that tells us that n prime of 0 is equal to 2. And so now I know the function value. I know the derivative. So now I want to write the equation of the tangent line. And if I want to approximate n of 0.1, it's going to be... Let's talk some limits. What's the limit as x goes to infinity of n prime of x? And what does this tell us about the graph of y equals n of x? So again, by the fundamental theorem of calculus, because I know that n of x is 2 times the definite integral, then I know... Now as x goes to infinity, this expression goes to 0. And since n prime of x is the slope of the line tangent to the graph of y equals n of x, this suggests that as x goes to infinity, the tangent lines have slope going to 0, which means they're going to become more and more horizontal. So let's put together everything we know and everything we can determine and see if we can try to sketch a graph of y equals n of x. So first, remember we already found that n of 0 equals 0, and n prime of 0 is equal to 2. We also found that as x goes to infinity, n prime of x goes to 0. We could also try to find the limit as x goes to minus infinity of the derivative. And since this is also 0, this suggests that as x goes to minus infinity, the graph also becomes more and more horizontal. Next, we were able to use some of the properties of the function and the fundamental theorem of calculus to find the derivative. Since we know the first derivative, let's make use of it. From the form n prime of x equals 2e to power minus x squared over 2, we see that because exponentials can never be 0 or negative, n prime of x is always positive. We can also find the second derivative. And since we know the second derivative, we can use it. So from our second derivative, we see that if x is less than 0, our second derivative is going to be positive, so the graph will be concave up. On the other hand, if x is greater than 0, our second derivative is negative, so the graph is concave down. So that means at x equal to 0, the graph changes from concave up to concave down. So let's put all of these pieces together. First of all, we know that the graph goes through the origin, and the slope of the line tangent to the graph at that point is 2, so the tangent line looks something like this. If we go way off to the right, our slopes are getting close to 0, so our tangent lines look horizontal. And similarly, if we go way off left, the derivative is always positive, so the graph is always rising. At x equals 0, the graph changes from concave up to concave down. So before x equals 0, our graph is concave up, and afterwards it's concave down. And at this point we can connect the dots to form a smooth curve.