The next example involves some quite intricate mathematics, so you might not be able to follow all the detail at first. But we defer the complete analysis until after next time when the product rule will be explained and illustrated. To do so, we employ the so called product rule, which tells us how to differentiate a product of expressions. We don't yet have the tools to find y double dash easily. To be sure about the way the curve behaves, and in particular, to find these inflections, we need to go further and investigate the second derivative y double dash. The curve has natural symmetry above the vertical axis because e to the minus x squared doesn't change if you replace x by minus x, an instance of an even function. To create this shape, there appear to be two natural inflections where the curve changes concavity both to the right and to the left. Because the curve is increasing from the left and decreasing to the right, it's natural to fill in the rest of the curve to obtain this bell shape. The asymptotic behavior of the x-axis, both to the far left and far right, and the fact that there is a turning point at the y-intercept when x equals zero. We can put this altogether so far drawing the axes and noting the y-intercept. We have a pattern of plus minus, indicating that the curve is increasing for x less than zero, decreasing for x greater than zero and achieving a maximum at x equals zero. We can then build a simple sign diagram by noting that y dashed is zero when x is zero. In this case, this only happens when x equals zero, noting that the factor e to the minus x squared is never zero. It's important to know when the derivative is zero. Which is minus 2x times e to the minus x squared. Hence, the derivative y dash, which is dydx in Leibniz notation becomes dydu times dudx by the chain rule, which now becomes e to the u times minus 2x. Put y equal to e to the u where u equals minus x squared, so dydu equals e to the u, and dudx equals minus 2x. This is where the chain rule comes in handy. We next look at properties of the derivative. So, the x axis is a horizontal asymptote. As to asymptotic behavior, observe that e to the minus x squared, which is the reciprocal of e to the x squared, tends to zero as x gets arbitrarily large and positive or negative. There are, in fact no, x-intercepts because any power of e, in particular, e to the minus x squared for any x is positive. We first find the y-intercept, which is one. In the course of doing this, you will see an application of the chain rule. I'd like to apply the methods of curve sketching that we've discussed in earlier videos to the function y equals e to the minus x squared. We'll meet another important curve in the shape of a bell in a later video known as the witch of Maria Agnesi, after famous female 18th century Italian mathematician Maria Agnesi. There are many curves you could say are in the shape of the bell, but this is the one most people have in mind when they refer to a bell-shaped curve. The description 'bell-shaped' is possibly slightly misleading. It's a common debating theme, who was a great a mathematician, Euler or Gauss. The curve is named after Carl Friedrich Gauss, another giant in the development of modern mathematics who lived and worked, from towards the end of the 18th century into the middle of the 19th century. So, it's a fundamental importance in applications to science and many other disciplines. It's actually the graph is something called a probability density function for the normal distribution used in statistics. Our first example is a Gaussian curve, y equals e to the minus x squared, which is often described as bell-shaped, in the shape of the bell. In today's video, we demonstrate two contrasting applications of the Chain Rule, to begin to understand the behavior of the Gaussian curve, y equals e to the minus x squared, related to the normal probability distribution used in statistics, and to predict how long it takes for an ice cube to melt away completely.
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