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. 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. It's actually the graph is something called a probability density function for the normal distribution used in statistics. So, it's a fundamental importance in applications to science and many other disciplines. 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. It's a common debating theme, who was a great a mathematician, Euler or Gauss. The description 'bell-shaped' is possibly slightly misleading. 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. 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. But more about that later. 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. In the course of doing this, you will see an application of the chain rule. We first find the y-intercept, which is one. 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. 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. So, the x axis is a horizontal asymptote. We next look at properties of the derivative. This is where the chain rule comes in handy. 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. 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. Which is minus 2x times e to the minus x squared. It's important to know when the derivative 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. We can then build a simple sign diagram by noting that y dashed is zero when x is zero. 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 put this altogether so far drawing the axes and noting the y-intercept. 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. 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. To create this shape, there appear to be two natural inflections where the curve changes concavity both to the right and to the left. 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 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. But y dashed is quite complicated. We don't yet have the tools to find y double dash easily. To do so, we employ the so called product rule, which tells us how to differentiate a product of expressions. But we defer the complete analysis until after next time when the product rule will be explained and illustrated. The next example involves some quite intricate mathematics, so you might not be able to follow all the detail at first. I hope you'll nevertheless persevere, maybe using the pause and replay buttons on the video from time to time, because the mathematics describes a very beautiful physical phenomenon with surprising accuracy, and involves an elegant and powerful application of the chain rule. In this next example, we've created an ornamental sculpture in the form of a perfect ice cube, which we've suspended from the ceiling in a warm room. One hundred liters of water were used to create the original ice cube. We notice after three hours, that 10 liters of water have accumulated in a tray below the dripping ice cube. Now problem is, to use mathematics to estimate the number of hours it takes the ice cube to melt away to nothing. We begin by denoting the side length of the cube by x units. What the units turn out to be is unimportant. We don't even need to know what the original value of x is or even to know what exact volume of frozen ice corresponds to, say, one liter of water. It's very important however, to think of the value of x as a function that changes with time t. The units of time will be hours, since the question asks "How many hours it takes for the cube to melt away to nothing?" You might notice in this example, we're using t as the independent variable and x as the dependent variable, similar to the conventions used in describing displacement functions. There are two quantities that turn out to be important in the modeling that precedes the analysis. These are the volume of the cube, denoted by V, which is also a function of time t and takes the value x cubed, cube units, for whatever value x happens to be at that particular time. The surface area of the cube denoted by a, another function of time t and takes the value six x squared. Since there are six faces to the cube, each of area x squared square units. So, we have these three quantities associated with the melting ice cube, x, v and a, all functions of time t. In fact, we want to know when the ice cube disappears. So, we want the time t at which all of these will be zero. The key physical fact, which starts our mathematical modelling, is that the rate of melting of the volume of the cube is proportional to the surface area of the cube. This makes good intuitive sense that the surface of the cube is where the cube interfaces directly with a warm air in the room. So, the more area there is on the surface of the cube, the more associated substance of the cube measured by volume should disintegrate from solid into liquid and drip into the tray below. The rate at which the volume changes with time is the derivative dv/dt and this fact states that dv/dt is a constant multiple of the surface area A, say k times times A for some constant K. But A equals 6(x) squared so dv/dt becomes 6k(x) squared. We're developing a list of important facts. Note also, that dv/dx, the derivative of x cubed with respect to x is 3x squared. We'd like information about the width x of the cube and how it changes with time t. So, it's natural to ask what the derivative dx/dt might be. The link between this and the other derivatives is the chain rule, which in this case says that dv/dt equals dv/dx times dx/dt. But dv/dt is 6k(x) squared and dv/dx is 3x squared. So, this says, that 6k(x) squared equals 3x squared times dx/dt. So, dividing through gives dx/dt equals 6k(x )squared divided by 3x squared which simplifies to 2k. Hence, dx/dt is simply two times the constant k which is another constant. If the rate of change of x with respect to time t is constant, then the function x must be a linear function of t, the slope 2k, so that x must have the form x equals 2kt plus C for some constant C. This is an important step going backwards from a derivative to a function. It's a fact and part of a more general theory that we'll explore in the final module, that if the derivative is constant then the function that it comes from must be a linear function. If you think about this geometrically, if the slopes of all the tangent lines of a curve are constant, then the curve must be a straight line. That's what's happening here. Our variables are t and x, rather than the usual x and y. This is significant progress to see that V is x cubed and x is 2kt plus C where K and C are some constants. We haven't yet used the given information that 10 liters of water dripped off the cube in the first three hours which represents 10 percent of the volume leaving intact 90 percent or 9/10ths of the original volume. So, V of three equals 9/10ths of V of zero. We can use the fact that V equals the cube of x to rewrite this as, the cube of x evaluated at three is 9/10ths of the cube of x evaluated at zero. But we know that x evaluated at zero is just the constant C and x evaluated at three becomes 6k plus C. So, we get a nice equation linking the constants k and C. We can rewrite this as 6k plus C cubed equals 0.9 of C cubed. Now, solve for C, first by taking cube roots to get 6k plus C equals the cube root of 0.9 times C and some rearrangements gives finally C equals 6k divided by the cube root of 0.1 minus one. We have an elegant relationship between the constant C and k and can now ask the time t at which the cube disappears by melting away. Which is when the width x becomes zero. That is, when 2kt plus C equals zero. We can sub in the formula for C in terms of k. We therefore want t such that this equation holds. Rearranging the information and noticing that k cancels out, we get the t equals three divided by one minus the cube root of 0.9 which our calculator tells us is approximately 86.9 to one decimal place. Thus, the answer to our original problem is that it takes almost 87 hours for the cube to melt away completely. If you look carefully at the mathematical analysis, you'll see that the final answer depends only on the proportion of the cube that had melted after three hours. Clearly, the warmer the temperature, the larger would be this proportion. One could generalize the problem to include this proportion as a variable parameter and deduce a general formula. It's interesting that the model predicts the cube disappears completely. There is a time at which the width and the volume actually becomes zero. This is in contrast to exponential decay models with the amount of the given substance tends towards zero. In fact, very rapidly, but never actually reaches zero. In the case of our melting ice cube, the width turns out to be a decreasing linear function of time. In today's video, we discussed two contrasting examples that relied on the chain rule in order to progress towards a solution. The first example looked at the Gaussian bell-shaped curve related to the normal probability distribution used in statistics. The Chain Rule assisted us in finding the derivative of the associated function. In a future video, I will go further with this example to find and analyze the second derivative. But to achieve this, we'll use the product rule which is introduced in the next video. In the second example, we performed a detailed analysis of a melting ice cube. We used the chain rule to connect together different aspects of the problem expressed in terms of derivatives. We obtained a model in which the main measure, the width of the cube, becomes a decreasing linear function so that the cubed disappears completely. This is in contrast to exponential decay models where the value of the function decreases and gets arbitrarily close to zero, but that's not actually reach zero. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
