Welcome to week 7 of Science Friday! For the last two weeks, I’ve done a miniseries on evolution. This week, I’d like to discuss the basic concepts of calculus. This branch of mathematics is tremendously useful in many different fields—physics, chemistry, economics, population ecology, and the list goes on. Often times, we like to differentiate (no pun intended) between math and science. However, as we see from the applications of calculus, these two branches of study are inextricably linked. I often put it this way: If science is the tome to understanding the Universe, then mathematics is the language in which it is written. So what is calculus? On its most basic level, calculus is the study of change and limits. While there was no single creator of calculus since it is such a broad field, its early development is largely attributed to Isaac Newton and Gottfried Leibniz. The best introduction to calculus, in my opinion, is attempting to find the slope of a tangent line to a curve. In algebra, we often drew linear curves and found their slopes by determining the change in y and dividing it by the change in x. For nonlinear curves, we could find slopes by drawing [i]secants[/i]—lines that share two points with a curve. In fact, this is often used in algebra, and even calculus, to approximate the slope of a curve at a given point. The biggest problem with finding the slope at a point is that we have no change in y and no change in x. This quantity is 0/0, which is what mathematicians call an [b]indeterminate[/b] form. Put simply, you cannot divide by zero (I may just do an entire week on why this has to be true). So how do we remedy this problem? The answer is limits. First, we approximate the slope at a point by drawing a secant line with the one point on either side of the point we are trying to pinpoint. The distance between the point we want the slope at and the point that forms the secant with the first point is denoted [i]h[/i]. We note that the slope of a line is still change in y over change in x. The change in x is now h and the change in y is f(x+h) - f(x). m(slope) = f(x+h) - f(x) / h This slope (m) is the slope of the secant line. Notice that, as we make h smaller, the slope of our secant resembles the slope of the tangent more closely. This is where the idea of a limit comes in. If we take the limit as h approaches 0 of the difference quotient I set up above, we now have the slope at a point. This limit is the [b]definition of the derivative.[/b] But how does this work? More specifically, how does creating a limit aid us in finding the slope at a point? This is best shown using a concrete example. Consider the function y = f(x) = x^2. This is a parabola whose vertex is at the origin and opens upward. Let’s say we want to find the slope of this parabola [i]at the point[/i] x = 1. m(slope) = lim (h->0) of f(x+h) - f(x) / h m(slope) = lim(h->0) of (x+h)^2 - x^2 / h At this point, we need to evaluate this limit. Plugging h = 0 directly into the expression gives an indeterminate form. We are going to have to manipulate this expression algebraically. Let’s start out by expanding the square: (x+h)^2 = x^2 + 2xh + h^2 The x^2 terms will cancel in the numerator (subtract out). We will be left with lim (h->0) of 2xh + h^2 / h Now, let’s factor out an h from the numerator so we get lim (h->0) of h(2x-h) / h The h’s cancel (divide out), and we get lim (h->0) of 2x+h Plug in h = 0 and we get 2x. We have just found the formula for the slope at any point of the function y = x^2. In other words, [b]the derivative of f(x) = x^2 is f ‘ (x) = 2x[/b]. We wanted the slope at x = 1. Plugging in x = 1 to the derivative, we get 2(1) = 2. In the example case here, finding the derivative of x^2 (with respect to x) was fairly simple using the limit definition. Unfortunately, most functions we encounter would be extremely difficult to [i]differentiate[/i] (take the derivative of) by this method. As a result, there are many different shortcuts to differentiate various functions. I encourage you to look up these shortcuts as I will not discuss them in detail here. Some of them include Power rule (used to differentiate power functions) Product rule (used to differentiate a function that is a product of two functions) Quotient rule (used to differentiate a function that is a quotient of two functions) Chain rule (used to differentiate a function that is a composite of two functions) Next week, I will talk about how these abstract mathematical concepts can be used to solve real world problems. As we will see, the power of the derivative is immense. I hope you enjoyed this week’s SciFri. If you have any questions or comments, please leave them below. I look forward to reading your thoughts about the post and answering any questions that may arise. As always, you can check out previous weeks of Science Friday by clicking the Science Friday tag on the thread and changing the filter settings to see all threads.