More than a few people see calculus as a branch of mathematics that is so intensely complicated that only the brightest of the bright understand it. However, many college students are at least able to grasp the most important points, so it surely isn’t as bad as it’s made out to be. In fact, it might even come in handy someday. Whether you’re on Team Newton or Team Leibniz (the two founders of calculus), these are the three concepts with which you should be familiar.
Limits are a fundamental part of calculus and are among the first things that students learn about in a calculus class. In short, finding the limit of a function means determining what value the function approaches as it gets closer and closer to a certain point. For example, finding the limit of the function f(x) = 3x + 1 as x nears 2 is the same thing as finding the number that f(x) = 3x + 1 approaches as x gets closer and closer to 2.
For many functions, finding the limit at a point p is as simple as determining the value of the function at p. However, in cases where f(x) does not exist at point p, or where p is equal to infinity, things get trickier. Overall, though, you should just know what a limit is, and that limits are necessary for calculus because they allow you to estimate the values of certain things, such as the sum of an infinite series of values, that would be incredibly difficult to calculate by hand.
Derivatives are similar to the algebraic concept of slope. In algebra, the slope of a line tells you the rate of change of a linear function, or the amount that y increases with each unit increase in x. Calculus extends that concept to nonlinear functions (i.e., those whose graphs are not straight lines). In other words, it lets you find the slope, or rate of increase, of curves.
The catch is that the slopes of these nonlinear functions are different at every point along the curve. That means that the derivative of f(x) usually still has a variable in it. For example, the derivative, or rate of change, of f(x) = x2 is 2x. Therefore, to find the rate of change of f(x) at a certain point, such as x = 3, you have to determine the value of the derivative, 2x, when x = 3. The answer, of course, is 2x = (2)(3) = 6.
The easiest way to define an integral is to say that it is equal to the area underneath a function when it is graphed. For example, integrating the function y = 3, which is a horizontal line, over the interval x = [0, 2] is the same as finding the area of the rectangle with a length of 2 and a width (height) of 3 and whose southwestern point is at the origin. That is an easy example, of course, and the areas calculus is interested in calculating can’t be determined by resorting to the equation A = l x w.
Instead, calculus breaks up the oddly shaped space under a curve into an infinite number of miniature rectangular-shaped columns. Each miniature rectangle has a height of f(x) and a width that is called dx. While dx is always constant, f(x) is different for each rectangle. Therefore, the area of a single miniature rectangle at x = p is equal to the product [dx][f(x(p))], so the sum of the areas, or the integral, is equal to [dx][f(x(a))] + [dx][f(x(b))] + [dx][f(x(c))] + . . . + [dx][f(x(infinity))]. In other words, integrating, or finding the area under a curve, can be more formally defined as calculating the limit of an infinite series (i.e., calculating the sum of the areas of the miniature rectangles).
It sounds complicated, but it is just a way of modifying the algebraic concept of area to work with weird shapes comprised of “wavy” curves instead of straight edges. Finally, another cool and useful feature of integrals is the derivation of the integration of f(x) = f(x). In other words, deriving a function and integrating a function are opposite operations. To “undo” a derivative, you just have to integrate it (and vice versa).
Calculus is tricky, so don’t feel bad if you don’t understand everything here. If you take away nothing else, however, let it be these three things:
- Limits predict the value of a function at given point.
- Derivatives give the rate of change of a function.
- Integrals calculate area, and they are the opposite of derivatives.