![]() ![]() While dx is always constant, f(x) is different for each rectangle. Each miniature rectangle has a height of f(x) and a width that is called dx. Instead, calculus breaks up the oddly shaped space under a curve into an infinite number of miniature rectangular-shaped columns. 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. For example, integrating the function y = 3, which is a horizontal line, over the interval x = 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. The easiest way to define an integral is to say that it is equal to the area underneath a function when it is graphed. The answer, of course, is 2x = (2)(3) = 6. 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. ![]() For example, the derivative, or rate of change, of f(x) = x2 is 2x. That means that the derivative of f(x) usually still has a variable in it. The catch is that the slopes of these nonlinear functions are different at every point along the curve. In other words, it lets you find the slope, or rate of increase, of curves. ![]() Calculus extends that concept to nonlinear functions (i.e., those whose graphs are not straight lines). 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. 2) Derivativesĭerivatives are similar to the algebraic concept of slope. 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. However, in cases where f(x) does not exist at point p, or where p is equal to infinity, things get trickier. 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.įor many functions, finding the limit at a point p is as simple as determining the value of the function at p. 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. Limits are a fundamental part of calculus and are among the first things that students learn about in a calculus class. With that in mind, let’s look at three important calculus concepts that you should know: 1) Limits In fact, it might even come in handy someday. 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. Many people see calculus as an incredibly complicated branch of mathematics that only the brightest of the bright understand. The Three Calculus Concepts You Need to Know ![]()
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