The Calculus and Levels of Math Education - Essay Example

During the elementary level of math education, one merely learns that divisibility by zero is not in any way valid or possible and becomes content with treating such a case as closed without entertaining its logic any further. In Calculus, however, though the concept widely recognizes that functions do have domains and ranges within which they remain defined, the subject goes beyond a such point as extending concern to the limits of a function. By using the principle behind the asymptotes for instance whenever a function is rational, logarithmic, or exponential, Calculus enables a student to strive to grasp and appreciate the idea regarding how a function may come close to or approach a value at least even if it is never meant to cross its exact location.

            A function, according to Calculus, is said to be continuous in an interval [a, b] if it is continuous and defined at any point within this same interval. If this initial condition is not satisfied, then the non-continuity implies that the function is also non-differentiable within [a, b]. Equivalently, the two-sided limits are stated in theorems that guide the study of whether a function’s limit does exist or not as x approaches a certain value and this requires tests to be conducted before the conclusion.

            These accounts are essential in discerning the significance of derivatives which pertains to an instantaneous rate of change or the slope of a tangent line to a curve at a point. While the concept of finding a derivative is useful in applications that relate displacement, velocity, and acceleration, the reverse process of getting the antiderivative proves to be of crucial advantage in determining an area under the curve or a volume of a solid generated by revolving a strip of an element in a bounded region about a particular axis or line of the revolution. Since functions come in different types and complexities, Calculus thereby promotes a variety of ways or techniques by which integration may be carried out yet where no method seems adequate in evaluating a definite integral, numerical means of approximations are employed as a more flexible alternative.

            Limits are a basic tool in setting up the framework of differentiation. Solving for derivatives in turn establishes applicability with problems on related rates which are extensively used even in physics, chemistry, and other hard sciences that identify relevance in differentiating a function given a system with multiple variables. It then serves as a device for estimating how one rate varies concerning the dependent or independent variation of the other whereas the rest of the properties are held constant.