Calculus? Calculus Is a Vast

¶ … Calculus? Calculus is a vast topic and it also forms the basic foundation of any calculations that are based on math. Calculus is divided into two branches, one being differential, and the other being integral. Differential calculus deals with the study of the rates of changes that may happen in functions. For example, it teaches how to find the derivative of a particular function when the function is to determine the angle of the slope of a graph, which deals with that function, at a particular point. Integral calculus generally helps the beginner to deals with certain primitive functions, like for example, indefinite integrals, and also for finding, for example, the area under a curve, known as a definite integral. (Quick math, Automatic Math Solution)

In other words, calculus can be defined as a branch of mathematics that pertains to the rates of change. Although its basic roots can be traced back to Ancient Greece and to Ancient China, its actual origins are indicated in the time of Newton and Leibnitz in the seventeenth century. In the modern world, calculus has proved its usefulness time and again, and it is used extensively in many different areas of science. The there ideas of 'limit', 'derivative', and 'integral' are interwoven into the principles of calculus, wherein while derivative indicates the instantaneous rate of change, in comparison to something else, the integral generally indicates the area under its graph, or a sort of total 'over time'. For example, when the derivative of 'height' in respect to its position, is 'slope', then the derivative of 'position' with respect to 'time', is velocity', and when 'velocity' is taken with respect to 'time', then the derivative would be 'acceleration', and so on. As far as integrals are concerned, the area under the graph is taken into consideration, and this means that when the integral of 'slope', in respect to a constant, is 'height', then the integral of 'velocity', up to a constant, 'position', and thereafter, the integral of 'acceleration', with respect to 'time', is 'velocity'. It is quite evident therefore, that derivatives and integrals are inter-related and are also at times, complete opposites. (Preparing for University Calculus: At Atlantic Canadian Universities)

Why is 'calculus' important in today's world? Science today studies many processes that involve change, and since calculus deals with changes, it is very important. (Preparing for University Calculus: At Atlantic Canadian Universities) One particular High School teacher gave her students concrete examples of how calculus can be applied in real life to solve real problems. The links between the physics concepts of position and velocity and acceleration and the totally calculus concepts of function, derivative and anti-derivative were found. (Dosemagen; Schwalbach, 54) Another teacher toot found that when physics is applied to calculus, the problems would be solved easily. When compared to the way in which these students found it difficult to use mathematics for their calculations, and how they preferred to use physics as a better option, it can be seen that calculus, when combined with the basic principles of physics, would be easier to apply. (Marrongelle, 30)

There are many more teachers who feel that all students must compulsorily learn something of the history of mathematics and of calculus so that they would be able to better understand the advances that have been made in the area, so that in the long run they would be able to apply and to relate the old concepts to their new ways of thinking. This would also mean that they would be able to apply these principles to other areas as well, in other words, when the student begins to think about the most important developments in the history of, say, differential calculus, then the notion of 'limit', and its final elucidation would come to mind. (Hilton, 26) in general, no mathematician would be willing to accept the solution to a problem without some sort of proof, and in the same way, no student of calculus would be ready to accept the resolution of a problem without the necessary proof. (Cadena; Travis; Norman, 77)

It must be stated that Newton's mathematics that involved 'fluxions' was one of the first forms of the area defined as 'differential calculus'. Although Newton used and preferred to use geometrical methods to algebraic equations, calculus methods had come into importance. However, calculus was not widely accepted at the time, and there were several philosophical objections to the science, but the fact remains that these objections over the years have made no difference to the application of the science. This is mainly because of its abstract nature, and also the logically sufficient nature of the science. The mathematician, Karl Popper, has stated that scientific theories are definitely 'sufficient conditions', but are not 'necessary conditions', of the observation of scientific phenomena. (Philosophical problems with calculus)

This means that the metaphysics of mathematics belongs to the branch called meta-mathematics, and not to math proper. There have been numerous approaches to the question of infinitesimals over the years, but all the approaches have been replaced by talk about limits. When, for example, an individual talked about justifying the existence of these minute quantities, he was faced with a number of difficulties, and the solution to this was the theory of limits. This theory became so popular that most mathematicians felt that infinitesimals must be actually banished from their science. Infinitesimals had in fact addressed the originally made objections to calculus, and it was eventually discovered that calculus had in fact worked for several centuries before. When this is taken in a scientific context, it is a very good occurrence, and most physicists and scientists and mathematicians agree with the idea. (Philosophical problems with calculus)