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Overview of The Calculus

Joseph-Louis Lagrange
Joseph-Louis Lagrange

The calculus is the study of two kinds of operations on functions. While many practical problems involving the calculus seek an answer that is a number, the calculus part of calculus problems is fundamentally about functions.

The operations of the calculus are differentiation and integration. Differentiation operates on a function to produce another function, but it is a many-to-one operation so differentiation on many different functions will produce the same result. Integration is an operation that reverses differentiation, but it is a one-to-many operation so integration on a function produces many different functions.

The many different functions that produce a single function in differentiation differ only by a constant. Likewise, the many functions that are produced by integration of a single function differ only by a constant. Integration then is not perfectly the inverse operation of differentiation, since differentiating a function and then integrating the result does not produce the original function uniquely. In particular problems, however, the value of the constant term is known or can be found or is not relevant to the information wanted.

While integration is not the perfect inverse of differentiation, It has an important quality of an inverse: the result of a differentiation can be integrated (at least in theory). The implication of this is that the result of a differentiation is the starting point of an integration whose outcome will be the starting point of the differentiation up to a constant.

Integration, here represented by L, is almost the inverse operation of differentiation, here represented by D, while C is an indefinite constant:

{$$ {\Large \left. \begin{matrix} x^2 + 5 \cr x^2 + 3 \cr x^2 - 17 \cr x^2 + \sqrt 2 \end{matrix} \right\} \Rightarrow {\Large \boxed{\;\operatorname{D}\;}} \Rightarrow 2x \Rightarrow {\Large \boxed{\;\operatorname{L}\;}} \Rightarrow x^2 + C }$$}

The implication for learning is that when differentiation produces a result it is worth while to look backwards because, aside from any constant, an integration can be learned at the same time.

{$$ {\Large 2x \Rightarrow {\Large \boxed{\;\operatorname{L}\;}} \Rightarrow x^2+C \Rightarrow {\Large \boxed{\;\operatorname{D}\;}} \Rightarrow 2x }$$}

In practical problems, of course, the answer sought is almost always a value or a set of values. Only a little of this part of the problem is calculus. Much of the problem is devising a mathematical model of the problem. That part of the problem is science, be it mechanics, sociology, actuary, economics, chemistry, or whatever. But the horrible part is usually arithmetic and algebra.

Because the problem often contains plenty of error-prone arithmetic and algebra apart from the calculus, the calculus is better when it involves as little algebra and arithmetic as possible. So while the beginning of calculus often involves much algebra, considerable emphasis is placed on learning short cuts and formulas that obviate the drudgery. As a result, the calculus part of a calculus problem is usually the easiest part.

A few calculus problems have no known method of solution or no practical means of calculating a solution that is known to exist. Only a few of these will be encountered here, but are mentioned just to make clear that calculus is not yet entirely cut and dried, but that discoveries may still be made.

The calculus requires, as a minimum, a grasp of basic algebra, familiarity with graphing and reading graphs in the x-y plane, and some exposure to the concept of limits. A few basic trigonometry relations become necessary very soon. The calculus tends to revisit, expand upon, and explain nearly all the mathematics that comes before it, so while advanced work in the previous subjects is certainly no handicap, gaps and sketchiness in some of the previous material can be overcome.

Next: Differentiation




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December 24, 2018

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