DIFFERENTIAL & INTEGRAL CALCULUS. Mathematics & Economics Essay

DIFFERENTIAL & INTEGRAL CALCULUS
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Introduction
This paper discusses the relationship between mathematics and economics. It specifically relates calculus and economics with an objective to construe the use of mathematical formula and concepts in the economic field. Mathematics is useful in economics as its application is paramount in to represent many theoretical frameworks and assist in problem analysis. The use of mathematics plays a great role in guiding economists to formulate meaningful and testable propositions in respect to broad, in-depth and complex subjects that could turn hard to express informally. With mathematical language, economists are given a way through which they prove positive claims are regards to contentious and controversial areas (Spivak, 2018). To mention the least, most economical theories are well described in mathematical summaries and models to create a relationship that would be hard to construe in the absence of mathematics.
Economics as applicable with Calculus
It may be described as a special area in the society that is a social science involving the production, distribution and the consumption of goods and services. It gives an understanding on the human behavior and also the interaction of economic agents critically showing how economics would work in such a situation. The production of goods requires the planning, estimation and the optimization of resources to be used. Producers will use various mix of resources so as to arrive at a required product. According to Spivak (2018), the logistics behind production, distribution and consumption spirals around calculus through its functions and derivatives. An in-depth understanding of calculus is prerequisite in handling complex areas of expertise such as economics. As well, distribution will involve strategic logistics that will be used to deliver the produced goods to its intended consumers. Before these two activities take place, they are preceded by evaluation of the consumption patterns of the final users of commodities being produced and distributed. This therefore depicts the in-depth interaction that economics has with the society. Economics can hence be ruled to be a society.
Economics divides into areas of microeconomics and macroeconomics. Microeconomics provides an analysis on the very basic areas of economy which include markets and individual agents, how these two elements interact and the expected results of such interactions. This is simply the market interaction of firms, households, buyers and sellers and the results of their relationships which are mutually beneficial. On the other hand, macroeconomics deals with a bigger economy such as that of a country hence gives results at an aggregated level. It would touch on aggregate savings, investments, production and consumption (Graham, 2018). Macroeconomics explores other factors that would affect these aggregates such as inflation rates, under-employment of capital, labour and land resources and the remedy to these factors which involve public measures such as fiscal and monetary policies.
The analysis of economics has a wide and total application to the society. It covers major sectors of business, demographics, real estate, finance, health care provision and the control of the government. Others areas of economic application involve political affairs, warfare, environmental management and conservation, rule of law in a country, country’s education and not limited to social institutions. It emerges out that economics controls are a population should live and its absence may lead to nuisance in the society due to loss of significant control.
Principles of Economics/Calculus
Limits and infinitesimals which is a method used to develop calculus by applying very small quantities. The small quantities are objects which are like small number but are greater than zero. In this way, calculus only used to manipulate objects and there boasts a collection of techniques. Taking symbols of dx and dy as the infinitesimals, then their ratio would be dx/dy.
Principles of economics are important in depicting how rational people would be when they face scarcity. Economics involves the solving of scarcity of resources to attain optimality and efficiency while endeavoring to provide satisfaction to human needs. One of the principles is the tradeoff principle. In tradeoff, people should be made aware that there exist no such things as free things. It stipulates that people have to give up on a thing to gain on another. For instance, if one wants leisure time, they have to forego work and the reverse case applies.
Economics applies this principle to award resource accordingly. This principle compliments another principle where the cost of something is whatever one has to give up getting it. The principle gives a relationship between cost and benefit that people look at when making choices. If one is in need of something of benefit, then the benefited anticipated is equivalent of the cost payable in the process of acquiring that good or service. This serves as an opportunity cost as represented in the production possibility frontier below. This production possibility frontier would be clearly managed by use of differential calculus to ascertain what good choice to select so as to optimize and still stand a chance of not losing (Stroock, 2018). Graphing this idea borrowed its knowledge from relationships only possible through calculus without which it would be undoubtedly impossible.
Figure SEQ Figure \* ARABIC 1: Calculus in Opportunity Cost relationships
Integral calculus would be taken a development of calculus though a division but it is mostly applied in the calculation of area under the curves of functions. It is critical in the calculation of area, mass, weight and any other related items so long as they are represented under the curve as portions or areas (Magnus & Neudecker, 2019). The graphical representation below gives a sample of what integral calculus deals with in the least.
Figure SEQ Figure \* ARABIC 2: Integral Calculus
The shows variable x and y and their function y = f(x). Calculus would be applied in calculating the area under curve marked as S.
Calculus
This may be defined as a mathematical study of continuous change. It is divided into differential and integral calculus. Differential calculus involves an instant rate of change while integral is about amassing of quantities and areas under or between curves. Both branches of calculus are well related through the fundamental theorem of calculus where they use central convergence notions of infinite series and consequences. This theorem of calculus stipulates that differentiation and integration processes are opposite to each other in that, to get to the initial state of a function, one has to carry out a reverse operation. This base on the anti-derivatives of any certain integral and fundamental theorem of calculus offers an easy approach of making calculations for such. According to Bolza (2018), in mathematical derivation of the theorem, it states that if a function denoted f is deemed continuous at an interval of (a,b), and F function has a derivative f in the same interval, therefore for every x the interval will be constant.{\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}
Calculus was developed with the idea to solve various life challenging situations. War lords in the past were tactical enough to find weapons and chariots for example so as to win their battles. Managers or what could have termed as the rulers, had problems relating to spending of resources and this called for utilization of the same in a manner that was purely optimal (Bolza, 2018). It is through calculus functions that they would derive optimal solutions and were able to strive within challenging means. This work was keenly developed by Newton and Wilhelm and their basis of references were from scanty ideas that had earlier been put forth by one scientist Isaac Barrows.
History of calculus
Contemporary calculus was established in the seventeenth era in Europe by Gottfried Wilhelm and Isaac Newton with each publishing independent work. Elements of calculus contrary appeared for the first time in Greece, China, Middle East and later again were exercised in Europe and India. Integral calculus was developed from ideas in the ancient period which seemed not to have been rigorously and systematically developed.
Significance of calculus
Many of calculus ideas being developed in Greece, India, China and Japan, the initial use of it was in Europe after its introduction in the seventeenth century by Isaac Newton and Gottfried Wilhelm. Its basic principles were based on models of instant wave and areas beneath curves. Differential calculus has its application in the computation of velocity, acceleration, slope of a curve and optimization. Calculus has tremendous applicability in many fields including economics. The differential calculus gives instantaneous rate of change useful in the study of functions, slopes and gradient with respect to relationships between variables. These prices may involve prices, demands and supply of goods and services. The important part in the involvement of calculus is the notion behind the rate of change in functions (Lerner & Nazarov, 2019). This would help solve and analyze various theoretical problems.
Calculus was historically meant to analyze ideas, theories and concepts to formally depict the intended purposes. This discipline gives a wide and in-depth knowledge on the formulation of mathematical relationships in solving major and critical crisis. These crises would have resulted from the scarcity of resources warranting a superior encounter in solving the same. In the periods of warfare, troops and their commanders had to apply calculus knowledge in the process of coming up with solutions.
Relationship between Calculus and Economics
The introductory economics course works involves very little mathematics but to develop a deep understanding of economics, one must have an equivalent deep understanding of mathematics more so calculus. Calculus provides amateurs and experts with a communication tool of economics and offers a means through which economists get to solve problems. It is calculus that significantly illustrates what economics experts refer as principles of economics such as cost –benefit and scarcity principles.
Calculus being an advanced division in mathematics, it heavily centers on functions and derivatives. The functions shows the relationship between two or more variables in the form of X and Y which represent the variables being related. Change in one variable will have a significant effect on the other causing a less, equivalent or huge change. There occurs an indirect and direct relationship in between the variables. For example an increase in variable X may cause a less, equivalent or more increase in variable Y in what is referred to as a direct proportionality in their relationship.
In other cases, there will occur an indirect relationship between variables with an increase in one variable causing a less, equivalent or more decrease in the other variable. Derivatives are highly depended on the rate of change and they enable calculation of gradients, slopes, instantaneous changes (Trepte, Scharkow & Dienlin, 2020). This can be evidenced by graphical representation below which shows the calculation of an arbitrary point from the function f(x) and the use tangent line to achieve the same. Tangent line on touching a slope of a curve gives rise to a point shown here as the arbitrary point. Getting a derivative from a curve is also another way of achieving that arbitrary point in the form of (x, f(x)).
Figure SEQ Figure \* ARABIC 3: Arbitrary Point from a slope and tangent line
Research by economists and mathematics experts always use calculus to examine relationships in functions. A good example on dependent variable such as income and independent variable such as medication and treatment relationship would show that, if income goes up with increase in the level of medication, there exists a positive direct relationship between the two variables. Using differential calculus, economists would use procedures in obtaining derivatives so as to measure average income changes in relation to a particular period’s medication or treatment.
Application of Calculus
The beauty of Calculus is not in the making of mathematical computations. Its pretty side is in the way it is capable of building connections and relationships and a language useful in describing changes taking place in the world. This helps i...