Differential calculus is a subfield of Calculus that focuses on derivates, which are used to describe rates of change that are not constants. The term ‘differential’ comes from the process known as differentiation, which is the process of finding the derivative of a curve. Differential calculus is a major topic covered in calculus. According to Interactive Mathematics, “We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).” Not only are derivatives used to determine how to maximize or minimize functions, but they are also used in determining how two related variables are changing over time in relation to each other. Eight different differential rules were established in order to assist with finding the derivative of a function. Those rules include chain rule, the differentiation of the sum and difference of equations, the constant rule, the product rule, the quotient rule, and more. In addition to these differential rules, optimization is an application of differential calculus used today to effectively help with efficiency. Also, partial differentiation and implicit differentiation are subgroups of differential calculus that allow derivatives to be taken to more challenging and difficult formulas. The mean value theorem is applied in differential calculus. This rule basically states that there is at least one tangent line that produces the same slope as the slope made by the endpoints found on a closed interval. Differential calculus began to develop due to Sir Isaac Newton’s biggest problem: navigation at sea. Shipwrecks were frequent all due to the captain being unaware of how the Earth, planets, and stars mov...
... middle of paper ...
... mathematics, would not be able to exist to the extent that it is today.
Works Cited
William, Walter, and Rouse Ball. Differentiation Rules: Chain Rule, Inverse Functions and Differentiation, Sum Rule in Differentiation, Constant Factor Rule in Differentiation. New York City, NY: General Books LLC, 1888. Print.
Kouba, Duane. "Implicit Differentiation Problems." Collection of Lectures. (1998): Print.
Rusin, Dave. "Partial differential equations." Mathematical Atlas. 35.1 (2000): Print.
Foster, Niki. "Who is Gottfried Leibniz." Brief and Straightforward Guide (2011): n. pag. Web. 14 Apr 2011. .
Bourne, M. "Applications of Differentiation." Interactive Mathematics. N.p., 25 02 2011. Web. 14 Apr 2011. .
Fig. 1. A graph of the marginal value theorem from Krebs (1993). The asymptotic curve represents food intake. The optimal number of food items to take is found by drawing a line from the travel time to the patch to the steepest point possible on the curve.
Look, B. (2007, December 22). Gottfried Wilhelm Leibniz. Stanford University. Retrieved May 2, 2014, from http://plato.stanford.edu/entries/leibniz/
The contemporary world is full of marvels. Technological advances have enabled mankind to fly in the heavens, instantaneously communicate with distant relatives thousands of miles away, construct buildings that are able to withstand many natural disasters, cure deadly diseases, and even travel to and study areas beyond the confines of planet Earth. While there are many factors that contributed to man’s ability to overcome what many once thought were impossible feats, it is the study of engineering that has enabled one to study the elements and leverage all that they have to offer. Mathematics lies at the heart of all science, including engineering. Without progressions in mathematical concepts, engineering principles and applications would not have advanced as quickly as they have throughout history.
I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every math student will recognize these periods of frustration and helplessness.
The argument in this paper that even though the onus of the discovery of calculus lies with Isaac Newton, the credit goes to Leibniz for the simple fact that he was the one who published his works first. Appending to this is the fact that the calculus wars that ensue was merely and egotistic battle between humans succumbing to their bare primal instincts. To commence, a brief historical explanation must be given about both individuals prior to stating their cases.
The derivative of a function is the rate of change of that function. It shows how fast or how slow the function is changing. This can be useful in determining things such as instantaneous rates of change, velocity, acceleration and maximum profits. A good way to explain the concept of a derivative is to do it graphically. To illustrate, think of a drag car race. The track is only ¼ of a mile long, or 1320 feet. The dragster crosses the finish line in six seconds. How fast was the dragster going when it crossed the finish line? The dragster traveled 1320 feet in 6 seconds, so the average speed of the dragster is 1320 divided by 6 which equals 220 feet per second, or 150 miles per hour. The following graph represents the dragster’s position function as the red curve. The position function for the dragster is 36 2/3 x^2. The green line is the secant line connecting the dragster’s starting point and end point. The slope of this secant line is the average speed of the dragster, 220 feet per second, or 150 miles per hour.
Finite difference scheme can be categorized and formulated in Taylor series expansions. When a function and its derivatives are single valued, finite and the continuous functions then the Taylor series expansion for function may be written at particular point as:
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Because to solve a problem analytically can be very hard and spend a lot of time, global, polynomial and numerical methods can be very useful. However, in last decades, numerical methods have been used by many scientists. These numerical methods can be listed like The Taylor-series expansion method, the hybrid function method, Adomian decomposition method, The Legendre wavelets method, The Tau method, The finite difference method, The Haar function method, The...
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
...derstand the behavior of a non-linear system you need in principle to study the system as a whole and not just its parts in isolation.
This evaluation has not only allowed me explore calculus more in depth, but also physics, and the way the world works. This has personally allowed me to explore the connections between math and real-world situations, which is hard to find in textbooks.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
...d a better understanding of differentiation, I have had several of my students tell me that I am the best math teacher they have ever had. They express their happiness by telling me that I teach math in a way they understand. They state, “You do not stand in front of the classroom and explain how to do the problem, give us homework, and move on to the next topic”. I take pride in this. I try very hard to help each of my students understand the necessary standards so when they leave my room, they are able to take a real-world problem and find solutions to them.
Through out my life, nothing has ever fascinated me like the application of mathematics in real life situations. In real life, the application of mathematics acts as a foundation from which economics progresses as it offers a person with different knowledge on how they are supposed to carry out different life applications. Mathematics offers an individual with the right differentiation skills that are highly required in making different calculations of elasticity coefficients that are applied in different sectors of the world. My greatest inspiration in the application of mathematics in real life experience has been “ The Pleasure of Counting”. By T. W. Korner’s. Through the work of Korner, I have been enabled to increase the rate to which I apply the mathematical knowledge that I have acquired over the years to real world situations. Some of the areas that I have increasingly applied my mathematical knowledge includes in the economics as well as in the field of finance. The application of mathematics in this filed has contributed towards my critical thinking and analytical skills progress.