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CHAPTER ONE

1.1  INTRODUCTION

From the beginning of time man has been interested in the rate at which physical and non physical things change. Astronomers, physicists, chemists, engineers, business enterprises and industries strive to have accurate values of these parameters that change with time. The mathematician therefore devotes his time to understudy the concepts of rate of change. Rate of change gave birth to an aspect of calculus know as 'differentiation'. There is another subject known as 'integration'. Integration and Differentiation in broad sense together form subject called 'calculus'. Hence in a bid to give this research project an excellent work, which is of  great utilitarian value to the students in science and social science, the research project is divided into four chapters, with each of these chapters broken up into sub units. Chapter one contains the introduction, scope of study, purpose of study, review of related literature and  limitation. Chapter two dwells on the fundamental of calculus which has to do with functions of single real variable and their graph, limits and continuity. Chapter three deals properly with differentiation which also include gradient of a line and a curve, gradient function also called the derived function. Chapter four contains the application of differentiation, summary and conclusion.

1.2  Scope of the Study and Limitation

This research work will give a vivid look at differentiation and its application. It will state the fundamental of calculus, it shall also deal with limit and continuity. For this work to be effectively done, there is need for the available of time, important related text book and financial aspect cannot be left out.

1.3 Purpose of the Study

The purpose of this project is to introduce the operational principles of differentiation in calculus. Also to analyse many problems that have long be considered by mathematicians and scientists.

1.4 Significance of the Study

The significance of this study cannot be over emphasized especially in this modern era where everything in the entire world is changing with respect to time, because the rate of change is an integral part of operation in science and technology, hence there is need to ascertain the origin of calculus and its application. Finally, the goal of this work is to review the application of differentiation in calculus.

1.5  LITERATURE REVIEW

Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivations, integrals and infinite series. Ideas leading up to the notion of function, derivatives and integral were developed through out the 17th century but the decisive step was made by Isaac Newton and Gottfried Leibniz. Ancient Greek Precursors (Forerunners)  Of The Calculus. Greek mathematicians are credited with a significant use of infinitesimals. Democritus is the first  person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-section  with a cone’s  smooth slope prevented him from accepting the idea, at  approximately the same time. Zeno of Elea  discredited infinitesimals further by his articulation of the paradoxes which they create.

Antiphon and later Eudoxus are generally creadited with implementing the method of exhaustion which implementing the method of exhaustion which made it possible to compute the area and volume of regions and solids by breaking them up  into an into an infinite number of recognizable shapes. Archimedes of Syracuse developed this method further, while also inventing heuristic method which resemble modern day concept some what. It was not until the time of Newton that these methods  were incorporated into a general framework of integral calculus. It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. 

Pioneers of modern calculus

In the 17th century, European mathematicians Isaac barrow, Rene Descartes, Pierre deferment the idea of a deferment. Blaise Pascal, john Wallis and others discussed the idea of a derivative. In particular, in method sad disquirendam maximum et minima and in De tangetibus linearism Curvarum, Fermat developed an adequality method for  determining maxima, minima and tangents to various curves that was equivalent to differentiation. Isaac Newton would latter write that his own early ideas about calculus came directly from formats way of drawing tangents. On the integral side cavalieri developed his method of in divisibles in the 1630s and 40s, providing a modern form of the ancient Greek method of exhaustion and computing cavalierr’s quadrate formula, the area under the curves Xn of higher degree, which had previously only been computed for the parabola by Archimedes. Torricili extended this work to other curves such as cycloid and then the formula was generalized to fractional and negative powers by Wallis in 1656.

In an 1659 treatise, fermat is credited with an ingenious trick for evaluating the integral of any  power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. James Gregory influenced by fermat’s contributions both to tangency and to quadrature, was then able to prove a restricted version on the second fundamental theorem of calculus in the mid -17th century. The first full proof of fundamental theorem of calculus was given by Isaac barrow. Newton and Leibniz building on this work independently developed the surrounding theory of infinitesimal calculus in the late 17 century. Also, leibniz did a great deal of work with developing consistent and useful notation and concepts.

Newton provided some of the most important applications to physics, especially of integral calculus. Before Newton and Leibniz the word “calculus” was a general term used to refer to any body of mathematics, but in the following years, “calculus”. Became a popular term for a field of mathematics based upon their insight. The work of both Newton and Leibniz is reflected in the notation used today. Newton introduced he notation f for the derivative of function f. Leibniz introduced the symbol  for the integral and wrote the derivative of a function y of the variable x as both of which are still in use today.