“The Contribution of Calculus in the Social Progress”

The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[1][2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2]

In AD 499 the Indian mathematician Aryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method that is readily generalizable to finding the formula for the sum of any integral powers, which was fundamental to the development of integral calculus.[6] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Madhava of Sangamagrama, along with other...

...No 1. 2. 3. 4. 5. 6. 7. 8. Code: UCCM1153 Status: Credit Hours: 3 Semester and Year Taught:
Information on Every Subject Name of Subject: Introduction to Calculus and Applications
Pre-requisite (if applicable): None Mode of Delivery: Lecture and Tutorial Valuation: Course Work Final Examination 40% 60%
9. 10.
Teaching Staff: Objective(s) of Subject: • Review the notion of function and its basic properties. • Understand the concepts of derivatives. • Understand linear approximations. • Understand the relationship between integration and differentiation and continuity. Learning Outcomes: After completing this unit, students will be able to: 1. describe the basic ideas concerning functions, their graphs, and ways of transforming and combining them; 2. use the concepts of derivatives to solve problems involving rates of change and approximation of functions; 3. apply the differential calculus to solve optimization problems; 4. relate the integral to the derivative; 5. use the integral to solve problems concerning areas.
11.
12.
Subject Synopsis: This unit covers topics on Functions and Models, Limits and Derivatives, Differentiation Rules, Applications of Differentiation and Integrals.
13.
Subject Outline and Notional Hours: Topic Learning Outcomes 1 L 4 T 1.5 P SL 6.25 TLT 11.75
Topic 1: Functions and Models
• • • • • • Functions Models and curve fitting Transformations, combinations, composition and graphs of...

...Calculus
is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integralcalculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem ofcalculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-definedlimit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.
Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
BRANCHES OF CALCULUSCalculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If...

...How the calculus was invented?
Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was made by Isaac Newton and Gottfried Leibniz. Publication of Newton's main treatises took many years, whereas Leibniz published first (Nova methodus, 1684) and the whole subject was subsequently marred by a priority dispute between the two inventors of 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-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Elea discredited infinitesimals further by his articulation of the paradoxes which they create.
Antiphon and later Eudoxus are generally credited with 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 infinite number of recognizable shapes.
Archimedes of Syracuse developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See...

...History of Calculus
The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[1][2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2]
In AD 499 the Indian mathematician Aryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around AD 1000, the Islamic...

...English life and laws,
of England’s social organization, earned her a reputation
Line as a “subversive.” Her novels contain some of the earliest
5 literary attacks on the English legal system. In comparison
to later exposés by nineteenth-century novelists such as
Charles Dickens, Charlotte Smith’s attacks appear somewhat timorous. However, it cannot be denied that it was
Smith who introduced such a target for later novelists and
10 that when she did, her action was considered so audacious
that it laid her open to the charge of being a “menace.”
6. The reference to “clean linen” (line 2) primarily
serves to
(A)
(B)
(C)
(D)
(E)
8. The passage suggests that many of Smith’s critics
considered her novels to be
explain a course of action
evoke a particular sensation
describe an unexpected development
show nostalgia for a past experience
point out a pressing obligation
(A)
(B)
(C)
(D)
(E)
7. The word “colorful” in line 8 conveys a sense
of something
(A)
(B)
(C)
(D)
(E)
marred by frivolous and pointless descriptions
riddled with historical inaccuracies
harmful to the established social order
indifferent to questions of morality
disrespectful of British literary traditions
9. The author of the passage mentions Charles Dickens
primarily as an example of a novelist who
garish
robust
subtly hued
vividly descriptive
eye-catching
(A) found success by simply exposing rather than
attacking...

...CALCULUSCalculus is the study of change which focuses on limits, functions, derivaties, integrals, and infinite series. There are two main branches of calculus: differential calculus and integral calculus, which are connected by the fundamental theorem of calculus. It was discovered by two different men in the seventeenth century. Gottfried Wilhelm Leibniz – a self taught German mathematician – and Isaac Newton - an English scientist - both developed calculus in the 1680s. Calculus is used in a wide variety of careers, from credit card companies to a physicist use calculus in their work. In general, it is a form of mathematics which was developed from algebra and geometry.
Integration and differentiation are an important concept in mathematics, and are the two main operations in calculus. Differential calculus is a subfield of calculus which concentrates over the study of how functions change when their inputs are changed. The main focus in a differential calculus is the derivative which can be thought of as how much one quantity is changing in response to changes in some other quantity. The process to find the derivative is called differentiation, the fundamental theorem of calculus states that the differentiation is the reverse process to integration. Derivatives are mainly...

...Sustainability and Calculus
Introduction and Preview
Calculus is all about change. Calculus provides the mathematical tools to examine important questions about dynamic behavior; e.g. how fast is the world population increasing? If we continuously release a pollutant into a lake at a known rate, what's the total amount of pollutant that will be dumped into the water in the next five years? How long will the nonrenewable supplies of coal and oil last if we maintain the current per capita use but population continues to grow? How long will supplies last if industrialization and a "rising standard of living" push per capita usage ever higher? If we disturb the natural population dynamics of a salmon species by fishing, how should we regulate our removal of fish to provide a maximal sustainable yield in the future?
Although we will study many different applications of calculus in this introductory course, we will place a particular emphasis on the problem of sustainability.
Sustainability means meeting the needs of the present without compromising the ability of future generations to meet their own needs.
Our definition of sustainability comes from the 1987 publication Our Common Future by the United Nations World Commission on Environment and Development (also known as the Brundtland commission after its chair, Norwegian diplomat Gro Harlem Brundtland)
We will highlight two...

...History of CalculusCalculus is an integral part of the mathematics world. Various mathematicians coming from all parts of the world have shaped this theorem but the two main contributors are Sir Isaac Newton and Wilhelm Von Leibniz. The reason they are considered the inventors of Calculus is because they were able to give a unified approach to tangent and area problems unlike the others who used specific methods. Both of these mathematicians developed general concepts Newton was associated with the fluxion and the fluent as for Leibniz, he produced the differential and the integral.
Isaac Newton was a self-taught mathematic student who studied at Trinity College in Cambridge starting in 1661. He shaped his work in optics, celestial mechanics and mathematics, including calculus. His early work consisted of Analysis with Infinite Series in 1669 but his most famous work is the Mathematical Principles of Natural Philosophy published in 1687. Newton only introduced his notions of calculus in detail until the years 1704 to 1736.
Gottfried Wilhelm Leibniz was a German who at first, concentrated on the topics of philosophy and law but was introduced to advanced mathematics during a brief stay at the University of Jena in 1663. He worked on his calculus from 1673 to 1676 and revealed his work on differential calculus in 1684 with the integral calculus in 1686....

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