THE HISTORY OF CALCULUS
The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x' and y', which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis.

It is interesting to note that Leibniz was very conscious of the importance of good notation and put a lot of thought into the symbols he used. Newton, on the other hand, wrote more for himself than anyone else. Consequently, he tended to use whatever notation he thought of on that day. This turned out to be important in later developments. Leibniz's notation was better suited to generalizing calculus to multiple variables and in addition it highlighted the operator aspect of the derivative and integral. As a result, much of the notation that is used in Calculus today is due to Leibniz.

The development of Calculus can roughly be described along a time line which goes through three periods: Anticipation, Development, and Rigorization. In the Anticipation stage techniques were being used by mathematicians that involved infinite processes to find areas under curves or maximize certain quantities. In the Development stage Newton and Leibniz created the foundations of Calculus and brought all of these techniques together under the umbrella of the derivative and integral....

...mathematician. With Leibniz’s ideas and theories, he became a prominent figure in mathematics and was a major mathematician that greatly contributed to calculus.
Leibniz’s contribution to the invention of infinitesimal calculus was monumental and is widely recognized as modern mathematics starting point. Leibniz’s Nova Methodus pro Maximis et Minimis, itemque Tangentibus… in Acta Eruditorum, in 1684, published Leibniz’s details of his ideas of differentialcalculus. The paper contained the d notation, the derivatives of powers rule, the quotient rule, and product rule. But the journal did not contain any proof of the ideas. In Acta Eruditorum, a paper was published by Leibniz which dealt with calculus integrals and had the first appearance of the integral notation. The next year Newton’s Principia appeared and contained Newton’s ‘method of fluxions’ but Newton failed to get it published in 1671 which resulted in a major dispute between Newton and Leibniz. Newton claimed Leibniz’s ideas were plagiarized from his work and ideas but there was no proof of the claim. Newton’s approach to it was geometrical, while Leibniz’s approach was algebraical. Leibniz’s language was appropriate for the ideas presented. Leibniz’s crystallization of infinitesimal calculus is considered as his designing of a new problem solving process. His discovery of that calculus extended the mathematical treatment to...

...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...

...N.E.D University of Engg. & Tech. CS-14
Integral Calculus:
Definition:
“The branch of mathematics that deals with integrals, especially the methods of
ascertaining indefinite integrals and applying them to the solution of differential
equations and the determining of areas, volumes, and lengths.”
History of Integral Calculus:
Pre-calculus integration:
The first documented systematic technique capable of determining integrals is
the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC),
which sought to find areas and volumes by breaking them up into an infinite number
of shapes for which the area or volume was known. This method was further
developed and employed by Archimedes in the 3rd century BC and used to calculate
areas for parabolas and an approximation to the area of a circle. Similar methods
were independently developed in China around the 3rd century AD by Liu Hui, who
used it to find the area of the circle. This method was later used in the 5th century by
Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the
volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).
The next significant advances in integral calculus did not begin to appear until the
16th century. At this time the work of Cavalieri with his method of indivisibles, and
work by Fermat, began to lay the foundations of modern calculus, with Cavalieri
computing the integrals of xn up to degree...

...History and the Importance of CalculusCalculus can be summed up as "the study of mathematically defined change"5, or the study of infinity and the infinitesimal. The basic concepts of it include: limits, derivatives, differentiation and integrals. The word "calculus" means "rock"; the reason behind the naming of it is that rocks were used to used to carry out arithmetic. This branch of mathematics is able to be rooted all the way back to around 450 B.C., when Zeno of Elea discovered infinite numbers and distances. Later, in 225 B.C., Archimedes developed a formula for a sum of infinite series and also created the area of a circle and the volume of a sphere by using "calculus thinking". Not much progress took place until the 17th century, Pierre de Fermat looked at parabolas' maximum and minimum and discovered the tangent. Mathematicians Torricelli and Barrow then decided to put that tangent on a curved line, which can be used to calculate instantaneous rate of change.
Although all of these steps are relating to calculus, the branch was not officially introduced to the world until the 1640's. It has been said that it was specifically founded by two people--Isaac Newton and Gottfried Wilhelm Leibniz. Despite this synonymous finding, both mathematicians came up with completely different methods and notations. Newton had ideas that were based on limits and concrete concepts while Leibniz's views were...

...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...

...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...

...1. ht= -4.9t2+ 450, where t is the time elapsed in seconds and h is the height in metres.
a) Table of Values
t(s) | h(t) (m) |
0 | ht= -4.9(0)2+ 450= 450 |
1 | ht= -4.9(1)2+ 450= 445.1 |
2 | ht= -4.9(2)2+ 450= 430.4 |
3 | ht= -4.9(3)2+ 450= 405.9 |
4 | ht= -4.9(4)2+ 450=371.6 |
5 | ht= -4.9(5)2+ 450=327.5 |
6 | ht= -4.9(6)2+ 450= 273.6 |
7 | ht= -4.9(7)2+ 450= 209.9 |
8 | ht= -4.9(8)2+ 450= 136.4 |
9 | ht= -4.9(9)2+ 450=53.1 |
10 | ht= -4.9(10)2+ 450= -40 |
b) Average velocity for the first 2 seconds after the ball was dropped=
h2-h02-0 = 430.4-4502-0 = -19.62 = -9.8 m/s
c) Average velocity for the following time intervals:
i. 1≤t≤4 = h4-h14-1 = 371.6-445.14-1 = -73.53 = -24.5 m/s
ii. 1≤t≤2 = h2-h12-1 = 430.4-445.12-1 = -14.71 = -14.7 m/s
iii. 1≤t≤1.5 = h1.5-h11.5-1 = h1.5=-4.91.52+450-445.11.5-1 = 438.975-445.11.5-1
= -6.1250.5 = -12.25 m/s = -12.3 m/s
d) Instantaneous velocity at t= 1 second = h1-h0.751-0.75 = 445.1- h0.75= -4.90.752+ 4501-0.75 = 445.1-447.21-0.75 = -2.10.25 = -8.4 m/s
FIND GRAPH ON FOLLOWING PAGE
2. M=10.5-0.4t2, where M is the mass in grams and t is the time in seconds.
a) All the sugar has dissolved when M= 0 g
M=10.5-0.4t2
0=10.5-0.4t2
0.4t2=10.5
t2=10.50.4
t2=26.25
t2=√26.25
t=±5.12 s
Since t is time, the negative value cannot be considered and therefore M is 0 g when t= 5.12 s.
b) Average rate...

...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...