1. Calculus is split into two branches, differential calculus and integral calculus. Differential calculus is used to find the rates of change for geometric curves. This means that differential calculus is used to find the slope or tangent along a specific direction of a geometric curve. This relates directly to change, because finding the slope or tangent of a geometric curve is essentially finding the rate of change for that geometric curve. The other branch of calculus, integral calculus, is concerned with finding the area under a curve. This is accomplished by using small towers, to find the closest area of the curve. This relates to change because you can find the difference in area of the curve, depending on the equation’s curve’s restrictions. The relationship between these two branches is that they are inverse operations. This means that taking a derivative after finding an integral, would leave you with the original equation. 2. The method of exhaustion is a process that is still used today in order to find the area of irregular shapes that typically have curved bounded areas. The process uses regular polygons to approximate the area of irregular polygons. This is done by the process of circumscription, which means that a regular polygon is placed around the irregular polygon, but each of the corners of the regular polygon will touch the edge of the circle. Then, another regular polygon is inscribed within the irregular polygon. This means that the regular polygon is placed within the irregular polygon, with each of the corners just touching the edges of the irregular polygon. The word exhaustion is accurately used in the description of this method, because by adding more and more regular polygons to inscribe and circumscribe around the irregular polygon, the extra space is actually exhausted so that the closest area approximation can be found.

3. Isaac Barrow was an English mathematician, and was the teacher of Isaac Newton. Barrow is...

...History of Differential Calculus
Universidad Iberoamericana
September 20, 2013
Ever since men felt the need to count, the history of calculus begins, which together with Mathematics is one of the oldest and most useful science. Since men felt that need for counting objects, this need led to the creation of systems that allowed them to maintain control of their properties. They initially did it with the use of fingers, legs, or stones. But as humans continued developing intellectually, they achieved to implement systems or more advanced forms that allowed them to solve problems.
The Egyptians were the first civilization to develop mathematical knowledge. They devised numeral systems through hieroglyphs, representing the numbers 1, 10 and 100 through sticks and human figures. This system evolved into what we know today as the Roman system. Other important civilizations in history, such as the Babylonians, created other numeral systems, where the solution to the problem of counting the objects was solved with the implementation of a sexagesimal method. Civilizations as ancient China and ancient India used a hieroglyph decimal system, with the characteristic that these implemented the number zero.
The progress achieved ever since each culture implemented their numeral system are still used today. The algebraic advance of the Egyptians resulted in the resolution to significant equations. The correct...

...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...

...discoveries, calculus was the culmination of centuries of work rather than an instant epiphany. Mathematicians all over the world contributed to its development, but the two most recognized discoverers of calculus are Isaac Newton and Gottfried Wilhelm Leibniz. Although the credit is currently given to both men, there was a time when the debate over which of them truly deserved the recognition was both heated and widespread. Evidence also shows that Newton was the first to establish the general method called the "theory of fluxions" was the first to state the fundamental theorem of calculus and was also the first to explore applications of both integration and differentiation in a single work (Struik, 1948). However, since Leibniz was the first to publish a dissertation on calculus, he was given the total credit for the discovery for a number of years. This later led, of course, to accusations of plagiarism being hurled relentlessly in the direction of Leibniz. It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics (Boyer, 1968). In fact, Newton first described his methods, formulas and concepts of calculus, including his binomial theorem, fluxions and tangents, in letters he wrote to Leibniz (Ball, 1908). However an examination of Leibniz' unpublished manuscripts provided evidence that despite his correspondence...

...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...

...
1a. h=-4.9t^2+450
1b. h(t)=-4.9t^2+450
(h(2)-h(0))/(2-0)
((-4.9(〖2)〗^2+450)-(-4.9(0)^2+450))/2
=(430.4-450)/2
=-19.6
∴The average velocity for the first two seconds was 19.6 metres per second.
c. i)
i)
=
=-24.5
∴ The average velocity from is 24.5 metres/s.
ii)
= -14.7
iii)
= -12.25
∴ The average velocity from is 12.25 metres/s.
d) Instantaneous velocity at 1s:
=-9.8
∴ The instantaneous velocity at 1s is 9.8 metres/s.
2a)
=
=
=
=
=
b)
=
∴ The average rate of change from is -0.4g/s.
C)
∴The instantaneous rate at t = 2 seconds is -1.6g/s
3)
b)
=
=
=22
∴ from seconds the car moves at an average of 22m/s
c)
t=4
=
=16
∴ The instantaneous rate at 4s is 16m/s
4a) In order to determine the instantaneous rate of change of a function using the methods discussed in this lesson, we would use the formula where h will approach 0, and the closer it gets to 0 the more accurate our answer will be.
4b)
∴
=1
Therefore, = 1
5a)
Therefore the instantaneous rate at x=2 is 0.
5b)
Therefore at t=4 the instantaneous rate is 0 and the particle is at rest.
6a)
Rate of change is positive when:
Rate of change in negative when:
6b)
Rate of change is 0 when:
X=-1, x=1
6c)
Local Maximum: (-1,2)
Local Minimum: (1,-2)...

...“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...

...X 410 “Business Applications of Calculus”
PROBLEM SET 1 [100 points]
PART I
As manager of a particular product line, you have data available for the past
11 sales periods. This data associates your product line’s units sold “x” and
total PROFIT “P” results for these sales periods.
Product
Red03
Units [x]
Profit [P]
10
20
100
130
190
240
300
320
380
430
500
-33986
-31792
-9200
790
21418
37728
54000
58208
65840
65050
50000
1
Section A: 1st Order Model
1. [4] Use Microsoft Excel’s Chart feature to graph a plot of the data,
assuming P = (x). Add the most appropriate 1st order “trend line”, the
equation of this line, and the equation’s coefficient of determination—its
“[(R2)]”.
Profit [P]
P(x)= 215.51x - 26052
R² = 0.8623
100000
Profit P(x) (dollars)
80000
60000
40000
Profit [P]
20000
Linear (Profit [P])
0
-20000
-40000
0
200
400
600
Units (x)
2. Answer the following questions using this 1st order model. Assume that,
unless otherwise indicated, the restricted domain for “x” is 0 ≤ x ≤ 510 units.
a. [4] Estimate Profit “P” @ “x” = 0 units and “x” = 70 units.
( )
( )
( )
(
)
(
(
)
)
b. [4] Estimate how many units “x” of the product must be sold in order
to generate a PROFIT of $0.00 and a PROFIT of $35,000.
( )
( )
2
( )
( )
( )
( )
c. [4] Calculate how many product units “x” should be sold per sales
period to...