1. Use the midpoint rule, the trapezium rule and Simpson’s rule to approximate the integral of the function f between and , using the values of f given in the following table. In each case use the maximum number of strips that you can with the given information.

2. Let f (x) be a function and consider . Let be the approximations given by the midpoint rule, the trapezium law and Simpson’s rule respectively all with n strips.

(i) Write down a formula which gives in terms of and .

(ii) Write down a formula which gives in terms of and .

(iii) Calculate and for the integral .

(iv) Calculate directly for the same integral.

(v) Verify that your answers to (iii) and (iv) agree with the formula you gave in (ii).

(20 marks)

3. Consider the integral .

(i) What is the exact value of this integral?

Let .

(ii) Calculate .

(iii) Compare your answers to (ii) to your answer to (i). Calculate the absolute error in each case. (iv) Complete the following table.

No of strips, to calculate Mn and Tn
4
8
10
Strip Width, h
1.25
0.625
0.5
Absolute Error, in

(v) From your table, to which power of h would you say the absolute error is proportional?

(25 marks)
Topic assessment Solutions

1. The midpoint rule is:

The maximum number of strips we can use with the available information is 5, each of width .

The trapezium rule is:

The maximum number of trapezia we can use with the available information is 10, each of width .

Simpson’s rule is:

The maximum number of strips is 10, each of width .

...History of mathematics
A proof from Euclid's Elements, widely considered the most influential textbook of all time.[1]
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available arePlimpton 322 (Babylonian mathematics c. 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-calledPythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greekμάθημα (mathema), meaning "subject of instruction".[4]Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning andmathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Chinese...

...Week 5
Final Exam
Continuous schedule from Friday , November 1st. 9am until Saturday , November 2nd., 23:59pm.
Monday, November 4, 2013
20%
100%
To obtain the opportunity to take your final exam you should have delivered at least 6 activities.
Please keep this Agenda at hand so that you can deliver you assignments on time.
Greetings,
Blanca Alanís
Posted by: BLANCA HILDA ALANIS PENA
Posted to: CEL.HI09107V.343.13320 Inglés VII
Bibliography
Posted on: Thursday, October 3, 2013
Hello guys,
The books we are going to use are:
Text book:
Richards, Jack C. & Sandy, Chuck (2009). Passages 2 (2nd ed.). New York, N.Y. Cambridge University Press.
ISBN 978-0-521-68391-3
Workbook:
Richards, Jack C. & Sandy, Chuck (2009). Passages 2 (2nd ed.). New York, N.Y. Cambridge University Press.
ISBN 978-0-521-68393-7
Make sure they are the 2nd. edition, because the 1st. edition is completely different.
In your course, in the Bibliography Section you have a link of a bookstore where you can buy the books. You can try other bookstores in your city, of course, but they don't usually have the book in stock.
Greetings,
Blanca Alanís
Posted by: BLANCA HILDA ALANIS PENA
Posted to: CEL.HI09107V.343.13320 Inglés VII
Grading in the courseWeek 5
Final Exam
Continuous schedule from Friday , November 1st. 9am until Saturday , November 2nd., 23:59pm.
Monday, November 4, 2013
20%
100%
To...

...concrete model.
Looking on the locality of the paper, I highly acknowledge the fact that the researchers described the current state of math education in the Philippines. They emphasized the fact that we are more focused on procedural knowledge rather than the more desired conceptual knowledge. That is our disadvantage because we usually train students to perform math without understanding or making connections on what they are doing. By mentioning this, the readers would really have an idea that the paper itself could be a solution to the problem mentioned. Moreover, it makes the thesis more realistic.
To sum up everything that was tackled, I could say that the thesis served to have an important purpose in the current state of Mathematics Education in the Philippines. It is very informative and feasible. Since it is a small study because it only involved 6 average students, we could propose more studies rooting from this which would have a bigger scope such as implementing the same study but now comparing it to the results gathered from high and low performing students....

...The numbers are overwhelming: Over the next 17 years, 350 million rural residents (more than the entire U.S. population today) will leave the farm and move to China’s cities. That will bring the Chinese urban population from just under 600 million today to close to 1 billion, changing China into a country where more than two-thirds of its people are city dwellers, says Jonathan Woetzel, a director in McKinsey’s Shanghai office. The change will reverse China’s centuries-old identity as a largely rural country. Thirty years ago, when China started modernizing its economy, more than 80% of Chinese lived in the countryside. And just six years ago it still was about 60%. Today China is just under 50% urban.
The newly urbanized population will live in eight megacities, those with a population of more than 10 million, as well 15 big cities with populations between 5 million and 10 million. In addition, by 2025 China will probably have at least 221 cities with a population over 1 million, estimates Woetzel. That compares with 35 cities of that scale across all of Europe today. These new urbanites are expected to be a powerful booster of growth: Urban consumption as a share of gross domestic product will most likely rise from 25% today to roughly 33% by 2025. “Urbanization is the engine of the Chinese economy—it is what has driven productivity growth over the last 20 years,” says Woetzel. “And China has the potential to keep doing this for the next 20 years.”...

...HISTORY OF MATHEMATICS
The history of mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. It has evolved from simple counting, measurement and calculation, and the systematic study of the shapes and motions of physical objects, through the application of abstraction, imagination and logic, to the broad, complex and often abstract discipline we know today.
From the notched bones of early man to the mathematical advances brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary developments of ancient Greece and its Hellenistic empire, the story of mathematics is a long and impressive one.
Prehistoric Mathematics
The oldest known possibly mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It consists of 29 distinct notches cut into a baboon's fibula. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old, suggest early attempts to quantify time.
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known...

... I. Number Systems in Mathematics:
A Number system (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing number of a given set, using graphemes or symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for othernumbers in different bases.
Ideally, a number system will:
* Represent a useful set of numbers (e.g. all integers, or rational numbers)
* Give every number represented a unique representation (or at least a standard representation)
* Reflect the algebraic and arithmetic structure of the numbers.
For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits. However, when decimal representation is used for the rational or real numbers, such numbers in general have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999... etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown....

...finishing respectively.
The company has 300 hours available for cutting, 240 hours for assembling and
120 hours finishing . Each english type earns $85 in profit while the italian type earns $75.
a. If you are required to find the combinations of both types of space-savers that will
[6 mks]
maximize profits, list the inequalities that would be necessarry.
Let x be number of english type space-savers made and sold
Let y be number of italian type space-savers made and sold
3x + 2y ≤ 300
2x + y ≤ 240
2x + 4y ≤ 120
x ≥ 0, y ≥ 0
b. From a graph of the inequalities in part 𝑎. , − 𝐲𝐨𝐮 𝐚𝐫𝐞 𝐧𝐨𝐭 𝐫𝐞𝐪𝐮𝐢𝐫𝐞𝐝 𝐭𝐨 𝐬𝐮𝐛𝐦𝐢𝐭 𝐭𝐡𝐞 𝐠𝐫𝐚𝐩𝐡
[6 mks]
𝐢𝐧 𝐲𝐨𝐮𝐫 𝐚𝐧𝐬𝐰𝐞𝐫𝐬 − list the vertices of the solution set.
The vertices of the solution set will be (0,0); (0,30); (60,0)
c.
Which of the combinations of the two types of space-savers will maximize profits?
[2 mks]
P = 85x + 75y
𝐴𝑡 (0,0); P = 85(0) + 75(0) = $0
𝐴𝑡 (0,30);
P = 85(0) + 75(30) = $2250
At (60,0); P = 85(60) + 75(0) = $5100
At (60,0) profits are maximised , produce 60 english type and no italian type.
d.
How much will be the maximum profit?
[2 mks]
$5100
Question 2 - Matrices
A security firm borrowed $900,000 for business expansion. Some of the money was borrowed at 5%,
some at 6%, and some at 7% simple annual interest. The total annual interest is $52,000 and
the amount borrowed at 6% is one and a half of that borrowed at 7%.
a. Create a system of equations which captures the essence of...

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