15 Integration 2 Further Techniques
Johann Bernoulli, Jakobs
brother, was born on 27 July
1667 and was the tenth child
of Nicolaus and Margaretha
Bernoulli. Johanns father
wished him to enter the
family business, but this did
not suit Johann and in the
en
17 Complex Numbers
Abraham de Moivre was born in
Vitry-le-Franois in France on 26
May 1667. It was not until his late
teenage years that de Moivre had
any formal mathematics training. In
1685 religious persecution of
Protestants became very serious in
Fra
11 Matrices
The concept of matrices and determinants was probably first understood by the
Babylonians, who were certainly studying systems of linear equations. However, it was
Nine Chapters on the Mathematical Art, written during the Han Dynasty in China
3 Functions
Calculators are an integral part of school mathematics, and this course requires the
use of a graphing calculator. Although much of the content of this chapter is
ancient mathematics in that it has been studied for thousands of years, the use
9 Differentiation 2 Further
Techniques
Leonhard Euler is considered
to be one of the most
important mathematicians of
all time. He was born on 15
April 1707 in Basel,
Switzerland, and died on 18
September 1783 in St
Petersburg, Russia, although
he spent m
13 Vectors, Lines and Planes
Stefan Banach was born in Krakw,
Poland (at the time part of the AustroHungarian Empire) on 30 March
1892. During his early life Banach was
brought up by Franciszka Plowa, who
lived in Krakw with her daughter
Maria. Marias gua
7 Trigonometry 2
Consider the equation below. If all angles are in radians
arctan112 arctan122 arctan132 p
Can you prove this?
The mathematics behind the fact
The hint is in the diagram, since we can check that the angles of the three triangles at
their c
1 Trigonometry 1
Although most people
connect trigonometry
with the study of
triangles, it is from the
circle that this area of
mathematics originates.
The study of trigonometry is not new. Its roots come from the Babylonians around
300 BC. This area of m
5 Exponential and Logarithmic
Functions
In this chapter we will meet logarithms,
which have many important applications,
particularly in the field of natural science.
Logarithms were invented by John Napier
as an aid to computation in the 16th
century.
Jo
20 Probability
Have you ever asked around a group of your classmates and been surprised to find
that two of them share the same birthday? At first thought it seems likely that for two
people in the room to share a birthday there will need to be a lot of p
Answers
6 a Consistent. Lines intersect giving unique solution.
7 p3
b Consistent. Same line giving infinite solutions.
Chapter 11
Exercise 4
1 a x 1, y 9, z 13
b x 10, y 10, z 36
c x 2, y 4, z 3
d x 2, y 1, z 2
2 b x 2, y 1, z 4
c x 1, y 1, z 2
d x 4, y
8 Differential Calculus 1 Introduction
The ideas that are the basis for calculus have been with us for a very long time.
Between 450 BC and 225 BC, Greek mathematicians were working on problems that
would find their absolute solution with the invention of
22 Continuous Probability
Distributions
Binomial distributions for 2, 4 and 12 throws
0.7
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Number of sixes
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Number of sixes
3
4
0.35
0.30
0.25
Probability
In Chapter 21,
14 Integration 1
Jakob Bernoulli was a Swiss
mathematician born in Basel,
Switzerland, on 27 December 1654.
Along with his brother, Johann, he is
considered to be one of the most
important researchers of calculus after
Newton and Leibniz. Jakob studied
th
16 Integration 3 Applications
When students study integral
calculus, the temptation is to see it
as a theoretical subject. However,
this is not the case. Pelageia
Yakovlevna Polubarinova Kochina,
who was born on 13 May 1899 in
Astrakhan, Russia, spent muc
2 Quadratic Equations, Functions
and Inequalities
The first reference to quadratic equations
appears to be made by the Babylonians in
400 BC, even though they did not actually
have the notion of an equation. However,
they succeeded in developing an algori
12 Vector Techniques
Bernard Bolzano was born in 1781 in
Prague in what is now the Czech Republic.
During Bolzanos early life there were two
major influences. The first was his father,
who was active in caring for others and
the second were the monks who
18 Mathematical Induction
Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji was born on 13 April 953 in
Baghdad, Iraq and died in about 1029. His importance in the field of mathematics is
debated by historians and mathematicians. Some consider that he only re
4 Polynomials
The Italian mathematician Paolo Ruffini,
born in 1765, is responsible for
synthetic division, also known as
Ruffinis rule, a technique used for the
division of polynomials that is covered
in this chapter.
Ruffini was not merely a mathematici
6 Sequences, Series
and Binomial Theorem
Pascals triangle is constructed by
1
adding together the two numbers
1
1
above as shown (with a 1 on the end
of each row). There are many
1
2
1
interesting results and applications
related to this triangle. For exa
10 Differentiation 3 Applications
Differential calculus is widely used in both the natural sciences and the human sciences. In
physics, if we want to investigate the speed of a body falling under gravity, the force that
will give a body a certain accelera