Name:
Student Number:
MAT271F
Term Test
October
10:10 am 11:00 am
No Aids allowed
It is not necessary to simplify numerical answers.
Answer all questions on the question paper.
50 minutes; 50 marks.
1. [8 marks] A bag contains a large number of marbles, o

- perspective drawing projects the scene into the plane of the paper
- projection lines, which represent parallel rays of light passing from objects of the scene, appear to converge at a certain point
as their go into the plane of the paper
- represents t

Keplers Laws
How do the planets move around the sun?
The problem is made much more dicult because we observe
it from a moving pla=orm.
horocycles
Astronomy was
considered a branch of
MathemaBcs, not part of
wh

Exponen'al Growth
John Napier,
Baron of
Merchistoun
(1550 1617)
Napiers tenants suspected that he was a
sorcerer, and he encouraged this idea.
He had a pet black rooster, and they thought it
was his fami

Proofs in Mathema/cs
In this sec/on, we discuss the proofs of a number of interes/ng
results in mathema/cs.
The main goal is to see how mathema/cal proofs are constructed.
But rst, we need some terminolog

Probability
Pierre de Fermat (1601 - 1665)
Blaise Pascal (1623 - 1662)
Much of the original discussion was framed in terms of
gambling, especially cards and dice.
Experiment
Take a coin.
Flip it ve Kmes.

The Mbius Strip
August Ferdinand Mbius (1790-1868)
Klein Bo4le
Klein Bo4le
In fact, if you cut the Klein bo4le in two, each
half is a Mbius strip.
In fact, if you cut the Klein bo4le in two, each
h

Music
Das Wohltemperierte
Klavier
1722
Physics
Sound is vibrations in the air.
In a musical instrument, something
vibrates (a string, a reed, a membrane,
etc).
The frequency of the vibration
determines the pitch
(high frequency ~ high pitch).
The frequenc

Applications of Game Theory
Human societies frequently have to make strategic decisions. In
primitive cultures, there were questions of where to hunt, where
to live, whether to engage in warfare, etc.
Many of these decisions could be analyzed in terms of

Game Theory
(continued)
A common type of problem involves finding an equitable division of
something.
A simple example is cutting a cake into two pieces.
A clever solution to that problem is to let one participant cut the
cake, and then let the other choo

Regression
Data
width
length
Data
width
3ng a curve
to the data
length
Data
width
length
width
length
width
length
Pheidole dentata
The larvae of this species of
ants are a favourite food of
arm

Perspec've
It is generally accepted that linear perspec've was invented by
Brunelleschi around 1413.
His discovery certainly revolu'onized art in Florence, and then
quickly spread throughout Italy and more slowly

INSIGHTS FROM MATHEMATICS
1. Lesson 2. Slices
In the first section, we explored the boundary of the hypercube, as illustrated in
the painting by Salvador Dali. Recall that the boundary consists of eight threedimensional cubes, attached in pairs across mut

TheCrucixion:
CorpusHypercubus
(1954)
Segments,Squares,Cubes
0
1
2
3
1
0
1
2
3
1
0
1
2
3
1
Segment
0x1
0
1
2
3
y
(x,y)
,y)
y
x
x
y
1
0x1
0y1
0
0
1
x
y
y=1
0x1
1
x=0
0y1
x=1
0y1
0
0
y=0
0x1
1
x
(x,y,z)
,y,z)
z
x
y
z
Cube
0x1
0y1
0z1
y
x
z
z=1
0x1
0y1
y=1

Gene$cs
Each individual has gene$c informa$on encoded in DNA
molecules, one strand inherited from each parent. Each DNA
molecule is a double helix, but for our purposes, it can be
regarded as a long chain.

Sickle Cell Anemia
Sickle cell anemia is a serious gene2c disease. It appears in
individuals who have two copies of an allele c, but not in those
whose genotype is CC or Cc. The disease is caused by the

Johann Carl Friedrich Gauss (1777 1855)
Gausss father owned a butcher shop. The rst sign of his
sons unusual gi9s occurred when he was three. His father
was adding up a column of numbers and made a mistake;
young

Perspec've
(con'nued)
Perspec've drawing addresses the issue of producing a two-
dimensional image of a three-dimensional object.
The simplest way to make such an image is simply to project the
object onto a plane,

Perspec've
Similar triangles
ra'os of corresponding sides are
equal
vanishing
point
Mount Edith Cavell
Shen Zhou:
Poet on a Mountain (c. 1500)
Claude
Lorrain:

The Bridges of Knigsberg
(Kaliningrad)
Is it possible to devise a path which
crosses each bridge once and not
more than once?
Leonhard Paul Euler (1707 1783),
Swiss mathema=cian and physicist
L
Leonhard Eu

Arrows Impossibility Theorem
(Arrows Paradox)
Many forms of
Government have been
tried, and will be tried in
this world of sin and
woe. No one pretends
that democracy is
perfect or all-wise.
Indeed, it has been said
that democracy is the
worst form of
Gov

Game Theory
Game theory is an attempt to model the way decisions are made
in competitive situations.
It has obvious applications in economics.
But it has also been applied to a huge range of other areas,
including politics, philosophy, biology, and comput

Sex Ra*o
(the ra*o of males to females)
Chromosomes
In humans and most other mammals, sex is determined by the X
and Y chromosomes.
Females have two X chromosomes, and males have one X and
one Y c

Sex Ratio
(the ratio of males to females)
Chromosomes
In humans and most other mammals, sex is determined by the X
and Y chromosomes.
Females have two X chromosomes, and males have one X and
one Y chromosome.
When offspring are produced, there is a 50/50

The Binomial Theorem
(which really has something to do
with probability)
The Binomial Theorem has to do with taking a binomial,
something like (a+b), and raising it to a power: (a+b)n
(a+b)2 = a2 + 2ab + b2
3
(a+b)3 = a(a+b)
+ 3a32b
= ?+ 3ab2 + b3
3
(a+b)

The Binomial Theorem
(which really has something to do
with probability)
The Binomial Theorem has to do with taking a binomial,
something like (a+b), and raising it to a power: (a+b)n
(a+b)2 = a2 + 2ab +

Proofs in Mathematics
In this section, we discuss the proofs of a number of interesting
results in mathematics.
The main goal is to see how mathematical proofs are constructed.
But first, we need some terminology.
The natural numbers are the positive inte

Keplers Laws
How do the planets move around the sun?
The problem is made much more difficult because we observe
it from a moving platform.
horocycles
Astronomy was
considered a branch of
Mathematics, not part of
what we would call
Physics.
Early astronome

Name:
Student Number:
MAT271F
Term Test
October
10:10 am 11:00 am
No Aids allowed
It is not necessary to simplify numerical answers.
Answer all questions on the question paper.
50 minutes; 50 marks.
1. [8 marks] A bag contains a large number of marbles, o