Handbook Contents |
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Module
Descriptions |
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MATHEMATICS
The
aim of this module is to provide a foundation of competence
in Mathematics that will be relevant for entry into undergraduate
science and engineering degrees. Studying and practicing Maths
is the BEST way to learn how to:
• think logically and carefully, and
• present ideas and arguments clearly and concisely.
These are skills that you will use every day in your career
as a scientist or engineer.
Module details
Language: English
& Chinese
Lectures: 30 lectures (2 hours lecture)
Tutorials: 10 (1 hour/tutorial)
Attendance: 12 hours per week
Duration
The
economics module is composed of 36 lectures, which are delivered
over a six week term. Each week their will be 12 hours of
lecture and 6 hours tuitorial. At the completion of the
coursework, sudents will have a two week study recess followed
by the final examination.
Useful
website
Online resources are available at www.mathscentre.ac.uk
They
have short leaflets, teach yourself booklets, on-line exercises.
There are even video tutorials that you can access from
University machines.
Text
Book
This
book has the best coverage of the
Foundation modules and lots and lots of exercises.
ISBN: 0748755098
Price: £16.00 @ Amazon
Croft
and Davison:
Foundation Mathematics
Particularly useful for those with little recent maths
practice.
ISBN: 0130454265
Price: £23.99 @ Amazon
Syllabus
1 |
Functions
and Graphs
The terms function, domain, range and one-one function.
• Composite functions and inverse of functions.
• Forms of the graphs of y = kxn where n is a positive
or negative
• integer or a simple rational, ax + by = c, x2/a2
+ y2/b2 = 1.
• Relationships between the graphs of y = f(x),
y = a f(x), y= f(x)+a, y= f(x+a), y= f (ax), where a is
a constant, expressed in terms of translations, reflections
and scalings.
• Relation of the equation of a graph to its symmetries.
• Coordinates of a point on a curve in terms of
a parameter.
Conic Sections
• Standard Equations of: (i) Parabola with Vertex;
(ii) Ellipse; (iii) Hyperbola;
• Standard Equations for Translated Conics |
2 |
Partial Fractions
• Expression of rational functions in partial fractions,
where the denominator is no more complicated than
(ax + b)(cx + d)(ex + f), (ax + b)(cx + d )2, (ax + b)(x2
+ c2). |
3 |
Inequalities; The
Modulus Function
• Properties of inequalities.
• Solution of inequalities reducible to the form
f(x) > 0, where f(x) can be factorised, and graphical
illustration of such solutions.
• The meaning of | x |.
• The relations |x – a | < b ? a –
b < x < a + b and | a | = | b | ? a2 = b2.
• Graphs of functions of the form y = | ax + b |. |
4 |
Logarimthmic and
Exponential Functions
• Laws of logarithms (including change of base)
• Graphs of simple logarithmic and exponential functions
• The definition ax = ex ln a
• Solutions of equations reducible to the form ax
= b. |
5 |
Matrices and Determinants
• Matrix Product
• Inverse of Square Matrix
• Basic Properties of Matrices
• Using Inverse Methods to Solve Systems of Equations
• Value of a Third-Order Determinant
• Summary of Determinant Properties
• Cramer's Rule for Three Equations and Three Variables |
6 |
Sequences and Series
Arithmetic and Geometric Progressions.
• The ? notation.
Binomial Expansions of
(i) (a + b)n, where n is a positive integer.
(ii) (1 + x)n, where n is rational.
• The notation n! (with 0! = 1) and |
7 |
Trignometry Relations
• Sine and cosine formulae.
• Angle between a line and a plane, between two
planes, and between two skew lines in simple cases. |
8-9 |
Trignometry Functions
and Calculus
• Geometric definitions from a right triangle
• Units of angle
• Trigonometric functions for all angles
• Special values and periodicity of trigonometric
functions
• Inverse trigonometric functions
• Some common applications of trigonometric functions
• Limiting values of trigonometric functions
• Derivatives of trigonometric functions
• Series expansions of trigonometric functions
• Hyperbolic functions |
10-11 |
Differentiation
and Limits
• The idea of a limit and the derivative defined
as a limit, including geometrical interpretation.
• The standard notations
• Derivatives of xn (for any rational n), sin x,
cos x, tan x, ex, ax,
• ln x, sin-1x, cos-1x, tan-1x; together with constant
multiples, sums, differences, products, quotients and
composites.
• Derivative of a function defined implicitly or
parametrically.
• Applications
• Stationary points.
• Tangents and normals to curves.
• Connected rates of change.
• Small increments and approximations.
Maclaurin Series
• First few terms of the Maclaurin series for a
function |
12-13 |
Integration
• Indefinite integration as the reverse process
of differentiation.
• Integration of xn (including the case where n
= -1), ex, sin x, cos x, sec2x, together with sums, differences
and constant multiples, linear substitution, partial fractions,
and trigonometrical identities.
• Integration by parts.
• Integration by substitution.
• Definite integrals.
• Areas and Volumes.
• Area under a curve.
• Plane areas and volumes of revolution in simple
cases.
• The Trapezium Rule. |
14 |
Calculus Applications
• Finding Extreme Values
• Equations Between Functions
• Inequalities Between functions
• Limit Computations and l’Hospital’s
Rule
• Areas of domains between the graph of a function
and the x-axis
• Areas of domains between the graphs of two functions
• Solids of revolution |
15 |
Permutations and
Combinations
• Arrangements of objects in a line or in a circle,
including those involving repetitions and restrictions. |
16-17 |
Vectors
• Rectangular cartesian coordinates in three dimensions
• The standard notations for vectors.
• Addition and subtraction of vectors, multiplication
of a vector by a scalar, and their interpretation in geometrical
terms.
• Unit vectors, position vectors and displacement
vectors.
• The magnitude of a vector and the scalar product
of two vectors. Angle between two directions and perpendicularity
of vectors. Equation of a straight line in vector and
in Cartesian forms.
• Point of intersection of two lines when it exists,
perpendicular distance from a point to a line, and angle
between two lines.
• The ratio theorem in geometrical applications |
18 |
Mathematical Induction
• The steps in the method of induction to establish
a given result, e.g. the sum of a finite series, or the
form of an nth derivative. |
20 |
Complex Numbers
• The idea of a complex number.
• Addition, subtraction, multiplication and division
of two complex numbers expressed in cartesian form.
• The relation z z* = | z |2.
• For a polynomial equation with real coefficients,
non-real roots occur in conjugate pairs.
• Argand diagrams.
• The polar form r(cos? + i sin?)= rei? .
• Geometrical effects of conjugating, adding, subtracting,
multiplying, and dividing two complex numbers.
• Loci in an Argand diagram. |
21 |
First Order Differential
Equations
• Formulation of a simple statement involving a
rate of change as a differential equation.
• Solution of a differential equation with separable
variables.
• Solution of a differential equation which is linear
by means of an integrating factor.
• Reduction of a differential equation to one with
separable variables or one which is linear by means of
a given simple substitution.
• General solution of a differential equation in
graphical terms as a family of curves.
• Particular solution to a differential equation,
and its interpretation in terms of the problem modeled
by a differential equation |
22 |
Numerical Methods
• Location of an approximate root of an equation
by means of graphical considerations and/or searching
for a sign change.
• The method of linear interpolation.
• The idea of, and the notation for, a sequence
of approximations converging to a root of an equation.
• Simple iterative formula of the form xn+1 = F(xn
).
• Determination of a root to a prescribed degree
of accuracy.
• The Newton-Raphson method in geometrical terms.
• Derivation and application of iterations based
on the Newton-Raphson method.
• Appreciation that an iterative method may fail
to converge to the required root. |
23 |
Forces and Equilibrium
• Forces acting in a given situation.
• Representation of forces by vectors, and by resultants
and components.
• Equilibrium of a particle under the action of
coplanar forces.
• Representation of the contact force between two
surfaces by the 'normal component' and the 'frictional
component'.
• The model of a 'smooth' contact and its limitations.
• Limiting friction and limiting equilibrium, the
definition of coefficient of friction, and the relationship
F = µR or F = µR.
• Newton's third law. |
24 |
Linear Motion
• Kinematics of Motion in a Straight Line
• The distance and speed as scalar quantities, the
displacement, velocity and acceleration as vector quantities,
and the relationships between them.
• The x-t and v-t graphs.
• Formulae for motion with constant acceleration
in a straight line. Newton's Laws of Motion
• Newton's first and second laws of motion applied
to the linear motion of a particle of constant mass moving
under the action of constant forces.
• The motion of two particles, connected by a light
inextensible string which may pass over a fixed smooth
light pulley or peg.
• Linear Motion under a Variable Force
• The linear motion of a particle of constant mass
moving under the action of variable forces.
• Solution of an appropriate differential equation
(first order with separable variables). |
25 |
Energy Work Power
• Work done by a constant force when its point of
application undergoes a displacement not necessarily parallel
to the force.
• Gravitational potential energy and kinetic energy.
• Relationship between the change in energy of a
system and the work done by the external forces; and the
principle of conservation of energy.
• Power as the rate at which a force does work,
and the relationship between power, force and velocity.
• Resistance to motion.
Hooke's Law
• Hooke's law as a model relating the force in an
elastic string or spring to the extension or compression,
and the term 'modulus of elasticity'.
• Elastic potential energy.
• Work and energy involving elastic strings and
springs |
26 |
Motion of Projectile
• The motion of a projectile as a particle moving
with constant acceleration, and the limitations of this
model.
• Horizontal and vertical equations of motion of
projectiles, the magnitude and direction of the velocity
at a given time or position, and the range on a horizontal
plane.
• The cartesian equation of the trajectory of a
projectile.
Uniform Circular Motion
• Angular speed for a particle moving in a circle
with constant speed.
• Acceleration of particle moving in a circle with
constant speed.
• Newton's second law applied to the motion of a
particle moving in a circle with constant speed. |
27 |
Probability
• Addition and multiplication of probabilities in
simple cases, and the representation of events by means
of tree diagrams.
• Mutually exclusive and independent events, and
conditional probabilities in simple cases.
• The notations P(A), P(A U B), P(A n B), P(A |
B).
• The equations P(A U u B) = P(A) + P(B) —
P(A n B) and
P(A n B) = P(A) P(B | A) = P(B) P(A | B). |
28 |
Discrete Random
Variables
• Concept of a discrete random variable.
• Probability distribution table, expectation and
variance.
• The uniform distribution and the Binomial distribution
B(n,p) as the probability model.
• The Poisson distribution Po(p) as the probability
model.
• Poisson distribution as an approximation to a
Binomial distribution. |
29 |
Continuous Random
Variables
• The probability density function and its properties.
• The mean, mode and variance of a distribution,
and the result
• Relationship between the probability density function
and the distribution function
• The median, quartiles and other percentiles
• The probability density function or the distribution
function in the context of a probability model.
Linear Combination of Random Variables
• Expectations, variance and distribution of linear
combinations of independent random variables |
30 |
The Normal Distribution
• The general shape of a Normal curve
• Standardised Normal variables and the Normal distribution
tables
• The normal distribution N(µs2) as the probability
model
• The Normal distribution of an approximation to
a bionomial distribution or a Poisson distribution |
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