CAEM Syllabus Effective Apr 2012

Diploma Plus in CAEM Certificate in Advanced Engineering Mathematics
Revised 18 Mar 2012

Ngee Ann Polytechnic / School of Interdisciplinary Studies 1

Diploma Plus in
Certificate in Advanced Engineering
Mathematics

Syllabus from April 2012
Revised 18 Mar 2012
Ngee Ann Polytechnic / School of Interdisciplinary Studies
2
APPENDIX A
I. SYNOPSIS

This course of advanced standing is designed for top Ngee Ann students who wish
to further strengthen their mathematical foundation for entry into a university
engineering degree programme.

A student who has successfully completed three Maths modules will be awarded the
Certificate in Advanced Engineering Mathematics under the framework of Diploma
Plus Programme.

II. MODULE OBJECTIVES

Upon completion of the programme, the student will be considered proficient to gain
exemption from NUS MA1301 Proficiency Test and the level of proficiency will be on
par with or better than an A-level student. The course aims to broaden and deepen
students knowledge in these areas:

Functions and graphs
Sequences and Series
Vectors
Trigonometry
Plane Analytic Geometry
Complex Numbers
Matrices
Differentiation and Applications
Integration and Applications
Differential Equations

III. TEACHING AND LEARNING

The teaching and learning applied in each module include lectures cum consultation
classes with a maximum of 3 contact hours per week. Students will also be able to
use NPs Maths OnLine for self-learning on campus as well as off-campus.

Revised 18 Mar 2012
3
IV. ASSESSMENT

NUS assessment scheme for engineering mathematics is adopted by the
programme to ensure that adequate rigour is given in preparing the students. The
assessment will consist of the following components:
Common Test (1. 5 hours) 20%
The Common Test will be conducted by the 8
th
week.

Assignments 10%
Each assignment must be submitted within one week after the completion of every
chapter.

Final Examination (2 hours) 70%
The final examination ties together all the elements of the mathematics studies in
that semester. The emphasis in the final examination is on applications of
mathematics.

Revised 18 Mar 2012
4
Module 1: Advanced Engineering Mathematics 1 (CAEM1)

Level: 2.1

Lecture Hours: 30
Tutorial Hours: 15

Prerequisite: Minimum entry requirements:
(i) GPA: 3.0; and
(ii) EG2: B

Syllabus

1. Functions and graphs

1.1 Functions, inverse functions and compose functions

concepts of functions, domain and range
use of notations such as ()
() () and
()
finding inverse functions and composite functions
conditions for the existence of inverse functions and composite functions
domain restriction to obtain an inverse function
relationship between a function and its inverse as reflection in the line

1.2 Graphing techniques

relating the following equations with their graphs
characteristics of graphs such as symmetry, intersections with the axes, turning
points and asymptotes
determining the equations of asymptotes, axes of symmetry, and restrictions on
the possible values of x and/or y
effect of transformations on the graph of () as represented by
() () ( ) () , and combinations of
these transformations
relating the graphs of () ()
()
and
() to the graph
of ()
simple parametric equations and their graphs
use of CAS to verify graphing of a given function

1.3 Equations and inequalities

interval notation
solving inequalities of the form
()
()
where ()and () are quadratic
expressions that are either factorisable or always positive
solving inequalities by graphical methods
formulating an equation or a system of linear equations from a problem situation
finding the numerical solution of equations (including system of linear equations),
using CAS to verify results.

Revised 18 Mar 2012
5
2. Trigonometry

Angles and their measures
Trigonometric functions : unit circle approach
Properties of trigonometric functions
Graphs of trigonometric functions
The inverse trigonometric functions
Sum and difference formulas
Double-angle and half-angle formulas
Product-to-sum and sum-to-product formulas
Solving acosx + bsinx = c
Proof of trigonometric identities
Solutions of trigonometric equations
Right triangle trigonometry
The law of sine
The law of cosine
Problems in three dimensions

3. Plane Analytic Geometry

Straight lines and conic sections
Polar coordinates
Equations and graphs in polar coordinates
Plane curves and parametric equations

4. Matrices

Finding the Inverse of a 3x3 Matrix
Matrices and Linear Equations
Gaussian Elimination

5. Arithmetic and geometric series

formula for the th term and the sum of a finite arithmetic series
formula for the th term and the sum of a finite geometric series
condition for convergence of an infinite geometric series
formula for the sum to infinity of a convergent geometric series
solving practical problems involving arithmetic

Revised 18 Mar 2012
6
Level: 2.2

Lecture Hours: 30
Tutorial Hours: 15

Prerequisite: Pass in Advanced Engineering Mathematics 1

Syllabus

1. Sequences and series

1.1 Summation of series

concepts of sequence and series
relationship between
(the n th term) and
(the sum to n terms)

sequence given by a formula for the th term
sequence generated by a simple recurrence relation of the form
()
use of notation
summation of series by the method of differences
convergence of a series and the sum to infinity
binomial expansion of ( )
for any rational

condition for convergence of a binomial series
proof by the method of mathematical induction

2. Complex Numbers

complex roots of quadratic equations
conjugate roots of a polynomial equation with real coefficients
representation of complex numbers in Argand diagram
Eulers formula
geometrical effects of conjugating a complex number and of adding, subtracting,
multiplying, dividing two complex numbers
loci such as and arg ( )
use of de Moivres theorem to find the powers and th root

3. Vectors

3.1 Vectors in two and three dimensions

addition and subtraction of vectors, multiplication of a vector by a scalar, and
their geometrical interpretations
use of notations such as (
) (

position vectors and displacement vectors
magnitude of a vector
unit vectors
distance between two points
Revised 18 Mar 2012
7
angle between a vector and the - , - or -axis
use of the ratio theorem in geometrical applications

3.2 The scalar and vector products of vectors

concept of scalar product and vector product of vectors
calculation of the magnitude of a vector and the angle between two directions
calculation of the area of triangle or parallelogram
geometrical meanings of and where is a unit vector

3.3 Three-dimensional geometry

vector and Cartesian equations of lines and planes
finding the distance from a point to a line or to a plane
finding the angle between two lines, between a line and a plane, or between two
planes
relationships between
two lines ( coplanar or skew)
a line and a plane
two planes
three planes
find the intersections of lines and planes

Revised 18 Mar 2012
8


Level: 3.1

Lecture Hours: 30
Tutorial Hours: 15

Prerequisite: Pass in Advanced Engineering Mathematics 2

Syllabus

1. Differentiation and Applications

Overview of Differentiation
Implicit differentiation
Increments and differentials
existence and non-existence of limits and continuity
Newtons iteration
Indeterminate form and LHopitals rule
Rolles theorem
Mean value theorem
Differential of arc length
Curve sketching
Extreme values and inflection points

2. Integration and Applications

Anti-differentiation
Fundamental theorem of calculus
Basic rules of integration
Integration of polynomial and trigonometric functions
Integration of exponential and logarithmic functions
Integration by substitution
Integration by parts
Riemann sum and area under a curve
Finding the area under a curve defined parametrically
Trapezoidal rule and Simpsons rule
Volume of solid of revolution
convergence/divergence of improper integrals

3. Differential Equations
Overview of differential equations
Formulating a differential equation from a problem situation
Use of a family of solution curves to represent the general solution of a differential
equation
Interpretation of a solution in terms of the problem situation

Revised 18 Mar 2012
9
4. Sequences and Series

Limits of sequences and series
Test of convergence and divergence including Cauchys criteria
Power series in one variable
Interval of convergence
Taylor series including Maclaurin
Taylors theorem with remainder
Fourier series: formulae for coefficients of a function
Fourier series: half range expansion
Fourier series: complex form

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Diploma Plus in CAEM Certificate in Advanced Engineering Mathematics

Revised 18 Mar 2012

(the n th term) and

(the sum to n terms)

for any rational

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