Diploma Plus in CAEM. Certificate in Advanced Engineering Mathematics. Ngee
Ann Polytechnic / School of Interdisciplinary Studies. 1. Diploma Plus in.
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011
Ngee Ann Polytechnic / School of Interdisciplinary Studies
1
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
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 student’s knowledge in these areas:
• • • • • • • • • • III.
Functions and graphs Sequences and Series Vectors Trigonometry Plane Analytic Geometry Complex Numbers Matrices Differentiation and Applications Integration and Applications Differential Equations 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 NP’s Maths OnLine for self-learning on campus as well as off-campus.
Ngee Ann Polytechnic / School of Interdisciplinary Studies
2
Diploma Plus in CAEM
IV.
Certificate in Advanced Engineering Mathematics
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 8th 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.
Ngee Ann Polytechnic / School of Interdisciplinary Studies
3
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
Module 1: Advanced Engineering Mathematics 1 (CAEM1) Level:
2.1
Lecture Hours: Tutorial Hours:
30 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 ݂ ሺݔሻ = ݔଶ + 5, ݂: ݔ → |ݔଶ + 5, ݂ ିଵ ሺݔሻ, ݂݃ሺݔሻ and ݂ ଶ ሺݔሻ 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 ݔ = ݕ
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 ݂ = ݕሺ ݔሻ as represented by = ݕ ݂ܽ ሺ ݔሻ , ݂ = ݕሺ ݔሻ + ܽ , ݂ = ݕሺ ݔ+ ܽሻ ܽ݊݀ ݂ = ݕሺܽݔሻ , and combinations of these transformations ଵ relating the graphs of |݂ሺݔሻ| , ݂ = ݕሺ||ݔሻ , = ݕ and ݕଶ = ݂ሺݔሻ to the graph ሺ௫ሻ
of ݂ = ݕሺݔሻ 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
• • •
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.
ሺ௫ሻ
ሺ௫ሻ
> 0 where ݂ሺ ݔሻand ݃ሺ ݔሻ are quadratic
Ngee Ann Polytechnic / School of Interdisciplinary Studies
4
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
2. Sequences and series 2.1. Summation of series • • • • • • • • • •
concepts of sequence and series relationship between ݑ (the n th term) and ݏ (the sum to n terms) sequence given by a formula for the ݊ th term sequence generated by a simple recurrence relation of the form ݔାଵ = ݂ሺݔሻ use of Σ notation summation of series by the method of differences convergence of a series and the sum to infinity binomial expansion of ሺ1 + ݔሻ for any rational ݊ condition for convergence of a binomial series proof by the method of mathematical induction
2.2. 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
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 ቀݕቁ , ቆݕቇ , ݅ݔ+ ݆ݕ, ݅ݔ+ ݆ݕ+ ݇ݖ,ሬሬሬሬሬሬሬሬԦ ܤܣ, ܽ
• • • • • •
ݔ
ݔ ݖ
position vectors and displacement vectors magnitude of a vector unit vectors distance between two points angle between a vector and the ݔ- , ݕ- or ݖ-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
Ngee Ann Polytechnic / School of Interdisciplinary Studies
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Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
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
Ngee Ann Polytechnic / School of Interdisciplinary Studies
6
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
Module 2: Advanced Engineering Mathematics 2 (CAEM2) Level:
2.2
Lecture Hours: Tutorial Hours:
30 15
Prerequisite:
Pass in Advanced Engineering Mathematics 1
Syllabus 1. 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
2. Plane Analytic Geometry & Complex Numbers • • • • • • • • • • •
Straight lines and conic sections Polar coordinates Equations and graphs in polar coordinates Plane curves and parametric equations complex roots of quadratic equations conjugate roots of a polynomial equation with real coefficients representation of complex numbers in Argand diagram Euler’s formula geometrical effects of conjugating a complex number and of adding, subtracting, multiplying, dividing two complex numbers loci such as | ݖ− ܿ| ≤ ݎ, | ݖ− ܽ| = | ݖ− ܾ| and arg ሺ ݖ− ܽ ሻ =∝ use of de Moivre’s theorem to find the powers and ݊th root
3. Matrices • • •
Finding the Inverse of a 3x3 Matrix Matrices and Linear Equations Gaussian Elimination
Ngee Ann Polytechnic / School of Interdisciplinary Studies
7
Diploma Plus in CAEM
Certificate in Advanced Engineering Mathematics
Module 3: Advanced Engineering Mathematics 3 (CAEM3) Level:
3.1
Lecture Hours: Tutorial Hours:
30 15
Prerequisite:
Pass in Advanced Engineering Mathematics 2
Syllabus 1.
Differentiation and Applications • • • • • • • • • • •
2.
Integration and Applications • • • • • • • • • • • •
3.
Overview of Differentiation Implicit differentiation Increments and differentials existence and non-existence of limits and continuity Newton’s iteration Indeterminate form and L’Hopital’s rule Rolle’s theorem Mean value theorem Differential of arc length Curve sketching Extreme values and inflection points
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 Simpson’s rule Volume of solid of revolution convergence/divergence of improper integrals
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
Ngee Ann Polytechnic / School of Interdisciplinary Studies
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Diploma Plus in CAEM
4.
Certificate in Advanced Engineering Mathematics
Sequences and Series • • • • • • • • •
Limits of sequences and series Test of convergence and divergence including Cauchy’s criteria Power series in one variable Interval of convergence Taylor series including Maclaurin Taylor’s theorem with remainder Fourier series: formulae for coefficients of a function Fourier series: half range expansion Fourier series: complex form
Ngee Ann Polytechnic / School of Interdisciplinary Studies
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