Download mathematics department 2003/2004

ST. FRANCIS' CANOSSIAN COLLEGE
MATHEMATICS DEPARTMENT 2003/2004
CURRICULUM PLANNING
Subject: Pure Mathematics
Class: 6Sc
Subject Teacher(s): Ms Q. Kwok
MONTH UNIT
BASIC CONTENT / OBJECTIVES
Sept 2003 1
Preliminary Chapter
- to introduce terminologies and notations
Sept
2
Mathematical Induction
- to introduce the principle and use of the method
of mathematical induction in mathematical proof
- to pave the foundation of logical argument and
reasoning
Sept
3
Indices and Logarithm
Oct
4
Binomial Theorem
- to learn binomial expansion with positive
exponent
- to acquire some methods in summing series
DETAILED CONTENT
1. Terminologies and the Number system
2. Logic and set notations
1. First principle of mathematical induction
2. Second principle of mathematical induction
1. changing base
1. Binomial theorem for positive integral
exponents
2. properties of binomial coefficients
3. methods of summing general series
- by differentiation
- by integration
- method of difference
-1-
Remarks
MONTH
Nov
Nov
Nov
UNIT
BASIC CONTENT / OBJECTIVES
5
Determinant and Matrices
- to introduce the idea of matrices and
determinants
- to acquire skills in solving system of linear
equations
- to understand of idea of transformation and to
equip with skills in relating transformation
with matrices
DETAILED CONTENT
1. idea of determinants
2. expansion of determinants
3. properties of determinants
4. simple treatment in solving system of linear
equations - use of Cramer's rule
5. idea of matrices
6. addition, subtraction, scalar multiplication and
multiplication of matrices
7. inverse of matrices
8. matrices as transformation
- translation, reflection, rotation and shear
9. more detailed treating in solving system of linear
equations - Gaussian elimination
COMMON TEST
6
Polynomials
- to introduce the idea of polynomial
- to study the properties of polynomial
- to equip with more abstract way of thinking
and reasoning through the study of the
abstract theorems and proofs in polynomial
1. idea of polynomial
2. theorems concerning polynomials
- Division algorithm
- Remainder theorem
- Euclidean algorithm: use of the algorithm in
finding the GCD of polynomials
-2-
Remarks
MONTH
Nov/Dec
Dec/Jan
2004
Jan 2004
Feb
UNIT
BASIC CONTENT / OBJECTIVES
7
Simple Partial Fraction
- to acquire skills in resolving partial fraction
DETAILED CONTENT
1. idea of rational functions
3. methods of resolving rational functions
into partial fractions
HALF-YEAR EXAMINATION
8
9
Polynomial Equation
- to learn the properties of polynomial equations
- to study the condition for multiple roots
Inequalities
- to study the various techniques used in proving
inequalities
1. relations between roots and coefficients of
an equation
2. multiple roots
3. properties of polynomial equations
1. elementary properties of inequalities
2. idea of intervals
3. triangle inequalities
4. A.M.  G.M.
5. Cauchy-Schwarz inequality
6. general techniques in proving inequality
- by proving increasing / decreasing
function
- by locating max / min of function
- working backwards
-3-
Remarks
Feb/
March
10
Sequence
- to introduce the idea of limit of sequence
- to study the properties of limits
- to lay the foundation of calculus
March
11
Limit of function and Continuity
- to study the idea of the limit of a function
- to study the properties of continuous functions
March.
1. limit of sequence
2. idea of
- convergent / divergent sequence
- bounded sequence
- monotonic / strictly increasing /
decreasing sequence
3. general theorems on limits of sequence;
Sandwich Theorem
4. some important limits of sequence – e
1. idea of functions; types of functions
2. limit of a function
3. theorems on limits of function
4. idea of continuity / discontinuity
5. properties of continuous functions
TEST
-4-
Microsoft Excel
Spreadsheets for
properties of
sequences
Excel
Spreadsheets
March/
April
12
Differential Calculus
- to study the idea of differentiability
- to study the method in finding the derivative of
function from first principle
- to study the methods of finding higher order
derivative
- to study limits in indeterminate form
- to study the application of differentiation
- to acquire skills in curve sketching
- to study and learn to apply the mean-value
theorem
1. differentiability; idea of derivative
2. finding derivative from first principle
3. logarithmic differentiation
4. higher order derivative - Leibnitz' Rule
5. Limit in indeterminate form - L'Hopital Rule
6. concave / convex function
7. maxima and minima
8. point of inflexion
9. application to
- small increase
- curve sketching; idea of asymptote
10. Rolle's theorem
11. Mean-value theorem; its application
-5-
- Calculus
Learning
packages will also
be used packages
- IT Graphical
sketching programs will
be used to sketch graphs
of functions
MONTH
April/
May
UNIT
BASIC CONTENT / OBJECTIVES
13
Integral Calculus
- to study the idea of integrability
- to study the various methods of integration
- to study the fundamental theorem of calculus
- to introduce the idea of improper integral
- to apply integration technique in solving
practical problems
DETAILED CONTENT
1. Riemann Sum / integrability
2. use of Riemann sum in evaluating
infinite series
3. properties of definite integral
4. indefinite integral
5. First and Second Fundamental
Theorem of Calculus
6. integration by substitution /
integration by parts
7. integration of rational functions
8. reduction formula
9. Improper integral
10. application to the finding of
- plane area
- arc length
- volume of solids of revolution
- area of surface of revolution
11. Integral mean-value theorem; its
application
-6-
Remarks
Calculus Learning
packages will also be
used
MONTH
May
June
UNIT
BASIC CONTENT / OBJECTIVES
14
Complex Number
- to investigate the various properties of complex
number
- to study the De Moivre's theorem and to use it
in factorizing polynomials
- to study complex value function as an example
of function mapping from a two dimensional
space to another two dimensional space
DETAILED CONTENT
1. idea of complex number
2. operations with complex numbers
3. conjugate / properties of conjugates
and its relation with modulus
4. polar form
5. geometric representation of complex
number / Argand diagram
6. geometric interpretation of the
operations of complex number
7. Complex value function
8. De Moivre's Theorem
9. Nth root of unity / of a complex
number
10. Factorization of polynomial
FINAL EXAMINATION
-7-
Remarks
Use of complex numbers
in “daily life”
engineering examples