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Transcript
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