# Mathematics

1. The IUT pre-university mathematics curriculum offers its students chances to study mathematics more thoroughly and systematically, supplementing their high school class work.

2. The pre-university courses in mathematics are designed to help students to understand fundamental concepts and work out more challenging problems.

3. The IUT pre-university mathematics program offers three levels of curriculum depending on the students’ mathematical knowledge and problem-solving abilities.

Basic Level

– The basic level is designed for high school students with a working knowledge of about 1th or higher grade mathematics.

– In particular, students who register at this level should know in advance how to manipulate real numbers and polynomial expressions, draw graphs of linear, quadratic and trigonometric functions, understand geometry on Euclidean plane, solve quadratic equations, and prove mathematical statements from a given set of conditions, etc.

– The curriculum at this level provides the students with fundamental mathematical concepts including complex numbers, matrices, exponents, logarithms, sequence of numbers, and infinite series and basic techniques in calculus involving polynomial functions only

– The courses in the basic level will help students to progress to the intermediate or advanced level.

 № Subject Contents 1 Complex numbers Definition of complex numbers, their additions, multiplications, polar form r(cos a + i sin b) 2 Matrices and their operations Definition of matrices, their additions and multiplications, multiplication of matrices with a real number, and other properties 3 Matrices and system of linear equations Inverse matrix of matrix, formula of inverse matrix for a 2×2 matrix, Use of matrices to solve system of 2 linear equations with 2 variables 4 Exponents Manipulation of exponents and n-th root, how to define a real power of a positive real number 5 Exponential functions Properties of exponential functions and their graphs 6 Logarithms Definition of a logarithm, common logarithm, characteristic, mantissa 7 Logarithmic functions Properties of logarithmic functions and their graphs 8 Arithmetic progressions Definition of a sequence and arithmetic progressions 9 Geometric progressions Definition of a geometric progressions 10 Other sequences Progression of differences and other type of sequences 11 Mathematical induction, limits of Sequences Mathematical induction, intuitive introduction to limit of a sequence 12 Infinite series Partial sum and infinite series, geometric series 13 Mid-term examination 14 Solving equations Solving rational and irrational equations 15 Solving inequalities Solving cubic, quartic and rational inequalities 16 Trigonometric functions Manipulation of trigonometric functions using various formulas 17 Limits of functions Intuitive understading of limits of a function in terms of shape of the graph of a function 18 Continous functions Definition of a continuous function, intermediate value theorem, extreme value theorem 19 Derivative and differentiation Definition of a derivative, differentiation of polynomial functions, sum and product of two functions 20 Tangent lines to a curve Application of derivative to find the tangent line to a curve, increasing and decreasing functions 21 Derivative and graph of a function Many worked out examples of graph drawing of cubic or quartic polynomial functions using differentiation 22 Moving particle, velocity, acceleration Application of derivative to physics of a moving particle. 23 Anti-derivative (Indefinite integral) Definition of indefinite integrals and formulas for indefinite integral of polynomial functions 24 Definite integral Definition of definite integral and fundamental theorem of calculus 25 Areas and other applications of integral Computation of area enclosed by curves, how to find a position of a moving particle when its velocity is known 26 Term-end examination

Intermediate

– The intermediate level is designed for high school students with a working knowledge of 2nd or higher grade mathematics.

– The courses in this level provide students more advanced topics in high school and prepare them to go on to the advanced level seamlessly.

– The intermediate level differs from the basic level in that it deals with trigonometric, exponential, logarithmic functions and utilize chain rule in calculus contents whereas the basic level only deals with polynomial functions.

 № Subject Contents 1 Complex numbers Definition of complex numbers, their additions, multiplications, polar form r(cos a + i sin b) 2 Matrices and their operations Definition of matrices, their additions and multiplications, multiplication of matrices with a real number, and other properties 3 Matrices and system of linear equations Inverse matrix of matrix, formula of inverse matrix for a 2×2 matrix, Use of matrices to solve system of 2 linear equations with 2 variables 4 Exponents and exponential functions Manipulation of exponents and n-th root, how to define a real power of a positive real number, graphs of exponential functions 5 Logarithms and logarithmic functions Definition of a logarithm, common logarithm, characteristic, mantissa, graphs of logarithmic functions 6 Arithmetic and geometric progressions Definition of a sequence, arithmetic progressions, and geometric progressions 7 Other sequences and mathematical induction Progression of differences and other type of sequences, how to prove a theorem using a mathematical induction 8 Limits of Sequences and infinite series Intuitive introduction to limit of a sequence, infinite series, geometric series 9 Solving equations Solving rational and irrational equations 10 Solving inequalities Solving cubic, quartic and rational inequalities 11 Trigonometric functions Manipulation of trigonometric functions using various formulas 12 Trigonometric equations Solving equations involving trigonometric functions 13 Mid-term examination 14 Limits of a function Intuitive understading of limits of a function in terms of shape of the graph of a function, limits of sum, product, quotient of 2 functions 15 Continuous function Definition of a continuous function, intermediate value theorem, extreme value theorem 16 Derivative and differentiation Definition of a derivative, differentiation of polynomial functions, sum and product of two functions 17 Differentiation techniques Differentiation of trigonometric functions, exponential functions 18 More differentiation techniques Chain rule, differentiation of inverse functions 19 Application of derivative Concavity, many worked out examples of graph drawing of various functions using its first and second derivative 20 Anti-derivative (Indefinite integral) Definition of indefinite integrals and formulas for indefinite integral, integration by parts, integration by substition 21 Definite integral Definition of definite integral and fundamental theorem of calculus 22 Areas and other applications of integral Computation of area enclosed by curves, motion of a particle (position, velocity, acceleration) 23 Basic combinatorics and binomial theorem Permutations, combinations, binomial theorem 24 Probability, conditional probability Probability, conditional probability 25 Probability distribution Discrete and continuous probability distribution, expected value, variance, standard deviation 26 Term-end examination

– The advanced level is designed to provide students with enough mathematical knowledge and maturity to study the IUT mathematics courses.

– In an early stage of the advanced level, the students will go over matrices which are also treated in the intermediate level but their relation with linear transformations are emphasized.

– Then, the program introduces single variable calculus involving more complicated functions than the previous levels.

– This level also provides vector treatment of space geometry. Note also that the language of instruction in the advanced level is English.

 № Subject Contents 1 Matrices and linear equations Matrix, sum and product of matrices, determinant, inverse matrix, use of matrices to solve system of 2 linear equations with 2 variables 2 Matrices and linear transformation Linear transformation and its associated matrices, rotation, reflection about x and y axis 3 Composition of linear transformations and inverse transformation Composition of linear transformations and inverse transformation and their associated matrices 4 Mathematical induction and sequence of numbers How to prove a theorem using a mathematical induction, sequences, arithmetic and geometric progressions 5 Limits of sequences and infinite series Intuitive introduction to limit of a sequence, infinite series, geometric series 6 Trigonometric functions and equations Manipulation of trigonometric functions using various formulas, inverse trigonometric functions, equations involving trigonometric functions 7 Limits of a function Intuitive understading of limits of a function in terms of shape of the graph of a function, limits of sum, product, quotient of 2 functions 8 Continuous function Definition of a continuous function, intermediate value theorem, extreme value theorem 9 Derivative and differentiation Definition of a derivative, differentiation of polynomial functions, sum and product of two functions 10 Differentiation techniques Differentiation of trigonometric functions, exponential functions 11 More differentiation techniques Chain rule, differentiation of inverse functions, related rate real-life problems 12 Application of derivative Concavity, many worked out examples of graph drawing of various functions using its first and second derivative 13 Mid-term examination 14 Anti-derivative (Indefinite integral) Definition of indefinite integrals and formulas for indefinite integral, integration by parts, integration by substitution 15 Definite integral Definition of definite integral and fundamental theorem of calculus 16 Integration techniques Partial fractions, trigonometric substitutions, many worked out examples of integrations 17 Area on a plane and volume of a rotated solid Computation of area enclosed by curves, volume of a rotated solid 18 Arc length and other applications of integral Arc length, motion of a particle (position, velocity, acceleration) 19 Some basic differential equations Introduction to differential equations and how to solve first order ordinary differential equations 20 Basic combinatorics and binomial theorem Permutations, combinations, binomial theorem 21 Probability, conditional probability probability, conditional probability 22 Probability distribution discrete and continuous probability distribution, expected value, variance, standard deviation 23 Euclidean coordinates in 3 dimensional space Equation of lines, planes and spheres, angle between 2 planes, orthogonal projection onto a plane, 24 Vectors, inner and cross product of two vectors Vectors, sum of 2 vectors, a scalar product of a vector, angle between 2 vectors, inner and cross product of 2 vectors 25 Equations of lines and planes in 3 dimensional space Vector equation of lines, planes and spheres, solving 3 dimensional geometry problems 26 Term-end examination