TRANSMATH SYLLABUS There are sixteen Transmath modules covering nine topics. A list of sub- sections within each module is given here as a brief indication of the syllabus covered. Algebra Elementary Algebra (5 exercises) 1. Pre-requisites (Negative Numbers, Order of Operations, HCF and LCM, Fractions, Powers & Indices) 2. Symbols and Numbers 3. Like Terms (Ex.1 & Ex.2) 4. Multiplying Expressions (Ex.3) 5. Factorising Expressions (Ex.4) 6. Algebraic Fractions (Ex.5*) 7. Summary Calculus Introduction to Differentiation (5 exercises) 1. The Tangent Problem 2. Rates of Change (Ex.1) 3. Combined Rates of Change (Ex.2) 4. Gradient of Curves 5. Derivatives of Polynomials (Ex.3) 6. Negative and Fractional Powers (Ex.4) 7. Trig, Exp and Log Functions (Ex.5*) 8. Summary Techniques of Differentiation (5 exercises) 1. Max. and Min. (Ex.1*) 2. Compound Functions (Product Rule (Ex.2*), Quotient Rule (Ex.3*), Chain Rule (Ex.4*)) 3. Implicit Differentiation (Ex.5*) 4. Summary Indefinite Integration (4 exercises) 1. Introduction 2. Standard Integrals (Ex.1*) 3. Fractional and Negative Powers (Ex.2) 4. Change of Variable (Ex.3*) 5. Integration by Parts (Ex.4*) 6. Other Tricks (adding on zero, multiplying by one) 7. Summary Techniques of Integration (5 exercises) 1. Introduction 2. Trigonometric Integrals (Ex.1*) 3. Trigonometric Substitution (Ex.2*) 4. Partial Fractions (Ex.4*) (incl. Completing the Square and Polynomial Long Division (Ex.3*)) 5. Reduction Formulas (Ex.5) 6. Summary Definite Integration (2 exercises) 1. Introduction 2. Limits of an Integral (Ex.1*) 3. Area under a Curve (including Area under Speed-Time graph) 4. Volumes of Revolution (Ex.2*) 5. Summary Ordinary Differential Equations (5 exercises) 1. Introduction 2. 1st Order Separable Equations (Ex.1*) 3. 1st Order Linear Equations (Integrating Factor) (Ex.2*) 4. Applications (Population Dynamics, Motion of a Rocket) 5. 2nd Order Linear Equations (Complementary Function (Ex.3*), Particular Integral (Ex.4*), General Solution, Particular Solution (Ex.5*)) 6. Summary Complex Numbers Introduction to Complex Numbers (4 exercises) 1. Introduction 2. Imaginary Numbers (Ex.1) 3. The Argand Diagram 4. Modulus and Argument Form (Ex.2) 5. De Moivre's Theorem (Ex.3*) 6. nth Roots of a Complex Number (Ex.4*) 7. Summary Functions Introduction to Functions (3 exercises) 1. Introduction 2. Definitions 3. Algebraic Functions (Ex.1) 4. Function of a Function (Ex.2*) 5. Inverse Functions (Ex.3) 6. Summary Matrices Introduction to Matrices (4 exercises) 1. Introduction 2. Applications 3. Definitions (Transpose and Symmetry (Ex.1)) 4. Matrix Algebra (Addition (Ex.2), Multiplicative Condition (Ex.3), Multiplication (Ex.4)) 5. Summary Transformations (1 exercise) 1. Introduction 2. Transformations in the Plane 3. Rotation and Reflection 4. Stretching and Shearing 5. Combinations of Transformations 6. Inverse Transformations 7. Investigation* and Exercises (Ex.1) 8. Summary Advanced Matrices (4 exercises) 1. Introduction 2. Determinants (Ex.1) 3. Inverse of a Matrix (Ex.2) 4. Systems of Linear Equations (Ex.3*) 5. Eigenvalues and Eigenvectors (Ex.4*) 6. Summary Numerical Methods Introduction to Numerical Methods (3 exercises) 1. Introduction 2. Definitions 3. Root Finding (Bisection Method, Newton-Raphson Method (Ex.1)) 4. Numerical Integration (Trapezoidal Rule, Simpson's Rule (Ex.2)) 5. Solution of First Order ODEs (Euler's Method (Ex3.)) 6. Summary Trigonometry Introduction to Trigonometry (5 exercises) 1. Introduction 2. The Triangle (Ex.1) 3. Pythagoras' Theorem (Ex.2) 4. Trigonometric Ratios (Ex.3) 5. Trigonometric Identities (Ex.4*) 6. Sine and Cosine Rules (Ex.5) 7. Trigonometric Functions 8. Summary Sequences and Series Introduction to Sequences and Series (3 exercises) 1. Introduction 2. Sequences (Algebraic, geometric, harmonic, quadratic, (Ex.1)) 3. Partial Sums (Ex.2) 4. Infinite Geometric Series (Ex.3) 5. Summary Vectors Introduction to Vectors (6 exercises) 1. Introduction 2. Vector Algebra (Ex.1) 3. Vectors in terms of i, j and k (Ex.2) 4. Equation of a Straight Line (Ex.3) 5. Scalar Product (Ex.4) 6. Equation of a Plane (Ex.5) 7. Vector Product (Ex.6) 8. Summary Total number of exercises = 64 * Exercises and other sections marked with an asterisk require Mathematica.