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Course notes

Advanced Higher Maths Notes

Start at the beginning
01

Binomial Theorem

4 topics
1. Pascal's Triangle & Binomial Coefficients 4 examples2. Binomial Expansion 2 examples3. General Term & Coefficients 3 examples4. Approximating Powers 3 examples
02

Partial Fractions

3 topics
1. Algebraic Long Division 2 examples2. Proper Rational Functions 4 examples3. Improper Rational Functions 2 examples
03

Matrices

3 topics
1. Matrix Algebra 4 examples2. The 3×3 Inverse 2 examples3. Transformations of the Plane 2 examples
04

Systems of Equations

1 topic
1. Gaussian Elimination 6 examples
05

Sequences and Series

3 topics
1. Arithmetic Sequences & Series 3 examples2. Geometric Sequences & Series 4 examples3. Summation Formulae 2 examples
06

Maclaurin Series

2 topics
1. Maclaurin Series 2 examples2. Combining Expansions 3 examples
07

Complex Numbers

5 topics
1. Algebraic Operations 5 examples2. Polar Form & Loci 4 examples3. Roots of Polynomial Equations 3 examples4. De Moivre's Theorem & Roots 3 examples5. Multiple Angle Formulae 3 examples
08

Number Theory

2 topics
1. Euclid's Algorithm 2 examples2. Number Bases 2 examples
09

Methods of Proof

5 topics
1. Direct Proof & Counterexample 2 examples2. Proof by Contrapositive 2 examples3. Proof by Induction 2 examples4. Proof by Contradiction 3 examples5. Proof by Exhaustion 2 examples
10

Differentiation

10 topics
1. Differentiation from First Principles 3 examples2. Rules & Standard Derivatives 4 examples3. Higher Derivatives 3 examples4. Inverse Trigonometric Differentiation 4 examples5. Implicit Differentiation 4 examples6. Logarithmic Differentiation 4 examples7. Parametric Differentiation 4 examples8. Motion in a Plane 3 examples9. Rectilinear Motion 3 examples10. Related Rates of Change 4 examples
11

Functions and Graphs

7 topics
1. Asymptotes 2 examples2. The Modulus Function 3 examples3. Odd & Even Functions 3 examples4. Stationary Points & Inflection 2 examples5. Maximum & Minimum Values 2 examples6. Optimisation 3 examples7. Curve Sketching & Related Graphs 2 examples
12

Integration

5 topics
1. Standard Integrals & Substitution 4 examples2. Integrating Rational Functions 2 examples3. Integration by Parts 5 examples4. Volumes of Revolution 2 examples5. Areas Under & Between Curves 3 examples
13

Differential Equations

5 topics
1. First-Order Separable 2 examples2. First-Order Linear 2 examples3. Second-Order Homogeneous 3 examples4. Second-Order Non-Homogeneous 2 examples5. Applications & Modelling 3 examples
14

Vectors

4 topics
1. Vector & Scalar Triple Product 3 examples2. Equations of Lines 2 examples3. Equations of Planes 3 examples4. Intersections of Lines and Planes 3 examples