Advanced Higher Mathematics

Course Overview

The course is based on : Calculus, Algebra, proof and number theory, Matrices, vectors and complex numbers
An external SQA exam is sat at the end of the course, with calculator and non-calculator papers

Scholar Advanced Higher Maths Past Papers SQA Advanced Higher Maths Check List

Higher Maths Advanced Higher Mechanics

Prerequisite Knowledge

You must have a good grasp of the following :

Differentiation.
Integration.
Manipulating Vectors.
Algebraic manipulation.

Alphabetical Listing of Topics

Binomial Theorem Complex Numbers Differentiation Differential Equations Functions Gaussian Elimination Integration Matrices Number Theory Partial Fractions Proof Sequences and Series Vectors

Algebra, Proof & Number Theory

Binomial Theorem

Key subskills
Expand expressions using the binomial theorem.

Factorials Permutations Combinations
Pascal’s Triangle. Binomial Co-efficients Binomial Co-efficient equations Binomial Expansions
Binomial Theorem Probability and the Binomial Theorem Binomial Expansions and e

Functions

Key subskills
Finding the asymptotes of rational functions.
Investigating features of graphs and sketching graphs of functions, including appropriate analysis of stationary points.

Refresher of functions
Modulus function Inverse functions
Polynomials Extrema
Concavity and points of inflexion Odd and Even functions
Asymptotes

Number Theory

Key subskills
Using Euclid’s algorithm to find the greatest common divisor of two positive integers.

The division algorithm
The euclidean algorithm
Diophantine equations
Pythagorean Triples
Number bases

Partial Fractions

Key subskills
Expressing proper rational functions as a sum of partial fractions.

Rational fractions Partial fractions
Distinct linear factors Repeated linear factors
Irreducible quadratic factor

Proof

Key subskills
Disproving a conjecture by providing a counter-example.
Using direct and indirect proof in straightforward examples.

Mathematical proof
Direct proof
Proof by contradiction
Proof by contrapositive
Proof by induction

Sequences and Series

Key subskills
Finding the general term and summing arithmetic and geometric sequences.
Using the Maclaurin series expansion to find a stated number of terms of the power series for a simple function.

Recurrence relations
Fixed points
Arithmetic sequences Arithmetic series
n^th term of arithmetic sequence Sum to n terms of an arithmetic sequence
Geometric sequences Geometric series
Sum to n terms of a geometric sequence Infinite series
Sum to infinity - geometric series Sum first n natural numbers
Sigma notation - Rules Common series - sigma notation

Power Series

Power series D'alembert's ratio test
Absolute convergence Fibonacci series
Centre of convergence Taylor's series
Maclaurin's Series Maclaurin expansion

Application of summation formulae

Standard series Examples
Iteration Iteration - graphical method
First order process Second order process
Rule of false position Newton-Raphson iteration

Calculus

Differentiation

Key subskills
Differentiating functions given in the form of a product and in the form of a quotient.
Differentiating exponential and logarithmic functions.
Differentiating inverse trigonometric functions.
Finding the derivative of functions defined implicitly.
Finding the derivative of functions defined parametrically.

First Principles

First principles x^n ax^n
Derivatives of sums (x + a)^n (ax + b)^n
The chain rule The product rule
The quotient rule Trig functions

Differentiation Refresher

Differentiation Refresher Trig formulae refresher
seca coseca cota Higher Derivatives
Formulae for differentiation

Rules

Chain rule. Product rule. Quotient rule
Exponentials & Logs Differentiating inverse functions
Differentiating inverse trig functions Inverse trig functions:Recap
Implicit & explicit functions Logarithmic functions

Parametric Differentiation

Parametric equations Parametric equations:circles
Parametric constraint equations Differentiating parametric
Second Derivative: parametric equations Uses of parametric equations
Equations of motion

Integration

Key subskills
Integrating expressions using standard results.
Integrating by substitution.
Integrating by parts.

Integration refresher
Areas under curves Area between curve and the x-axis
Area between curve and the y-axis
Standard integrals e^x and 1/x
Infinite integrals (discontinuities) sec²x
Integrating Inverse trig functions Integration of rational functions
Volumes of revolution
Integration by substitution. Integration by parts

Differential Equations

Key subskills
Solving a first order differential equation with variables separable.
Solving a first order linear differential equation using the integrating factor.
Solving second order differential equations.
Applying differentiation to problems, in context where appropriate.
Applying integration to problems, in context where appropriate.

Differential equations First-order differential equations
Second-order differential equations
Applications of differential equations

Applications of Calculus

Rectilinear motion Distance, velocity & acceleration
Extrema of functions
Approximating roots
Uses of parametric equations
Equations of motion

Matrices, vectors and complex numbers

Complex Numbers

Key subskills
Performing geometric operations on complex numbers

Complex numbers Complex conjugate
Arithmetic operations Argand diagrams
Loci
Polar form
De moivre's theorem
Roots of a complex number

Matrices

Key subskills
Using Gaussian elimination to solve a 3x3 system of linear equations.
Performing matrix operations of addition, subtraction and multiplication.
Calculating the determinant of a matrix.
Finding the inverse of a matrix.

Matrices Matrix equality
Matrix addition and subtraction Matrix notation
Matrix elements Matrix Scalar multiplication
Transpose of a matrix Matrix multiplication

Matrices - Determinant

2 x 2 matrix 3 x 3 matrix

Matrices - Inverse

Cofactor of a 3 x 3 matrix Adjoint of a 3 x 3 matrix
Inverse of a square matrix Product of a square matrix and its inverse
Using Gaussian elimination to find inverse of matrix

Special Matrices

Special matrices
Null matrix Unit matrix
Transformation matrices
Reflections Rotations Scalings

Gaussian Elimination

Vedic way of solving simultaneous equations
Augmented matrix
Elementary Row Operations
Gaussian elimination Gaussian elimination : Excel solver
Using Gaussian elimination to find inverse of matrix

Vectors

Key subskills
Calculating a vector product.
Finding the equation of a line in three dimensions.
Finding the equation of a plane.

Vector Product

Direction Ratios and Cosines Vector product
Right Hand Screw Rule Vector Product properties
Components Scalar triple product

Vector equations of lines

Intersection of two lines Intersection of two planes
The distance from a point to a plane
The distance from a point to a line Equations of a line

Vector equation of planes

Planes Planes in space
Vector equations The angle between two planes
The distance between parallel planes Coplanar vectors
Vector equation of a plane

Course Overview

Check List

Algebra, Proof & Number Theory

Decomposing a rational function into a sum of partial fractions (denominator of degree at most three)
Finding the asymptotes to the graphs of rational functions
Investigating features of graphs and sketching graphs of functions
Expanding expressions using the binomial theorem
Finding the general term and summing arithmetic and geometric progressions
Applying summation formulae
Using the Maclaurin expansion to find specified terms of the power series for simple functions
Disproving a conjecture by providing a counterexample
Using indirect or direct proof in straightforward examples
Using proof by induction
Using Euclid’s algorithm to find the greatest common divisor of two positive integers

Older Learning Outcomes

Apps 1.1 Applying algebraic skills to the binomial theorem and to complex numbers.
Apps 1.2 Applying algebraic skills to sequences and series.
Apps 1.3 Applying algebraic skills to summation and mathematical proof .
Apps 1.3 Applying algebraic skills to summation and mathematical proof .
Apps 1.4 Applying algebraic and calculus skills to properties of functions.
GPS 1.4 Applying algebraic skills to number theory.
GPS 1.5 Applying algebraic and geometric skills to methods of proof.
MAC 1.1 Applying algebraic skills to partial fractions

Unit 1 LO1 Use algebraic skills
Unit 1 LO4 Use properties of functions.
Unit 2 LO4 Understand and use sequences and series.
Unit 2 LO5 Use standard methods to prove results in elementary number theory.
Unit 3 LO3 Understand and use further aspects of sequences and series.
Unit 3 LO5 Use further number theory and direct methods of proof.

Calculus

Differentiation

Differentiating exponential and natural logarithmic functions
Differentiating functions using the chain rule
Differentiating functions given in the form of a product and in the form of a quotient
Finding the derivative where relationships are defined implicitly
Finding the derivative where relationships are defined parametrically
Applying differentiation to problems in context

Integration

Integrating expressions using standard results
Integrating by substitution
Integrating by parts
Applying integration to problems in context

Differential Equations

Solving first-order differential equations with variables separable
Solving first-order linear differential equations using an integrating factor
Solving second-order differential equations
Applying integration to problems in context

Older Learning Outcomes

MAC 1.2 Applying calculus skills through techniques of differentiation.
MAC 1.3 Applying calculus skills through techniques of integration.nit 1 LO2 Use the rules of differentiation on elementary functions.
MAC 1.4 Applying calculus skills to solving differential equations.
Apps 1.5 Applying algebraic and calculus skills to problems.

Unit 1 LO3 Integrate using standard results and the substitution method.
Unit 2 LO2> Use further integration techniques.
Unit 3 LO4 Solve further ordinary differential equations.

Matrices,vectors and complex numbers

Using Gaussian elimination to solve a 3x3 system of linear equations
Understanding and using matrix algebra
Calculating the determinant of a matrix
Finding the inverse of a matrix
Using transformation matrices
Calculating a vector product
Working with lines in three dimensions
Working with planes
Performing algebraic operations on complex numbers
Performing geometric operations on complex numbers

Older Learning Outcomes

GPS 1.1 Applying algebraic skills to matrices and systems of equations.
GPS 1.2 Applying algebraic and geometric skills to vectors.
GPS 1.3 Applying geometric skills to complex numbers.
Apps 1.1 Applying algebraic skills to the binomial theorem and to complex numbers.

Unit 1 LO5 Use matrix methods to solve systems of linear equations.
Unit 2 LO3 Understand and use complex numbers .
Unit 3 LO1 Use vectors in three dimensions.
Unit 3 LO2 Use matrix algebra.

Books

A selection of resources available at Amazon

Maths Mutt at Amazon

New Advanced Higher Mathematics
Revision Notes Differentiation
By First Principles Quick Forty
Integration and Differential Equations

Amazon: Advanced Higher Maths MM Amazon Store UK school subjects Maths Mutt Publications

As an Amazon Associate I earn from qualifying purchases.