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

Prerequisite Knowledge

You must have a good grasp of the following :

  • Differentiation.
  • Integration.
  • Manipulating Vectors.
  • Algebraic manipulation.




Algebra, Proof & Number Theory


Functions


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

Number Theory


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

Partial Fractions


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

Proof


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


Calculus

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.


Matrices, vectors and complex numbers

Complex Numbers


Key subskills
Performing geometric operations on complex numbers

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.

Gaussian Elimination


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.


© Alexander Forrest