ImperialX: A-level Mathematics for Year 13 - Course 1: Functions, Sequences and Series, and Numerical Methods

By the end of this course, you'll be able to:

• Define a mapping and a function
• Define the domain and range for a function
• Combine functions to create a composite function
• Find the inverse of a function
• Define a sequence using an nth term formula and an inductive definition
• Define an arithmetic and a geometric sequence
• Use sigma notation to define a series
• Expand a binomial expression for both a positive integer index and for an index which is not a positive integer
• Use radians as a measure of angle
• Find arc lengths, areas of sectors and areas of segments on circles where angles are in radians
• Use small angle approximations for sine, cosine and tangent functions
• Find multiple solutions to trigonometric equations
• Use the reciprocal trigonometric functions
• Use the inverse trigonometrical functions
• Use trigonometrical identities
• Derive and use the trapezium rule to find the area under a curve
• Approximate the root of an equation using a sign change method
• Approximate the root of an equation using the Newton-Raphson method
• Approximate the root of an equation by fixed point iteration

Module 1: Algebra and Functions

• The difference between a mapping and a function
• Function notation
• Domain and range of a function
• Composition of functions
• Inverse functions

Module 2: Sequences and Series 1

• How to define a sequence by an nth term rule
• How to define a sequence by an inductive rule
• Arithmetic sequences
• Geometric sequences
• The sum of n terms of an arithmetic and a geometric sequence
• Series and the sigma notation

Module 3: Sequences and Series 2

• The binomial expansion for positive integer n
• The general binomial expansion
• Properties of sequences

Module 4: Trigonometry 1

• Radian measure
• Circle calculations
• Small angle approximations
• Circular functions

Module 5: Trigonometry 2

• The reciprocal trigonometric functions
• Inverse trigonometric functions
• Addition and double angle formulae
• Trigonometric identities

Module 6: Numerical Methods 1

• An introduction to numerical methods
• The trapezium rule
• Numerical solution of equations

Module 7: Numerical Methods 2

• The Newton-Raphson method
• Fixed point iteration