13年级高级数学 1:函数、序列和级数以及数值方法

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

Develop your thinking skills, fluency and confidence to aim for an A* in A-level maths and prepare for undergraduate STEM degrees.

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伦敦帝国理工学院
edX
  • 完成时间大约为 7
  • 中级
  • 英语
注:因开课平台的各种因素变化,以上开课日期仅供参考

你将学到什么

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

课程概况

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level maths exams.
The course is most appropriate to the Edexcel, AQA, OCR and OCR(MEI) papers. You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

Fluency – selecting and applying correct methods to answer with speed and efficiency
Confidence – critically assessing mathematical methods and investigating ways to apply them
Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied
Over seven modules, covering an introduction to functions and their notation, sequences and series and numerical methods testing your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A-level course.

You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

课程大纲

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

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