13年级高级数学 2:一般运动、力矩和平衡、正态分布、矢量、微分法、积分法和微分方程

A-level Mathematics for Year 13 - Course 2: General Motion, Moments and Equilibrium, The Normal Distribution, Vectors, Differentiation Methods, Integration Methods and Differential Equations

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

Use calculus in kinematics for motion in a straight line

Use differentiation and integration of a vector with respect to time for motion in two dimensions

Solve projectile motion problems using both calculus/vector methods and constant acceleration formulae

Use a standard model for friction

Calculate moments understanding what they mean and how they might be used

Solve problems involving parallel and nonparallel coplanar forces

Apply an understanding of moments to statics problems involving rigid bodies

Use the Normal distribution as a model for continuous data

Conduct a hypothesis test of the mean using a Normal distribution

Use a Normal distribution as an approximation of a Binomial distribution

Add vectors diagrammatically

Perform the algebraic operations of vector addition and multiplication by scalars

Apply vector calculations to problems in pure mathematics

Use methods for differentiating a function of a function, differentiating a product and differentiating a quotient

Differentiate trigonometric and inverse trigonometric functions

Use implicit and parametric differentiation

Identify integrals that can be dealt with “by sight”

Use a substitution method to integrate a function

Use partial fractions to integrate rational functions

Use the method of integration by parts

Use the method of separating the variable to solve differential equations

find the family of solutions for a differential equation

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.

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 general motion in a straight line and two dimensions, projectile motion, a model for friction, moments, equilibrium of rigid bodies, vectors, differentiation methods, integration methods and differential equations, 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: Calculus in Kinematics and Projectile Motion

Using calculus for kinematics for motion in a straight line:
Using calculus in kinematics for motion extended to 2 dimensions using vectors.
Modelling motion under gravity in a vertical plane using vectors; projectiles.
Composition of functionsInverse functions

Module 2: Friction, Moments and Equilibrium of rigid bodies

Understanding and using the F≤μR model for friction
The coefficient of friction motion of a body on a rough surface limiting friction
Understanding and using moments in simple static contexts.
The equilibrium of rigid bodies involving parallel and nonparallel coplanar forces

Module 3: The Normal Distribution

Understanding and using the Normal distribution as a model
Finding probabilities using the Normal distribution
Conducting statistical hypothesis tests for the mean of a Normal distribution with known, given or assumed variance
Interpreting the results of hypothesis tests in context

Module 4: Vectors

Using vectors in two dimensions and in three dimensions
Adding vectors diagrammatically
Performing the algebraic operations of vector addition and multiplication by scalars
Understanding the geometrical interpretations of vector calculations
Understanding and using position vectors
Calculating the distance between two points represented by position vectors.
Using vectors to solve problems in pure mathematics

Module 5: Differentiation Methods

Differentiation using the product rule, the quotient rule and the chain rule
Differentiation to solve problems involving connected rates of change and inverse functions.
Differentiating simple functions and relations defined implicitly or parametrically

Module 6: Integration Methods

Integrating e^kx, 1/x, sin⁡kx, cos⁡kx and related sums, differences and constant multiples
Integration by substitution
Integration using partial fractions that are linear in the denominator
Integration by parts

Module 7: Differential Equations

The analytical solution of simple first order differential equations with separable variables
Finding particular solutions
Sketching members of a family of solution curves
Interpreting the solution of a differential equation in the context of solving a problem
Identifying limitations of the solution to a differential equation

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