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

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Overview

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.

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

• 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
• 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

Syllabus

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
• 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

Authored by

Imperial College London

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Beginner
English
7 weeks
Self-paced
Online