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

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.

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