A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices from edX

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further 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, the 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 eight modules, you will be introduced to

Analytical and numerical methods for solving first-order differential equations
The nth roots of unity, the nth roots of any complex number, geometrical applications of complex numbers.
Coordinate systems and curve sketching.
Improper integrals, integration using partial fractions and reduction formulae
The area enclosed by a curve defined by parametric equations or polar equations, arc length and the surface area of revolution.
Solving second-order differential equations
The vector product and its applications
Eigenvalues, eigenvectors, diagonalization and the Cayley-Hamilton Theorem.

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A -level further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

What you'll learn

How to find the general or particular solution to a first-order differential equation by inspection or by using an integrating factor.

How to find a numerical solution to a differential equation using the Euler method or an improved Euler method..

How to find the nth roots of unity

How to find the nth roots of a complex number in the form

How to use complex roots of unity to solve geometrical problems.

How to identify the features of parabolas, rectangular hyperbolae, ellipses and hyperbolae defined by Cartesian and parametric equations.

How to identify features of graphs defined by rational functions.

How to define a parabola, ellipse or hyperbola using focus-directrix properties and eccentricity.

How to evaluate improper integrals.

How to integrate using partial fractions

How to derive and use reduction formulae

How to find areas enclosed by curves that are defined parametrically.

How to find the area enclosed by a polar curve.

How to calculate arc length.

How to calculate the surface area of revolution.

How to find the auxiliary equation for a second order differential equation.

What's inside

Syllabus

Module 1: First Order Differential Equations

Solving first order differential equations by inspection

Solving first order differential equations using an integrating factor

Finding general and particular solutions of first-order differential equations

Euler’s method for finding the numerical solution of a differential equation

Improved Euler methods for solving differential equations.

Module 2: Further Complex Numbers

The nth roots of unity and their geometrical representation

The nth roots of a complex number

and their geometrical representation

Solving geometrical problems using complex numbers.

Module 3: Properties of Curves

Cartesian and parametric equations for the parabola and rectangular hyperbola, ellipse and hyperbola.

Graphs of rational functions

Graphs of

for given

The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity.

Module 4: Further Integration Methods

Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.

Integrate using partial fractions including those with quadratic factors

in the denominator

Selecting the correct substitution to integrate by substitution.

Deriving and using reduction formula

Module 5: Further Applications of Integration

Finding areas enclosed by curves that are defined parametrically

Finding the area enclosed by a polar curve

Using integration methods to calculate the arc length

Using integration methods to calculate the surface area of revolution

Module 6: Second Order Differential Equations

Solving differential equations of form y″ + ay′ + by = 0 where a and b are constants by using the auxiliary equation.

Solving differential equations of form y ″+ a y ′+ b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function

Module 7: The Vector (cross) Product

The definition and properties of the vector product

Using the vector product to calculate areas of triangles.

The vector triple product.

Using the vector triple product to calculate the volume of a tetrahedron and the volume of a parallelepiped

The vector product form of the vector equation of a straight line

Solving geometrical problems using the vector product

Module 8: Matrices - Eigenvalues and Eigenvectors

Calculating eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices.

Reducing matrices to diagonal form.

Using the Cayley-Hamilton Theorem

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Focuses on strengthening an existing foundation for intermediate learners, developing skills to comprehend A level further mathematics

Instructors have extensive knowledge and expertise in further mathematics

Teaches essential skills for A level further mathematics exams, focusing on fluency, confidence, and critical thinking

Builds a strong foundation for advanced mathematical topics, expanding knowledge beyond A level further mathematics

Requires learners to come in with a background in mathematics, potentially creating a barrier for some learners

Course materials may not align with the latest industry practices and versions of software

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices with these activities:

Organize and review class materials

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Review the lecture notes, assignments, and other materials from the course to reinforce your understanding of the concepts covered and identify areas where you need additional support.

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Gather all the materials from the course.
Organize the materials into a logical order.
Review the materials and make notes.

Review 'Further Pure Mathematics 2' by Frank Wood

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Review 'Further Pure Mathematics 2' by Frank Wood to supplement your learning. This book covers advanced topics in further mathematics, providing additional insights and examples to reinforce your understanding of the concepts covered in this course.

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Read the relevant chapters.
Take notes and highlight important concepts.
Solve practice problems.

Practice first order differential equations

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Practice solving first order differential equations by inspection and using an integrating factor to strengthen your understanding of the techniques covered in Module 1.

Browse courses on First Order Differential Equations

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Find the general solution of a first-order differential equation by inspection.
Solve a first-order differential equation using an integrating factor.
Find the particular solution of a first-order differential equation given initial conditions.

One other activity

Expand to see all activities and additional details

Show all four activities

Calculate arc length and surface area of revolution

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Practice calculating arc length and surface area of revolution using integration techniques to reinforce your understanding of the concepts covered in Module 5.

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Set up the integral for arc length.
Evaluate the integral to find the arc length.
Set up the integral for surface area of revolution.
Evaluate the integral to find the surface area of revolution.

Career center

Learners who complete A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices will develop knowledge and skills that may be useful to these careers:

College Mathematics Teacher

A College Mathematics Teacher at a community college or university educates students about foundational and advanced mathematical concepts. Instructing students on these concepts will be a lot easier after taking this comprehensive A-Level Further Mathematics course. The course covers a wide range of mathematical topics, including differential equations, complex numbers, vectors, and matrices. Those enrolling in this course will greatly benefit from the focus on analytical and numerical methods for solving complex mathematical problems, which will be invaluable as an educator of college-level mathematics students.

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Operations Research Analyst

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Investment Banker

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Financial Analyst

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Data Scientist

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Teacher

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Statistician

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Market Research Analyst

A Market Research Analyst collects and analyzes data to help businesses understand their target market. This course in A-Level Further Mathematics provides a strong foundation in the statistical and mathematical techniques commonly used in market research. The course's focus on problem solving and critical thinking will also be beneficial for a Market Research Analyst, who must be able to identify trends and patterns in data.

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Software Engineer

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Data Analyst

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Economist

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Consultant

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Market Researcher

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Risk Manager

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Financial Planner

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