Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors 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, 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

* Simple harmonic motion and damped oscillations.

* Impulse and momentum.

* The work done by a constant and a variable force, kinetic and potential energy (both gravitational and elastic) conservation of energy, the work-energy principle, conservative and dissipative forces, power.

* Oblique impact for elastic and inelastic collision in two dimensions.

* The Poisson distribution, its properties, approximation to a binomial distribution and hypothesis testing.

* The distribution of sample means and the central limit theorem.

* Chi-squared tests, contingency tables, fitting a theoretical distribution and goodness of fit.

* Type I and type II errors in statistical tests.

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 derive and solve a second order differential equation that models simple harmonic motion.

How to derive a second order differential equation for damped oscillations.

The meaning of underdamping, critical damping and overdamping.

How to solve coupled differential equations.

How to calculate the impulse of one object on another in a collision.

How to use the principle of conservation of momentum to model collisions in one dimension.

How to use Newton’s experimental law to model inelastic collisions in one dimension.

How to calculate the work done by a force and the work done against a resistive force.

How to calculate gravitational potential energy and kinetic energy.

How to calculate elastic potential energy.

How to solve problems in which energy is conserved.

How to solve problems in which some energy is lost through work against a dissipative force.

How to calculate power and solve problems involving power.

How to model elastic collision between bodies in two dimensions.

How to model inelastic collision between two bodies in two dimensions.

How to calculate the energy lost in a collision.

How to calculate probability for a Poisson distribution.

How to use the properties of a Poisson distribution.

How to use a Poisson distribution to model a binomial distribution.

How to use a hypothesis test to test for the mean of a Poisson distribution.

How to estimate a population mean from sample data.

How to estimating population variance using the sample variance. How to calculate and interpret the standard error of the mean.

How and when to apply the Central Limit Theorem to the distribution of sample means.

How to use the Central Limit Theorem in probability calculations, using a continuity correction where appropriate.

How to apply the Central Limit Theorem to the sum of n identically distributed independent random variables.

How to conduct a chi-squared test with the appropriate number of degrees of freedom to test for independence in a contingency table and interpret the results of such a test.

How to fit a theoretical distribution, as prescribed by a given hypothesis involving a given ratio, proportion or discrete uniform distribution, to given data.

How to use a chi-squared test with the appropriate number of degrees of freedom to carry out a goodness of fit test.

How to calculate the probability of making a Type I error from tests based on a Poisson or Binomial distribution.

How to calculate probability of making Type I error from tests based on a normal distribution.

How to calculate P(Type II error) and power for a hypothesis test for tests based on a normal, Binomial or a Poisson distribution (or any other A level distribution).

What's inside

Syllabus

Module 1: Applications of Differential Equations

Using differential equations in modelling in kinematics and in other contexts.

Hooke’s law.

Simple harmonic motion (SHM).

Damped oscillatory motion.

Light, critical and heavy damping.

Coupled differential equations.

Module 2: Momentum and Impulse

Momentum and the principle of conservation of momentum.

Impulse.

Newton’s experimental law (restitution)

Impulse for variable forces.

Module 3: Work, Energy and Power

The work-energy principle.

Conservation of mechanical energy.

Gravitational potential energy and kinetic energy.

Elastic potential energy.

Conservative and dissipative forces.

Power

Module 4: Oblique Impact

Modelling elastic collision in two dimensions.

Modelling inelastic collision in two dimensions.

The kinetic energy lost in a collision.

Module 5: Expectation and Variance and the Poisson Distribution

The Poisson distribution.

Properties of the Poisson distribution.

Approximating the binomial distribution.

Testing for the mean of a Poisson distribution.

Module 6: The Central Limit Theorem

The distribution of a sample mean.

Underlying normal distributions.

The Central Limit Theorem.

Module 7: Chi-Squared Tests

Chi-squared tests and contingency tables.

Fitting a theoretical distribution.

Testing for goodness of fit.

Module 8: Type I and Type II Errors

What are type I and type II errors?

A summary of all probability distributions encountered in A level maths and further maths.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Prepares learners to advance to more complex mathematical concepts, which is useful in pursuing mathematical degrees in college

Supports learners in developing skills in fluency, confidence, problem solving, mathematical argumentation, and more

Instructs learners on a wide range of relevant subtopics, equipping them with comprehensive knowledge

Builds a strong foundation in mechanics for individuals pursuing further mathematics in the college setting

Save this course

Save Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors to your list so you can find it easily later:

Save

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 Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors with these activities:

Review Trigonometric Identities

Show steps

Strengthen foundational understanding of trigonometric identities, which are essential for success in oblique impact.

Browse courses on Trigonometry

Show steps

Revisit the definitions and properties of trigonometric functions.
Practice applying trigonometric identities to simplify expressions and solve equations.
Identify and utilize common trigonometric identities, such as the Pythagorean identity, double-angle formulas, and sum-to-product formulas.

Differential Equations Modelling Practice

Show steps

Build strong problem-solving skills, apply problem-solving skills developed in preceding units, and consolidate foundational understanding of differential equations.

Browse courses on Differential Equations

Show steps

Select an appropriate problem related to kinematics or other contexts that can be modelled using differential equations.
Derive the differential equation that models the problem.
Solve the differential equation to obtain a solution.
Interpret the solution in the context of the problem.
Repeat the process for different problems, exploring various scenarios and applications.

Probability and Statistics Textbook Review

Show steps

Enhance comprehension of probability and statistics concepts by reviewing a comprehensive textbook.

View Probability & Statistics for Engineers &... on Amazon

Show steps

Read and understand the fundamental concepts of probability and statistics presented in the textbook.
Review specific chapters relevant to the topics covered in the course, such as Poisson distribution, Central Limit Theorem, and chi-squared tests.
Solve practice problems and exercises to solidify understanding and application of the concepts.

Five other activities

Expand to see all activities and additional details

Show all eight activities

Practice Problems for Impulse and Momentum

Show steps

Enhance understanding and problem-solving abilities in impulse and momentum topics, improving overall comprehension.

Browse courses on Impulse

Show steps

Solve a variety of practice problems involving the concepts of impulse and momentum.
Analyze real-world scenarios and apply principles of impulse and momentum to solve problems.
Identify different types of collisions and apply appropriate equations to determine outcomes.
Develop a deeper understanding of the relationship between impulse and momentum, and their applications in various scenarios.

Poisson Distribution Problem-Solving Group

Show steps

Foster collaborative learning and enhance problem-solving abilities by engaging in peer-led discussions and solving problems related to Poisson distribution.

Browse courses on Poisson Distribution

Show steps

Form study groups with peers.
Identify and collect Poisson distribution problems of varying difficulty levels.
Work together to solve the problems, discussing different approaches and sharing insights.
Present solutions to the group, explaining the reasoning and methodology.

Work, Energy, and Power Problems Compilation

Show steps

Solidify understanding by compiling problems and solutions, fostering a deeper grasp of the interplay between work, energy, and power.

Browse courses on Work

Show steps

Gather and organize practice problems related to work, energy, and power.
Solve the problems, providing detailed solutions and explanations.
Analyze the patterns and relationships within the problems, identifying common concepts and techniques.
Create a comprehensive compilation that serves as a valuable resource for future reference and problem-solving.

Central Limit Theorem Simulations

Show steps

Gain a deeper understanding of the Central Limit Theorem through hands-on simulations, solidifying its implications and applications.

Browse courses on Central Limit Theorem

Show steps

Explore online simulations or create your own using software or programming languages.
Conduct simulations for different sample sizes and distributions to observe the convergence towards a normal distribution.
Analyze the results and compare them with the theoretical expectations of the Central Limit Theorem.
Apply the insights gained from the simulations to solve related problems.

Chi-Squared Goodness-of-Fit Test Project

Show steps

Develop advanced analytical skills by conducting a comprehensive goodness-of-fit test using chi-squared distribution.

Browse courses on Chi-squared Test

Show steps

Identify a dataset and formulate a hypothesis about its distribution.
Perform a chi-squared goodness-of-fit test to determine if the data fits the hypothesized distribution.
Analyze the results, calculate the test statistic, and interpret the p-value to draw conclusions.
Write a detailed report summarizing the findings, including any implications or recommendations.

Career center

Learners who complete Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors will develop knowledge and skills that may be useful to these careers:

Mining Engineer

Mining Engineers design and operate mines and quarries. They use their knowledge of engineering principles to ensure that these mines and quarries are safe and efficient. This course may be useful for Mining Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of mining systems. Additionally, the course can help Mining Engineers learn about momentum, energy, and power, which are all important concepts in mining engineering.

See salaries and explore the career path for Mining Engineer

Software Engineer

Software Engineers design, develop, and test software applications. They use their knowledge of engineering principles to ensure that these applications are reliable and efficient. This course may be useful for Software Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of software systems. Additionally, the course can help Software Engineers learn about momentum, energy, and power, which are all important concepts in software engineering.

See salaries and explore the career path for Software Engineer

Mechanical Engineer

Mechanical Engineers are responsible for designing, manufacturing, and maintaining a wide array of mechanical systems. They use their knowledge of engineering principles to solve problems and improve the performance of these systems. This course may be useful for Mechanical Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of mechanical systems. Additionally, the course can help Mechanical Engineers learn about momentum, energy, and power, which are all important concepts in mechanical engineering.

See salaries and explore the career path for Mechanical Engineer

Chemical Engineer

Chemical Engineers design and operate chemical plants and processes. They use their knowledge of engineering principles to ensure that these plants and processes are safe and efficient. This course may be useful for Chemical Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of chemical processes. Additionally, the course can help Chemical Engineers learn about momentum, energy, and power, which are all important concepts in chemical engineering.

See salaries and explore the career path for Chemical Engineer

Petroleum Engineer

Petroleum Engineers design and operate oil and gas wells. They use their knowledge of engineering principles to ensure that these wells are safe and efficient. This course may be useful for Petroleum Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of oil and gas reservoirs. Additionally, the course can help Petroleum Engineers learn about momentum, energy, and power, which are all important concepts in petroleum engineering.

See salaries and explore the career path for Petroleum Engineer

Biomedical Engineer

Biomedical Engineers design and develop medical devices and systems. They use their knowledge of engineering principles to ensure that these devices and systems are safe and effective. This course may be useful for Biomedical Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of biological systems. Additionally, the course can help Biomedical Engineers learn about momentum, energy, and power, which are all important concepts in biomedical engineering.

See salaries and explore the career path for Biomedical Engineer

Civil Engineer

Civil Engineers design and construct infrastructure projects, such as roads, bridges, and buildings. They use their knowledge of engineering principles to ensure that these projects are safe and efficient. This course may be useful for Civil Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of structures. Additionally, the course can help Civil Engineers learn about momentum, energy, and power, which are all important concepts in civil engineering.

See salaries and explore the career path for Civil Engineer

Nuclear Engineer

Nuclear Engineers design and operate nuclear power plants and other nuclear facilities. They use their knowledge of engineering principles to ensure that these facilities are safe and efficient. This course may be useful for Nuclear Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of nuclear systems. Additionally, the course can help Nuclear Engineers learn about momentum, energy, and power, which are all important concepts in nuclear engineering.

See salaries and explore the career path for Nuclear Engineer

Aerospace Engineer

Aerospace Engineers design, develop, and test aircraft, spacecraft, and other aerospace vehicles. They use their knowledge of engineering principles to ensure that these vehicles are safe and efficient. This course may be useful for Aerospace Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of aerospace vehicles. Additionally, the course can help Aerospace Engineers learn about momentum, energy, and power, which are all important concepts in aerospace engineering.

See salaries and explore the career path for Aerospace Engineer

Electrical Engineer

Electrical Engineers design and develop electrical systems and devices. They use their knowledge of engineering principles to ensure that these systems and devices are safe and efficient. This course may be useful for Electrical Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of electrical systems. Additionally, the course can help Electrical Engineers learn about momentum, energy, and power, which are all important concepts in electrical engineering.

See salaries and explore the career path for Electrical Engineer

Industrial Engineer

Industrial Engineers design and improve industrial processes and systems. They use their knowledge of engineering principles to ensure that these processes and systems are efficient and effective. This course may be useful for Industrial Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of industrial processes. Additionally, the course can help Industrial Engineers learn about momentum, energy, and power, which are all important concepts in industrial engineering.

See salaries and explore the career path for Industrial Engineer

Systems Engineer

Systems Engineers design and operate complex systems, such as aircraft, spacecraft, and power plants. They use their knowledge of engineering principles to ensure that these systems are safe and efficient. This course may be useful for Systems Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of complex systems. Additionally, the course can help Systems Engineers learn about momentum, energy, and power, which are all important concepts in systems engineering.

See salaries and explore the career path for Systems Engineer

Materials Engineer

Materials Engineers develop and test new materials for use in a variety of applications. They use their knowledge of engineering principles to ensure that these materials are safe and effective. This course may be useful for Materials Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of materials. Additionally, the course can help Materials Engineers learn about momentum, energy, and power, which are all important concepts in materials engineering.

See salaries and explore the career path for Materials Engineer

Metallurgical Engineer

Metallurgical Engineers develop and test new metals and alloys for use in a variety of applications. They use their knowledge of engineering principles to ensure that these materials are safe and effective. This course may be useful for Metallurgical Engineers because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of metals and alloys. Additionally, the course can help Metallurgical Engineers learn about momentum, energy, and power, which are all important concepts in metallurgical engineering.

See salaries and explore the career path for Metallurgical Engineer

Data Scientist

Data Scientists use data to solve problems and make decisions. They use their knowledge of statistics, machine learning, and other data science techniques to analyze data and extract insights. This course may be useful for Data Scientists because it can help them develop the skills they need to understand and apply differential equations, which are used to model the behavior of data. Additionally, the course can help Data Scientists learn about momentum, energy, and power, which are all important concepts in data science.

See salaries and explore the career path for Data Scientist