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ImperialX: 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
Develop your thinking skills, fluency and confidence in the applied mathematics content of A-level further maths and prepare for undergraduate STEM degrees.
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7 weeks at 2-4 hours per week
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Online - Self Paced
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ImperialX
About ImperialX: 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
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 Will 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)..