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Pure Mathematics 4
In Pure Mathematics 4 we finish looking at the core knowledge required to effectively use mathematics in various fields.
Chapter 1 - Proof
We look at the concept of negation before moving onto proof by contradiction. Famous examples, including the irrationality of the square-root of 2, and that there are an infinite number of primes, are studied
Chapter 2 - Partial Fractions
Here we separate fractions of rational functions into individual terms based on the linear factors of the denominator. We look at dealing with repeated factors, and also top-heavy fractions.
Chapter 3 - Coordinate Geometry
We introduce the concept of a parametric equation, as opposed to a cartesian equation, and the uses of these. We also look at sketching these curves.
Chapter 4 - Binomial Expansion
While the binomial expansion for a positive integer exponent is easy to find, when the exponent is negative, or a non-integer, we need to instead use an infinite series to represent the Binomial Expansion. We look at using these series to find approximations to surds.
Chapter 5 - Differentiation
Using the chain-rule, we first differentiate parametric functions, before looking at the differentiation of functions defined implicitly. We then see how we can use the chain rule to relate rates of change in real world situations.
Chapter 6 - Integration
Continuing from Pure Mathematics 3, we look at finishing the key integration techniques, such as substitution and integration by parts. We look at using integrals to find volumes of revolution, before finally looking at differential equations, and combining the chain rule with integration to solve real world problems.
Chapter 7 - Vectors
The world of vectors is introduced, including position and displacement vectors in three dimensions. We look at geometric aspects of vectors, moving onto intersections of straight lines in three dimensions. Finally, the scalar product is introduced and we look at solving problems involving the angles between vectors.
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