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Algebra (Metric)

This class covers the definition of the derivative from first principles, differentiation of key functions sums, products, quotients and composites, and using differentiation to sketch curves and solve problems. If you've done A-level Maths, this is mostly a revision module.

Quadratics

This unit covers topics at Higher GCSE / beginning of A-level; Science and Engineering courses at Imperial will assume you know them. This unit introduces topics such as factorising quadratics, completing the square, using the quadratic formula, solving quadratic equations.

Factorising quadratics such as \(x^2-5\,x+6\) and \(6\,x^2+7\,x-3\), and solving quadratic equations using this technique.

Using the formula \(x = (-b\pm \sqrt{b^2-4\,a\,c})/(2\,a)\) to solve quadratic equations.

Rational Expressions

Simplifying rational expressions, partial fractions, polynomial division, graphs of rational functions. This is an A-level topic; Science and Engineering courses at Imperial will assume you know it.

A rational expression is one polynomial divided by another; simplifying sums of rational expressions by putting them over a common denominator.

Resolving rational expressions into partial fractions. (You meet simple cases at A-level, and more complicated ones in Further Mathematics.)

What happens if the degree of the numerator is greater than than of the denominator?

Sketching graphs of rational functions.

Series

Arithmetic and geometric sequences and series (this is an A-level topic, and Science and Engineering courses at Imperial will assue you know it). Convergence of series (this is a university level topic, which you'll meet in a First Year course).

\(n\)th term, and sum of first \(n\) terms, of series like \(2 + 5 + 8 + 11 + \dots\)

\(n\)th term, sum of first \(n\) terms, and sum to infinity of series like \(4 + 2 + 1 + 1/2 + \dots\)

Investigating the convergence of series by considering the ratio of successive terms.

Investigating the convergence of series whose terms alternate in sign.

Using the ratio test to calculate the values of x for which a series of the form \(a_0+a_1\,x+a_2\,x^2=\dots\) converges.