The calculus is one of mankind’s most significant scientific achievements, transforming previously intractable physical problems into often routine calculations. Although its roots trace back into antiquity, it was developed in the late 17th century by Newton, when developing his laws of motion and gravitation, and Leibniz, who developed the notation we still use today. Analysis is the branch of mathematics that underpins the theory behind the calculus, placing it on a firm logical foundation through the introduction of the notion of a limit. This module introduces differentiation and integration from this rigorous point of view. The notion of a function of a real variable and its derivative are formalized. The familiar techniques and applications of differentiation and integration are reviewed and extended. Simple first and second order ordinary differential equations are studied. The theory of infinite sequences and series, including Taylor series, is introduced.

This is a 20 credit version of the 30 credit module ‘Real Analysis and the Calculus’ specifically for students taking 40 only credits of Mathematics at Level C. It will be delivered alongside the 30 credit version. Material not required for the 20 credit version will be clearly indicated.

Learning Outcomes

By the end of the module students should be able to:

State the definition of a function and related notions and be able to sketch graphs of functions of a real variable.

Solve basic inequalities, including those involving quadratic terms and moduli.

Calculate derivatives and integrals of functions of a real variable using standard techniques.

Apply differentiation and integration in standard situations.

State the definition of the derivative and calculate simple derivatives from first principles.

Solve simple examples of first and second order ordinary differential equations.

State the definition of convergence for sequences and series and determine the convergence of standard sequences and series using standard techniques

State the Taylor series of common functions and calculate standard variants.