# Real Analysis and the Calculus

This module is taught over intensively by Birmingham academics over 12 weeks. There is continuous assessment and a final examination.

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 rigourous point of view.

The notion of a function of a real variable and its derivative are formalised. 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.

## 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 appropriate situations.
- State the definition of the derivative and calculate derivatives from first principles. State the Fundamental Theorem of Calculus and have an appreciation of its proof.
- Solve simple examples of first and second order ordinary differential equations.
- State the definition of convergence for sequences and series.
- Determine the convergence of various sequences and series using the algebra of limits and other standard techniques.
- State the Taylor series of common functions and calculate Taylor series of functions.
- Construct simple proofs from definitions and standard results.
- Typeset simple text and mathematics using the LaTeX package.