## MATH101: Calculus

**Terms Taught:**Michaelmas Term Only**US Credits:**2 semester credits**ECTS Credits:**4 ECTS**Pre-requisites:**A year of general mathematics, including basic calculus.

### Course Description

The course covers: complex numbers, functions and graphs; limits of sequences and sums of infinite series; differentiation, product and chain rules; Taylor series; integration: fundamental theorem of calculus; integration by parts and substitution.

### Educational Aims

This course aims to provide the student with an understanding of functions, limits, and series, and a knowledge of the basic techniques of differentiation and integration. The purpose of this course is to study functions of a single real variable. Some of the topics will be familiar, others will be studied more thoroughly in subsequent courses.

The module begins by introducing examples of functions and their graphs, and techniques for building new functions from old. We consider rational functions and the exponential function. We then consider the notion of a limit, sequences and series and then introduce the main tools of calculus. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The rules for calculating derivatives are obtained from the definition of the derivative as a rate of change. Taylor series are calculated for functions such as sin, cos and the hyperbolic functions.

We then introduce the integral and review techniques for calculating integrals. We learn how to add, multiply and divide polynomials and introduce rational functions and their partial fractions. Rational functions are important in calculations, and we learn how to integrate rational functions systematically. The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parameterize geometrical curves.

### Outline Syllabus

- Arithmetic of complex numbers;
- Polynomials;
- Rational functions and partial fractions;
- Exponential and hyperbolic functions;
- Compositions and inverses;
- Induction;
- Sequences and limits;
- Differentiation;
- Product and Chain rules;
- Maxima and minima;
- Taylor series;
- Complex exponentials and trigonometric functions;
- Definite integral as areas;
- Fundamental theorem of calculus;
- Integration by parts and by substitution;

### Assessment Proportions

- Coursework: 50%
- Exam: 50%