# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 22 Hours. (22 x 1 hour lectures) |

Seminars / Tutorials | 5 Hours. (5 x 1 hour tutorials) |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 80% |

Semester Assessment | (coursework) | 20% |

Supplementary Assessment | 2 Hours (written examination) | 100% |

### Learning Outcomes

On completion of this module, a student should be able to:

1. determine solution sets of elementary inequalities;

2. determine whether or not a set of real numbers is bounded;

3. determine the supremum and infimum of bounded sets;

4. describe the notion of a sequence of real numbers and determine whether sequences are convergent or divergent;

5. apply the standard theorems on convergence of sequences;

6. manipulate sequences defined by recurrence relationships;

7. use the basic tests for convergence of series;

8. describe the notions of continuity and differentiability for real-valued functions and determine whether functions have these properties;

9. state and use the mean-value theorem of the differential calculus, L'Hopital's rule, Taylor's theorem and Maclaurin's theorem.

### Brief description

A first course in Mathematical Analysis aims to tackle some of the issues which are glossed over in the development of calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. These ideas play a basic part in the subsequent development of mathematics.

### Aims

This module aims to tackle some of the issues which are glossed over in the development of the calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. The theoretical aspects of the subject will be developed in conjunction with the techniques required to solve problems.

### Content

2. BOUNDED SETS: Upper bound, lower bound, infimum, supremum. Completeness axiom for the real numbers.

3. SEQUENCES: Limit of a convergent sequence of real numbers. Formal derivation of some limit theorems. The sandwich theorem. Sequences defined by recurrence relationships. Increasing and decreasing sequences and related convergence theorems. Boundedness of convergent sequences. Subsequences.

4. APPLICATIONS OF THE DIFFERENTIAL CALCULUS: Rolle's theorem. Mean-value theorem of the differential calculus. L'Hopital's rule. Taylor's theorem, Maclaurin's theorem.

5. INFINITE SERIES: Partial sums. Convergence of infinite series. Examples of convergent and divergent series, including geometric series. Tests for convergence of series of positive terms: comparison test, ratio test, integral test.

### Reading List

**Recommended Text**

J Stewart (2001) Calculus: concepts and contexts 2nd edition Brooks/Cole Primo search Weir, M D, Haas, J and Giordano, F R (2005) Thomas' Calculus Addison Wesley Primo search

**Supplementary Text**

K E Hirst (1995) Numbers, Sequences and Series Arnold Primo search R Adams (1999) Calculus - a Complete Course 4th Addison-Wesley Primo search R G Bartle & D R Sherbert (1992) Introduction to Real Analysis 2nd Wiley Primo search R Haggarty (1993) Fundamentals of Mathematical Analysis 2nd Addison-Wesley Primo search

### Notes

This module is at CQFW Level 4