### General Information

Course Code | XB_0009 |
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Credits | 6 EC |

Period | P4+5 |

Course Level | 100 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | prof. dr. J. Hulshof |

Examiner | prof. dr. J. Hulshof |

Teaching Staff |
prof. dr. J. Hulshof |

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods | Seminar, Lecture |
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Target audiences

This course is also available as:

### Course Objective

After this course, the student...1. ... knows basic definitions concerning limits and continuity

(convergence, Cauchy sequence, limit, completeness, continuity, uniform

continuity) and is able to determine whether a sequence, series or

function satisfies these definitions;

2. ... knows the definition of differentiability (i.e., that a function

can be approximated by a linear one) and can determine whether a

function (and in particular, a power series) is differentiable;

3. ... knows the definition of Riemann integrability and can prove that

certain functions (in particular, polynomials, monotone and uniformly

continuous functions) are Riemann integrable;

4. ... knows the definition of basic concepts from metric topology

(metric, convergence, completeness, Banach space) and can prove that

simple examples satisfy these definitions;

5. ... knows the statement of the Banach Fixed Point Theorem, and can

apply this theorem to solve fixed point equations (in particular

integral and differential equations).

### Course Content

This course treats the rigorous mathematical theory behind Calculus:limits, continuity, linear approximation, differentiability,

integrability, and the mutual relation between these concepts. The

mathematical tools that are necessary for formulating and proving the

essential results of Calculus are first presented in the context of real

valued sequences and real valued functions of a real variable, in such a

way that everything can later be generalised (to Y-valued functions of

variables in X, with X and Y Banach spaces). The space C[a,b] of real

valued continuous functions on an interval [a,b] will appear as the

first example of such a Banach space.

Starting point of the course are an ancient iterative scheme for solving

equations, and the fundamental properties of (the set of) real numbers.

Highlights: a fairly complete exposition of power series directly based

on a systematic algebraic approach for monomials, and an early

introduction of the Implicit Function Theorem via a contraction argument

and the Banach Fixed Point Theorem.

Topics covered:

1. Cauchy sequences, convergence, limits;

2. Completeness of the real numbers; theorem of Bolzano-Weierstrass;

3. Continuity and uniform continuity;

4. The concept of differentiability (including differentiability of

power series);

5. The concept of Riemann integrability (including Riemann integrability

of monotone and uniformly continuous functions);

6. The language of metric topology;

7. Completeness of the space C[a,b]; uniform convergence;

8. The Banach Fixed Point Theorem (with applications to integral and

differential equations, and the implicit function theorem).

### Teaching Methods

Lectures, study sessions and tutorials (2+1+2 hours per week).You are also required to hand in a homework assignment every other week.

We expect you to dedicate in total about 10 hours per week to this

course.

### Method of Assessment

Your final grade is built up as follows:A written midterm exam [40%];

A written final exam [50%];

Six written assignments [10%].

To pass the course...

... your final score must be no less then 55% (all students);

... you must have been present at 70% of the study sessions and

tutorials (full time students only);

If you don't fulfil these requirements, then you must take the resit

exam. The resit exam then counts for 100% (i.e. the partial grades that

you obtained during the course will no longer be valid).

### Entry Requirements

Basic Concepts in Mathematics (or another course on general mathematicallanguage, notation, and concepts, including proof by induction and

elementary combinatorics);

### Literature

Course notes will be made available through Canvas.### Target Audience

Bachelor Mathematics Year 1### Additional Information

Participation in 70% of study sessions and tutorials is mandatory (forfull time students) in order to pass the course.