Analysis I

COURSE OBJECTIVE
At the end of this course the student
a) is able prove a theorem with mathematical induction.
b) knows the definition of limit of a sequence and a function and is able to calculate limits, using various calculus techniques (e.g. squeeze law and l'Hospitals rule).
c) knows the definition of continuity and is able to prove or disprove the continuity of a function.
d) knows the definition of derivative of a single variable function and is able to calculate (higher) derivatives and a Taylor polynomial of a function.
e) knows the definition of a Riemann integral and is able to prove if a function is Riemann integrable.
f) is able to calculate an integral, using various calculus techniques (e.g. substitution method, integration by parts, partial fraction
decomposition).
g) is able to determine if an improper integral is convergent, and calculate its value.
h) is able to work with complex numbers.

COURSE CONTENT
In this course we present a thorough introduction of the theory of real analysis for single variable functions. Theorems and their proofs form
an important part of this course. In addition sufficient attention is paid to various calculus techniques. We will treat the following topics:
a) Natural numbers and mathematical induction.
b) Rational and real numbers and the completeness of the real numbers.
c) Sequences of real numbers (convergence, subsequences, Cauchy sequences).
d) Continuity and limits of real functions. Uniform continuity.
e) Differentiation (derivative of a function, mean value theorems, L'Hospital's rule, Taylor's theorem).
f) Integration (Riemann integral, improper integral, integration techniques)
g) Complex numbers

TEACHING METHODS
Lectures (2x2 hours per week) and tutorials (1x2 hours per week)

TYPE OF ASSESSMENT
There will be a midterm exam at the end of period 1 and a final exam at the end of period 2. Details about the topics treated in each exam and the calculation of the final grade will be published in Canvas. If your grade is not sufficient, it is possible to make the resit about all
topics in the spring semester.