Advanced calculus

MAT11006 Calculus IA: Limits and differentiation
MAT11007 Calculus IB: Integration

After successfully completing this course, students will learn to

• Operate with Axioms of the real number system, especially the completeness of the ordering of the real numbers and their consequences.
• Rigorously study convergence phenomena of sequences of real numbers.
• Prove statements about real numbers, sequences of real numbers, and real valued functions based on axioms, exact definitions, and with various proof techniques, such as estimating quantities with triangle inequality, mastering the so called epsilon-delta method, using induction, and proof by contradiction.
• Understand the proofs of key theorems of Calculus, such as, the Bolzano-Weirstrass, Extreme Value, and Intermediate Value Theorems.
• Handle sequences of functions and determine their convergence either point-wise or uniformly as well as find their limit functions.
• Understand why the limit function of uniformly convergent sequence of continuous functions is continuous, differentiable, and integrable under certain conditions.

All necessary study material can be found in the course area. Books on Introductory Real Analysis may be used.

In addition, the book "Analyysiä reaaliluvuilla" (Analysis with Real Numbers) by Harjulehto, et al. is suitable course material for those who know Finnish.

Equivalences to other courses

This is part in a series of online Calculus courses. The collection of the three courses

• MAT11006: Calculus IA: Limits and Differentiation •
• MAT11007: Calculus IB: Integration
• MAT11008: Advanced Calculus

correspond to the collection of the three Finnish analysis courses

• MAT11003 Raja-arvot
• MAT11004 Differentiaalilaskenta
• MAT11005 Integraalilaskenta

However, no single course in either collection can substitute a course from the other.

Completion methods

This is an online course. All course material and activities can be found on the online course area. There is a final proctored paper and pencil exam at the end of the course.

Assessment practices and criteria

The lecture course will be evaluated based on course work and/or course exam.

Activities and methods in support of learning

Peer-evaluated homework. Discussion area for asking help to homework and guidance to course topics.

Schedule

The course is offered every year in spring, period 3.

Recommended time or stage of studies for completion

Study module

MAT110 Mathematics, Basic Studies
BSMS1000 Mathematics and Statistics, Basic Studies

The course is suitable for several other study modules in mathematics.

Expiry of studies

The course expires according to the university's instructions. (Expiry of studies(opens in a new tab)(opens in a new tab)(Avautuu uuteen välilehteen))

Languages of instruction

English (Students may take the final assessment in Finnish and Swedish also)