Polynomial Division can be used to find Taylor polynomials of rational functions easily and quickly. The polynomial division that does this trick is inverted in the sense that, instead of canceling out the leading terms at each step, one cancels out the trailing terms. Otherwise everything goes just like in the ordinary long division of polynomials.

Testing the repository

This is a test

This video explains why the geometric series 1/2 + 1/4 + 1/8 + ... converges using an elementary argument. The video is meant to be preparatory materials for students to view before the discussion of geometric series takes place in a class room. The video finishes with an intriguing example.

This is a short video that discusses the paradox of Zeno. It turns out that this paradox is not a paradox but rather an easy way to show that a geometric series converges (if |q| < 1) and to find a formula for the sum. Suitable for high schools.

English version in YouTube:

The resource is the complete diagnostic testbank used for testing students entering the first calculus course at the University of Helsinki. This diagnostic test covers a large part of the normal high school curriculum in Finland. The test was originally written by Andre Heck and later slightly edited by Mika Seppälä. Actual diagnostic tests used contain a part of these questions, not all of them.

This set of exercises will teach the student to compute limits of functions. A WebALT service.

The Online Single Variable Calculus is a full course at university level covering topics from Real Numbers, Functions, Differentiation, Integration, Series and Differential Equations. Each topic is organized in a module consisting of a Ten Minute Talk accompanied by Open Problems and Solved Problems. The material has been designed especially for online delivery.

This is a series of calculus podcasts in English based on the materials of the Single Variable Calculus Lecture Notes by M. Seppälä.

How do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. This website contains interactive materials to support the book How Round Is Your Circle?, by John Bryant and Chris Sangwin. This book invites readers to explore many of the same fundamental questions that working engineers deal with every day. It also illustrates how physical models are created from abstract mathematical ones.

Riemann Surfaces and Teichmüller Spaces by Mika Seppälä and Tuomas Sorvali. A graduate level book. Originally published by North Holland. Now out of print.