The Math Tutor material is arranged according to topics and points of view that correspond to different needs of students.

**Chapters.**

**Sequences**covers basic properties of sequences (boundedness, monotonicity) and above all limit. Some methods use knowledge of functions, in particular the L'Hospital rule.**Functions**introduces real functions. It covers topics that do not require differentiation: domain, boundedness, symmetry, periodicity, continuity and limits. It includes a survey of elementary, useful and weird functions. It also covers monotonicity and concavity, but from a theoretical point of view - definitions, properties, and some examples on how to determine monotonicity by definition. Practical approach requires derivatives and so it is covered in the appropriate chapter. Similarly, in the section on limits we talk about asymptotes generally, practical approach to asymptotes is in the section Derivatives - Graphing functions.**Derivatives**covers derivative and its properties, important theorems that use derivative (derivative and monotonicity/concavity, Mean Value Theorem) and applications (optimization, Taylor polynomial). One section focuses on graphing functions, which - when done properly - collates information covered in chapter Functions (domain, limits, symmetry) and information obtained using derivatives (monotonicity, concavity).**Integrals**covers integrals (indefinite and definite, proper and improper including testing convergence) and their applications (e.g. area, volume)**Series**covers series of real numbers (convergence, absolute convergence, tests), then branches out to function series including the Taylor series and the Fourier series.

**Points of view.**

**Theory**is a description of the topic; the definitions and important properties are briefly covered, illustrated by simple examples. Important theorems are quoted, also introduced are important examples and the most important theoretical tools.

But Theory is not the core of the Math Tutor, it definitely does not offer a concise and logical exposition, which you could read and learn the material with all depth. That you can find in textbooks, here we focus on what textbooks traditionally lack: Understandable explanation showing how to solve problems. This among other things means that we sometimes refer to things that are covered later, since when solving problems we often draw on methods from different areas; and there is no question that you will not find proofs here.**Methods Survey**offers an overview of techniques of calculation (based on tools covered in the Theory part) and advice on their use; while in Theory, the methods are covered from the point of view of theory, in Methods Survey a different approach is chosen: Given a problem from the chosen area, how can I solve it and how do I recognize what to actually do with it? You will also meet favourite tricks there. In the exposition it is assumed that the reader has a basic idea concerning the notions used and mastered basic tools of calculation.**Solved Problems**are sorted by topics and they try to cover all typical problems and tricks. We also discuss alternative solutions and blind valleys, the focus is not on techniques of calculation but on the decision-making process: How do you recognize which calculation should be performed?**Exercises**offer problems without solutions, but with step-by-steps hints, sorted by topics and difficulty.

Math Tutor can be used for instance like this: If you have troubles with understanding the notions, Theory with its explanatory remarks could help. You can also find there explanation of methods that are basic in the area and that form building blocks for further understanding and applications. Basically you find there everything you need to know before you start solving problems.

If you have a feeling that you have some idea what is going on, but have trouble with practical application of your knowledge, try Methods Survey. Here we attempt to systematize knowledge, from mastered basic methods we develop algorithms for solving specific types of problems. For instance, if you are studying integrals, it is assumed that when reading Methods Survey, you already know how to perform (mechanically) substitution, integration by parts and partial fractions decomposition (they are covered in Theory); this knowledge is then applied to various types of integral.

Solved problems are really useful if you already know something about the methods (best if you have already learend the material in Methods Survey), so when you are reading a solution, you can compare the chosen procedure with what you learned theoretically.

Exercises are chosen to cover typical problems of varying difficulty. It is strongly recommended that you first attempt to solve them on your own. You learn best if you first try to solve a problem, and only after finishing the calculation (or getting stuck) ask what to do (the frustration and other strong emotions will then etch the solution in your memory). If you do not have enough time, it may still help if you just loook at a problem, think of the method you would apply and then check whether you were correct; in this way you will strengthen the association between types of problems and corresponding methods.