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