## Структура за темами

• ### Course materials

The materials will be added here as the course proceeds. The videos do not cover all the course material; they only contain the main definitions and some examples. Short videos (with varying length):

1. Sequences (25:01)
2. Series (35:43; please skip blackboard cleaning from 10:00 to 13:45 :)
3. Continuity (33:09)
4. Derivative (30:11)
5. Taylor polynomials and series Part 1: Taylor (35:50), Part 2: Power series (23:24)
6. Elementary functions Part 1: Inverse (18:49); Part 2: Inverse trig. (18:58); Part 3: exp and ln (21:34)
7. The concept of area in the plane (14:22)
8. Integral Part 1: Definition (32:06); Part 2: Applications (13:12); Part 3: Improper integrals (10:58); Part 4: Methods of integration (41:51)
9. ODE of the 1st order  Part 1: Separable (24:55); Part 2: Linear (14:24)
10. ODE of the 2nd order Part 1: General theory (14:46); Part 2: Homogeneous case (18:46); Part 3: Non-homogeneous case (21:25)
• • Some parts of the material were or will be edited from these Beamer slides for the corresponding Finnish version of the course. These may be helpful for some participants.

• Сторінка
###### Chapter 1:
• Basics of sequences
• Some important sequences
• Convergence, divergence and limits
• Сторінка
###### Chapter 2:
• Convergence
• Basic results
• Absolute convergence
• Convergence tests
• Сторінка

Chapter 3:

• Limit of a function
• Limits and continuity
• Properties of continuous functions
• Сторінка

Chapter 4:

• Derivative
• Properties of derivative
• Derivatives of Trigonometric Functions
• The Chain Rule
• Extremal Value Problems
• Сторінка

Chapter 5:

• Taylor polynomial
• Taylor polynomial and extereme values
• Newton's method
• Taylor series
• Power series
• Chapter 6:

• Functions
• Inverse functions
• Transcendental functions
• Trigonometric functions
• Arcus functions
• Exponential function
• Logarithms
• Hyperbolic functions
• Сторінка

Chapter 7:

• Area in the plane
• Starting point
• General case
• Сторінка

Chapter 8:

• From sum to integral
• Integration of continuous functions
• Piecewise-defined functions
• Important properties
• Fundamental Theorem of Calculus
• Geometric applications
• Area of a plane region
• Arc length
• Surface of revolution
• Solid of revolution
• Integrals of elementary functions
• Improper integral
• Comparison test
• Integration techniques
• Logarithmic integration
• Partial fraction decomposition
• Integration by Parts
• Integration by Substitution
• Chapter 9:

• Introduction
• Solutions of a differential equation
• Initial condition
• Direction field
• 1st Order Ordinary Differential Equations
• Linear 1st order ODE
• Solving a 1st order linear ODE
• Separable equation
• The trivial solutions of a separable ODE
• $$\star$$ Equations expressible as separable>
• $$\star$$ Euler's method
• 2nd and higher order ODEs
• Solving a homogeneous ODE
• Equations with constant coefficients
• Euler's differential equation
• Nonhomogeneous linear differential equations

Note. Extracurricular chapters are marked with $$\star$$ !