### 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):

- Sequences (25:01)
- Series (35:43; please skip blackboard cleaning from 10:00 to 13:45 :)
- Continuity (33:09)
- Derivative (30:11)
- Taylor polynomials and series Part 1: Taylor (35:50), Part 2: Power series (23:24)
- Elementary functions Part 1: Inverse (18:49); Part 2: Inverse trig. (18:58); Part 3: exp and ln (21:34)
- The concept of area in the plane (14:22)
- Integral Part 1: Definition (32:06); Part 2: Applications (13:12); Part 3: Improper integrals (10:58); Part 4: Methods of integration (41:51)
- ODE of the 1st order Part 1: Separable (24:55); Part 2: Linear (14:24)
- 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.

###### Chapter 1:

- Basics of sequences
- Some important sequences
- Convergence, divergence and limits

###### Chapter 2:

- Convergence
- Basic results
- Absolute convergence
- Convergence tests

Chapter 3:

- Limit of a function
- Limits and continuity
- Properties of continuous functions

Chapter 4:

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

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

Chapter 7:

- Area in the plane
- Starting point
- General case

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 \(\star\) !