Differential and Integral Calculus 2021
Topic outline

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: Nonhomogeneous 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.

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Chapter 1:
 Basics of sequences
 Some important sequences
 Convergence, divergence and limits

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Chapter 2:
 Convergence
 Basic results
 Absolute convergence
 Convergence tests

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Chapter 3:
 Limit of a function
 Limits and continuity
 Properties of continuous functions

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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

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Chapter 7:
 Area in the plane
 Starting point
 General case

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Chapter 8:
 From sum to integral
 Integration of continuous functions
 Piecewisedefined 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 \(\star\) !
 Sequences (25:01)