Differential and Integral Calculus 2021

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)
      

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    Differential and Integral Calculus Книга
  • 

    Slides in Finnish Файл

    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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    1. Sequences Сторінка

    Chapter 1:

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

    2. Series Сторінка

    Chapter 2:

    • Convergence
    • Basic results
    • Absolute convergence
    • Convergence tests
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    3. Continuity Сторінка

    Chapter 3:

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

    Chapter 4:

    • Derivative
    • Properties of derivative
    • Derivatives of Trigonometric Functions
    • The Chain Rule
    • Extremal Value Problems
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    5. Taylor polynomial Сторінка

    Chapter 5:

    • Taylor polynomial
    • Taylor polynomial and extereme values
    • Newton's method
    • Taylor series
    • Power series
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    6. Elementary functions Сторінка

    Chapter 6:

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

    Chapter 7:

    • Area in the plane
    • Starting point
    • General case
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    8. Integration 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
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    9. Differential equations Сторінка

    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 \(\star\) !