Differential and Integral Calculus 2021

Topic outline

• 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)
      

  • 

    Differential and Integral Calculus Book
  • 

    Slides in Finnish File

    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.
    

  • 

    1. Sequences Page

    Chapter 1:

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

    2. Series Page

    Chapter 2:

    • Convergence
    • Basic results
    • Absolute convergence
    • Convergence tests
  • 

    3. Continuity Page

    Chapter 3:

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

    4. Derivative Page

    Chapter 4:

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

    5. Taylor polynomial Page

    Chapter 5:

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

    6. Elementary functions Page

    Chapter 6:

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

    7. Area Page

    Chapter 7:

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

    8. Integration Page

    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
  • 

    9. Differential equations Page

    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 \(\star\) !