Center for Fundamental Sciences

• 1-1. Sequences and Series
  • Examples of sequences and pattern observation_Arithmetic sequences
  • Series_Σk _Properties of series
  • Arithmetic series_Geometric sequences and geometric series
  • Series_Σk² sum
• 1-2. Limits of Sequences and Infinite Series
  • Intuitive understanding of sequence limits_Three example discussions
  • Sequence limits_Three additional examples
  • Properties of limits
  • Applying limit properties to compute sequence limits
  • Sequence limits and the squeeze theorem
  • Supplement_Two incorrect reasonings about limits
• 1-3. Limits of Functions
  • Intuitive understanding
  • Examples where function value is unrelated to limit
  • Gaussian function
  • Properties of function limits (theorems)
  • Three example problems and additional squeeze theorem problems
  • Squeeze theorem examples
  • Rational function examples
  • Horizontal and vertical asymptotes
• 1-4. Continuous Functions and Their Properties

• 2-1. Derivatives and Derivative Functions
  • Definition of derivative
  • Differentiation from definition
  • Differentiability implies continuity, but continuity does not imply differentiability
• 2-2. Basic Differentiation Formulas
  • Differentiation formula_xⁿ
  • Sum, difference, product, and quotient rules
  • Chain rule and deriving quotient rule using product rule
• 2-3. Higher-order Derivatives
• 2-4. Derivatives of Trigonometric Functions
  • Sum and difference formulas and half-angle formulas
  • lim(sin(x)/x, x → 0) Part 1 (area estimation of three figures)
  • lim(sin(x)/x, x → 0) Part 2 (limit = 1 via squeeze theorem)
  • The derivative of sin(x) is cos(x)
  • Using derivative of sin(x) to find derivatives of other trigonometric functions
• 2-5. Inverse Trigonometric Functions and Their Derivatives
  • Functions and inverse functions
  • Definition and derivative of tan^-1(x)
  • Definition and derivative of sin^-1(x)
  • Definition of sec^-1(x) and notes on inverse trigonometric functions
  • Derivative formula of sec^-1(x)
• 2-6. Derivatives of Exponential and Logarithmic Functions
  • Exponential functions and rules (includes overview of exponentials and logarithms)
  • Logarithmic functions and rules
  • Proof of logarithmic rules
  • Euler's number e and natural logarithm_lim(ln(1+h)/h, h→0)
  • Differentiation of logarithmic and exponential functions
• 2-7. Implicit Differentiation
  • Implicit differentiation_Circle equation
  • Implicit differentiation and tangent slope
  • Differentiation of inverse functions
  • Implicit differentiation with parameters

Exercise 1-1 No.4(6) and Exercise 1-2 No.4(5)
https://youtu.be/gWbcmfiLCjc

Exercise 1-1 No.4(7) and Exercise 1-2 No.4(6)
https://youtu.be/1DsUQu-vVU0

Exercise 2-1 No.1(7)
https://youtu.be/-OBNRWt6nFo

Exercise 2-1 No.3
https://youtu.be/VJ0zB6DI9pE

Exercise 2-1 No.4
https://youtu.be/2i_c5rTRzIE

Exercise 2-1 No.5
https://youtu.be/npnNjBl5KWM

Exercise 2-2 No.15
https://youtu.be/0bIV9vvtWhk

Exercise 2-2 Similar to No.17
https://youtu.be/cOQ8o5BBtxM

Exercise 2-2 No.26
https://youtu.be/HfAFqXWk2KE

Exercise 2-2 No.27
https://youtu.be/OXoHuVA34RY