CALCULUS

SEQUENCES

• Introduction to Sequences
• Basic Definitions
• Limit of a Sequence
• More on Limit of a Sequence
• Some Special Limits
• More Challenging Limits
• More Problems on Sequences
• Stirling's Formula

SERIES

• Introduction to Series
• Convergence of Series
• The Geometric Series
  • Application: A Bouncing Ball
• The Particular Case of Positive Series
• The Root and Ratio Tests
• More Examples on Series
• Bertrand Series
• Absolute Convergence
• Conditional Convergence
• More Problems on Series

LIMIT AND CONTINUITY

• Introduction and Basic Definitions
• Some Basic Properties
• The Pinching or Sandwich Theorem
• Limits and Infinity
• Continuity
• The Intermediate Value Theorem
• The Bisection Method

DIFFERENTIATION

• The Definition of the Derivative
• Using the Definition to Compute the Derivative
• Techniques of Differentiation
• Derivatives of Trigonometric Functions
• The Chain Rule
• Implicit Differentiation
• Linear Approximations
• The Newton-Raphson Method
• Differentiating Inverse Functions
• Vertical Tangents and Cusps
• Differentiation and Continuity
• The Mean-Value Theorem
• Monotonicity and the Sign of the Derivative
• Critical Points
• Global Extrema
• Concavity and Points of Inflection
• More Problems on the Derivative

INTEGRATION

• The Area Problem and the Definite Integral
• Properties of the Definite Integral
• More on the Area Problem
• The Fundamental Theorem of Calculus
• Mean Value Theorems for Integrals

TECHNIQUES OF INTEGRATION

• Integration by Parts
• Integration of Rational Functions
• Substitution
• Trigonometric Substitution
• Rational Expressions of Trigonometric Functions
• Integrating Powers and Product of Trigonometric Functions
• More on Product of Sines and Cosines
• Other Trigonometric Powers
• Numerical Integration
• Problems on Techniques of Integration

LOCAL BEHAVIOR of FUNCTIONS

• Taylor Polynomials
• Indeterminate Forms: Introduction
• Indeterminate Quotient Forms
• Other Indeterminate Quotient Forms

   

• Improper Integrals
  • Introduction and Basic Definitions
  • Convergence and Divergence of Improper Integrals
  • Tests of Convergence
  • Absolute Convergence of Improper Integrals
  • Improper Integrals and Series: The Integral test
  • Problems on Improper Integrals
  • Special Functions
    • The Gamma Function
    • The Beta Function

POWER SERIES

• The Geometric Series
• Power Series
• The Radius of Convergence of Power Series
• Taylor Series

FOURIER SERIES

• Fourier Series: Basic Results
• Fourier Sine and Cosine Series
• Convergence of Fourier Series
• More on Convergence of Fourier Series
• Gibbs Phenomenon
• Bessel's Inequality and Parseval Formula: The Energy Theorem
• Operations on Fourier Series
• Application of Fourier Series to Differential Equations