Course Notes

Here is an interactive text to accompany the course. We invite you to take advantage of the many demonstrations to explore Calculus more deeply.

• Definite Integrals and the Fundamental Theorem of Calculus
  • Introduction
  • Overview of calculus from a historical perspective
  • Some special sums
  • How Archimedes found the area of a circle
  • The area under a curve
  • The average value
  • The definite integral
  • The Fundamental Theorem of Calculus
  • How to use the Fundamental Theorem

• Applications of the Definite Integral
  • Acceleration, Velocity, and Position
  • Average value of a function
  • Calculating Simple Volumes
  • Arc Length and Surface Area
  • Work

• Techniques for Antidifferentiation
  • Indefinite Integrals
  • Change of Variables
  • Integration by Parts

• More Applications
  • Differential Equations
    • Separable Differential Equations
    • The Logistic Equation and Integration by Partial Fractions
    • The Logistic Equation with Harvesting and Qualitative Methods
    • More about Qualitative Methods for Differential Equations
    • Equilibrium Solutions and Stability
    • An Application: Can cats control a rat population?
  • Weighted Averages
    • Centers of Mass
    • Probability Densities
    • Means, Medians, and Averages
    • Moments of a Probability Distribution
    • The Normal distribution

• More Techniques for Integration
  • Numerical Integration
  • Improper Integrals

• Taylor polynomials and series
  • Introduction to series
  • Convergence of series
  • Finding Taylor Series
  • Why do Taylor series work?
  • What can we do with Taylor series?
  • More applications of series

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