EXPLORATORY CALCULUS I AND II

Instructor: Thomas Bieske

I can be reached via tbieske at math dot usf dot edu.

This is the homepage to Exploratory Calculus I and II, a new approach to Calculus instruction at the University of South Florida. This sequence is designed for mathematics majors, mathematics education majors, and physical science majors. It aims to teach calculus using interactive methods, self-exploration, and technology. Lecturing will be minimized, as students will be encouraged to explore new topics as scientists.

Topics to be explored in Calculus I (Semester 1)

• Functions
  • Characteristics of linear, exponential, power, logarithmic, and trigonometric functions
  • Composition of functions and inverse functions
• Derivatives
  • Instantaneous velocity vs. average velocity
  • Derivative at a point
    • Algebraic computation-limit definition, instantaneous rate of change of a function
    • Numerical approximation-using tables of data
    • Graphical information-slope of the tangent line versus slope of secant line
  • Derivative as a function
    • Rules for computing: power, chain, product, and quotient rules
    • Increasing/decreasing vs. derivative
    • Estimating derivative from graphs and tables of data
  • Second derivative
    • Interpretation with respect to the original function
    • Interpretation with respect to the derivative
    • Using first and second derivatives to approximate functions
    • Optimization problems
  • Anti-derivatives
    • Algebraic and graphical reconstruction of functions given its derivative
    • Determine properties of a function given its derivative, either algebraically or graphically

Topics to be explored in Calculus II (Semester 2)

• Definite Integrals
  • Geometric and algebraic interpretation
  • Fundamental theorem of calculus
  • Numerical methods
    • Left and right hand sums
    • Trapezoid, midpoint, simpson methods
    • Applications
  • Indefinite Integrals and integration techniques
    • Substitution
    • Parts
    • Improper integrals
  • Sequences and series
    • Sequences
    • Series
      • Geometric series
      • Taylor series
      • Ratio, root, integral, and comparison tests
      • Fourier series