“Never Believe The Math Teacher”

An Approach to Calculus

by Lawrence Spector

Borough of Manhattan Community College,
City University of New York


The Lessons


  • 1.  Continuous versus discrete

    • The definition of a “continuous” quantity.

  • 2. Limits

    •  A sequence of rational numbers. 
    •  The definition of the limit of a variable. 
    •  The limit of a function.
    •  Theorems on limits. Limits of polynomials.
  • 3. Continuous functions

    • The definition of “a function is continuous at a value of x.”
  • 4.  Infinity (∞)

    •  The definition of “a variable becomes infinite.” 
    •  Limits of rational functions.
  • 5.The Derivative

    • The slope of a tangent line to a curve.
    • The difference quotient and the definition of the derivative. 
    • Notations for the derivative.
    • The equation of a tangent to a curve.
  • 6. Rules for derivatives

    • The derivative of a constant. The derivative of y = x
    • The product rule.
    • The power rule.
    • The derivative of the square root function.
  • 7.  The chain rule

    • The derivative of “a function of a function.”

Appendix 1.  Are the real numbers really numbers?

Appendix 2.  Is a line really composed of points?