Contents

• Contents
• An introduction to calculus
  • Constant functions
  • Linear functions
  • Factoring differences of powers
  • Squaring functions
  • Cubing functions
  • The general case
  • The fundamental theorem of calculus
• The rates of change of some other functions
• Some extensions of our results
  • Positive rational powers
• More on rates of change
  • Rates of change of sums
  • Two applications
• What do we mean by ``approaching''?
• The funnelling theorem:
• Formalizing the concept of ``approaching''
  • Constant functions:
  • Linear functions
  • Quadratic functions
  • The sum of two functions
  • Other properties
  • An abuse of notation and terminology
• Homework, due 1/28:
• Infinity
• Continuity
• More rates of change for familiar functions
  • The tangent function
  • Negative integer powers
  • Rates of change for inverse functions
  • The natural logarithm
  • The arctangent function:
  • The arcsine function:
• Another extension of the power rule
  • The algebraic computation
  • The geometric approach
  • Some gaps in our reasoning
• Information from derivatives
• Product rule, quotient rule, and chain rule
  • Differentiating products:
  • Differentiating quotients
  • Composite functions:
• More on composite functions
  • Continuity composite functions
  • Differentiabilty and continuous difference quotients
  • Another application of the same idea
• Summary of differentiation
  • The definition
  • The interpretation
  • General rules
  • Specific derivatives:
  • Additional derivatives:
  • A note on the power rule
• Some simple applications of differentiation
  • Growth and Decay Problems
  • Hooke's Law
  • Implicit Differentiation
  • Related rates
  • Numerical solutions of equations
• Homework, due 2/23
• Two important properties of functions
  • The range of a continuous function
  • Where maxima are not located
  • Applications of Rolle's Theorem
    • The mean value theorem
• Some examples of searching for extreme values
• The fundamental theorem of calculus
• More on limits
  • Related facts about the natural logarithm
  • Cauchy's Mean Value Theorem and L'Hopital's Rule
    • Examples
  • Polynomial Approximations
    • Approximating sine for small values
    • Approximating the natural logarithm
• Linear approximations