AP Calculus AB  

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Derivatives (1st Semester)

1. Concept of the Derivative

  • Graphically, numerically and algebraically
  • Instantaneous rate of change
  • Limit of the difference quotient: the definition
  • Relationship between differentiability and continuity

2. Derivative at a point

  • Slope of a curve at a point
  • Tangent line and linear approximation
  • Instantaneous rate of change
  • Approximate rate of change calculated from tables and charts

3. Derivative as a function

  • Comparison of graphs of a function and its derivative
  • The Mean Value Theorem and its consequences
  • Equations involving derivatives; verbal descriptions of functions and their derivatives

4. Second derivatives

  • Comparisons of the graphs of a function with its first and second derivative
  • Concavity
  • Inflection points

5. Applications of derivatives

  • Analysis of curves
  • Optimization, global and local
  • Modeling rates of change, including related rates
  • Implicit differentiation, including its use for inverse functions
  • The derivative as a rate of change in applied contexts, including motion, velocity speed and acceleration

6. Computation of the derivative

  • Derivatives of power, trigonometric, logarithmic, exponential, and inverse trigonometric functions
  • Sum, product, quotient and chain rules

 

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