II. Derivatives

A.  Concept of the derivative.

• Derivative presented graphically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.

• Slope of a curve at a point.  Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
• Tangent line to a curve at a point and local linear approximation.
• Instantaneous rate of change as the limit of average rate of change.
• Approximate rate of change from graphs and tables of values.

• Corresponding characteristics of graphs of f and f'.
• Relationship between the increasing and decreasing behavior of f and the sign of f'.
• The Mean Value Theorem and its geometric consequences.
• Equations involving derivatives.  Verbal descriptions are translated into equations involving derivatives and vice versa.

• Corresponding characteristics of the graphs of f, f', and f''.
• Relationships between the concavity of f and the sign of f''.
• Points of inflection as places where concavity changes.