In this, the second section of the course, the topics include:
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.
Derivative at a point
• 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.
Derivative as a function
• Corresponding characteristics of graphs of ƒ and ƒ’.
• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.
• The Mean Value Theorem and its geometric interpretation.
• Equations involving derivatives. Verbal descriptions are translated into equations
involving derivatives and vice versa.
• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ?’’.
• Relationship between the concavity of ƒ and the sign of ƒ?’’.
• Points of inflection as places where concavity changes.