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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.Second derivatives• 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.
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