* Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.

II. Derivatives

Concept of the derivative.

  • Derivative presented geometrically, 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 consequences.

  • 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.

Applications of derivatives.

  • Analysis of curves, including the notions of monotonicity and concavity.

  • Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.

  • Optimization, both absolute (global) and relative (local) extrema.

  • Modeling rates of change, including related rates problems.

  • Use of implicit differentiation to find the derivative of an inverse function.

  • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

  • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.

  • Numerical solution of differential equations using Euler’s method.

  • L’Hô pital’s Rule and its use in determining convergence of improper integrals and series.

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