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