Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin

Chapter 11: Parametric Equations and Polar Coordinates Parametrizations of Plane Curves Calculus with Parametric Curves Polar Coordinates Graphing in Polar Coordinates Areas and Lengths in Polar Coordinates Conic Sections Conics in Polar Coordinates

Chapter 11 Overview Parametric equations are used to describe the position of an object in space by making each coordinate a function of time. This allows each coordinate motion to be considered separately. Polar coordinates allow the description of many curves to be written more simply than in the usual rectangular system. This greater simplicity of description allows the standard Calculus operations to become simpler as well.

11.01: Parametrizations of Plane Curves 1 Definition of a Parametric Curve: A parameter t (usually time) is assigned an ordered pair, (x, y), by the relations x[t] and y[t]. Example 1 The Lion and the Ranger problem. Eliminating the parameter. Examples 2 & 4 Using a TI-84 calculator to graph parametric equations and observe the direction of motion along the curve. Example 3 Finding ‘a’ parametrization of f[x] = (x, y). Examples 5 & 7

11.01: Parametrizations of Plane Curves 2 A point on the rim of a wheel traces out the Cycloid curve as it rolls along the ground. The starting position of the cycloid curve:

Deriving the parametric equations for the Cycloid : Parametrizations of Plane Curves 3

11.01: Parametrizations of Plane Curves 4 The parametric equation of the cycloid: Example 8

11.02: Calculus with Parametric Curves 1 First derivative of a parametric curve: Tangent to a parametric curve. Example 1 Area under a parametric curve. Example 3 Length of a parametric curve. Examples 4 & 5

11.03: Polar Coordinates 1 Polar functions are written in the form r[θ] but polar plane points are written as (r, θ). Examples 1 – 3 Graphing with software. Polar coordinates and Cartesian coordinates are related by the following equations:. Examples 4 – 6

11.04: Graphing in Polar Coordinates 1 Symmetry tests for Polar graphs. Examples 1 – 3 The slope of is not given by but by.

11.05: Areas and Lengths in Polar Coordinates 1 Polar areas: Examples 1 & 2 Lengths of Polar curves: Example 3

11.06: Conic Sections 1 This section is not covered.

11.07: Conics in Polar Coordinates 1 This section is not covered.