Chapter 10 – Parametric Equations & Polar Coordinates 10.1 Curves Defined by Parametric Equations 1Erickson
Parametric Equations 10.1 Curves Defined by Parametric Equations2 Imagine a particle that moves along the curve C as shown below. It is impossible to define C by an equation of the form y = f (x) because C fails the vertical line test. But the x- and y- coordinates of the particle are functions of time so we can write x = f (t) and y = g(t). Erickson
Definitions 10.1 Curves Defined by Parametric Equations3 Suppose that both x and y are given as functions of a third variable t, called a parameter, by the equations x = f(t) and y = g(t) These equations are called parametric equations. As t varies, the point (x, y) = (f(t), g(t)) varies and traces out a curve C which is called a parametric curve. NOTE: The parameter t does not necessarily represent time. Erickson
Definitions 10.1 Curves Defined by Parametric Equations4 In general, the curve with the parametric equations x = f(t) and y = g(t) a t b has an initial point (f(a), g(a)) and terminal point (f(b), g(b)). Erickson
Sketching Parametric Curves Erickson10.1 Curves Defined by Parametric Equations5 The following app will demonstrate how to sketch parametric curves: Graphing Parametric Curves Graphing Parametric Curves
Example 1 – pg 641 # Curves Defined by Parametric Equations6 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. Erickson
Example Curves Defined by Parametric Equations7 (a) Eliminate the parameter to find the Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. Erickson
Example 3 – pg 642 # Curves Defined by Parametric Equations8 Describe the motion of a particle with position (x, y) as t varies in the given interval. Erickson
Book Resources Erickson10.1 Curves Defined by Parametric Equations9 Video Examples Example 2 – pg. 637 Example 2 – pg. 637 Example 5 – pg. 638 Example 5 – pg. 638 Example 8 – pg. 640 Example 8 – pg. 640 More Videos Sketch the curve given by parametric equations Sketch the curve given by parametric equations Wolfram Demonstrations Comparing Parameterizations Comparing Parameterizations Parametric Trace Parametric Trace Construction of a Bezier Curve Construction of a Bezier Curve Cycloid Curves Cycloid Curves Brachistochrone Problem Brachistochrone Problem