1. Chapter 10 – Parametric Equations & Polar Coordinates 10.1 Curves Defined by Parametric Equations 1Erickson

2. 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

3. 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

4. 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

5. 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

6. Example 1 – pg 641 # 4 10.1 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

7. Example 2 10.1 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

8. Example 3 – pg 642 # 22 10.1 Curves Defined by Parametric Equations8  Describe the motion of a particle with position (x, y) as t varies in the given interval. Erickson

9. 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