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ch10 - 206 Chapter 10 Parametric And Polar Curves Conic...

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Chapter 10: Parametric And Polar Curves; Conic Sections Summary: This chapter begins by introducing the idea of representing curves using parameters. These parametric equations of the curves can then be used to graph, find tangent lines, and find arc lengths of the given curves. Polar co- ordinates and their relationship to Cartesian or rectangular coordinates are then described. Polar coordinates may also be used as a particular parameterization of curves. Then tangent lines, arc length, and area may be found using polar coordi- nates as the parameterization of the curve. The latter half of the chapter discusses conic sections and how they may be graphed. Then the idea of conic sections is extended by representing these curves using polar coordinates.x OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Represent a function as a parametric curve (§10 . 1). 2. Find tangent lines of curves that are either defined parametrically (§10 . 1) or defined using polar coordinates (§10 . 3). 3. Find the arc length of a parametric curve (§10 . 1) or a polar curve (§10 . 3). 4. Convert between rectangular and polar coordinates (§10 . 2). 5. Graph functions in polar coordinates (§10 . 2). 6. Find the area between curves that have been defined in polar coordinates (§10 . 3). 7. Identify different conic sections and their graphs (§10 . 4). 8. Rotate conic sections and determine their resulting quadratic equations (§10 . 5). 9. Describe conic sections and their graphs using polar coordinates (§10 . 6). 207
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208 10.1 Parametric Equations; Tangent Lines And Arc Length For Parametric Curves PURPOSE: To discuss tangent lines and arc length of parametric curves and curves in polar coordinates. One of the main idea in this section is to come up with functions for both the x and y coordinates of a curve. In this way, curves that are generally not considered as functions may be graphed quite easily. For example, this gives a great way to graph the inverse of a function. If x = f ( t ) and y = g ( t ) then the inverse can be graphed by reversing the definitions. x = g ( t ) and y = f ( t ) would result in the graph of the inverse. The parameter t used in the definitions of x and y also parameter parametric curve results in the parametric curve having a definite orientation . If t is followed in orientation a positive direction, for example, then this will take you in one direction along a curve while a negative direction for t would take you in the opposite direction. Tangent lines were first introduced in Chapters 1 and 2 where the idea of a deriva- tive was first developed. The same kinds of tangent lines to curves are discussed in this section. The new approach taken here is to represent the curves parametri- cally by defining a function to each coordinate of a point on the curve. or as polar curves.
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