Introduction
636
- 9.1Parametric Equations
- 9.2Tangent Lines; Arc Length
- 9.3Surface Area of a Solid of Revolution
- 9.4Polar Coordinates
- 9.5Polar Equations; Parametric Equations of Polar Equations; Arc Length of Polar Equations
- 9.6Area in Polar Coordinates
- 9.7The Polar Equation of a Conic
- Chapter Review
- Chapter Project
Polar Graphs and Microphones
If you've ever shown up early to a concert or performance and seen the staff setting up the sound system, you understand a bit about the work that goes into obtaining optimal sound quality. Different microphones are designed to pick up sound from different directions, and sound technicians know that the selection and placement of microphones and monitors (speakers that are aimed back at the musicians so that they can hear the other instruments) vary depending on the specific purpose and venue. The same choices factor into the construction of cellphones and Bluetooth devices, which need to amplify the sound coming out of your mouth and minimize the barking dogs, loud trucks, and random conversations that may be happening around you.
The choice of the right microphone and speaker and their placement to achieve a specific goal requires an understanding of the direction of the sound and the area where it is being picked up. We can use polar coordinates and polar equations to model these goals.
CHAPTER 9 PROJECT
The Chapter Project on p. 692 examines the design of several different microphones and the directions from where the sounds they amplify originate.
Until now, we have worked primarily with explicitly defined functions \(y=f( x)\), where \(x\) and \(y\) represent rectangular coordinates. Because the graphs of functions satisfy the Vertical-line Test, many graphs, such as circles, cannot be represented using functions. To address this issue, we introduce parametric equations, a way to represent graphs that are not necessarily those of functions. Additionally, parametric equations are particularly useful because they allow us to model motion along a curve, and to determine not only the location of an object (a point) but also the time it is there.
The chapter continues with a discussion of an alternate coordinate system, polar coordinates. Using polar coordinates, we can represent graphs that would be extremely complicated, or even impossible, to represent in rectangular coordinates.