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5: Parametric Equations and Polar Coordinates

Last updated

May 14, 2023
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- 4.5: Chapter 4 Review Exercises
- 5.1: Parametric Equations

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Parametric equations define a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization. The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth

5.1: Parametric Equations
This section introduces parametric equations, where two separate equations define $x$ and $y$ as functions of a third variable, usually $t$ . It explains how to graph parametric curves, eliminate the parameter to find Cartesian equations, and analyze motion along a path. Examples illustrate the flexibility of parametric equations in describing curves that are difficult to express in standard Cartesian form.
- 5.1E: Exercises
5.2: Calculus of Parametric Curves
This section covers the calculus of parametric curves, including finding derivatives and integrals for curves defined parametrically. It explains how to compute the slope of the tangent line, arc length, and area under a parametric curve. Examples demonstrate how to apply these techniques to analyze the geometric and physical properties of parametric equations.
- 5.2E: Exercises
5.3: Polar Coordinates
This section introduces polar coordinates, where points are defined by a radius and angle relative to the origin. It explains how to convert between polar and Cartesian coordinates and how to plot polar equations. The section highlights the unique features of polar graphs, such as symmetry and periodicity, and provides examples to illustrate these concepts.
- 5.3E: Exercises
5.4: Area and Arc Length in Polar Coordinates
This section covers calculating area and arc length in polar coordinates. It explains how to compute the area enclosed by a polar curve using the formula $\frac{1}{2} \int r^2 \, d\theta$ and how to find the arc length of a polar curve using the appropriate integral. Examples illustrate these calculations for various polar equations, emphasizing their geometric applications.
- 5.4E: Exercises
5.5: Conic Sections
Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle.
- 5.5E: Exercises
5.6: Chapter 5 Review Exercises

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