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Chapter 10: Problem 65

State the definition of a plane curve given by parametric equations.

Short Answer

Expert verified
A plane curve given by parametric equations is defined as a set of points (x, y) in the plane determined by two functions of a variable (t), written as x = f(t) and y = g(t). Here, 't' is the parameter, usually considered to be time.

Step by step solution

01

Understand the term 'plane curve'

A plane curve is a curve in a two-dimensional space. These curves are usually described in a Cartesian coordinate system (x, y). It can be a closed curve or an open one. Examples include circles, ellipses, parabolas, and hyperbolas.
02

Introduce Parametric Equations

Parametric equations use a variable (usually denoted as t) to define the coordinates x and y of a curve. For instance, equations x = f(t), y = g(t) where f and g are any function of t, represent a parametric representation of a curve. Here, t is called the parameter. It can take any real value which gives different points on the curve.
03

State the Definition

A plane curve given by parametric equations is defined as a set of points (x, y) in a plane that are determined by two functions of a variable t, typically noted as x = f(t) and y = g(t). The parameter t usually represents time.

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Most popular questions from this chapter

Finding Points of Intersection In Exercises \(25-32,\) find the points of intersection of the graphs of the equations. $$ \begin{array}{l}{r=2-3 \cos \theta} \\ {r=\cos \theta}\end{array} $$

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. One petal of \(r=\cos 5 \theta\)

Finding the Area of a Polar Region Between Two Curves In Exercises \(35-42,\) use a graphing utility to graph \(h\) the polar equations. Find the area of the given region analytically. Inside \(r=2 \cos \theta\) and outside \(r=1\)

Finding the Arc Length of a Polar Curve In Exercises \(51-56,\) find the length of the curve over the given interval. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=4 \sin \theta} & {0 \leq \theta \leq \pi}\end{array} $$

Area of a Region In Exercises \(57-60\) , use the integration capabilities of a graphing utility to approximate, to two decimal places, the area of the region bounded by the graph of the polar equation. $$ r=\frac{3}{2-\cos \theta} $$
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