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Chapter 10: Problem 71

What are plane curves and parametric equations?

Short Answer

Expert verified
A plane curve is a curve that lies entirely in one plane and it's visually represented in a 2D space. Parametric equations express quantities as explicit functions of a number of independent variables, called parameters. These equations are used extensively to describe the positional coordinates of a point on a plane curve.

Step by step solution

01

Definition of Plane Curves

A plane curve is a curve that lies entirely in one plane. These planes can be the x-y, x-z, y-z for instance. Plane curves are visually represented in a 2-dimensional space.
02

Defintion of Parametric Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. For instance, in a 2-dimensional space, a parametric equation allows us to represent a curve by a pair of functions \( x=f(t) \) and \( y=g(t) \) where \( x \) and \( y \) are coordinates on the curve and \( t \) is the parameter.
03

Connection between Plane Curves and Parametric Equations

Parametric equations are used to specify the positional coordinates of a point on a plane curve in terms of a single parameter. This makes it possible to 'draw' the curve in terms of this parameter, which typically represents time.

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Most popular questions from this chapter

Exercises \(95-97\) will help you prepare for the material covered in the next section. Divide both sides of \(4 x^{2}-9 y^{2}=36\) by 36 and simplify. How does the simplified equation differ from that of an ellipse?

What is an ellipse?

Expand: $$ \log _{b}\left(x^{3} \sqrt{y}\right) $$

Exercises 105鈥�107 will help you prepare for the material covered in the next section. Simplify and write the equation in standard form in terms of \(x^{\prime}\) and \(y^{\prime}\) $$ \left[\frac{\sqrt{2}}{2}\left(x^{\prime}-y^{\prime}\right)\right]\left[\frac{\sqrt{2}}{2}\left(x^{\prime}+y^{\prime}\right)\right]=1 $$

Use a graphing utility to graph the equation. Then answer the given question. $$ \begin{aligned} &r=\frac{4}{1-\sin \left(\theta-\frac{\pi}{4}\right)} ; \text { How does the graph differ from the }\\\ &\text { graph of } r=\frac{4}{1-\sin \theta} ? \end{aligned} $$
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