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Chapter 10: Problem 64

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Short Answer

Expert verified
A plane curve from parametric equations is sketched by plotting points \( (x(t), y(t)) \) for selected values of \( t \). The orientation of the curve, represented by the direction in which the curve is traced as \( t \) increases, is determined by the order of plotted points.

Step by step solution

01

Understanding Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points in the plane as functions of a variable, called a parameter. A point in the plane is expressed as \( (x(t), y(t)) \), where \( x(t) \) and \( y(t) \) are functions of the parameter \( t \).
02

Sketching the Plane Curve

To sketch the plane curve represented by parametric equations, it can be helpful to create a table of values for \( t \), \( x(t) \), and \( y(t) \). Plotting the points \( (x(t), y(t)) \) for selected values of \( t \) will help visualize the curve in the plane. The curve should be smooth and any discontinuities should be denoted.
03

Orientation of the Curve

The orientation of the plane curve is the direction in which the curve is traced as the parameter \( t \) increases. The orientation can be deduced from the order in which the points are plotted for increasing values of \( t \). If \( t \) is time, the orientation of the curve represents the path of an object moving along the curve.

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