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Chapter 10: Problem 66

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 given by parametric equations can be sketched by plotting points derived from the equations and joining them. The orientation of this curve is the direction in which it is traversed as the parameter increases.

Step by step solution

01

鈥� Understanding 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 example, the equations \(x = a \cos t\) and \(y = b \sin t\) are a pair of parametric equations where \(x\) and \(y\) are functions of the parameter \(t\).
02

鈥� Sketching the curve

To sketch the curve represented by the parametric equations, first isolate the parameter \(t\) in each equation. Then combine the two equations into one single equation, by eliminating \(t\). This will give a single equation which can be sketched on a plane. Sketch the curve by plotting points for selected values of \(t\), then joining these points smoothly.
03

鈥� Understanding the Orientation of the Curve

The orientation of a curve described by a set of parametric equations refers to the 'direction' in which the curve is traversed as the parameter \(t\) increases. For instance, if the curve is described by the equations \(x = a \cos t\) and \(y = b \sin t\) for \(0 \leq t \leq 2\pi\), the curve is traversed in a counterclockwise direction as \(t\) increases from \(0\) to \(2\pi\). This direction is the orientation of the curve.

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Most popular questions from this chapter

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. Common interior of \(r=5-3 \sin \theta\) and \(r=5-3 \cos \theta\)

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. $$ r=\frac{9}{4+\cos \theta} $$

Area Sketch the strophoid $$r=\sec \theta-2 \cos \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}$$ Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Finding the Arc Length of a Polar Curve In Exercises \(51-56,\) find the length of the curve over the given interval. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=2 a \cos \theta } & {-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}}\end{array} $$

Writing In Exercises 33 and \(34,\) use a graphing utility to graph the polar equations and approximate the points of intersection of the graphs. Watch the graphs as they are traced in the viewing window. Explain why the pole is not a point of intersection obtained by solving the equations simultaneously. $$ \begin{array}{l}{r=4 \sin \theta} \\ {r=2(1+\sin \theta)}\end{array} $$
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