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Chapter 10: Problem 52

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Involute of a circle } x=\cos t+t \sin t, y=\sin t-t \cos t$$

Short Answer

Expert verified
Question: Explain in your own words how to graph the involute of a circle using the given parametric equations, $$x = \cos t + t\sin t,\, y = \sin t - t\cos t$$. Answer: To graph the involute of a circle using the parametric equations given, first understand that the equations define the curve. Next, since the equations involve trigonometric functions (cosine and sine), recall that they have a period of \(2\pi\), which affects the graphing process. In order to capture the full details of the curve, select an interval for the parameter t that covers at least one period of the trigonometric functions - for example, \(0 \leq t \leq 4\pi\). Finally, use a graphing utility to plot the curve using the parametric equations and the chosen interval for t. This will result in a detailed graph of the involute of a circle.

Step by step solution

01

Define the parametric equations

We are given the parametric equations for the involute of a circle: $$ x = \cos t + t\sin t,\, y = \sin t - t\cos t $$ These equations define the curve.
02

Examine properties of the trigonometric functions

Since we are dealing with trigonometric functions, let's examine their properties. The cosine and sine functions have a period of \(2\pi\). This means that when t increases by \(2\pi\), their values will repeat. However, the linear term is without any periodic behavior.
03

Choose an interval for t

As the value of t increases, the functions diverge from the origin because of the linear t term. So, the value of t should cover at least one period of the trigonometric functions. We can choose an interval of \(0 \leq t \leq 4\pi\). This interval covers two periods of the trigonometric functions giving out a more detailed picture of how the curve evolves.
04

Graph the curve using a graphing utility

Using a graphing utility, plot the parametric equations for the involute of a circle using the chosen interval for t: $$ x = \cos t + t\sin t,\, y = \sin t - t\cos t,\, 0 \leq t \leq 4\pi $$ The plotted curve will show the detailed structure of the involute of a circle in the given interval.

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Most popular questions from this chapter

Consider the polar curve \(r=\cos (n \theta / m)\) where \(n\) and \(m\) are integers. a. Graph the complete curve when \(n=2\) and \(m=3\) b. Graph the complete curve when \(n=3\) and \(m=7\) c. Find a general rule in terms of \(m\) and \(n\) (where \(m\) and \(n\) have no common factors) for determining the least positive number \(P\) such that the complete curve is generated over the interval \([0, P]\).

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola \(x^{2} / 4-y^{2} / 9=1\) has no \(y\) -intercepts. b. On every ellipse, there are exactly two points at which the curve has slope \(s,\) where \(s\) is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex.

Consider the polar curve \(r=2 \sec \theta\). a. Graph the curve on the intervals \((\pi / 2,3 \pi / 2),(3 \pi / 2,5 \pi / 2)\) and \((5 \pi / 2,7 \pi / 2) .\) In each case, state the direction in which the curve is generated as \(\theta\) increases. b. Show that on any interval \((n \pi / 2,(n+2) \pi / 2),\) where \(n\) is an odd integer, the graph is the vertical line \(x=2\).
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