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Chapter 6: Problem 50

Use a graphing utility to graph the curve represented by the parametric equations. Cycloid: \(x=\theta+\sin \theta, y=1-\cos \theta\)

Short Answer

Expert verified
The graph of cycloid with the given equations will resemble a series of arches, each one corresponding to a full revolution of the circle, formed by dropping the vertical axis (y), moving \(\theta\) over the horizontal axis (x). It's essential to ensure your graphing parameters for \(\theta\) range cover, at least, \(0\leq \theta \leq 2\pi\) to fully represent one complete cycloid.

Step by step solution

01

Understand the Parametric Equation

Parametric equations define a group of quantities in terms of independent parameters. For the given cycloid, the parametric equations are: \(x=\theta+\sin \theta\) and \(y=1-\cos \theta\). Here, \(\theta\) is the parameter - the angle subtended at the centre of the rolling circle.
02

Plot the Parametric Equations

Use a graphing utility to sketch the curve defined by the given parametric equations. While entering the equations into the utility, ensure that both x and y are defined in terms of the same parameter \(\theta\), with an appropriate range of \(\theta\) (for instance, \(0\leq \theta \leq 2\pi\)) set to generate a complete cycloid. The utility will generate a cycloid curve.
03

Analyse the Graph

After the graph is plotted, analyze the shape and properties. A cycloid curve has a distinct, loop-like shape, with a series of arches, each created when the point traces a full revolution of the rolling circle. The highest points (cusps) on the graph represent the moments when the rolling circle completes a rotation, and the point is at the top of the revolution.

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Most popular questions from this chapter

Let \(\left(x_{1}, y_{1}\right)\) be the coordinates of a point on the parabola \(x^{2}=4 p y .\) The equation of the line tangent to the parabola at the point is \(y-y_{1}=\frac{x_{1}}{2 p}\left(x-x_{1}\right)\) What is the slope of the tangent line?

Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(5 x^{2}+3 y^{2}=15\)

The equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. \(y^{2}-8 x=0 \quad x-y+2=0\)

Find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. \(x^{2}+4 x+6 y-2=0\)

Find the standard form of the equation of the parabola with the given characteristics. Focus: (2,2)\(;\) directrix: \(x=-2\)
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