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Chapter 10: Problem 62

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Folium of Descartes: \(x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}}\)

Short Answer

Expert verified
The parametric curve, plotted using the given equations, moves in a positive direction. There seems to be a cusp at the origin (0,0), hence the curve is not smooth at this point.

Step by step solution

01

Understand the given parametric equations

The given parametric equations are: \(x=\frac{3 t}{1+t^{3}}\) and \(y=\frac{3 t^{2}}{1+t^{3}}\). The parameter here is \(t\). These equations describe the horizontal (\(x\)) and vertical (\(y\)) coordinates of the points on the curve as \(t\) varies.
02

Graph the parametric equations

Input the given parametric equations into a graphing utility, and plot the curve. The graph represents the path of a point moving in the plane. With each value of \(t\), there corresponds a pair of coordinates \((x, y)\) from the equations, which are plotted on the coordinate plane to form the curve.
03

Indicate the direction of the curve

The direction of the curve can be determined by following the sequence of \((x, y)\) pairs for increasing \(t\) values. The curve starts near the origin for negative \(t\) values and moves towards positive \(x\) values as \(t\) increases.
04

Identify nonsmooth points

A smooth curve is one without corners or cusps. By inspecting the graph we generated, we can identify that the curve has a cusp at (0,0), so it's not smooth at this point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a tool that allows us to visualize mathematical equations and curves, especially when dealing with parametric equations like the ones mentioned in the exercise. These tools can be software programs or calculator features that draw graphs based on the equations provided.
  • To use a graphing utility, you input the parametric equations. For our example, that's \(x = \frac{3t}{1+t^3}\) and \(y = \frac{3t^2}{1+t^3}\).
  • The utility plots the points \(x, y\) as \(t\) varies. Therefore, each \(t\) value gives a coordinate pair \(x, y\), creating a path or curve.
The benefit of using a graphing utility is it helps us understand the behavior of the curve visually. It not only plots the shape of the curve but can also show us the direction of movement when the parameter \(t\) changes.
Folium of Descartes
The Folium of Descartes is a famous curve in mathematics, defined by the equations \(x=\frac{3t}{1+t^3}\) and \(y=\frac{3t^2}{1+t^3}\). It's known for having a loop and a cuspidal point at the origin \(0, 0\). This point is interesting because it displays the unique feature where the curve is not smooth.
  • The folium is an example of how parametric equations describe complex curves that cannot easily be expressed in simple polar or rectangular forms.
  • This specific folium includes a cusp, which is a place where the curve suddenly changes direction.
Understanding the Folium of Descartes allows us to appreciate how historical mathematicians conceived intricate curves and shapes that today we can easily explore with technology.
Smooth Curve
A smooth curve is one that does not have any sharp corners, breaks, or discontinuities. In mathematics, a curve is generally defined as smooth if it has continuous derivatives or slopes at all points along the curve.
  • A smooth curve will have a well-defined tangent at each point, meaning there are no abrupt changes in direction.
  • For the Folium of Descartes, the curve is smooth except at the point \(0, 0\), where there is a cusp.
To determine smoothness visually or mathematically, you might check for continuity and differentiability across the curve. Graphing tools can aid in visually identifying smooth and nonsmooth regions, which is crucial for understanding the behavior of the entire curve.
Parametric Graphing
Parametric graphing involves plotting equations that express coordinates as functions of a parameter, often denoted by \(t\). Unlike traditional graphs that relate \(x\) and \(y\) directly, parametric graphs use a third variable to define both coordinates.
  • This method of graphing is particularly useful for representing curves that have a direction or progression, as seen in animations or the path of a moving object.
  • The parametric form gives us more flexibility and a comprehensive description of motion and paths on a graph.
For the Folium of Descartes, parametric graphing helps graph the unique loop and identify the points where the motion is not smooth, something that would be challenging with standard Cartesian coordinates alone.

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Most popular questions from this chapter

A person moves from the origin along the positive \(y\) -axis pulling a weight at the end of a 12-meter rope. Initially, the weight is located at the point \((12,0)\). (a) In Exercise 86 of Section \(8.7\), it was shown that the path of the weight is modeled by the rectangular equation $$ y=-12 \ln \left(\frac{12-\sqrt{144-x^{2}}}{x}\right)-\sqrt{144-x^{2}} $$ where \(0

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=4 t, \quad y=t+1& \quad 0 \leq t \leq 2 \end{array} $$

(a) Use a graphing utility to graph the curve given by $$ x=\frac{1-t^{2}}{1+t^{2}}, y=\frac{2 t}{1+t^{2}}, \quad-20 \leq t \leq 20. $$ (b) Describe the graph and confirm your result analytically. (c) Discuss the speed at which the curve is traced as \(t\) increases from \(-20\) to 20 .

The path of a projectile is modeled by the parametric equations \(x=\left(90 \cos 30^{\circ}\right) t \quad\) and \(\quad y=\left(90 \sin 30^{\circ}\right) t-16 t^{2}\) where \(x\) and \(y\) are measured in feet. (a) Use a graphing utility to graph the path of the projectile. (b) Use a graphing utility to approximate the range of the projectile. (c) Use the integration capabilities of a graphing utility to approximate the arc length of the path. Compare this result with the range of the projectile.

It Use a graphing utility to graph the polar equation \(r=6[1+\cos (\theta-\phi)]\) for (a) \(\phi=0,(\) b) \(\phi=\pi / 4\) and (c) \(\phi=\pi / 2\). Use the graphs to describe the effect of the angle \(\phi\). Write the equation as a function of \(\sin \theta\) for part (c).
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