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Chapter 8: Problem 54

Give an example of: A polar curve \(r=f(\theta)\) that is symmetric about neither the \(x\) -axis nor the \(y\) -axis.

Short Answer

Expert verified
Polar curve \( r = 2 + \cos(\theta)\sin(\theta) \) is an example.

Step by step solution

01

Understand Polar Symmetry

In polar coordinates, a curve given by the function \( r = f(\theta) \) can exhibit symmetry in different ways. A curve is symmetric about the x-axis if \( f(\theta) = f(-\theta) \), about the y-axis if \( f(\theta) = f(\pi - \theta) \), and about the pole if \( f(\theta) = -f(\theta + \pi) \). To find a curve that is not symmetric about the x-axis or y-axis, it must not satisfy these conditions.
02

Choose a Non-Symmetric Function

Consider the polar curve given by the function \( r = 2 + \cos(\theta) \sin(\theta) \). This function does not satisfy the conditions for symmetry because substituting \(-\theta\) or \(\pi - \theta\) does not result in the same function.
03

Verify Non-Symmetry for the x-axis

Substitute \(-\theta\) into the curve: \( r = 2 + \cos(-\theta) \sin(-\theta) = 2 - \cos(\theta) \sin(\theta) \). Since \( 2 + \cos(\theta)\sin(\theta) eq 2 - \cos(\theta)\sin(\theta) \), this function is not symmetric about the x-axis.
04

Verify Non-Symmetry for the y-axis

Substitute \(\pi - \theta\) into the curve: \( r = 2 + \cos(\pi - \theta) \sin(\pi - \theta) = 2 - \cos(\theta) \sin(\theta) \). Since \( 2 + \cos(\theta)\sin(\theta) eq 2 - \cos(\theta)\sin(\theta) \), this function is not symmetric about the y-axis.

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Key Concepts

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

Polar Curves
Polar curves are a unique way to represent mathematical functions and curves in the coordinate system. Instead of the traditional Cartesian coordinates (x, y), polar coordinates use a radius and an angle, denoted as \(r\) and \(\theta\). The function of a polar curve is typically represented as \(r = f(\theta)\).

In polar coordinates:
  • \(r\) is the distance from the pole (equivalent to the origin in Cartesian coordinates) to a point on the curve.
  • \(\theta\) is the angle measured from the positive x-axis.
These curves can take various shapes and patterns, creating a wide variety of graphs with different properties. This makes them particularly interesting for analyzing symmetries and other geometric properties.
Symmetry in Polar Coordinates
To determine if a polar curve is symmetric, we check how the function \(r = f(\theta)\) behaves when we apply changes to \(\theta\). There are distinct tests to assess symmetry:
  • Symmetric about the x-axis: If substituting \(-\theta\) into the function gives back the original function, i.e., \( f(\theta) = f(-\theta) \), the curve is symmetric about the x-axis.
  • Symmetric about the y-axis: If substituting \(\pi - \theta\) into the function results in the same function, i.e., \( f(\theta) = f(\pi - \theta) \), the curve is symmetric about the y-axis.
  • Symmetric about the pole: If \( f(\theta) = -f(\theta + \pi) \), the curve is symmetric about the pole (origin).
These tests allow mathematicians to categorize curves and identify their intrinsic properties. Understanding symmetry in polar coordinates helps in predicting the behavior of a curve and simplifying calculations.
Non-Symmetric Functions
Sometimes, a polar curve does not exhibit any of the familiar symmetries. A non-symmetric function is a great example of this.

In examining such functions, like \( r = 2 + \cos(\theta) \sin(\theta) \), it becomes clear how asymmetrical curves behave. This particular function doesn't meet any of the symmetry conditions outlined in polar coordinates.
  • Substituting \(-\theta\): \( r = 2 - \cos(\theta) \sin(\theta) \) is not equal to \( r = 2 + \cos(\theta) \sin(\theta) \), disproving x-axis symmetry.
  • Substituting \(\pi - \theta\): \( r = 2 - \cos(\theta) \sin(\theta) \) again gives a different result, proving no y-axis symmetry.
These results confirm that the curve has no simple reflection across the x or y axes, allowing students to see examples of curves with unique paths and characteristics.

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Most popular questions from this chapter

A particle moves with velocity \(d x / d t\) in the \(x\) -direction and \(d y / d t\) in the \(y\) -direction at time \(t\) in seconds, where $$\frac{d x}{d t}=3 t^{2} \quad \text { and } \quad \frac{d y}{d t}=12 t$$ (a) Find the change in position in the \(x\) and \(y\) coordinates between \(t=0\) and \(t=3\). (b) If the particle passes through (-7,11) at \(t=0,\) find its position at \(t=3\). (c) Find the distance traveled by the particle from time \(t=0\) to \(t=3\).

Old houses may contain asbestos, now known to be dangerous; removal requires using a special vacuum. A contractor climbs a ladder and sucks up asbestos at a constant rate from a \(10 \mathrm{m}\) tall pipe covered by \(0.2 \mathrm{kg} / \mathrm{m}\) using a vacuum weighing \(14 \mathrm{kg}\) with a \(1.2 \mathrm{kg}\) capacity. (a) Let \(h\) be the height of the vacuum from the ground. If the vacuum is empty at \(h=0,\) find a formula for the mass of the vacuum and the asbestos inside as a function of \(h.\) (b) Approximate the work done by the contractor in lifting the vacuum from height \(h\) to \(h+\Delta h\) (c) At what height does the vacuum fill up? (d) Find total work done lifting the vacuum from height \(h=0\) until the vacuum fills. (e) Assuming again an empty tank at \(h=0,\) find the work done lifting the vacuum when removing the remaining asbestos.

Explain what is wrong with the statement. The point with Cartesian coordinates \((x, y)\) has polar coordinates \(r=\sqrt{x^{2}+y^{2}}, \theta=\tan ^{-1}(y / x)\)

A hemispherical bowl of radius \(2 \mathrm{ft}\) contains water to a depth of \(1 \mathrm{ft}\) at the center. Let \(y\) be measured vertically upward from the bottom of the bowl. Water has density \(62.4 \mathrm{lb} / \mathrm{ft}^{3}.\) (a) Approximately how much work does it take to move a horizontal slice of water at a distance \(y\) from the bottom to the rim of the bowl? (b) Write and evaluate an integral giving the work done to move all the water to the rim of the bowl.

Find the volume of the solid whose base is the region in the first quadrant bounded by \(y=4-x^{2},\) the \(x\) -axis, and the \(y\) -axis, and whose cross- section in the given direction is an equilateral triangle. Include a sketch of the region and show how to find the area of a triangular crosssection. Perpendicular to the \(y\) -axis.
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