Problem 46 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r= \dfrac{6}{2\ \sin\ \theta -\ 3\ \cos\ \theta}\)

Step-by-Step Solution

Answer

The symmetry, zeros, maximum \(r\)-values and additional points all contribute to the sketch of the graph of the polar function. Due to the complexity of the given equation, this process involves multiple steps, usage of different areas of mathematics like calculus, algebra and geometry. This solution however provides the overall methodology to approach such problems.

Step 1: Checking Symmetry

To check for symmetry, replace \(\theta\) with -\(\theta\), \(\pi - \theta\), and \(\pi + \theta\) in the original equation and see if it reduces to the original form: \n\n- Symmetric with respect to x-axis if the function does not change when \(\theta\) is replaced by -\(\theta\), \n\n- Symmetric with respect to y-axis if the function does not change when \(\theta\) is replaced by \(\pi - \theta\), \n\n- Symmetric with respect to the origin if the function does not change when \(\theta\) is replaced by \(\pi + \theta\).

Step 2: Finding Zeros

Next, set \(r = 0\) to find the zeros of the function. If there are any values of \(\theta\) that can make \(r = 0\), they should be found.

Step 3: Determine Maximum r-values

The maximum value of \(r\) can be found by setting the derivative of \(r\) with respect to \(\theta\) to zero and solving for \(\theta\). Due to the complexity of the given equation, this step will likely involve the use of calculus and algebra.

Step 4: Find Additional Points

Now, the polar function should be evaluated at different values of \(\theta\) in order to get more points. A good strategy can be to start with values of \(\theta\) that make the denominator zero, as these will give extreme values for \(r\), and then choose other \(\theta\) values incrementally, until the overall shape of the function is revealed.

Key Concepts

Polar Coordinates

The concept of polar coordinates is an alternative to the more familiar Cartesian coordinate system. Instead of using x and y coordinates to define a point's location, polar coordinates use the distance from a reference point (called the pole, typically analogous to the origin in the Cartesian system) and an angle from a reference direction (usually the positive x-axis).

In mathematical terms, a polar coordinate is expressed as \(r, \theta\), where \(r\) is the radius or the direct distance from the pole, and \(\theta\) is the angle measured in radians from the reference direction. To convert polar coordinates to Cartesian coordinates, you can use the equations \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). This system is particularly useful when dealing with problems that have circular symmetry or involve rotations.

Symmetry in Polar Equations

Symmetry in polar equations can often simplify graphing, as recognizing symmetry allows one to plot fewer points and still produce a complete graph. There are three types of symmetry to consider in polar graphs:

• Line symmetry with respect to the x-axis occurs if, when \(\theta\) is replaced with \( -\theta\), the polar equation remains unchanged.
• Line symmetry with respect to the y-axis occurs if, when \(\theta\) is replaced with \(\pi - \theta\), the polar equation remains unchanged.
• Point symmetry with respect to the pole (origin) occurs if, when \(\theta\) is replaced with \(\pi + \theta\), the polar equation remains unchanged.

When checking for symmetry, it's like looking in a mirror along these lines or points and seeing if the graph reflects perfectly across.

Polar Zeros

Polar zeros are points where the radius \(r\) is zero. In a graph, these points correspond to the pole itself, since the distance from the pole is zero. To find the polar zeros of an equation, you set \(r\) to zero and solve for \(\theta\).

Identifying the zeros can help to pinpoint where the graph intersects the pole. It's important to remember that for certain angles \(\theta\), there might not be a real solution for \(r\), implying that the graph does not intersect the pole at these angles.

Maximum r-values in Polar Equations

Maximum r-values correspond to the furthest points from the pole in the graph of a polar equation. These points are significant as they can define the outer boundary of the graph. To find these values, one may need to use calculus, specifically by finding the derivative of \(r\) with respect to \(\theta\) and setting it to zero to solve for \(\theta\).

This process identifies the critical points, which can then be tested to determine whether they represent maximum values. Understanding how to calculate and interpret maximum \(r\)-values is crucial for effectively sketching complex polar graphs and envisioning the shape that the equation defines.