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Chapter 10: Problem 30

Sketch the graph of the equation and label the vertices. $$r=\frac{10}{3+2 \cos \theta}$$

Short Answer

Expert verified
Based on the solution, create a short answer that includes the polar equation's symmetry, vertices, and a brief overview of sketching the graph. The polar equation $$r = \frac{10}{3 + 2 \cos \theta}$$ has symmetry around the polar axis. The vertices are found at 胃 = n蟺, where n 鈭� 鈩�. To sketch the graph, restrict the range of 胃 to [0, 蟺], plot polar points by calculating the cartesian coordinates (x, y) for selected 胃 values, and connect the points to create the graph. Ensure it correctly displays the identified symmetries and vertices.

Step by step solution

01

Identify Symmetry

We will determine if the polar equation has symmetry with respect to the polar axis, the pole, or the line(theta=蟺/2). 1. Polar axis symmetry: plug in -胃 for 胃, if r stays the same, there's symmetry. $$r = \frac{10}{3 + 2 \cos(-\theta)} \Rightarrow r = \frac{10}{3 + 2 \cos(\theta)}$$ Since r doesn't change, it has polar axis symmetry. 2. Pole symmetry: plug in (胃+蟺) for 胃, if r stays the same, there's symmetry. $$r = \frac{10}{3 + 2 \cos(胃+蟺)} \Rightarrow r = \frac{10}{3 - 2 \cos(\theta)}$$ r changes, so there's no symmetry about the pole. 3. Line symmetry(胃=蟺/2): plug in (蟺/2-胃) for 胃, if r stays the same, there's symmetry. $$r = \frac{10}{3 + 2 \cos(蟺/2-胃)} \Rightarrow r = \frac{10}{3 - 2 \sin(\theta)}$$ r changes, so there's no symmetry about the line(胃=蟺/2).
02

Find the Vertices

Next, we'll find the vertices by locating the values of 胃 for which r is either a maximum or a minimum. For that, we can differentiate the equation w.r.t 胃 and find the critical points by setting that equal to zero. $$r = \frac{10}{3 + 2 \cos \theta}$$ Differentiate r w.r.t 胃: $$\frac{dr}{d\theta} = \frac{20 \sin \theta}{(3+2 \cos \theta)^2}$$ Set the derivative equal to zero to find the critical points: $$\frac{20 \sin \theta}{(3+2 \cos \theta)^2} = 0$$ This implies: $$\sin \theta = 0 \Rightarrow \theta = n\pi, \, n \in \mathbb{Z}$$ Now, to find the vertices, plug the critical points into the original equation to get the r-values: $$r = \frac{10}{3 + 2 \cos(n\pi)}$$ For the vertices, cartesian coordinates (x, y) can be calculated using x = rcos(胃) and y = rsin(胃).
03

Sketch the Graph

To sketch the graph, follow these steps: 1. Determine the range for 胃: Since the graph shows symmetry around the polar axis and there are vertices at 胃 = n蟺, we can restrict the range of 胃 to [0, 蟺]. 2. Plot polar points: Calculate the (x, y) coordinates for selected 胃 values in the range [0, 蟺] using x = rcos(胃) and y = rsin(胃). Make sure to include the vertices calculated earlier. 3. Sketch the graph: Connect the polar points to create the final graph. Double-check if the graph displays the symmetries determined earlier. Finally, label the vertices found earlier on the sketch for a complete graph of the polar equation.

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Key Concepts

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

Symmetry in Polar Graphs
When dealing with polar coordinates, symmetry can help simplify the graphing process. In polar equations, symmetry can be about:
  • Polar Axis: This is equivalent to the horizontal axis in Cartesian coordinates. If an equation remains unchanged when we substitute \( -\theta \) for \( \theta \), it has polar axis symmetry. This means if we flip the graph over this axis, it remains the same.
  • Pole: This point is like the origin in Cartesian coordinates. By checking if the equation stays the same when \( \theta \) is substituted with \( \theta + \pi \), we test for symmetry about the pole. For the exercise above, there is no pole symmetry.
  • Line \( \theta = \pi/2 \): This is a vertical line in polar graphs. If substituting \( \pi/2 - \theta \) for \( \theta \) leaves the equation unchanged, the graph is symmetric about this line. In our problem, the graph lacks this symmetry as well.
Identifying symmetry simplifies graphing since you only need to plot a segment and then reflect it across the symmetrical line.
Derivative of Polar Functions
To understand changes in a polar function, differentiating with respect to \( \theta \) is crucial. The derivative helps determine critical points where the function could reach maximums or minimums, like the tips of petals in a polar rose or the vertices of a lima莽on.

In the given exercise, the derivative \( \frac{dr}{d\theta} = \frac{20 \sin \theta}{(3 + 2 \cos \theta)^2} \) helps us find points where the rate of change is zero (\( \sin \theta = 0 \)). This identifies locations where the function stabilizes, marking changes in direction on the graph.

  • Compute the derivative to identify critical points.
  • Solve \( \frac{dr}{d\theta} = 0 \) for \( \theta \).
  • Substitute these \( \theta \) values back into \( r(\theta) \) to find corresponding \( r \) values or radii at these points.
These steps will determine key features of the graph, providing insights into its shape and helping locate extrema, essential for sketching an accurate curve.
Graphing Polar Equations
Graphing in polar coordinates requires understanding both the direction and distance from the pole. Unlike Cartesian graphs, polar graphs display points as determined by \( (r, \theta) \): the radius \( r \) denotes the distance from the origin, and angle \( \theta \) determines the direction.

To embark on graphing a polar equation like \( r = \frac{10}{3 + 2 \cos \theta} \), follow these simple steps:
  • Identify Symmetry: As earlier discovered, our example is symmetric regarding the polar axis. This knowledge reduces plotting effort.
  • Select \( \theta \) Range: Often, ranging from \( 0 \) to \( 2\pi \) is necessary for thoroughness, but symmetry here suggests \( 0 \) to \( \pi \) suffices.
  • Calculate Coordinates: Use \( x = r \cos \theta \) and \( y = r \sin \theta \) to transform polar coordinates to Cartesian, helping visualize plotting.
  • Plot Points: Use calculated coordinates to mark points; connect logically, reflecting symmetry.
  • Label Special Points: Don鈥檛 forget to mark vertices and other notable features.
Following these steps provides a comprehensive approach to graphing polar equations, yielding curves representing radial dynamics vividly and accurately. Adding labels simplifies interpretation, ensuring clarity in the representation.

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Most popular questions from this chapter

Locate all local maxima and minima (other than endpoints of the curve. $$x=4 t-6, \quad y=3 t^{2}+2, \quad-10 \leq t \leq 10$$

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