Problem 62 - Calculus Early Transcendentals

Question

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+2 \cos \theta}$$

Step-by-Step Solution

Answer

Based on the solution provided, the short answer can be written as follows: The polar equation \(r = \frac{1}{1+2 \cos \theta}\) creates a curve formed by two loops as \(\theta\) increases from 0 to \(2 \pi\). The first smaller loop is formed when \(\theta\) ranges from 0 to \(\pi\), and the second larger loop, symmetric to the first one, is formed when \(\theta\) ranges from \(\pi\) to \(2 \pi\). The curve is generated in a counterclockwise direction, as indicated by arrows on the graph.

Step 1: Analyze the Polar Equation

We begin by analyzing the equation \(r = \frac{1}{1+2 \cos \theta}\). The main element here is the cosine function, which varies between -1 and 1 as \(\theta\) increases from 0 to \(2 \pi\). Let's analyze the equation as follows: - For \(\theta = 0\): \(\cos \theta = 1\), so \(r = \frac{1}{1+2 (1)} = \frac{1}{3}\) - For \(\theta = \frac{\pi}{2}\): \(\cos \theta = 0\), so \(r = \frac{1}{1+2 (0)} = 1\) - For \(\theta = \pi\): \(\cos \theta = -1\), so \(r = \frac{1}{1+2(-1)} = -1\) - For \(\theta = \frac{3 \pi}{2}\): \(\cos \theta = 0\), so \(r = \frac{1}{1+2 (0)} = 1\) Since the cosine function repeats itself in the range from 0 to \(2 \pi\), the curve will have a loop shape.

Step 2: Plot the Polar Equation

To plot the polar equation \(r = \frac{1}{1+2 \cos \theta}\), we first create a polar grid. Next, we can choose values for \(\theta\) from 0 to \(2 \pi\) and calculate the corresponding \(r\) values using the polar equation. As a result, we obtain a series of points in polar coordinates. Connect the points one by one, following the order in which they appear as \(\theta\) increases. Observe that for \(\theta\) in the range from 0 to \(\pi\), the curve forms a smaller loop, while for \(\theta\) in the range from \(\pi\) to \(2 \pi\), the curve forms a larger, symmetric loop in relation to the first one.

Step 3: Add Arrows and Labeled Points

Now that we have graphed the curve and identified how it evolves as \(\theta\) increases from 0 to \(2 \pi\), we can add arrows to indicate the direction in which the curve is generated. Following the sequence of points, place an arrowhead at the endpoint of each segment, pointing towards the next point. Label specific points on the curve corresponding to key angles, such as \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}\), and \(2 \pi\). For each labeled point, indicate its polar coordinates, \((r, \theta)\), alongside the label. This completes the graphical representation of the polar equation, with arrows indicating how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\).