Question

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

Short answer

Answer: The polar curve of the given equation is a limaçon that starts from the point \((0.5, \frac{\pi}{2})\), passes through the maximum point \((1, \frac{3\pi}{2})\), and returns to the starting point as \(\theta\) reaches \(2\pi\). The curve follows a counterclockwise direction as \(\theta\) increases from \(0\) to \(2\pi\).

Step-by-step solution

Analyze the behavior of r

Let's analyze how \(r\) behaves as \(\theta\) increases from \(0\) to \(2\pi\). This will help us to better understand the evolution of the curve. In particular, pay attention to any special points where \(r\) might be at its maximum or minimum values and analyze how the equation behaves near these points.

Determine the special points on the curve by finding extrema of r

To determine the special points, we'll find the extrema of \(r\) by differentiating \(r(\theta)\) with respect to \(\theta\): $$\frac{d[r(\theta)]}{d\theta}=0$$ Take the derivative of \(r(\theta)\): $$\frac{d}{d\theta}\left(\frac{1}{1+\sin{\theta}}\right)=\frac{-\cos{\theta}}{(1+\sin{\theta})^2}$$ Now we need to find the values of \(\theta\) that make \(\frac{d[r(\theta)]}{d\theta} = 0\), which happens at the extremes of \(r\): $$\frac{-\cos{\theta}}{(1+\sin{\theta})^2}=0$$ Solving for \(\theta\), we find: $$\theta = \frac{\pi}{2},\frac{3\pi}{2}$$ These are the two extrema, which correspond to the minimum and maximum values of \(r\). To find the actual value of r, substitute the extrema back into the equation for r: - At \(\theta = \frac{\pi}{2}\), $$r(\theta)=\frac{1}{1+\sin{\left(\frac{\pi}{2}\right)}}=\frac{1}{1+1}=0.5$$ - At \(\theta = \frac{3\pi}{2}\), $$r(\theta)=\frac{1}{1+\sin{\left(\frac{3\pi}{2}\right)}}=\frac{1}{1-1}=1$$ Now, we have two special points: \((0.5, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\).

Graph the equation by connecting the special points

Plot the points \((0.5, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\) on polar graph paper. Keep in mind that the polar coordinates are given as \((r, \theta)\). Next, connect these points to create the curve and trace it as \(\theta\) increases from \(0\) to \(2\pi\). You will notice that the curve is a limaçon that starts from the point \((0.5, \frac{\pi}{2})\), passing through \((1, \frac{3\pi}{2})\), and finally returning to its starting point as \(\theta\) reaches \(2\pi\).

Indicate the direction of the curve

In order to indicate how the curve is generated as \(\theta\) increases from \(0\) to \(2\pi\), use arrows along the curve. The arrows should point counterclockwise from the starting point \((0.5, \frac{\pi}{2})\) to the maximum point \((1, \frac{3\pi}{2})\), and then back down to the starting point. This demonstrates that as \(\theta\) increases, the curve follows a counterclockwise direction.

Labeling the points on the curve

Finally, label the points \((0.5, \frac{\pi}{2})\) and \((1, \frac{3\pi}{2})\) on the graph so that anyone studying the graph can easily identify these special points. This will also help in understanding the direction of the curve.