Problem 21 - Essential Calculus

Question

In Exercises 21 and 22, use a graphing utility to approximate the points of intersection of the graphs of the polar equations. Confirm your results analytically. $$ \begin{array}{l} r=2+3 \cos \theta \\ r=\frac{\sec \theta}{2} \end{array} $$

Step-by-Step Solution

Answer

To approximate the intersection points of the two polar curved equations \( r=2+3 \cos \theta \) and \( r=\frac{\sec \theta}{2} \), one needs to graph the equations and observe the intersection points. The exact intersection points will be verified by substituting the values of \( \theta \) back into the original equations and checking if \( r \) comes out the same.

Step 1: Graph the Polar Equations

Sketch the graphs of the polar equations using a graphing utility. This will identify the approximate points where the curves intersect.

Step 2: Approximate the Points of Intersection

Observing the graph, identify the points where the two curves seem to intersect and note down the approximate coordinates. These are depending on both \( r \) and \( \theta \).

Step 3: Analytical Verification

Substitute the approximate points from Step 2 into both original polar equations and verify that they satisfy both equations. This would clarify that the points calculated earlier are indeed the intersection.

Step 4: Analytical Method

To confirm the results analytically, equate the two polar equations and solve for \( \theta \). Then substitute the solution of \( \theta \) into any original polar equations to get the value of \( r \) and these are the exact intersection points.

Key Concepts

Graphing Utility

Solving mathematical problems often involves visual representation, and this is where a graphing utility becomes an invaluable tool. When dealing with polar equations like
\( r = 2 + 3 \cos \theta \) and \( r = \frac{\sec \theta}{2} \),
a graphing utility can help students approximate the points of intersection of the graphs. To use a graphing utility effectively:

• Enter the given polar equations into the system.
• Adjust the viewing window to ensure that all relevant parts of the graph are visible.
• Use the utility's features to trace or directly identify the points of intersection.

These tools offer a dynamic visual approach that can support understanding the behavior of polar equations. Approximating the points where the curves intersect provides an initial check before proceeding to analytical verification. To enhance the exercise, educators can encourage students to manipulate the graphing utility settings, exploring the effects of variations in \( \theta \) on the graph, which can deepen their understanding of polar equations.

Analytical Verification

While graphing utilities are powerful, reliance on visual approximations alone isn't sufficient in mathematics. Analytical verification is a method to confirm mathematically that our graphically obtained results are correct. After graphing the polar equations
\( r = 2 + 3 \cos \theta \) and \( r = \frac{\sec \theta}{2} \),
we systematically apply analytical methods to verify the intersection points:

• Substitute the approximate \( \theta \) values into each equation to ensure both are satisfied, confirming the intersection points are likely correct.
• Equate the two equations to find exact \( \theta \) values, solving the resulting trigonometric equation.
• Insert the solved \( \theta \) values back into either equation to find the corresponding \( r \) values for precise coordinates.

This process eliminates the guesswork, providing a concrete verification that the points found graphically are indeed the intersections of our polar equations. It reinforces the importance of combining visual tools with rigorous mathematical procedures.

Polar Coordinates

Understanding polar coordinates is essential when working with polar equations. Polar coordinates express points in a plane using a distance from a reference point, called the pole (analogous to the origin in Cartesian coordinates), and an angle from a reference direction, typically the positive x-axis.

• In polar coordinates, \( r \) represents the radius, or the distance from the pole, and \( \theta \) represents the angle in radians or degrees.
• The polar equations \( r = f(\theta) \) describe a relationship where each angle \( \theta \) has an associated radial distance \( r \).
• Transformations can be applied to polar coordinates, similar to Cartesian coordinates, changing the appearance of the curve represented by the equation.

For the exercise given, \( r = 2 + 3 \cos \theta \) and \( r = \frac{\sec \theta}{2} \),
we see the radial component changes with \( \theta \), creating curves that can intersect. Understanding these relationships and transformations in polar coordinates is foundational to analyzing and graphing polar equations effectively.