Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Investigate the family of curves given by \(r=a+b \cos (n(\theta+\phi))\) where \(a, b\), and \(\phi\) are real numbers and\(n\) is a positive integer. As you answer the following questions, be sure that you graph a sufficient number of examples to justify your conclusions. (a) How are the graphs for \(\phi=0\) related to those for which \(\phi \neq 0 ?\) (b) How does the graph change as \(n\) increases? (c) How do the relative magnitude and sign of \(a\) and \(b\) change the nature of the graph?

Step-by-Step Solution

Answer

(a) Graphs with \(\phi\neq 0\) are rotated versions of those with \(\phi=0\). (b) Increasing \(n\) adds more petals to the graph. (c) Difference in \(|a|\) and \(|b|\) affects whether it's a simple limaçon, cardioid, or one with loops.

Step 1: Understanding the Polar Equation

The polar equation given is \( r = a + b \cos(n(\theta + \phi)) \). This form is a generalized version of polar equations that can create a variety of curves. Depending on the values of \( a \), \( b \), \( n \), and \( \phi \), you can get different types of rose curves and limaçon.

Step 2: Effect of Phi on the Graph

The term \( \phi \) represents a phase shift in the angle \( \theta \). When \( \phi = 0 \), the curve has no phase shift and starts at a reference position on the polar plot. When \( \phi eq 0 \), the graph is essentially the same shape but rotated by \( \phi \) radians. Thus, the graphs with non-zero \( \phi \) are rotated versions of those with \( \phi = 0 \).

Step 3: Analyzing Changes as n Increases

The parameter \( n \) affects the number of petals or loops in the graph. Specifically, if \( n \) is odd, the graph will have \( n \) petals; if \( n \) is even, the graph will display \( 2n \) petals. So, as \( n \) increases, the number of petals increases, making the graph more complex.

Step 4: Impact of a and b on Curve Shape

The values of \( a \) and \( b \) determine the size and nature of the curve. If \( |a| > |b| \), the curve will always be limaçon without an inner loop. If \( |a| = |b| \), it forms a cardioid. If \( |a| < |b| \), there's a loop inside the limaçon. The sign of \( b \) will further determine if these features are inverted.

Step 5: Conclusion and Visualization

Graphing several examples with varied parameters can demonstrate the curve changes effectively. Use the values of \( a \), \( b \), and different \( \phi \)s and \( n \)s to show the changes in petals, size, and rotation. Visualizing these graphs will confirm the analytical conclusions derived above.

Key Concepts

Rose Curves

Rose curves are a fascinating family of polar graphs named for their resemblance to rose petals. With their distinct and symmetrical petal shapes, they are a popular subject in polar coordinate studies. These curves are described by the polar equation \( r = a + b \cos(n(\theta + \phi)) \). Here, the parameter \( n \) significantly influences the appearance of the curve.

• If \( n \) is an odd integer, the rose curve will have exactly \( n \) petals.
• If \( n \) is an even integer, the number of petals doubles, resulting in \( 2n \) petals.

These petals give the rose curve its signature look, with more petals being added as \( n \) increases. In scenarios where \( n = 1 \), the result is a simple circle rather than a rose. Increasing \( n \) adds to the complexity and beauty of the graph.

Limaçon

Limaçon is another intriguing polar curve that emerges under certain conditions of the given polar equation \( r = a + b \cos(n(\theta + \phi)) \). The shape of the limaçon is dependent on the relationship between \( a \) and \( b \). The limaçon has a unique form characterized by its loops or dimpled shapes.

• A limaçon without an inner loop occurs if \( |a| > |b| \).
• If \( |a| = |b| \), the curve transforms into a cardioid, a limaçon with a single cusp.
• A limaçon with an inner loop appears when \( |a| < |b| \).

This variability makes the limaçon a particularly interesting curve to explore, as slight changes in the values of \( a \) and \( b \) can lead to markedly different shapes. Visualizing these differences through graphing can help cement understanding.

Phase Shift

The concept of phase shift is crucial when dealing with the polar equation \( r = a + b \cos(n(\theta + \phi)) \). The term \( \phi \) accounts for the phase shift of the curve. Essentially, it determines how the entire graph is rotated around the pole (origin). When \( \phi = 0 \), no rotation occurs, and the curve's position is determined solely by \( a \), \( b \), and \( n \). If \( \phi eq 0 \), the curve is rotated through an angle of \( \phi \) radians. This means:

• The same curve shape can be rotated to start at a different position.
• Every different \( \phi \) results in a symmetrical counterpart, displaced by \( \phi \).

Understanding how phase shift affects polar curves can help in predicting the orientation of the graph without visualizing it graphically first.

Petal Count

In polar equations that describe rose curves, the number of petals is an essential feature often dictated by the value of \( n \). The parameter \( n \) directly influences the petal count in the graph of the polar equation \( r = a + b \cos(n(\theta + \phi)) \).

• If \( n \) is odd, the curve will display \( n \) petals.
• If \( n \) is even, the amount increases to \( 2n \) petals.

This predictable expansion in petal count as \( n \) increases benefits from visualization. By sketching several versions of the function with varying \( n \) values, one can intuitively understand the petal configuration. This not only helps in recognizing the equations but also assists in mastering the polar coordinate system.