Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Polar Equation Curves The image contains four graphs depicting curves generated from polar equations. **1. Curve with the Equation \( r = 4 \cos(2\theta) \):** - **Description:** This curve features a "four-leaf clover" shape, symmetrical about the origin, with petals extending along both the x-axis and y-axis. - **Axes:** The x-axis and y-axis range from -2 to 2. - **Characteristics:** Each petal reaches out exactly to r = 2 units at the furthest points, reflecting the nature of a polar "rose" curve with an even number of petals. **2. Second Curve Pattern (Top Right):** - **Description:** This curve depicts a "two-petal" rose. - **Axes:** The x-axis and y-axis range from -3 to 3. - **Characteristics:** The petals are broader, appearing on opposite quadrants. The maximum radius achieved by each petal extends to r = 3 units. **3. Third Curve Pattern (Bottom Left):** - **Description:** This is another variation showing a "four-leaf" pattern. - **Axes:** The x-axis and y-axis range from -3 to 3. - **Characteristics:** Similar to the first graph, but larger petal radius. Symmetrical along both axes, demonstrating complete symmetry across both dimensions. **4. Fourth Curve Pattern (Bottom Right):** - **Description:** A smaller, more intricate "six-leaf" pattern. - **Axes:** The x-axis and y-axis range from -1.5 to 1.5. - **Characteristics:** Displays more frequency, with smaller petals compared to the others. Each petal reaches a maximum near r = 1.5 units, indicating higher frequency and intricate design. These curves highlight the diversity of shapes formed through variations in polar equations, showcasing symmetrical petal patterns characteristic of "rose" curves with unique configurations based on the polar angle coefficient, \( \theta \).
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Transcribed Image Text:### Polar Equation Curves The image contains four graphs depicting curves generated from polar equations. **1. Curve with the Equation \( r = 4 \cos(2\theta) \):** - **Description:** This curve features a "four-leaf clover" shape, symmetrical about the origin, with petals extending along both the x-axis and y-axis. - **Axes:** The x-axis and y-axis range from -2 to 2. - **Characteristics:** Each petal reaches out exactly to r = 2 units at the furthest points, reflecting the nature of a polar "rose" curve with an even number of petals. **2. Second Curve Pattern (Top Right):** - **Description:** This curve depicts a "two-petal" rose. - **Axes:** The x-axis and y-axis range from -3 to 3. - **Characteristics:** The petals are broader, appearing on opposite quadrants. The maximum radius achieved by each petal extends to r = 3 units. **3. Third Curve Pattern (Bottom Left):** - **Description:** This is another variation showing a "four-leaf" pattern. - **Axes:** The x-axis and y-axis range from -3 to 3. - **Characteristics:** Similar to the first graph, but larger petal radius. Symmetrical along both axes, demonstrating complete symmetry across both dimensions. **4. Fourth Curve Pattern (Bottom Right):** - **Description:** A smaller, more intricate "six-leaf" pattern. - **Axes:** The x-axis and y-axis range from -1.5 to 1.5. - **Characteristics:** Displays more frequency, with smaller petals compared to the others. Each petal reaches a maximum near r = 1.5 units, indicating higher frequency and intricate design. These curves highlight the diversity of shapes formed through variations in polar equations, showcasing symmetrical petal patterns characteristic of "rose" curves with unique configurations based on the polar angle coefficient, \( \theta \).
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