Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Sketch the level curves** of the function \( f(x, y) = \frac{x^2}{9} - y^2 \) for \( c = -4, -1, 0, 1, 4 \). **Graph Explanation:** The image shows a coordinate plane with both the x-axis and y-axis ranging from -8 to 8. The grid lines are divided into one-unit increments. This blank coordinate plane is used to sketch the level curves of the given function for different values of \( c \). Each level curve represents the set of points \((x, y)\) that satisfies the equation \( \frac{x^2}{9} - y^2 = c \) for a specific value of \( c \). To sketch these curves: 1. **For \( c = -4 \):** This value will typically result in hyperbolas opening along the y-axis. 2. **For \( c = -1 \):** Similar to \( c = -4 \), expect hyperbolas opening along the y-axis. 3. **For \( c = 0 \):** This will represent the degenerate case, usually a pair of intersecting lines. 4. **For \( c = 1 \):** Expect hyperbolas opening along the x-axis. 5. **For \( c = 4 \):** Similar to \( c = 1 \), hyperbolas opening along the x-axis. Each curve will illustrate how the function \( f(x, y) \) behaves for different constant outputs.
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Transcribed Image Text:**Sketch the level curves** of the function \( f(x, y) = \frac{x^2}{9} - y^2 \) for \( c = -4, -1, 0, 1, 4 \). **Graph Explanation:** The image shows a coordinate plane with both the x-axis and y-axis ranging from -8 to 8. The grid lines are divided into one-unit increments. This blank coordinate plane is used to sketch the level curves of the given function for different values of \( c \). Each level curve represents the set of points \((x, y)\) that satisfies the equation \( \frac{x^2}{9} - y^2 = c \) for a specific value of \( c \). To sketch these curves: 1. **For \( c = -4 \):** This value will typically result in hyperbolas opening along the y-axis. 2. **For \( c = -1 \):** Similar to \( c = -4 \), expect hyperbolas opening along the y-axis. 3. **For \( c = 0 \):** This will represent the degenerate case, usually a pair of intersecting lines. 4. **For \( c = 1 \):** Expect hyperbolas opening along the x-axis. 5. **For \( c = 4 \):** Similar to \( c = 1 \), hyperbolas opening along the x-axis. Each curve will illustrate how the function \( f(x, y) \) behaves for different constant outputs.
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Step 1: Introduction of the given problem

We have to sketch the level curve of f open parentheses x comma y close parentheses equals x squared over 9 minus y squared for c equals negative 4 comma negative 1 comma 0 comma 1 comma 4

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