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Sketch the level curves of the function f(x, y) = = - y² for c= -4,-1, 0, 1, 4. 9

Sketch the level curves of the function f(x, y) = = - y² for c= -4,-1, 0, 1, 4. 9

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Instructions** Sketch the level curves of the function \( f(x, y) = \frac{x^2}{9} - y^2 \) for \( c = -4, -1, 0, 1, 4 \). **Graph Description** The image contains a coordinate plane with the x-axis and y-axis ranging from -8 to 8. The graph is set up to plot the level curves of the given function for different constant values \( c \). Each level curve represents the set of points \((x, y)\) that satisfy the equation \(\frac{x^2}{9} - y^2 = c\). **Explanation** For each value of \( c \), solve the equation \(\frac{x^2}{9} - y^2 = c\) to find the corresponding level curves: - If \( c = 0 \), the equation simplifies to \(\frac{x^2}{9} = y^2\), which gives two intersecting lines. - If \( c > 0 \), the equation represents a hyperbola. - If \( c < 0 \), the equation doesn't have real solutions, indicating no ellipse or real level curve on the plane. These concepts can be represented graphically, considering the intersections and types of conic sections formed by the function.
Transcribed Image Text:**Instructions** Sketch the level curves of the function \( f(x, y) = \frac{x^2}{9} - y^2 \) for \( c = -4, -1, 0, 1, 4 \). **Graph Description** The image contains a coordinate plane with the x-axis and y-axis ranging from -8 to 8. The graph is set up to plot the level curves of the given function for different constant values \( c \). Each level curve represents the set of points \((x, y)\) that satisfy the equation \(\frac{x^2}{9} - y^2 = c\). **Explanation** For each value of \( c \), solve the equation \(\frac{x^2}{9} - y^2 = c\) to find the corresponding level curves: - If \( c = 0 \), the equation simplifies to \(\frac{x^2}{9} = y^2\), which gives two intersecting lines. - If \( c > 0 \), the equation represents a hyperbola. - If \( c < 0 \), the equation doesn't have real solutions, indicating no ellipse or real level curve on the plane. These concepts can be represented graphically, considering the intersections and types of conic sections formed by the function.
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