Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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(a) Circle the equation that generates the surface shown below. *The image shows a 3D graph of a hyperbolic paraboloid. This surface is a saddle-shaped figure that extends infinitely.* Equations to choose from: a) \( y - z^2 = 0 \) b) \( 2x + 3y - z = 5 \) c) \( 4x^2 + \frac{y^2}{2} + z^2 = 9 \) d) \( x^2 + \frac{y^2}{2} - z^2 = 9 \) e) \( x^2 + \frac{y^2}{2} = z^2 \) f) \( x^2 + \frac{z^2}{2} - y^2 = 9 \) g) \( z^2 + \frac{y^2}{2} = x^2 \) h) \( z = x^2 - y^2 \) i) \( 9 = z^2 + y^2 \) *See next page for part (b)* The graph is color-coded, with green and orange portions indicating the different sections of the saddle surface. The axes are labeled as \( x \), \( y \), and \( z \).
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Transcribed Image Text:(a) Circle the equation that generates the surface shown below. *The image shows a 3D graph of a hyperbolic paraboloid. This surface is a saddle-shaped figure that extends infinitely.* Equations to choose from: a) \( y - z^2 = 0 \) b) \( 2x + 3y - z = 5 \) c) \( 4x^2 + \frac{y^2}{2} + z^2 = 9 \) d) \( x^2 + \frac{y^2}{2} - z^2 = 9 \) e) \( x^2 + \frac{y^2}{2} = z^2 \) f) \( x^2 + \frac{z^2}{2} - y^2 = 9 \) g) \( z^2 + \frac{y^2}{2} = x^2 \) h) \( z = x^2 - y^2 \) i) \( 9 = z^2 + y^2 \) *See next page for part (b)* The graph is color-coded, with green and orange portions indicating the different sections of the saddle surface. The axes are labeled as \( x \), \( y \), and \( z \).
**Contour Map Sketching** **Problem 9(b):** Given the function \( f(x, y) = 2x - y \), sketch a well-labeled contour map for the contour levels \( z_0 = 0, 1, 2, 3 \). --- ### Explanation In the image, there is a grid provided for sketching the contour map. To create a contour map for the given function \( f(x, y) = 2x - y \), follow these steps: 1. **Contour Lines:** For each contour level \( z_0 \), set the equation \( f(x, y) = z_0 \). - For \( z_0 = 0 \): \( 2x - y = 0 \) → \( y = 2x \) - For \( z_0 = 1 \): \( 2x - y = 1 \) → \( y = 2x - 1 \) - For \( z_0 = 2 \): \( 2x - y = 2 \) → \( y = 2x - 2 \) - For \( z_0 = 3 \): \( 2x - y = 3 \) → \( y = 2x - 3 \) 2. **Sketching:** Using the grid, plot each line based on the equations derived for each \( z_0 \). These lines will be linear and parallel since they have the same slope (2). 3. **Labeling:** Clearly label each line with its respective \( z_0 \) value for clarity. By sketching these lines on the provided grid, a contour map representing different elevation levels (z-values) for the function can be visualized.
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Transcribed Image Text:**Contour Map Sketching** **Problem 9(b):** Given the function \( f(x, y) = 2x - y \), sketch a well-labeled contour map for the contour levels \( z_0 = 0, 1, 2, 3 \). --- ### Explanation In the image, there is a grid provided for sketching the contour map. To create a contour map for the given function \( f(x, y) = 2x - y \), follow these steps: 1. **Contour Lines:** For each contour level \( z_0 \), set the equation \( f(x, y) = z_0 \). - For \( z_0 = 0 \): \( 2x - y = 0 \) → \( y = 2x \) - For \( z_0 = 1 \): \( 2x - y = 1 \) → \( y = 2x - 1 \) - For \( z_0 = 2 \): \( 2x - y = 2 \) → \( y = 2x - 2 \) - For \( z_0 = 3 \): \( 2x - y = 3 \) → \( y = 2x - 3 \) 2. **Sketching:** Using the grid, plot each line based on the equations derived for each \( z_0 \). These lines will be linear and parallel since they have the same slope (2). 3. **Labeling:** Clearly label each line with its respective \( z_0 \) value for clarity. By sketching these lines on the provided grid, a contour map representing different elevation levels (z-values) for the function can be visualized.
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Solution for 9 (a):

The equation that generates the given graph is choice (e):

x2+y22=z2

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