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Match the following functions with the appropriate cross-section: I) z = (y²-1) cos(x) II) z=(x²-1) sin(y) III) z=sin(x) cos(y) a) Which of the above functions has the following cross-section to the x-axis? Why? b) Which of the above functions has the following cross-section to the y-axis? Why?

Match the following functions with the appropriate cross-section: I) z = (y²-1) cos(x) II) z=(x²-1) sin(y) III) z=sin(x) cos(y) a) Which of the above functions has the following cross-section to the x-axis? Why? b) Which of the above functions has the following cross-section to the y-axis? Why?

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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**Match the following functions with the appropriate cross-section:** I) \( z = (y^2 - 1) \cos(x) \) II) \( z = (x^2 - 1) \sin(y) \) III) \( z = -\sin(x) \cos(y) \) **a) Which of the above functions has the following cross-section ⊥ to the x-axis? Why?** *Graph Explanation:* The graph shown is a parabola opening upwards. This indicates that the cross-section is quadratic. **b) Which of the above functions has the following cross-section ⊥ to the y-axis? Why?** *Graph Explanation:* The graph shown is a sinusoidal wave. This indicates that the cross-section is a sine wave. **Answer:** a) The function that has the cross-section perpendicular to the x-axis, which is a parabola, is \( z = (y^2 - 1) \cos(x) \). This is because when \( x \) is held constant and \( y \) varies, the \( y^2 - 1 \) term produces a parabolic shape. b) The function that has the cross-section perpendicular to the y-axis, which is a sinusoidal wave, is \( z = -\sin(x) \cos(y) \). This is because when \( y \) is held constant and \( x \) varies, the \(-\sin(x)\) term produces a sinusoidal shape.
Transcribed Image Text:**Match the following functions with the appropriate cross-section:** I) \( z = (y^2 - 1) \cos(x) \) II) \( z = (x^2 - 1) \sin(y) \) III) \( z = -\sin(x) \cos(y) \) **a) Which of the above functions has the following cross-section ⊥ to the x-axis? Why?** *Graph Explanation:* The graph shown is a parabola opening upwards. This indicates that the cross-section is quadratic. **b) Which of the above functions has the following cross-section ⊥ to the y-axis? Why?** *Graph Explanation:* The graph shown is a sinusoidal wave. This indicates that the cross-section is a sine wave. **Answer:** a) The function that has the cross-section perpendicular to the x-axis, which is a parabola, is \( z = (y^2 - 1) \cos(x) \). This is because when \( x \) is held constant and \( y \) varies, the \( y^2 - 1 \) term produces a parabolic shape. b) The function that has the cross-section perpendicular to the y-axis, which is a sinusoidal wave, is \( z = -\sin(x) \cos(y) \). This is because when \( y \) is held constant and \( x \) varies, the \(-\sin(x)\) term produces a sinusoidal shape.
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