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Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u,v): vs 1-u, u 20, v≥ 0); T: x=-u, y=v² Sketch S. Choose the correct answer below. SOA. -2 Sketch R. Choose the correct answer below. О А. Q B. OB. u E G O D. O D.

Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u,v): vs 1-u, u 20, v≥ 0); T: x=-u, y=v² Sketch S. Choose the correct answer below. SOA. -2 Sketch R. Choose the correct answer below. О А. Q B. OB. u E G O D. O D.

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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Homework # 15
The problem involves finding the image \( R \) in the xy-plane of the region \( S \) using a given transformation \( T \). The task is to sketch both regions \( R \) and \( S \). ### Problem Statement: - **Region \( S \):** - Defined by \((u,v)\) such that: - \( v \leq 1 - u \) - \( u \geq 0 \) - \( v \geq 0 \) - **Transformation \( T \):** - Defined by: - \( x = -u \) - \( y = v^2 \) ### Sketch \( S \): Choose the correct sketch from the provided diagrams for region \( S \), illustrated on the \((u, v)\)-plane. - Option A: Displays a triangular region in the upper right quadrant. - Option B: Shows a triangular region with a horizontal base along \( u \)-axis extending leftward. - Option C: Depicts a triangular region in the lower half-plane. - Option D: Shows a triangular region extending from the origin into the upper right quadrant. ### Sketch \( R \): Choose the correct sketch from the provided diagrams for region \( R \) after the transformation, illustrated on the \((x, y)\)-plane. - Option A: Shows a transformed shape in the lower left quadrant. - Option B: Displays a shape along the positive \( y \)-axis. - Option C: Depicts a shape in the lower left quadrant, inverse of a triangle. - Option D: Shows a curve extending in the first quadrant. Each sketch should be carefully evaluated to determine which option accurately represents the regions based on the conditions and transformation applied.
Transcribed Image Text:The problem involves finding the image \( R \) in the xy-plane of the region \( S \) using a given transformation \( T \). The task is to sketch both regions \( R \) and \( S \). ### Problem Statement: - **Region \( S \):** - Defined by \((u,v)\) such that: - \( v \leq 1 - u \) - \( u \geq 0 \) - \( v \geq 0 \) - **Transformation \( T \):** - Defined by: - \( x = -u \) - \( y = v^2 \) ### Sketch \( S \): Choose the correct sketch from the provided diagrams for region \( S \), illustrated on the \((u, v)\)-plane. - Option A: Displays a triangular region in the upper right quadrant. - Option B: Shows a triangular region with a horizontal base along \( u \)-axis extending leftward. - Option C: Depicts a triangular region in the lower half-plane. - Option D: Shows a triangular region extending from the origin into the upper right quadrant. ### Sketch \( R \): Choose the correct sketch from the provided diagrams for region \( R \) after the transformation, illustrated on the \((x, y)\)-plane. - Option A: Shows a transformed shape in the lower left quadrant. - Option B: Displays a shape along the positive \( y \)-axis. - Option C: Depicts a shape in the lower left quadrant, inverse of a triangle. - Option D: Shows a curve extending in the first quadrant. Each sketch should be carefully evaluated to determine which option accurately represents the regions based on the conditions and transformation applied.
**Problem Statement:** Find the image R in the xy-plane. Choose the correct answer below. **Options:** - **A.** The image is the region bounded by the x-axis, the y-axis, and the line \( y = 1 - x \). - **B.** The image is the region bounded by the x-axis, the y-axis, and the curve \( y = (1-x)^2 \). - **C.** The image is the region bounded by the x-axis, the y-axis, and the line \( y = -1 - x \). - **D.** The image is the region bounded by the x-axis, the y-axis, and the curve \( y = (1+x)^2 \).
Transcribed Image Text:**Problem Statement:** Find the image R in the xy-plane. Choose the correct answer below. **Options:** - **A.** The image is the region bounded by the x-axis, the y-axis, and the line \( y = 1 - x \). - **B.** The image is the region bounded by the x-axis, the y-axis, and the curve \( y = (1-x)^2 \). - **C.** The image is the region bounded by the x-axis, the y-axis, and the line \( y = -1 - x \). - **D.** The image is the region bounded by the x-axis, the y-axis, and the curve \( y = (1+x)^2 \).
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