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Consider the unit circle C: u² + v² = 1. a) What is the tangent line to C at a point (u, v) E C with u> 0 and v> 0? Sketch a graph of the situation. b) Define A as the area enclosed by the triangle with vertices (0, 0), (x, 0) and (0, y), where for the line found in a), x is its x-intercept and y its y-intercept. The function A should depend on u and v. c) Using the method of Lagrange multipliers, find the point (u, v) E C that makes A minimum. Explain why there is no absolute maxima for A.

Consider the unit circle C: u² + v² = 1. a) What is the tangent line to C at a point (u, v) E C with u> 0 and v> 0? Sketch a graph of the situation. b) Define A as the area enclosed by the triangle with vertices (0, 0), (x, 0) and (0, y), where for the line found in a), x is its x-intercept and y its y-intercept. The function A should depend on u and v. c) Using the method of Lagrange multipliers, find the point (u, v) E C that makes A minimum. Explain why there is no absolute maxima for A.

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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2. Consider the unit circle C: u² + v² = 1. a) What is the tangent line to C at a point (u, v) ≤ C with u > 0 and v> 0? Sketch a graph of the situation. b) Define A as the area enclosed by the triangle with vertices (0, 0), (x, 0) and (0, y), where for the line found in a), x is its x-intercept and y its y-intercept. The function A should depend on u and v. c) Using the method of Lagrange multipliers, find the point (u, v) E C that makes A minimum. Explain why there is no absolute maxima for A.
Transcribed Image Text:2. Consider the unit circle C: u² + v² = 1. a) What is the tangent line to C at a point (u, v) ≤ C with u > 0 and v> 0? Sketch a graph of the situation. b) Define A as the area enclosed by the triangle with vertices (0, 0), (x, 0) and (0, y), where for the line found in a), x is its x-intercept and y its y-intercept. The function A should depend on u and v. c) Using the method of Lagrange multipliers, find the point (u, v) E C that makes A minimum. Explain why there is no absolute maxima for A.
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