8. In this problem we'll find the points on the curve y = 1.5 – 2² that are closest to the origin. (a) Here is a graph of the curve y = 1.5 – 2². Visually estimate the coordinates of the points where the curve is nearest the origin. 0.5 -2.5 -2 -1.5 -1 -0.5 0.5 1.5 2 2,5 -0.5 -1 (b) Let f(x, y) = x² + y² represent the square of the distance from (0, 0) to a point (x, y). Use the method of Lagrange multipliers to minimize f subject to the constraint y = 1.5 – x². Hint: what is g(x, y) in this problem?
8. In this problem we'll find the points on the curve y = 1.5 – 2² that are closest to the origin. (a) Here is a graph of the curve y = 1.5 – 2². Visually estimate the coordinates of the points where the curve is nearest the origin. 0.5 -2.5 -2 -1.5 -1 -0.5 0.5 1.5 2 2,5 -0.5 -1 (b) Let f(x, y) = x² + y² represent the square of the distance from (0, 0) to a point (x, y). Use the method of Lagrange multipliers to minimize f subject to the constraint y = 1.5 – x². Hint: what is g(x, y) in this problem?
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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REFER TO IMAGE PLEASE ANSWER THE FIRST THREE PARTS
Transcribed Image Text:8. In this problem we'll find the points on the curve y = 1.5 – a² that are closest to the origin.
(a) Here is a graph of the curve y = 1.5 – 2². Visually estimate the coordinates of the points where the
curve is nearest the origin.
15
0.5
-2.5
-2 -1.5/
-0.5
0.5
1
1.5
2.5
-0.5
-1
(b) Let f(x, y) = x² + y² represent the square of the distance from (0, 0) to a point (x, y).
Use the method of Lagrange multipliers to minimize f subject to the constraint y = 1.5 – x².
Hint: what is g(r, y) in this problem?
(c) If D represents the minimum distance from the curve to the origin, sketch the circle f(x, y) = D²
on the graph. What do you notice about the points you solved for above?
(d) Calculate Vf at the points you found above, and sketch the vector Vf at each point. What is the
relationship between Vf and the circle f(x, y) = D²?
2.
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