Consider the function F: R² – R defined by: F(x, y) = xy – 3y, for all (x, y) E R². (a) Find the range of F. Your answer should be fully justified. (Hint. Fix a c E R: can you explicitly find one (x, y) E R?, meaning, can you explicitly find one value for x and one value for y, such that F(x, y) = c? If you can do that, then you have showed that the range of F is...) (b) Find all level sets of F. Give a qualitative description of each level set (e.g., say if they are circles, lines, parabolas, etc.) (c) Find the curve on the graph of F that lies above the curve y: R → R?. y(t) = (cos(t), cos(t) – sin(t)).
Consider the function F: R² – R defined by: F(x, y) = xy – 3y, for all (x, y) E R². (a) Find the range of F. Your answer should be fully justified. (Hint. Fix a c E R: can you explicitly find one (x, y) E R?, meaning, can you explicitly find one value for x and one value for y, such that F(x, y) = c? If you can do that, then you have showed that the range of F is...) (b) Find all level sets of F. Give a qualitative description of each level set (e.g., say if they are circles, lines, parabolas, etc.) (c) Find the curve on the graph of F that lies above the curve y: R → R?. y(t) = (cos(t), cos(t) – sin(t)).
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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Transcribed Image Text:4.
Consider the function F: R² → R defined by:
F(*, у) — ху — Зу,
for all (x, y) E R².
-
(a) Find the range of F. Your answer should be fully justified. (Hint. Fix a c e R: can you explicitly
find one (x, y) E R², meaning, can you explicitly find one value for x and one value for y, such that
F(x, y) = c? If you can do that, then you have showed that the range of F is ..
(b) Find all level sets of F. Give a qualitative description of each level set (e.g., say if they are circles,
lines, parabolas, etc.)
(c) Find the curve on the graph of F that lies above the curve
y: R → R².
r(t) = (cos(t), cos(t) – sin(t)).
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