Let f(x, y) = (x - xy²). In this problem, you will find the absolute maximum and minimum values of f(x, y) = x - xy² such that (x, y) is within the unit circle; i.e., x² + y² ≤ 1. (a) Find critical points (ro, yo) of the function f(x, y) = x - xy². (b) Recall that the unit circle can be parameterized by r(t) = cos(t) and y(t) = sin(t) for 0≤t≤ 2. Substitute these in for the function f(x, y) = x - xy². (c) In (b) you are now left with a single-variable function f(t) in terms of t. Find the maximum and minimum values of f(t) on the closed interval [0, 27]. This will give you the maximum and minimum values of f(x, y) that lie on the unit circle. Hint: you may want to use a trig identity to write f(t) just in terms of cos(t). Please list the corresponding points (ro, yo) where the maximum and minimum values occur. (d) Combine your answers to (a) and (c) to find the absolute maximum and minimum values of f(x, y) = x - xy² such that (ro, yo) is in the unit disk (x² + y² ≤ 1). (e) Let g(x, y) = 1- x² - y². Note that the level curve corresponding to g(x, y) = 0 is the same unit circle you parameterized in (b). Compute Vg(xo, yo) and Vf(xo, yo) for the point (s) (ro, yo) you found in (c) that are NOT the critical point(s) that you find in (a). What do you notice?
Let f(x, y) = (x - xy²). In this problem, you will find the absolute maximum and minimum values of f(x, y) = x - xy² such that (x, y) is within the unit circle; i.e., x² + y² ≤ 1. (a) Find critical points (ro, yo) of the function f(x, y) = x - xy². (b) Recall that the unit circle can be parameterized by r(t) = cos(t) and y(t) = sin(t) for 0≤t≤ 2. Substitute these in for the function f(x, y) = x - xy². (c) In (b) you are now left with a single-variable function f(t) in terms of t. Find the maximum and minimum values of f(t) on the closed interval [0, 27]. This will give you the maximum and minimum values of f(x, y) that lie on the unit circle. Hint: you may want to use a trig identity to write f(t) just in terms of cos(t). Please list the corresponding points (ro, yo) where the maximum and minimum values occur. (d) Combine your answers to (a) and (c) to find the absolute maximum and minimum values of f(x, y) = x - xy² such that (ro, yo) is in the unit disk (x² + y² ≤ 1). (e) Let g(x, y) = 1- x² - y². Note that the level curve corresponding to g(x, y) = 0 is the same unit circle you parameterized in (b). Compute Vg(xo, yo) and Vf(xo, yo) for the point (s) (ro, yo) you found in (c) that are NOT the critical point(s) that you find in (a). What do you notice?
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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This is a calc 3 question. Please try and solve to the best of your ability, thank you!
![Let f(x, y) = (x - xy2). In this problem, you will find the absolute
maximum and minimum values of f(x, y) = x-xy² such that (x, y) is within the unit circle;
i.e., x² + y² ≤ 1.
(a) Find critical points (ro, yo) of the function f(x, y) = x - xy².
(b) Recall that the unit circle can be parameterized by r(t) = cos(t) and y(t) = sin(t)
for 0≤t≤ 27. Substitute these in for the function f(x, y) = x - xy².
(c) In (b) you are now left with a single-variable function f(t) in terms of t. Find the
maximum and minimum values of f(t) on the closed interval [0, 27]. This will give
you the maximum and minimum values of f(x, y) that lie on the unit circle. Hint:
you may want to use a trig identity to write f(t) just in terms of cos(t). Please list
the corresponding points (ro, yo) where the maximum and minimum values occur.
(d) Combine your answers to (a) and (c) to find the absolute maximum and minimum
values of f(x, y) = x - xy² such that (xo, yo) is in the unit disk (x² + y² ≤ 1).
(e) Let g(x, y) = 1- x² - y². Note that the level curve corresponding to g(x, y) = 0 is
the same unit circle you parameterized in (b). Compute Vg(zo, yo) and Vf(xo, yo)
for the point (s) (xo, yo) you found in (c) that are NOT the critical point(s) that you
find in (a). What do you notice?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37da68b3-0faa-4aa4-865e-e958f15b2b06%2Fe6e94f0a-c235-49c4-9898-c521fd4b3b01%2F9rka2yj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let f(x, y) = (x - xy2). In this problem, you will find the absolute
maximum and minimum values of f(x, y) = x-xy² such that (x, y) is within the unit circle;
i.e., x² + y² ≤ 1.
(a) Find critical points (ro, yo) of the function f(x, y) = x - xy².
(b) Recall that the unit circle can be parameterized by r(t) = cos(t) and y(t) = sin(t)
for 0≤t≤ 27. Substitute these in for the function f(x, y) = x - xy².
(c) In (b) you are now left with a single-variable function f(t) in terms of t. Find the
maximum and minimum values of f(t) on the closed interval [0, 27]. This will give
you the maximum and minimum values of f(x, y) that lie on the unit circle. Hint:
you may want to use a trig identity to write f(t) just in terms of cos(t). Please list
the corresponding points (ro, yo) where the maximum and minimum values occur.
(d) Combine your answers to (a) and (c) to find the absolute maximum and minimum
values of f(x, y) = x - xy² such that (xo, yo) is in the unit disk (x² + y² ≤ 1).
(e) Let g(x, y) = 1- x² - y². Note that the level curve corresponding to g(x, y) = 0 is
the same unit circle you parameterized in (b). Compute Vg(zo, yo) and Vf(xo, yo)
for the point (s) (xo, yo) you found in (c) that are NOT the critical point(s) that you
find in (a). What do you notice?
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