Let f(x, y) = xy + x − y be defined on the closed disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} of radius 2. (a) Find the maximum and minimum of Duf at (0, 0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} and all unit vectors u. (Hint: Think of |∇f|2 as a function of x and y in the disk.)

Let f(x, y) = xy + x − y be defined on the closed disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} of radius 2. (a) Find the maximum and minimum of Duf at (0, 0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} and all unit vectors u. (Hint: Think of |∇f|2 as a function of x and y in the disk.)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.3: Subspaces Of Vector Spaces

Problem 49E

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Question

Let f(x, y) = xy + x − y be defined on the closed disk {(x, y) ∈ R² : x² + y² ≤ 4} of radius 2.

(a) Find the maximum and minimum of D_uf at (0, 0) over all unit vectors u.

(b) Find the maximum and minimum of D_uf over all points in the disk {(x, y) ∈ R² : x² + y² ≤ 4} and all unit vectors u. (Hint: Think of |∇f|² as a function of x and y in the disk.)

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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