Consider the function f : R³ R defined by f(x, y, z) = tanh(5x² + 2y² − 3z²) and the point P : (x, y, z) = (1, 1, 1). (a) In what direction is ƒ most rapidly increasing at the point P? (b) Calculate the directional derivative of f in the direction of u = (c) Determine an expression for the linear approximation ƒ of ƒ at point P. (d) Use the linear approximation to approximate the value of ƒ(1,-1, 1). N = (2, -1, 2) at the point P.
Consider the function f : R³ R defined by f(x, y, z) = tanh(5x² + 2y² − 3z²) and the point P : (x, y, z) = (1, 1, 1). (a) In what direction is ƒ most rapidly increasing at the point P? (b) Calculate the directional derivative of f in the direction of u = (c) Determine an expression for the linear approximation ƒ of ƒ at point P. (d) Use the linear approximation to approximate the value of ƒ(1,-1, 1). N = (2, -1, 2) at the point P.
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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![Consider the function f : R³ → R defined by
f(x, y, z) = tanh(5x² + 2y² - 3z²)
and the point P: (x, y, z) = (1, 1, 1).
(a) In what direction is f most rapidly increasing at the point P?
(b) Calculate the directional derivative of f in the direction of u = (2, -1, 2) at the point P.
(c) Determine an expression for the linear approximation f of f at point P.
(d) Use the linear approximation to approximate the value of f(-).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aad4bd6-f67d-4613-88de-9ea67499279b%2F362e80be-fda4-4353-818a-f0664dc9716d%2Fr12nm75_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function f : R³ → R defined by
f(x, y, z) = tanh(5x² + 2y² - 3z²)
and the point P: (x, y, z) = (1, 1, 1).
(a) In what direction is f most rapidly increasing at the point P?
(b) Calculate the directional derivative of f in the direction of u = (2, -1, 2) at the point P.
(c) Determine an expression for the linear approximation f of f at point P.
(d) Use the linear approximation to approximate the value of f(-).
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