1. For the given function f(x, y, z), first evaluate the gradient vector f(p) at the given point p, and then evaluate the directional derivative (p) using the given vector v: af f(x, y, z) = x² + 2y²+3z² at p = (1,1,0), with ʊ= (1,-1,2) 2. A differentiable function f(x, y) has, at a point p, a directional deriva- tive +2 with respect to the unit vector in the direction of the vector (2, 2), and a directional derivative -2 with respect to the unit vector in the direction of the vector (-1, 1). ⚫ Determine the gradient vector at p. • Compute the directional derivative at p with respect to the vec- tor (4, 6). 3. Consider the function with undetermined constants a, b, c: Cz²x³ f(x, y, z) = axy²+byz + cz Find values of a, b, c such that the directional derivative of f(x, y, z) at the point (1,2, -1) has a maximum value of 64 in a direction parallel to the z-axis.
1. For the given function f(x, y, z), first evaluate the gradient vector f(p) at the given point p, and then evaluate the directional derivative (p) using the given vector v: af f(x, y, z) = x² + 2y²+3z² at p = (1,1,0), with ʊ= (1,-1,2) 2. A differentiable function f(x, y) has, at a point p, a directional deriva- tive +2 with respect to the unit vector in the direction of the vector (2, 2), and a directional derivative -2 with respect to the unit vector in the direction of the vector (-1, 1). ⚫ Determine the gradient vector at p. • Compute the directional derivative at p with respect to the vec- tor (4, 6). 3. Consider the function with undetermined constants a, b, c: Cz²x³ f(x, y, z) = axy²+byz + cz Find values of a, b, c such that the directional derivative of f(x, y, z) at the point (1,2, -1) has a maximum value of 64 in a direction parallel to the z-axis.
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:1. For the given function f(x, y, z), first evaluate the gradient vector
f(p) at the given point p, and then evaluate the directional derivative
(p) using the given vector v:
af
f(x, y, z) = x² + 2y²+3z² at p = (1,1,0), with ʊ= (1,-1,2)
2. A differentiable function f(x, y) has, at a point p, a directional deriva-
tive +2 with respect to the unit vector in the direction of the vector
(2, 2), and a directional derivative -2 with respect to the unit vector
in the direction of the vector (-1, 1).
⚫ Determine the gradient vector at p.
• Compute the directional derivative at p with respect to the vec-
tor (4, 6).
3. Consider the function with undetermined constants a, b, c:
Cz²x³
f(x, y, z) = axy²+byz + cz
Find values of a, b, c such that the directional derivative of f(x, y, z) at
the point (1,2, -1) has a maximum value of 64 in a direction parallel
to the z-axis.
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