point P are given. Let 0 correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of 0; call this function g. d. Find the value of 0 that maximizes g(0) and find the maximum value. e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. |38. (х, у) — 8 + х + 3у?; Р(-3, —1)

point P are given. Let 0 correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of 0; call this function g. d. Find the value of 0 that maximizes g(0) and find the maximum value. e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. |38. (х, у) — 8 + х + 3у?; Р(-3, —1)

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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Question

point P are given. Let 0 correspond to the direction of the directional
derivative.
a. Find the gradient and evaluate it at P.
b. Find the angles 0 (with respect to the positive x-axis) associated
with the directions of maximum increase, maximum decrease, and
zero change.
c. Write the directional derivative at P as a function of 0; call this
function g.
d. Find the value of 0 that maximizes g(0) and find the maximum
value.
e. Verify that the value of 0 that maximizes g corresponds to the
direction of the gradient. Verify that the maximum value of g equals
the magnitude of the gradient.
|38. (х, у) — 8 + х + 3у?; Р(-3, —1)

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