Please give a full explanation to the answer and each of the steps. 2. Suppose f is a continuous function, and the following facts hold:(8,14,11)I|({8,14,11)||f (2, 10,9) = 8, and Duf(2,10,9) = -3, where û :If, from point (2,10,9), one moves a distance of .003 units along vector û to a new point, then whatwill the approximate value of f be at this new point (give your answer to 3 decimal places)?

Name: Please give a full explanation to the answer and each of the steps. 2.
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Description: Please give a full explanation to the answer and each of the steps. 2. Suppose f is a continuous function, and the following facts hold:(8,14,11)I|({8,14,11)||f (2, 10,9) = 8, and Duf(2,10,9) = -3, where û :If, from point (2,10,9), one moves a distan

Answered: Suppose f is a continuous function, and…

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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Please give a full explanation to the answer and each of the steps.

2. Suppose f is a continuous function, and the following facts hold:
(8,14,11)
I|({8,14,11)||
f (2, 10,9) = 8, and Duf(2,10,9) = -3, where û :
If, from point (2,10,9), one moves a distance of .003 units along vector û to a new point, then what
will the approximate value of f be at this new point (give your answer to 3 decimal places)?

Expert Solution

Solution:

Definition:

Directional derivative is the rate of change of $f (x, y, z)$ in the direction of the unit vector $\vec{u} = ⟨ a, b ⟩$ .

The definition of the directional derivative is, $D_{\hat{u}} (x, y, z) = \lim_{h \to 0} \frac{f (x + a h, y + b h, z + c h) - f (x, y, z)}{h}$

Given: $f (2, 10, 9) = 8 and D_{u} f (2, 10, 9) = - 3, where \hat{u} = \frac{<8, 14, 11>}{||<8, 14, 11>||}$

$D_{\hat{u}} (2, 10, 9) = \lim_{h \to 0.003} \frac{f (2 + \frac{8}{\sqrt{381}} h, 10 + \frac{14}{\sqrt{381}} h, 9 + \frac{11}{\sqrt{381}} h) - f (2, 10, 9)}{h} 8 = \frac{f (2 + \frac{8}{\sqrt{381}} (0.003), 10 + \frac{14}{\sqrt{381}} (0.003), 9 + \frac{11}{\sqrt{381}} (0.003)) + 3}{0.003}$

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