Answered: f(x, y, z) = x² + 2y² + 4z² + 10;…

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

Gradients in three dimensions Consider the following functions ƒ, points P, and unit vectors u.
a. Compute the gradient of ƒ and evaluate it at P.
b. Find the unit vector in the direction of maximum increase of ƒ at P.
c. Find the rate of change of the function in the direction of maximum
increase at P.
d. Find the directional derivative at P in the direction of the given vector.

f(x, y, z) = x² + 2y² + 4z² + 10; P(1,0, 4);
V2

Expert Solution

Step 1: Given:

$f left parenthesis x comma y comma z right parenthesis equals x squared plus 2 y squared plus 4 z squared plus 10 space semicolon space space P left parenthesis 1 comma 0 comma 4 right parenthesis space semicolon space space left angle bracket 1 comma 0 comma 4 right angle bracket$

Step 2: a. Computing the gradient of ƒ and evaluate it at P.

Differentiate f with respect to $x comma space y$ and z.

$table attributes columnalign right center left columnspacing 0px end attributes row cell f subscript x end cell equals cell 2 x end cell row cell f subscript y end cell equals cell 4 y end cell row cell f subscript z end cell equals cell 8 z end cell end table$

Thus, the gradient of f is given by:

$nabla f equals open angle brackets 2 x comma 4 y comma 8 z close angle brackets$

At the point P,

$table attributes columnalign right center left columnspacing 0px end attributes row cell nabla f end cell equals cell open angle brackets 2 open parentheses 1 close parentheses comma 4 open parentheses 0 close parentheses comma 8 open parentheses 4 close parentheses close angle brackets end cell row blank equals cell open angle brackets 2 comma 0 comma 32 close angle brackets end cell end table$

Step 3: b. Finding the unit vector in the direction of maximum increase of ƒ at P.

The unit vector in the direction of maximum increase of ƒ at P is given by:

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator nabla f open parentheses p close parentheses over denominator vertical line vertical line nabla f open parentheses p close parentheses vertical line vertical line end fraction end cell equals cell fraction numerator open angle brackets 2 comma 0 comma 32 close angle brackets over denominator square root of 2 squared plus 0 squared plus 32 squared end root end fraction end cell row blank equals cell fraction numerator open angle brackets 2 comma 0 comma 32 close angle brackets over denominator 2 square root of 257 end fraction end cell row blank equals cell open angle brackets fraction numerator 2 over denominator 2 square root of 257 end fraction comma fraction numerator 0 over denominator 2 square root of 257 end fraction comma fraction numerator 32 over denominator 2 square root of 257 end fraction close angle brackets end cell row blank equals cell open angle brackets fraction numerator 1 over denominator square root of 257 end fraction comma 0 comma fraction numerator 16 over denominator square root of 257 end fraction close angle brackets end cell end table$

The unit vector in the direction of maximum increase of ƒ at P: $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open angle brackets fraction numerator 1 over denominator square root of 257 end fraction comma 0 comma fraction numerator 16 over denominator square root of 257 end fraction close angle brackets end cell end table$

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