Q. 27 In Exercises 21鈥�28, find the d... [FREE SOLUTION]

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Chapter 12: Q. 27 (page 953)

In Exercises 21鈥�28, find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (2, 鈭� 2, 2), u = <\frac{3}{5}, 0, \frac{4}{5})$

Short Answer

Expert verified

The directional derivative of function is $鈭� \frac{36}{5}$

Step by step solution

Given data

The function is $f (x, y, z) = x^{2} + y^{2} 鈭� z^{3}$

The given points is $P = (x_{0}, y_{0}, z_{0}) = (2, 鈭� 2, 2) and u = (伪, 尾, 纬) = (\frac{3}{5}, 0, \frac{4}{5})$

Solution

Consider directional derivative

$D_{u} f (x_{0}, y_{0}, z_{0}) = {Lim}_{h 鈫� 0} 鈦� \frac{f (x_{0} + 伪 h, y_{0} + 尾 h, z_{0} + 纬 h) 鈭� f (x_{0}, y_{0}, z_{0}}{h}$

$D_{u} f (2, 鈭� 2, 2) = {Lim}_{h 鈫� 0} 鈦� \frac{f (2 + \frac{3}{5} h, 鈭� 2 + 0 h, 2 + \frac{4}{5} h) 鈭� f (2, 鈭� 2, 2)}{h}$ $(1)$

Therefore,

$f (2 + \frac{3}{5} h, 鈭� 2 + 0 h, 2 + \frac{4}{5} h) = {(2 + \frac{3}{5} h)}^{2} + (鈭� 2 + 0 h)^{2} 鈭� {(2 + \frac{4}{5} h)}^{3}$

$= 4 + \frac{12}{5} h + \frac{9}{25} h^{2} + 4 鈭� 8 鈭� \frac{48}{5} h 鈭� \frac{96}{25} h^{2} 鈭� \frac{64}{125} h^{3}$

$= 鈭� \frac{36}{5} h 鈭� \frac{87}{25} h^{2} 鈭� \frac{64}{125} h^{3}$ $((2))$

And

$f (x_{0}, y_{0}, z_{0}) = f (2, 鈭� 2, 2) = 2^{2} + (鈭� 2^{2}) 鈭� 2^{3}$

$f (2, 鈭� 2, 2) = 0$ $(3)$

Substitute

Substituting

$D_{w} f (2, 鈭� 2, 2) = {Lim}_{h 鈫� 0} 鈦� \frac{鈭� \frac{36}{5} h 鈭� \frac{87}{25} h^{2} 鈭� \frac{64}{125} h^{3} 鈭� 0}{h}$

$= {Lim}_{h 鈫� 0} 鈭� \frac{36}{5} 鈭� \frac{87}{25} h 鈭� \frac{64}{125} h^{2}$

$D_{w} f (2, 鈭� 2, 2) = 鈭� \frac{36}{5}$

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91影视

In Exercises 21鈥�28, find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (2, 鈭� 2, 2), u = <\frac{3}{5}, 0, \frac{4}{5})$

Short Answer

Step by step solution

Given data

Solution

Substitute

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91影视

In Exercises 21鈥�28, find the directional derivative of the givenfunction at the specified point Pand in the direction of thegiven unit vectoru.f(x,y,z)=x2+y2鈭�z3atP=(2,鈭�2,2),u=35,0,45

Short Answer

Step by step solution

Given data

Solution

Substitute

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Geometry

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 21鈥�28, find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (2, 鈭� 2, 2), u = <\frac{3}{5}, 0, \frac{4}{5})$