Chapter 12: Q. 27 (page 953)

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

Short Answer

Expert verified

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

Step by step solution

Given data

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

The given points is $P = (x_{0}, y_{0}, z_{0}) = (2, - 2, 2) and u = (α, β, γ) = (\frac{3}{5}, 0, \frac{4}{5})$

Solution

Consider directional derivative

$D_{u} f (x_{0}, y_{0}, z_{0}) = {Lim}_{h \to 0} \frac{f (x_{0} + α h, y_{0} + β h, z_{0} + γ h) - f (x_{0}, y_{0}, z_{0}}{h}$

$D_{u} f (2, - 2, 2) = {Lim}_{h \to 0} \frac{f (2 + \frac{3}{5} h, - 2 + 0 h, 2 + \frac{4}{5} h) - f (2, - 2, 2)}{h}$ $(1)$

Therefore,

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

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

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

And

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

$f (2, - 2, 2) = 0$ $(3)$

Substitute

Substituting

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

$= {Lim}_{h \to 0} - \frac{36}{5} - \frac{87}{25} h - \frac{64}{125} h^{2}$

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

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

Short Answer

Step by step solution

Given data

Solution

Substitute

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In Exercises 21–28, 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−z3atP=(2,−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

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Discrete Mathematics

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