Chapter 12: Q. 28 (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.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} - z^{3} at P = (2, - 2, 2) \\ u = \frac{2}{3} i - \frac{2}{3} j + \frac{1}{3} k \end{aligned}$

Short Answer

Expert verified

The directional derivative of the

function is $- \frac{24}{3}$

Step by step solution

Given data

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

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

Solution

Consider directional derivative

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

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

Therefore,

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

$= 4 + \frac{8}{3} h + \frac{4}{9} h^{2} + 4 - \frac{8}{3} h + \frac{4}{9} h^{2} - 8$

$= - \frac{24}{3} h - 24 h^{2} - \frac{8}{27} h^{3}$

$= - \frac{24}{3} h - \frac{208}{9} h^{2} - \frac{8}{27} h^{3}$

And

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

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

substitute

Substituting

$D_{u} f (2, - 2, 2) = {Lim}_{h \to 0} \frac{- \frac{24}{3} h - \frac{208}{9} h^{2} - \frac{8}{27} h^{3} - 0}{h}$

$= {Lim}_{h \to 0} - \frac{24}{3} - \frac{208}{9} h - \frac{8}{27} h^{2}$

$D_{u} f (2, - 2, 2) = - \frac{24}{3}$

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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.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} - z^{3} at P = (2, - 2, 2) \\ u = \frac{2}{3} i - \frac{2}{3} j + \frac{1}{3} k \end{aligned}$

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 P and in the direction of thegiven unit vector u.f(x,y,z)=x2+y2−z3atP=(2,−2,2)u=23i−23j+13k

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

Discrete Mathematics

Theoretical and Mathematical Physics

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Geometry

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

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.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} - z^{3} at P = (2, - 2, 2) \\ u = \frac{2}{3} i - \frac{2}{3} j + \frac{1}{3} k \end{aligned}$