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

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

Short Answer

Expert verified

The directional derivative of the

function is $鈭� \frac{24}{3}$

Step by step solution

Given data

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

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

Solution

Consider directional derivative

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

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

Therefore,

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

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

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

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

And

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

$f (2, 鈭� 2, 2) = 0$

substitute

Substituting

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

$= {Lim}_{h 鈫� 0} 鈭� \frac{24}{3} 鈭� \frac{208}{9} h 鈭� \frac{8}{27} h^{2}$

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

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Short Answer

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

Short Answer

Step by step solution

Given data

Solution

substitute

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Most popular questions from this chapter

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Statistics

Probability and Statistics

Geometry

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Discrete Mathematics

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