Q. 38 In Exercises 35-38聽find the dir... [FREE SOLUTION]

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Chapter 12: Q. 38 (page 953)

In Exercises $35 - 38$ find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (0, 鈭� 2, 5) \\ v = 鈭� i + 3 j + 5 k \end{aligned}$

Short Answer

Expert verified

The directional derivative of function is $鈭� \frac{137}{\sqrt{35}}$

Step by step solution

Given data

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

$P = (x_{0}, y_{0}, z_{0}) = (0, 鈭� 2, 5) and v = 鈭� i + 3 j + 5 k$

Solution

Consider directional derivative

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

Therefore

$鈭� v 鈭� = \sqrt{35}$

$鈭� u = (伪, 尾, 纬) = (\frac{鈭� 1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}})$

Consider

$f (x_{0} + 伪 h, y_{0} + 尾 h, z_{0} + 纬 h) = f (0 鈭� \frac{1}{\sqrt{35}} h, 鈭� 2 + \frac{3}{\sqrt{35}} h, 5 + \frac{5}{\sqrt{35}} h)$

$= {(0 鈭� \frac{1}{\sqrt{35}} h)}^{2} + {(鈭� 2 + \frac{3}{\sqrt{35}} h)}^{2} 鈭� {(5 + \frac{5}{\sqrt{35}} h)}^{3}$

$= \frac{1}{35} h^{2} + (4 鈭� \frac{12}{\sqrt{35}} h + \frac{9}{35} h^{2}) 鈭� (125 + \frac{375}{\sqrt{35}} h + \frac{375}{35} h^{2} + \frac{125}{(\sqrt{35})^{3}} h^{3})$

$= \frac{1}{35} h^{2} + 4 鈭� \frac{12}{\sqrt{35}} h + \frac{9}{35} h^{2} 鈭� 125 鈭� \frac{375}{\sqrt{35}} h 鈭� \frac{375}{35} h^{2} 鈭� \frac{125}{(\sqrt{35})^{3}} h^{3}$

$= 鈭� 121 鈭� \frac{365}{35} h^{2} 鈭� \frac{137}{\sqrt{35}} h 鈭� \frac{125}{(\sqrt{35})^{3}} h^{3}$

And

$f (x_{0}, y_{0}, z_{0}) = 0^{2} + (鈭� 2)^{2} 鈭� 5^{3} = 鈭� 121$

substitute

Substituting

$D_{w} f (0, 鈭� 2, 5) = {Lim}_{h 鈫� 0} 鈦� \frac{鈭� 121 鈭� \frac{137}{\sqrt{35}} h 鈭� \frac{365}{35} h^{2} 鈭� \frac{125}{(\sqrt{35})^{3}} h^{3} 鈭� 121}{h}$

$= {Lim}_{h 鈫� 0} 鈭� \frac{137}{\sqrt{35}} 鈭� \frac{365}{35} h 鈭� \frac{125}{(\sqrt{35})^{3}} h^{2}$

$D_{w} f (0, 鈭� 2, 5) = 鈭� \frac{137}{\sqrt{35}}$

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91影视

In Exercises $35 - 38$ find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (0, 鈭� 2, 5) \\ v = 鈭� i + 3 j + 5 k \end{aligned}$

Short Answer

Step by step solution

Given data

Solution

substitute

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91影视

In Exercises 35-38find the directional derivative of the givenfunction at the specified point P and in the direction of thegiven vector v.f(x,y,z)=x2+y2鈭�z3atP=(0,鈭�2,5)v=鈭�i+3j+5k

Short Answer

Step by step solution

Given data

Solution

substitute

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Theoretical and Mathematical Physics

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises $35 - 38$ find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$\begin{aligned} f (x, y, z) = x^{2} + y^{2} 鈭� z^{3} at P = (0, 鈭� 2, 5) \\ v = 鈭� i + 3 j + 5 k \end{aligned}$