Q. 53. Use Theorem 12.32 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 53. (page 965)

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$蝇 = x \sin y \cos z, P = (3, \frac{蟺}{4}, - \frac{蟺}{2}), v = i - 2 j + 3 k$

Short Answer

Expert verified

The required directional derivative of the function is $鈭� z 路 u = 1.7$

Step by step solution

Given information

Think about the following function.

$蝇 = x \sin y \cos z$

The objective is to find the directional derivative of the function at the point P=3,π4,-π2

$鈭� 蝇 = i \frac{鈭� 蝇}{鈭� x} + j \frac{鈭� 蝇}{鈭� y} + k \frac{鈭� 蝇}{鈭� z} 蝇 = x \sin y \cos z \frac{鈭� 蝇}{鈭� x} = \sin y \cos z 鈬� \frac{鈭� 蝇}{鈭� y} = x \cos y \cos z 鈬� \frac{鈭� 蝇}{鈭� z} = - x \sin y \sin z$

When it comes to the next step,

$鈭� 蝇 = i \frac{鈭� 蝇}{鈭� x} + j \frac{鈭� 蝇}{鈭� y} + k \frac{鈭� 蝇}{鈭� z} 鈬� 鈭� 蝇 = i (\sin y \cos z) + j (x \cos y \cos z) + k (- x \sin y \sin z)$

Consider the vector below.

$v = i - 2 j + 3 k$

Along the vector $v = i - 2 j + 3 k$ , there is a unit vector.

$u = \frac{i - 2 j + 3 k}{\sqrt{(1)^{2} + (- 2)^{2} + 3^{2}}} 鈬� u = \frac{i - 2 j + 3 k}{\sqrt{14}} 鈬� u = \frac{i - 2 j + 3 k}{\sqrt{14}}$

The directional derivatives of the function z in the direction

$鈭� 蝇路 u = (i (\sin y \cos z) + j (x \cos y \cos z) + k (- x \sin y \sin z)) 路 (\frac{i - 2 j + 3 k}{\sqrt{14}}) 鈬� 鈭� 蝇路 u = \frac{(1 (\sin y \cos z) - 2 (x \cos y \cos z) + 3 (- x \sin y \sin z))}{\sqrt{14}}$

At point $P = (3, \frac{蟺}{4}, - \frac{蟺}{2})$

$鈭� 蝇路 u = \frac{(1 (\sin (\frac{蟺}{4}) \cos (- \frac{蟺}{2})) - 2 (3 \cos (\frac{蟺}{4}) \cos (- \frac{蟺}{2})) + 3 (- 3 \sin (\frac{蟺}{4}) \sin (- \frac{蟺}{2})))}{\sqrt{14}} 鈬� 鈭� z 路 u = \frac{6.36}{\sqrt{14}}$

Hence, the directional derivative of the function is

$鈭� z 路 u = 1.7$

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91影视

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$蝇 = x \sin y \cos z, P = (3, \frac{蟺}{4}, - \frac{蟺}{2}), v = i - 2 j + 3 k$

Short Answer

Step by step solution

Given information

The objective is to find the directional derivative of the function at the point P=3,π4,-π2

The directional derivatives of the function z in the direction

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Most popular questions from this chapter

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91影视

Use Theorem 12.32 to find the indicated derivatives in Exercises21鈥�26. Express your answers as functions of a single variable蝇=xsinycosz,P=3,蟺4,-蟺2,v=i-2j+3k

Short Answer

Step by step solution

Given information

The objective is to find the directional derivative of the function at the point P=3,π4,-π2

The directional derivatives of the function z in the direction

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$蝇 = x \sin y \cos z, P = (3, \frac{蟺}{4}, - \frac{蟺}{2}), v = i - 2 j + 3 k$