Q. 51 In Exercises 49鈥�54, find the d... [FREE SOLUTION]

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Chapter 12: Q. 51 (page 964)

In Exercises 49鈥�54, find the directional derivative of the given function at the specified point P and in the specified direction v. Note that some of the direction vectors are not unit vectors.
$z = \ln (\frac{x}{y^{2}}), P = (7, 1), v = - i - 4 j$

Short Answer

Expert verified

The directional derivative of the given function is $\frac{55 \sqrt{17}}{119}$ .

Step by step solution

Step 1. Given information.

The given function is

$z = \ln (\frac{x}{y^{2}})$

Step 2. Calculation.

The given vector is $v = - i - 4 j$ .

First we find the magnitude of the given vector. The magnitude of the given vector is:

$\sqrt{{(- 1)}^{2} + {(- 4)}^{2}} = \sqrt{17}$

so, the unit vector is $\hat{n} = \frac{1}{\sqrt{17}} (- i - 4 j)$
Now we have to find the gradient of the function.

$鈭� z = \frac{鈭� (\ln (\frac{x}{y^{2}}))}{鈭� x} i + \frac{鈭� (\ln (\frac{x}{y^{2}}))}{鈭� y} j = \frac{1 路 (\frac{1}{y^{2}})}{(\frac{x}{y^{2}})} i + \frac{1 路 (\frac{- 2 x}{y^{3}})}{(\frac{x}{y^{2}})} j = \frac{1}{x} i - \frac{2}{y} j$

Therefore the required directional derivative is equal to:

$\hat{n} 路 (鈭� z) = \frac{1}{\sqrt{17}} (- i - 4 j) 路 (\frac{1}{x} i - \frac{2}{y} j) = - \frac{1}{x \sqrt{17}} + \frac{8}{y \sqrt{17}}$

Step 3. Calculation.

Now we find directional derivative of the given function at the point $(7, 1)$ .

localid="1650435868672" $\hat{n} 路 {鈭� z}_{(7, 1)} = - \frac{1}{x \sqrt{17}} + \frac{8}{y \sqrt{17}} = - \frac{1}{7 \sqrt{17}} + \frac{8}{\sqrt{17}} = \frac{55}{7 \sqrt{17}} = \frac{55 \sqrt{17}}{119}$

Step4. Conclusion.

The directional derivative of the given function is $\frac{55 \sqrt{17}}{119}$ .

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91影视

In Exercises 49鈥�54, find the directional derivative of the given function at the specified point P and in the specified direction v. Note that some of the direction vectors are not unit vectors.
$z = \ln (\frac{x}{y^{2}}), P = (7, 1), v = - i - 4 j$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Calculation.

Step4. Conclusion.

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91影视

In Exercises 49鈥�54, find the directional derivative of the given function at the specified point P and in the specified direction v. Note that some of the direction vectors are not unit vectors.z=lnxy2,P=7,1,v=-i-4j

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Calculation.

Step4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 49鈥�54, find the directional derivative of the given function at the specified point P and in the specified direction v. Note that some of the direction vectors are not unit vectors.
$z = \ln (\frac{x}{y^{2}}), P = (7, 1), v = - i - 4 j$