Given the function:f(x,y)=3-(x/3)-(y/2)Solve the questions below, Thank you **Problem Statement:**(b) Find $ D_{\mathbf{u}} f(3, 2) $, where $ \mathbf{u} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j} $, using each given value of $ \theta $.**Given Values of $ \theta $:**(i) $ \theta = \frac{\pi}{4} $\[ \text{(Box to display result)} \](ii) $ \theta = \frac{2\pi}{3} $\[ \text{(Box to display result)} \](iii) $ \theta = \frac{4\pi}{3} $\[ \text{(Box to display result)} \](iv) $ \theta = -\frac{\pi}{6} $\[ \text{(Box to display result)} \]**Explanation:**For each given angle $ \theta $, determine the directional derivative $ D_{\mathbf{u}} f(3, 2) $ by substituting $ \theta $ into the unit vector formula:\[ \mathbf{u} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j} \]Calculate the cosine and sine components of $ \mathbf{u} $ for each case and find the directional derivative using these values. ### Directional Derivative and Gradient Calculation**(c)** Calculate $ D_u f(3, 2) $, where $ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} $, using each given vector $ \mathbf{v} $.- **(i)** $ \mathbf{v} = \mathbf{i} + \mathbf{j} $ - [Answer Box]- **(ii)** $ \mathbf{v} = -3\mathbf{i} - 4\mathbf{j} $ - [Answer Box]- **(iii)** $ \mathbf{v} $ is the vector from (1, 2) to (-2, 6) - [Answer Box]- **(iv)** $ \mathbf{v} $ is the vector from (3, 2) to (4, 5) - [Answer Box]**(d)** Find $ \nabla f(x, y) $.- $ \nabla f(x, y) = $ - [Answer Box]**(e)** Find the maximum value of the directional derivative at (3, 2).- [Answer Box]**(f)** Identify a unit vector $ \mathbf{u} $ orthogonal to $ \nabla f(3, 2) $ and compute $ D_u f(3, 2) $.- $ D_u f(3, 2) = $ - [Answer Box]This section requires understanding of vector calculus, focusing on the computation of directional derivatives using various vectors.

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Find D„f(3, 2), where u = cos(0)i + sin(0)j, using each given value of 0. (i) 4 (ii) 0 3 (iii) 0 3. (iv) Ө — 6.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

Given the function:

f(x,y)=3-(x/3)-(y/2)

Solve the questions below,

Thank you

**Problem Statement:**

(b) Find $ D_{\mathbf{u}} f(3, 2) $, where $ \mathbf{u} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j} $, using each given value of $ \theta $.

**Given Values of $ \theta $:**

(i) $ \theta = \frac{\pi}{4} $

\[ \text{(Box to display result)} \]

(ii) $ \theta = \frac{2\pi}{3} $

\[ \text{(Box to display result)} \]

(iii) $ \theta = \frac{4\pi}{3} $

\[ \text{(Box to display result)} \]

(iv) $ \theta = -\frac{\pi}{6} $

\[ \text{(Box to display result)} \]

**Explanation:**

For each given angle $ \theta $, determine the directional derivative $ D_{\mathbf{u}} f(3, 2) $ by substituting $ \theta $ into the unit vector formula:
\[ \mathbf{u} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j} \]

Calculate the cosine and sine components of $ \mathbf{u} $ for each case and find the directional derivative using these values.

### Directional Derivative and Gradient Calculation

**(c)** Calculate $ D_u f(3, 2) $, where $ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} $, using each given vector $ \mathbf{v} $.

- **(i)** $ \mathbf{v} = \mathbf{i} + \mathbf{j} $
- [Answer Box]

- **(ii)** $ \mathbf{v} = -3\mathbf{i} - 4\mathbf{j} $
- [Answer Box]

- **(iii)** $ \mathbf{v} $ is the vector from (1, 2) to (-2, 6)
- [Answer Box]

- **(iv)** $ \mathbf{v} $ is the vector from (3, 2) to (4, 5)
- [Answer Box]

**(d)** Find $ \nabla f(x, y) $.

- $ \nabla f(x, y) = $
- [Answer Box]

**(e)** Find the maximum value of the directional derivative at (3, 2).

- [Answer Box]

**(f)** Identify a unit vector $ \mathbf{u} $ orthogonal to $ \nabla f(3, 2) $ and compute $ D_u f(3, 2) $.

- $ D_u f(3, 2) = $
- [Answer Box]

This section requires understanding of vector calculus, focusing on the computation of directional derivatives using various vectors.

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