Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. 1. At the point (0, 2) in the direction of 7, 2. At the point (-1, 1) in the direction of ? ? (-i+j)/√2, ? (i - 2j)/√5, ? (-i - j)/√2, ? ? 3. At the point (0, -2) in the direction of 4. At the point (-1, 1) in the direction of 5. At the point (-2, 2) in the direction of 7, 6. At the point (1, 0) in the direction of-j, 2.4 1.6 0.8 o -0.8 -1.6 -2.4 12.0 12.0 10.0 10.0 6.0 -0.8 0.8 4.0 1.6 12.0 10.0 10.0 12.0 2.4 (Click graph to enlarge)
Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. 1. At the point (0, 2) in the direction of 7, 2. At the point (-1, 1) in the direction of ? ? (-i+j)/√2, ? (i - 2j)/√5, ? (-i - j)/√2, ? ? 3. At the point (0, -2) in the direction of 4. At the point (-1, 1) in the direction of 5. At the point (-2, 2) in the direction of 7, 6. At the point (1, 0) in the direction of-j, 2.4 1.6 0.8 o -0.8 -1.6 -2.4 12.0 12.0 10.0 10.0 6.0 -0.8 0.8 4.0 1.6 12.0 10.0 10.0 12.0 2.4 (Click graph to enlarge)
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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![**Educational Website Content: Directional Derivatives using a Contour Diagram**
Use the contour diagram of \( f \) to determine if the specified directional derivative is positive, negative, or approximately zero.
1. **At the point** \( (0, 2) \) **in the direction of** \( \vec{j} \):
- [Dropdown: Positive, Negative, Approximately Zero]
2. **At the point** \( (-1, 1) \) **in the direction of** \( (-\vec{i} + \vec{j})/\sqrt{2} \):
- [Dropdown: Positive, Negative, Approximately Zero]
3. **At the point** \( (0, -2) \) **in the direction of** \( (\vec{i} - 2\vec{j})/\sqrt{5} \):
- [Dropdown: Positive, Negative, Approximately Zero]
4. **At the point** \( (-1, 1) \) **in the direction of** \( (-\vec{i} - \vec{j})/\sqrt{2} \):
- [Dropdown: Positive, Negative, Approximately Zero]
5. **At the point** \( (-2, 2) \) **in the direction of** \( \vec{i} \):
- [Dropdown: Positive, Negative, Approximately Zero]
6. **At the point** \( (1, 0) \) **in the direction of** \( -\vec{j} \):
- [Dropdown: Positive, Negative, Approximately Zero]
**Contour Diagram Explanation:**
The contour diagram on the right depicts concentric circles with values ranging from 2.0 to 12.0, increasing uniformly as you move outward. These circles represent constant values of the function \( f \). Darker shades suggest lower values, while lighter shades represent higher values. The center of the diagram appears to be the lowest point, indicating a valley-like shape. Changes in the function \( f \) are observed through the gradient, which is perpendicular to the contour lines.
Click the graph to enlarge for a more detailed view.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2Fabf540c7-ab78-4e55-aea2-fdc2f171d3a0%2F6nmyujg_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Website Content: Directional Derivatives using a Contour Diagram**
Use the contour diagram of \( f \) to determine if the specified directional derivative is positive, negative, or approximately zero.
1. **At the point** \( (0, 2) \) **in the direction of** \( \vec{j} \):
- [Dropdown: Positive, Negative, Approximately Zero]
2. **At the point** \( (-1, 1) \) **in the direction of** \( (-\vec{i} + \vec{j})/\sqrt{2} \):
- [Dropdown: Positive, Negative, Approximately Zero]
3. **At the point** \( (0, -2) \) **in the direction of** \( (\vec{i} - 2\vec{j})/\sqrt{5} \):
- [Dropdown: Positive, Negative, Approximately Zero]
4. **At the point** \( (-1, 1) \) **in the direction of** \( (-\vec{i} - \vec{j})/\sqrt{2} \):
- [Dropdown: Positive, Negative, Approximately Zero]
5. **At the point** \( (-2, 2) \) **in the direction of** \( \vec{i} \):
- [Dropdown: Positive, Negative, Approximately Zero]
6. **At the point** \( (1, 0) \) **in the direction of** \( -\vec{j} \):
- [Dropdown: Positive, Negative, Approximately Zero]
**Contour Diagram Explanation:**
The contour diagram on the right depicts concentric circles with values ranging from 2.0 to 12.0, increasing uniformly as you move outward. These circles represent constant values of the function \( f \). Darker shades suggest lower values, while lighter shades represent higher values. The center of the diagram appears to be the lowest point, indicating a valley-like shape. Changes in the function \( f \) are observed through the gradient, which is perpendicular to the contour lines.
Click the graph to enlarge for a more detailed view.
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