Consider the contour map below. (a) Sketch and label the following contour plots f (2x, y) f(x, 2y) 2 f (x, y) (b) Sketch the graph of the surface z = f(x,y). (c) Use the contour plot to estimate ● ● -10 ● af əx -3) 16 25 af dy (0,-3) 10
Consider the contour map below. (a) Sketch and label the following contour plots f (2x, y) f(x, 2y) 2 f (x, y) (b) Sketch the graph of the surface z = f(x,y). (c) Use the contour plot to estimate ● ● -10 ● af əx -3) 16 25 af dy (0,-3) 10
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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#### Task Details:
**(a) Sketch and label the following contour plots:**
- \( f(2x, y) \)
- \( f(x, 2y) \)
- \( 2f(x, y) \)
**(b) Sketch the graph of the surface \( z = f(x, y) \).**
**(c) Use the contour plot to estimate:**
\[ \left. \frac{\partial f}{\partial x} \right|_{(0, -3)} \]
\[ \left. \frac{\partial f}{\partial y} \right|_{(0, -3)} \]
### Explanation of the Contour Map:
The contour map provided is a graphical representation of a function \( f(x, y) \). It shows levels (contours) of the function where \( f(x, y) \) is constant. Each contour line represents a different value of \( f(x, y) \), with the spacing of the lines indicating the gradient of the function. Closely spaced contour lines indicate a steep gradient, while widely spaced lines indicate a gentle gradient.
#### Analyzing Contour Plots:
1. **Contour Plot \( f(2x, y) \):**
- This represents the function where the x-coordinate is scaled by a factor of 2.
2. **Contour Plot \( f(x, 2y) \):**
- This represents the function where the y-coordinate is scaled by a factor of 2.
3. **Contour Plot \( 2f(x, y) \):**
- This doubles the value of the function \( f(x, y) \), resulting in a steeper gradient for the same contour interval.
#### Sketching the Graph of \( z = f(x, y) \):
To graph \( z = f(x, y) \), we visualize the 3D surface where the height of the surface at any point \((x, y)\) corresponds to the value of \( f(x, y) \). This type of graph helps in understanding the topography of the function and its gradients.
#### Estimating Partial Derivatives:
To estimate the partial derivatives using the contour plot:
\[ \left. \frac{\partial f}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9601c05a-c9e7-4fb3-a6ea-9012c5e99927%2F8f3e391b-f577-4e30-b922-394198e9999d%2Fqb07nrh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Contour Map Analysis
#### Consider the contour map below.
#### Task Details:
**(a) Sketch and label the following contour plots:**
- \( f(2x, y) \)
- \( f(x, 2y) \)
- \( 2f(x, y) \)
**(b) Sketch the graph of the surface \( z = f(x, y) \).**
**(c) Use the contour plot to estimate:**
\[ \left. \frac{\partial f}{\partial x} \right|_{(0, -3)} \]
\[ \left. \frac{\partial f}{\partial y} \right|_{(0, -3)} \]
### Explanation of the Contour Map:
The contour map provided is a graphical representation of a function \( f(x, y) \). It shows levels (contours) of the function where \( f(x, y) \) is constant. Each contour line represents a different value of \( f(x, y) \), with the spacing of the lines indicating the gradient of the function. Closely spaced contour lines indicate a steep gradient, while widely spaced lines indicate a gentle gradient.
#### Analyzing Contour Plots:
1. **Contour Plot \( f(2x, y) \):**
- This represents the function where the x-coordinate is scaled by a factor of 2.
2. **Contour Plot \( f(x, 2y) \):**
- This represents the function where the y-coordinate is scaled by a factor of 2.
3. **Contour Plot \( 2f(x, y) \):**
- This doubles the value of the function \( f(x, y) \), resulting in a steeper gradient for the same contour interval.
#### Sketching the Graph of \( z = f(x, y) \):
To graph \( z = f(x, y) \), we visualize the 3D surface where the height of the surface at any point \((x, y)\) corresponds to the value of \( f(x, y) \). This type of graph helps in understanding the topography of the function and its gradients.
#### Estimating Partial Derivatives:
To estimate the partial derivatives using the contour plot:
\[ \left. \frac{\partial f}{
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