Match each contour map with its graph
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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Transcribed Image Text:### Match Each Contour Map with Its Graph
Below are several contour maps alongside 3D surface plots. Your task is to match each contour map with its corresponding 3D surface plot.
#### Contour Maps:
There are four contour maps, each containing lines that represent points of equal value (contours) on a given surface. The maps are labeled and displayed on the left side:
1. **Contour Map 1**:
- Features curved, vertical lines that indicate a series of concentric parabolas, open towards the bottom.
2. **Contour Map 2**:
- Consists of straight, parallel, horizontal lines, indicating a plane surface.
3. **Contour Map 3**:
- Displays concentric circles which suggest a surface with radial symmetry around a central peak or valley.
4. **Contour Map 4**:
- Shows straight, parallel, vertical lines, suggesting another planar but angled surface.
#### 3D Surface Plots:
Each 3D surface plot represents a three-dimensional graph of a function. They are labeled a, b, and c on the right side:
1. **3D Surface Plot a**:
- The graph depicts a plane surface inclined diagonally within the 3D space. The values increase linearly along both the x and y axes.
2. **3D Surface Plot b**:
- A saddle-shaped surface that goes from low to high values along one axis and from high to low values along the other, creating a distinct curve known as a hyperbolic paraboloid.
3. **3D Surface Plot c**:
- Shows a surface with a parabolic shape opening upwards. This represents a parabolic cylinder, where the graph forms a 'bowl'.
### Task:
Use the dropdowns next to each contour map to select the corresponding 3D surface plot based on the given features and descriptions.
Correct Matching:
1. **Contour Map 1** → 3D Surface Plot **b** (saddle-shaped surface)
2. **Contour Map 2** → 3D Surface Plot **a** (inclined plane surface)
3. **Contour Map 3** → 3D Surface Plot **c** (parabolic bowl-shaped surface)
4. **Contour Map 4** (Not shown in graphs provided)
Transcribed Image Text:# Visualization of Mathematical Functions
## 1. Contour Plots and Surface Plots
In this section, we provide an educational overview of contour plots and 3D surface plots for three different mathematical functions. Understanding these visualizations can significantly aid in comprehending the behavior of functions in various dimensions.
### a. Contour Plot 1 and 3D Surface Plot
**Contour Plot:**
This is a contour plot representing the function \(z = x^2 + y^2\). The contour lines are equidistant circles centered at the origin, signifying that the function value increases as we move away from the origin. The labels on the contour lines indicate the function value \(z\).
- **X-axis:** ranges from -2 to 2.
- **Y-axis:** ranges from -2 to 2.
- **Contour levels:** -2, -1, 0, 1, 2.
**3D Surface Plot:**
This 3D plot shows the function \(z = x^2 + y^2\). The graph displays a parabolic surface that opens upwards.
- **X-axis (x):** ranges from -1.5 to 1.5.
- **Y-axis (y):** ranges from -1.5 to 1.5.
- **Z-axis (z):** ranges from 0 to 4.
- The surface is color-coded, indicating the increasing function values as moving away from the origin along the z-direction.
**Figure Label:** (a)
### b. Contour Plot 2 and 3D Surface Plot
**Contour Plot:**
This contour plot represents the function \(z = x^2 - y^2\). The contour lines are hyperbolas centered along the x and y axes, signifying that the function value changes as moving along these axes.
- **X-axis:** ranges from -2 to 2.
- **Y-axis:** ranges from -2 to 2.
- **Contour levels:** -2, -1, 0, 1, 2.
**3D Surface Plot:**
This 3D plot illustrates the function \(z = x^2 - y^2\). The graph displays a saddle surface where the function value decreases in one direction and increases in the perpendicular direction.
- **X-axis (x):** ranges from -1.5 to 1.5.
- **Y-axis (y):
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