https://r2.erweima.ai/mathbot/image/1726423517527oj2oyeid.jpg

The following contour plots illustrate a function f(x,y) and its partial derivatives fx(x,y) and fy(x,y). Match each contour plot to the correct option.

To match each contour plot (A, B, and C) with the correct function (either \( f(x, y) \), \( f_x(x, y) \), or \( f_y(x, y) \)), we need to analyze the behavior of the contour lines for the function and its partial derivatives.

### Plot A
- The contour lines in Plot A seem symmetric and focused around both the \( x \)- and \( y \)-axes.
- This pattern is typical of the **original function \( f(x, y) \)** itself because the contours represent the values of the function at various levels.
- The symmetry suggests this plot is the primary function, as it is smooth and does not indicate directional changes typical of derivatives.

### Plot B
- The contour lines in Plot B show variation primarily along the \( x \)-axis. The changes are more rapid in the \( x \)-direction compared to the \( y \)-direction.
- This behavior suggests it represents the **partial derivative \( f_x(x, y) \)**, which would highlight how the function changes as \( x \) changes.

### Plot C
- In Plot C, the contour lines are tightly packed around the \( y \)-axis, indicating steeper changes along this axis.
- This suggests that this plot represents the **partial derivative \( f_y(x, y) \)**, showing how the function changes with respect to \( y \).

### Final Answer:
- **Plot A**: \( f(x, y) \) (the original function)
- **Plot B**: \( f_x(x, y) \) (partial derivative with respect to \( x \))
- **Plot C**: \( f_y(x, y) \) (partial derivative with respect to \( y \))

Would you like a more detailed explanation of how to identify such patterns in contour plots?

Here are 5 related questions you might find helpful:
1. How do contour plots visually represent the behavior of a function?
2. What differences can we expect between contour plots of a function and its partial derivatives?
3. Can we use contour plots to find critical points of a function?
4. How does the density of contour lines affect our understanding of a function’s gradient?
5. What is the significance of symmetry in contour plots?

**Tip**: When analyzing contour plots, pay attention to the direction and spacing of the lines—closer lines indicate a steeper gradient!

This problem involves matching contour plots with a function f(x, y) and its partial derivatives fx(x, y) and fy(x, y). Learn how to differentiate between the original function and its derivatives using contour lines, gradient patterns, and symmetry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation