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To analyze the contour map of the function \( f \), we can estimate the function values at the specified points by observing the contours passing through or near each point. Each contour line represents a specific value of \( f \), indicated by the labels on the contours. Let's proceed with the estimation.

### Solution:

1. **Estimate \( f(-3, 3) \):**
   - Locate the point \((-3, 3)\) on the map.
   - This point is on or near the contour labeled \( 70 \).
   - So, \( f(-3, 3) \approx 70 \).

2. **Estimate \( f(3, -2) \):**
   - Locate the point \( (3, -2) \) on the map.
   - This point is close to the contour labeled \( 10 \).
   - So, \( f(3, -2) \approx 10 \).

3. **Estimate \( f_x(3, 2) \):**
   - The partial derivative \( f_x(3, 2) \) represents the rate of change of \( f \) with respect to \( x \) at \( (3, 2) \).
   - Locate the point \( (3, 2) \) on the map.
   - Identify the contours near this point to observe how \( f \) changes with small movements in the \( x \)-direction.
   - Moving in the \( x \)-direction (right or left) around \( (3, 2) \), there is a visible change in contour values from 20 to 30, indicating a positive slope.
   - The estimated value for \( f_x(3, 2) \) should be positive, and based on the contour spacing, \( f_x(3, 2) \) is approximately a small positive value (likely around 5, but this is rough).

### Summary of Estimated Values:
- \( f(-3, 3) \approx 70 \)
- \( f(3, -2) \approx 10 \)
- \( f_x(3, 2) \approx \) small positive value (approx. 5)

Would you like further details on any specific part? 

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### Further Exploration Questions:
1. How would we calculate \( f_y(3, 2) \) using the contour map?
2. What does the spacing of contours indicate about the gradient's magnitude?
3. How can we estimate \( f_x \) at a different point, such as \( (-3, -3) \)?
4. Why is the gradient perpendicular to the contour lines?
5. How does the map change if \( f \) represents a temperature field?

### Tip:
To approximate partial derivatives from contour maps, look at the spacing and direction of contours around your point; closer contours indicate steeper gradients.

A contour map for a function f is shown below. Use it to estimate the value of (a) f(-3,3), (b) f(3,-2), (c) fx(3,2).

This solution explains how to estimate the values of f(-3,3), f(3,-2), and the partial derivative fx(3,2) from a contour map of a function f. Key concepts include reading contour maps and understanding partial derivatives, suitable for undergraduate calculus students.

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