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?

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.