To estimate the value of $f(-3, 3)$ using the contour map:

1. Locate the point $(-3, 3)$ on the map. This corresponds to moving three units left (in the negative $x$-direction) and three units up (in the positive $y$-direction).

2. Observe the contour line that passes through or is closest to $(-3, 3)$. Each contour line represents a constant value of the function $f(x, y)$, and these values are indicated by the numbers on the map.

• The contour lines are labeled with values such as 40, 50, 60, and 70.
• The point $(-3, 3)$ appears to be between the 40 and 50 contour lines, but closer to the 40 line.

Thus, the estimate for $f(-3, 3)$ is likely between 40 and 50, and based on its position, a reasonable estimate would be approximately 48.

Let me know if you need more details or further clarification.

5 Follow-Up Questions:

1. What do the contour lines on this map represent mathematically?
2. How can contour maps be used to estimate gradient values?
3. What is the significance of the spacing between contour lines?
4. How would the value of $f(-3, 3)$ change if the point were on a different contour?
5. Can you explain how this contour map relates to partial derivatives?

Tip:

When interpreting contour maps, remember that closely spaced contour lines indicate steep regions, while widely spaced lines suggest flatter regions of the function.