To estimate the partial derivative $f_y(3, 29)$ from the contour map, we will consider how the function changes as we move vertically near the point $(3, 29)$.

Here's a step-by-step breakdown of how to approach this:

1. Locate the point $(3, 29)$ on the graph:

   • The $x$-coordinate is 3, and the $y$-coordinate is 29.
   • On the graph, this point lies between the contour lines for $f = 270$ (at $y = 30$) and $f = 360$ (above, around $y = 40$).

2. Determine the approximate rate of change in the vertical direction:

   • Moving vertically from $y = 30$ to $y = 40$, the function changes from 270 to 360.
   • The change in $f$ is approximately $360 - 270 = 90$, over a vertical distance of $40 - 30 = 10$.

3. Estimate the partial derivative $f_y(3, 29)$:

   • The partial derivative $f_y$ represents the rate of change of the function with respect to $y$.
   • The change of 90 in the function over a distance of 10 in $y$ gives a rate of $\frac{90}{10} = 9$.

Thus, $f_y(3, 29) \approx 9$, not 2 as suggested in the initial answer.

Would you like more details on how contour maps are used for partial derivatives?

Here are 5 questions related to this problem:

1. How is the concept of partial derivatives related to contour maps?
2. What do the contour lines represent in terms of the function values?
3. How can we estimate the partial derivative $f_x(3, 29)$ using this contour map?
4. What does it mean for the function when the contour lines are closer together?
5. Why do we need both $f_x$ and $f_y$ to fully understand the behavior of the function at a point?

Tip: When estimating partial derivatives from contour maps, always check how quickly the function value changes in the desired direction (horizontal or vertical).