In the image, the question asks for the estimation of the directional derivative of a function $f(x, y)$ at various points using a contour diagram. Here are the specific cases mentioned:

1. Part (a):

   • Point: $P = (2, 2)$
   • Direction: $\vec{v} = \hat{i}$ (along the positive $x$-axis)
   • Directional derivative: Approximately 2

2. Part (b):

   • Point: $P = (3, 2)$
   • Direction: $\vec{v} = -\hat{i}$ (along the negative $x$-axis)
   • Directional derivative: Approximately −4

3. Part (c):

   • Point: $P = (4, 1)$
   • Direction: $\vec{v} = \frac{\hat{i} + \hat{j}}{\sqrt{2}}$ (this direction is a unit vector at a 45° angle to the positive $x$- and $y$-axes)
   • Directional derivative: Approximately 6

4. Part (d):

   • Point: $P = (4, 0)$
   • Direction: $\vec{v} = -\hat{i}$ (along the negative $x$-axis)
   • Directional derivative: Approximately −5

These directional derivatives are estimated using the slope of the contour lines in the respective directions at the given points.

Would you like further clarification on how directional derivatives are computed or more details on contour lines?

Here are some related questions to explore:

1. How is the gradient vector related to the directional derivative?
2. Why is the direction $\vec{v}$ normalized in part (c)?
3. How would the directional derivative change if the direction was reversed?
4. What is the significance of the sign of the directional derivative?
5. How do contour lines indicate the steepness of a function at a point?

Tip: The magnitude of the directional derivative indicates the rate of change of the function in that direction, while its sign shows whether the function is increasing or decreasing.