To answer the questions about the directional derivatives using the contour diagram:

1. At the point (-2, 2) in the direction of $\vec{i}$:

   • Moving in the direction of $\vec{i}$ (positive x-direction), we can determine if the function increases or decreases by observing if the contour values increase or decrease in that direction. Based on the contour diagram, if the function value increases to the right (positive x-direction), the directional derivative would be positive. Here, it appears positive.

2. At the point (0, 2) in the direction of $\vec{j}$:

   • Moving in the direction of $\vec{j}$ (positive y-direction), we should look if the function values increase upwards. If the contour values decrease as we go up, the directional derivative is negative. Here, it appears negative.

3. At the point (-1, 1) in the direction of $\frac{-\vec{i} + \vec{j}}{\sqrt{2}}$:

   • This direction represents a movement diagonally left-up. If the contour values remain approximately constant in that diagonal direction, the directional derivative would be near zero. Here, it appears zero.

4. At the point (0, -2) in the direction of $\frac{\vec{i} - 2\vec{j}}{\sqrt{5}}$:

   • Moving in the direction $\vec{i} - 2\vec{j}$ means a movement to the right and downward. If the contour values increase in that direction, the directional derivative is positive. Here, it appears positive.

5. At the point (1, 0) in the direction of $-\vec{j}$:

   • Moving downward in the negative y-direction. If the contour values decrease as we go down, the directional derivative is negative. Here, it appears negative.

6. At the point (-1, 1) in the direction of $\frac{-\vec{i} - \vec{j}}{\sqrt{2}}$:

   • Moving diagonally down-left. If the contour values decrease as we go in that direction, the directional derivative is negative. Here, it appears negative.

If you would like further details on any specific part, let me know.

Here are 5 questions that expand on this information:

1. How do contour intervals indicate the gradient and its direction?
2. Why is the directional derivative zero in some directions?
3. What role does the unit vector play in finding the directional derivative?
4. How would we compute the directional derivative analytically, given a function?
5. Can the contour diagram also help identify points of maximum or minimum gradient?

Tip: Remember that the gradient vector always points in the direction of the greatest rate of increase and is perpendicular to the contour lines.