The image shows a contour plot with concentric circles, and the question asks about the directional derivative $f_u$ at various points in different directions, based on the contour lines.

Directional derivatives describe how a function changes as you move in a certain direction. The key is to observe how the function (which could represent elevation or some other scalar field) behaves as you move in the given direction relative to the contour lines.

1. Interpreting the contour plot:

   • The circles represent levels of the function. Moving outward along the radius increases the function's value, meaning that the function increases as you move away from the center.
   • Contours closer together imply steeper slopes, while widely spaced contours suggest flatter regions.

2. Directional derivative considerations:

   • If you move perpendicular to the contour lines (towards or away from the center), the derivative will be most positive (increasing) or negative (decreasing).
   • If you move along the contour lines, the derivative will be zero since you're not moving across levels of the function.

Let’s break down each part of the question:

(a) At point $(-2, 2)$, in the direction of $\hat{i}$:

• $\hat{i}$ points to the right (positive x-direction).
• Moving right from $(-2, 2)$, you're moving towards higher contour values (closer to the center).
• Therefore, $f_u$ is negative.

(b) At point $(0, -2)$, in the direction of $\hat{j}$:

• $\hat{j}$ points upwards (positive y-direction).
• Moving up from $(0, -2)$ brings you closer to the center, where the contour values increase.
• Therefore, $f_u$ is positive.

(c) At point $(-1, 1)$, in the direction of $-\hat{i} + \hat{j}$:

• This direction is a diagonal, moving left and up.
• Based on the plot, this movement is roughly along a contour line, so the change in the function value will be small.
• Therefore, $f_u$ is approximately zero.

(d) At point $(-1, 1)$, in the direction of $\hat{i} + \hat{j}$:

• This direction is a diagonal, moving right and up.
• Moving in this direction takes you slightly towards higher values, but not significantly.
• Therefore, $f_u$ is negative.

(e) At point $(0, -2)$, in the direction of $\hat{i} + 2\hat{j}$:

• This direction is a steeper diagonal moving right and up.
• Moving in this direction brings you towards the center, where values are higher.
• Therefore, $f_u$ is positive.

(f) At point $(0, -2)$, in the direction of $-\hat{i} - 2\hat{j}$:

• This direction is the opposite diagonal, moving left and down.
• Moving away from the center decreases the function's value.
• Therefore, $f_u$ is negative.

Would you like more details on the directional derivative concept?

Here are 5 questions to expand on this:

1. How is the gradient of a function related to the directional derivative?
2. Why does moving along a contour line give a zero directional derivative?
3. How do you compute the magnitude of the directional derivative from the gradient vector?
4. In what real-world applications do we use contour plots and directional derivatives?
5. How would the answers change if the contour lines were elliptical instead of circular?

Tip: The gradient vector always points in the direction of the steepest ascent, and its magnitude indicates how steep that ascent is.