Questions: Directional derivative: Problem 4 (1 point) The figure below shows some level curves of a differentiable function f(x, y). Based only on the information in the figure, estimate the directional derivative: fu(3,3) where u=(-i-j) / sqrt(2) fu(3,3) ≈ 0 (Hint to obtain the best estimate, find the average rate of change over equal increments in the u and -u directions, then find the mean of these two estimates.)

Solution Steps

The problem asks to estimate the directional derivative of the function \( f(x, y) \) at the point \( (3, 3) \) in the direction of the unit vector \( \mathbf{u} = \frac{-\mathbf{i} - \mathbf{j}}{\sqrt{2}} \).

The gradient of \( f \) at \( (3, 3) \), denoted as \( \nabla f(3, 3) \), is needed to compute the directional derivative. The gradient is a vector of partial derivatives \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).

From the contour plot, estimate the partial derivatives:

• \( \frac{\partial f}{\partial x} \) at \( (3, 3) \)
• \( \frac{\partial f}{\partial y} \) at \( (3, 3) \)

Combine the partial derivatives to form the gradient vector \( \nabla f(3, 3) \).

Use the formula for the directional derivative: \[ D_{\mathbf{u}} f(3, 3) = \nabla f(3, 3) \cdot \mathbf{u} \]

Final Answer