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