The problem asks to estimate the directional derivative of the function f(x,y) f(x, y) f(x,y) at the point (3,3) (3, 3) (3,3) in the direction of the unit vector u=−i−j2 \mathbf{u} = \frac{-\mathbf{i} - \mathbf{j}}{\sqrt{2}} u=2−i−j.
The gradient of f f f at (3,3) (3, 3) (3,3), denoted as ∇f(3,3) \nabla f(3, 3) ∇f(3,3), is needed to compute the directional derivative. The gradient is a vector of partial derivatives (∂f∂x,∂f∂y) \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) (∂x∂f,∂y∂f).
From the contour plot, estimate the partial derivatives:
Combine the partial derivatives to form the gradient vector ∇f(3,3) \nabla f(3, 3) ∇f(3,3).
Use the formula for the directional derivative: Duf(3,3)=∇f(3,3)⋅u D_{\mathbf{u}} f(3, 3) = \nabla f(3, 3) \cdot \mathbf{u} Duf(3,3)=∇f(3,3)⋅u
Duf(3,3)≈0 D_{\mathbf{u}} f(3, 3) \approx 0 Duf(3,3)≈0
(Note: The exact values of the partial derivatives are not provided in the image, so the final answer is based on the given hint and the assumption that the directional derivative is approximately zero.)
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