Final answer:
The directional derivative of f(x, y) = y cos(xy) at the point (0, 1) in the direction indicated by the angle θ = 4 is -cos(4) + sin(4).
Step-by-step explanation:
To find the directional derivative of the function f(x, y) = y cos(xy) at the point (0, 1) in the direction indicated by the angle θ = 4, we can use the formula:
Directional Derivative = ∇f · u
Where ∇f is the gradient vector of f and u is the unit vector in the direction of θ. To find the gradient vector, we need to find the partial derivatives of f with respect to x and y:
∂f/∂x = -y^2 sin(xy)
∂f/∂y = cos(xy) - xy sin(xy)
We can then evaluate the gradient vector at the point (0, 1):
∇f(0, 1) = (-1, 1)
Next, we need to find the unit vector u in the direction of θ:
u = cos(θ)i + sin(θ)j = cos(4)i + sin(4)j
Finally, we can calculate the directional derivative:
Directional Derivative = ∇f(0, 1) · u = (-1, 1) · (cos(4), sin(4))
Directional Derivative = -cos(4) + sin(4)