Final answer:
The directional derivative of the function f(x, y) = 7eˣ sin(y) at the point (0, π/3) in the direction of the vector v = (-5, 12) is 0.282.
Step-by-step explanation:
The directional derivative of a function represents the rate at which the function changes along a given direction. To find the directional derivative of the function f(x, y) = 7eˣ sin(y) at the point (0, π/3) in the direction of the vector v = (-5, 12), we can use the formula:
directional derivative = ∇f · u = ∂f/∂x * u₁ + ∂f/∂y * u₂
where ∇f is the gradient of f and u is the unit vector in the direction of v.
To calculate the gradient of f, we need to find ∂f/∂x and ∂f/∂y:
∂f/∂x = 7eˣ sin(y)
∂f/∂y = 7eˣ cos(y)
Next, we normalize the vector v to find the unit vector u:
u = v/|v| = (-5/13, 12/13)
Finally, we substitute the values into the formula to find the directional derivative:
directional derivative = ∂f/∂x * u₁ + ∂f/∂y * u₂ = (7e⁰ sin(π/3)) * (-5/13) + (7e⁰ cos(π/3)) * (12/13) = 0.282