Final answer:
To find the directional derivative of the function f(x,y) = 2xy at the point (12,1), we use the formula: Directional Derivative = (∇f) · d. The gradient vector of f at the point (12,1) is (2, 24). We need to find the value of θ to calculate the directional derivative.
Step-by-step explanation:
To find the directional derivative of the function f(x,y) = 2xy at the point (12,1), we use the formula:
Directional Derivative = (∇f) · d
where ∇f is the gradient vector of f, and d is the direction vector.
The gradient vector ∇f = (∂f/∂x, ∂f/∂y)
= (2y, 2x)
At the point (12,1), the gradient vector becomes ∇f = (2, 24).
The direction vector d = (cosθ, sinθ)
We need to find the value of θ.
The directional derivative of the function f(x,y) = 2xy at the point (12,1) can be calculated as:
Directional Derivative = (∇f) · d
= (2, 24) · (cosθ, sinθ)
= 2cosθ + 24sinθ