Question

Calculate D u f ( − 4 , − 4 , 4 ) in the direction of → v = − 3 → i + → j − → k for the function f ( x , y , z ) = 3 x² + 3 x y + 2 y² − x + y z + 5 z² − 5 x z . Round your answer to four decimal places.

Answer 1

To calculate Duf at the point (-4, -4, 4) in the direction of v = -3i + j - k for the function f(x, y, z) = 3x² + 3xy + 2y² - x + yz + 5z² - 5xz, we first find the gradient vector ∇f at that point. The gradient vector is (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k. Evaluating the partial derivatives at (-4, -4, 4), we compute the dot product of the gradient vector and the direction vector v to find the directional derivative Duf as 79.0000.

Step-by-step explanation:

To calculate Duf at the point (-4, -4, 4) in the direction of v = -3i + j - k for the function f(x, y, z) = 3x² + 3xy + 2y² - x + yz + 5z² - 5xz, we first need to find the gradient vector ∇f at that point. The gradient vector is given by ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k. Differentiating f with respect to each variable, we get ∂f/∂x = 6x + 3y - 5z - 1, ∂f/∂y = 3x + 4y + z, and ∂f/∂z = y + 10z - 5x. Evaluating these partial derivatives at (-4, -4, 4), we find ∂f/∂x = -23, ∂f/∂y = 6, and ∂f/∂z = -4.

The directional derivative Duf in the direction of v is given by Duf = ∇f · v, where · represents the dot product. Substituting the values, we have Duf = (-23)(-3) + 6(1) + (-4)(-1) = 69 + 6 + 4 = 79. Rounded to four decimal places, the directional derivative Duf is 79.0000.