Question

If f(x,y)=−2x^2+4y^2, find the value of the directional derivative at the point (3, 4) in the direction given by the angle θ. More specifically, find the directional derivative of f at the point (3, 4) in the direction of the unit vector determined by the angle θ in polar coordinates. Please show work! I’ve tried this multiple times and not sure where I’m going wrong.

Answer 1

To find the directional derivative of f(x,y) = -2x^2 + 4y^2 at the point (3, 4) in the direction of the unit vector determined by angle θ, we need to find the gradient of f and then take the dot product with the unit vector.

Step-by-step explanation:

To find the directional derivative of f(x,y) = -2x^2 + 4y^2 at the point (3, 4) in the direction of the unit vector determined by angle θ, we need to find the gradient of f and then take the dot product with the unit vector.

Step 1: Find the gradient of f.
∇f = (∂f/∂x, ∂f/∂y) = (-4x, 8y)

Step 2: Convert the angle θ to rectangular coordinates.
x = cos(θ)
y = sin(θ)

Step 3: Evaluate the gradient at the point (3, 4).
∇f(3, 4) = (-12, 32)

Step 4: Compute the dot product between the gradient and the unit vector.
Directional derivative = ∇f(3, 4) · (cos(θ), sin(θ)) = (-12, 32) · (cos(θ), sin(θ)) = -12cos(θ) + 32sin(θ)