Question

Find a normal vector to the level curve f(x, y)=c at P given f(x, y)= x²+y², c=25, P(3,4).

Answer 1

The normal vector to the level curve f(x, y) = 25 at the point P(3, 4) is found using the gradient of f, resulting in the vector (6, 8).

Step-by-step explanation:

To find a normal vector to the level curve f(x, y) = c at a given point P, we use the gradient of f. Since f(x, y) = x² + y² and c = 25, the level curve represents a circle of radius 5 centered at the origin.

The gradient of f is given by the vector of partial derivatives (∂f/∂x, ∂f/∂y). Thus, we calculate the gradient of f at the point P(3, 4) to find the normal vector.

1. Calculate partial derivatives: ∂f/∂x = 2x and ∂f/∂y = 2y.
2. Evaluate at P(3, 4): ∂f/∂x = 2(3) = 6 and ∂f/∂y = 2(4) = 8.
3. The gradient at P is (6, 8), which is also the normal vector to the level curve at P.

The normal vector to the level curve f(x, y) = 25 at the point P(3, 4) is (6, 8).