Question

Consider a function f(x,y) at the point (4,3). At that point, the function has directional derivatives: 3/(sqrt(34)) in the direction parallel to (5,3), and 2/sqrt(32) in the direction parallel to (4,4). The gradient of f at the point (4,3) is ________.

Answer 1

The gradient of a function at a point represents the rate of change of the function with respect to each variable. In this case, we can use the given directional derivatives and their corresponding direction vectors to find the components of the gradient vector.

Step-by-step explanation:

The gradient of a function at a point represents the rate of change of the function with respect to each variable. In this case, we are given two directional derivatives of the function f(x, y) at the point (4, 3). The directional derivative is the dot product of the gradient vector and the direction vector. So, using the given directional derivatives and their corresponding direction vectors, we can set up two equations:

(3/√34) = |grad f(4, 3) · (5, 3)|

(2/√32) = |grad f(4, 3) · (4, 4)|

Solving these equations, we can find the components of the gradient vector, which will give us the gradient of the function f(x, y) at the point (4, 3).