Question

Consider a function at the point . At that point, the function has directional derivatives: in the direction (parallel to) , and in the direction (parallel to) . What is the gradient of at the point?

Answer 1

The gradient of a function at a point represents the slope of the function at that point and gives the direction in which the function is increasing the fastest. In this case, we are given the directional derivatives of the function in two different directions. The gradient of the function at the point is a vector that points in the direction of the steepest increase in the function.

Step-by-step explanation:

The gradient of a function at a point represents the slope of the function at that point and gives the direction in which the function is increasing the fastest. In this case, we are given the directional derivatives of the function in two different directions. The gradient of the function at the point is a vector that points in the direction of the steepest increase in the function. To find the gradient, we can use the partial derivatives of the function with respect to each variable.

Let's denote the function as f(x, y) and the point as (a, b). Given that the directional derivative in the direction of the vector i is a and the directional derivative in the direction of the vector j is b, we have:

• Partial derivative of f with respect to x, evaluated at (a, b): a
• Partial derivative of f with respect to y, evaluated at (a, b): b

Therefore, the gradient of the function at the point (a, b) is the vector ∇f = (a, b).