Question

Consider the following function f(x,y)=(x−2)² −(y−2)² - Gradient vector of f at the point (0,0) is < - Directional derivative of f at the point (0,0) in the northwest direction is

Answer 1

The gradient vector of the function f(x,y)=(x-2)² −(y-2)² at point (0,0) is −4​4. To find the directional derivative in the northwest direction, calculate the dot product of the gradient vector at (0,0) with the northwest unit vector.

Step-by-step explanation:

The student has presented a problem involving multivariable calculus, specifically finding the gradient vector and the directional derivative of the function f(x,y)=(x−2)² −(y−2)² at the point (0,0).

To find the gradient vector, we need to take the partial derivatives of the function with respect to x and y. For the given function, the partial derivative with respect to x is 2(x-2), and with respect to y, it is −2(y-2). At the point (0,0), these values are −4 and 4, respectively. Thus, the gradient vector at the point (0,0) is −4​4.

For the directional derivative in the northwest direction, we need to first obtain a unit vector in that direction. The northwest direction can be represented by the vector −1,1, and the corresponding unit vector is −− −−. The directional derivative is then the dot product of the gradient vector and this unit vector, which is −− −− at point (0,0).