Find the rate of change of f at p in the direction of the vector u.

Question

(a) Find the gradient of f.

(b) Evaluate the gradient at the point p.

(c) Find the rate of change of f at p in the direction of the vector u.

Answer 1

Step 1:

The function is , point is  and vector is .

(a)

Find gradient of .

The gradient of the function is the vector function of ,

then .

Consider .

Apply partial derivative on each side with respect to .

Apply partial derivative on each side with respect to .

Then the gradient vector of is

.

Step 2:

(b)

Find the gradient vector at a point .

Substitute in the gradient vector.

The gradient vector at a point is .

Step 3:

(c)

The rate of change of the function in the direction of a vector u is .

The vector and a point is .

The rate of change of the function at a point in the direction of a vector u is

Solution :

(a) .

(b) .

(c) .