Question

(a) What is the rate of change of f(x,y)=3xy+y2 at the point (3,1) in the direction v=−3i−3j? fv= (b) What is the direction of maximum rate of change of f at (3,1)? direction = (Give your answer as a vector.) (c) What is the maximum rate of change? maximum rate of change =

Answer 1

The rate of change of f(x,y)=3xy+y^2 at the point (3,1) in the direction v=-3i-3j can be found by calculating the dot product of the gradient of f and the normalized direction vector v. The direction of maximum rate of change of f at (3,1) is in the direction of the gradient vector ∇f. The maximum rate of change is equal to the magnitude of the gradient of f at (3,1).

Step-by-step explanation:

(a) To find the rate of change of the function f(x,y)=3xy+y^2 at the point (3,1) in the direction v = -3i - 3j, we calculate the dot product of the gradient of f at (3,1) and the normalized direction vector v.

(b) The direction of maximum rate of change of f at (3,1) is in the direction of the gradient vector ∇f.

(c) The maximum rate of change is equal to the magnitude of the gradient of f at (3,1).