Question

Suppose f(x,y)=x/y, P=(−1,4) and v=4i−1j.A. Find the gradient of f.∇f=i+ jNote: Your answers should be expressions of x and y; e.g. "3x - 4y"B. Find the gradient of f at the point P.(∇f)(P)=i+ jNote: Your answers should be numbersC. Find the directional derivative of f at P in the direction of v.Duf=Note: Your answer should be a numberD. Find the maximum rate of change of f at P.Note: Your answer should be a numberE. Find the (unit) direction vector in which the maximum rate of change occurs at P.u=i+ j

Answer 1

Answer:

1) gradient of f = (1/y)i+(-x/y^2)j

2) rate of change of f at P in direction of v = i - 1/16 j

3) maximum rate of change at P= (√17)/16

4) unit vector of maximum rate of change = 4/√17 i + 1/√17 j

Step-by-step explanation:

The explanation can be found in the attached picture

Note:

Since the gradient vector represents the direction of maximum change:

3) The maximum rate of change is the modulus of the gradient vector

4) The unit direction vector of maximum rate of change is the unit gradient vector.