What is the maximum rate of change of f(x,y) = ln(x^2 + y^2) at the point (-1, 1) and the direction in which it occurs?

I got gradf = (2/x, 2/y)

gradf(-1,1)=(-2,2)

I thought abs(gradf(-1,1)) was maximum gradiant change...Which would be sqrt(8)?

Check your(my) math...gradf=((2x)/(x^2+y^2), (2y)/(x^2+y^2))

1 Answer
Jan 25, 2018

When given two dimensional function f(x,y) and a point (x_0,y_0), the maximum rate of change is:

|gradf(x_0,y_0)|

and the direction is:

gradf(x_0,y_0)

Explanation:

Compute gradf(x,y):

gradf(x,y) = (del(f(x,y)))/(delx)hati+(del(f(x,y)))/(dely)hatj

gradf(x,y) = (2x)/(x^2+y^2)hati+(2y)/(x^2+y^2)hatj

Evaluate at the point (-1,1):

gradf(-1,1) = (2(-1))/((-1)^2+(1)^2)hati+(2(1))/((1)^2+(1)^2)hatj

gradf(-1,1) = -hati+hatj

Compute the magnitude:

|gradf(-1,1)| = |-hati+hatj|

|gradf(-1,1)| = sqrt((-1)^2+1^2)

|gradf(-1,1)| = sqrt2

The maximum rate of change is sqrt2 and its direction is -hati+hatj