Find the maximum rate of change of f(x,y)=ln(x^2+y^2) at the point (3, 1) and the direction in which it occurs.

Question

Find the maximum rate of change of f(x,y)=ln(x^2+y^2) at the point (3, 1) and the direction in which it occurs.

Maximum rate of change: ?

Direction (unit vector) in which it occurs: ⟨ ? , ? ⟩

Answer 1

Hi,

f(x,y)=ln(x^2+y^2) you know ( Ln u )' = u'/u

first, find df/dx = 2x / ( x^2 + y^2) and df/dy = 2y / ( x^2 + y^2)

gradient of f(x,y) = (df/dx = 2x / ( x^2 + y^2),df/dy = 2y / ( x^2 + y^2))

gradient of f( 3, 1 ) = < 6/10 , 2/10 > = <3/5, 1/5> = 3/5i + 1/5j

|| gradient of f( 3, 1 )|| = √ (3/5)^2 + (1/5)^2 = √10/25 = √10/5 ≅0.6325

I hope it is useful,

Minoo