# directional derivative

• Apr 29th 2007, 09:32 AM
littlemu
directional derivative
let f(x,y) = 4x^3y^2.
a)Find a unit vector in the direction in which f decreases most rapidly from the point (2,1).
b) what is the rate of change of f in the direction given in (a).
• Apr 29th 2007, 10:57 AM
Jhevon
Quote:

Originally Posted by littlemu
let f(x,y) = 4x^3y^2.
a)Find a unit vector in the direction in which f decreases most rapidly from the point (2,1).

a function f(x,y) increases most rapidly at a point (x0,y0) in the direction of gradf(x0,y0) and decreases most rapidly in the direction of -gradf(x0,y0)

gradf(x,y) = <fx , fy> = <12(y^2)(x^2) , 8(x^3)y>

u = <-48, -64>/(sqrt(48^2 + 64^2)) = <-48/80 , -64/80> = <-3/5 , -4/5> ....unit vector in the direction of most rapid decrease

Quote:

b) what is the rate of change of f in the direction given in (a).
the rate of change in the direction of u is given by
• Apr 29th 2007, 11:29 AM
littlemu
Hi Jhevon,

I got -80 for my part B answer which is slightly different from your answer 80. I am checking why?

Thank you again.
• Apr 29th 2007, 11:38 AM
Jhevon
Quote:

Originally Posted by littlemu
Hi Jhevon,

I got -80 for my part B answer which is slightly different from your answer 80. I am checking why?

Thank you again.