# Thread: Average and instantaneous rate of change of f.

1. ## Average and instantaneous rate of change of f.

Another one I'm currently stuck on:

"Let f(x,y)=x^2+ln(y). Find the average rate of change of f as you go from (3,1) to (1,2). Find the instantaneous rate of change of f as you leave the point (3,1) heading toward (1,2)."

What exactly am I supposed to do in order to solve this problem?

2. I believe the average rate of change is given by:

$\dfrac{f(1,2) - f(3,1)}{\sqrt{(1-3)^2 + (2-1)^2}}$

I believe the instantaneous rate of change is given by the directional derivative. If $\mathbf u = <1-3,2-1> = <-2,1> = $, then the directional derivative is given by:

$\mathbf{D_u}f(3,1) = f_x(3,1)u_1 + f_y(3,1)u_2 = f_x(3,1)(-2) + f_y(3,1)(1) = -12 + 10 = -2$