## anonymous 5 years ago f(x,y) = x/y, P=(2,1) and v= -1i -1j. Find the maximum rate of change of f at P. Find the (unit) direction vector in which the maximum rate of change occurs at P.

1. anonymous

Find the gradient of f(x,y), evaluate it at P, and then find the dot product with v

2. anonymous

Grad(f)=<1/y, -x/y^2> at P: <1, -2>

3. anonymous

so the dot product is: <1, -2>*<-1, -1> = -1 + 2 = 1

4. anonymous

oops. I skipped a step

5. anonymous

<$<-1/\sqrt{2}, -1/\sqrt{2}>$

6. anonymous

$<1, -2>*<-1/\sqrt{2}, -1/\sqrt{2}>$, where * is a dot product

7. anonymous

so the final is $1/\sqrt{2}$

8. anonymous

I'm not asking for a directional derivative of f at P in the direction of V. I'm asking maximum rate of changing f at P and unit vector in which the maximum rate of changing occurs at P.

9. anonymous

ah, sorry. The magnitude of the gradient is the greatest rate of change

10. anonymous

$<1, -2>=$=$\sqrt{5}$

11. anonymous

where that is the magnitude (greatest rate)

12. anonymous

then the unit vector would be $<1/\sqrt{5}, -2/\sqrt{5}>$