Question

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.

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

oops. I skipped a step

Answer 5

<\[<-1/\sqrt{2}, -1/\sqrt{2}>\]

Answer 6

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

Answer 7

so the final is \[1/\sqrt{2}\]

Answer 8

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.

Answer 9

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

Answer 10

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

Answer 11

where that is the magnitude (greatest rate)

Answer 12

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