anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Find the gradient of f(x,y), evaluate it at P, and then find the dot product with v
anonymous
  • anonymous
Grad(f)=<1/y, -x/y^2> at P: <1, -2>
anonymous
  • anonymous
so the dot product is: <1, -2>*<-1, -1> = -1 + 2 = 1

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anonymous
  • anonymous
oops. I skipped a step
anonymous
  • anonymous
<\[<-1/\sqrt{2}, -1/\sqrt{2}>\]
anonymous
  • anonymous
\[<1, -2>*<-1/\sqrt{2}, -1/\sqrt{2}>\], where * is a dot product
anonymous
  • anonymous
so the final is \[1/\sqrt{2}\]
anonymous
  • 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.
anonymous
  • anonymous
ah, sorry. The magnitude of the gradient is the greatest rate of change
anonymous
  • anonymous
\[<1, -2>=\]=\[\sqrt{5}\]
anonymous
  • anonymous
where that is the magnitude (greatest rate)
anonymous
  • anonymous
then the unit vector would be \[<1/\sqrt{5}, -2/\sqrt{5}>\]

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