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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.
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.

This Question is Closed

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Find the gradient of f(x,y), evaluate it at P, and then find the dot product with v

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Grad(f)=<1/y, x/y^2> at P: <1, 2>

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so the dot product is: <1, 2>*<1, 1> = 1 + 2 = 1

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oops. I skipped a step

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0<\[<1/\sqrt{2}, 1/\sqrt{2}>\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[<1, 2>*<1/\sqrt{2}, 1/\sqrt{2}>\], where * is a dot product

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so the final is \[1/\sqrt{2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I'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
 5 years ago
Best ResponseYou've already chosen the best response.0ah, sorry. The magnitude of the gradient is the greatest rate of change

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[<1, 2>=\]=\[\sqrt{5}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where that is the magnitude (greatest rate)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0then the unit vector would be \[<1/\sqrt{5}, 2/\sqrt{5}>\]
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