Please Provide correct answer Suppose f(x,y) = x/y, P=(-2,3)and v = 2i - 1j. Fin

Question

Please Provide correct answer

Suppose f(x,y) = x/y, P=(-2,3)and v = 2i - 1j. Find the gradient of f. Note: Your answers should be expressions of x and y; e.g. "3x-4y" Find the gradient of f at the point P. Note: Your answers should be numbers Find the directional derivative of f at P in the direction of v. Note: Your answer should be a number Find the maximum rate of change of f at P. Note: Your answer should be a number Find the (unit) direction vector in which the maximum rate of change occurs at P. Note: your answers should be numbers

Explanation / Answer

f(x,y) = x/y

fx(x,y) = 1/y

fy(x,y) = -x/y^2

A. gradient of f

(1/y) i + (-x/y^2) j

B. gradiesnt at point ( -2,3)

(1/3) i + ( 2/9) j

C. directional derivaitve of f at P in direction of v v = 2i - j

= <1/3 , 2/9 > . < 2 , -1>

= 2/3 - 2/9

= 6/9 - 2/9

= 4/9

D. max rate of change of f at P.

sqrt( 1/ 9 + 4/81) = sqrt( 9/81 + 4/81) = sqrt(13)/ 9

E. unit direction vector is

1/3 / ( sqrt(13)/9 i + 2/9 sqrt(13)/9 j

= 3/sqrt(13) i + 2/sqrt(13) j

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