Suppose f(x, y) = x/y, P = (0, 3) and v = -1i + 2j. Find the gradient of f. nabl

Question

Suppose f(x, y) = x/y, P = (0, 3) and v = -1i + 2j. Find the gradient of f. nabla f = i+ j Your answers should be expressions of x and y; e.g. ''3x - 4y'' Find the gradient of f at the point P. (nabla f) (P) = i+ j Your answers should be numbers Find the directional derivative of f at P in the direction of V. Du f = Your answer should be a number Find the maximum rate of change of f at P. Your answer should be a number Find the (unit) direction vector in which the maximum rate of change occurs at P. u = i+ j Your answers should be numbers

Explanation / Answer

First two parts u have solved correctly.

3.Directional derivative =( grad f (P) ) dot * unit vector of v

v (cap) = (-1 i + 2 j) /( 1^2 + 2^2) = (-1 i + 2 j) /sqrt 5

so,

Dir. derivative = (1/3 i ) * ((-1 i + 2 j) /sqrt 5 = -1/(3*sqrt5)

4. the maximum value occurs when the angle between the gradient and the vector at that poitn is zero.

so ,maximum change will occur in direction of gradient a that point.

therefore, answer to part 5 ..is grad f ( P) = 1/3 i

And as it is a direction so answer must be unit vector and that is 'i'

hence, direction vector in which maximum change occurs = i

for part (4),

and maximum chane will be |  grad f . ( P)| = 1/3

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