High School

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Find the directional derivative of the function [tex]f(x, y) = 8xy[/tex] at [tex]P(2, 2)[/tex] in the direction from [tex]P[/tex] to [tex]Q(-1, 6)[/tex].

A. 24
B. -24
C. 16
D. -16

Answer :

Final answer:

The student needs to calculate the directional derivative of f(x, y) = 8xy at point P(2, 2) towards Q(-1, 6). The calculation involves finding the unit vector in the desired direction and the gradient of the function. The final directional derivative comes out to 16/5, which is not a provided option, indicating potential error in the options or calculation. None of the options are valid.

Explanation:

To find the directional derivative of the function f(x, y) = 8xy at P(2, 2) in the direction from P to Q(-1, 6), you must first determine the unit vector in the direction from P to Q. The vector from P to Q is given by the difference Q - P = (-1, 6) - (2, 2) = (-3, 4). The magnitude of this vector is √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5. Therefore, the unit vector in the direction from P to Q is (-3/5, 4/5).

The gradient of f is given by the partial derivatives with respect to x and y, ∇f = (8y, 8x). Evaluating the gradient at P(2, 2) gives us ∇f(P) = (16, 16). The directional derivative at P in the direction of the unit vector is the dot product of this gradient and the unit vector: ∇f(P) ⋅ unit vector = (16, 16) ⋅ (-3/5, 4/5) = 16(-3/5) + 16(4/5) = -48/5 + 64/5 = 16/5, which is not an option given in the question, suggesting there may be an error.

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