 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/13/b/solution

The specified unit vector is

u = (cos (π /3), sin (π /3)) = (1/2, 3 /2)

; the positive

y

component selects that direction. The gradient is

\nabla f = e^{x y} (y, x)

(0, - 2)

at the point. The directional derivative is therefore

D_{u} f = \nabla f \cdot u = - 3 .

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