 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/12/e/iii/solution

(1, 2)

, the gradient is

(4, 2)

. The requested direction must be normalized to the unit vector

(1, 1) / 2

D_{(1, 1) / 2} f (1, 2) = \nabla f (1, 2) \cdot \frac{( 1 , 1 )}{2} = 32 .

Using

(1, 1)

without normalization would instead measure change per unit of a differently scaled path parameter.

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