Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 19 / b

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- Solution b

Solution

The partial derivatives are

f_{x} = e^{x y} (1 + x y)

and

f_{y} = x^{2} e^{x y}

. At

(1, 1)

they have values

2 e

and

e

. The first-order Multivariate Taylor polynomial therefore gives the change

δ f = 2 e δ x + e δy + O (δ x^{2} + δ y^{2}) .

Thus the requested linear approximation is

e (2 δ x + δy)

; it is the change from

f (1, 1) = e

, not the new function value itself.

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