Unformatted text preview: variables the partial derivatives of order k determine the part of the Taylor polynomial which is homogeneous of degree k . Here, we will concentrate on degree two. ±or a C 2 function f ( x ) of one variable, the Taylor polynomial of degree two T 2 f ( −→ a ) −→ x := f ( a ) + f ′ ( a )( x − a ) + 1 2 f ′′ ( a )( x − a ) 2 has contact of order two with f ( x ) at x = a , and hence is a closer approximation to f ( x ) (for x near a ) than the linearization (or degree one Taylor polynomial). To obtain the analogous polynomial for a function f of two or three variables, given −→ a and a nearby point −→ x , we consider the restriction of f to the line segment from −→ a to −→ x , parametrized as g ( t ) = f ( −→ a + t △ −→ x ) , ≤ t ≤ 1...
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 Calculus

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