Engineering Calculus Notes 362

# Engineering Calculus Notes 362 - variables the partial...

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350 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION exponent sum 6. We group the terms of a polynomial according to their exponent sums: the group with exponent sum k on its own is a homogeneous function of degree k . This means that inputs scale via the k th power of the scalar. We already saw that homogeneity of degree one is exhibited by linear functions: ( c −→ x ) = cℓ ( −→ x ) . The degree k analogue is ϕ ( c −→ x ) = c k ϕ ( −→ x ) ; for example, ϕ ( x,y,z ) = 3 x 2 yz 3 + 2 xyz 4 + 5 x 6 satisFes ϕ ( cx,cy,cz ) = 3( cx ) 2 ( cy )( cz ) 3 + 2( cx )( cy )( cz ) 4 + 5( cx ) 6 = c 6 (3 x 2 yz 3 + 2 xyz 4 + 5 x 6 ) so this function is homogeneous of degree 6. In general, it is easy to see that a polynomial (in any number of variables) is homogeneous precisely if the exponent sum of each term appearing in it is the same, and this sum equals the degree of homogeneity. ±or functions of one variable, the k th derivative determines the term of degree k in the Taylor polynomial, and similarly for a function of several
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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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