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Engineering Calculus Notes 361

Engineering Calculus Notes 361 - 5 Taylor Polynomials The...

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3.7. HIGHER DERIVATIVES 349 where ˜ x = x 0 + t x is between x 0 and x 0 + x , and ˜ y = y 0 + ˜ t y lies between y 0 and y 0 + y . Now, if we divide both sides of the equation above by x y , and take limits, we get the desired result: lim ( x, y ) (0 , 0) y x f x y = lim ( x, y ) (0 , 0) 2 f ∂x∂y x, ˜ y ) = 2 f ∂x∂y ( x 0 ,y 0 ) because (˜ x, ˜ y ) ( x 0 ,y 0 ) as ( x, y ) (0 , 0) and the partial is assumed to be continuous at ( x 0 ,y 0 ). But now it is clear that by reversing the roles of x and y we get, in the same way, lim ( x, y ) (0 , 0) x y f y x = 2 f ∂y∂x ( x 0 ,y 0 ) which, together with our earlier observation that y x f = x y f completes the proof. At first glance, it might seem that a proof for functions of more than two variables might need some work over the one given above. However, when we are looking at the equality of two specific mixed partials, say 2 f ∂x i ∂x j and 2 f ∂x j ∂x i , we are holding all other variables constant, so the proof above goes over verbatim, once we replace x with x i and y with x j (Exercise
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Unformatted text preview: 5 ). Taylor Polynomials The higher derivatives of a function of one variable can be used to construct a polynomial that has high-order contact with the function at a point, and hence is a better local approximation to the function. An analogous construction is possible for functions of several variables, however more work is needed to combine the various partial derivatives of a given order into the appropriate polynomial. A polynomial in several variables consists of monomial terms, each involving powers of the di±erent variables; the degree of the term is the exponent sum : the sum of the exponents of all the variables appearing in that term. 15 Thus, each of the monomial terms 3 x 2 yz 3 , 2 xyz 4 and 5 x 6 has 15 The variables that don’t appear have exponent zero...
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