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# goldch4 - Contents IV Approximation of Rational Functions...

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Contents IV Approximation of Rational Functions 1 IV.A Constant Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 IV.B Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 IV.C Bounding (Rational) Functions on Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 IV.D The Extreme Value Theorem (EVT) for Rational Functions . . . . . . . . . . . . . . . . . . 7

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IV Approximation of Rational Functions IV.1 IV Approximation of Rational Functions Suppose f ( x ) is a rational function. An approximating function, A ( x ), is another function — usually much simpler than f — whose values are “approximately” the same as the values of f . The difference f ( x ) - A ( x ) = E ( x ) is the error , and measures how accurate or inaccurate our approximation is. Thus f ( x ) = A ( x ) + E ( x ) (approximation) (error) The two simplest functions are constant functions and linear functions and if A ( x ) is constant or linear we talk about a constant approximation , or a linear approximation for f . When we choose our approximation we want the error to be small. Now we know what it means for a number to be small, but our error, E ( x ), is a function and it is less clear what it means for a function to be small. In fact there are different notions of “smallness” for functions and depending on which one uses, one gets different kinds of approximations. In this section we shall fix a domain ( f ) and we shall say our error is small if E ( x ) is very small for all x very close to a . Thus our approximation will be good if A ( x ) is very close to f ( x ) for all x very close to a . Accordingly, we want to find an approximation for a rational function f ( x ) on a “small” interval [ a - h, a + h ] dom( f ) by some simpler function A ( x ). Then we want to determine the accuracy of the approximation by bounding the absolute value of the error | E ( x ) | = | f ( x ) - A ( x ) | . In order to use the approximation we need to have an idea of how big the error is. (If the error is 25 times the value of the function, the approximation is clearly no good!) The MVT and EMVT allow us to get a bound on the error. IV.A Constant Approximation The best constant approximation of f is A 0 ( x ) = f ( a ). f(a) a x graph of f(x) graph of A 0 (x) (a, f(a)) E 0 (x) A 0 (x) f(x) We write the error term as E 0 ( x ) = f ( x ) - A 0 ( x ) = f ( x ) - f ( a ) . The MVT tells us that f 0 ( z ) = f ( x ) - f ( a ) x - a for some z between a and x . Therefore f 0 ( z )( x - a ) = f ( x ) - f ( a ) = E 0 ( x ).
IV.A Constant Approximation IV.2 So to bound the error on some interval [ a - h, a + h ] around a , we need to bound | E 0 ( x ) | = | f 0 ( z ) || x - a | on [ a - h, a + h ]. Observe that | x - a | ≤ h and so | E 0 ( x ) | ≤ max z | f 0 ( z ) | · h . We can thus bound the error | E 0 ( x ) | as soon as we can bound | f 0 ( z ) | on [ a - h, a + h ]. Example . Let f ( x ) = x 3 - 4 x + 5 and a = 2. Find A 0 ( x ) and a number h so that | E 0 ( x ) | ≤ 1 20 = δ when 2 - h x 2 + h . Solution : A 0 ( x ) = f (2) = 5. How to find h when given a and δ ? 1. Choose h 0 so that [ a - h 0 , a + h 0 ] dom( f ). ( h 0 is an initial guess for h ) 2. Find M so that | f 0 ( z ) | ≤ M on [ a - h 0 , a + h 0 ]. 3. Choose h in (0 , h 0 ] so that M · h δ . (Remember we want h > 0, thus the “(” in (0 , h 0 ].) If we succeed with 1.–3., we get: If x [ a - h, a + h ], then | E 0 ( x ) | = | f 0 ( z ) || x - a | ≤ Mh δ . Summarized as a diagram we have the implications a - h x a + h ?

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goldch4 - Contents IV Approximation of Rational Functions...

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