Lagrange Error Bound.docx - The Lagrange Error Bound In our...

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The Lagrange Error Bound In our previous studies, we have approximated curves with various polynomial functions. We noticed that as the degree of the functions increased, so did the _________________ in approximating the curves. The Lagrange error bound (today’s lesson, yippee!!) focuses on determining how accurate these Taylor Polynomial approximations are. Let us consider the form of any function, f ( x ) , in terms of its n th degree Taylor Polynomial, T n ( x ) , and the function that represents the error (or remainder) in using the that Taylor Polynomial, R n ( x ) . We can then express the function as ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ f ( x ) = ¿ ¿ . It follows, then, that ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ R n ( x ) = ¿ ¿ . So, this theorem states that the error associated with using the n th degree Taylor Polynomial centered about x = a to approximate f ( x ) near a is at most the value of the __________ ____________ in the Taylor series. The Lagrange Error Bound (Remainder Estimation Theorem) Suppose a function f ( x ) is differentiable on an interval containing a number a , and M is a value such that | f ( n + 1 ) ( x ) | ≤M for all x

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