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We now use integration by parts to determine just how good of an approximation is given by the Taylor polynomial of degree n , p _{n}(x) . By the fundamental theorem of calculus,
f (b) - f (a) = f'(t)dt |
Integrating by parts, choosing - (b - t) as the antiderivative of 1 , we have
f'(t)dt | = | - f'(t)(b - t)|_{a} ^{b} + f ^{(2)}(t)(b - t)dt | |
= | f'(a)(b - a) + f ^{(2)}(t)(b - t)dt |
Again, integrating by parts yields
f ^{(2)}(t)(b - t)dt | = | - f ^{(2)}(t) + f ^{(3)}(t) dt | |
= | f ^{(2)}(a) + f ^{(3)}(t) dt |
Putting these equations together, we have
f (b) = f (a) + f'(a)(b - a) + f ^{(2)}(a) + f ^{(3)}(t) dt |
Continuing the process, we arrive at
f (b) = f (a) + f'(a)(b - a) + ^{ ... } + f ^{(n-1)}(a) + f ^{(n)}(t) dt |
Substituting x for b , we have an expression for f (x) , called Taylor's formula at x = a , involving the familiar Taylor polynomial of degree n - 1 for f and an integral called the remainder term and denoted by r _{n}(x) :
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