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The Remainder Term

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The Remainder Term

The Remainder Term

The Remainder Term

The Remainder Term

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) :