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^{(n1)}(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):
f (x)  =  f (a) + f'(a)(x  a) + ^{ ... } + f^{(n1)}(a) + f^{(n)}(t)dt 

 =  p_{n1}(x) + r_{n}(x) 

Therefore, in order to compute how close p_{n}(x) is to f (x), we need to find the
magnitude of the remainder term. Fortunately, there is a simpler way to express
r_{n}(x).
Letting x be fixed for a moment, choose numbers m and M in the interval [a, x]
so that f (m) is the minimum value of f on the interval and f (M) the maximum
value. Then for any t in [a, x],
f^{(n)}(m)≤f^{(n)}(t)≤f^{(n)}(M) 

The corresponding integrals must satisfy similar inequalities:
f^{(n)}(m)dt≤r_{n}(x)≤f^{(n)}(M)dt 

or
f^{(n)}(m)≤r_{n}(x)≤f^{(n)}(M) 

By the intermediate value theorem,
f^{(n)}(t) 

takes on all values between its minimum and maximum in the interval [a, x], so there
exists some c in [a, x] such that
r_{n}(x) = (x  a)^{n} 

The expression on the right looks very much like the nth term of the Taylor
polynomialthe only difference is that the derivative is evaluated at some number c in
the interval [a, x] rather than at a.
Now that the remainder term is in a more manageable form, we can try to bound it.
Suppose we have a bound, B_{n}, for the absolute value of the nth derivative of f on
the interval [a, x]. That is,
for all c in [a, b]. Then we have the bound
 r_{n}(x)≤ x  a^{n} 

To conclude, we restate the results of this section in the case a = 0. The Taylor formula
in this case reads
f (x) = f (0) + f'(0)x + ^{ ... } + f^{(n1)}(0) + f^{(n)}(t)dt 

with remainder term bounded by
 r_{n}(x)≤ x^{n} 
