Taylor s Theorem

In the last few lectures we discussed the mean value theorem (which basically relates a function
and its derivative) and its applications. We will now discuss a result called Taylor s Theorem which
relates a function, its derivative and its higher derivatives. We will see that Taylor s Theorem is
an extension of the mean value theorem. Though Taylor s Theorem has applications in numerical
methods, inequalities and local maxima and minima, it basically deals with approximation of
functions by polynomials. To understand this type of approximation let us start with the linear
approximation or tangent line approximation.
Linear Approximation : Let f be a function, di®erentiable at x0 2 R. Then the linear polynomial
P1(x) = f(x0) + f0(x0)(x ? x0)
is the natural linear approximation to f(x) near x0. Geometrically, this is clear because we approx-
imate the curve near (x0; f(x0)) by the tangent line at (x0; f(x0)). The following result provides
an estimation of the size of the error E1(x) = f(x) ? P1(x):
Theorem 10.1: (Extended Mean Value Theorem) If f and f0 are continuous on [a; b] and
f0 is di®erentiable on (a; b) then there exists c 2 (a; b) such that
f(b) = f(a) + f0(a)(b ? a) + f00(c)
2
(b ? a)2: