 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-2/17/a/solution

A cubic version of Taylor's theorem requires the corresponding higher differentiability, rather than only one derivative. For example, if

f

has four continuous derivatives on the interval between

a

and

x

, then

f (x) = f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \frac{f ^{(3)} ( a )}{3 !} (x - a)^{3} + R_{3} (x),

where the Lagrange form of the Taylor remainder is

R_{3} (x) = \frac{f ^{(4)} ( ξ )}{4 !} (x - a)^{4}

for some

ξ

strictly between

a

and

x

when they differ. At

x = a

, the remainder is zero. These hypotheses apply to both functions used below.

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