2013-61.md

course	course_year	question_number	tags	title	year
Numerical Analysis	IB	61	IB 2013 Numerical Analysis	Paper 3, Section II, C	2013

$$f^{\prime}(0) \approx a_{0} f(0)+a_{1} f(1)+a_{2} f(2)=: \lambda(f)$$

be a formula of numerical differentiation which is exact on polynomials of degree 2 , and let

$$e(f)=f^{\prime}(0)-\lambda(f)$$

be its error.

Find the values of the coefficients $a_{0}, a_{1}, a_{2}$.

Using the Peano kernel theorem, find the least constant $c$ such that, for all functions $f \in C^{3}[0,2]$, we have

$$|e(f)| \leqslant c\left|f^{\prime \prime \prime}\right|_{\infty} .$$