grandes-ecoles 2011 QI.A.1

grandes-ecoles · France · centrale-maths1__mp Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences

Let $f$ be a real function, defined, continuous and decreasing on $[ a , + \infty [$, where $a \in \mathbb { R }$. Show that for every integer $k \in \left[ a + 1 , + \infty \left[ \right. \right.$, we have $\int _ { k } ^ { k + 1 } f ( x ) d x \leqslant f ( k ) \leqslant \int _ { k - 1 } ^ { k } f ( x ) d x$.

Let $f$ be a real function, defined, continuous and decreasing on $[ a , + \infty [$, where $a \in \mathbb { R }$. Show that for every integer $k \in \left[ a + 1 , + \infty \left[ \right. \right.$, we have $\int _ { k } ^ { k + 1 } f ( x ) d x \leqslant f ( k ) \leqslant \int _ { k - 1 } ^ { k } f ( x ) d x$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2