grandes-ecoles 2014 QIB2

grandes-ecoles · France · centrale-maths1__pc Taylor series Taylor's formula with integral remainder or asymptotic expansion

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Deduce that in a neighbourhood of 0 $$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by
$$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$
Deduce that in a neighbourhood of 0
$$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$

Paper Questions

QIA QIB1 QIB2 QIC1 QIC2 QIC3 QIIA QIIB1 QIIB2 QIIB3 QIIC1 QIIC2 QIID1 QIID2 QIID3 QIIIA QIIIB1 QIIIB2 QIIIB3 QIIIB4 QIIIC QIVA1 QIVA2 QIVA3 QIVB QIVC