\( \begin{array}{ll}\text { (b) }[n=5] & \int_{0}^{1} \sin x^{2} d x \\ \text { (c) }[n=4] & \int_{0}^{2} x\left(e^{x}\right) d x\end{array} \)

Find the area under the curve defined by \( f(x) = 2x + 1 \) between the points \( x = 1 \) and \( x = 3 \).

25) \( \lim _{x \rightarrow 0} \frac{|x-2|}{x-2}= \) ? \( \begin{array}{lllll}\text { A) }-\infty & \text { B) Limit yoktur } & \text { C) } 1 & \text { D) } \infty & \text { E) }-1\end{array} \)

PROBLEME 1 Partie A Soit g la fonction numérique définie sur \( ] 0 ;+\infty\left[\right. \) par \( g(x)=x(1+\ln x)^{2}-1 \) 1. On admettra que g est dérivable sur \( ] 0 ;+\infty[ \) a) Calculer \( g^{\prime}(x) \) pour tout \( \left.x \in\right] 0 ;+\infty[ \) b) Vérifier que pour tout \( x \in] 0 ;+\infty\left[g^{\prime}(x)=(1+\ln x)(3+\ln x)\right. \) c) Etudier le signe de \( g^{\prime}(x) \) suivant lês valeurs de \( x \) et en déduirele sens de variation de g.

Use Lagrange interpolation polynomial of degree 3 to estimate \( f(8.4) \) if \( f(8.1)=16.9441, f(8.3)=17.5649, f(8.6)=18.5052 \), and \( f(8.7)=18.8209 \) Verify that \( \sum_{i=0}^{3} L_{i}(8.4)=1 \)

11) \( f(x)=\left(x^{2}-4\right)^{2} \) fonksiyonunun artan olduğu aralıklar hangisidir? A) \( (-2,0) \) ve \( (2, \infty) \) B) \( (-2,0) \) ve \( (-\infty,-2) \) C) \( (-2,0) \) ve \( (0,2) \) D) \( (2, \infty) \) ve \( (-\infty,-2) \) E) \( (2, \infty) \) ve \( (0,2) \)