Exercice 4 Soit \( f \) la fonction définie sur \( \mathbb{R} \) par \( f(x)=x^{2}-x+7 \) si \( x \leq 1 \) \( f(x)=-x^{2}+3 x+5 \) si \( \quad x>1 \) 1) Montrer que \( f \) est dérivable en 1 . 2) Déterminer une équation de \( T \), la tangente en 1 à \( \mathcal{C} \), la courbe représentative de 3) Etudier la position de \( \mathcal{C} \) par rapport à \( T \).

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{2}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{7}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

What is the average rate of change on the function \( f(x)=x^{2} \) between \( x= \) 1 and \( x=3 \) ? -1

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. (Simplify your answer.) 7 \( \int_{2} 5 g(x) d x=20 \) (Simplify your answer.) 7 \( \int[g(x)-f(x)] d x=6 \) 2 \( (S i m p l i f y \) 7 \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) 2

Suppose \( \int^{4} f(x) d x=6, \int^{7} f(x) d x=-2 \), and \( \int^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{2}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) 7 \( \int_{2}[g(x)-f(x)] d x=6 \) 2 (Simplify your answer.) 7 \( \int_{2}[5 g(x)-f(x)] d x=\square \) (Simplify your answer.)