Exercice 3 Soit \( f \) la fonction définie sur \( \mathbb{R}^{*} \) par \( f(x)=4 x+\frac{1}{x} \) Soit \( C \) la courbe représentative de la fonction \( f \) dans un repère \( (O ; \vec{l}, \vec{j}) \). a) Déterminer l'équation réduite de \( T \), la tangente à \( C \) au point d'abscisse - 2 . b) Déterminer la position relative de \( T \) et de \( C \).

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

5. a) The function \( f(x)=x \sin (1 / x) \) is continuous for all \( x \neq 0 \). Show that if we define \( f(0)=0 \), then the function is continuous for all reals Hint: Squeeze Theorem b) Let \( g(x)=x^{2} \sin (1 / x) \) for \( x \neq 0 \) and \( g(0)=0 \). Show that \( g(x) \) is differentiable for all \( x \) but that \( g^{\prime}(x) \) is not continuous. Hint: sequences c) Suppose we have \( g(x) \) as defined in \( b) \) but \( g(0) \) was not given. Instead, we are given that \( g(x) \) is differentiable for all \( x \). Show that \( g^{\prime}(x)=0 \) for some point in the interval \( [-1 / \pi, 1 / \pi] \) Hint; Rolle's theorem d) Let \( H(x)=\int|t| \) dt where the limits are -17 to \( x \). Show that \( H(x) \) is continuously differentiable for all reals, but \( H^{\prime \prime}(x) \) does not exist at \( x=0 \). Hint: sequences

Evaluate the integral. \( \int_{1}^{4}\left(x^{2}-1\right) d x=\square \) (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.)

The president of a company that manufactures toy cars thinks that the function \( P(c)=-2 c^{2}+14 c-20 \) represents the company's profit, where \( c \) is the number of cars produced, in thousands, and \( P(c) \) is the company's profit, in hundreds of thousands of dollars. Determine the maximum profit the company can earn.