\( 1 \leftarrow \quad \begin{array}{l}\text { The function } A=A_{0} e^{-0.00866 x} \text { models the amount in pounds of a particular radioactive } \\ \text { material stored in a concrete vault, where } x \text { is the number of years since the material was } \\ \text { put into the vault. If } 200 \text { pounds of the material are placed in the vault, how much time will } \\ \text { need to pass for only } 46 \text { pounds to remain? Round to nearest whole number. } \\ \square \text { years will need to pass for only 46pounds to remain. } \\ \text { (Round to nearest whole number.) }\end{array} \).

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.)

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. (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 \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), 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.)

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