A marshy region used for agricultural drainage has become contaminated with a particular element. It has been determined that flushing the area with clean water will reduce the element for a while, but it will then begin to build up again. A biologist has found that the percent of the element in the soil \( x \) months after the flushing begins is given by the function below. When will the element be reduced to a minimum? What is the minimum percent? \( f(x)=\frac{x^{2}+16}{2 x}, 1 \leq x \leq 12 \) Find \( f^{\prime}(x) \). \( f^{\prime}(x)=\frac{1}{2}-\frac{8}{x^{2}} \) When will the element be reduced to a minimum? The element will be reduced to a minimum after 4 month(s). (Type an integer or a decimal.) What is the minimum percent of the element?

Al calcular el \( \lim _{x \mapsto-\infty} \frac{2 x}{\sqrt{x^{2}+1}} \) se obtiene: \( \begin{array}{l}\text { a. }-2 \\ \text { b. } 1 \\ \text { c. } 2 \\ \text { d. }-1\end{array} \)

\( \lim _ { x \rightarrow \infty } \frac { e ^ { x } \cdot \operatorname { ch } ( 2 x ) } { \operatorname { sh } ( 3 x ) } \)

Al calcular el \( \lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan x} \) se obtiene: a. 0 b. 4 c. \( 1 / 4 \) d. No existe

The cost to produce \( x \) items is modeled by \( C(x)=0.003 x^{2}+3.2 x+60 \) dollars and the revenue from selling \( x \) items is given by \( R(x)=6 x \) dollars. a. Find the marginal profit when 359 items are sold using the formula \( P(x+1)-P(x) \). (Round your answer to three decimal places.) b. Use the derivative to estimate the marginal profit when 359 items are sold. (Round your answer to three decimal places.)

\( \lim _ { x \rightarrow \infty } \frac { e ^ { x } \cdot \operatorname { ch } ( 2 x ) } { 1 h ( 3 x ) } \)