A tank contains water with a volume changing based on rain falling into it and water leaking out. If the rate of inflow is given by \( R(t) = 5 + 2\sin(t) \) liters per hour and the leak occurs at a constant rate of 3 liters per hour, formulate a differential equation for the volume of water in the tank over time.

7.- Resuelve el siguiente límite * \( \operatorname{lí}_{t \rightarrow 1}\left(\frac{t^{2}+2 t+1}{2 t-4}\right) \) 2 -3 -2 3

Find \( (A) f^{\prime}(x) \), (B) the partition numbers for \( f^{\prime} \), and \( (C) \) the critical numbers of \( f \). \( f(x)=x^{3}-12 x-8 \) (A) \( f^{\prime}(x)=\square \) (B) Find the partition numbers for \( f^{\prime} \). Select the correct choice below and, if necessary, fill in the answer box to complete your choic A. The partition pumber(s) is/are \( x=\square \). (Use a comma to separate answers as needed.) B. There are no partition numbers. (C) Find the critical numbers for \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical number(s) is/are \( x=\square \). (Use a comma to separate answers as needed.)

10.- Calcula la derivada de la función * \( y=12 x^{4}+5 x^{3}-3 x^{2}+5 \) \( \begin{array}{ll}\text { A) } \frac{d y}{d x} 48 x^{3}+15 x^{2}-6 x & \text { B) } \frac{d y}{d x} 48 x^{4}+15 x^{2}-6 x \\ \text { C) } \frac{d y}{d x} 48 x^{3}+15 x^{2}+6 x & \text { D) } \frac{d y}{d x} 48 x^{2}+15 x^{2}-6 x\end{array} \)

1. \( \lim _{x \rightarrow \infty}(\overline{3 x-5}) \) 2. \( \lim _{x \rightarrow 0}(1-\tan x)^{\frac{4 x}{\sin ^{2} 3 x}} \)

\( \int \frac { 3 / \sqrt { a ^ { 3 } y ^ { 4 } } x d x } { 4 / 3 a ^ { 2 } b ^ { 3 } x ^ { 2 } + 6 } \)