4. Материальная точка движется прямолинейно по закону \( x(t)=\frac{1}{3} t^{3}-3 t^{2}-5 t+3 \) (где \( x- \) расстояние чки отсчета в метрах, \( t- \) время в секундах, измеренное с начала движения). В какой момент времени (в секу x) ее скорость была равна 2 м/с?

дите точку максимума функции \( y=2 \ln (x+4)^{3}-8 x-19 \).

Fasendo \( f\left(-\frac{\pi}{2}\right)=-1, f\left(\frac{\pi}{2}\right)=0 \), deve risultare \( b-a=-1 \) e \( a+b=0 \). Da cuii \( a=\frac{1}{2} \) \( \left.b=-\frac{1}{2}\right] \) \[ f(x)=\left\{\begin{array}{ll}{[\log (1+x)] / x} & \text { se } x \in(-1,0) \\ \left(e^{x}-1\right) / x \quad \text { se } \quad x \in(0,1)\end{array}\right. \] è continua in \( (-1,1)-\{0\} \), ammette prolungamento continuo in \( (-1,1) \), ma la funzione \( f(x) \) definita da non su \( [-1,1] \)

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.

9.17 Determinare \( a \) e \( b \) in modo che la funzione \( f(x) \) definita in \( \mathbb{R} \) da \[ f(x)=\left\{\begin{array}{ll}\operatorname{sen} x & \text { se } x \leq-\frac{\pi}{2} \\ a \text { sen } x+b & \text { se }-\frac{\pi}{2}<x<\frac{\pi}{2} \\ \cos x & \text { se } x \geq \frac{\pi}{2}\end{array}\right. \] sia continua in \( \mathbb{R} \). \( \left[\right. \) Essendo \( f\left(-\frac{\pi}{2}\right)=-1, f\left(\frac{\pi}{2}\right)=0 \), deve risultare \( b-a=-1 \) e \( a+b=0 \). Da cui \( a=\frac{1}{2} \) e \( \left.b=-\frac{1}{0}\right] \)

Evaluate \( \int_{1}^{2}\left(3 x^{2}-2 x+1\right) \mathrm{d} x \)