Use integration by parts to evaluate the integral \( \int x e^{x} \, dx \).

\( \left. \begin{array} { l } { f ( x ) = - x ^ { 4 } - 3 x ^ { 3 } } \\ { h ( x ) = - 2 ( x - 2 ) ^ { 2 } ( x + 2 ) } \end{array} \right. \)

El material de la base de una caja abierta cuesta 1.5 veces lo que cuesta el de los laterales Hallar las dimensiones de la caja de volumen maximo que se puede construir con un costo fijo considere que v \( =x y z ; \quad c=1.5 x y+2 x z+2 y z \)

Find the rate of change of total revenue, cost, and profit with respect to time. Assume that \( R(x) \) and \( C(x) \) are in doll \( R(x)=45 x-0.5 x^{2}, C(x)=4 x+10 \), when \( x=35 \) and \( \frac{d x}{d t}=20 \) units per day The rate of change of total revenue is \( \$ \square \) per day.

18. Fie funcția \( f: R \rightarrow R, f(x)=e^{x} \cdot\left(a x^{2}+b x+c\right) \). Determinaţi numerele reale \( a, b \) şi \( c \), ştiind că \( f(0)=0, f^{\prime}(0)=1 \) şi \( f^{\prime \prime}(0)=4 \).

3. \( h(x)=x^{2}+5 x-2 \) \( \frac{d h(x)}{d x}=\lim _{\Delta x \rightarrow 0}=\frac{h(x+\Delta x)-h(x)}{\Delta x} \)