Partie B Pour la suite de l'exercice, on admet que la fonction \( f \) est définie sur par \( \left[0 ;+\infty\left[: f(x)=(x+1) e^{-\frac{1}{2} x}\right.\right. \). 1) a] Justifier que, pour tout réel \( x \) positif, \( f(x)=2\left(\frac{\frac{1}{2} x}{e^{\frac{1}{2} x}}\right)+e^{-\frac{1}{2} x} \).

Calcular \( y^{\prime}=\frac{d y}{d x} \) de la función implicita \( \frac{x^{2}}{4}+\frac{x^{2}}{9}=3 x y \) y simplificar:

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

\( x^{3} y+y^{2}-x^{2}=5 \) Encuentra el valor de \( \frac{d y}{d x} \) en el punto \( (2,1) \)

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 \( f^{\prime \prime}(0)=4 \).