Exercice 6: Supposons que \( D(p, s)=s-p \) est la demande du bien de la qualité \( s \) au prix \( p \). Supposons aussi que \( C(q, s)=q \cdot c(s) \) est le coût de production de \( q \) unités du bien de qualité \( s \) où \( c(0)=0, c^{\prime}(s)>0 \) et \( c^{\prime \prime}(s)>0 \). Donc \( c(s) \) est égal au coût marginal de production du bien de qualité \( s \). (i) Donnez la qualité \( s \), calculez le prix de monopole \( p^{m}(s) \). Montrez en- suite que pour la qualité optimal du monopoleur, \( s^{m} \), on a \( c^{\prime}\left(s^{m}\right)=1 \). (ii) Déterminez la qualité \( s^{c} \) pour l'optimum social \( s^{c} \). Ensuite calculez le bien-être social, et comparez votre résultat de la qualité \( s^{c} \) avec celui de \( s^{m} \) trouvé dans (i).

Divide. \[ \left(9 u^{5} z-17 u^{5} z^{4}\right) \div\left(-3 u^{4} z^{2}\right) \] Simplify your answer as much as possible. \( \square \)

If \( f(x)=\ln (5 x+1)^{6} \) Find the slope of the tangent line to \( f(x) \) at \( x=-1 \)

4. Im conducting a study about the heights and weights of Rutgers undergraduates. To do this, I send out a survey by email to 500 randomly selected undergraduates and ask them for both their height and their weight. (a) ( 2 points) After collecting my data, I find that only 152 of the people in my sample returned the survey. Do you think the topic of the survey contributes to the non-response? What about the method of survey distribution? (b) ( 1 point) In addition, a large proportion of those who responded gave their weights in multiples of 5 (e.g., 165, 170, 185). Can you identify what might be happening here?

\( x^{4}-2 x^{2}+3 \), quando \( x=\frac{1}{\sqrt{2}} \),

ite: \( \lim _{x \rightarrow-\infty} \frac{|x|}{x} \)