Question 3 (4 points) For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of \( 191^{\circ} \) Fahrenheit was taken off the stove to cool in a \( 66^{\circ} \mathrm{F} \) room. After 16 minutes, the internal temperature of the soup was \( 94^{\circ} \mathrm{F} \), how long will it take the soup models this situation too cool to \( 79^{\circ} \mathrm{F} \) ? (round to 2 decimal place)

Find the producers' surplus at a price level of \( \overline{\mathrm{p}}=\$ 64 \) for the price-supply equation below. \( \mathrm{p}=\mathrm{S}(\mathrm{x})=15+0.1 x+0.0003 \mathrm{x}^{2} \) The producers' surplus is \( \$ \square \). (Round to the nearest integer as needed.)

Let \( R \) be the region enclosed by the graphs of \( f(x)=4 \cos (x)-3 \) and \( g(x)=4 \sin (x)-4 \). Find the length of the boundary of region \( R \). Use a calculator and round your answer to 3 decimal places.

importante de l'appréciation des copies. Les résultats doivent être encadrés. L'usage de la calculatrice n'est pas autorisé au cours de l'épreuve. Exercice 1. \( \quad \) 1. Étudier la nature des séries suivantes : (a) \( \sum_{n} \frac{n}{2 n^{3}+1} ; \sum_{n} \frac{\ln ^{3}(n)}{n^{2}} ; \sum_{n} \frac{\ln (n+1)}{n \sqrt{n+1}} ; \sum_{n} \sqrt{n} e^{-n^{2}} ; \sum_{n} \ln (n) e^{-n} \).

La derivada de \( e^{x^{2}+2 x+2} \) es: A) \( (2 x-2) e^{x^{2}+2 x+2} \) B) \( (2+2) e^{x^{2}+2 x+2} \) C) \( (2 x+2) e^{x^{2}+2 x+2} \) D) \( -(2 x+2) e^{x^{2}+2 x+2} \)

Let \( R \) be the region enclosed by the graphs of \( f(x)=4 \cos (x)-1 \) and \( g(x)=4 \sin (x)-2 \). Find the length of the boundary of region \( R \). Use a calculator and round your answer to 3 decimal places.