8. A spherical balloon is released from rest and expands as it Answer: rises. After rising for \( t \) seconds its radius is \( r \mathrm{~cm} \), and its \( 402 \mathrm{~cm}^{2} / \mathrm{s} \) surface area is \( A \mathrm{~cm}^{2} \), where \( A=4 \pi r^{2} \). The initial radius of the balloon is 16 cm . Given that the rate of increase of the radius is constant and has the value \( 0.8 \mathrm{~cm} \mathrm{~s}^{-1} \), find the rate of increase of \( A \) when \( t=5 \).

4. (8 points) Calculate the following limit. Write DNE if the limit does not exist \( \lim _{x \rightarrow-1}\left(x^{2}-\frac{3 x}{x+2}\right)= \) 5. (16 points) Use the following phrases to complete the statements below to give

(10 points) Use the Fundamental Theorem of Calculus to find the exact values of the definite integral. \( \int_{1}^{3} 3\left(2^{x}\right) d x= \)

7. (8 points) Calculate the following limits. Write DNE if the limit does not exist. (a) \( \lim _{x \rightarrow 0^{+}} \frac{4}{x^{3}}= \) (b) \( \lim _{x \rightarrow 0^{-}} \frac{4}{x^{3}}= \) (c) \( \lim _{x \rightarrow 0} \frac{4}{x^{3}}= \) 8. ( 20 points) Suppose \( f \) and \( g \) are functions such that

38. Use integration formulas to solve each integral. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. (a) (10 points) \( \int\left(x^{2}-3 x^{8}-7\right) d x= \) (b) (10 points) \( \int 3 \cos (x) d x= \) (c) (10 points) \( \int x^{2}\left(x^{3}+1\right) d x= \)

\begin{tabular}{|l|}\hline 3. \( \begin{array}{l}\text { Calcula el volumen } \\ \text { del sólido de } \\ \text { revolución que se } \\ \text { forma de girar sobre } \\ \text { el eje } x \text { las funciones } \\ f(x)=2 x^{2}+\frac{1}{2} y \\ g(x)=x \text { con límites } \\ {[0,3]}\end{array} \) \\ \hline\end{tabular}