The rates of change in population for two cities are \( P^{\prime}(t)=45 \) for Alphaville and \( Q^{\prime}(t)=105 e^{0.04 t} \) for Betaburgh, where \( t \) is the number of years since 1990 , and \( P^{\prime} \) and \( Q^{\prime} \) are measured in people per year. In 1990, Alphaville had a population of 5500 , and Betaburgh had a population of 3500 . Answer parts a) through \( c \) ).

Sketch the region enclosed by \( y=e^{2 x}, y=e^{4 x} \), and \( x=1 \). Find the area of the region.

\( \int \sqrt[ \infty ] { x ^ { r } } d x \)

Evaluate the \( \lim _{x \rightarrow-2}\left(\frac{x^{3}+6 x^{2}+8 x}{x+2}\right) \) Solution:

A factory selling cell phones has a marginal cost function \( C(x)=0.01 x^{2}-3 x+229 \), where \( x \) represents the number of cell phones, and a marginal revenue function given by \( R(x)=429-2 x \). Find the area between the graphs of these curves and \( x=0 \).

Set up the definite integral required to find the area of the region between the graph of \( y=13-x^{2} \) and \( y=24 x+148 \) over the interval \( -5 \leq x \leq 3 \).