\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{enumerate}
\begin{document}
\flushleft\underline{Homework 2 --- MATH 1586 Spring 2018} \\
\begin{enumerate}[1.]
\item Compute $\displaystyle\int e^{12x} \mathrm{d}x$.
\item Compute $\displaystyle\int (t^3-2t)^{25}(3t^2-2) \mathrm{d}t$.
\item Compute $\displaystyle\int \dfrac{z^4}{1-z^5} \mathrm{d}z$.
\item Compute $\displaystyle\int \dfrac{5}{w-3} \mathrm{d}w$.
\item Compute $\displaystyle\int q^2 e^{q^3-1} \mathrm{d}q$.
\item In calm waters, the oil spilling from the ruptured hull of a grounded tanker forms an oil slick that is circular in shape. If the radius $r$ of the circle is increasing at a rate of
$$r'(t) = \dfrac{30}{\sqrt{2t+4}} \hspace{5pt} \dfrac{\mathrm{ft}}{\mathrm{min}}$$
$t$ minutes after the rupture occurs, find an expression for the radius at any time $t$. How large is the polluted area $16$ minutes after the rupture occurred? (\textit{note: $r(0)=0$})
\item Suppose that in a certain country, the life expectancy at birth of a female is changing at the rate of
$$g'(t)=\dfrac{5.45218}{(1+1.09t)^{0.9}} \hspace{5pt} \dfrac{\mathrm{years}}{\mathrm{year}}.$$
Here $t$ is measured in years with $t=0$ corresponding to the beginning of the year $1900$. Find an expression $g(t)$ giving the life expectancy at birth (in years) of a female in that country if the life expectancy at the beginning of $1900$ is $50.02$ years. What is the life expectancy at birth of a female born in the year $2000$ according to this model?
\end{enumerate}
\end{document}