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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

Solutions

February 14, 2003}
\end{center}

\begin{enumerate}

\item  Compute the following limits:
	\begin{enumerate}
	\item  \[ \lim_{t \ra \infty} 25-\frac{18}{t^3} \]
	\item  \[ \lim_{x \ra \infty} \frac{5x^3-2x}{16x^3+1} \]
	\item  \[ \lim_{b \ra \infty} xe^{-x} \]
	\item  \[ \lim_{t \ra \infty} \frac{x}{\ln x} \]
	\item  \[  \lim_{t \ra \infty} \ln t \]
	\end{enumerate}

\newpage
\lecture

\item  Consider the function defined below:
	\[
	G(t) = \int_0^t e^{-x} \; dx
	\]
	\begin{enumerate}
	\item  Find a ``simple'' expression for $G(t)$ (i.e., without an integral).
	\item  Fill out the following chart:
		\[
		\begin{array}{|c||c|c|c|c|c|}
		\hline
		t & 0 & 1 & 2 & 3 & 10 \\
		\hline
		G(t) & & & & &  \\
		\hline
		\end{array}
		\]
	\item  Find the limit:
		\[
		\lim_{t \ra \infty} G(t)
		\]
	\item  What does this limit correspond to in terms of area under the curve?
		\\\sol This means that the area under the curve $y=e^{-x}$ from $0$
		to $\infty$ is equal to $1$.
	\end{enumerate}


\newpage

\item  Compute the following improper integrals.
	What makes these integrals ``improper''?
	Are the integrals divergent or convergent?
	\begin{enumerate}
	\item  \[ \int_1^\infty \frac{1}{x^5} \; dx \]
	\item  \[ \int_1^\infty \frac{1}{\sqrt{x}} \; dx \]
	\item  \[ \int_1^\infty \frac{1}{x^{1/3}} \; dx \]
	\item  \[ \int_1^\infty \frac{1}{x^{1.2}} \; dx \]
	\item  \[ \int_0^1 \frac{1}{x^5} \; dx \]
	\item  \[ \int_0^1 \frac{1}{\sqrt{x}} \; dx \]
	\item  \[ \int_0^1 \frac{1}{x^{1/3}} \; dx \]
	\item  \[ \int_0^1 \frac{1}{x^{1.2}} \; dx \]
	\end{enumerate}

\newpage

\item  Determine the values of $p$ for which the following integral converges:
	\[
	\int_1^\infty \frac{1}{x^p} \; dx 
	\]

\item  Determine the values of $p$ for which the following integral converges:
	\[
	\int_0^1 \frac{1}{x^p} \; dx
	\]


\end{enumerate}



\end{document}