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

February 12, 2003}
\end{center}

\begin{enumerate}

\item Suppose you are given the following inequality:
	\[
	\frac{K(b-a)^5}{180n^4} < 0.1
	\]
	Suppose you know the following: $K=10$, $a=1$, $b=5$.
	\begin{enumerate}
	\item What are some possible values for $n$ that make the inequality hold?
	\item  What is the smallest value of $n$ that will make this inequality hold?
	\end{enumerate}

\item  Suppose you are given the following inequality:
	\[
	E \leq \frac{K(b-a)^5}{180n^4}
	\]
	Again, you have $K=10$, $a=1$ and $b=5$ and you have control over $n$.
	You now want to ensure that $E<0.001$.
	\begin{enumerate}
	\item  What are some values of $n$ what will force $E<0.001$?
	\item  What is the smallest value of $n$ that will force $E<0.001$?
	\item  Will all values of $n$ larger then the value you found in the previous
		part ensure that $E<0.001$?  Why?
	\end{enumerate}


\newpage

\textbf{Lecture Problems}

\item  Given the error bound formula for Simpson's rule:
	\[
	|E_S| \leq \frac{K(b-a)^5}{180n^4}
	\]
	Where $K$ is some number such that $|f^{(4)}(x)| \leq K$ for all $a\leq x \leq b$.
	We want to investigate what this means for estimating:
	\[
	\int_2^4 \ln x \; dx
	\]
	\begin{enumerate}
	\item  Find a suitable $K$.
	\item  Determine $n$ so that the error in Simpson's rule is less then $0.00001$.
	\end{enumerate}

\newpage

\item  Consider the following integral (and function!):
	\[
	G(t) = \int_1^t \frac{1}{x^3} \; dx
	\]
	\begin{enumerate}
	\item  Find a simple expression for $G(t)$ by using the Fundamental Theorem of 
		Calculus (the evaluation part!)
	\item  What happens to $G(t)$ as $t \ra \infty$?
		(In other words, what is the limit: $\lim_{t\ra \infty} G(t)$?)
	\item  What does $t \ra \infty$ correspond to in terms of area under the curve?
	\end{enumerate}


\end{enumerate}



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