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%  Spring semester, 2026, Wei-Cheng Wang.


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\begin{document}


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%          The title and header.

\noindent
{\scriptsize Calculus II, Spring 2026 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\noindent
{\scriptsize Thomas' Calculus Early Transcendentals 13ed} \hfill 

\begin{center}
\large
Study guide for quiz 01
\normalsize
\end{center}

\noindent

\vspace{-.3cm}
Quiz problems include both the lecture contents and homework problems.

\begin{enumerate}

%  8.8: Improper integrals.
\item 
Section 8.8: 

Memorize the definitions of type I and type II 
improper integrals and study how to verify
convergence/divergence of improper integrals
from these definitions. 
% You may need to review some techniques of integration, such as "integration by parts", etc..

%  8.8: Improper integrals.
\item 
Section 8.8: 

Study the convergence/divergence of the improper integrals
$\displaystyle{\int_a^\infty {1\over x^{p}} dx}$ ($a>0$),
$\displaystyle{\int_0^a {1\over x^{p}} dx}$,
$\displaystyle{\int_c^{c+a} {1\over (x-c)^{p}} dx}$,
and
$\displaystyle{\int_{c-a}^c {1\over |x-c|^{p}} dx}$
for $p>1$, $p=1$ and $0<p<1$,
respectively by direct computation.
Then memorize the results.
Plotting the graph of $f(x) = \ds{1\over x^p}$
for various $p$'s as in page 11 of lecture 01
would likely help you memorize the results.

%  8.8: Improper integrals.
\item
Section 8.8:
Study the comparison Theorems (Theorem 2, Theorem 3) used to decide convergence/divergence
of improper integrals. Sample problems: section 8.8, problems 35-64.

%  8.8: Improper integrals.
\item
Section 8.8:
Study how to determine convergence/divergence of an improper integral
with multiple `improper' parts,
such as the domain contains both $x \to \pm \infty$,
and/or points of discontinuity within the domain.

% 10.1: Sequences.
\end{enumerate}

\end{document}
