
%  This is a LaTex file.

%  Spring semester, 2024, Wei-Cheng Wang.


%   A latex format for making homework assignments.

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{centernot}
\usepackage{graphicx}
\usepackage{float}
\DeclareSymbolFontAlphabet{\Bbb}{AMSb}


%          The page format, somewhat wider and taller page than in art12.sty.

\topmargin -0.1in \headsep 0in \textheight 9.05in \footskip 0.6in
\oddsidemargin 0in  \evensidemargin 0in  \textwidth 6.5in

\def\V{\mathcal V}
\def\L{\mathcal L}
\def\S{\mathcal S}
\def\T{\mathcal T}
\def\U{\mathcal U}
\def\A{\boldmath A}
\def\M{\boldmath M}
\def\l{\mbox{\boldmath $l$}}
\def\r{\mbox{\boldmath $r$}}
\def\v{\mbox{\boldmath $v$}}
\def\ds{\displaystyle}

\newtheorem{Remark}{Remark}
\newcommand{\bp}{{\noindent \it proof }: \\}
\newcommand{\ep}{\hfill $\Box$}
\newcommand{\bR}{\begin{Remark}}
\newcommand{\eR}{\end{Remark}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}

\begin{document}


%          Definitions of commonly used symbols.



%          The title and header.

\noindent
{\scriptsize Calculus II, Spring 2024 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

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

\begin{center}
\large
Study guide for quiz 07
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

\item % 14.3: Partial Derivatives.
Section 14.3:

Study how to evaluate 
$\displaystyle{\partial f \over \partial x}$,
$\displaystyle{\partial f \over \partial y}$
at $(x_0,y_0)$
when $f(x,y)$ is given explicitly,
and
$\displaystyle{\partial z \over \partial x}$,
$\displaystyle{\partial z \over \partial y}$
at $(x_0,y_0,z_0)$
when $z(x,y)$ is given implicitly by $F(x,y,z)=0$.

\item % 14.3: Partial Derivatives.
Section 14.3:

Study the definition of differentiability 
for functions of two or more variables in section 14.3.
Study why the four statements (1)-(4) on page 1 of Lecture 15 are identical
(see also problem 2 of homework 07).

%%  Study the relation of differentiability with "the tangent plan"
%% (which is the same as "the linear approximation" (section 14.6))
\item % 14.3: Partial Derivatives.
Section 14.3:

Study why "existence of partial derivatives % and directional derivatives 
at a point" does not imply "differentiability at a point" (find a counter example).
Does this contradict Theorem 3?

\item % 14.3: Partial Derivatives.
Section 14.3:

Study why differentiability at a point implies continuity at that point
(study the proof of Theorem 4).
Try to find a function of two variables that is continuous at a point
but not differentiable at that point.

\item % 14.3: Partial Derivatives.
Section 14.3:

The proofs of Theorem 2 and Theorem 3 are in Appendix 9.
They will not appear in any exam in this course.


\end{enumerate}

\end{document}
