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%  Spring semester, 2024, Wei-Cheng Wang.


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\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 10
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

\item % 14.10: Partial Derivatives with Constrained Variables.
Section 14.10: 

Review partial derivatives with constrained variables.
In particular, study how to identity the independent
variables and dependent variables from the all the
variables and and the number of equations/constraints.
Study how to 
find partial derivatives with constrained variables
when it is difficult to
solve the  
dependent variables underline{explicitly} as functions of the independent
variables such as problem 12.

% For example, partial derivatives of 
% \begin{itemize}
% \item
% functions of 3 variables, with 1 constraint.
% \item
% functions of 4 variables, with 2 constraints.
% \item
% functions of 3 variables, with 2 constraints.

% 15.1: Double and Iterated Integrals over Rectangles.
% 15.2: Double Integrals over General Regions.
\item 
Section 15.1, 15.2: 

Study how to identify the limits of integration
in 
$\displaystyle{\int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \;dx dy} $ 
and $\displaystyle{ \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \; dy dx}$ for
general domains (that is, not rectangles).

% 15.2: Double Integrals over General Regions.
% 15.3: Area by Double Integration.
\item 
Section 15.2: 

Study how to interchange between
$\displaystyle{\int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \;dx dy} $ 
and $\displaystyle{ \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \; dy dx}$ for
general domains
as in problems 33-56 of section 15.2.

\end{enumerate}

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