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%  Spring semester, 2024, Wei-Cheng Wang.


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\begin{document}


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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 08
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

\item % 14.4: The Chain Rule.
Section 14.4: Study the Chain rule 
for composition of differentiable
functions of one or more independent variables
(along with one or more intermediate variables)
such as Theorem 5 (1 independent, 2 intermediate variables),
Theorem 6 (1 independent, 3 intermediate variables),
Theorem 7 (2 independent, 3 intermediate variables)
and so on for more general cases.
% For example,  
% Cartesian coordinate $(x,y)$ and polar coordinate $(r,\theta)$
% where $x = r \cos \theta$,  $y = r \sin \theta$.

\item % 14.4: The Chain Rule.
Section 14.4:
Study how to evaluate
$$
{d \over dx} \int_{g_1(x)}^{g_2(x)} f(x,t) \;dt.
$$

%
\begin{figure}[H]
 \centering
 \includegraphics[width=4.5in]{./diff_int-1}
 \caption{Guide to item 2}
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 \includegraphics[width=3.0in]{./diff_int-2}
 \caption{Guide to item 2, continued}
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 \includegraphics[width=3.0in]{./diff_int-3}
 \caption{Guide to item 2, continued}
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%

\item % 14.5: Directional Derivatives and Gradient Vectors.
Section 14.5:
Study the definition of directional derivative
% Practice with some examples to 
and how to compute it from definition,
and alternatively how to compute it using partial derivatives
when the function is differentiable.

\item % 14.5: Directional Derivatives and Gradient Vectors.
Section 14.5:
Study the geometric meaning of the gradient vector.
Study how to find the tangent line and normal line of a
level curve of $f(x,y)$ 
(i.e., $\{(x,y)\; | \; f(x,y) = c \})$ using gradient of $f$.
\end{enumerate}

\end{document}
