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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 09
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

\item % 14.6: Tangent Planes and Differentials.
Section 14.6:
Review the definition and properties of the gradient vector
and its application in finding the tangent plane and normal line
at a point $(x_0,y_0,z_0)$ on the level surface $F(x,y,z)= c$
of a differentiable function $F$.
Review how to find
the tangent line and normal plane
at a point $(x_0,y_0,z_0)$ on a curve in space
defined as the intersection of two surfaces
$F(x,y,z)= C_1$
and
$G(x,y,z)= C_2$.

\item % 14.6: Tangent Planes and Differentials.
Section 14.6:
Study and memorize the error estimate of
$E(x,y) = f(x,y)-L(x,y)$ 
for a function $z=f(x,y)$ and its linearization
$z=L(x,y)$ at a point $(x_0,y_0)$. 
Study its generalization to functions of more than two 
variables (i.e. $g(x,y,z)$, $h(x,y,z,w)$, $\cdots$, etc.).

% Generalize the result to functions of
% more than two variables (such as $w=f(x,y,z)$).

\item % 14.6: Tangent Planes and Differentials.
Section 14.6: SKIP the 'differential' part on page 858, which reads:
$$
df = f_x(x_0,y_0)dx + f_y( x_0,y_0) dy.
$$
Instead, you need to memorize the formula below and understand its meaning:
$$
\Delta f \approx f_x(x_0,y_0)\Delta x + f_y( x_0,y_0) \Delta y.
$$

\item % 14.7: Extreme Values and Saddle Points (and Gradient Analysis).
Section 14.7:
Study the
'First Derivative Test', 'Second Derivative Test', 
how the sign of the determinant 
$ f_{xx} f_{yy} -  f_{xy}^2 $ is related
to whether 
$f_{xx} \Delta x^2  
+2f_{xy} \Delta x \Delta y
+f_{yy} \Delta y^2
$
can be rewritten as 'sum of squares' or 'difference of squares',
which in term determines whether a critical
point is local min, local max, or neither.
Review the procedure of finding potential local minima
and/or local maxima of a differentiable function $z=f(x,y)$.

\item % 14.7: Extreme Values and Saddle Points (and Gradient Analysis).
Section 14.7: % (study guide for midterm 02)
Review the gradient analysis that helps to classify the
critical points when the second derivative test is 
inconclusive, and how it can be applied to find
absolute max and absolute min on a bounded region.
\end{enumerate}

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