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%  Homework for the course "Calculus II", 
%  Spring semester, 2024, Wei-Cheng Wang.

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

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\large
Homework 10   
\normalsize
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\vspace{.3in}

%           The questions!

\begin{enumerate}

% 14.6:  Tangent Planes and Differentials.
\item
% Section 14.6: Problems                      33, 39(a), 45. 
% Section 14.6: Problems 5, 9, 17, 19, 25(b),                55, 57, 58.
  Section 14.6: Problems 5, 9, 17, 19, 25(b), 33, 39(a), 45, 55, 57, 58.

Remark: 
Suppose that $f(x,y)$ is differentiable at $(x_0,y_0)$.
Both "linearization" and "linear approximation" of $f(x,y)$ at $(x_0,y_0)$ 
refer to the tangent plane of the surface $z=f(x,y)$ at
$(x_0,y_0,f(x_0,y_0))$.
That is, the plane $z=L(x,y)$ given on page 856.

Problem 55 provides
another way of finding the tangent plane $z=L(x,y)$. 
That is, apply the definition and equation (1) on page 853 to the
function $F(x,y,z) = z-f(x,y)$:
$$
\nabla F(x_0,y_0,z_0)\cdot(x-x_0,y-y_0,z-z_0) = 0
$$
Verify that this will lead to the equation $z=L(x,y)$.

Hint for problem 39, 45:
See page page 859 (item 1 and 2) for
generalization to functions of 3 variables. 

% 14.7: Extreme Values and Saddle Points (and Gradient Analysis).
\item
% Section 14.7: Problems 1, 19,         39, 43,     49, 51. %
% Section 14.7: Problems 31, 35, 44.
  Section 14.7: Problems 1, 19, 31, 35, 39, 43, 44, 49, 51. %

Alternative method for problems 31, 35:
To find absolute minimum and absolute maximum,
find all local minima and local maxima first, by
plotting the gradient vectors inside the domain
and tangential component of the gradient vectors
on the boundary as done in Lecture 20.
Then compare all the local minima and local maxima
to find the absolute ones.

Hint for problem 44:
Do this problem using the gradient analysis. That is,
plot $\nabla f$ near the critical point to
determine whether the critical point is a local
minimum, local maximum or neither.

In some cases, there may be other methods other than
plotting $\nabla f$, but plotting $\nabla f$
is more systematic and guaranteed to work.
 
Hint for problem 49, 51:
Minimizing/maximizing distance is the same as minimizing/maximizing
$({\rm distance})^2$. The latter is easier to compute.

\end{enumerate}
 
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