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\begin{document}


\begin{center}
   {\bf Mathematics 108}
    \end{center}


\noindent
{\bf  Exercises 6.2}

\vskip .25 in
\noindent
In Exercises 1 through 4 estimate the first-order partial
derivatives from the given values of each function $f(x,y)$, at
the given point.

\vskip .25 in
${\bf 1.} \hskip.10 in f(0,0)=0, \ f(0.1, 0)=0.3, \ f(0,0.2)=-0.4; \
(0,0)$

\vskip .25 in
${\bf 2.} \hskip.10 in f(1,2)=8, \ f(1.5,2)=10, \ f(1,2.6)=8.3; \
(1,2)$

\vskip .25 in
${\bf 3.} \hskip.10 in f(-4,5)=-7, \ f(-3.8,5)=-6, \ f(-4,5.4)=-7.2; \
(-4,5)$

\vskip .25 in
${\bf 4.} \hskip.10 in f(4.5,6)=20, \ f(5,5.7)=23, \ f(5,6)=21; \
(5,6)$

\vskip .25 in
\noindent
In Exercises 5 through 8 compute the indicated partial
derivatives of each function at the given points.

\vskip .25 in
${\bf 5.} \hskip.10 in \displaystyle{\frac{\partial f}{\partial x}} \
(2,1)$ if $f(x,y)=x^2 +xy-3y^2 +4$

\vskip .25 in
${\bf 6.} \hskip.10 in \displaystyle{\frac{\partial f}{\partial y}} \
(1,0)$ if $f(x,y)=x^3 e^{-y^{2}} -x$

\vskip .25 in
${\bf 7.} \hskip.10 in \displaystyle{\frac{\partial f}{\partial x}} \
(25,16)$ if $f(x,y)=100x^{\frac{1}{2}} y^{\frac{1}{2}}$

\vskip .25 in
${\bf 8.} \hskip.10 in \displaystyle{\frac{\partial f}{\partial y}} \
(0,0)$ if $f(x,y)= \ln \ (x^2 +y^2 +1).$

\vskip .25 in
\noindent
In each Exercise 9 through 22 find the first order partial
derivatives of the fiven function.  Assume that the variables are
restricted on a domain on which the function is well defined.

\vskip .25 in
${\bf 9.} \hskip.10 in f(x,y)=5-8x+5y \hskip 1.0 in {\bf 10.}
\hskip.10 in f(x,y)=3x^2 + xy-2$

\vskip .25 in
${\bf 11.} \hskip.10 in f(x,y)=x^3 y^2 -2x^2 y+y \hskip 1.0 in {\bf
12.} \hskip.10 in f(x,y) = y^3 e^{2xy} - x^5$

\vskip .25 in
${\bf 13.} \hskip.10 in  M(r,t)=1000 e^{rt} \hskip 1.0 in {\bf 14.}
\hskip.10 in f(x,y) = \sqrt{x^2 + y^2}$

\vskip .25 in
${\bf 15.} \hskip.10 in R(s,t)= \frac{5 t^{2}}{s} \hskip 1.0 in {\bf
16.} \hskip.10 in d(a,b)=(3a+b-5)^2$

\vskip .25 in
${\bf 17.} \hskip.10 in l(x,y)= \ln \ (x^2 -5y+3) \hskip 1.0 in {\bf
18.} \hskip.10 in f(x,y)= \ln \ (x^2 +1)e^{-xy^{2}}$

\vskip .25 in
${\bf 19.} \hskip.10 in Q(K,L)=10K^{0.4} L^{0.6} \hskip1.0 in {\bf
20.} \hskip.10 in f(x,y)=16 x^{\frac{1}{8}} y^{\frac{7}{8}}$

\vskip .25 in
${\bf 21.} \hskip.10 in M(t,r)=1000 \ (1+ \frac{r}{360})^{360t}
\hskip 1.0 in {\bf 22.} \hskip.10 in P(t,K)= \frac{K}{1+e^{-0.02t}}$

\vskip .25 in
\noindent
For each function in Exercises 23 through 28 find the equation of
the tanger + plane to its graph at the given point, as well as it's
linear aproximation at the same point.

\vskip .25 in
${\bf 23.} \hskip.10 in f(x,y)=3x^2 +5y^2 ; \ (1,2) \hskip 1.0 in {\bf
24.} \hskip.10 in f(x,y)=x^2 -3y^2 ; \ (-1,2)$

\vskip .25 in
${\bf 25.} \hskip.10 in f(x,y)=e^{x^{2}-y^{4}}; \ (1,-1) \hskip1.0 in
{\bf 26.} \hskip.10 in f(x,y) = \ln \ (x^2 +3y^2 +2); \ (0,0)$

\vskip .25 in
${\bf 27.} \hskip.10 in f(x,y)=8x^{\frac{1}{2}} y^2 ; \ (4,1)
\hskip1.0 in {\bf 28.} \hskip.10 in f(x,y)=x^2 e^{yx} +y^3 ; \ (0,-2)$

\newpage

\noindent
{\bf  Exercises 6.3}

\vskip .25 in
\noindent
In Exercises 1 through 6 find the critical points of the given
function.

\vskip .25 in
${\bf 1.} \hskip.10 in f(x,y)=x^2 +y^2 -2x+4y+6 \hskip 1.0 in {\bf
2.} \hskip.10 in f(x,y)=2y^2 +x^2 +xy+3x-2y+1$

\vskip .25 in
${\bf 3.} \hskip.10 in f(x,y)=y^2 -5xy-2x^2 +3 \hskip1.0 in {\bf 4.}
\hskip .10 in f(x,y)=x^2 =3y^2 =3y-2x+8$

\vskip .25 in
${\bf 5.} \hskip.10 in f(x,y)=3xy-x^3 -y^3 +10 \hskip1.0 in {\bf 6.}
\hskip.10 in f(x,y)=y^3 +3x^2 -6xy-9y-2$

\vskip .25 in
\noindent
In Exercises 7 through 18 use the first and second derivative
tests to find the local minima, maxima, and saddle points of the
given function.  If the second derivative test is inconclusive, so
state.

\vskip .25 in
${\bf 7.} \hskip.10 in g(x,y)=x^2 -xy+y^2 +3y-1 \hskip1.0 in
{\bf8.} \hskip.10 in f(x,y)=2y^2 -xy+x^2 -7x+4$

\vskip .25 in
${\bf 9.} \hskip.10 in f(x,y)=x^2-xy+2y+x+3 \hskip1.0 in {\bf10.}
\hskip.10 in f(x,y)=8+2y-x^2 -y^2$

\vskip .25 in
${\bf 11.} \hskip.10 in f(x,y)=x^3 +y^3 -3xy+4 \hskip1.0 in {\bf
12.} \hskip.10 in f(x,y)=y^3 -x^3 -3xy+5$

\vskip .25 in
${\bf 13.} \hskip.10 in f(x,y)=8+4x+2y-x^2 -y^2 \hskip1.0 in {\bf
14.} \hskip.10 in f(x,y)=xy- \frac{1}{2} x^2 - \frac{1}{3} y^3$

\vskip .25 in
${\bf 15.} \hskip.10 in f(x,y)=x^4 +y^4 \hskip2.0 in {\bf 16.}
\hskip.10 in f(x,y)=x^4 -y^4$

\vskip .25 in
${\bf 17.} \hskip.10 in f(x,y)=1-x^4 -y^4 \hskip1.75 in {\bf 18.}
\hskip.10 in f(x,y)=1+x^4 +y^4$

\vskip .25 in
\noindent
In exercises 19 through 22 the given $P(x,y)$ is the profit
function of a company selling $x$ units of a product $X$ and $y$
units of another product $Y$.  In each case find the values of $x$
and $y$ that maximize the company's profit, as well as the
maximum profit.

\vskip .25 in
${\bf 19.} \hskip.10 in P(x,y)=8x+11y-900-0.01(x^2 +xy+2y^2)$

\vskip .25 in
${\bf 20.} \hskip.10 in P(x,y)=12x+9y-500-0.01(4x^2 +xy+y^2)$

\vskip .25 in
${\bf 21.} \hskip.10 in P(x,y)y-10,000-0.01(2x^2 +xy+3y^2)$

\vskip .25 in
${\bf 22.} \hskip.10 in P(x,y)=150x+320y-15,000-0.01(3x^2
+xy+5y^2)$

\newpage
\vskip .25 in
{\bf Answers of Exercises 6.3}

\vskip .25 in
${\bf 1.} \hskip.10 in (1,-2) \hskip .5 in {\bf 2.} \hskip.10
in (-2,1) \hskip.5 in {\bf 3.} \hskip.10 in (0,0) \hskip.5 in {\bf 4.}
\hskip.10 in (1,- \frac{1}{2}) \hskip.5 in {\bf 5.} \hskip.10 in (0,0),
\ (1,1)$

\vskip .25 in
${\bf 6.} \hskip.10 in (-1,-1), \ (3,3) \hskip.5 in {\bf 7.} \hskip.10
in$ min of $-4$ at $(-1, -2) \hskip.5 in {\bf 8.} \hskip.10 in$ min of
$-10$ at $(4, 1)$

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${\bf 9.} \hskip.10 in$  saddle point at $(5,2)$ $\hskip.5 in {\bf 10.}
\hskip.10 in$ max of 9 at $(0,1)$

\vskip .25 in
${\bf 11.} \hskip.10 in$ min of 3 at $(1,1)$, and saddle point at
$(0,0)$

\vskip .25 in
${\bf 12.} \hskip.10 in$ max of 10 at $(1,-1)$, and saddle point at
$(0,0)$

\vskip .25 in
${\bf 19.} \hskip.10 in x=300, \ y=200 \hskip.5 in {\bf 20.}
\hskip.10 in x=100, \ y=400 \hskip.5 in {\bf 21.} \hskip.10 in
x=2000, \ y=1000$

\vskip .25 in
${\bf 22.} \hskip.10 in x=2000, \ y=3000$

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