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\topmatter

\vfil\eject\output={}
\vbox{{\noindent
{\bf Name:}  \vrule height .4pt depth 0 pt width 1.4
in  \hskip .5 in {\bf Instructor:}  Viorel Nitica
\bigskip
\bigskip
\bigskip

\centerline{\bf {Math 108.02, Calculus II for Business}}
\centerline{\bf Fall Semester 1998}
\centerline{\bf  Exam 3 }
\centerline{\bf {Wednesday, November 18, 3:00-3:55 PM}}
\bigskip
\hrule
\bigskip
This Examination contains {\bf 15} problems, worth a total of  100  points,
on (10) sheets of paper including the front cover.
The first {\bf 11} problems are multiple choice with no partial credit, and
each is worth 5 points. Record your answers to these problems by placing
an $\times$ through one letter for each problem below:

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\bigskip
\bigskip
The last {\bf four} problems are partial credit problems worth
a total of 45 points. For these problems, {\bf show} your
computations and {\bf clearly} mark your answers on the page.
Books and notes are not allowed. You may use your calculator.
\bigskip
\hrule
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{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Exam}":
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\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
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\centerline{\bf GOOD LUCK}

\newpage

\noindent {\bf 1.} (5 pts.) Find the linear function f(x,y) which satisfies
$f(0,1)=5, f(1,0)=3, f(1,-1)=0$. 

\vskip .2in

a) \;\;$f(x,y)=3x+5y$;

\vskip .1in

b) \;\;$f(x,y)=3y+2x+2$;

\vskip .1in

c) \;\;$f(x,y)=x+3y+2$;

\vskip .1in

d) \;\;$f(x,y)=x+3y-2$;

\vskip .1in

e) \;\;$f(x,y)=3x+2y-2$.

\vskip 2in

\noindent {\bf 2.} (5 pts.) Assume that $x,y,\lambda$ are the dimensions of 
a box. Solve the following system of equations for $x,y,\lambda$:
$$
\gather
	xy+y\lambda=15\\
	xy+x\lambda=7\\
	\lambda x+\lambda y=16.
\endgather
$$

\vskip .2in

a)\;\;(2,3,4);\;(b)\;\;(1,2,3);\;\;(c)\;\;(1,3,4);\;\;(d)\;\;(1,2,4);\;\;(e)\;\;(2,3,2).

\newpage

\noindent {\bf 3.} (5 pts.) Write the equation of the plane that contains the
points $(0,3,0)$, $(3,0,0)$, $(0,0,-3)$.
\vskip .2in

a)\;\;$z=x+y+3$

\vskip .1in

b)\;\;$z=-x-y+3$

\vskip .1in

c)\;\;$z=x+y-3$

\vskip .1in

d)\;\;$z=-3x-3y-3$

\vskip .1in

e)\;\;None of the above.

\vskip 2in

\noindent {\bf 4.} (5 pts.) Write the equation of the plane that contains the
points $(3,4,-1)$, $(2,4,0)$, $(3,-1,-1)$.

\vskip .2in

a)\;\;$z=-x+2$

\vskip .1in

b)\;\;$z=-y+2$

\vskip .1in

c)\;\;$z=3x+4y-1$

\vskip .1in

d)\;\;$z=x-y$

\vskip .1in

e)\;\;None of the above.
 
\newpage

\noindent {\bf 5.} (5 pts.) For which value(s) of $k$ will the following system
of equations have infinitely many solutions:
$$
\gather
	       x+y+z=3\\
        x+2y+2z=4\\
        2x+2y+2kz=6
\endgather
$$

\vskip .2in

a)\;\;k=0,-1;

\vskip .1in

b)\;\;k=1;

\vskip .1in

c)\;\;k=-1;

\vskip .1in

d)\;\;k=2;

\vskip .1in

e)\;\;None of the above.

\vskip 2in

\noindent {\bf 6.} (5 pts.) Find the equation of the tangent plane to the graph
of the function $f(x,y)=x^2+ye^x$ at (1,1).

\vskip .2in

a) \;\;$z=x-2y-1$

\vskip .1in

b) \;\;$z=3x+y-2$

\vskip .1in

c) \;\;$z=3x-y+1$

\vskip .1in

d) \;\;$z=x-2y+1$

\vskip .1in

e) \;\;None of the above.

\vskip .1in

\newpage

\noindent {\bf 7.} (5 pts.) Which of the following is the function that must be
minimized in order to find the Least Squares Fit straight line ($y=ax+b$)
through the three points $(0,1), (1,3), (2,4)$?
\vskip .2in

a)\; $E(a,b)=(b-1)^2+(a+b-3)^2+(2a+b-4)^2$;

\vskip .1in

b)\; $E(a,b)=(a+b-1)^2+(2a+b-3)^2+(2a+b-4)^2$;

\vskip .1in

c)\; $E(a,b)=(b-1)^2+(2a+b-2)^2+(3a+b-3)^2$;

\vskip .1in

d)\; $E(a,b)=(2a+b)^2+(a+b-1)^2+(2a+b-3)^2$;

\vskip .1in

e)\; $E(a,b)=(b-2)^2+(a+b-3)^2+(2a+b-3)^2$.

\vskip 2in

\noindent {\bf 8.} (5 pts.) Suppose that $f(x,y)=xe^y+ye^x+ln(xy)$. Find 
$\frac{\partial^2f}{\partial x^2}$.

\vskip .2in
 
a)$\frac{\partial^2f}{\partial x^2}=ye^x-\frac{1}{x^2}$ 

\vskip .1in

b)$\frac{\partial^2f}{\partial x^2}=xe^y-\frac{1}{y^2}$ 

\vskip .1in

c)$\frac{\partial^2f}{\partial x^2}=ye^x+\frac{1}{x}$ 

\vskip .1in

d)$\frac{\partial^2f}{\partial x^2}=xe^y+\frac{1}{y}$ 

\vskip .1in

e)\; None of the above.

\newpage

\noindent {\bf 9.} (5 pts.) A factory makes two kinds of widgets:$\alpha$-type
and $\beta$-type. The profit from selling $x$ widgets of $\alpha$-type, and $y$
widgets of $\beta$-type is a function $P(x,y)$. Assume that
$P(100,200)=\$80,000, \frac{\partial P}{\partial x}(100,200)=500, \frac{\partial
P}{\partial y}(100,200)=1000$. Estimate the profit of the company if it sells
80 widgets of $\alpha$-type and $180$ widgets of $\beta$-type. 

\vskip .2in

a)\;\; 80,000;

\vskip .1in

b)\;\; 90,000;

\vskip .1in

c)\;\; 100,000;

\vskip .1in

d)\;\; 110,000;

\vskip .1in

e)\;\; 120,000.

\vskip 1.8in

\noindent {\bf 10.} (5 pts.)\;A function $f(x,y)$ satisfies $\frac{\partial f}{\partial x}=
0= \frac{\partial f}{\partial y}$ at the point $(-3, 4)$. Suppose also that
$\frac{\partial^2f}{\partial x^2}=3$, $\frac{\partial^2f}{\partial y^2}=4$,
$\frac{\partial^2f}{\partial x \partial y}=3$ at the point $(-3,4)$. Which of the following is
true?

\vskip .2in

a)\; $(-3,4)$ is a relative maximum point;

\vskip .1in

b)\; $(-3,4)$ is a saddle point;

\vskip .1in

c)\; $(-3,4)$ is a relative minimum point;

\vskip .1in

d)\; $(-3,4)$ is not a critical point;

\vskip .1in

e)\; $(-3,4)$ is a critical point, but the second derivative test is inconclusive.

\newpage


\noindent {\bf 11.} (5 pts.)\;\;At which point is the $x$-slope of the surface
$z=x^2+xy$
{\bf not} equal to 3?

a)\;\;(1,1)\;\;\;b)\;\;(0,3)\;\;\;c)\;\;(2,-1)\;\;\;d)\;\;(-2,4)\;\;\;e)\;\;(3,-3).

\vskip 2in

\noindent {\bf 12.} (5 pts.)\;\; Four of the vertices of a cube are $(1,1,1)$, 
$(4,4,4)$, $(1,1,4)$, and $(4,1,1)$. Find the coordinates of the center of
the cube.

\newpage


\noindent {\bf 13.} (10 pts.)\;\; Consider the function $f(x,y)=2x^2+2y^2$. Sketch the
level curves for $f(x,y)$ at the values $z=-4, 0, 8, 32$ on the following graph.
Label each curve. If a level curve does not exist, say so.  


\newpage


\noindent {\bf 14.} (15 pts.)\;\; The volume of a cylindrical beverage 
can {\bf with open top} is 340 cubic centimeters.

\;\;a) Assume that you want to find the radius of the basis and the height of the can which
minimize its surface area. Write the related constrained optimization problem.

\vskip 1.8in

\;\;b) Using the method of Lagrange multipliers, set up a system of equations whose
solution gives the possible points at which the minimum could occur. 

\vskip 1.8in

\;\;c) Solve the system of equations.

\newpage

\noindent {\bf 15} (15 pts.)\;\; A firm manufactures and sells two products, $X$ and $Y$,
that sells for $\$10$ and $\$13$ each, respectively. The cost of producing $x$ units of $X$,
and $y$ units of $Y$ is
$$
	C(x,y)=100-2x+5y+0.01(2x^2+xy+2y^2).
$$
Find the values of $x$ and $y$ that maximize the firm's profit. 

\end

