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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
\bigskip
\bigskip
\bigskip


{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Exam}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}

\newpage

\noindent {\bf 1.} (5 pts.) The rate of change of an investment is $R(t)=20t+5$. 
If $\$1000$ was initially invested, find the value after 5 years. 

\vskip .2in

a) \;\; 1325;

\vskip .1in

b) \;\; 1300;

\vskip .1in

c) \;\; 1025;

\vskip .1in

d) \;\; 1275;

\vskip .1in

e) \;\; 1225.

\vskip 2in

\noindent {\bf 2.} (5 pts.) Solve the initial value problem
$$
		\frac{dy}{dx}=3x^2+1,\ \ y(1)=2.
$$

\vskip .2in

a)\;\;$y=x^3+x+2$;

\vskip .1in

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

\vskip .1in

(c)\;\;$y=x^3+x$;

\vskip .1in

(d)\;\;$y=3x^3+x$;

\vskip .1in

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

\newpage

\noindent {\bf 3.} (5 pts.) Write the equation of the plane that contains the
points $(0,0,3)$, $(2,0,1)$, $(0,2,1)$.
\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.) Which of the following methods would you use to compute
$$
	\int xe^{2x} dx.
$$

\vskip .2in

a)\;\;Integration by parts with $u=e^{x^2}$ and $v'=x$.

\vskip .1in

b)\;\;Integration by parts with $u=x$ and $v'=e^{x^2}$.

\vskip .1in

c)\;\;Partial fractions.

\vskip .1in

d)\;\;Substitution with $u=x^2$.

\vskip .1in

e)\;\;Direct integration using formulas.
 
\newpage

\noindent {\bf 5.} (5 pts.) For which value(s) of $k$ will the following system
of equations have no solution:
$$
\gather
	       x+2y+z=3\\
        2x+5y+2z=5\\
        2x+4y+kz=5
\endgather
$$

\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 $f(x,y)=x^2-y$ at (1,0).

\vskip .2in

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

\vskip .1in

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

\vskip .1in

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

\vskip .1in

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

\vskip .1in

e) \;\;$z=x-2y$.

\vskip .1in

\newpage

\noindent {\bf 7.} (5 pts.) Compute $\int x\ln x dx$ using integration by parts.
\vskip .2in

a)\; $x^2\ln x - \frac{x^2}{2}+C$;

\vskip .1in

b)\; $\frac{x^2}{2}\ln x+\frac{x^2}{4}+C$;

\vskip .1in

c)\; $\frac{x^2}{2}\ln x-\frac{x^2}{4}+C$;

\vskip .1in

d)\; $\frac{x^2}{2}-\frac{x^2}{4}\ln x +C$;

\vskip .1in

e)\; None of the above.

\vskip 2in

\noindent {\bf 8.} (5 pts.) Suppose the function $f(x)$ takes 
the following values:

\vskip 1in

Estimate $\int^1_{-1}f(x) dx$ by computing the Riemann sum with 4 
subintervals and left endpoints. 

\vskip .2in
 
a)\;1 

\vskip .1in

b)\;1.9

\vskip .1in

c)\;2

\vskip .1in

d)\;1.1

\vskip .1in

e)\; 1.45

\newpage

\noindent {\bf 9.} Find the partial fraction decomposition of 
$$
	\frac{1}{x^2-6x+5}.
$$
That is find the numbers $A$ and $B$ such that:
$$
	\frac{1}{x^2-6x+5}=\frac{A}{x-5}+\frac{B}{x-1}.
$$

\vskip .2in

a)\;\; $A=1,B=-1$

\vskip .1in

b)\;\; $A=\frac{-1}{4},B=\frac{1}{4}$

\vskip .1in

c)\;\; $A=\frac{1}{4},B=\frac{-1}{4}$

\vskip .1in

d)\;\; $A=\frac{-1}{3},B=\frac{1}{3}$

\vskip .1in

e)\;\; $A=\frac{1}{3},B=\frac{-1}{3}$

\vskip 1.8in

\noindent {\bf 10.} (5 pts.) Find the solution of the following 
initial value problem:
$$
	\frac{dy}{dt}=4y^2t^3, \ \ y(0)=-1.
$$

\vskip .2in

a)\; $y=-\frac{1}{4t^4+1}$;

\vskip .1in

b)\; $y=-\frac{1}{t^4+1}$;

\vskip .1in

c)\; $y=\frac{1}{4t^4+1}$;

\vskip .1in

d)\; $y=\frac{1}{t^4+1}$;

\vskip .1in

e)\; None of the above.

\newpage


\noindent {\bf 11.} (5 pts.) Compute the improper integral:
$$
	\int^{\infty}_{1}\frac{2x}{(x^2+1)^2}dx.
$$

\vskip .2in

a)\;\;1/2;

\vskip .1in

b)\;\;1/4;

\vskip .1in

c)\;\;1;

\vskip .1in

d)\;\;0;

\vskip .1in

e)\;\;None of the above.

\vskip 2in

\noindent {\bf 12.} (5 pts.)\;\; Find the present value of a perpetual 
income stream flowing continuously at a rate of $\$20,000$ per year, with 
interest rate compounded continuously at $8\%$.

\vskip .2 in

a)\; 140,000

\vskip .1in

b)\; 160,000

\vskip .1in

c)\; 180,000

\vskip .1in

d)\; 200,000

\vskip .1in

e)\; 14,000

\newpage

\noindent {\bf 13.} (5 pts.) Solve the following equation for $p$ in terms of $t$.
$$
	\frac{1}{2}\ln \left(\frac{p-1}{p}\right )=2t+1.
$$


\vskip .2 in

a)\; $p=\frac{1}{1-e^2e^{4t}}$;

\vskip .1in

b)\; $p=\frac{1}{1+e^{4t}}$;

\vskip .1in

c)\; $p=\frac{1}{1-e^{4t}}$;

\vskip .1in

d)\; $p=\frac{1}{1+e^2e^{4t}}$;

\vskip .1in

e)\; $p=e^{-2-4t}$.

\vskip 2in

\noindent {\bf 14.} (5 pts.) The level curve for the function $f(x,y)=(2x^2+2y^2)^2$
with height $z=1$ is which of the following? 

\vskip .2 in

a)\; a line;

\vskip .1in

b)\; a circle;

\vskip .1in

c)\; a point;

\vskip .1in

d)\; a parabola;

\vskip .1in

e)\; does not exist.

\newpage


\noindent {\bf 15.} An individual opens an account with initial amout of 
$\$100,000$, and then makes continuous withdrawals at the rate of $\$2,000$ per year. 
We assume an interst rate of $\%5$ compounded continuously. Find the differential
equation that models the amount of money $M(t)$ in the account.

\vskip .2in

a)\; $\frac{dM}{dt}=.05M+2000,\; M(0)=100,000$

\vskip .1in

b)\;  $\frac{dM}{dt}=-.05M-2000,\; M(0)=100,000$

\vskip .1in

c)\; $\frac{dM}{dt}=.05M+2000,\; M(0)=98,000$

\vskip .1in

d)\; $\frac{dM}{dt}=-.05M+2000,\; M(0)=98,000$

\vskip .1in

e)\; $\frac{dM}{dt}=2000M-50000,\; M(0)=100,000$

\vskip 2in


\noindent {\bf 16.}According to the method of Lagrange multipliers, at which of the 
following points $(x,y)$ could the function $f(x,y)=2x^2-y^2$ possibly obtain a 
maximum value subject to constraint $x+y=2$.

\vskip .2in

a)\; $(-2, 4)$

\vskip .1in

b)\;  $(2, -4)$

\vskip .1in

c)\; $(0,-2)$

\vskip .1in

d)\; $(1,-2)$

\vskip .1in

e)\; $(-1,2)$


\newpage


\noindent {\bf 17.} A function $f(x,y)$ satisfies $\frac{\partial f}{\partial x}=0=
\frac{\partial f}{\partial y}$ at the point $(0,1)$. Suppose also that 
$\frac{\partial^2 f}{\partial x^2}=1, \frac{\partial^2 f}{\partial y^2}=4, 
\frac{\partial^2 f}{\partial x \partial y}=5$ at the point $(0,1)$. Which of the 
following is true?

\vskip .2in

a)\; (0,1) is a relative maximum point

\vskip .1in

b)\;  (0,1) is a saddle point

\vskip .1in

c)\; (0,1) is a relative minimum point

\vskip .1in

d)\; (0,1) is not a critical point

\vskip .1in

e)\; (0,1) is a critical point, but the second derivative is 
inconclusive

\vskip 2in


\noindent {\bf 18.} Assume that $f(1,2)=4, \frac{\partial f}{\partial x}(1,2)=2,
\frac{\partial f}{\partial y}(1,2)=3$. Find an aprroximation for $f(.8,2.1)$.
\vskip .2in

a)\; 4.1

\vskip .1in

b)\; 3.9

\vskip .1in

c)\; 4.2

\vskip .1in

d)\; 3.8

\vskip .1in

e)\; 4.3



\newpage

\noindent {\bf 19.} For a certain item the demand curve is $D(q)=-2q+12$, and the supply curve is
$S(q)=3q+2$. 

\vskip .1in

a)\; Find the equilibrium price and the equilibrium quantity.

\vskip 1.5in

b)\; Find the consumer surplus.

\vskip 1.5in

c)\; Find the producer surplus.                              


\newpage

\noindent {\bf 20.} Consider the functions $f(x)=2x, g(x)=x^2-x$.

\vskip .1in

a)\; Find the values of the points where the graph of $f(x)$ and $g(x)$ intersect.

\vskip 1.5in

b)\; Draw the graphs of $f(x)$ and $g(x)$ using the same coordinate system. Shade 
the region enclosed by the graphs of $f(x)$ and $g(x)$.

\vskip 2in

c)\; Write the definite integral that gives the area of the region
enclosed by the graphs of $f(x)$ and $g(x)$.                               


\end

