%First version of exam 2

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\def \sm {\smallskip\noindent}
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\def \hs {~~~~~~}
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\def \ds {\displaystyle}

\begin{document}


\centerline{\bf MULTIPLE CHOICE SECTION - EACH PROBLEM IS WORTH  5 POINTS }
\centerline{\bf (In Problems 1-12 mark the correct answer on the front cover)}
\bigskip

{\bf 1.} If \$12,000 is deposited in an account earning 8\% continuously
compounded
interest, how much money is in the account 8 years later?

\mm
(a) $8000e^{-.64}$ \hfil (b) $12,000e^{.8}$ \hfil (c) $8000e^{.96}$ \hfil
(d) $12,000e^{.64}$ \hfil (e) $12,000e^{-.8}$

\vskip1.5in

{\bf 2.} Find the constant $k$ such that the function $y = x^4+k$
is a solution to the differential equation $xy'- 4y = -16$.

\mm
(a) $-6$ \hfil (b) 6 \hfil (c) $4$ \hfil (d) $-2$ \hfil (e) $-4$

\vskip1.5in

{\bf 3.}
A certain investment pays 5\% interest compounded continuously.  Suppose
that deposits
are made into the account in the following manner.  An initial deposit of
\$5000 is
made.  After 2 years, additional money is deposited continuously into the
account at the rate of \$1000 per year.  This lasts for 4  years.  How much
money is in the account 6 years after the initial investment?

\mm
(a) $5000e^{.3} + \int_0^61000e^{.05(6-t)}dt$ \hfil
(b) $\int_0^65000e^{.05(6-t)}dt$

(c) $5000e^{.3} + \int_0^41000e^{.05(4-t)}dt$ \hfil \hh
(d) $e^{.1}\int_0^41000e^{.05(4-t)}dt$

(e) $5000e^{.3}\int_0^41000e^{.05(4-t)}dt$

\newpage
{\bf 4.} Widgets-R-Us makes a profit of \$500,000 per day when they produce
(and sell) 10 widgets.  The company finance director estimates
that the marginal profit function for widget production is
$ -2x + 20$ in {\em thousands} of dollars per day.
What will be the change in profit (in thousands of dollars) if the company
increases
production from 10 to 20 widgets?

\mm
(a) $-\$100$ \hfil (b) \$600 \hfil (c) \$100 \hfil (d) $-\$400$ \hfil
(e) \$400

\vskip2in

{\bf 5.}  Compute the improper integral $\ds\int_{-\infty}^{-1}
\frac{1}{x^3}dx$.

\mm
(a) $-1$ \hfil (b) $-1/2$ \hfil (c) 1/2 \hfil (d) 1 \hfil (e) the integral
diverges

\vskip2in

{\bf 6.}
Compute the average value of $f(x) = 3x^2$ on $ 1 \le
x \le 4$.

\mm
(a) 189 \hfil (b) 63 \hfil (c) 42 \hfil (d) 21 \hfil (e) 9

\newpage
{\bf 7.} Find the present value of a perpetual income stream
flowing continuously at a rate  of \$32,000 per year,  and with interest
compounded continuously at the rate of $ 8\%$.

\mm
(a) \$400,000 \hfil (b) \$380,000 \hfil (c) \$390,000
  \hfil (d) \$410,000\hfil (e) \$420,000

\vskip1.7in

{\bf 8.}  Use the trapezoidal rule with 3 subintervals to approximate
the value of $\int_0^1 e^{x^2}dx$.

\mm
(a)  $\frac13(1 + 2e^{\frac19} + 2e^{\frac49} + e)$ \hfil
(b) $\frac13(e^{\frac19} + e^{\frac49} + e)$ \hfil
(c) $\frac12(1 + 2e^{\frac19} + 2e^{\frac49} + e)$

(d) $\frac13(1 + e^{\frac19} + e^{\frac49})$ \hs\hs\hs
(e)  $\frac16(1 + 2e^{\frac19} + 2e^{\frac49} + e)$

\vskip2in

{\bf 9.} After graduating from ND, Jane Q.\ Student begins to pay back her
\$60,000
student loan. She pays money continuously at the rate of \$4000 per
year and is charged 4\% interest (continuously compounded).  Let $M(t)$ be
the amount of money she still owes on the loan. Set up an initial value
problem which
models this situation.

\mm
(a) $\displaystyle\frac{dM}{dt} = .04M + 8000, \hh M(10) = 0$ \hfil \hs
(b) $\displaystyle\frac{dM}{dt} = .04M - 4000, \hh M(0) = 60,000$

(c) $\displaystyle\frac{dM}{dt} = .04M - 60,000, M(0) = 4000$ \hfil
(d) $\displaystyle\frac{dM}{dt} = .04M + 10,000, M(10) = 0$

(e) $\displaystyle\frac{dM}{dt} = .04M + 4000, M(0) = 60,00$

\newpage
{\bf 10.} A person deposits money into an account continuously at the rate of
$s(t) = 1000 + 100t$ dollars per year for 12 years.  Assuming that the
account pays
5\% continuously compounded interest, what is the present value of this
investment?

\mm
(a) $\int_0^{12}1000e^{.05(12 - t)}dt$ \hfil \hs\hh
(b) $\int_0^{12}(1000t + 50t^2)e^{-.05t}dt$

(c) $\int_0^{12}(1000 + 100t)e^{.05(12 - t)}dt$ \hfil
(d) $\int_0^{12}(1000 + 100t)e^{-.05t}dt$

(e) $\int_0^{12}1000e^{-.05t}dt$ \hfil


\vskip1.3in

{\bf 11.} To solve the differential equation $\frac{dp}{dt} = 3p - 4p^2$, which
of the following pairs of integrals would you need to compute?

\mm
(a) $\ds\int\ln(3p-4p^2)dp$ \hh and $\ds\int1dt$ \hfil
(b) $\ds\int(3p-4p^2)dp$ \hh and $\ds\int1dt$

(c) $\ds\int\frac{1}{3p-4p^2}dp$ \hh and $\ds\int t dt$ \hfil
(d) $\ds\int\frac{1}{3p}dp$ \hh and $\ds\int 4t^2dt$

(e) $\ds\int\frac{1}{3p-4p^2}dp$ \hh and $\ds\int1dt$


\vskip1.3in

{\bf 12.} Your little sister opens her first bank account with \$100.  The
account pays 6\% interest compounded continuously and she wants to know how
many years it will be until her money doubles?

(a) $\ds\frac{10}{3}$ \hfil (b) $\ds\frac{\ln2}{.06}$ \hfil (c) $\ds100e^{.6}$
\hfil (d) $\ds\frac{\ln(.5)}{.06}$ \hfil (e) $\ds100e^{-.6}$

\newpage

\centerline{\bf PARTIAL CREDIT SECTION}
\centerline{\bf (In Problems 13-15 show all work on the paper)}
\bigskip

{\bf 13.}   (Partial credit, 20 points)
The supply and demand curves for a certain item are
$$p = S(q) = q+1 \hs \text{and} \hs p = D(q) = \frac{64}{q + 1}.$$

(a) Find the equilibrium price and equilibrium quantity.

\vskip1.5in

(b) Set up but do {\bf not} evaluate the expression which gives the
consumer  surplus.

\vskip1in

(c)  Set up but do {\bf not} evaluate the expression which gives the
producer surplus.

\vskip1in
(d) Give a rough sketch of the graphs of the demand and supply curves,
indicating
the equilibrium price and quantity, and the regions whose area represent the
consumer and producer surplus.

\newpage
{\bf 14.} (Partial credit, 10 points)
Consider the differential equation $\ds\frac{dy}{dt} = 3t^2y^2$.

(a) Find the general solution.

\vskip4in
(b) Find the particular solution which satisfies $y(1) = -1/8$.

\newpage
{\bf 15.} (Partial credit, 10 points)
Consider the Logistic differential equation $\ds\frac{dp}{dt} = 2p - 6p^2$.
To solve the differential equation, from the method of separation of
variables, we get the equation
$$
\ln\Bigl|\frac{p}{1-3p}\Bigr| = 2t + C.
$$
Find the general solution to the differential equation by solving this equation
for $p$ in terms of $t$.

\end{document}

