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\def\ans{\vskip 4cm \hfill Answer:\vrule
height0pt depth.5pt width6cm}




 {\bf \centerline{ Part I: Multiple Choice
Questions {\sl (5 points each)}}}
\bigskip

1.   \rm   Which of the following
expressions is equivalent to $e^{\ln t + \ln
(1/t)}$?

$$\hbox{(a)} \quad 1 \qquad  \hbox{(b)} \quad
t\ln{1\over t} \qquad  \hbox{(c)} \quad 0
\qquad  \hbox{(d)}
\quad t+{1\over t} \qquad  \hbox{(e)} \quad
t+\ln({1\over t})
$$
\vfill

2.   \rm
  Given a function $y=f(x)$, we can calculate
the derivative by using the formula
$$\qquad\hbox{(a)}\quad f'(x) = \lim_{h
\to 0}{f(x+h)+f(x)\over h}
\qquad\qquad
\hbox{(b)} \quad f'(x) = \lim_{h \to
0}{f(x)-f(h)\over h} \qquad\qquad
$$
$$\quad\ \ \hbox{(c)}  \quad
f'(x) =
\lim_{h \to 0}{f(x+h)-f(x)\over h}
 \qquad\qquad
\hbox{(d)}
\quad f'(x) = \lim_{h \to 0}{f(x+h)-f(x)\over
x-h}
\qquad  $$
$$\hbox{(e)}\quad f'(x) = \lim_{h \to
0}{f(h)-f(x)\over h-x} \qquad$$
\vfill












3.  \rm   If you deposit \$500
in an account paying $4\%$ interest compounded
continuously, how much will be in the account
after 20 years?
$$\hbox{(a)} \quad 4e^{20} \qquad
\hbox{(b)}\quad 500e^{80} \qquad  \hbox{(c)}
\quad 500(1.04)^{20}
\qquad \hbox{(d)} \quad 4e^{10000} \qquad
\hbox{(e)}\quad 500e^{0.8}
$$\vfill\eject


4.  \rm   Which of the following
limits would you obtain when using the
definition of the derivative to compute
$f'(x)$ where $f(x) = x^2-x$?

$$\qquad\hbox{(a)} \quad \lim_{h \to 0}
{2xh+h^2-h-xh^2
\over 2}
  \qquad\qquad\qquad\qquad \hbox{(b)}   \quad
\lim_{h
\to 0} {x^2h-xh-h^2+h \over h}$$
$$\quad\hbox{(c)} \quad \lim_{h \to 0}
{2xh+h^2-h
\over h}
 \qquad\qquad\qquad\qquad\qquad\quad \hbox{(d)}
\quad
\lim_{h
\to 0} {h^2-h-x^2+x \over h-x}$$
$$ \hbox{(e)} \quad \lim_{h \to 0} {2xh+h^2-h
\over x-h}  $$
\vfill


\bigskip  { 5.} What is  the derivative
$f'(x)$ of the function
$f(x)=3x^4-{4\over{\sqrt x}}\ $?\bigskip
\hskip 10pt { (a)}\hskip 10pt $12x^3-2x^{-3/2}
$
 \hskip 40pt {(b)}
\hskip 10pt $3x^3-4x^{-3/2}$\hskip 40pt
{(c)}\hskip 10pt $12x^4-2x^{-1/2}$
\bigskip \hskip 70pt { (d)}\hskip 10pt
$3x^3+4x^{-3/2}$  \hskip 40 pt {(e)}
\hskip 10pt $12x^3+2x^{-3/2} $

\vfill

{ 6.}  A company finds that if it produces
and sells  $x$ boxes of assorted chocolates
per week, its profit (in dollars) is $-1000+15x
+0.02 x^2$. What is its marginal profit (in
dollars) at a production level of $100$ boxes
per week?
\bigskip
\hskip 10pt { (a)}\hskip 10pt \$15.02
 \hskip 70pt {(b)}
\hskip 10pt $ \$19.00 $ \hskip 70pt
{(c)}\hskip 10pt $\$15.04$
\bigskip \hskip 70pt { (d)}\hskip 10pt
$\$17.00 $  \hskip 70pt
{(e)}\hskip 10pt $\$15.20$
 \vfill\eject




 { 7.} If $y=e^{2t}-2e^t$, which of the
following functions is equal to ${dy\over dt}\
$?
\bigskip
\hskip 20pt { (a)}\hskip 10pt $2te^t-2 $
 \hskip 40pt {(b)}
\hskip 10pt $2te^{2t-1}-2te^{t-1}$\hskip 40pt
{(c)}\hskip 10pt $e^{2t}-2e^t$
\bigskip \hskip 70pt { (d)}\hskip 10pt
$ 2e^t-2$  \hskip 60 pt {(e)}
\hskip 10pt $2e^{2t}-2e^t$
\vfill

8. If $g(x)=3x^4-2x^3+5x-\ln x$, which of the
following functions is equal to $g''(x)$ ?
\bigskip
\hskip 10pt (a)\hskip 10pt
$36x^4-12x^3+{1\over{x^2}}$
 \hskip 20pt (b)\hskip 10pt
$3x^2-2x-{1\over{x^2}}$ \hskip 20pt(c) \hskip
10 pt $36x^2-12x+{1\over
{x^2}}$
\bigskip
\hskip 70pt (d) \hskip 10pt
$36x^4-12x^3-{1\over{x^2}}$\hskip 30pt
(e)\hskip 10pt $3x^2-2x+{1\over{x^2}}$
\vfill

{ 9.}  What is the   point on the curve
$y=\ln x$ where the tangent line is parallel
to the line $3y=2x+1$?
\bigskip
\hskip 10pt { (a)}\hskip 10pt
$(e^{3/2},
{3\over 2})$
 \hskip 70pt
{(b)}
\hskip 10pt $({2\over 3}, \ln {2\over3})$
\hskip 70pt  {(c)}\hskip 10pt
$(e^{2/3},{2\over 3})$
\bigskip \hskip 70pt { (d)}\hskip 10pt
$ ({3\over 2}, \ln {3\over2}) $  \hskip 70 pt {(e)}\hskip 10pt
$(e^{-3/2},-3/2)$
 \vfill

 { 10.}  With an inflation rate of
 $2\%$ per annum, prices are described by the
formula $P=P_0(1.02)^t$ where $P$ is the price
in dollars after $t$ years and $P_0$ is the
price when $t=0$.
How fast are prices rising in dollars per year
when $t=3$?
\bigskip
\hskip 10pt { (a)}\hskip 10pt
$P_0 (1.02)^3$
 \hskip 40pt {(b)}
\hskip 10pt
$P_0 (1.02)^3\ln (1.02) $\hskip
40pt  {(c)}\hskip 10pt $P_0\ln (1.02)$
\bigskip \hskip 70pt { (d)}\hskip 10pt
$3P_0 (1.02)^2\ln (1.02) $  \hskip 40 pt
{(e)}
\hskip 10pt $
 0.02 P_0  $
\vfill
\eject \bigskip
{\bf {\centerline{ Part II: Partial Credit
Questions}}}
\bigskip

Show all work and put your final answer in the
space provided. You will receive no credit if
the answer is not in the space provided and no
partial credit for a wrong answer if you do
not show your work.

Answers may be approximated by decimals if
desired, but full points will also be awarded
for correct answers left in exact form in
terms of logarithms or exponentials.\bigskip

11.  \sl (5 points) \rm
Find
all  values of $x$ which satisfy the
equation $4+\ln (x^3) =
10$.
\ans
\vfill

12.  \sl (5 points) \rm   You are considering
investing some money with Investments 'R' Us.
When you do some research, you find that they
plan to invest your money in bonds paying 7\%
interest compounded continuously. How long
will it take Investments 'R' Us to triple your
investment?
\ans\vfill\eject

13.  \sl (6 points) \rm   Tritium, a synthetic
form of hydrogen, satisfies the exponential
decay law, $y=Ae^{-kt}$, where $t$ is the time
(in years) and
$y$ is the amount (in grams) at time $t$.
 If an absent-minded scientist receives a
shipment of 200 grams of tritium and finds that
only 50 grams of it are left when he remembers
the sample 25 years later, find a function
that gives the amount of tritium remaining in
the sample $t$ years after the scientist
received the shipment.
\ans \vfill




{ 14.} {\sl (6 points)} Find the equation of
the tangent line to the curve $y=2x^3-2x$ at
the point
$(1,0)$.
\ans\vfill\eject

{ 15.} A toy rocket launched straight up from
the top of a building  has a height $s$ in feet
above ground given by the formula $s=96
+80t-16t^2$, where $t$ denotes the time in
seconds after launch.
\bigskip { (a)} {\sl (4 points)}  When is the
velocity of the rocket zero?\ans
\bigskip{(b)} {\sl (4 points)} What is
the  velocity of the rocket when it strikes the
ground?\ans
\vfill { 16.}
  {\sl (5 points)}
Let $P(x)$ be the profit
made by a company selling $x$ midsize
cars per year. Suppose that
$P(100)=60,000$ and $P'(100)=1,000$. Estimate
the profit from the sale of $102$ cars per
year.\ans\vfill\eject

17. {\sl (5 points)}
Calculate the
derivative of the function
$y=\ln{\root 3 \of x}$.\ans

\vfill



{ 18.}  The distance $s$ in
feet travelled by a car during the first ten
seconds after it starts from rest is given by
the formula
$s={5\over 2}t^3-{4\over 7}t^{7/2}$, where $t$
is the time in seconds.
\bigskip (a) {\sl (4 points)} What is the
average velocity of the car in the first
second. \ans
\bigskip (b) {\sl (6 points)} What is the
 acceleration of the car after
$4$ seconds?\ans
 \vfill\eject\bye
