\documentstyle[12pt]{article}

\parindent 0cm

\parskip .5cm

\begin{document}

{\bf 1. } Find the domain of the function \ \ $\displaystyle {1
\over {\sin(x-3)}}$.

(a) $x \neq 3+\pi n, \ n = 0,\pm 1, \dots$

(b) $x \neq \pi$

(c) all $x$

(d) $3 < x < 3+\pi$

(e) $3 < x < 6+2\pi$

\bigskip

{\bf 2. } If $f(x) = x^3$, $g(x) = \sin(x-2)$, find the composite
function $f(g(x))$.

(a) $\sin^3 (x-2)$

(b) $\sin (x^3-2)$

(c) $x^3 \sin(x-2)$

(d) $x^3 + \sin (x-2)$

(e) $\sqrt[3] {\sin(x-2)}$

\pagebreak

{\bf 3. } Find $\displaystyle \lim_{x \rightarrow 0^{-}} f(x)$ if
\[
f(x) =
\left \{
\begin{array}{rl}
1-x & \hbox{ if } x \leq 0 \\
2   & \hbox{ if } x>0
\end{array}
\right.
\]

(a) 1

(b) 2

(c) does not exist

(d) $\displaystyle {3 \over 2}$

(e) 0

\bigskip

{\bf 4. } Evaluate the limit if it exists:
\[
\lim_{x \rightarrow 2} {{x-2} \over {x^2-3x+2}}
\]

(a) $\displaystyle 1$

(b) $\displaystyle -{1 \over 3}$

(c) $\displaystyle {1 \over 3}$

(d) $-1$

(e) The limit does not exist

\pagebreak


{\bf 5. } Evaluate the limit if it exists:
\[
\lim_{t \rightarrow 1} \left [ {1 \over {t-1}} -
{1 \over {t^2 -1}} \right ]
\]

(a) the limit does not exist

(b) 0

(c) $\displaystyle {1 \over 2}$

(d) $\displaystyle -{1 \over 2}$

(e) 1

\bigskip

{\bf 6. } Find a constant $c$ that makes
$g(x)$ continuous on $(-\infty,\infty)$.
\[
g(x) = \left \{
\begin{array}{ll}
cx+1 & \hbox{if } x \leq 2 \\
x+c & \hbox{if } x>2
\end{array}
\right.
\]

(a) $c=1$

(b) No value of $c$ makes $g(x)$ continuous

(c) $c=2$

(d) $c = 3$

(e) $\displaystyle c = -{1 \over 2}$

\pagebreak

{\bf 7. } Which one of the following is true about
$\displaystyle  \lim_{h \rightarrow 0}
{{\sqrt{4+h} -2} \over h}$?

(a) It is the derivative of $f(x) = \sqrt{4+x}$ at
$x=0$.

(b) It is undefined.

(c) It is the derivative of $f(x) = \sqrt{4+x}$ at
$x=4$.

(d) It is the derivative of $f(x) = \sqrt{4+x}$ at
$x=16$.

(a) It is the derivative of $f(x) = \sqrt{4+x}$ at
$x=2$.

\bigskip

{\bf 8. }  Differentiate $\displaystyle f(x) = {{x+2}
\over {3x+4}}$.

(a) $\displaystyle {{-2} \over {(3x+4)^2}}$

(b) $\displaystyle {{2x+2} \over {(3x+4)^2}}$

(c) $\displaystyle {{6x-2} \over {(3x+4)^2}}$

(d) $\displaystyle {{10} \over {(3x+4)^2}}$

(e) $\displaystyle {{6x+10} \over {(3x+4)^2}}$

\bigskip

{\bf 9. } Find the limit
$\;\;\;\displaystyle{\lim_{x\rightarrow0}\frac{\sin^2(5x)}
{x^2}}\;.$

\vskip .1 in
\noindent
(a)  $25$ \hfil
(b) $5$ \hfil
(c)  $\displaystyle{\frac 15}$ \hfil
(d)  $1$  \hfil
(e)  doesn't exist

\bigskip

{\bf 10. } If $f(x) =3\tan x$, what is $f'(x)$ at $x=0$?
\vskip .1 in
\noindent
(a)  $3$ \hfil
(b) $0$ \hfil
(c)  $\displaystyle{\frac 13}$ \hfil
(d)  $9$  \hfil
(e) $\displaystyle{\frac 43}$

\bigskip

{\bf 11. }  Let $f$ and $g$ be two functions such that
$f'(3)=7\;,\;f'(5)=1\;,\;g(3)=5\;,\;g'(3)=2$ . If
$F(x)=f(g(x))$ find
$F'(3)$ .

\vskip .1 in
\noindent
(a) 2 \hfil
(b)  14 \hfil
(c)  35 \hfil
(d)  5 \hfil
(e)  10

\bigskip

{\bf 12. } Find $\displaystyle{\frac{dy}{dx}}\;,$  given that
$y$ is defined implicitly as a function of $x$ by the equation
$$x^2+xy+y^3=1$$

\vskip .1 in
\noindent
(a) $\displaystyle{-\frac{2x+y}{x+3y^2}}$  \hfil
(b) $\displaystyle{-\frac{x+2y}{3x^2+y}}$  \hfil
(c) $\displaystyle{\frac{2x-y}{x+3y^2}}$  \hfil
(d)  $\displaystyle{-\frac{x+y}{3x+y^2}}$ \hfil
(e) $\displaystyle{\frac{x+y^2}{x+3y^2}}$

\bigskip

{\bf 13. } The position function of a particle moving along a
straight line is
\ \ $\displaystyle{s=2t^3-9t^2}\;.$ Find its acceleration when
$t=1$.

\vskip .1in
\noindent
(a) $-6$ \hfil
(b) $-12$ \hfil
(c) $-7$ \hfil
(d) $-3$ \hfil
(e) $12$

\bigskip

{\bf 14. } If $xy=1$ and $\displaystyle{\frac{dx}{dt}}=4$, find
$\displaystyle{\frac{dy}{dt}}$
when $x=2$.

\vskip .1 in
\noindent
(a) $-1$ \hfil
(b) $1$ \hfil
(c) $0$ \hfil
(d) $-2$ \hfil
(e) $2$

\pagebreak

{\bf 15. } An ice cube is melting and the length $x$ of its side
changes at a rate $\displaystyle{\frac{dx}{dt}=-0.1}\;{\rm cm}/{\rm min}$. At
what rate (in
${\rm cm}^3/{\rm min}$) is the volume
of the ice cube changing at the time when $x=1\;{\rm cm}$ ?

\vskip .1in
\noindent
(a) $-0.3$ \hfil
(b) $-0.1$ \hfil
(c) $0$ \hfil
(d) $-0.6$ \hfil
(e) $-0.9$

\bigskip

{\bf 16. } The absolute maximum value of $f(x)=x^3-12x+1$ over
the  interval $[-3,5]$ is

\vskip .1in
\noindent
(a) 66 \hfil
(b) 17 \hfil
(c) 10 \hfil
(d) 93 \hfil
(e) 42

\bigskip

{\bf 17. } Compute
\[
\lim_{x\to \infty} \frac{3x^3-3x}{\sqrt{4x^6+x}}
\]

(a)  $\displaystyle \frac{3}{2}$

(b) $\displaystyle \frac{3}{\sqrt{2}}$

(c) $0$

(d)  $\displaystyle \frac{3}{4}$

(e) does not exist

\pagebreak

{\bf 18. } Let
\[
y=f(x)=-4x^3+12x^2+36x-43.
\]
Find the interval(s) where $y=f(x)$ is increasing.

(a) $[-1,3]$

(b) $(-\infty,-1]$ and $[3,\infty)$

(c) $(-\infty,\infty)$

(d) $(-\infty,1]$ and $[3,\infty)$

(e) $[-3,1]$

\bigskip

{\bf 19. } Let
\[
y=f(x)=x^2+\frac{2}{x}.
\]
Which of the following is true?

\begin{itemize}
\item[(a)] $f(x)$ has a local minimum at $x=1$ and its graph is
concave up at $x=1$.

\item[(b)] $f(x)$ has a local minimum at $x=1$ and its graph is
concave down at $x=1$.

\item[(c)] $f(x)$ has a local maximum at $x=1$ and its graph is
concave up at $x=1$.

\item[(d)] $f(x)$ has a local maximum at $x=1$ and its graph is
concave down at $x=1$.

\item[(e)] $f(x)$ has neither a local maximum nor a local minimum
at $x=1$.
\end{itemize}

\pagebreak

{\bf 20. } Which of the following graphs has exactly one
inflection point?

(a) \hskip 3in (b)

\vskip 1.2in

(c) \hskip 3in (d)

\vskip 1.2in

(e)

\bigskip

{\bf 21. } Lilliputian postal requirements specify that
rectangular packages must have height plus the perimeter of the
base at most 12 inches. Find the dimensions (in inches) of the
rectangular box with square base of greatest volume that is
mailable in Lilliput.

(a) $2 \times 2 \times 4$

(b) $6 \times 6 \times 0$

(c) $6 \times 6 \times 2$

(d) $2 \times 2 \times 8$

(e) $4 \times 4 \times 4$

\pagebreak

{\bf 22. } Find the most general antiderivative of $f(x)=2\cos
x+5$.

(a) $2 \sin x + 5x +C$

(b) $-2 \sin x + 5x + C$

(c) $\cos^2 x + 5x + C$

(d) $- \cos^2 x + 5x + C$

(e) $-2\sin x +C$

\bigskip

{\bf 23. } If $f'(x)=2x-3$ and $f(0)=6$ find $f(2)$.

(a) 4

(b) 1

(c) 6

(d) 5

(e) $-2$

\bigskip

{\bf 24. } Find $\displaystyle \sum_{i=1}^{10} (2i-3)$.

(a) 80

(b) 25

(c) 77

(d) 107

(e) 52

\pagebreak

{\bf 25. } Find $\displaystyle \int_1^6(2x-4)dx$.

(a) 15

(b) 16

(c) 32

(d) 30

(e) 24


\end{document}

