 \documentclass[12pt]{article}

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\textheight=24cm

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\begin{document}



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

\next
1. What is the equation of the line which passes through the point $(0,2)$ and is 
parallel to the line $4x+2y=3$ ?

\vskip 3mm
(a) $y=-2x+2$ \hfil (b) $y=-2x$ \hfil (c) $y=2x+2$ \hfil (d)
$y=0.5x+2$ \hfil (e) $y=-0.5x+2$

\vskip 6cm
2. How much should you deposit in a savings account with an interest rate
of 7\%, compounded annually, in order to have \$300,000  at
the end of 20 years?

\vskip 3mm
(a) $\ds \frac{300,000}{(1.07)^{20}}$
\hfil
(b) $300,000\cdot (1.07)^{20}$
\hfil
(c) $\ds\frac{300,000}{(.07)^{20}}$

\bigskip
\hfil
(d) $300,000\cdot (1.7)^{20}$
\hfil
(e) $300,000\cdot (.07)^{20}$ 

\vskip 4cm 
3. Evaluate $\displaystyle{\lim_{n\to\infty} (1-\frac{1}{2n})^{n}}$.

\vskip 3mm
(a) $e^{-1/2}$
\hfil
(b) $\ln(1/2)$
\hfil
(c) $e^{1/2}$
\hfil
(d) $e^2$
\hfil
(e) $e^{-2}$

\newpage
4. Find the slope of the line between the points $(-2,2)$ and $(3, -2)$.

\vskip 3mm
(a) $-\frac{4}{5}$ \hfil (b) $-\frac{5}{4}$ \hfil (c) 4 \hfil (d)
$\frac{4}{5}$ \hfil (e) $\frac{5}{4}$

\vskip 4cm
5. A certain radioactive substance decays exponentially with decay equation $$y = Ab^t$$ 
for some constant $b$. An initial amount of 400 grams of this substance decays to
approximately 100 grams in 2 years. How much of this initial amount will be 
left in 4 years?

\vskip 3mm
(a) $25$ g
\hfil
(b) $5$ g
\hfil
(c) $10$ g
\hfil
(d) $50$ 
\hfil
(e) $20$ g


\vskip 5cm
6. If ball is thrown upward from a height of 100 feet with an initial
velocity of 64 feet per second, its height is given by the formula
$$h = -16t^2 +64t + 100,$$
where $h$ is the height in feet and $t$ is the time in seconds.
What is the maximum height the ball reaches?

\vskip 3mm
(a) 164 ft. \hfil (b) 144 ft.\hfil (c) 196 ft. \hfil (d) 200 ft. \hfil (e) 150 ft.

\newpage
7. Let $f(x) = \cases{2x - 2 &if $x < -1$ \cr  3x-1 &if $-1 \le x \le 0$ \cr 4\sqrt{x}-1
&if $0 < x \le1$ \cr 2x^2 &if $1 < x$ \cr}$.

\bigskip \noindent
Where does $f(x)$ have
discontinuities, if any?

\vskip 3mm
(a) At $x = 1$ \hfil (b) At $x= 0$ and $x=1$ \hfil (c) At $x = -1$ and $x=0$

\bigskip \noindent
\hfil (d) At $x=-1$, $x=0$ and $x=1$ \hfil (e) No discontinuities

\vskip 5cm
8. Joe deposits \$30,000 in an account with an annual interest rate of
5\%, compounded monthly. How much money will he have in this account at the 
end of 6 years?

\vskip 3mm
(a) $\ds 30,000\cdot \left(1+\frac{.05}{12}\right)^{72}$
\hfil
(b) $\ds 30,000\cdot \left(1+\frac{.05}{12}\right)^6$

\bigskip
(c) $30,000\cdot (1+.05)^6$
\hfil
(d) $30,000\cdot (1+.05)^{72}$

\bigskip \hfil
(e) $\ds 30,000 \cdot\left(1+\frac{.05}{72}\right)^{6}$ \hfil

\vskip 3cm
9. What is the value of $\ds \log_{8}2$ ?

\vskip 3mm
(a) $1/3$ \hfil (b) $1/2$ \hfil (c) 1 \hfil (d) 2 \hfil (e) 0

\newpage
10. Suppose you open an account with \$10,000, at an interest
rate of 8\% compounded continuously. How much will you have in
your account at the end of 20 years?

\bigskip
(a) $10,000\,e^{1.6}$
\hfil
(b) $\ds \left(\frac{10,000}{e^{.08}}\right)^{20}$
\hfil
(c) $\ds 10,000\left(1+\frac{.08}{20}\right)^{20}$

\bigskip \hfil
(d) $10,000\,(1+.08)^{20}$
\hfil
(e) $10,000\,e^{-1.6}$ 

\vskip 3cm
11. Which of the following expressions is equal to $e^{2\ln(5x)}$?

\bigskip
(a) $25x^2$
\hfil
(b) $5x$
\hfil
(c) $5xe^2$
\hfil
(d) $10x$
\hfil
(e) $2e^{5x}$ 

\vskip 3cm
12. Consider the function $\ds f(x)=\frac{x^3}{x^2-2x+1}$. Which of the following 5 
statements is true for this function ?

\vskip 3mm
(a) $f(x)$ has a vertical asymptote and no horizontal asymptotes. \\
(b) $f(x)$ has a vertical asymptote and a horizontal asymptote.\\
(c) $f(x)$ has no vertical asymptotes. \\
(d) $f(x)$ has two vertical asymptotes.\\
(e) none of the above statements is true about $f(x)$.

\newpage
13. Which of the following is the equation of an asymptote for the function 
$$\ds f(x)=\frac{6x^4-x^2+1}{2x^4+2}\;\;?$$

\vskip 3mm
(a) $y=3$ \hfil (b) $x=-1$ \hfil (c) $y=6$ \hfil (d) $y=2$ \hfil (e) $x=1$

\vskip 5cm
14. Using the properties of the logarithms and the approximations 
$$\log_23\approx 1.6\;\;\;{\rm and}\;\;\;\log_25\approx 2.3$$
find an approximation of 
$$\ds \log_2\left(\frac{10}{3}\right)\;.$$

\vskip 3mm
(a) 1.7 \hfil (b) 0.7 \hfil (c) 3.9 \hfil (d) 4.9 \hfil (e) 0.9

\newpage
{\bf {\centerline{ Part II: Partial Credit
Questions}}}

\smallskip 
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.

\next
15. (10 pts.) A small company assembling computers has fixed costs of \$6,000 per
month. Each computer costs \$800 to make and sells for \$1200.  How many computers 
do they have to make and sell per month to break even?

\vskip1.25in \ans

\vskip 2cm
16. (10 pts.) A colony of bacteria grows exponentially, as given by an equation of 
the form 
$$P=Ab^t\;.$$
A biologist determined that there were 1000 bacteria initially and 27,000 bacteria 
3 hours later.

\bigskip
(a) Write a formula giving the number of bacteria at any time $t$.

\vskip 5cm
\bigans

\bigskip
(b) How many bacteria will be in the colony in 4 hours?

\vskip 3cm
\ans

\newpage
17. (10 pts.) A movie theater sells on average 120 tickets a day, at a price of 
\$6 per ticket. They estimate that each 50 cents increase in the ticket price will 
result in losing on average 10 customers per day.

\bigskip
(a)  Find the formula for the number of tickets sold per day, $q$, as a function of
the price per ticket, $p$.

\vskip 5cm
\bigans

\vskip 5mm
(b) Find the formula for the daily revenue, $R$, as a function of the price per 
ticket, $p$.

\vskip 4cm
\bigans



\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% exam 2
\documentclass[12pt]{article}

\everymath{\displaystyle}
\hbadness = 10000
\tolerance = 10000

\hoffset=-2cm
\voffset=-2cm
\textwidth=17cm
\textheight=24cm

\parindent=0pt

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\def\ans{ \hbox{\hskip2.5in Ans. \vrule height .4pt depth 0 pt width 2.5in}}
\def\bigans{ \hbox{\hskip1in Ans. \vrule height .4pt depth 0 pt width 4in}}
\def\page{\vfill \eject \noindent}
\def\bigno{\bigskip \noindent}
\def\medno{\medskip \noindent}

\begin{document}

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

\next
\parindent=0mm
1. The position of a particle (in meters) which moves along a straight line is given by
\[
s(t) = t\ln t+t^3+71.
\]
What is the acceleration (in $m/sec^2$) at the end of 1 second?

\vspace{2mm}
(a) $7$ \hfill (b) $4$  \hfill (c) $72$ \hfill (d) $3$  \hfill (e) $71$

\vspace{3in}

2. Find the derivative $f'(x)$ of the function $\ds f(x)=\frac{3}{x^{1/3}}+7x^4$.

\vspace{2mm}
 (a) $\ds -x^{-4/3}+28x^3$ \hfill (b)  $\ds x^{-2/3}+28x^3$ \hfill 
 (c) $\ds \frac{1}{3}\,x^{-2/3}+28x^3$

\vspace{2mm}
(d) $\ds -\frac{1}{3}\,x^{-4/3}+28x^3$ \hfill (e) $\ds -x^{-4/3}+7x^3$  


\newpage

3. You open an account with \$15,000 at a continuous interest rate which doubles the 
money in 12 years. How much money will you have in this account after 36 years? 

\vspace{2mm}
(a) $\$120,000$ \hfil (b) $\$30,000$ \hfil (c)  $\$60,000$ \hfil (d) $\$240,000$ \hfil 
(e) $\$45,000$

\vspace{2.5in}

4. The height above the ground (in feet) of an object thrown downward from the top 
of a 744 feet tower is given by the formula
$$h(t) = -16t^2 -100t + 744\;.$$
What is the {\bf velocity} of the object at time $t=3$ seconds?

\vspace{2mm}
(a) $-196$ \hfill (b) $-96$ \hfill (c) $-148$ \hfill
(d) $300$ \hfill (e) $4$


\newpage

5. Compute the derivative of  $f(x)= \frac{x-5}{2x-1}$ .

\vspace{3mm}
(a) $\frac{9}{(2x-1)^2}$  \hfill (b) $\frac{4x-11}{(2x-1)^2}$ \hfill
(c) $\frac{1}{(2x-1)^2}$ \hfill
(d) $\frac{7}{(2x-1)^2}$  \hfill (e) $\frac{4x+9}{(2x-1)^2}$

\vspace{3.2in}

6. Given two functions $f(x)$ and $g(x)$ such that
\[
f(0)=2,\;\;g(0)=1,\;\;f'(0)=3,\;\;g'(0)=4
\]
determine the slope of the graph of $y=f(x)g(x)$ at $x=0$ .

\vspace{3mm}
(a) $11$ \hfill (b) $12$ \hfill (c) $2$ \hfill (d) $10$  \hfill (e) $14$



\newpage
7. Which of the following limits represents the derivative of the function 
$f(x)=\sqrt{x+9}$ at $x=16$ ?

\vspace{3mm}
(a) $\lim_{h\rightarrow 0}\frac{\sqrt{25+h}-5}{h}$ \hfill
(b) $\lim_{h\rightarrow 0}\frac{\sqrt{25-h}-5}{h}$ \hfill
(c) $\lim_{h\rightarrow 0}\frac{5+\sqrt{h}}{h}$

\vspace{3mm}
(d) $\lim_{h\rightarrow 16}\frac{\sqrt{h}-4}{h}$ \hfill
(e) $\lim_{h\rightarrow 0}\frac{\sqrt{25+h}+5}{h}$


\vspace{2.7in}

8.  Compute the derivative of $f(x)= x^{15}\ln x$.

\vspace{3mm}
(a) $15x^{14}\ln x+x^{14}$ \hfill (b) $15x^{14}\ln x+x^{15}$  \hfill (c) $15x^{13}$

\vspace{3mm}
(d) $15x^{14}\ln x$ \hfill (e)  $x^{14}$



\newpage

9. The revenue of a company selling $x$ units of a product is
\[
R(x)=1515x - 2x^3
\]
(in dollars). Find the marginal revenue when the company sells 10 units of this product.

\vspace{2mm}
(a) $915$ \hfil (b)  $13150$ \hfil (c) $9500$ \hfil (d) $1315$ \hfil (e) $550$



\vspace{3in}

10.  Compute the derivative of  $f(x)=\ln(x^4-x^2+1)$ .

\vspace{3mm}
(a) $\frac{4x^3-2x}{x^4-x^2+1}$  \hfill (b) $\frac{1}{x^4-x^2+1}$ \hfill 
(c) $(4x^3-2x)\ln(x^4-x^2+1)$

\vspace{3mm}
(d) $\frac{4x^3}{x^4-x^2+1}$ \hfill (e) $4x^3-2x$  




\newpage

11. Find the $x$-coordinate of the point where the tangent line to the graph of 
$$ f(x)=e^{2x^2+8x}$$ 
is horizontal.

\vspace{3mm}
(a) $x=-2$ \hfill (b) $x=0$  \hfill (c) $x=-4$ \hfill (d) $x=1$ \hfill (e) $x=2$

\vspace{2.7in}
12.  If $\;\;\;y= 3e^{4x}-2\ln(9x)\;\;\;$ find $\;\;\;\frac{dy}{dx}\;$.

\vspace{3mm}
(a) $12e^{4x}-\frac{2}{x}$ \hfill (b) $3e^{4x}-\frac{2}{x}$ \hfill 
(c) $12e^{4x}-\frac{2}{9x}$

\vspace{3mm}
(d) $12e^{4x}-\frac{18}{x}$ \hfill (e) $ 3e^{4x}+2\ln(9x)$


\newpage

13. If $f(x)=xe^x$ then $f''(x)$ equals

\vspace{3mm}
(a) $(x+2)e^x$ \hfil (b) 0 \hfil (c) $e^x$ \hfil (d) $xe^x$ \hfil (e) $(x+1)e^x$

\vspace{3.3in}

14. Let $f(x)$ and $g(x)$ be two functions such that 
$$f(5)=2,\;f'(5)=4,\;f'(7)=6,\;g(5)=7,\;g'(5)=3,\;g'(7)=8.$$
Find the derivative of $f(g(x))$ at $x=5$.

\vspace{3mm}
(a) $18$ \hfill (b) $12$ \hfill (c) $6$ \hfill (d) $16$  \hfill (e) $28$

\page
{\bf {\centerline{ Part II: Partial Credit Questions}}}

\smallskip \noindent
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.

\next
15. (10 pts.) Mary buys a cup of coffee and leaves it on her desk. 
The initial temperature of the coffee was $63^\circ$ Celsius and the 
temperature in her office is $21^\circ$ Celsius.  According to Newton's
law of cooling, the temperature $H(t)$ of the coffee at time $t$
is given by an equation of the form
$$H(t)=21+Ae^{kt}\;.$$
After $15$ minutes  the temperature of the coffee drops to  $42^\circ$. What is the
temperature of the coffee after $30$ minutes?

\vspace{6in}

\ans

\newpage

16 . (10 pts.)  Consider the function $y=\sqrt{3x^2+1}$.

\bigskip

(a)  Find $\frac{dy}{dx}$.

\vspace{3.5in}
\ans

\vspace{5mm}
(b)  Write the equation of the tangent line to the graph of this function at the 
point where $x=1$.

\vspace{3in}
\ans


\newpage

17. (10  pts.)  Consider the function $f(x)=\sqrt{x+1}$. Compute $f'(0)$
{\bf using the definition of the derivative}, and no other method.

\vspace{8in}
\ans


\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% exam 3
\documentclass[12pt]{article}

\everymath{\displaystyle}
\hbadness = 10000
\tolerance = 10000

\hoffset=-2cm
\voffset=-2cm
\textwidth=17cm
\textheight=24cm

\parindent=0pt

\font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5
\newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb \def\Bbb#1{{\fam\msbfam #1}}
\font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5
\newfam\eufmfam
\textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm
\scriptscriptfont\eufmfam=\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}}
\let\goth\frak

\def\next{\vskip.25in \noindent}
\def\ds{\displaystyle}
\def\smallans{ \hbox{\hskip2.5in Ans. \vrule height .4pt depth 0 pt width
1.5in}}
\def\ans{ \hbox{\hskip2.5in Ans. \vrule height .4pt depth 0 pt width 2.5in}}
\def\bigans{ \hbox{\hskip1in Ans. \vrule height .4pt depth 0 pt width 4in}}
\def\page{\vfill \eject \noindent}
\def\bigno{\bigskip \noindent}
\def\medno{\medskip \noindent}

\begin{document}

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

\next
\parindent=0mm
1. How many times does the derivative of the function 
$$f(x)=\frac{3}{2}\;x^{2/3}-x$$ 
change its sign?

\vspace{2mm}
(a) never \hfill (b) 1 time \hfill (c) 2 times \hfill (d) $3$ times  \hfill (e) $4$ times

\vspace{3.8in}

2. Let $f(x)$ be a function satisfying $f'(1)=0$, $f'(7)=0$, $f''(1)=15$ and $f''(7)=-10$. 
Which of the following statements is true? 

\vspace{2mm}
(a) $f$ has a relative minimum at $x=1$ and a relative maximum at $x=7$ \\
(b) $f$ has relative minimums at $x=1$ and at $x=7$ \\ 
(c) $f$ has relative maximums at $x=1$ and at $x=7$ \\ 
(d) $f$ has a relative maximum at $x=1$ and a relative minimum at $x=7$ \\ 
(e) $f$ has no relative minimum points 
 


\newpage

3. What number $x$ yields the absolute minimum value of the function 
$f(x)=2x^2-12x+10$ ?

\vspace{2mm}
(a) $x=1$ \hfil (b) $x=0$ \hfil (c)  $x=2$ \hfil (d) $x=3$ \hfil 
(e) $x=5$

\vspace{2in}

4. How many critical points does the function $f(x)=x^4-2x^2+2000$ have?

\vspace{2mm}
(a) $4$ \hfill (b) $3$ \hfill (c) $2$ \hfill
(d) $1$ \hfill (e) $0$


\vspace{2in}

5. On which of the following intervals is the function $\ds f(x)=\frac{x^4}{4}-x^3+12$ 
concave down?

\vspace{2mm}
(a) $(0,3)$  \hfill (b) $(-\infty,0)$ \hfill
(c) $(0,+\infty)$ \hfill
(d) $(2,+\infty)$  \hfill (e) $(0,2)$

\newpage

6. Find the number $x$ where the function $\ds f(x)=\frac{x^3}{3}-2x^2+3x$ 
changes from decreasing to increasing.

\vspace{3mm}
(a) $x=0$ \hfill (b) $x=1$ \hfill (c) $x=3$ \hfill (d) $x=-1$  \hfill (e) $x=2$

\vspace{3.5in}

7. For which number $x$ does $f(x)=xe^x$ have an inflection point?

\vspace{3mm}
(a) $x=2$ \hfil (b) $x=1$ \hfil (c) $x=0$ \hfil (d) $x=-1$ \hfil (e) $x=-2$  


\newpage

8.  Which of the following corresponds to the region where the function 
$f(x)=-x^3+3x^2+1$ is decreasing?

\vspace{3mm}
(a) $(0,2)$ \hfil (b) $(-\infty,0)$ and $(2,+\infty)$  \hfil (c) $(1,3)$ \hfil
(d) $(-\infty,0)$ \hfil (e)  $(2,+\infty)$

\vspace{3.5in}

9. What is the absolute maximum value of the function $f(x)=\ln x-x+3$ on the interval 
$(0,+\infty)$?

\vspace{2mm}
(a) $\ln 2+1$ \hfil (b)  $1$ \hfil (c) does not exist \hfil (d) $2$ \hfil (e) $3$



\newpage

10.  What value of the constant $A$ makes the formula \ \ \ \ $\int x^{-2}dx=Ax^{-1}+C$ 
\ \ \ \ true?

\vspace{3mm}
(a) $A=-1$  \hfil (b) $\ds A=-\frac{1}{2}$ \hfil 
(c) $A=1$ \hfil (d) $\ds A=\frac{1}{2}$ \hfil (e) $A=-2$  

\vspace{3in}

11. Consider the points marked below on the graph of a function $f(x)$. At which of these 
points are the values of $f(x)$, $f'(x)$ and $f''(x)$ all negative?

\vspace{4in}
(a) $A$ \hfill (b) $B$  \hfill (c) $C$ \hfill (d) $D$ \hfill (e) $E$


\newpage

12. Let $f(x)$ be the function with derivative $f'(x)=x$ and such that $f(0)=1$. 
Find $f(2)$.

\vspace{3mm}
(a) $1$ \hfill (b) $2$ \hfill (c) $3$ \hfill (d) $4$ \hfill (e) $5$


\vspace{4in}

13. How many inflection points does $\;\;\;f(x)=5x^4-10x+5\;\;\;$ have?

\vspace{3mm}
(a) one \hfil (b) none \hfil (c) three \hfil (d) four \hfil (e) two


\newpage

14. Consider the function 
$$f(x)=-2x^3-3x^2+12x \;.$$
Which of the following curves most closely resembles the graph of this function? 


\newpage
{\bf {\centerline{ Part II: Partial Credit Questions}}}

\smallskip \noindent
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.

\next
15. (10 pts.) Let $f(x)$ be a function defined for all $x\neq1$, with the following 
properties: 

\vspace{2mm}
(i) $x=1$ is a vertical asymptote, 

\vspace{2mm}
(ii) $\lim_{x\rightarrow+\infty}f(x)=2$ and $\lim_{x\rightarrow-\infty}f(x)=2$, 

\vspace{2mm} 
(iii) $f'(x)>0$ on $(0,1)$ and $f'(x)<0$ on $(-\infty,0)$ and $(1,+\infty)$, 

\vspace{2mm} 
(iv) $f''(x)<0$ on $(-\infty,-1)$ and $f''(x)>0$ on $(-1,1)$ and $(1,+\infty)$, 

\vspace{2mm} 
(v) $f(0)=0$. 

\vspace{2mm}
Sketch the graph of $f(x)$ and include all asymptotes this function may 
have. Mark on your graph all relative minimums, relative maximums and inflection 
points that $f(x)$ may have.

\newpage

16 . (10 pts.)  Find the antiderivative $F(x)$ of $\;\;f(x)=e^{x/3}-3x^2\;$, \ given 
that $F(0)=7$.

\vspace{3.8in}
\ans


\vspace{9mm}

17. (10  pts.)  Find the absolute maximum value of $f(x)=x^3-3x+10$ on the interval 
$[-2,2]$. Specify all the numbers $x$ in $[-2,2]$ at which this absolute maximum value 
occurs.

\vspace{4.3in}
\hskip 5mm
The absolute maximum value of $f(x)$ on $[-2,2]$ is \rule[-.01in]{0.75in}{.01in} and it 
occurs at $x=$ \rule[-.01in]{0.75in}{.01in}


\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% final
\documentclass[12pt]{article}

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\begin{document}

\parindent=0mm
1. How many asymptotes (both vertical and horizontal) does the function
$\ds f(x)=\frac{x^2}{x^2-1}$ have?

\vspace{2mm}
(a) none \hfill (b) 1  \hfill (c) 2  \hfill (d) 3  \hfill (e) 4

\vspace{7cm}

2. What amount should you deposit in an account with a 7\% continuously compounded interest 
rate, in order to have \$10000 after 4 years?

\vspace{2mm}
(a) $\$10000e^{0.28}$ \hfil (b) $\$10000(1.07)^{-4}$ \hfil (c) $\$5000$ \hfil 
(d) $\$10000e^{-0.28}$ \hfil (e) $\$2500$ \hfil 


\vspace{7cm}

3. The limit $\ds\lim_{n\ra+\infty}\left(1-\frac{2}{n}\right)^n$ is equal to

\vspace{2mm}
(a) $0$ \hfil (b) $e^2$ \hfil (c) $e^{-2}$ \hfil (d) 1 \hfil 
(e) $e^{-1/2}$

\newpage

4. The area under the graph of the function $f(x)=3x^2-2x+2$ from $x=-1$ to $x=1$ is 

\vspace{2mm}
(a) $4$ \hfill (b) $3$ \hfill (c) $6$ \hfill
(d) $2$ \hfill (e) $0$


\vspace{7cm}

5. The half-life of a radioactive substance is 20 years. How much of an initial amount of 
88 grams will be left in 40 years?

\vspace{2mm}
(a) $66$ grams  \hfill (b) $33$ grams \hfill
(c) $11$ grams \hfill
(d) $44$ grams  \hfill (e) $22$ grams

\vspace{7cm}

6. The value of \ \ \ $\ln(e^7e^2)$ \ \ \ is 

\vspace{2mm}
(a) $14$ \hfill (b) $9$ \hfill (c) $5$ \hfill (d) $0$  \hfill (e) $3.5$

\newpage

7. The equation of the line which passes through the points $(-1,-1)$ and $(1,3)$ is 

\vspace{2mm}
(a) $y=2x+1$ \hfil (b) $y=2x$ \hfil (c) $y=-2x-3$ \hfil (d) $y=x-2$ \hfil (e) $y=2x-1$  


\vspace{7cm}

8. If \$250 is invested in an account which pays 8\% interest compounded 4 times a year, 
how much will it grow to in 5 years?

\vspace{2mm}
(a) $250(1.02)^{20}$ \hfil (b) $250(1.08)^5$  \hfil (c) $250(1.02)^5$  \hfil
(d) $250(1.32)^{1.25}$  \hfil (e)  $250(1.08)^{20}$ 

\vspace{7cm}

9. The height (in feet) of a falling object is given by $h(t)=-16t^2+128t+320$, where the 
time $t$ is in seconds. At what moment of time is the velocity of the object equal to $0$?

\vspace{2mm}
(a) $t=10$ \hfil (b)  $t=4$ \hfil (c) $t=0$ \hfil (d) $t=2$ \hfil (e) $t=6$

\newpage

10. If \ $\ds y=\frac{x^2-1}{x^2+1}$ \ then \ $\ds \frac{dy}{dx}$ \ equals 

\vspace{3mm}
(a) $\ds\frac{4x^3}{(x^2+1)^2}$ \hfil (b) $\ds\frac{4}{(x^2+1)^2}$ \hfil 
(c) $\ds\frac{2x}{(x^2+1)^2}$ \hfil (d) $\ds\frac{-4x}{(x^2+1)^2}$ \hfil 
(e) $\ds\frac{4x}{(x^2+1)^2}$    

\vspace{7cm}

11. Find the equation of the tangent line to the graph of \ $f(x)=x\ln x$ \ at the point 
where $x=e$. 

\vspace{2mm}
(a) $y=2x+e$ \hfill (b) $y=2x-e$  \hfill (c) $y=x-e$ \hfill (d) $y=x+e$ \hfill (e) $y=ex$

\vspace{7cm}

12. If $g(x)=x^2f(x)$, $f(1)=2$ and $f'(1)=1$ find $g'(1)$.

\vspace{2mm}
(a) $1$ \hfill (b) $2$ \hfill (c) $5$ \hfill (d) $4$ \hfill (e) $3$


\newpage

13. Which of the following is an antiderivative of \ $f(x)=xe^{-x}$ ?

\vspace{2mm}
(a) $F(x)=xe^{-x}-e^{-x}$ \hfil (b) $F(x)=-xe^{-x}+e^{-x}$ \hfil (c) $F(x)=xe^{-x}$ 

\vspace{2mm} 
(d) $F(x)=-xe^{-x}-e^{-x}$ \hfil (e) $F(x)=xe^{-x}+e^{-x}$ 

\vspace{7cm}

14. The cost function of a small business making $x$ hats per day is $C(x)=200+4x+5x^2$. 
What is the marginal cost at a production level of $x=10$ hats per day?

\vspace{2mm}
(a) $104$ \hfill (b) $740$ \hfill (c) $304$ \hfill (d) $54$ \hfill (e) $254$

\vspace{7cm}

15. At what number $x$ is the slope of the curve $y=5^x$ equal to $25\ln5$ ?

\vspace{2mm}
(a) $x=1$ \hfill (b) $x=3$ \hfill (c) $x=\ln5$ \hfill (d) $x=2\ln5$ \hfill (e) $x=2$

\newpage 

16. The derivative of \ $\ds f(x)=\ln(e^{2x}+e^{-x})$ \ is  

\vspace{3mm}
(a) $\ds\frac{2e^{2x}-e^{-x}}{e^{2x}+e^{-x}}$ \hfil (b) $\ds\frac{1}{e^{2x}+e^{-x}}$ \hfil 
(c) $2e^{2x}-e^{-x}$ \hfil (d) $\ds\frac{1}{2e^{2x}-e^{-x}}$ \hfil 
(e) $\ds\frac{e^{2x}-e^{-x}}{e^{2x}+e^{-x}}$    

\vspace{7cm}

17. At what number $x$ does the function \ $f(x)=x^3+6x^2-7$ \ have a relative maximum?

\vspace{2mm}
(a) $x=1$ \hfill (b) $x=0$ \hfill (c) $x=-6$ \hfill (d) $x=-2$ \hfill (e) $x=-4$


\vspace{7cm}

18. How many inflection points does the function \ $\ds f(x)=-\frac{x^4}{4}+x^3-2x+1$ \ 
have?

\vspace{2mm}
(a) $1$ \hfill (b) $3$ \hfill (c) $2$ \hfill (d) $4$ \hfill (e) none

\newpage

19. Which of the following could be the graph of the function \ $f(x)=x^4-2x^2$ ?

\newpage

20. Calculate the area under the graph of \ $\ds f(x)=\frac{5}{x}$ \ from $x=1$ to $x=e^2$.

\vspace{2mm}
(a) $10$ \hfill (b) $5$ \hfill (c) $2$ \hfill (d) $15$ \hfill (e) $1$

\vspace{8cm}

21. On which of the following intervals is the function 
$$f(x)=\frac{1-\ln x}{x}\;,\;\;x>0\;,$$
decreasing?

\vspace{2mm}

(a) $(e^2,+\infty)$ \hfil (b) $(0,e^2)$ \hfil (c) $(1,+\infty)$ \hfil 
(d) $(0,1)$ \hfil (e) $(0,+\infty)$ 

\newpage

22. On which of the following intervals is the function \ 
$\ds f(x)=\frac{x^2}{2}+\ln x\;$, $x>0$, \ concave up?

\vspace{2mm}

(a) $(e^{-1},+\infty)$ \hfil (b) $(0,1)$ \hfil (c) $(0,e)$ \hfil 
(d) $(1,+\infty)$ \hfil (e) $(0,+\infty)$ 

\vspace{11cm}

23. The revenue function of a company making bicycles is \ $R(p)=-2p^3+600p^2$, where 
$p$ is the price per bicycle and $0\leq p\leq300$. What price $p$ should the company 
charge in order to maximize its revenue?

\vspace{2mm}
(a) $p=100$ \hfill (b) $p=300$ \hfill (c) $p=200$ \hfill (d) $p=150$ \hfill (e) $p=250$

\newpage

24. If $f(x)$ is a function such that $f'(x)=6x^2+4x$ and $f(1)=5$, find $f(-1)$.

\vspace{2mm}
(a) $2$ \hfill (b) $0$ \hfill (c) $1$ \hfill (d) $-1$ \hfill (e) $-2$

\vspace{10cm}

25. Find \ $\int 3x\sqrt{x^2+1}\;\;dx$ .

\vspace{3mm}
(a) $\ds\frac{1}{3}\,(x^2+1)^{3/2}+C$ \hfil 
(b) $2(x^2+1)^{3/2}+C$ \hfil 
(c) $3(x^2+1)^{3/2}+C$ 

\vspace{3mm} 
(d) $(x^2+1)^{3/2}+C$ \hfil 
(e) $3(x^2+1)^{1/2}+C$ 


\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% handout
\documentstyle[12pt]{article}


\begin{document}

\noindent
{\bf Information for Students} \hfil {\bf Math 105}

\vskip.25in \noindent
\underbar{\bf Instructor:} Dan Coman

\noindent
\underbar{\bf Office:}  Room 302, CCMB

\noindent
\underbar{\bf Phone:} 631-6832
\hfil \underbar{\bf e-mail:}  coman.2@nd.edu 

\noindent
http://www.nd.edu/ $\widetilde{ }$ dcoman/Teach.html

\smallskip \noindent
\underbar{\bf Office hours:} Monday 3:30-4:30, Tuesday 2-3.

\medskip \noindent
\underbar{\bf Textbook:} \  {\bf Calculus for Business and Social Sciences}

 by Himonas and Howard, available at the Notre Dame bookstore.

We will cover chapters 1 through 4, plus part of chapter 5.

\medskip \noindent
\underbar{\bf Tests:} There will be {\bf three one-hour exams}.  All of
them will be held {\bf outside of class}, in {\bf Nieuwland 118}, as follows: 

\centerline{\bf Thursday, February 10, 8-9:15 AM}

\centerline{\bf Tuesday, March 7, 8-9:15 AM}

\centerline{\bf Thursday, April 13, 8-9:15 AM}

\noindent
There will also be a two-hour {\bf final exam}, at a place to be announced, on

\centerline {\bf Friday, May 12, from 1:45 to 3:45 PM}

\smallskip \noindent
An unexcused absence from any of these tests will result in a grade of zero
for the test.  If you are absent with a valid excuse you will be given a
make-up test at
a later date.  Excused absences must be arranged through the offices of the
Vice
President of Student Affairs or the Dean of First Year.  Contacting your
instructor on
the night before or morning of an exam to say you are dizzy, nauseated,
etc. will not be
sufficient grounds for an excused absence.

\smallskip \noindent
Please take these dates into account when making travel plans for the Spring 
break and the Summer vacation.  Having an airline ticket or an opportunity for a ride
home that leaves before an exam is not a valid excuse for missing the exam.

\medskip \noindent
\underbar{\bf Grades:} Grades will be based on 450 points, distributed as
follows:

 \quad Hour tests during semester \hskip.4in 100 points each

 \quad Final exam \hskip1.5in 150 points

\noindent
After each test, a scale for assigning letter grades will be announced.
These are
intended as guidelines only, so that you can judge your progress.  The
scale for
assigning letter grades for the course will be decided by the instructors
at the end of
the semester.  In borderline cases, your homework record may influence your
grade, as
explained in the next paragraph.

\medskip \noindent
\underbar{\bf Homework} will be assigned and collected regularly.  Doing it
diligently
and on time is a major factor in how well you learn and how well you do on
tests.  It is
in your interest to attempt every homework problem and do as much as you can,
whether or not you can completely solve it.  Students are encouraged to
work together
and help one another.  Collaboration on homework is not an honor code
violation, but
you will not learn anything by simply copying someone else's finished product.

\smallskip \noindent
A record of your homework grades will be kept.  After the final grading
scale is
determined (based, as explained above, on 450 points), your instructor will add
between 0 and 5 points to your total, based on your homework record.

\medskip \noindent
\underbar{\bf Calculators:} A scientific calculator (with exp, log and ln
keys) will be
necessary for some homework problems. The use of calculators will not be
permitted
on tests.

\medskip \noindent
\underbar{\bf First Year Learning Resource Center} is a great source of
help for
first-year students.  It offers a weekly Math 105 workshop to answer
questions and
help students with the course material and problems.  In addition, there are
small-group tutoring sessions and collaborative learning sessions.  Times
and places
for these sessions and for the workshop will be announced in class later,
or you can
ask your first-year advisor for more information about them.





\end{document}
