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\title{Brief Article}
\author{The Author}
\begin{document}
\maketitle

 
 
\title {Math 105 \qquad Final Exam}
\date {May 6, 1998}
\noprofessor
\twocolumn{p.6}
%\comment {{\bf Part II, consisting of one partial credit problem is on back
%of this sheet}} 
\answersheetfootline {I have observed the honor code
\hskip 10pt 
  \underbar{\hskip 3 in}}

%%%1
\problem
Suppose that when a busy restaurant charges $\$9$ for their octopus
appetizer, an average of 50 people order the dish each night.  When
they raise the price of the appetizer to $\$12$, 41 people order it
each night.  Assuming that the demand $q$ is a linear function of the price $p$, find an equation for $q$ in terms of
$p$.

\correct $q=-3p+77$
\wrong $q=-3p^{2}+5p$
\wrong $q=50p-9$
\wrong $q=3p+23$
\wrong $q=3p+5$
\perm1:cbade
\perm2:edbca

%%%2
\problem
The same restaurant also serves sea cucumber.  The sea cucumber demand
curve---that is, the number of people who order sea cucumber when the
price of the dish is $p$ dollars---is given by $q=80-2p$.  What price
should the restaurant charge per dish to maximize revenue?

\correct $\$20$
\wrong $\$40$
\wrong $\$10$
\wrong $\$30$
\wrong $\$80$
\perm1:ebdac
\perm2:cadbe

%%%3
\problem
Which of the following is equal to $\ln(3\sqrt{e})$?

\correct $\displaystyle {1 \over 2} + \ln 3$ 
\wrong $(\ln 3) (\sqrt{\ln e})$ 
\wrong $\displaystyle {1 \over 2}(\ln 3 + \ln e)$ 
\wrong $\displaystyle {1 \over 3 \sqrt{e}}$ 
\wrong $\displaystyle (\sqrt{e})^{3}$
\perm1:edcba
\perm2:cbade

\pagebreak
\problem
Find the area under the curve $y=\sqrt{x}$  from $x=0$ to $x=4$.

\correct $\displaystyle {16 \over 3}$
\wrong $\displaystyle {2 \over 3}$
\wrong $\displaystyle 12$
\wrong $\displaystyle {4 \over 3} + C$
\wrong undefined
\perm1:abcde
\perm2:bcdae

\problem
Suppose that the number of fruit flies in a biology experiment grows
exponentially, so that the population $P$ after $t$ days is given by
the equation $P=Ae^{kt}$ for some constants $A$ and $k$.  If the
population triples in 8 days, find the growth constant $k$.

\correct $\displaystyle {\ln 3 \over 8}$  \hfill 
\wrong $\displaystyle {\ln 3 \over \ln 2}$  \hfill
\wrong $\displaystyle {\ln 2 \over 8}$  \hfill
\wrong $\displaystyle {\ln (2A) \over 8}$ \hfill
\wrong $\displaystyle {\ln (3A) \over 8}$
\perm1:cdabe
\perm2:abcde

\problem
  Let $f(x) = x - \ln x$ for $x > 0$.  Where is $f(x)$ increasing and where is it decreasing?

\correct decreasing on $(0,1)$, increasing on $(1, \infty)$ 
\wrong increasing on $(0,1)$, decreasing on $(1, \infty)$
\wrong increasing on $(0,1)$ and $(e,\infty)$, decreasing on $(1,e)$
\wrong decreasing on $(0,1)$ and $(e,\infty)$, increasing on $(1,e)$
\wrong increasing for all $x > 0$
\perm1:bacde
\perm2:ecdab

\pagebreak

\problem
 Find $\displaystyle{d\over{dx}}\>\left({{x^3}\over{x^2+1}}\right)$.

\correct $\displaystyle{{x^4 + 3x^2}\over{(x^2 + 1)^2}}$
\wrong$\displaystyle{{3x^2}\over{2x}}$
\wrong $\displaystyle{{3x^2}\over{(x^2+1)^2}}$
\wrong$\displaystyle{{x^3}\over{(x^2+1)^2}}$ 
\wrong $\displaystyle{{x^4 + 4x^2}\over{x^2 + 1}}$
\perm1:abcde
\perm2:bdaec

\problem
Which of these is equal to $\displaystyle \int (2e^{2x} + 3x - {3
\over x^{2}}) \ dx$?

\correct $\displaystyle e^{2x} + {3 \over 2}x^{2} + {3 \over x} + C$
\wrong $\displaystyle 4e^{2x} + 6 x^{2} + {3 \over x} + C$
\wrong $\displaystyle 4e^{2x} + 3 x^{2} + {1 \over x^{3}} + C$
\wrong $\displaystyle 2e^{2x} + 3x^{2} - {3 \over x} + C$
\wrong $\displaystyle e^{2x} + {3 \over 2}x^{2} - 3 \ln (x^{2}) + C$
\perm1:bcead
\perm2:debca

\problem
The formula for the area of a circle with radius $r$ is $A = \pi  r^{2}$.  Suppose that when the radius is 6 feet,
the  area is increasing at a rate of 10 square feet per second. How fast is the radius increasing  when the 
radius is 6 feet?

\correct $\displaystyle {5 \over 6 \pi}$ ft./sec.
\wrong $120 \pi$ ft./sec.
\wrong $3$ ft./sec.
\wrong $\displaystyle \sqrt{10 \over \pi}$ ft./sec.
\wrong $12\pi$ ft./sec.
\perm1:dbcea
\perm2:acdbe

\pagebreak

\problem
Find the value of $x$ where the tangent line to the graph of $y = x^2 - 3x + 6$ is parallel to the line $x+y = 7$.

\correct $x=1$ 
\wrong  $x=-1$ 
\wrong $x=0$ 
\wrong  $x = 7/3$ 
\wrong  $x=-3/7$ 
\perm1:dbcae
\perm2:eacbd

\problem
Find all vertical asymptotes, if any, for the function 
$\displaystyle {\ln x \over x-2}$.

\correct $x=0$ and $x= 2$
\wrong $y=0$ and $y=2$
\wrong $x=0$
\wrong  $x=2$
\wrong no vertical asymptotes
\perm1:cdabe
\perm2:eacdb

\problem
The natural log of 2 is approximately $0.693$.  Using linear
approximation, what is the approximate value of $\ln(2.2)$? ({\bf Warning:} the answer your calculator gives may not be the
correct answer to this problem.)
\correct 0.793  
\wrong 0.788  
\wrong 0.743  
\wrong 0.801  
\wrong 0.593  
\perm1:dabce
\perm2:ecbad

\pagebreak

\problem Here is the graph of a function $f(x)$.  Where does $f(x)$ have
critical points and inflection points?

\bigskip
\centerline{\picture 3in by 1.75in (graph scaled 450)}

\bigskip

\correct critical points at $x=-1$ and $x=2$, inflection points at $x=0$
and $x=4$.
\wrong critical points at $x=0$ and $x=4$, inflection points at $x=-1$
and $x=-2$.
\wrong no critical points, no inflection points
\wrong critical points at $x=-1$, $x=0$, $x=2$, and $x=4$; inflection
points at $x=-2$, $x=10$
\wrong critical points at $x=-2$ and $x=0$, inflection points at $x=-1$
and $x=2$.
\perm1:abcde
\perm2:cedba

\problem
Find $\displaystyle{d\over{dx}}\,\left(10^x\right)$.

\correct $10^x\ln10$ 
\wrong $10^x$ 
\wrong $x10^{x-1}$ 
\wrong $10^x\ln x$ 
\wrong $x10^{x-1}\ln10$
\perm1:bceda
\perm2:abcde

\pagebreak

\problem
 Find all horizontal asymptotes, if any, for the function $\displaystyle f(x) = {{2x^2 + 3x - 7}\over{x^2 -1}}$.
\correct $y = 2$
\wrong $y = 7$
\wrong $y=0$
\wrong $x=1$ and $x=-1$ 
\wrong no horizontal asymptote
\perm1:abcde
\perm2:ecadb

\problem
Which of the following limits equals the derivative
of $e^{3x}$?

\correct $\displaystyle \lim_{h\rightarrow 0} {e^{3x+3h}-e^{3x} \over h}$
\wrong $\displaystyle \lim_{h\rightarrow 0} {e^{3x-3h}-e^{3x} \over h}$
\wrong $\displaystyle \lim_{h\rightarrow 0} {e^{3x+3h}+e^{3x} \over h}$
\wrong $\displaystyle \lim_{h\rightarrow 0} {e^{3x+h}-e^{3x} \over h}$
\wrong $\displaystyle \lim_{h\rightarrow 0} {e^{3x+3h}-e^{3x} \over 3h}$
\perm1:daebc
\perm2:abcde

\problem
Let $\displaystyle f(x) = {x^{3} \over 3} - 2x^{2} + 3x-1$.  Where
does $f(x)$ have its absolute minimum and maximum on the interval
$[-3,3]$?

\correct minimum at $x=-3$; maximum at $x=1$
\wrong minimum at $x=3$; maximum at $x=1$
\wrong minimum at $x=-3$; maximum at $x=3$
\wrong minimum at $x=-3$; maximum at $x=-1$
\wrong no minimum; maximum at $x=-1$
\perm1:cbaed
\perm2:cdeba

\pagebreak

\problem
Suppose \$2,000 is deposited in an account paying 5\% annual interest compounded
continuously.  Assuming no further deposits or withdrawals are made, how many years will it
take for the balance to reach \$6,000?

\correct $\displaystyle{{\ln3}\over{0.05}}$ 
\wrong $\displaystyle{3\over{\ln(0.05)}}$
\wrong $\displaystyle{{\ln 3}\over5}$
\wrong $e^{1.5}$ 
\wrong $e^{0.05\ln3}$ 
\perm1:badce
\perm2:cdaeb

\problem
 Suppose \$500 is deposited in an account that pays 6\% annual interest, compounded 4
times per year.  Assuming no further deposits or withdrawals are made, how much will be in
the account at the end of 3 years?

\correct $500(1.015)^{12}$ 
\wrong $500(1.18)^4$ 
\wrong $500(1.06)^{12}$
\wrong $500(1.02)^4$ 
\wrong $500(1.24)^3$
\perm1:edcba
\perm2:bacde

\problem
Which of the following is the derivative of $\displaystyle {e^{3x^{2}} 
\over x-2}$?

\correct $\displaystyle {(6x^{2}-12x-1)e^{3x^{2}} \over (x-2)^{2}}$
\wrong $\displaystyle {{(x-3)e^{3x^{2}}} \over (x-2)^{2}}$
\wrong $\displaystyle {{(6x^2-12x-1)e^{3x^2}} \over x-2}$
\wrong $\displaystyle {{(13 - 3x)e^{3x^2}} \over 
(x-2)^{2}}$
\wrong $\displaystyle {{(6x^2 - 12x -2)e^{3x^2}} \over
(x-2)^{2}}$
\perm1:badec
\perm2:deabc

\pagebreak

\problem
The function $f(x) = (x-1)^{4}$ has
only one critical point, at $x=1$.  Apply the first and second
derivative tests for a relative extremum at that critical point.  Which of the following
statements is true?

\correct The first derivative test says that $f(x)$ has a relative minimum
at $x=1$; the second derivative test is inconclusive.
\wrong The first derivative test is inconclusive; the second derivative
test says that $f(x)$ has a relative minimum at $x=1$.
\wrong The first derivative test says that $f(x)$ has a relative minimum
at $x=1$; the second derivative test says that $f(x)$ has a relative
maximum at $x=1$.
\wrong Both tests say that $x=1$ is a relative minimum.
\wrong The first derivative test says that $f(x)$ has a relative minimum
at $x=1$; the second derivative test says that at $x=1$, $f(x)$ has
neither a relative maximum nor a relative minimum.
\perm1:cbade
\perm2:bcdae

\problem
 If $F(x)$ is an antiderivative of $\displaystyle {1\over{\sqrt{x}}}$ and $F(1) = 0$, what is 
$F(9)$?

\correct 4
\wrong 6 
\wrong 13/27 
\wrong 1/54 
\wrong 8
\perm1:dbcae
\perm2:acedb


\problem
  Where does the function $f(x) = x^5 - 5x^4 + x -2$ have inflection points, if any?

\correct Only at $x=3$ 
\wrong Only at $x=0$ 
\wrong Only at $x=-1$
\wrong At $x=0$ and $x=3$
\wrong At $x=0$ and $x=-1$
\perm1:debca
\perm2:bacde

\pagebreak

\problem  A certain function $f(x)$  has first and second derivatives as
follows:
 $$f'(x) = {{-2x}\over{(x^2-1)^2}} \hbox{\quad and \quad}
f''(x) = {{6x^2 + 2}\over{(x^2-1)^3}}\,.$$

Which of the following most closely resembles the graph of $f(x)$? (Note: the dashed lines are
not part of the graph.  They are only there to indicate asymptotes.)

\smallskip \noindent
Use the letter to the lower left of the picture.

\correct \picture 1.75in by 1.75in (graph1 scaled 450) 
\wrong \picture 1.75in by 1.75in (graph2 scaled 450)
\wrong \picture 1.75in by 1.75in (graph3 scaled 450)
\wrong \picture 1.75in by 1.75in (graph4 scaled 450)
\wrong \picture 1.75in by 1.75in (graph5 scaled 450)
\perm1:abcde
\perm2:dbeca

\pagebreak

\problem
 If the {\bf derivative} $f'(x)$ is decreasing for $x < a$ and increasing for $x > a$, which of the following
must be true about the {\bf function} $f(x)$?

\correct It is concave down for $x < a$ and concave up for $x > a$.
\wrong It is concave up for $x < a$ and concave down for $x > a$.
\wrong It is decreasing for $x < a$ and increasing for $x > a$.
\wrong It is increasing for $x < a$ and decreasing for $x > a$.
\wrong None of the other statements is necessarily true.
\perm1:cdbae
\perm2:badce

\problem
 Suppose $f(x)$ and $g(x)$ are functions satisfying
$$f(0) = 6, \quad g(0) = 3, \quad f'(0) = 5, \quad g'(0) = -1.$$
Let $h(x) = \ln\left(f(x) + g(x)\right)$.  What is $h'(0)$?

\correct 4/9 
\wrong 1/4 
\wrong 1/9 
\wrong 4
\wrong 2
\perm1:abcde
\perm2:edcba

\problem  An object whose initial temperature is $5 ^{\rm o}$C is placed in a large room whose temperature is kept at a
steady $20^{\rm o}$C.  According to Newton's law of cooling, the temperature is given by the formula $H = 20 + Ae^{kt}$, where
$H$ is the temperature (in degrees Celsius) and $t$ is the time in hours.  At the end of 1 hour the object has warmed to
$10^{\rm o}$C.  Find the constants $A$ and $k$.

\correct $A = -15, \ k = \ln(2/3)$ 
\wrong $A = -15, \ k = \ln(3/2)$ 
\wrong $A = 5, \ k = -\ln2$ 
\wrong $A = -15, \ k = \ln2$ 
\wrong $A = 5, \ k = -0.15$
\perm1:cabde
\perm2:cdbae

\pagebreak

\problem
 If you apply linear approximation to the function $f(x) = x^{-1/2}$ to approximate 
$(4+h)^{-1/2}$, where $h$ is a very small number, what do you get?

\correct $\displaystyle{1\over2} - {h\over16}$ 
\wrong $\displaystyle{1\over2} + {h\over4}$
\wrong $\displaystyle{1\over{2+h}}$
\wrong  $\left(2 + {h\over16}\right)^{-3/2}$ 
\wrong $\left(2 + \sqrt h\right)^{-1}$
\perm1:dbcae
\perm2:adbec

\problem
Where are the absolute maximum and minimum, if any, of the function $f(x) = xe^{-x}$?

\correct absolute maximum at $x=1$, no absolute minimum
\wrong no absolute maximum, absolute minimum at $x=1$
\wrong absolute maximum at $x=1$, absolute minimum at $x=0$
\wrong absolute maximum at $x=0$, absolute minimum at $x=1$
\wrong no absolute maximum or minimum 
\perm1:cdeba
\perm2:bcade

\problem
Find the equation of the line tangent to the curve $y=2x\ln x$ at the
point $(1,0)$.

\correct $y=2x-2$
\wrong $y=0$
\wrong $y=4x-4$
\wrong $y=2\ln x$
\wrong $y=-2x$
\perm1:cbade
\perm2:dceab



\end{document}


