\lastversion =2
\hbadness = 10000
\hfuzz = 20 in

\title {Math 105 \qquad Test 1}
\date {September 24, 1996}
\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}}



\problem If $f(x) = 2^x$, which of the following limits is equal to $f'(0)$?
\correct $\lim_{h\to0}{{2^h - 1}\over h}$
\wrong $\lim_{h\to0} 2^h$
\wrong $\lim_{h\to0} h2^{h-1}$
\wrong $\lim_{h\to0}{{2^h}\over h}$
\wrong $\lim_{h\to0}{{2^h + 1}\over 2}$
\perm1:edcba
\perm2:baced

\problem If $f(x) = {x\over{x-3}}\,,$ \ for which value(s) of $x$ does $f(x)$ have relative
extrema?
\correct none
\wrong $x=-1$ and $x=-3$
\wrong $x=0$
\wrong $x=1$
\wrong $x=3$
\perm1:bacde
\perm2:cdbea

\problem Suppose $y = e^x$ and $x=t^2$. \ Find ${{dy}\over{dt}}$ when $t=10$.
\correct $20e^{100}$
\wrong $e^{100}$
\wrong 20
\wrong 200
\wrong $100e^{20}$
\perm1:cabde
\perm2:debac

\pagebreak

\problem Let $f(x) = {{\ln x}\over{x}}$.  On which of the following intervals is $f(x)$
increasing?
\correct $(0,e)$
\wrong $(1, 2e)$
\wrong $(\ln2, 4)$
\wrong $(e, \infty)$
\wrong no interval
\perm1:dbace
\perm2:edcba

\problem Calculate $\int_1^2 x^3 dx$.
\correct 15/4
\wrong 13/4
\wrong 11/4
\wrong 3 
\wrong 4
\perm1:dcbae
\perm2:eabcd

\problem Simplify ${{\sqrt{x}}\over{x^{-2}}}\,.$
\correct $x^{5/2}$
\wrong $x^{7/2}$
\wrong $x^{3/2}$
\wrong $x^3$
\wrong $x^{-3}$
\perm1:dcabe
\perm2:adbec

\pagebreak

\problem If $f(x) = x + {1\over x}$ for $x > 0$, the first and second derivatives are as follows:
$$f'(x) = {{(x-1)(x+1)}\over{x^2}} \hbox{\qquad and \qquad}
       f''(x) = {2\over{x^3}}\,.$$
Which of the following might be the graph of $f$? (Use the letter to the lower left of the
correct picture.)
\correct \picture 1.75in by 1.75in (newgraph1 scaled 450) 
\wrong \picture 1.75in by 1.75in (newgraph2 scaled 450)
\wrong \picture 1.75in by 1.75in (newgraph3 scaled 450)
\wrong \picture 1.75in by 1.75in (newgraph4 scaled 450)
\wrong \picture 1.75in by 1.75in (newgraph5 scaled 450)
\perm1:bdeac
\perm2:cbade

\pagebreak

\problem Find the derivative of  $f(x) = x^2 + {1\over{x^2}}\,.$
\correct $2x - {2\over{x^3}}$
\wrong $2x + {2\over{x^3}}$
\wrong $2x - {2\over x}$
\wrong $2x + {2\over x}$
\wrong $x - {1\over x}$
\perm1:bceda
\perm2:adecb

\problem Solve for $x$: \quad $\Big(2^x2^{-3}\Big)^2 = 2^2\,.$
\correct 4
\wrong 3
\wrong 2
\wrong 5
\wrong 1
\perm1:dabce
\perm2:ecbad

\problem The growth rate of a certain bacteria culture is proportional to its size.  Suppose
that the culture contains 10,000 bacteria at noon, and 40,000 bacteria at 2:00 PM.  How many
bacteria will it contain at 3:00 PM?
\correct 80,000
\wrong 60,000
\wrong $10,000\cdot2^{3/2}$
\wrong ${{10,000}\over{\ln4}}$
\wrong 120,000
\perm1:bcade
\perm2:bdeca

\pagebreak

\problem \hangindent =-2.25in \hangafter=0
Suppose  that for a certain value of $x$,
$$\sin x > 0 \hbox{\ and \ } \cos x < 0\,.$$
If $$0 \le x \le 2\pi, $$ which of the following can you conclude about $x$?

\par  \hskip 3.5 in  \special{picture axes scaled 300}
\correct $\pi/2 \le x \le \pi$
\wrong $0 \le x \le \pi/2$
\wrong $\pi \le x \le 3\pi/2$
\wrong $3\pi/2 \le x \le 2\pi$
\wrong There is no $x$ satisfying $\sin x > 0$ and $\cos x < 0$.
\perm1:bacde
\perm2:ebacd

\problem Suppose $\ f(x) = \cases{2x-2 &if $x \ne 0 \hbox{\ and\ } x \ne 2$ \cr
                          0 &if $x=2$ \cr  -2 &if $x=0$}$.
\bigskip \noindent
At which points does $f(x)$ fail to be continuous?
\correct only $x=2$
\wrong only $x = 0$
\wrong only $x = -2$
\wrong both $x=0$ and $x=2$
\wrong none
\perm1:abcde
\perm2:cdabe

\problem Let $f(x) = x^2 + 2x + 3$. Where does $f(x)$ have an inflection point?
\correct nowhere
\wrong $x=0$
\wrong $x=1$
\wrong $x=2$
\wrong $x=3$
\perm1:abcde
\perm2:bcdea

\pagebreak

\problem Suppose that $x$ and $y$ are related by the equation $\ y = xy^2 + x$. \ 
Find ${{dy}\over{dx}}$.
\correct ${{y^2 + 1}\over{1 - 2xy}}$
\wrong $y^2 + 1$
\wrong $y^2 + 2xy + 1$
\wrong ${{-y^2-1}\over{2xy}}$
\wrong none of the other answers is correct
\perm1:dbace
\perm2:dbcae

\problem Let $y = 2^x$ and compute ${{dy}\over{dx}}$.
\correct $(\ln2)2^x$
\wrong $2^x$
\wrong $x2^{x-1}$
\wrong 0
\wrong 2
\perm1:bdcea
\perm2:eacbd

\problem \hangindent =-3.5in \hangafter=0
What is the radian measure of the angle shown on the right?

\bigskip
\ 
\bigskip

\par  \hskip 3.5 in  \special{picture angle scaled 450}
\correct $5\pi/2$
\wrong $\pi/2$
\wrong $3\pi$
\wrong $5\pi$
\wrong $\pi$
\perm1:abcde
\perm2:cdeba

\pagebreak

\problem Find the slope of the graph of $y = \sqrt{x}$ at the point $(9,3)$.
\correct 1/6
\wrong 1/3
\wrong 1/2
\wrong 3 
\wrong 27
\perm1:dbcae
\perm2:aebdc

\problem If $f(x) = {x \over{1 + x^2}}$, \ what is $f'(x)\,$?
\correct ${{1-x^2}\over{(1+x^2)^2}}$
\wrong ${1\over{2x}}$
\wrong ${1\over{(1+x^2)^2}}$
\wrong ${{1+3x^2}\over{1+x^2}}$
\wrong ${1\over{1+x^2}}$
\perm1:edcba
\perm2:debac

\problem Which of the following is equal to $e^{(\ln3)/2}\>$?
\correct $\sqrt3$
\wrong $2^{1/3}$
\wrong $2^{-1/3}$
\wrong $\sqrt{e^3}$
\wrong $3/2$
\perm1:bdeac
\perm2:cadbe

\pagebreak

\problem \hangindent =-2.25in \hangafter=0
The graph of $y = f(x)$ is shown on the right.
\smallskip
 Which of the following statements is true? 
\bigskip
\bigskip
 \par \hskip3.5in \vbox{\special{picture pict scaled 400}}
\correct $f(x)$ is decreasing and $f'(x)$ is increasing.
\wrong $f(x)$ is increasing and $f'(x)$ is decreasing.
\wrong $f(x)$ is increasing and $f'(x)$ is increasing.
\wrong $f(x)$ is decreasing and $f'(x)$ is decreasing.
\wrong None of the other statements is true.
\perm1:dabce
\perm2:cbade

\problem \hangindent =-2.25in \hangafter=0
A farmer with 800 feet of fencing wants to enclose a rectangular region and then
divide it into three pens with fencing parallel to one side of the rectangle (as shown on the
right).
What is the maximum total area of the region?
 \par \hskip3.5in \vbox{\special{picture fence scaled 400}}
\correct 20,000 square feet
\wrong 160,000 square feet
\wrong 50,000 square feet
\wrong 10,000 square feet
\wrong 30,000 square feet
\perm1:cbeda
\perm2:daecb

\pagebreak

\problem Find the equation of the line passing through the points $(3,7)$ and $(2,2)$.
\correct $y = 5x - 8$
\wrong $y = 7x - 14$
\wrong $y = 7x - 12$
\wrong $y = {1\over5}x + {8\over5}$
\wrong $y = 5x + 2$
\perm1:bedac
\perm2:ebcda

\problem The derivative of $(z^4 + 3z^2 + 1)^2$ \ is
\correct $2(z^4 + 3z^2 + 1)(4z^3 + 6z)$
\wrong $2(z^4 + 3z^2 + 1)$
\wrong $(4z^3 + 6z)$
\wrong $(4z^3 + 6z)^2$
\wrong $2(4z + 6z)$
\perm1:abcde
\perm2:deacb

\problem Suppose you put \$250 into a bank account that receives 8\% annual interest
compounded continuously. How many years will it take before your bank account has 
\$1,000?
\correct ${{\ln4}\over{0.08}}$
\wrong $(0.08)\ln4$
\wrong ${{\ln(0.08)}\over4}$
\wrong $4\ln(0.08)$
\wrong $250e^{0.08}$
\perm1:bcead
\perm2:abecd

\pagebreak

\problem Find the derivative of the function $f(x) = (\sin x)^2\,.$
\correct $2\sin x \cos x$
\wrong $2\sin x$
\wrong $-2\cos x$
\wrong $-(\cos x)^2$
\wrong $(\cos x)^2$
\perm1:abcde
\perm2:deabc

\problem If $\ f(x) = \cases{\sqrt{x} &if $0 \le x < 2$ \cr \ \cr 1+x &if $2 \le x < \infty$ \cr}$
\ find the value of the product $f(1)\cdot f(2)$.
\correct 3
\wrong $\sqrt2$
\wrong $2\sqrt2$
\wrong $3\sqrt2$
\wrong 0
\perm1:edcba
\perm2:cebad

\problem Find the equation of the line tangent to the curve
$$y = 1 + \ln x$$
at the point $(e,2)$.
\correct $y = {x\over e} + 1$
\wrong $y = ex + 2$
\wrong $ey = x - 2$
\wrong $y -2 = x - e$
\wrong $y = ex - 2$
\perm1:bacde
\perm2:cabed

\pagebreak

\problem A ball is thrown up into the air so that its height $h(t)$ after $t$ seconds is given by
the formula
$$h(t) = -16t^2 + 128t + 5\,.$$
Find the maximum height that the ball reaches. 
\correct 261
\wrong 256
\wrong 266
\wrong 278
\wrong 237
\perm1:ebacd
\perm2:abcde

\problem Consider the function $\ f(x) = (8-2x)\,e^{(x+7)}$. Which one of the following
statements is true?
\correct $x = 3$ gives an absolute maximum.
\wrong $x=3$ gives a relative minimum.
\wrong $x=4$ gives a relative maximum.
\wrong The function is everywhere concave down.
\wrong There are no relative maxima or minima.
\perm1:abcde
\perm2:edcba

\problem If $\ f'(x) = x^2 + x^{1/2}$ \ and $\ f(4) = 80/3$,  then $f(1)$ is equal to
\correct 1
\wrong 1/3
\wrong 2/3
\wrong 4/3
\wrong 8/3
\perm1:ebadc
\perm2:bcdae



\end 

%%%%%%%%

\hangindent =-2.25in \hangafter=0
Suppose you know that for a certain value of $x$,
$$\sin x > 0 \hbox{\ and \ } \cos x < 0\,.$$
If $$0 \le x \le 2\pi, $$ which of the following can you conclude about $x$?

\par  \hskip 3.5 in  \special{picture axes scaled 300}




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