\documentclass[12pt]{report}
%\addtolength{\textheight}{0.8in}
%\addtolength{\topmargin}{-0.4in}
\everymath{\displaystyle}
\hbadness = 10000
\tolerance = 10000

\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\name{\hbox{\hskip2.25in  {\bf Name:} \vrule height .4pt depth 0 pt
width 3in}
               \vskip.25in}
\def\med{\medskip}
\def\small{\smallskip}
\def\superskip{\bigskip \  \medskip}
\def\c{\centerline}
\def\eq{\ = \ }
\def\and{\hbox{\quad and \quad}}
\def\andd{\hbox{\qquad and \qquad}}
\def\page{\vfill \eject \noindent}
\def\next{\vskip.25in \noindent}

\begin{document}
\noindent
{\bf Math 105}  \hfil {\bf Review questions --- final}

\vskip.25in \noindent
1. If $f(x) = {x\over{x-3}}\,,$ \ for which value(s) of $x$ does $f(x)$
have a rel. max or min?

\medskip \noindent
(a) none \hfil (b)  $x=-1$ and $x=-3$ \hfil (c) $x=0$ \hfil (d) $x=1$ \hfil
(e) $x=3$

\bigskip \noindent
2. Suppose $y = e^x$ and $x=t^2$. \ Find ${{dy}\over{dt}}$ when $t=10$.

\medskip \noindent
(a) $20e^{100}$ \hfil (b) $e^{100}$ \hfil (c) 20 \hfil (d) 200 \hfil (e)
$100e^{20}$

\bigskip \noindent
3.  Let $f(x) = (\ln x)/x$.  On which of the following intervals is $f(x)$
increasing?

\medskip \noindent
(a) $(0,e)$ \hfil (b) $(1, 2e)$ \hfil (c) $(\ln2, 4)$ \hfil (d) $(e,
\infty)$ \hfil (e) no interval

\bigskip \noindent
4. 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.

\smallskip \noindent
(a) 261  \hfil (b)  256 \hfil (c) 266 \hfil (d) 278 \hfil (e)  237

\bigskip \noindent
5. Where does the function $f(x) = e^{-x^2}$ have an absolute maximum?

\medskip \noindent
(a) $x=0$ \hfil (b) $x=-1/\sqrt{2}$ \hfil (c) $x=1/\sqrt{2}$ \hfil (d)
$x=-1$ \hfil (e) no abs. max.

\bigskip \noindent
6. Where does the function $f(x) = e^{-x^2}$ have an absolute minimum?

\medskip \noindent
(a) no abs. min. \hfil (b) $x=-1/\sqrt{2}$ \hfil (c) $x=1/\sqrt{2}$ \hfil
(d) $x=1$ \hfil (e) $x=0$

\medskip \noindent
7. Find the derivative of  $f(x) = x^2 + (1/x^2).$

\medskip \noindent
(a) $2x - {2\over{x^3}}$ \hfil (b) $2x + {2\over{x^3}}$ \hfil (c) $2x -
{2\over x}$

\smallskip
\hfil (d) $2x + {2\over x}$ \hfil (e) $x - {1\over x}$ \hfil

\bigskip \noindent
8. Let $f(x) = x^2 + 2x + 3$. Where does $f(x)$ have an inflection point?

\medskip \noindent
(a) nowhere \hfil (b) $x=0$ \hfil (c) $x=1$ \hfil (d)  $x=2$ \hfil (e)  $x=3$

\bigskip \noindent
9. Let $y = 2^x$ and compute $dy/dx$.

\smallskip \noindent
(a) $(\ln2)2^x$ \hfil (b)  $2^x$ \hfil (c) $x2^{x-1}$ \hfil (d)  0 \hfil (e) 2

\pagebreak \noindent
10. The growth rate of a certain bacteria culture is proportional to its
size, which means that it satisfies an exponential
growth equation
$y = Ae^{kt}$
where $t$ is the time and $y$ is the size of the population..  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?

\smallskip \noindent
(a)  80,000 \hfil (b) 60,000 \hfil (c) $10,000\cdot2^{3/2}$ \hfil (d)
${{10,000}\over{\ln4}}$ \hfil (e) 120,000

\bigskip \noindent
11.  If $\displaystyle f(x) = {1\over {x^2}}$, which of the following
limits equals $f'(x)$?

\medskip \noindent
(a) $\displaystyle \lim_{h\to0} {1\over h} \left({1\over {(x+h)^2}} -
{1\over {x^2}}\right)$ \hfil
(b) $\displaystyle \lim_{h\to0} {h\over{(x+h)^2 - x^2}}$

\medskip \noindent
(c) $\displaystyle \lim_{h\to0} h\left({1\over {(x+h)^2}} - {1\over
{x^2}}\right)$ \hfil
(d) $\displaystyle \lim_{h\to0} {{1\over{x^2}} - {1\over{h^2}}\over h}$ \hfil
(e) $\displaystyle \lim_{h\to0} {1\over h} \left({1\over{x^2 - h^2}}\right)$

\bigskip \noindent
12. Find the slope of the graph of $y = \sqrt{x}$ at the point $(9,3)$.

\medskip \noindent
(a) 1/6 \hfil (b) 1/3 \hfil (c) 1/2 \hfil (d) 3  \hfil (e) 27

\bigskip \noindent
13. If $f(x) = {x \over{1 + x^2}}$, \ what is $f'(x)\,$?

\medskip \noindent
(a) ${{1-x^2}\over{(1+x^2)^2}}$ \hfil (b) ${1\over{2x}}$ \hfil (c)
${1\over{(1+x^2)^2}}$

\smallskip
\hfil (d) ${{1+3x^2}\over{1+x^2}}$ \hfil (e)  ${1\over{1+x^2}}$ \hfil

\bigskip \noindent
14. Which of the following is equal to $e^{(\ln3)/2}\>$?

\medskip \noindent
(a) $\sqrt3$ \hfil (b) $2^{1/3}$ \hfil (c) $2^{-1/3}$ \hfil (d)
$\sqrt{e^3}$ \hfil (e) $3/2$

\bigskip \noindent
15. 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?

\smallskip \noindent
(a) ${{\ln4}\over{0.08}}$ \hfil (b)  $(0.08)\ln4$ \hfil (c)
${{\ln(0.08)}\over4}$
\hfil (d)  $4\ln(0.08)$ \hfil (e) $250e^{0.08}$

\next
16.  Apply linear approximation to the function $f(x) = 1/x$ to estimate
$1/2.016\,.$  Write your answer in decimal form.

\pagebreak \noindent
17. \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
 \par \hskip3.5in \vbox{\special{picture pict scaled 400}}

\smallskip
(a) $f(x)$ is decreasing and $f'(x)$ is increasing.

\smallskip (b) $f(x)$ is increasing and $f'(x)$ is decreasing.

\smallskip
(c) $f(x)$ is increasing and $f'(x)$ is increasing.

\smallskip (d) $f(x)$ is decreasing and $f'(x)$ is decreasing.

\smallskip
 (e)  None of the other statements is true.

\bigskip \noindent
18. Find the equation of the line passing through the points $(3,7)$ and
$(2,2)$.

\smallskip \noindent
(a)  $y = 5x - 8$ \hfil (b) $y = 7x - 14$ \hfil (c) $y = 7x - 12$

\medskip
\hfil (e) $y = {1\over5}x + {8\over5}$ \hfil (e) $y = 5x + 2$ \hfil

\bigskip \noindent
19. The derivative of $(z^4 + 3z^2 + 1)^2$ \ is

\medskip \noindent
(a)  $2(z^4 + 3z^2 + 1)(4z^3 + 6z)$ \hfil (b) $2(z^4 + 3z^2 + 1)$ \hfil (c)
$(4z^3 + 6z)$

\medskip
\hfil (d) $(4z^3 + 6z)^2$ \hfil (e) $2(4z + 6z)$ \hfil

\bigskip \noindent
20. Suppose $f(x)$ is a function satisfying the following:
$$f'(1) = -2; \qquad f'(2) = -1; \qquad f'(3) = 1; \qquad f'(4) = 3$$
If $h(t) = f(t^3 + 1)$, what is $h'(1)$?

\medskip \noindent
(a) $-3$ \hfil (b) 3 \hfil (c) 1 \hfil (d) $-6$ \hfil (e) $-1$

\bigskip \noindent
21. Find the equation of the line tangent to the curve
$y = 1 + \ln x$
at the point $(e,2)$.

\medskip \noindent
(a) $y = {x\over e} + 1$ \hfil (b) $y = ex + 2$ \hfil (c) $ey = x - 2$

\smallskip
\hfil (d) $y -2 = x - e$ \hfil (e) $y = ex - 2$

\bigskip \noindent
22.  A manufacturer's monthly revenue from the sale of $x$ items is $R(x) =
20\sqrt{100x - x^2}$.  Find the marginal revenue.

\pagebreak
\noindent
23. Consider the function $\ f(x) = (8-2x)\,e^{(x+7)}$. Which one of the
following
statements is true?

\smallskip (a) $x = 3$ gives an absolute maximum.

\smallskip (b) $x=3$ gives a relative minimum.

\smallskip (c) $x=4$ gives a relative maximum.

\smallskip (d) The function is everywhere concave down.

\smallskip (e) There are no relative maxima or minima.

\bigskip \noindent
24.  What is $\lim_{n\to\infty}\left(1 + {2\over n}\right)^n$?

\medskip \noindent
(a) $e^2$ \hfil (b) $\ln 2$ \hfil (c) 1 \hfil (d) 0 \hfil (e) 1/2

\bigskip \noindent
25.  For each of the following functions, determine where it is increasing,
decreasing, concave up, concave down, has rel. max
or min, has an inflection point.  Then draw a rough sketch of the graph.

\medskip \noindent
(i) $f(x) = x^2e^{-x}$ \hfil (ii) $f(x) = 2x^3 - 3x^2 - 12x +5$ \hfil (iii)
$f(x) = x - \ln x$

\next
26.  Find $f'(x)$ in each of the following cases.  Simplify your answer as
much as possible.

(i) $f(x) = xe^{x^2}$ \hfil (ii)  $f(x) = {{\ln x}\over {x^2}}$, \quad$x > 0$.

\next
27.  Where, if anywhere, is the following  function {\em not} continuous?
$$f(x) =  \cases{x-2 &if $x \le 0$ \cr
                            2x-2 &if $0 < x \le 1$ \cr
                           1 - x &if $1 < x < 2$ \cr
                          x &if $x \ge 2$ \cr }$$

\smallskip \noindent
(a) only at $x = 2$ \hfil (b) only at $x = 1$ \hfil (c) only at $x=0$

\medskip
\hfil (d) at $x=1$ and $x=2$ \hfil (e) at $x=0$ and $x=1$ \hfil




\pagebreak \noindent
29. Suppose $f(x)$ is a function defined for $x>0$  whose 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.)

\vskip1.5in \noindent
\begin{minipage}[b]{2in} (a) \  \special{picture newgraph1 scaled 450}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(b) \  \special{picture newgraph2 scaled 450}
\end{minipage}

\vskip1.75in \noindent
\begin{minipage}[b]{2in} (c) \  \special{picture newgraph3 scaled 450}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(d) \  \special{picture newgraph4 scaled 450}
\end{minipage}

\vskip1.75in \noindent
(e) \ \special{picture newgraph5 scaled 450}


\pagebreak \noindent
\next
30. What is the slope of the line connecting the points $(-1, 3)$ and
$(2,1)$?

\medskip  \noindent
(a) $-2/3$ \hfil (b) $-1/2$ \hfil (c) $3/2$ \hfil (d) 2 \hfil (e) $-1/4$

\next
31.  Let $f(x) = xe^x$.  Determine where the function is increasing,
decreasing, concave up, and concave down; and find any relative
maximum, relative minimum and inflection points.

\next
32. (Continuing 31) Which of the following could be the graph of $y =
xe^x$?  Use the letter to  the lower left of the correct picture.

\vskip1.25in \noindent
\begin{minipage}[b]{2in} (a) \  \special{picture graph0 scaled 500}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(b) \  \special{picture graph2 scaled 500}
\end{minipage}

\vskip1.75in \noindent
\begin{minipage}[b]{2in} (c) \  \special{picture graph3 scaled 500}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(d) \  \special{picture graph4 scaled 500}
\end{minipage}

\vskip1.75in \noindent
\begin{minipage}[b]{2in} (e) \  \special{picture graph1 scaled 500}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(f) \  \special{picture graph6 scaled 500}
\end{minipage}

\pagebreak \noindent
33. Which of the following numbers is equal to $e^{-3\ln2}$?

\smallskip \noindent
(a) $1/8$ \hfil (b) $-8$ \hfil (c) $1/6$ \hfil (d) $2^{1/3}$ \hfil (e) $-6$

\next
34.  Suppose you put \$100 into an account that pays 6\% annual interest,
compounded
$n$ times a year.  How much do you have at the end of 1 year?

\bigskip \noindent
(a) $100\left(1 + {{0.06}\over n}\right)^{n}$ \hfil (b) $100\left(1 +
e^{0.06}\right)^{n}$
   \hfil (c) $100\left(1 + 0.06n\right)$

\bigskip
\hfil (d) $100e^{0.06/n}$ \hfil \hfil (e)  $100(0.06)^n$

\next
35. Suppose you put \$100 into an account that pays 6\% annual interest,
compounded
continuously.  How much do you have at the end of 3 years?

\bigskip \noindent
(a) $100e^{0.18}$ \hfil (b) $100\left(1 + e^{0.06}\right)^{3}$
   \hfil (c) $100\left(1 + {{0.06}\over {3}}\right)^3$

\bigskip
\hfil (d) $100\left(1 + 0.06\right)^{3}$ \hfil \hfil (e)  $100\left(1 +
0.18\right)^3$

\next
36.  What is $\lim_{n\to\infty}\left(1 + {0.06\over n}\right)^n$?

\bigskip \noindent
(a) $e^{0.06}$ \hfil (b) 1 \hfil (c) $\ln 0.06$ \hfil (d) 1/(0.06) \hfil (e) 0

\next
37. What is $\log_3(1/9)$?

\medskip \noindent
(a) $-2$ \hfil (b) $-1/2$ \hfil (c) 2 \hfil (d) 1/2 \hfil (e) $1/3$

\next
38.  For what $x$, if any, does
$y = 2x^2 + 5x + 3$
have an absolute minimum?

\smallskip \noindent
(a) $-1.25$ \hfil (b) 2.75  \hfil (c) 1.75 \hfil (d) $-2.25$ \hfil (e) none

\next 
39. Given the following table of values:
 \hfil
\begin{tabular}{|c||c|c|c|c|} \hline
$x$ & $f(x)$ & $g(x)$ & $f'(x)$ & $g'(x)$ \\ \hline \hline
0 & 2 & 1 & $-1$ & $3$ \\ \hline
1 & 1/5 & 3 & 1/2 & 1/3 \\ \hline
2 & $-2/3$ & 2/5 & 1/4 & 3/2 \\ \hline
\end{tabular}  \hfil

\smallskip \noindent
Find $h'(0)$ in each of the following cases:

\smallskip \noindent
(i) $h(x) = f(g(x))$ \hfil (ii) $h(x) = g(f(x))$ \hfil (iii) $h(x) =
\ln\left(f(x) + g(x)\right)$

\pagebreak \noindent
40.  A certain radioactive substance satisfies the exponential decay law,
$y = Ae^{-kt}$, where $t$ is the time (in days) and $y$ is the amount (in
grams) at time $t$.
Suppose an inital amount of 100 grams decays to 20 grams at the end of 4
days.
What is the value of $k$?

\smallskip \noindent
(a) ${{\ln5}\over 4}$ \hfil (b) $-4\ln(0.2)$ \hfil (c) $-(1.2)^4$ \hfil (d)
$0.2$\hfil
(e) $\ln(1.2)$

\bigskip \noindent
41. Suppose $f'(a) = 0$ and $f''(a) > 0$.  Which of the following
statements must be
true about $f(x)$?

 \noindent
(a) It has a relative minimum at $x=a$.

\noindent
(b) It has a relative maximum at $x=a$.

 \noindent
(c) It has an inflection point at $x=a$.

\noindent
(d) It is increasing in an interval around $x=a$.

\noindent
(e) It is decreasing in an interval around $x=a$.

\bigskip \noindent
42. Suppose that $f'(x) < 0$ for all $x < a$ and $f'(x) > 0$ for all $x >
a$.  Which of the
following must be true about $f(x)$?

\noindent
(a) It has an absolute minimum at $x=a$.

\noindent
(b) It has an absolute maximum at $x=a$.

\noindent
(c) It has a relative maximum at $x=a$ but not necessarily an absolute one.

 \noindent
(d) It has a relative minimum at $x=a$ but not necessarily an absolute one.

\noindent
(e) It has an inflection point at $x=a$

\bigskip \noindent
43.  If $f'(x)$ is increasing for $x < a$ and decreasing for $x > a$, which
of the following
must be true about  $f(x)$?

\noindent
(a) It has an inflection point at $x=a$.

\noindent
(b) It is increasing for $x < a$ and decreasing for $x > a$.

\noindent
(c) It is decreasing for $x < a$ and increasing for $x > a$.

\noindent
(d) It is concave up for all $x$.
\hfil
(e) It is concave down for all $x$.


\pagebreak \noindent
44.  Suppose   $f'(x) > 0 \hbox{\ for all \ } x, \hbox{\ and}$ \hfil
$$f''(x) > 0  \hbox{\ if \ } x < 0 \hbox{ \ and \ } f''(x) < 0 \hbox{\ if \
} x > 0.$$
Which of the following could be the graph of $f(x)$? Use the letter to the
lower left of the correct picture.
\vskip1.5in \noindent
\begin{minipage}[b]{2in} (a) \  \special{picture inc1 scaled 400}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(b) \  \special{picture inc2 scaled 400}
\end{minipage}

\vskip1.75in \noindent
\begin{minipage}[b]{2in} (c) \  \special{picture inc4 scaled 400}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(d) \  \special{picture inc3 scaled 400}
\end{minipage}

\vskip1.75in \noindent
(e) \ \special{picture inc5 scaled 400}

\pagebreak \noindent
45.  Suppose   $f'(x) > 0$ if $x < 0$ and $f'(x) < 0$ if $x > 0$, and
$$f''(x) < 0  \hbox{\ if \ } -1 < x < 1 \hbox{\ and \ } f''(x) > 0 \hbox{\
if\ } x < -1 \hbox{\ or\ } x >
1.$$   Which of the following could be the graph of $f(x)$? Use the letter
to the lower left of
the correct picture. \vskip1.5in \noindent
\begin{minipage}[b]{2in} (a) \  \special{picture graph6 scaled 400}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(b) \  \special{picture graph1 scaled 400}
\end{minipage}

\vskip1.75in \noindent
\begin{minipage}[b]{2in} (c) \  \special{picture graph3 scaled 400}
 \end{minipage} \ \hskip.5in \
\begin{minipage}[b]{2in}
(d) \  \special{picture graph7 scaled 400}
\end{minipage}

\vskip1.75in \noindent
(e) \ \special{picture graph8 scaled 400}

\pagebreak \noindent
46. If $f(x) = 2^x$, which of the following limits is equal to $f'(0)$?

\medskip \noindent
(a) $\lim_{h\to0}{{2^h - 1}\over h}$
\hfil (b) $\lim_{h\to0} 2^h$
\hfil (c) $\lim_{h\to0} h2^{h-1}$

\medskip \noindent
\hfil (d) $\lim_{h\to0}{{2^h}\over h}$
\hfil (e)  $\lim_{h\to0}{{2^h + 1}\over 2}$

\next
47. If $h$ is a small number and you use linear approximation to estimate
$\ln(2+h)$, what do you get?

\medskip \noindent
(a) $\displaystyle \ln2 + {h\over2}$ \hfil (b) $\ln 2+\ln h$ \hfil (c)
$\displaystyle \ln 2+{{\ln h}\over{2}}$

\smallskip \noindent
\hfil (d) $2 + \ln(h/2)$ \hfil (e) $2 + 2\ln h$ \hfil

\next
48.  A company has been selling lamps for \$25 apiece, and consumers have
been buying
2,900 a month at that price.  The company wants to raise the price and
estimates that for each
\$1 increase in price they will sell 100 fewer lamps.
What price should the company charge in order to maximize their revenue?

\next
49. Suppose a beaker of hot water is put in a room whose temperature is
kept at 70 degrees
Fahrenheit.  According to Newton's law of cooling, the temperature is given
by the formula
$\ H = 70 + Ae^{kt}, \ $
where $H$ is the temperature, $t$ is the time in minutes, and $A$ and $k$
are constants.  At the
end of 20 minutes the temperature of the water is 170 degrees and it is
decreasing at the
rate of 1 degree per minute.  Find the constants $A$ and $k$.

\next
50.  Use linear approximation to estimate $\sqrt{99}$.

\addtolength{\textheight}{1in}
\addtolength{\topmargin}{-0.5in}
\pagebreak
\noindent
\underbar{\bf Answers}

\vskip.25in \noindent
All multiple-choice questions have (a) as answer.

\next 
16. 0.496 \hfil 
22. ${{1000 - 20x}\over{\sqrt{100x - x^2}}}$

\next
 \begin{minipage}[b]{2.5in}
25 (i) increasing: $0 < x < 2$

decreasing: $x < 0$ or $x > 2$

rel.~min.~at $x=0$, rel.~max.~at $x=2$

concave up: $x< 2 - \sqrt{2}$ or $x > 2 + \sqrt{2}$

concave down: $2 - \sqrt{2} < x < 2 + \sqrt{2}$

inflection pt.~at $x=2 + \sqrt{2}$ and $x=2-\sqrt{2}$
\end{minipage} \ \hskip.25in \
\begin{minipage}[b]{3in}
\special{picture sketch1 scaled 400}
\end{minipage}

\next
\begin{minipage}[b]{2.5in}
(ii) increasing: $x<-1$ or $x>2$

decreasing: $-1 < x < 2$

rel.~max.~at $x=-1$

rel.~min.~at $x=2$

concave up: $x > 1/2$

concave down: $x < 1/2$

inflection pt.~at $x=1/2$
\end{minipage} \ \hskip.25in \
\begin{minipage}[b]{3in}
\special{picture sketch2 scaled 300}
\end{minipage}

\next
\begin{minipage}[b]{2.5in}
(iii) increasing: $x>1$

decreasing: $0 < x < 1$

rel.~(and abs.) min.~at $x=1$

always concave up

no inflection pt.
\end{minipage} \ \hskip.25in \
\begin{minipage}[b]{3in}
\special{picture sketch3 scaled 300}
\end{minipage}

\next 
26. (i) $e^{x^2}(1 + 2x^2)$ \quad (ii) $\displaystyle{{1-2\ln x}\over{x^3}}$

\next
31. increasing for $x > -1$, decreasing for $x < -1$, concave up for $x >
-2$, down for $x < -2$, rel.~min.~at $x=-1$, inflection pt.~at $x=-2$.

\next
39. (i) 3/2 \hfil 
(ii) $-3/2$ \hfil 
(iii) 2/3

\next
48. \$27 \hfil 
49.  $k = -0.01, \ A = 100e^{0.2}$ \hfil
50. 9.95


\end{document}


