% This is a plain TeX file. I have tried to avoid any intricate or
% nonstandard constructions. The pictures in this file are included
% using the epsf package of macros which seems to be generally
% available. Each picture file must be imported separately and put in
% the same directory as this TeX file using the names indicated. There
% are five picture files, with names rev1-1.eps, rev1-2.eps,
% rev1-3.eps, rev1-4.eps, and (unremarkably!) rev1-5.eps. The problems
% will fit rather snugly on two pages of standard 8.5 by 11
% inches paper. Emphasize, please, that these problems are designed to
% be suggestions for review and do NOT preclude other problems or
% types of problems, especially including material you might have
% stressed in your lectures. Please note that the eps files are in the
% same directory as this tex file.

\font\largerbold = cmbx10 scaled \magstep1
\font\seventt=cmtt10 at 7 pt
\input epsf
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%\input amssym.def
\def\d{\displaystyle}
\nopagenumbers
\rightline{\sevenbf 2/8/99}
\medskip
\centerline{\largerbold Review Problems for the first exam in Math 135
in the Spring 1999 semester}

\bigskip

Please note that this collection of 18 problems is definitely longer
than any ``real'' exam. The problems are intended to be representative
of the type of problems which may appear on such exams.

\smallskip

Graphing calculators may be used on exams but calculators and
computers with QWERTY keyboards or symbolic differentiation and
integration programs are not allowed. Students may bring one sheet of
paper containing notes (using both sides!) up to 8.5 by 11 inches in
size to any exam.

\medskip
\overfullrule=0pt
\hbox{\hsize=3.7in \vtop{
1.~Here is a graph of $y=A+B\sin(x+C)$. Estimate values of $A$, $B$,
and $C$ from the graph.
\smallskip
Note that the graph displayed is in the window $0 \le x \le 10$ and $4
\le y \le 10$. 
}}
 
\vskip -2in
\hskip 3.7in{\epsfxsize=3in\epsfbox{rev1-1.eps}}
\vskip -1.1in


2.\rightline{\hbox{\hsize=3.85in \vtop{Suppose that the line segment
$\overline{AB}$ has length 12 and the line segment $\overline{DE}$ has
length 5 in the right triangle shown. What is the side length $x$ of
the square? (Use similar triangles.)  What is the length of the
largest hypotenuse, $\overline{AE}$? What is the angle $\theta$?}}}

\vskip -1.3in
{\quad\ \epsfxsize=2.3in\epsfbox{rev1-2.eps}}

\medskip

3.~a) Simplify $e^{3\ln \sqrt{x}}$ (give an expression not involving
any exponential or logarithmic function).
\smallskip

b) If $\ln A = 1.2 $ and $\ln B = 5.3$, compute $\ln\! \left(B^2
\sqrt{A}\right)$ exactly. Also, is $A^{9}$ greater than or less than
$B^2$?

\medskip

4.~a) Find an equation of the circle $C$ whose center is at $(2,0)$ and
whose radius is $3$.

b) Find an equation of the straight line $L$ which goes through the
points $(3,0)$ and $(0,6)$.

c) Find the coordinates of the points where the straight line $L$ and
the circle $C$ intersect. Sketch the line and the circle on the same
coordinate axes.

\medskip

5.~A function $Z$, defined for $x \ge 0$, is given by
$Z(x) = \cases{ 2x & if $0\le x \le 3$\cr 
x+3 & if $3 < x$\cr}$.

\smallskip

a) Sketch the graph of $y=Z(x)$. 

b) Compute $Z(2)$ and $Z(4)$. 

c) Suppose the function inverse to $Z$ is called $N$. What is $N(8)$?
What is $N(2)$?

d) Sketch the graph of $y=N(x)$. 

\medskip

6.~Suppose $p = 100 + 3\sqrt{n}$ represents the production cost, $p$,
in dollars of $n$ boxes of detergent when $n$ satisfies the
restriction $1 \le n \le 100$. Solve the equation for $n$ as a
function of $p$. What is the domain of this function of $p$? 
Describe what the formula you have obtained represents in terms of
detergent and production costs. Be as specific as you can.

\medskip

7.~Use algebraic methods developed in this course to find the
following limits, if they exist, and give evidence supporting your answers:
{$\d {\rm a)\ } \lim_{x\to 2} {{x-2} \over {x^2 + 4}} \quad {\rm b)
\ } \lim_{x\to 2} {{x-2} \over {x^2 - 4}} \quad {\rm c)\ }
\lim_{x\to 4} {{x-4} \over {\sqrt{x} + 2}} \quad {\rm d)\ }
\lim_{x\to 4} {{x-4} \over {\sqrt{x} - 2}}$.}

\medskip

8.~Write an equation of the line tangent to the graph of the function
$f(x)=3x^2 +4$ when $x=1$. 

\medskip

9.~Use the definition of derivative and familiar properties of limits
to find the derivative of $\d f(x)= {3\over x}$.

\medskip 

10.~Compute the derivative of these functions: 
{$\d{ {\rm a)\ }\d x^3-3x+17 \quad {\rm b)\ }e^x \sin x \quad {\rm c)\ }
{{2x+3} \over{x^2+1}} \quad {\rm d)\ } 3e^x + {x \over {2+ 5\cos
x}}}$.}

\medskip

11.~Suppose $f$ and $g$ are differentiable functions such that $f(g(x))
= 8x^2$ for all real numbers $x$. Assume that $f(2)=7$, $g(2)=4$, and
$f'(4)=2$. What is $g'(2)$? 
\medskip
\rightline{\sevenbf OVER}
\vfil\eject

{\hsize=3.3in \vtop{
12.~A flag pole which is 10 feet tall is placed vertically on flat
ground. 

\smallskip

a) If a ball on the ground is 15 feet away from the base of the flag
pole, what is the angle seen at the ball from the base of the flag
pole to the top of the flag pole?

b) If a box on the ground is $A$ feet from the base of the flag pole,
write an equation for the angle $\alpha$ seen at the box from the base
of the flag pole to the top of the flag pole in terms of $A$.

}}
\vskip -1.9in
\rightline{\epsfysize=2in \epsfbox{rev1-3.eps}}
%50% portrait produced by xfig

13.~Let $\d f(x)=\left\{\matrix{x^{2}+2,& x<0\cr
x+A,& 0\le x<2\cr
x^{2}+1,& x\ge 2,}\right.
$
where $A$ is a number to be determined.

a) Find $A$ such that for this particular $A$, $f(x)$ is continuous at
$x=0$. 

b) For the $A$ you obtained in part a), is $f$ also continuous at
$x=2$? Explain your answer clearly.

\medskip

\rightline{\qquad\epsfxsize=3in \epsfbox{rev1-4.eps}} 

\vskip -2.3in

\medskip

14.~The short-lived commercial venture 135.com\footnote*{\sevenrm
Simulated. Your returns may vary. Please consult your own investment
and tax advisors} existed for 

{\hsize=3.7in \vtop{
only one
week in January. Various events affected the share price of 135.com:

\smallskip

{\baselineskip=.8\baselineskip
\hsize=3.45in 
{\sevenrm \item{\sevenbf A}
A rumor that the course would be required
by {$\scriptstyle every$} major at Rutgers with a final grade of ``B'' or
better.}

{\sevenrm \item{\sevenbf B} 
Anything with ``.com'' in it is a great stock.}

{\sevenrm \item{\sevenbf C} Only majors in Criminal Forestry would
have to take Math 135.}


{\sevenrm \item{\sevenbf D} 
A company selling calculus can't be a very good investment.}

{\sevenrm \item{\sevenbf E} 
Identical final exams every semester, producing uniformly high grades
with consequences for enrollment and re-enrollment.}

}

When is the price per share continuous? When is it differentiable?
Briefly explain your answers. To which points on the stock chart would
each news item listed above be likely to correspond?

}}


\medskip\bigskip\bigskip

15.~Draw the graph of a function $h$ with {\it all} these properties;
write the equations of any vertical asymptotes.
\medskip
\centerline{
$\d{\left[\matrix{
{\hsize=6in \vtop{
The domain of $h$ is $0 < x < 3$ and $3< x < 5$; $h$ is continuous
everywhere in its domain; $\d\lim_{x \to 0^+} h(x) = +\infty$ and
$\d\lim_{x \to 3^-} h(x) = +\infty$ and $\d\lim_{x \to 3^+} h(x) =
-\infty$ and $\d\lim_{x \to 5^-} h(x) = 1$; $h$ is differentiable
everywhere in its domain, and the only $x$'s for which $h'(x)=0$ are
$2$ and $4$; $h(2)=1$ and $h(4)=2$.}}
}\right]
}$
}

\medskip
\vskip .15in
\rightline{\qquad\epsfxsize=3in \epsfbox{rev1-5.eps}}
%50% portrait produced by xfig
\vskip -1.25in  

{\hsize=3.4in \vtop{ 16.~This graph represents the concentration of
enzyme in the bloodstream as a function of time. For each of the
following statements, use the graph to find a time interval in which
the statement is true. Explain your assertions.

%\parindent = .16in

i) Enzyme concentration is increasing, but the
rate of increase is slowing.

ii) Enzyme concentration is increasing and the
rate of increase is also increasing.
}}

\medskip\smallskip

17.~Suppose that $f$ and $g$ are differentiable functions, and the following
information is known:

\centerline{$f(1)=2; \quad f'(1)=-3; \quad g(1)=5; \quad g'(1)=7$.}

If $F(x)= f(x)g(x)$ and $\d G(x)= {f(x) \over {g(x) +x^2}}$, compute
$F(1)$, $F'(1)$, $G(1)$, and $G'(1)$.

\medskip

18.~Find $\d {{dy}\over {dx}}$ when $y$ is each of the following:
$\d{
{\rm a)\ } y= e^{3\cos 2x}\quad
{\rm b)\ } y=\ln(x^3 +5x+9)\quad
{\rm c)\ } y=\sin\!\left({{x+1}\over{x^2+2}}\right)
}$.

\smallskip

\centerline{\vrule height 0pt depth 1pt width 3in}
\medskip
\centerline{On the web at\ \  {{\tt
http://www.math.rutgers.edu/math135.html}}}

\vfill\eject\end 
