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\rightline{\sevenbf 10/27/98}

\medskip
\centerline{\largerbold Review Problems for the second exam in Math 135}

\bigskip

This collection of 16 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. Many routine review problems
are available in the review sections of the text (for example, the
{\it Review Exercises for Chapter 3}\/ appear on pages 256 through 258).
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

1.~What is the rectangle with largest area that has sides parallel to
the coordinate axes and that can be enclosed between the $x$-axis and
the curve $y= 4-x^2$?

\bigskip\smallskip
{\hsize=3.8in\vtop{2.~Here is a graph of $y=h(x)$, a function whose
domain is $-1\le x < 1.5$ and $1.5 < x \le 3$. Use this graph to
sketch the graph of the derivative of $h$, $h'(x)$, as well as you
can. Does $h$ have any critical numbers? Does $h$ have any relative
maxima or minima? Does $h$ have any inflection points?}}

\vskip -1in 

\rightline{\epsfxsize=2in\epsfbox{rev2-0.eps}\qquad}
\smallskip

3.~Two non-negative numbers have sum equal to a positive number,
$P$. How should the numbers be selected so that the product of the
cube of one times the square of the other is largest?

\medskip

4.~Find $\d\lim_{x \to \infty}$ and $\d\lim_{x \to -\infty}$ for the
following functions:

\centerline{$\d{{\rm a)\ }{{e^{2x}}\over {3+e^{4x}}}\qquad\qquad
{\rm b)\ }{{e^{4x}}\over {3+e^{2x}}}\qquad\qquad
{\rm c)\ }{{x^3+1}\over {x^2+x+1}}\qquad\qquad
{\rm d)\ }{{x+1}\over {\sqrt{x^2+2x+3}}}}$}

\medskip

5.~At a certain time, the length of a rectangle is 3 feet and its
width is 5 feet. At that time, the length is {\it increasing} at .4
feet per second, and the width is {\it decreasing} at .5 feet per
second.

\smallskip

a) At that time, how fast is the 
{%
\raise -.47em\hbox{\vrule\vbox{\hrule
        \hbox spread 8pt{\hfil\vbox spread 8pt{\vfil
                \hbox {\bf area}%
        \vfil}\hfil}
\hrule}\vrule}%
}
of the rectangle changing? Is
the area increasing or decreasing?

\smallskip

b) At that time, how fast is the 
{%
\raise -.7em\hbox{\vrule\vbox{\hrule
        \hbox spread 8pt{\hfil\vbox spread 8pt{\vfil
                \hbox {\bf diagonal}%
        \vfil}\hfil}
\hrule}\vrule}%
}
of the rectangle changing?
Is the diagonal increasing or decreasing?

\medskip

6.~a) Graph $\d{f(x)= e^{x - x^2}}$ in the window $-2 \leq x \leq 2$
and $0 \leq y \leq 2$.

\smallskip

b) Calculate $f'(x)$. Find the roots of $f'(x)=0$ algebraically. Where
is $f'(x)$ positive and where is it negative?  Explain how your
results apply to the graph of $y=f(x)$.  

\smallskip

c) Calculate $f''(x)$. Find the roots of $f''(x)=0$
algebraically. Where is $f''(x)$ positive and where is it negative?
Explain how your results apply to the graph of $y=f(x)$.

\medskip

7.~Sketch a graph of a function $W(x)$ which has all of the following
properties:
\smallskip

{\parindent=.3in 
\item{$\bullet$} The domain of $W$ is all numbers
except $1$ and $W$ is differentiable everywhere in its domain.

\item{$\bullet$} The only root of $W(x)=0$ is at $0$.

\item{$\bullet$} The only roots of $W'(x)=0$ are at $-1$ and $2$, and
$W(-1)=2$ and $W(2)=1$.

\item{$\bullet$} The only root of $W''(x)=0$ is at $-2$ and
$W(-2)=1$.

\item{$\bullet$} $\d \lim_{x\to -\infty} W(x) = 0$,
$\d \lim_{x\to 1^-} W(x) = -\infty$,
$\d \lim_{x\to 1^+} W(x) = \infty$, and 
$\d \lim_{x\to \infty} W(x) = \infty$.

}
\smallskip
Indicate carefully any relative extrema, any inflection points, and
any horizontal or vertical asymptotes. 

\medskip

8.~A circular puddle on a flat concrete floor is spreading so that at a
certain time, the radius is increasing at .3 inches per second. If the
area of the puddle at that time is 25 square inches, how fast is the
area changing? 

\medskip

9.~a) Find the maximum and minimum values of the function
$Q(x)=x^3+(x-4)^2$ on the interval $[-4,4]$.

\smallskip

b) Find the maximum and minimum values of the function
$Q(x)=x^3+(x-4)^2$ on the interval $[0,4]$.

\smallskip

10.~a) Where is $\d{{\sin x}\over{1+x^2}}$ equal to 0?

\smallskip

b) Why am I {\it sure}\/ that $\d{{(1 +x^2) \cos x -2x \sin
x}\over{(1+x^2)^2}}$ has at least 5 different roots inside the
interval $0 < x < 20$?

{\bf Hint} 20 divided by $\pi$ is more than 6. This fact and
a) allow you to verify b) {\it without}\/ substantial work if you can
see a simple relationship between a)'s function and b)'s function. 

\medskip

11.~An open-topped box has sides costing \$10 per
square foot and a square bottom costing \$20 per square foot. What is
the cheapest cost for such a box whose volume is 10 cubic feet? 

\medskip


{\hsize=3.4in \vtop{12.~Here is a graph of $y=Q(x)$ which you should
use to answer the following questions \underbar{as well as you
can}. Some of your answers may have to be approximate.  Note that the
domain of $Q$ is {\it all numbers}.

\smallskip 

Where is $Q$ continuous? Where is $Q$ differentiable? What are $Q$'s
critical numbers? What are the equations of the vertical asymptotes of
$Q$, if any? What are the equations of the horizontal asymptotes of
$Q$, if any? What are the open intervals in which $Q$ is
$\{$in/de$\}$creasing? Locate and describe any relative max/min of
$Q$. In what intervals is $Q$ concave $\{$up/down$\}$?  Locate any
points of inflection of $Q$. Find these limits, if they exist:
$\d\lim_{x \to -\infty} Q(x)$, $\d\lim_{x \to 0^-} Q(x)$, $\d\lim_{x
\to 0^+} Q(x)$, and $\d\lim_{x \to +\infty} Q(x)$. } } 

\vskip -2.4in

\rightline{\epsfxsize=3in\epsfbox{rev2-1.eps}} 

\vskip +.08in
%\rightline{\sevenbf OVER}%\vfil\eject

13.~Consider the curve defined by the equation $5x^2+2xy + 2y^2
=7$. Find $\d{{dy}\over{dx}}$.

a) Which points on the curve have horizontal tangent lines?

\smallskip

b) Which points on the curve have vertical tangent lines?

\smallskip

{\hsize=4.5in \vtop{c) The graph of this equation is a tilted
ellipse. What are the corners of the smallest box with sides parallel
to the coordinate axes which contains this ellipse?}}

\vskip -.6in
\rightline{{\bf Hint} \raise
-.2in\hbox{\epsfxsize=.5in\epsfbox{rev2-2.eps}\qquad\quad}}
\bigskip


14.~Suppose $F(x)=\d{{x^3}\over{x^2-x-2}}$. What is the domain of $F$?
What are $\d\lim_{x\to\infty} F(x)$ and $\d\lim_{x\to-\infty} F(x)$?
If $a$ is a number {\it not}\/ in $F$'s domain, what are
$\d\lim_{x\to a^-} F(x)$ and $\d\lim_{x\to a^+} F(x)$?  If there is
more than one number not in $F$'s domain, compute these limits for
each number not in the domain of $F$. Does $F$ have any vertical or
horizontal asymptotes, and, if it does, what are they? Find $F'$ and
simplify it. Where is $F'(x)=0$? If $F$ has any relative extrema, what
and where are they? In what intervals is $F$ $\{$in/de$\}$creasing?
Find $F''$ and simplify it. Where is $F''(x)=0$? Find any inflection
points of $F$. In what intervals is $F$ concave $\{$up/down$\}$?  Now
\underbar{use all of this information} to sketch a reasonable graph of
$F$.

\medskip


15.~Find the range of $V(x)=e^{-2x}+ e^{3x}$.

\smallskip

{\bf Note} Your answer should {\it not} be approximate, but should be
written in terms of values of well-known functions. Expressions
involving $\ln$ and $\exp$ should be simplified as much as possible.

\medskip

16.~Suppose that $W$ is a differentiable function with $W(3)=4$,
$W'(3)=2$ and $W''(3)=-5$. Use linear approximation or differentials
to find approximate values for $W(3.03)$ and $W(2.98)$. Use the
information given to decide if these approximate values are likely to be
greater than or less than the true values of $W$. Briefly explain your
answer.

%\smallskip
\vskip -.05in
\centerline{\vrule height 0pt depth 1pt width 3in}
\smallskip
\centerline{On the web (including solutions!) at\ \  {{\tt
http://www.math.rutgers.edu/math135.html}}}

\vfill\eject\end 

Write algebraic restrictions on the appropriate derivatives which
mean the same as each of the following separate statements (you will
need to be specific about the function and the derivatives!). In each
case, sketch a graph of a function having the indicated behavior.

\smallskip

a) The rate of enzyme activity is increasing, but the rate of increase is
slowing. 

\smallskip

b) The rate of increase of enzyme activity continues to decrease, but
more slowly than before.


  I think in my review set I will ask them
to find out a lot about a given function; maybe
$(x-4)/(x^2-3x)$---asymptotes, roots, critical points, relative extrema,
and finally a graph showing all these features.  For the first part I would
emphasize that they have to show how they got the answers with the methods
of calculus. I may try to find also a similar problem involving a function
which is not rational - perhaps involving fractional powers.




\centerline{\vrule height 0pt depth 1pt width 3in} 

\medskip 

Material related to this course is on the web at

\centerline{\tt
http://www.math.rutgers.edu/math135.html}

\rightline{including a page of solutions for these problems!}
