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

\bigskip

Please note that 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).

\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

1.~Find $\d {{dy}\over {dx}}$ when 
$${\rm a)\ } y= e^{3\cos 2x}\qquad\quad
{\rm b)\ } y=\ln(x^3 +5x+9)\qquad\quad
{\rm c)\ } y=\sin\!\left({{x+1}\over{x^2+2}}\right)\qquad\quad
{\rm d)\ } x^2 +5xy = e^y +y$$
\bigskip

2.~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 $g'(2)$? 


\bigskip

3.~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.

\medskip

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?

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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?

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4.~Let $\d{f(x)= e^{x - x^2} } $

\smallskip

a) Graph $y=f(x)$ in the window $-2 \leq x \leq 2$ and $0 \leq y \leq 2$.

\medskip

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 $y=f(x)$.  

\medskip

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 $y=f(x)$.

\bigskip

5.~Suppose $f(x)$ is a differentiable function and you know the
following:

\smallskip

\centerline{
$f(1)=10$ and $-1 \leq f'(x)\leq 2$ for $1 \le x \le 4$. } 

\smallskip

a) Is it possible that $f(4) = 15$?  If so, give an example of such a
function; if not, give a reason.

\medskip

b) Is it possible that $f(4) = 3$?  If so, give an example of such a
function; if not, give a reason.

\bigskip

{\hsize=3.4in \vtop{6.~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}.

\medskip 

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 $\d\lim_{x \to +\infty} Q(x)$. } } 

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

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

\bigskip

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? 

\bigskip

9.~Consider the curve defined by the equation $5x^2+2xy + 2y^2 =7$.

\smallskip

a) Find $\d{{dy}\over{dx}}$.

\medskip

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

\medskip

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

\medskip

d) It is known that the points satisfying this equation form a tilted
ellipse. Specify a box which contains this ellipse.\raise
.35ex\hbox{*}

\vfootnote*{Hint:\ \raise
-.2in\hbox{\epsfxsize=.5in\epsfbox{rev2-2.eps}}} 
%30% portrait produced by xfig

\bigskip

10.~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. 

\medskip

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

\bigskip

11.~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?

\bigskip

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

\medskip

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

\bigskip

13.~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.

\bigskip

14.~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. 


\bigskip
15.~Find $\d\lim_{x \to \infty}$ and $\d\lim_{x \to -\infty}$ for the
following functions:
$${\rm a)\ }{{e^{2x}}\over {3+e^{4x}}}\qquad\qquad
{\rm b)\ }{{e^{4x}}\over {3+e^{2x}}}\qquad\qquad
{\rm c)\ }{{x^3+1}\over {x^2+x+1}}\qquad\qquad
{\rm d)\ }{{x+1}\over {\sqrt{x^2+2x+3}}}$$
\bigskip

16.~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? 

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\smallskip
On the web at {\tt
http://math.rutgers.edu/$\tilde{\hphantom{a}}$greenfie/currentcourses/math135/math135\underbar{\hphantom{a}}index.html}.

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