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\rightline{\sevenbf 9/23/97; revised 9/26/97}
\smallskip
\centerline{\largerbold Review Problems for the first exam in Math 135}

\medskip

Please note that this collection of 17 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

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

b) Simplify $\d \cos(\arcsin\left({\textstyle{ 7 \over {11}}}\right)
)$ (give an exact answer).

c) Simplify $\arctan(\tan\left((7{3 \over 4})\pi\right))$
(give an exact answer). 
\medskip

2.~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

3.~Suppose we know that $Q(10^{10})= 7$, $Q(10^{100})=9$, and that
$Q(x)$ is a linear function of logarithm: that is, $Q(x)= A (\ln x) +
B$. What are the constants $A$ and $B$?

\medskip
\vskip .25in
{\hsize=3.3in \vtop{
4.~Here is a graph of $y=f(x)$, a function whose domain is all numbers
and whose range is $0<y\le 2$. 
Sketch as well as you can graphs of the
functions $\d F(x)= {1\over{f(x)}}$ and $\d G(x)= 3 +
2^{f(x)}$. Specify the domain and ranges of each of these functions as
precisely as you can.
}}
\vskip -1.15in
\rightline{\epsfxsize=3in\epsfbox{rev1-1.eps}}
%50% portrait produced by xfig

\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 $p$? Write a
phrase which describes what the formula you have obtained represents.

\medskip

7.~Use algebraic methods developed in this course to find the
following limits, if they exist. Give evidence supporting your answer.

\centerline{$\d \lim_{x\to 2} {{x-2} \over {x^2 + 4}}\; ;\quad 
\lim_{x\to 2} {{x-2} \over {x^2 - 4}}\; ;\quad
\lim_{x\to 4} {{x-4} \over {\sqrt{x} + 2}}\; ;\quad
\lim_{x\to 4} {{x-4} \over {\sqrt{x} - 2}}$.}

\medskip

8.~Find the value of these limits, if they exist:

\centerline{$\d
\lim_{x \to 0} {1\over x}\; ; \quad
\lim_{x \to 0^-}\left(1\over {\arctan x}\right); \quad
\lim_{x \to 0^+} \arctan \left( {1\over x}\right); \quad
\lim_{x \to 0} \arctan \left({1\over {x^2}}\right)\; ; \quad
\lim_{x \to 0} \left(\arctan {1\over x}\right)^{\!2}$.}

\medskip

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

\medskip

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

\medskip
11.~Compute the derivative of these functions:

\centerline{$\d x^3-3x+17; \quad e^x \sin x; \quad {{2x+3} \over
{x^2+1}}; \quad 3e^x + {x \over {2+ 5\cos x}}$.}
\rightline{\sevenbf OVER}
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\vskip -1.9in
{\hsize=3.2in \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 $A$.

}}



\bigskip\medskip 

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

14.~Consider the function $g$ whose graph is shown below:
\smallskip
\centerline{\epsfysize=2in\epsfbox{rev1-3.eps}}
%50% portrait produced by xfig
 
Answer the following questions as precisely as possible for each of
the values $x=A$, $x=B$ and $x=C$. Give reasons for your answers.

\centerline{Is $g$ continuous at this value of $x$? Is $g$
differentiable at this value of $x$?}

\medskip

15.~Draw the graph of a function $h$ which has {\it all} of the
following properties:

{\parindent=.3in \narrower \noindent
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 $h(4)=2$.\par}

Write the equations of any vertical asymptotes of this graph.

\medskip

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\vskip -1.2in

{\hsize=3.2in \vtop{16.~For each of the following statements, use this
graph to find a time interval in which the statement is true. Be sure
to 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

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 $G'(1)$.

\smallskip

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

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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