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\nopagenumbers
\rightline{\sevenbf 11/30/97}
\bigskip
\centerline{\largerbold Review \raise .015em\hbox{\largerqq advice} for the final exam in Math 135}
\centerline{\largerbold (including some review problems)}
\bigskip

After this introductory paragraph there are some problems covering the
last 3 or 4 weeks of the course. The final is cumulative and may cover
all the material in the course. There will be more emphasis, though,
on the material of the last 3 or 4 weeks which has not yet been
tested. So please look at the problems below. You should also review
the following material, keeping in mind that if we thought it was
important earlier, we probably still think it is important.

\medskip\parindent=2\parindent

\item{$\bullet$} The earlier review sheets (with some answers available
on the web through the address below). 

\smallskip

\item{$\bullet$} The exams that you had earlier in this course. 

\smallskip

\item{$\bullet$} Two ``real'' exams given to one lecture class in this
course (with answers and grading guide available on the web through
the address below).

\smallskip

\noindent There's lots of Math 135 material which can be accessed through:

\medskip

\centerline{\tt
http://math.rutgers.edu/$\tilde{\hphantom{a}}$greenfie/currentcourses/math135/math135\underbar{\hphantom{a}}index.html}
\medskip
\noindent The time, date, and location of final exams in this course will be
announced by the instructors in each class.

\smallskip

\parindent = 0pt

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

\medskip

The problems below are intended to be representative of the type of
problems which may appear on exams. Please realize that many routine
review problems are available in the review sections of the text.

\smallskip

Graphing calculators may be used on the Math 135 final exam 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.

\bigskip

1.~Suppose $M(x)= x^5 +7x +6\,$. 
\medskip

a) Compute $\d\lim_{x\to +\infty} M(x)$ and  $\d\lim_{x\to -\infty}
M(x)\,$. Explain briefly why the results allow you to conclude that
$M(x)=0$ has at least one root. 

\smallskip

b) Explain how to apply Newton's method to find a root of $M(x)=0\,$:
that is, describe how Newton's method gets a new approximation,
$x_{\rm new}$, to a root from an old one, $x_{\rm old}$, using a
formula of the form $x_{\rm new} = G(x_{\rm old})$. Describe $G$
precisely for this $M$.

\smallskip

c) Start with an initial guess of $1$ and repeat Newton's three
times to try to get close to the root of $M(x)=0$ (which has been
reported by rumor to be approximately $-.8079567434$).

\bigskip

\hskip 5.35in
\vbox{
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\halign to \hsize{\hfil $ # $\strut\ \  \vrule\thinspace
& \hfil $ # $\hfil &\vrule\thinspace \ $ # $ \thinspace \cr
\hfil\  x \hfil&\  Q(x)\thinspace\thinspace\vphantom{j}
\hfil& Q'(x)\cr
\tablerule
\tablerule
\ 0 & 1 & -1  \cr
\tablerule
1& 2 & -2 \cr
\tablerule
2 & 0 & 5 \cr
\tablerule
3 & 1 & 2 \cr}}
\vskip -.91in

\hbox{\hsize = 5.2in \vtop{ 2.~Suppose $Q$ and its first derivative,
$Q'$, have the values indicated in the accompanying table.  Use this
table to find $Q(1)$ and $Q(Q(1))$ and $Q(Q(Q(1)))$ exactly. Use
linear approximation (also known as differentials) and the information
in this table to find approximate values of $Q(1.03)$, $Q(.98)$,
$Q(Q(1.03))$, $Q(Q(.98))$, $Q(Q(Q(1.03)))$, and $Q(Q(Q(.98)))$. {\bf
Note} You could define $R(x)=Q(Q(x))$ and $S(x)=Q(Q(Q(x)))$ and then
work with the functions $Q$, $R$, and $S$.}}

\bigskip

3.~Evaluate these indefinite integrals.

\medskip

a) $\d \int 7x^2-3e^x +{5\over x}\, dx$

\smallskip

b) $\d \int (x^3 +5)^2 \, dx$

\smallskip

c) $\d \int 5 \sin x + \cos (5x) \, dx$

\bigskip

\rightline{\sevenbf OVER}

\vfil\eject

4.~Find a solution to the differential equation $y'=2x^3-1$ passing
through the point $(2,3)$.

\bigskip

5.~What's the highest the curve $y=K(x)$ can be if $K''(x)$ is $-9$ always
and if also $K(0)=0$ and $K(2)=0$? 

\bigskip

6.~Evaluate these definite integrals using methods of calculus. 

\medskip

a) $\d{\int_1^2 \left(x^3 -{1\over{x^4}}\right) dx}$  

\smallskip

b) $\d{\int_0^{\ln 3} 4e^{2x} \, dx}$

\smallskip

c) $\d{\int_0^{2} x^2 \sqrt{1+3x^3} \, dx}$

\smallskip

d) $\d{\int_{\ln \pi}^{\ln 2\pi} e^x \sin(e^x) \, dx}$

\bigskip

7.~a) Compute the derivative of $e^x \sin x - e^x \cos x$. 

\smallskip

b) Use your answer to a) to compute $\d \int_0^\pi e^x \sin x \, dx$. 

\bigskip

8.~If $F(x)=\d\int_{-42}^x {{\sin (t^2)}\over {1+t^4}}\,dt$, compute
$F(-42)$, $F'(0)$, and $F'(\sqrt{\pi})$ exactly.

\bigskip

9.~Below is a graph of $y=f(t)$.
\bigskip
\centerline{\epsfbox{revf-1.eps}}
\smallskip
\centerline{Graph of $y=f(t)$}
\medskip
Note that each segment of the graph is a semicircle.
Suppose that $\d{g(x)=\int_{-1}^x f(t)\; dt}$.
\medskip

a) What are $g(-1)$ and $g(0)$ and $g(5)$? What are $g'(-1)$ and
$g'(0)$ and $g'(5)$?

\smallskip

b) Compute a Riemann sum estimate for $g(3)$ associated with the
partition $-1 < 0 < 1 < 3$ where the sample points are the left
endpoints of the subintervals. 

\bigskip

10.~What is the area enclosed by $y=x^5$, the $x-$axis, and the lines
$x=1$ and $x=2$?

\bigskip

11.~Find the area under the arch of $y=x(x-1)^2\,$. Stated more
 properly: what is the area of the bounded region in the plane bounded
 by the $x-$axis and $y=x(x-1)^2\,$?

\centerline{\vrule height 0pt depth 1pt width 3in}
\smallskip
On the web at {\tt
http://math.rutgers.edu/$\tilde{\hphantom{a}}$greenfie/currentcourses/math135/math135\underbar{\hphantom{a}}index.html}.

\vfill\eject\end 




