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\nopagenumbers
\rightline{\sevenbf 4/10/98}
\bigskip

\centerline{\largerbold Review \raise .015em\hbox{\largerqq advice}
for the final exam in Math 135}

\centerline{\largerbold (including some review problems)}
\medskip

After this introductory paragraph there are some problems mostly
addressing 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 may
also review the following material, keeping in mind that if we thought
it was important earlier, we probably still think it is important.

\medskip

\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

\item{$\bullet$} A ``real'' final exam given last semester in this
course (available on the web through the address below).

\smallskip

\noindent All this and other Math 135 material can be accessed through:

\smallskip

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

\noindent The final exam will be given {\bf Wednesday, May 6, from 4
to 7 PM} with location to be announced separately for each class.

\vskip -.05in

\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.

\medskip

\hskip 4.7in\vbox{
\def\tablerule{\noalign{\hrule depth0pt height.5pt width1.57in}}
\halign to \hsize{\hfil $ # $\strut\ \  \vrule\thinspace
& \hfil $ # $\hfil &\vrule\thinspace \ $ # $ \thinspace 
&\vrule\thinspace \ $ # $ \thinspace \cr
\hfil\  x \hfil&\  Q(x)\thinspace\thinspace\vphantom{j}
\hfil& Q'(x)& Q''(x)\cr
\tablerule
\tablerule
0 & 1 & -1 & -3 \cr
\tablerule
1& 2 & 0 & 7 \cr
\tablerule
2 & 0 & -5 & 2 \cr
\tablerule
3 & 1 & 0 & -2 \cr
\tablerule
4 & 5 & 2 & -2 \cr
\tablerule
5 & 8 & -1 & 0 \cr
\tablerule
6 & -2 & 5 & 3 \cr
\tablerule
7 & -2 & 0 & 0 \cr
}}

\vskip -1in

\hbox{\hsize = 4.3in \vtop{1.~Suppose $Q$ and its first derivative
$Q'$ and its second derivative $Q''$ have the values indicated in the
accompanying table.  

\medskip

Below are some pieces of graphs. Each value of $x$ matches with one
picture. Find the matches. 

\overfullrule=0pt
\bigskip\bigskip
\centerline{\hskip 2.2in\epsfxsize=6.5in\epsfbox{revf-2.eps}}

}}
\medskip

2.~Graph $y=(2x+1)e^{-x^2}$. Use calculus to give information as
precisely as possible about local extrema, intervals of increase and
decrease, inflection points, and intervals of concavity.

\medskip

3.~The line $y=3x+7$ is tangent to the graph of $y=f(x)$ when
$x=4$. What is $f(4)$? what is $f'(4)$?

\medskip

4.~Suppose $y$ is defined implicitly as a function of $x$ by
$x^2+Axy^2+By^3=1$ where $A$ and $B$ are constants to be
determined. If the tangent line to this curve has slope $-1$ at the
point $(3,2)$, find $A$ and $B$.

\medskip

5.~Evaluate these indefinite integrals.

\medskip

\line{
a) $\d \int 7x^2-3e^x +{5\over x}\, dx$\qquad
b) $\d \int (x^3 +5)^2 \, dx$\qquad
c) $\d \int 5 \sin x + \cos (5x) \, dx$\qquad
d) $\d \int {x\over {x^2+5}} \, dx$\hfil
}

\bigskip

%\rightline{\sevenbf OVER}

%\vfil\eject

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

\bigskip

7.~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

8.~The following two equations are true:

$$\int_1^4 {x\over{(x+1)(x+2)}} \, dx = \ln 2-2\ln3 - \ln 5 + 2\ln 6$$

$$\int_2^4 {x\over{(x+1)(x+2)}} \, dx = \ln 3-2\ln 4 - \ln 5 + 2\ln 6$$

Find the rational number $B$ (that is, a quotient of two integers) so
that $\d\int_1^2 {x\over{(x+1)(x+2)}} \, dx = \ln B\;$.

\bigskip


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

\medskip
\line{
a) $\d{\int_1^2 \left(x^3 -{1\over{x^4}}\right) dx}$ \hfil
b) $\d{\int_0^{\ln 3} 4e^{2x} \, dx}$ \hfil
c) $\d{\int_0^{2} x^2 \sqrt{1+3x^3} \, dx}$ \hfil
d) $\d{\int_{\ln \pi}^{\ln 2\pi} e^x \sin(e^x) \, dx}$ \hfil
e) $\d{\int_e^{e^e} {1\over {x\ln x}}\, dx}$\hfil}

\bigskip

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

\medskip

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

\bigskip

11.~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

12.~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 $g'(5)$?

\medskip

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

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

\bigskip

14.~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 $y=x(x-1)^2\,$?

\vfill\eject\end 




