\chapter*{Review Problem Sets : Second Midterm}

\subsection*{Review Problem Set \#1: Second Midterm }
(50 points in all, time = 1 hour)

\bigskip
\noindent
1.~(15 points) A shape is bounded by the curve $y=x^2/\sqrt{4-x^2}$, the
$x$-axis, and the line $x=1$.  Find the $x$-coordinate of the center of
mass of a uniform plate in that shape.  Please show all your work carefully
(including the triangle you use for the inverse trig substitution, and the
trig identities you use to integrate powers of sine or cosine).

\bigskip
\noindent
2.~(10 points) The shape in Problem 1 is rotated around the $x$-axis.  Find
the volume of the resulting solid of revolution.  Again please show all of
your work.

\bigskip
\noindent
\begin{rightindent}{3.45 truein}
\noindent
3.~(10 points) Find the maximum possible volume of the cylinder obtained by
rotating the rectangle shown at the right around the $x$-axis.  (The upper-right
corner of the rectangle is at the point $(x,e^{-x})$ on the curve
$y=e^{-x}$.  Volume of cylinder $=\pi r^2h$.)


\medskip


\noindent
4.~(15 points) At time $t=0$ a ball is thrown horizontally at 8 m/sec from
a point 40 meters directly above the observer's eye O.  (See the drawing
at the right, neglect air resistance, and take $g=9.8$ m/sec${}^2$.)  At the
same instant a 20-meter span of a bridge extends from $O$ at an angle of
30$^\circ$ above the horizontal.  The bridge is being lowered by a rope
(the vertical dotted line) which is pulled downward at the steady rate of 2
m/sec.

(a) Express the $x$- and $y$-coordinates of the ball as functions of time.

(b) Express the height of the point $B$ (the tip of the bridge span) in
terms of $t$.

(c) Express the angle $AOB$ (the angle between the observer's line
of vision to the ball and her line of vision along the bridge) in terms of
$t$.

(d) Find a formula for the rate in rad/sec at which the angle $AOB$
is decreasing at time $t$.
 \end{rightindent}
\hfill
\begin{rightindent}{2.0 truein}
%-figure NA -----------------------------
\begin{psfigure}{1.7in}{1.5in}{figNA.ps}
%\putcrosshair{0,0}
\puttext{-13.3, -33.3}{\small$x$}
\puttext{6.7, 6.7}{\small$\displaystyle y = e^{-x}$}
\end{psfigure}
%------------------------------


%-figure NB -----------------------------
\begin{psfigure}{1.7in}{2in}{figNB.ps}
%\putcrosshair{0,0}
\puttext{-30., 40.}{\small $40$}
\puttext{-15., -25.}{\small$\theta$}
\puttext{5., 15.}{\small$A$}
\puttext{7.5, -15.}{\small$B$}
\puttext{-30., -40.}{\small$O$}

\end{psfigure}
%------------------------------
\end{rightindent}

%\bigskip
\newpage

\subsection*{Review Problem Set \#2: Second Midterm }
(50 points in all, time = 1 hour)

\bigskip
\noindent
1.~(15 points) Let $y=f(x)=\sqrt{x}\,(\ln x)$ for $x>0$.

(a)~What are the values of $f(x)$ at $x=1$ and as $x\longrightarrow 0$?

(b)~Find the $x$- and $y$-coordinates of the minimum point of $f(x)$.  {\bf
Express your answer in terms of $e$ (not as a decimal).}

(c)~Find the $x$-coordinate of the center of mass of a uniform flat plate
bounded by $y=f(x)$ and the $x$-axis between $x=0$ and $x=1$.

\smallskip
\noindent
2.~(12 points) Find the integral of the following two functions from 0 to
$\infty$: $$\text{(a)~}\frac{1}{
x^2+5x+4},\qquad\qquad\text{(b)~}\frac{1}{(x+1)\sqrt{x^2+2x}}.$$

\smallskip
\noindent
3.~(8 points) The first region in the 1st quadrant bounded by the curve
$\sqrt{x}\,\cos x$ and the $x$-axis is revolved around the $x$-axis to form
a solid of revolution.  Find its volume.

\smallskip
\noindent 
4.~(15 points) At time $t=0$ an experimenter counts 5 million of bacteria A
and 15 million of bacteria B.  For each type of bacteria the rate of change
of its population at any time $t$ is proportional to its population at time
$t$; the constant of proportionality is the instantaneous birthrate minus
the instantaneous deathrate.  For bacteria A, the instantaneous birthrate
{\bf exceeds} its instantaneous death rate by 3\% per hour; but for
bacteria B, the instantaneous birthrate is {\bf less than} its
instantaneous deathrate by 5\% per hour.  Find the time $t$ when the {\bf
total} population of bacteria (A plus B) is minimal.

\bigskip
 

\subsection*{Review Problem Set \#3: Second Midterm }
(50 points in all, time = 1 hour)

\bigskip

\noindent
1.~(24 points) Evaluate the following integrals.  In parts (a) and (c)
express your answers in terms of $\pi$, $\sqrt{2}$, $\ln 2$, etc.~(not as
decimals).
\[
\text{(a)~}\int_0^1(1+2y)\text{Arctan}(y)\,dy;
\qquad\qquad
\text{(b)~}\int\frac{x\,dx}{\sqrt{x-x^2}};
\qquad\qquad
\text{(c)~}\int_\pi^{4\pi}\cos^3(t/4)dt.
\]

\smallskip
\noindent
2.~(8 points) Find the derivative of $(\ln x)^{e^x}$.

\smallskip
\noindent
3.~(18 points) A wheel 20 cm in radius rolls in a straight line without
slipping.  Its velocity at time 0 is 5 m/sec, but friction
decelerates the wheel at a rate proportional to the velocity.  Let $k$
denote the constant of proportionality.  The only force on the wheel is
friction.  That is, the instantaneous rate of change of the wheel's
velocity at time $t$ equals {\it minus} $k$ times the velocity at time $t$.

(a) Write a differential equation for the wheel's velocity $v(t)$.

(b) Find a formula for $v(t)$.  The constant $k$ should appear in your
formula.

(c) Find a formula for the distance $s(t)$ traveled in the first $t$ sec.
The constant $k$ should appear in your formula.

(d) If $k=0.1$, find all of the following at time $t=8$: (1) $v(8)$; (2)
$s(8)$; (3) the angle $\theta$ in radians through which the wheel rotates
in the first 8 sec; (4) the angular velocity in rad/sec at time $t=8$.

(e) Suppose that the wheel picked up a pebble at time $t=0$.  Use the
parametric formulas for a cycloid ($x=r(\theta-\sin\theta)$, $y=r(1-
\cos\theta)$) to determine the speed of the pebble at time $t=8$. 

\newpage

\subsection*{Review Problem Set \#4: Second Midterm }
(50 points in all, time = 1 hour)

\bigskip

\noindent
1.~(24 points) Evaluate the following integrals.  In part (a) express your
answer in terms of $\pi$ (not as a decimal), and in part (c) express your
answer in terms of $e$ (not as a decimal).
\begin{xalignat*}{2}
\text{(a)~}&\int_{10\sqrt{3}}^\infty\frac{dx}{ 100+x^2};
&\quad
\text{(b)~}&\int\frac{x+2}{ x^2-3x+2}\,dx; \notag\\
\text{(c)~}&\int_{-1/2}^{1/2}x\,e^{2(x+1)}\,dx;
&\quad
\text{(d)~}&\int_0^\pi\sin^5\left(\frac{x}{ 3}\right)\,dx. \notag
\end{xalignat*}

\smallskip
\begin{rightindent}{3in}
\noindent
2.~(10 points) Point $A$ moves at constant velocity from the point (3,7) to
the point (10,2) in 5 seconds, starting at $t=0$.  Meanwhile, point $B$
rotates counterclockwise at 1 revolution per second around the circle of
radius 2 centered at the origin.  At time $t=0$ the point $B$ is at (2,0).

(a) Find a formula in terms of $t$ for the angle above the horizontal of
the line of vision from $B$ to $A$.

(b) Find a formula in terms of $t$ for the rate in {\bf degrees} per second
that the angle in part (a) is decreasing.
\end{rightindent}
\hfill
\begin{psfigure}{2in}{2in}{figNC.ps}
%-figure NC -----------------------------
%%% figure of points moving on circle counterclockwise and other
%%% in straight line
%\putcrosshair{0,0}
\puttext{-85,-5}{\small$B$}
\puttext{-40,30}{\small$A$}

\end{psfigure}
%------------------------------



\smallskip
\noindent
3.~(16 points) Let $f(x)=\sqrt{x}-\ln x,~~~x>0$.

(a) Find $f'(x)$ and all points where $f'(x)=0$.

(b) Find $f''(x)$, and use the second derivative test to determine whether
the point(s) in part (a) is (are) maximum or minimum point(s).

(c) Find all points of inflection, and regions of upward and downward
concavity.

(d) What happens to $f(x)$ as $x\longrightarrow 0$ and as $x\longrightarrow
\infty$?

(e) Sketch the graph of $f(x)$.

