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

\bigskip
\noindent
1.~(13 points) A particle is constrained to move only in the $x$-direction.
A rotating force of constant magnitude causes it to accelerate in the
$x$-direction with $a(t)=\sin(t/3)$.  The particle's initial velocity is
1.5 units/sec {\bf to the left}.  Find the {\bf TOTAL} distance the
particle travels between $t=0$ and $t=5$ sec.

\smallskip
\noindent
2.~(12 points) Write each of the following in terms of definite integrals.
Do {\bf NOT} attempt to evaluate the integrals.

(a) The volume of the solid of revolution obtained by rotating around the
$x$-axis the shape in the first quadrant bounded by the line $2x+y=1$ and
the curve $y=\cos(\pi x)$.

(b) The center of mass of a uniform plate in the 1st and 4th quadrants that
is bounded by the line from the origin to $(4,3)$, the line from the origin
to $(4,-3)$, and the circumference of the circle of radius 5 centered at
the origin.

\smallskip
\noindent
3.~(12 points) An ellipse is widest in the $x$-direction, where it measures
10 units across, and narrowest in the $y$-direction, where it measures 6
units across.  The bottom half of this ellipse is rotated around its minor
axis to form a container, into which water is poured at the rate of 5 cubic
units/sec.

(a) Find the length of time $T$ before the container is filled.

(b) At time $T$ (i.e., the instant when the container is about to overflow)
find the rate at which the water level is rising.  Compare this with the
{\it average} rate at which the water level rose between time $t=0$ and
$t=T$.

\smallskip
\noindent
4.~(13 points) A pebble is caught half-way up the spoke of a wheel of
radius $r$. The wheel is rolling along a flat surface without slipping.

(a) Set up a definite integral for the distance along its path that the
pebble travels as the wheel goes through half a revolution, during which
time the pebble goes from a position directly below the center of the wheel
to a position directly above the center.

(b) Find this distance using the trapezoid rule with $n=6$ subdivisions.
Express your answer as a multiple of $r$.  To receive credit you must show
clearly the steps in applying the trapezoid rule, indicating what numbers you
are substituting into the rule.

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

\bigskip
\noindent
1.~(12 points) In this problem neglect air resistance, choose upward to be
the positive direction, and take $g=9.8$ m/sec${}^2$.  You know that a
ball was thrown down from a window at 10 m/sec, but you do not know the
angle $\alpha$ below the horizontal at which it was thrown.  You also do
not know the height $s_0$ of the window.  Suppose that the ball hits the
ground after 5 seconds at a point on the ground that is 25 meters from the
side of the building from which the ball was thrown.  Find $\alpha$ and
$s_0$.

\medskip

\noindent
2.~(12 points) Write each of the following in terms of definite integrals.
Do {\bf NOT} attempt to evaluate the integrals.

(a) The volume of the solid of revolution obtained by rotating around the
$y$-axis the shape bounded by the $x$-axis, the line $y=2$, and the curve
$y=3-(1/x^2)$.

(b) The distance along its path traveled in the first tenth of a second by
a pebble which at time $t=0$ is caught in the tread of a wheel of radius 40
cm that rolls along the pavement without slipping at 5 m/sec.

\medskip

\begin{rightindent}{3.25 truein}
\noindent
3.~(14 points) A bug is on the circumference of the ellipse pictured at the
right.  It crawls in such a way that its $x$-coordinate increases at the
rate of 0.20 units/sec.  The sun is shining from the left from an angle of
45${}^\circ$ above the horizontal, casting the bug's shadow on the
$x$-axis.  At the instant when the $x$-coordinate of the bug is 3, find:


(a) the rate at which its $y$-coordinate is changing, 

(b) the rate at which its shadow is moving along the $x$-axis,

(c) the speed of the bug along the ellipse.

\end{rightindent}
\hfill
%-figure HA -----------------------------
\begin{psfigure}{2in}{2in}{figHA.ps}
%\putcrosshair{0,0}
\puttext{35., -25.}{\small$5$}
\puttext{-37.5, -25.}{\small$-5$}
\puttext{-2.5, 12.5}{\small$4$}
\end{psfigure}
%------------------------------

\medskip

\begin{rightindent}{3 truein} 
\noindent
4.~(12 points) The graph to the right shows the function $f(x)=(\sin
x)/x$ between $x=0$ and $x=\pi$.  Actually, when $x=0$ the ratio $(\sin
x)/x$ is $0/0$, which is undefined; however, the expression $(\sin x)/x$
has the limit 1 as $x$ approaches 0, so we take $f(0)=1$.  A uniform plate
is bounded by $f(x)$ and the $x$- and $y$-axes.


(a) Write the $x$-coordinate $\overline{x}$ of its center of mass in terms
of definite integrals.  One of the two definite integrals is easy to
evaluate by the anti-derivative method.

\end{rightindent}
\hfill
%-figure HB -----------------------------
\begin{psfigure}{2in}{2in}{figHB.ps}
%\putcrosshair{0,0}
\puttext{27.5, -20.}{\small$\pi$}
\puttext{-55., 10.}{\small$1$}
\end{psfigure}
%------------------------------

(b) Use the trapezoid rule with $n=6$ subdivisions to evaluate the second of
the two definite integrals in part (a).  To receive credit you must show
clearly the steps in applying the trapezoid rule, indicating what numbers you
are substituting in the rule.

(c) Find $\overline{x}$.

\newpage 

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

\bigskip

\noindent
1.~(10 points) Give the equation of the sinusoidal function below, and find
the  area of the region between the portion of the curve shown 
and the line $y=1$.

\begin{center}
%-figure HC -----------------------------
\begin{psfigure}{3in}{1.5in}{figHC.ps}
%\putcrosshair{0,0}
\puttext{-25., 10.}{\small$2$}
\end{psfigure}\\
%------------------------------
\end{center}

\noindent
2.~(20 points) Set up each of the following in terms of definite integrals.
Do {\bf NOT} evaluate the integrals.

(a)~The distance along its trajectory traveled in the first second by an
object thrown at 15 m/sec directed $30^\circ$ {\it below} the horizontal
from a window 20 meters high.  (Neglect air resistance, take upward to be
the positive direction, and take $g=9.8$ m/sec${}^2$.)

(b)~The entire circumference of an ellipse with semimajor axis $a=10$ and
eccentricity $e=0.6$.  (First find $b$, and write the ellipse in parametric
form.)

(c)~The net distance traveled between time $t=0$ and $t=2$ if the
acceleration is $a(t)=1/(0.1t+1)^2$ and the initial velocity is 0.

\smallskip

\begin{rightindent}{3 truein}
3.~(20 points)~A wheel of radius 1 rotates clockwise at the rate of 1
revolution every four seconds as the center of the wheel moves along the
straight line joining the points $A=(0,2)$ and $B=(3,0)$ (see the drawing
at the right).  The center moves at constant velocity, starting at $A$ at time
$t=0$ and ending up at $B$ at time $t=1$.  $P$ is the point on the rim of
the wheel which is directly above the center at time 0.
\end{rightindent}
\hfill
%-figure HD -----------------------------
\begin{psfigure}{2in}{2in}{figHD.ps}
%\putcrosshair{0,0}
\puttext{-25., 32.5}{\small$P$}
\puttext{-10., 32.5}{\small$t=0$}
\puttext{10., 15.}{\small$P$}
\puttext{2.5, 5.}{\tiny$\theta$}
\puttext{-25., 12.5}{\small$A$}
\puttext{32.5, -32.5}{\small$B$}
\end{psfigure}
%------------------------------

(a)~Find parametric formulas in terms of $t$ for the motion of the center
of the wheel.

(b)~After the wheel has rotated through an angle of $\theta$, how far is
the point $P$ from the center in the $x$-direction and in the
$y$-direction?

(c)~Express $\theta$ in terms of $t$.

(d)~Find parametric formulas in terms of $t$ for the motion of $P$.

(e)~Write an integral for the distance traveled by the point $P$ between
time 0 and 1.

(f)~Using the trapezoid rule with $n=4$ subdivisions, evaluate the integral in
part (e).  Please show your work carefully.


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

\bigskip
\noindent
1.~(8 points) Find the area in the first quadrant between the graphs of
$y=\cos(\pi x/2)$ and $x+y=1$.

\smallskip
\noindent
2.~(16 points) Set up each of the following in terms of definite integrals.
Do {\bf NOT} evaluate the integrals.

(a)~The distance along its trajectory until reaching the peak of the
trajectory that is traveled by an object thrown at 9.7 m/sec directed
45${}^\circ$ above the horizontal.  (Neglect air resistance, take upward to
be the positive direction, and take $g=9.8$ m/sec${}^2$.)

(b)~The volume of the solid obtained by rotating around the $y$-axis the
region in the first quadrant bounded by the axes and the curve which is
given parametrically by the formulas $x=3\cos\theta$, $y=2\sin\theta$.

\smallskip
\noindent
3.~(12 points) Find the center of mass of a uniform flat plate bounded by
$y=\sin x$, $y=-\sin x$, and $x=\pi/2$.  Use the trapezoid rule with $n=6$
subdivisions to find the value of the top integral.  Please show your work
clearly.

\smallskip
\noindent
4.~(14 points) In this problem you will need the parametric formulas for a
cycloid: $x=r(\theta-\sin(\theta))$, $y=r(1-\cos(\theta))$.  A car
accelerates at the constant rate of 6 m/sec${}^2$.  When it reaches a speed
of 20 m/sec, it picks up a pebble in the tread of a wheel.  The wheel has
radius 20 cm.  Let $t$ denote the time measured from the instant the wheel
picks up the pebble.

(a)~Express in terms of $t$ the angle $\theta$ through which the pebble has
rotated around the center of the wheel.  ($\theta$ is measured in radians.)

(b)~Draw a circle with an arrow pointing to the position of the pebble on
the wheel when $t=0.5$ sec.

(c)~Find the speed of the pebble along its trajectory when $t=0.5$ sec.

 
\newpage

\subsection*{Review Problem Set \#5: First Midterm }
(50 points in all, time = 1 hour)

\bigskip

\noindent
1.~(12 points) Find the coordinates of the center of mass of a uniform flat
plate in the shape shown below to the left.

\medskip

\begin{centering}
%-figure HE -----------------------------
\begin{psfigure}{1.5in}{1.3in}{figHE.ps}
%\putcrosshair{0,0}
\puttext{50., 50.}{\small$(2,2)$}
\end{psfigure}
%------------------------------
\hfill
%-figure HG -----------------------------
\begin{psfigure}{3.5in}{1.1in}{figHF.ps}
%\putcrosshair{0,0}
\puttext{-140., -10.}{\small$-25$}
\puttext{140., -10.}{\small$25$}
\puttext{-5., 40.}{\small$7$}
\puttext{-10., -45.}{\small$-7$}
\puttext{-10., -20.}{\small$-4$}

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

\smallskip
\noindent
2.~(12 points) The ellipse pictured above to the right is centered at the
origin of the $xy$-plane.  Its semi-major axis is 25 and its semi-minor
axis is 7.  The units are meters.

(a)~Find the coordinates of its foci.

(b)~Find the eccentricity.

(c)~Suppose that the bottom half of the ellipse is rotated around the
$y$-axis to form a container.  Water is entering the container at the rate
of $0.7$ m${}^3$/sec.  How fast is the water level rising when the water is
3 meters deep at its deepest point (i.e., when the water level is 4 meters
below the top of the container)?

\smallskip
\noindent
3.~(12 points) A bicycle has a wheel of radius 0.4 meter.  It is traveling
at a constant speed of 6 m/sec.  A pebble is lodged in a spoke of the wheel
at a distance of 0.3 m from the center.  At time $t=0$ the pebble is at its
lowest point (0.1 m above the pavement).  Suppose that the wheel is moving
from left to right along the $x$-axis, and the origin is the point where
the wheel meets the pavement at time $t=0$.  In this problem, use the
parametric formulas $x=r\theta-a\sin\theta$, $y=r-a\cos\theta$ for the
coordinates of the pebble after the wheel has rotated through an angle of
$\theta$ radians.

(a)~Find formulas for the coordinates of the pebble at time $t$.

(b)~When $t=0.4$ sec, find the coordinates of the pebble, and draw a
diagram showing the relative positions of the center of the wheel and the
pebble at that instant.

(c)~When $t=0.4$ sec, find the speed of the pebble along its trajectory,
and draw a picture of the velocity vector showing the pebble's direction of
motion.

\smallskip
\noindent
4.~(14 points) At time $t=0$ a particle is at rest (zero velocity) at the
origin of the $xy$-plane.  In the horizontal direction its acceleration at
time $t$ is given by the formula $a_{\text{horiz}}(t)=t$, and in the vertical
direction its acceleration at time $t$ is given by the formula
$a_{\text{vert}}(t)=\cos(\pi t/6)$.

(a)~Find formulas for the horizontal velocity and the vertical velocity at
time $t$.

(b)~Express the arclength of the particle's path during the first 3 seconds
(the distance it has traveled) as a definite integral.

(c)~Use the trapezoid rule with $n=6$ to find the value of the integral in part
(b).  You do {\bf NOT} have to do the final numerical computations, but be
sure you show clearly the values that you are substituting into the trapezoid
rule formula.

 

