%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\chapter*{Practice Final Exams}

\subsection*{Practice Final Exam \#1}\

\medskip

\noindent
(100 points in all; time = 3 hours, checking your work carefully)

\medskip

\noindent
1.~(15 points) Evaluate: (a) $\int_0^3{\root 3 \of {1-3x}}\>dx\qquad$
(b) $\int\sin\left(\frac{\pi}{ 10}(t-1)\right)\cos\left(\frac{\pi}{ 10}(t-1)
\right)\,dt.$

\smallskip
\noindent
2.~(15 points) A light is at the top of a pole 14 meters high.  A ball is
dropped from the same height from a point 7 meters from the light.  How
fast is the shadow of the ball moving along the ground 0.8 sec later?
(Assume that the ball falls a distance $s=4.9t^2$ meters in $t$ seconds.) 

\smallskip
\noindent
3.~(15 points) At time $t=0$ an object is at rest.  It then accelerates
according to the graph at the right, where the acceleration $a(t)$ is measured
in m/sec${}^2$.
\begin{rightindent}{2.75in}
(a) Sketch the graph of the velocity function $v(t)$ during these 5 seconds.

(b) Find a formula for $v(t)$ for $t$ between 0 and 1, for $t$ between 1
and 2, and for $t$ between 2 and 3.

(c) How far does the object travel during the first 3 seconds?

\end{rightindent}
\hfill
%%%% figure TA (practice final #1 problem  1)%%%%%%%%%%%%%%%
\begin{psfigure}{2.70in}{1.4in}{figTA.ps}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%%%%%%%%%%%%%%%%%

\smallskip

\noindent
4.~(15 points) Cylindrical cans with circular tops and bottoms are to be
manufactured to contain a given volume.  There is no waste involved in
cutting the tin that goes into the vertical side of the can.  But the
two circular end pieces are cut from a sheet, as shown at the bottom
of the page.  If the circles for the tops and bottoms of the cans are
fit as snugly as possible, each uses up a hexagonal area of tin, as shown
in the picture above.  First find the area of one of the hexagons in
terms of $r$.  (Hint: It consists of 6 equal equilateral triangles.)
Now find the ratio of height to radius for the most economical cans.
In other words, minimize the total area of tin needed to make a can.

%%%% figure TB (practice final #1 problem 4)%%%%%%%%%%%%%%%
%
\ \hfill
\begin{psfigure}{5in}{1.2in}{figTB.ps}
\puttext{-65., 30.}{\Tiny$r$}
\puttext{-55., 25.}{\Tiny$30^{\circ}$}
\end{psfigure}
\hfill
\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%\ \hfill (Continue to Problems 5 and 6 on the next page)

\smallskip\noindent
5.~(20 points) A 4-centimeter rod is attached at one end A to a point on
a wheel of radius 2 cm.  The other end B is free to move back and forth
along a horizontal bar that goes through the center of the wheel.  At time $t=0$
the rod is situated as in the diagram at the left below.  The wheel rotates
counterclockwise at $3\>1/2$ rev/sec.  Thus, when $t=1/21$ sec, the rod
is situated as in the diagram at the right below.

(a) How far is the right end of the rod (the point B) from the center of
the wheel at time $t=1/21$ sec?

(b) Express the position of the right end of the rod as a function of $t$.

(c) Express the speed of the right end of the rod as a function of $t$.

(d) Use parts (a), (c) and the tangent line approximation to find the
distance from $B$ to the center of the wheel at time $t=1/20$ sec.

%%%% figure TC (practice final #1 problem 5)%%%%%%%%%%%%%%
%DIA
%
\begin{center}
\begin{psfigure}{6in}{2in}{figTC.ps}
\puttext{-115., 35.}{\Small$t=0$ sec}
\puttext{-25., 35.}{\Small$ t = \frac{1}{42}$ sec}
\puttext{75., 35.}{\Small$t = \frac{1}{21}$ sec}
\puttext{-95., 5.}{\Small$A$}
\puttext{-55., 2.5}{\Small$B$}
\puttext{0., 15.}{\Small$A$}
\puttext{37.5, 5.}{\Small$B$}
\puttext{90., 22.5}{\Small$A$}
\puttext{125., 5.}{\Small$B$}
\end{psfigure}\\
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip\noindent
6.~(20 points) A wheel 20 cm in radius rolls along a flat surface without
slipping at a speed of 5 cm/sec.  A particle is caught exactly half way
up one of the spokes.  At time $t=0$ the particle is at its lowest point
(a height of 10 cm above the surface).  The drawing below shows the
wheel at time $t=0$.

(a) Find an expression for the horizontal position of the particle
(its $x$-coordinate) at time $t$.

(b) What is the $x$-coordinate of the particle when $t=3$ sec?
In this part and the next, carry out your computations to five places
beyond the decimal point.

(c) Suppose that you want to find the time when the particle has moved
a horizontal distance of 10 cm, i.e., when its $x$-coordinate is 10.
Write a function $f(t)$ that equals zero at this instant.  Then use
Newton's method with $t_0=3$ to find successive approximations $t_1$,
$t_2$.  Also compute the horizontal position of the particle at time
$t_2$.  Please show your work clearly.

%%%- figure TD (practice final #1 problem 6)%%%%%%%%%%%%%%%
%DIA
%
\begin{center}
\begin{psfigure}{3in}{1.5in}{figTD.ps}
\puttext{-10., -10.}{\Small Particle}
\put(-15,-15){\vector(-3,-1){15}}
\end{psfigure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\subsection*{Practice Final Exam \#2}\

\medskip

\noindent
(100 points; time = 3 hours, checking your work carefully)

\bigskip
\noindent
1.~(15 points) Evaluate: (a) $\int\frac{dt}{(1-2t)^3}$~~~(b) $\int_0^{\pi/6}
\sqrt{\cos(3x)}\sin(3x)\,dx$.

\smallskip
\noindent
2.~(10 points) A wheel 9 inches in radius is rolling along a pavement without
slipping.  Its angular velocity at time $t$ seconds is
$60+20\sin(\pi t)$ rev/sec.

(a)~How many revolutions does the wheel go through between $t=0$ and $t=\frac{1}{
3}$ sec?

(b)~How many radians does the wheel rotate during this time?

(c)~How far does the wheel roll during this time?  (Give your answer in feet.)

\smallskip
\noindent
3.~(10 points) A girl flies a kite at a height of 240 ft, the wind carrying
the kite horizontally away from her at a rate of 20 ft/sec.  How fast must
she let out the string when the kite is 400 ft away from her?

\smallskip
\noindent
4.~(10 points) You are making a rectangular box with square bottom.  If
the bottom has side 2 ft and the height is 4 ft, then the total area of
all six sides is 40 ft${}^2$.  If you decrease the height to 3.9 ft,
what should the dimensions of the square bottom be in order for the
surface area to remain the same?  Use implicit differentiation and the
tangent line approximation.

\smallskip
\begin{rightindent}{3in}
\noindent
5.~(10 points) You are building a trough whose sides are $45^\circ-
45^\circ-90^\circ$ triangles (see the diagram at the right).  Those triangular
sides are three times more expensive (in cost per unit area) than the
two rectangular pieces that run the length of the trough.  You want the
trough to hold 6 ft${}^3$.  What dimensions $s$ and $\ell$ will minimize the
cost?
\end{rightindent}
\hfill
%%%- figure TE (practice final #2 problem  5)%%%%%%%%%%%%%%%
%
\begin{psfigure}{3in}{1.55in}{figTE.ps}
\puttext{0, -20.}{\Small$\ell$}
\puttext{-65., -10.}{\Small$s$}
\puttext{-30., -12.5}{\Small$s$}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip\noindent
6.~(25 points) The purpose of this problem is to sketch the function
$y=f(x)=x^2+\sin(2x)$ between $x=-\pi/2$ and $x=\pi/2$.

(a)~On the same graph, sketch both the curve $y=x^2$ and the curve
$y=\sin(2x)$ between $x=-\pi/2$ and $x=\pi/2$.

(b)~What equation must be satisfied in order for $y=f(x)=x^2+\sin(2x)$
to have a maximum or minimum at $x$?

(c)~Find the $x$- and $y$-coordinates of all points of inflection
of $y=f(x)=x^2+\sin(2x)$ between $x=-\pi/2$ and $x=\pi/2$.

(d)~Give a value of $x$ near a minimum of $y=f(x)=x^2+\sin(2x)$.  Do
this by finding a minimum of $\sin(2x)$.  Call this value $x_0$.

(e)~Use Newton's method with two iterations to get a more accurate value
$x_2$ for the $x$-coordinate of the minimum point of $y=f(x)=x^2+\sin(2x)$.

(f)~Do you think the function $y=f(x)=x^2+\sin(2x)$ has any other maxima
or minima besides the minimum you found in part (e)?  (You are not expected
to do any computations or take a long time on this part.)

(g)~Sketch the graph of $y=f(x)=x^2+\sin(2x)$.


\vfill

\ \hfill (Continue to Problem 7 on the next page)

\newpage

\newpage

\begin{rightindent}{2.75in}
\noindent
7.~(20 points) You are standing at $B$, a distance of 75
ft from the bottom of a ferris wheel which is 20 ft in radius.  Your arm is
at the same level as the bottom of the ferris wheel.  At time $t=0$ your
friend is at the point $A$ rotating counterclockwise at the rate of 1
revolution every 12 seconds.  At that instant you throw a ball to her.  You
throw the ball at 60 ft/sec at an angle of 60${}^\circ$ above the
horizontal.  Take $g=32$ ft/sec${}^2$, and neglect air resistance.
\end{rightindent}
\hfill
%%%- figure TF (practice final #2 problem  7)%%%%%%%%%%%%%%%
%
\begin{psfigure}{3in}{2in}{figTF.ps}
\puttext{0,0}{\Small$A$}
\puttext{50., -15.}{\Small$B$}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent

\smallskip
(a)~What is your friend's angular velocity in rad/sec?

(b)~Using the $x$- and $y$-coordinates in the diagram, write parametric
equations for your friend's position.

(c)~Find the horizontal and vertical components of the velocity of the ball at
time $t=0$.  Notice that the horizontal motion is in the {\it negative}
$x$-direction.  Round off your answers to the nearest ft/sec.

(d)~Write parametric equations for the $x$- and $y$-coordinates of the
position of the ball after $t$ sec.

(e)~At time $t=2$ sec, how far is the ball from your friend?

(f)~Express the distance from the ball to your friend at time $t$ in terms
of $t$.

(g)~Suppose you want to find the closest distance the ball gets to your
friend.  Find a function of $t$ that must be equal to zero at the moment
of this minimum distance.  This is the function that would be put into
the Newton's method formula (with initial guess $t_0=2$ sec) to find the
time when the ball is closest to your friend. 

\newpage

\subsection*{Practice Final Exam \#3}\

\medskip

\noindent
(100 points in all, time = 3 hours, checking your work carefully)

\medskip
\noindent
1.~(15 points) Evaluate: (a) $\disp\int\frac{dz}{\sqrt{1-7z}}$~~~(b) 
$\disp\int_0^\pi \sin(x/4)\cos^2(x/4)\,dx$.

\smallskip
\begin{rightindent}{2.75in}
\noindent
2.~(15 points) At time $t=0$ an object is moving to the right at 2 m/sec.
It then decelerates according to the graph below, which shows rightward
acceleration as a function of $t$. Here $a(t)$ is measured in m/sec${}^2$.

(a)~Draw the graph of the velocity function $v(t)$ during these 4 seconds.

(b)~Find formulas for $v(t)$ for $t$ in the intervals (0,1), (1,2), (2,3)
and (3,4).

(c)~How far does the object travel between $t=0$ and $t=4$?
\end{rightindent}
\hfill
%%%- figure TG (practice final #3 problem  2)%%%%%%%%%%%%%%%
\begin{psfigure}{2.70in}{1.75in}{figTG.ps}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip
\noindent 
\begin{rightindent}{3.0in}
3.~(10 points) A baseball diamond is a square 90 ft on a side.  A player
runs from first base to second base at 16 ft/sec.  At what rate is the
player's distance from third base decreasing when the player is 20 ft
from first base?
\end{rightindent}
\hfill
%%%- figure TH (practice final #3 problem  3)%%%%%%%%%%%%%%%
%DIA 
%
\begin{psfigure}{2.0in}{1.5in}{figTH.ps}
\puttext{0, -50.}{\Small Home}
\puttext{0, 50.}{\Small Second}
\puttext{-58.7, 0}{\Small Third}
\puttext{55, 0}{\Small First}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%

\medskip

\noindent
4.~(15 points) Two legs of a right triangle are 7 and 24 (so the hypotenuse
is 25).  If you want to shorten the shorter leg to 6.980 and keep the
perimeter of the right triangle (the sum of the three sides) constant,
how long should you make the longer leg?  Use the tangent line
approximation and implicit differentiation.

\smallskip

\begin{rightindent}{3.5in}
\noindent
5.~(20 points) In this problem you want to find the best ratio of height to
radius $x=h/r$ so that a cone whose side has fixed area holds maximum
volume.  Use the formulas $\frac{1}{ 3}\pi r^2h$ for the volume of a cone
and $\pi r\sqrt{r^2+h^2}$ for the side of a cone.

(a)~Express both the volume and area of side in terms of $r$ and $x$ (without
$h$).

(b)~Suppose the side area must be a constant $C$.  Solve for $r$ in terms of
$x$.  (The constants $\pi$ and $C$ should also appear in your formula.)

(c)~Find the ratio of height to radius for which the cone of side area $C$
holds the most volume.
\end{rightindent}
\hfill
%%%- figure TI (practice final #3 problem  5)%%%%%%%%%%%%%%%
\begin{psfigure}{1.95in}{2in}{figTI.ps}
\puttext{15., 42.5}{\Small$r$}
\puttext{32.5, -5.}{\Small$h$}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%

\vfill

\ \hfill (Continue to Problems 6 and 7 on the next page)


\newpage

\smallskip\noindent
6.~(10 points) Use Newton's method with two iterations to find a number
that is equal to the sum of its square root and its cube root.  3 is a good
initial guess, since $\sqrt{3}+{\root 3 \of 3}=3.1743$.

\bigskip


\begin{rightindent}{3in}
\noindent
7.~(15 points) At time $t=0$ an object $P$ is dropped from a height of 4 m
on the $y$-axis.  At the same instant an object $Q$ at the origin is released
so that it can oscillate back and forth on a horizontal spring.  The point $Q$
moves sinusoidally, returning to the origin each 1 sec.  When the spring is
most compressed, $Q$ is at 1 m on the $x$-axis.  Your purpose in this problem
is to find the closest distance between $P$ and $Q$.

(a)~Write a formula for $Q$'s position as a function of time in the form
\[
x=A\sin\left(\frac{2\pi}{ B}(t-C)\right)+D\,.
\]


(b)~Write a formula for the {\bf square} of the distance from $P$ to $Q$ as
a function of time.

(c)~Find a function of $t$ that is zero at the instant when $P$ and $Q$ are
closest.  This is the function that would be put into Newton's method to
find the time when $P$ and $Q$ are closest.

\end{rightindent}
\hfill
%%%- figure TJ (practice final #3 problem  7)%%%%%%%%%%%%%%%
%DIA 
%
\begin{minipage}[t]{2.45 truein}
\begin{psfigure}{3in}{3in}{figTJ.ps}
\puttext{0., 43.3}{\Small$P = (0, 4 - 4.9 t^2)$}
\puttext{0., -33.3}{\Small$Q=(x,0)$}
\puttext{-6.67, -56.7}{\begin{minipage}{0.7 truein}
                       \Small Range of\\
                        oscillation
                     \end{minipage} }
\end{psfigure}
\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\subsection*{Practice Final Exam \#4}\

\medskip

\noindent
(100 points in all, time = 3 hours, checking your work carefully)

\bigskip
\noindent
1.~(20 points) Evaluate (in part (b) express your answer in terms of $B$):

\[
\text{(a) }\int_0^3\frac{dt}{{\root 4 \of {1+5t}}}
\text{ (b) }
\int_0^{B/6}\cos\left(\frac{2\pi x}{ B}\right)\sin\left(\frac{2\pi x}{ B}\right)
\,dx.
\]

\smallskip

\begin{rightindent}{3in}
\noindent
2.~(15 points) At time $t=0$ an object is moving to the right at 3 m/sec.  It
then accelerates according to the graph at the right, which shows
rightward acceleration as a function of $t$ (measured in m/sec${}^2$).

(a)~Find a formula for $v(t)$ when $t$ is between 0 and 2 and when $t$ is
between 2 and 3.

(b)~How far does the object travel in the 3 seconds?
\end{rightindent}
\hfill
%%%- figure TK (practice final #4 problem  2)%%%%%%%%%%%%%%%
\begin{psfigure}{2in}{1.8in}{figTK.ps}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%

\medskip

\noindent
\begin{rightindent}{3.2in}
\noindent
3.~(15 points) The trough shown at the right is to be constructed from a piece
of metal 50 inches long and 22 inches wide.  The trough is to be made by
turning up strips of width 4 inches to make equal angles $\theta$ with
the vertical.  {\bf See the diagram at the right.}

(a)~Express the volume of the trough in terms of the angle $\theta$.

(b)~Let $X=\sin\theta$.  Find an expression of the form $aX^2+bX+c$
which must equal 0 if $\theta$ is chosen so as to maximize the volume of
the trough.

(c)~Find the maximum possible volume of the trough.  (In this problem you
will need to use the quadratic formula $(-b\pm\sqrt{b^2-4ac})/2a$ to solve
for $X=\sin\theta$.)
\end{rightindent}
\hfill
%%%- figure TL (practice final #4 problem  3)%%%%%%%%%%%%%%%
%DIA 
%
\begin{minipage}[b]{3 truein} 
\begin{psfigure}{3in}{1.5in}{figTLa.ps}
\puttext{0, -23.3}{\Small $14''$}
\puttext{80, 0}{\Small $4''$}
\puttext{-80,0}{\Small $4''$}
\puttext{67., 7.}{\Small $\theta$}
\puttext{-67.,7.}{\Small $\theta$}
\end{psfigure}

\begin{psfigure}{3in}{1.25in}{figTLb.ps}
\puttext{66.7, -10.}{\Small $50''$}
\puttext{-33.3, -35}{\Small $14''$}
%%\puttext{20., -15}{\Small $4''$}
\puttext{-100., -15.}{\Small $4''$}
\end{psfigure}

\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%


\smallskip
\noindent
4.~(10 points) Use Newton's method to find the side $s$ of a cube for
which the sum of its volume and its surface area is 1500.  Use enough
iterations so as to be sure that your answer is accurate to four
significant figures.

\vfill

\ \hfill (Continue to Problems 5 and 6 on the next page)

\newpage


\begin{rightindent}{3.5in}
\noindent
5.~(15 points) The point $A$ revolves counterclockwise around the unit
circle at the rate of 1 revolution every 9 seconds.  At $t=0$ it starts
at the point $(1,0)$.  The point $B$ starts at the point (2,0) and travels
in a straight line at constant velocity, reaching the point (0,2) after 2
seconds.

(a)~Find parametric equations for both points $A$ and $B$.

(b)~Find a function which must be equal to zero at the instant when $A$ and
$B$ are closest.  This is the function that you could put into Newton's
method to find the instant when $A$ and $B$ are closest.
\end{rightindent}
\hfill
%%%- figure TM (practice final #4 problem  5)%%%%%%%%%%%%%%%
%DIA 
%
\begin{minipage}[t]{1.90 truein}
\begin{psfigure}{2in}{1.75in}{figTM.ps}
\puttext{0, 0}{\Small$A$}
\puttext{20., 20.}{\Small$B$}
\end{psfigure}
\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%


\smallskip


\noindent
6.~(25 points) At time $t=0$ a ball is thrown out of a 50-meter high
window at a velocity of 10 m/sec at an angle of $\alpha$ radians above
the horizontal.  Take the ground to be the $x$-axis and the side of the
building from which the ball was thrown to be the $y$-axis.  Neglect air
resistance, and take $g=9.8$ m/sec${}^2$.

\begin{rightindent}{3.0in}
(a)~Find parametric equations for the $x$- and $y$-coordinates of the ball
at time $t$.  Some trig functions of $\alpha$ should appear in your equations.

(b)~Let $T$ be the time when the ball hits the ground.  Express $T$ as a
function of $\alpha$ (you will need the quadratic formula

\[
    (-b\pm\sqrt{b^2-4ac})/2a,
\]

\noindent
where the sign in front of the square root should
be chosen so as to give positive T).

(c)~Let $X$ be the distance of the ball from the building when it hits the
ground.  Express $X$ as a function of $\alpha$.

(d)~Suppose that $\alpha$ is measured to be $60^\circ$ with an error of
$\pm e$ degrees: $\alpha=60^\circ\pm e^\circ$.  Use the tangent line
approximation to determine how sensitive $X$ is to the error in $\alpha$.
In other words, find the constant of proportionality which relates the error
in $X$ (in meters) to the error $e$ in $\alpha$ (in degrees).
\end{rightindent}
\hfill
%%% figure TN (practice final #4 problem  6)%%%%%%%%%%%%%%%
%
\begin{psfigure}{2.45in}{3.0in}{figTN.ps}
\puttext{ 9,  27}{\Small $\alpha$}
\puttext{-2,  33}{\Small $10 \text{ m/sec}$}
\puttext{-23, 40}{\Small $\alpha$}
\puttext{ 38,-43}{\Small$x$}
\puttext{  5,-50}{\Small$(X,0)$}
\puttext{-30, 50}{\Small$y$}
\end{psfigure}
%%%%%%%%%%%%%%%%%%%%%%%%

