% practice 2nd midterms
\chapter*{Practice Second Midterms}

{\bf Practice Second Midterm \#1} (50 points in all, time = 1 hour)

\medskip
\begin{rightindent}{2.5in}
\noindent
1.~(13 points) A rectangle having dimensions $2a$ and $2b$ has semicircles
on three sides, as shown in the drawing at the right.
Suppose that at first $a=10$ and $b=20$, but then you want to increase $a$
to $10.100$ and decrease $b$ in such a way that the total area of the shape
(the rectangle plus the three semicircles) remains constant.  Use the
tangent line approximation to determine your new value for $b$.
\end{rightindent}
\hfill
%DIA 
%--- figure OA (practice second midterm #1, prob 1)--------------------
%
\begin{psfigure}{2.95in}{2.3in}{figOA.ps}
\puttext{16.7, 3.33}{\Small$b$}
\puttext{-23.3, -8.33}{\Small$a$}
\end{psfigure}
%------------------------------

\smallskip
\noindent
2.~(12 points) Let $y=f(x)=x-\bigl(1/(2x^2)\bigr)$.

(a)~Find the $x$- and $y$-coordinates of any upper and lower turning
points, and {\bf use the second derivative test} to determine if they are
max or min.

(b)~Find all intervals of values of $x$ where the function is {\bf increasing}.

(c)~Find all vertical asymptotes (if there are any).

(d)~Find all points of inflection (if there are any), and determine the
upward or downward concavity of each part of the curve.

(e)~Sketch the curve.  Show in your sketch what $f(x)$ looks like as
$x$ approaches $+\infty$ and $-\infty$, and as $x$ approaches the vertical
asymptote (if there is one) from the left and from the right.

\smallskip
\noindent
3.~(10 points) Suppose that in Wanganui, New Zealand the mean daily
temperature (averaged over many years) is a sinusoidal function of time.
Let $t$ be the time {\bf in months} from the beginning of the year, and let
$y$ be the temperature in ${}^\circ$F.  Suppose that the hottest temperature
is $75^\circ$F in mid-January (that is, $t=0.5$), and the coldest is $35^\circ$F
in mid-July.

(a)~Graph the temperature function $y=f(t)$.

(b)~Give its equation, in the form $y=A\sin\left(\frac{2\pi}{ B}(t-C)\right)+D$.

(c)~Suppose that the typical Wanganuian homeowner turns on the heat when
the mean daily temperature drops below $65^\circ$F.  In an average year,
find the time interval during the year when she has the heat turned on.

\smallskip
\noindent
4.~(15 points) According to the law of cosines for the triangle $ABC$ below,
we have: $c^2=a^2+b^2-2ab\cos C$.  Suppose that $C=120^\circ$, $b=50$ meters,
and you are running from $D$ to $C$ at 5.70 m/sec.  How fast is your distance
$c$ from $A$ decreasing when you are at a distance $a=160$ m from $C$?

\smallskip

\begin{center}
%--- figure OB (practice second midterm #1, prob 4)--------------------
\begin{psfigure}{4in}{1.0in}{figOB.ps}
\puttext{-127., -6.67}{\Small$A$}
\puttext{86.7, -16.7}{\Small$B$}
\puttext{-80., 46.7}{\Small$C$}
\puttext{127., -30.}{\Small$D$}
\puttext{0., -20.}{\Small$c$}
\puttext{-113., 20.}{\Small$b$}
\puttext{0., 13.3}{\Small$a$}
\end{psfigure}\\
%------------------------------
\end{center}

\bigskip
%\vspace{1in}

 
{\bf Practice Second Midterm \#2} (50 points in all, time = 1 hour)

\medskip
\noindent
1.~(13 points) Let $y=f(x)=x^2-\frac{1}{ 6}x^4$.

(a)~Find the $x$- and $y$-coordinates of any upper and lower turning points.

(b)~Find all intervals of values of $x$ where the function is {\bf
increasing}.

(c)~Find all points of inflection.

(d)~For what values of $x$ is the function concave {\bf upward}?

(e)~Sketch the curve.  Show in your sketch what $f(x)$ does as $x$
approaches $+\infty$ and $-\infty$.

\smallskip
\noindent
2.~(12 points) A bicycle has a wheel whose {\bf circumference} is exactly
2 meters.  It is traveling at 5 m/sec from left to right.  At time $t=0$
it picks up a pebble in the tread of the wheel.  Take $(0,0)$ to be the
point where the pebble attaches itself to the tread.  In this
problem use the parametric equations of the cycloid: $x=r\theta-r\sin\theta$,
$y=r-r\cos\theta$.

(a)~What is the angular velocity $d\theta/dt$ of the wheel in rad/sec?

(b)~Find formulas for the vertical velocity and the horizontal velocity
of the pebble in terms of $\theta$.

(c)~Find the slope of the direction of motion of the pebble when $t=1/3$
sec.  Draw a picture of the wheel showing the position of
the pebble when $t=1/3$ sec, and draw an arrow showing the direction of
motion of the pebble when $t=1/3$ sec.

\smallskip
\noindent
3.~(12 points) (a)~Write the implicit equation of the ellipse whose
bottom half is pictured below to the right, and solve for $x$ in terms of $y$.

\noindent
\begin{rightindent}{2.5in}
(b)~Suppose that the drawing is the side view of a container which has a
circular top (and circular cross-section at any level).  That is, the 
container is obtained by rotating the bottom half of the ellipse around the
$y$-axis.  Suppose the container is being 
\end{rightindent}
\hfill
%--- figure OC (practice second midterm #2, prob 3)--------------------
%
\begin{psfigure}{2.95in}{1.0in}{figOC.ps}
\puttext{-5, 12}{\Small$y$}
\end{psfigure}
%------------------------------
filled with water.
Let $A=f(y)$ be the {\bf surface area} of the top of the water
when it reaches the level $y$ on the $y$-axis.  The units are {\bf meters}.
Write a formula for $A$ in term of $y$.

(c)~If the water is 250 cm = 2.5 m deep at its deepest point and is rising
at 1 cm/sec, how fast is the surface area of the top of the water
increasing?
\smallskip


\begin{rightindent}{2.5in}
\noindent
4.~(13 points) The ellipse pictured to the right has foci at the
two points $(-5,0)$ and $(0,-5)$.  The distance of any point $(x,y)$ on
the ellipse from $(-5,0)$ plus its distance from $(0,-5)$ has the constant
value 20.

(a)~Write an equation for this ellipse in the form $F(x,y)=\text{Const}$.

(b)~Notice that the point $(0,-12)$ is on the ellipse.  Find the slope of
the tangent line to the ellipse at $(0,-12)$.
\end{rightindent}
\hfill
%--- figure OD (practice second midterm #2, prob 4)--------------------
%
\begin{psfigure}{2.95in}{2.0in}{figOD.ps}
\puttext{-4., 20.}{\Small$A$}
\puttext{20., -8.}{\Small$B$}
\puttext{64., 0.}{\Small$P = (x,y)$}
\puttext{48., 48.}{\Small$\overline{AP} + \overline{BP} = 20$}
\end{psfigure}
%------------------------------

 \bigskip


{\bf Practice Second Midterm \#3} (50 points in all, time = 1 hour)

\medskip

\noindent
1.~(12 points) Let $y=f(x)=0.25\,x^4-2\,x^3+4.5\,x^2$.

(a) Find the $x$- and $y$-coordinates of any upper and lower turning points.

(b) Find all points of inflection.

(c) Sketch the curve, labeling all of the points found in parts (a) and (b)
(``max,'' ``min,'' ``inflec'').

(d) Give the exact interval(s) where the curve is concave downward.

\smallskip

\begin{rightindent}{3.5in}
\noindent
2.~(12 points) In the triangle shown below, the sides $a$ and $b$ are pieces
of metal hinged at $C$.  Here $a$ is {\bf exactly} 1.50$\,m$ long and $b$ is
{\bf exactly} 2.50$\,m$ long.  We measure
the side $c$ to be $3.50\,m$, but our measurement of $c$ has an
error of up to a centimeter, i.e., $\pm0.01\,m$.  Using the law of cosines
\[
    c^2=a^2+b^2-2ab\cos C,
\] 
\end{rightindent}
%--- figure OE (practice second midterm #3, prob 2)--------------------
\begin{psfigure}{2in}{1.0in}{figOE.ps}
\puttext{110., -25.}{\Small$A$}
\puttext{-105., -25.}{\Small$B$}
\puttext{-40., 35.}{\Small$C$}
\puttext{-75., 5.}{\Small$a$}
\puttext{25., 12.}{\Small$b$}
\puttext{0., -40.}{\Small$c$}
\end{psfigure}
%------------------------------
we find that the angle $C$ is $120^\circ$.
Use the tangent line approximation and implicit differentiation to
determine the error {\bf in degrees} (not radians) in our value for the
angle $C$. 

\medskip
\begin{rightindent}{3in}
\noindent
3.~(12 points) You are 5 ft tall, and the sun is casting a 5 ft shadow.
(See the diagram at the right.)
If the sun is descending toward the horizontal at the rate of 1/4 degree per
minute, how fast is your shadow lengthening?  (First express the length
of your shadow as a function of the sun's angle above the horizontal.)
\end{rightindent}
\hfill
%--- figure OF (practice second midterm #3, prob 3)--------------------
%
\begin{psfigure}{2.0in}{1.5in}{figOF.ps}
\end{psfigure}
%------------------------------



\smallskip
\noindent
4.~(14 points) An object oscillates horizontally at the end of a spring.  Its
sinusoidal horizontal motion goes from a minimum position of $x=0$ to a
maximum position of $x=4\,$m.  It goes back and forth once every 2 seconds,
starting at $x=0$ at time $t=0$.  Meanwhile, at time $t=0$ 
the whole spring system is dropped from a height of 19.6 meters.  Neglect
air resistance, and take $g=9.8~$m/sec${}^2$.

(a) Write a formula for $y(t)$.

(b) Draw a graph of $x$ as a function of $t$, and find a formula for
$x(t)$.

(c) At the instant $t=1.25$ sec, find the horizontal and vertical
components of velocity, and the direction (slope) of the path.


{\bf Practice Second Midterm \#4} (50 points in all, time = 1 hour)

\medskip

\noindent
1.~(15 points) Let $y=f(x)=\frac{1}{ 120}\Bigl(2x^6-9x^4-108x^2\Bigr)$.

(a)~Find the $x$- and $y$-coordinates of any upper and lower turning points.

(b)~Find all intervals of values of $x$ where the function is {\bf increasing}.

(c)~Find all points of inflection.

(d)~For what values of $x$ is the function concave {\bf downward}?

(e)~Sketch the curve.  Show in your sketch what $f(x)$ does as $x$
approaches $+\infty$ and $-\infty$. 

\smallskip
\noindent
2.~(10 points) A circular oil slick of uniform thickness is caused by a spill
of 1 cubic meter of oil.  The thickness of the oil is {\bf decreasing} at
the rate of 0.1 cm/hr as the oil spreads (note: 1 cm = 0.01 m).  At what rate
is the radius of the slick increasing when it is 8 meters?  (Note: the volume
of a circular oil slick is equal to $\pi\times$(radius)${}^2\times$thickness.)

\smallskip

\begin{rightindent}{3.5in}
\noindent
3.~(10 points) In the triangle pictured at the right, let $A$, $B$, $C$
be the angles at the three vertices, and let $a,b,c\>$ be the sides opposite
those angles.  According to the ``law of sines,'' you always have:
\[
\frac{b}{ a}=\frac{\sin B}{\sin A}.
\]
\end{rightindent}
\hfill
%------------------------------
%DIA
%--- figure OG (practice second midterm #4, prob 3)--------------------
%psfix -width 2 -height 2 figOG > figOG.ps
%
\begin{psfigure}{1.95in}{1.5in}{figOG.ps}
\puttext{-55., -25.}{\Small$A$}
\puttext{55., -25.}{\Small$B$}
\puttext{5., 30.}{\Small$C$}
\puttext{0., -35.}{\Small$c$}
\puttext{-25., 5.}{\Small$b$}
\puttext{30., 5.}{\Small$a$}
\end{psfigure}
Suppose that $a$ and $b$ are pieces of metal which
are hinged at $C$.  At first the angle $A$ is $45^\circ$ and the angle $B$
is $60^\circ$.  You then widen $A$ to $46^\circ$, without changing the sides
$a$ and $b$.  What happens to the angle $B$? Use the tangent line
approximation.


\smallskip

\begin{rightindent}{3.0in}
\noindent
4.~(15 points) A conical pile of sand is exactly 10 ft high.  Suppose
that you have an instrument to measure the vertex angle $\theta$ (see
the diagram at the right).

(a)~Express the volume $V$ of sand in terms of $\theta$.

(b)~Suppose you measure $\theta$ to be $60^\circ$ with an error of $\pm
1^\circ$.  Find $V$, and use the tangent line approximation to determine
the error in your value for $V$. (Warning: use radians for angles.)
\end{rightindent}
\hfill
%---------------------------------------------------------------------
% DIA
%--- figure OH (practice second midterm #4, prob 4)--------------------
%
\begin{psfigure}{2in}{2.5in}{figOH.ps}
\puttext{10., 38.}{\Small $\theta/2$}
\puttext{10., 32.}{\Small $\theta$}
\puttext{-5., 0}{\Small$10\text{~ft}$}
\end{psfigure}
%---------------------------------------------------------------------

 



%---------------end of practice 2nd midterms --------------------------
