%% Math 131, Fall, 2003 Course Review Sheet
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\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Course Review Sheet}
Professor Hwang \\
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\paragraph{Coverage}

The final exam for Math~131 covers everything we've done in class this
semester. An itemized breakdown of concepts is given below. There will
be a review session on Friday, December~12, from Noon--2~PM, in 
O'Neil~112 (the usual place).

In addition to the odd-numbered questions from the homework and the
questions on the first group worksheet, the problems below will
provide good practice for the final exam. The \emph{only} way to
prepare is to work lots of problems until you're fluent with the
techniques. Trying to memorize formulas is not a good study strategy.

\bi

\item Linear, exponential, logarithmic, trig, inverse trig, and power
  functions

\item Limits and continuity, asymptotes

\item Rates of change: Slopes of secant lines, average rate of change,
  including units

\item Derivatives of specific functions, general formulas (product,
  quotient, chain rule, implicit differentiation)

\item Practical and geometric interpretations of derivatives: Sign
  diagrams and intervals of monotonicity, critical points, concavity
  and inflection points, applications to graphing; local and absolute
  extrema

\item Applications of the derivative: Related rates, optimization,
  linear approximation, l'H\^opital's rule, asymptotic rates of growth

\item Integration: Approximating sums, error estimates for increasing
  and decreasing functions, definition of the integral

\item Definite integrals: Interpretations as signed area and total
  change, properties of the integral

\ei

\be

\item What type of function (linear, exponential, power, trig) is the
  most appropriate for modeling short-term population growth (such as
  the US~population on a scale of a few years, or of a bacterial
  culture on a scale of ten minutes)? What about modeling long-term
  population growth? Write a few sentences to justify your answers.

\item Let $f(x)=\frac{\sin x}{x}$ for $x\neq0$.
  \bd
  \item{(a)} What value for~$f(0)$ makes~$f$ continuous at $x=0$?
  \item{(b)} Show that $\lim\limits_{x\to0} f'(x)$ exists.
  \item{(c)} Answer the same two questions for $g(x)=\frac{e^x-1}{x}$
  \ed

\item A stone is dropped from a high bridge. After $t$~seconds, the
  height of the stone is $y(t)=150-16t^2$ feet.

  \bd
  \item{(a)} Without using a graphing calculator, sketch the graph of
  height as a function of time. What is the domain of the
  function~$y$? 

  \item{(b)} Find the average speed of the stone (including units)
  over the time interval $0.5\leq t\leq 2$, and draw the secant line
  whose slope represents this average speed.
  \ed

\item A savings account is opened on January~1 with~\$1000 at a
  constant interest rate of~2\% per year, compounded continuously.

  \bd
  \item{(a)} Find a formula for the account balance as a function of
  time, including units.

  \item{(b)} How many years does it take for the balance to double? To
  grow to \$1,000,000?

  \item{(c)} Find the average rate of earnings for the first quarter
  of the third year; express your answer in dollars per day.

  \item{(d)} Use linear approximation to estimate the amount of
    interest that accrues on the first day of the tenth year.

  \ed

\item Let $f(x)=(x^2-1)^2$. Find the critical point(s) and inflection
  point(s) of~$f$ by making sign diagrams for the first and second
  derivatives. Classify the critical point(s) as local minima, local
  maxima, or neither. Use the information you find to make a careful
  sketch of the graph of~$f$ on the interval $[-2,2]$. Do not use a
  graphing calculator.

\item The period of a satellite's orbit is proportional to the
  3/2~power of the radius of the orbit, measured from the center of
  the earth. 

  \bd
  \item{(a)} If an orbit's radius is increased by~10\%, by about how
  much does the period change? Use linear approximation, and show all
  work.

  \item{(b)} The space shuttle orbits at an altitude of 150~miles, and
  takes 90~minutes to complete one orbit. Estimate the new orbital
  period if the altitude is increased to 200~miles. The radius of the
  earth is 3900~miles.

  \ed

\item Water drains at 3~cubic feet per minute from a rain collector
  whose shape is an inverted cone of radius 6~feet and height 4~feet.
  Assuming the tank starts full, how rapidly is the water level
  falling initially?  How rapidly is the level falling when the depth
  of the water is 1~foot? Express your answers in feet per minute,
  feet per second, and inches per second. The volume of a cone of
  radius~$r$ and height~$h$ is $V=\frac{1}{3}\pi r^2h$.

\item The cross-sectional area of a tree increases by about the same
  amount each year, since the supply of nutrients is roughly constant,
  and the cross-sectional area is proportional to the volume of new
  tissue added. 

  \bd

  \item{(a)} Do the annual growth rings get thicker or thinner as the
  tree ages?

  \item{(b)} A five-year-old tree has radius 2~inches. Approximately
  how thick will the next ring be? That is, what will be the radius
  of the tree when it is 6~years old? (Suggestion: Use related rates.)

  \item{(c)} About how old will the tree be when its radius is
  10~inches?
  \ed
  
\item A car drives along a straight highway. The horizontal position
  of a point on the tread of a tire is given by
  $x(t)=80t-1.25\sin(64t)$ feet after $t$~seconds. (The path traced by
  the point---a \emph{cycloid} curve---is shown below.) Find the
  maximum forward speed of a point on the tread. At what point of the
  rotation does this maximum forward speed occur? Show that the point
  comes to a stop once per revolution.
  \begin{center}
    \input{review4a.eepic}
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\item A rumor spreads in a population of $M$~people. After $t$~days,
  $P$~people have heard the rumor. To a good approximation,
  \[
  \frac{dP}{dt}=kP(M-P)
  \]
  for a positive constant~$k$ that measures how frequently the rumor
  is repeated. 

  \bd

  \item{(a)} Using the fact that $dP/dt$ is the number of people per
  day who hear the rumor for the first time, explain in words why the
  equation above holds.

  \item{(b)} How many people have heard the rumor when the rate of
  spread is largest?

  \item{(c)} Compute $d^2P/dt^2$; show that the graph of~$P$ as a
  function of~$t$ is concave up when $P<M/2$, and is concave down when
  $P>M/2$. Does this confirm your answer to part~(b)? Explain.

  \ed

\item A spherical balloon is inflated at a constant rate of
  $25~\textrm{in}^3\!/\textrm{sec}$. 

  \bd
  \item{(a)} When the radius is $4$~inches, how rapidly is the radius
  increasing?

  \item{(b)} Assuming the balloon never pops, what is the limit
  of~$dr/dt$ as $t\to\infty$?
  \ed

\item A cylindrical can of radius~$r$ and height~$h$ must hold 1~liter
  (1000~cc). Find the dimensions that minimize the surface area
  $A=2\pi r^2 + 2\pi rh$. The volume of a cylinder is $V=\pi r^2h$,
  and your answer should be given in both symbolic and numerical form.

\item Consider the curve $x^3+y^3-6xy=0$. Compute the slope of the
  tangent line at~$(x,y)$. Find the equations of the tangent lines at
  the points $(3,3)$~and $(4/3,8/3)$.

\item Find the tangent line to the curve $\sqrt{x}+\sqrt{y}=\sqrt{c}$
  at the point $(x_0,y_0)$. Show that the sum of the $x$- and the
  $y$-intercepts is~$c$, regardless of the point of tangency.

\item Use calculus to show that the function
  $f(x)=\arctan(x)+\arctan(1/x)$ is constant. Explain this fact
  geometrically. What is the constant?

\item Two cars start moving from the same point; the first travels
  south at 60~mph and the second travels west at 25~mph. At what rate
  is the distance between the cars increasing after two hours?

\item A street light is at the top of a 15~foot post. A person 6~feet
  tall walks along a straight sidewalk underneath at 5~feet/sec. When
  they are 40~feet from the post, how rapidly is the length of their
  shadow increasing?

\item The top of a 10-foot ladder rests against a vertical wall, and
  the base slides away from the wall at 2~feet/sec.

  \bd
  \item{(a)} When the base of the ladder is 6~feet from the wall, how
  rapidly is the top of the ladder dropping? 

  \item{(b)} When the angle between the ladder and the wall
  is~$45^\circ$, how rapidly is the angle changing? (Be careful with
  units!)
  \ed

\item A car traveling 60~mph brakes to a stop in 5~seconds. If we
  have measurements of the speed as a function of time, we can use
  left- and right-hand sums to estimate the stopping distance.

  \bd
  \item{(a)} Are the left-hand sums under- or over-estimates?
  Explain. 

  \item{(b)} Assuming the speed measurements are taken at equal
  intervals, how many measurements would be required to estimate the
  stopping distance to within one foot, that is, to make the
  difference between upper and lower estimates smaller than 2~feet?
  \ed

\item If we want to estimate $\int_0^1 e^{-x^2}\,dx$ to an accuracy
  of~$0.01$ using left- and right-hand sums with equally-spaced
  points, what is the smallest number of intervals we need?
  
\item Without evaluating the integral, determine whether $\int_0^\pi
  e^x\cos x\,dx$ is positive. (Hint: The interval contains two
  half-arches of cos. If you can, express the given integral in terms
  of $\int_0^{\pi/2}e^x\cos x\,dx$.)

\item Let $f'$~be the function below. On the grid provided, carefully
  sketch the function~$f$, assuming $f(0)=0$. Hint: The graph of
  $f$~has no corners.
  \begin{center}
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  \end{center}
  \begin{center}
    \input{review4c.eepic}
  \end{center}


\ee

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