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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Final Examination -- May 10, 2006}
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{\it Directions:\/}
Do all work in the blue exam booklet.
There are 200 regular points and 20 Extra Credit points, distributed
as marked.  Take a deep breath before plunging in and read each question
carefully.
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I.  
\item{A)}  (10) Give a precise statement of the Fundamental Theorem
of Calculus (both parts).
\item{B)}  (15) The following graph shows $y = f(x)$.  Let $F$
be the antiderivative of $f$ with $F(0) = 0$ and $F$ continuous. Sketch 
the graph $y = F(x)$.
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II.  Compute each of the following integrals.  You may use
the table of integrals anywhere on these.  If you do, say
which table entry you are using.
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\item{A)} (10)  $\int 3x^6 - 4\sqrt{x} + \sin(x)\ dx$
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\item{B)} (10)  $\int {x^3 \over x^4 + 1}\ dx$  
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\item{C)} (15)  $\int x^2 e^{-9x}\ dx$
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\item{D)} (10)  $\int {1 \over \sqrt{4x^2 + 9}}\ dx$
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\item{E)} (15)  $\int {x+1\over x^2 + 5x + 6}\ dx$
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III.  Let $R$ be the region bounded by $y = x$, the $x$-axis,
and $x = 1$, $x = 4$.
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\item{A)} (15)  Find the volume of the solid obtained by rotating
$R$ about the line $y = -2$.
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\item{B)} (10)  A thin plate has the shape of the region
$R$ ($x,y$ in cm) and density $\delta(x) = x^{-1/2}$ grams/cm${}^2$.
Find its total mass.
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IV.  At a particular location in Natick on the Mass Pike, a sensor
was set up to measure the passage of traffic.  The measurements
made were used to derive a probability density function for the
quantity $x = $ time gap between successive cars (in minutes).  The results
gave the following formula as a good fit for the pdf:
$p(x) = 11(1-x)^{10}$ if $0 < x < 1$, and zero otherwise.
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\item{A)} (10)  Show that $p$ satisfies the usual property for 
a probability density function:  $\int_0^1 p(x)\ dx = 1$.
\item{B)} (15)  What is the probability that the time gap
between successive cars is between $x = .1$ minute and $x = .2$ minute?
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V.  
\item{A)} (10) Using the definition of Taylor polynomials, compute the
Taylor polynomial of degree $n = 3$ for $f(x) = \sqrt{1 + 2x}$ at $a = 0$.  
\item{B)} (10) Use our shortcut methods to check your work in part A.
\item{C)}  (5) Use your polynomial from part A to compute an approximation
to $\sqrt{1.2}$.  What is the error in your approximation?
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VI. (15) Solve for $y$ by separation of variables:
${dy\over dx} = \cos(x)(1 + y^2)$ with  $y(0) = 1$.
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VII.  An avian flu epidemic has broken out in 
Birdsburgh, a large city with total population 10 million.  
Write $N$ for the number of people
who have been infected, as a function of time.  The Birdsburgh
Public Health department determines that:
$${dN\over dt} = k N (10 - N),\leqno(1)$$
or in words: the rate of change of $N$ is proportional to
the product of $N$ and the number of people not yet infected, 
where $N$ is in millions of people, $t$ in weeks, $k$ a positive
constant.
\item{A)} (10) Which of the following slope field plots matches (1)?
Explain how you can tell.
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\item{B)} (10) For what value of $k$ is $N(t) = 10/(1 + 1000e^{-t})$
a solution of (1)?
\item{C)}  (5)  If the epidemic proceeds according to the 
function $N(t)$ in part B, 
how many weeks will pass before the 
number of infected people reaches 1 million?

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\noindent
{\it Extra Credit} (20) 
The hull of a boat is 20 feet long.  At a distance
$s$ feet from the bow (the front), the cross section of the part
of the hull below $y = 0$ (the waterline) has the shape of the region 
in the $xy$-plane below $y = 0$ and above 
the parabola $y = ax^2 - b$, where $a,b$ are given in the following table:
$$\matrix{s & 5 & 10 & 15 & 20 \cr
          a & 2 & 3 & 4 & 5 \cr
          b & 2 & 3 & 4 & 4 \cr}$$
Estimate the {\it volume\/} enclosed by the hull below the water line.
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\centerline{\it Have an enjoyable, safe, and productive summer!}
\bye