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\leftline{Mathematics 134 -- Intensive Calculus for Science 2}
\leftline{Midterm Exam 3 -- April 28, 2006}
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{\it Directions:}  Do all work in the blue exam booklet.  You
may use an approved calculator, and the table of integrals provided.  
There are 100 possible points.
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I. 
\item{A)} (10) Which of the following series is geometric:
$${\rm series\ A:\ }  1 + {1\over 2!} + {1\over 3!} + {1\over 4!} + \cdots
\quad {\rm series\ B:\ }  {3\over 2} + {3\over 4} + {3\over 8} + {3\over 16}
+\cdots$$
Explain how you can tell.
\item{B)}  (10) The sum of the geometric series 
$$ (b/2) + (b/2)^2 + (b/2)^3 + (b/2)^4 + \cdots $$
is equal to 3.  What is the value of $b$?
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II.
\item{A)} (15) Using the definition, compute the Taylor polynomial
of degree $n = 4$ for $f(x) = \cos(2x)$ at $a = 0$.
\item{B)} (15) Using any applicable method, find the first three
non-zero terms in the Taylor series for $g(x) = (1 + x)^{1/3}$ at $a = 0$.
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III.  
\item{A)} (10) Which of the following two slope field plots corresponds
to the differential equation ${dy\over dx} = (y - 1)(y - 5)$.
Explain how you can tell.
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\item{B)} (10) Give qualitative sketches of the solutions with $y(0) = 3$
for each of the differential equations
whose slope fields are shown in part B.  (Show the graph in the 
blue book -- you {\it do not} need to recopy the slope field).
\item{C)}  (10) Find the general solution of the differential equation
${dy\over dx} = {y\over x(x+1)}$.
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IV.  A culture of bacteria contains 200 cells originally
and {\it grows at a rate proportional to its size}.  
After half an hour the population has increased to 360 cells.
\item{A)} (10) Find the function $P(t)$ giving the population
after $t$ hours.
\item{B)} (10) When will the population reach 10000 cells?
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