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Mathematics 132 -- Calculus for Physical and Life Sciences 2 \\
Review Sheet for Midterm 3\\
April 20, 2005\\
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\bigskip
\noindent 
{\it General Information}
\bigskip
The third midterm will be held the evening of Wednesday,
April~27, from 6:00--7:30. Material from Section~8.6 through
Section~10.4 will be covered.  We will review for the exam in class
on Tuesday and Wednesday next week.

\bigskip
\noindent
{\it Topics to be Covered}
\bigskip
Here is a brief list of topics:

\bi

\item Probability density and cumulative distribution functions; mean
  and median of a density.
  
\item Numerical series: partial sums and convergence; finite and
  infinite geometric series.  Convergence tests: integral, comparison,
  and ratio tests, alternating series.

\item Power series, radius and interval of convergence; 
  Taylor polynomials and series, series
  manipulations, the error estimate for Taylor polynomials.

\ei

\bigskip
{\it Some Sample Problems}
\bigskip
The following problems are typical of test-type questions. Note that a
single problem may call for techniques spanning the coverage of the
test.  Do not attempt to memorize how to do ``all the types of
problems''; instead, strive to understand what each idea is used for,
and select your approaches accordingly.

\be
\item The following graph shows either a pdf or a cdf.  Which
type of function is it, and why?  If it is a pdf, sketch the 
graph of the corresponding cdf; if it is a cdf, sketch
the graph of the corresponding pdf.
\vglue 1.0truein

\item Let~$x$ be the size of an email message in kilobytes~(KB). An
  ISP finds that the fraction of emails between $x$~and $x+\Delta x$
  KB~in size is about $cxe^{-0.01x}\Delta x$ (that is, the density
  function for $x$ is $p(x) = c x e^{-0.01x}$).

  (a) Find the value of~$c$
  
  (b) Calculate the fraction of email messages that are at most 100~KB
  in size, 
  
  (c) Calculate the fraction that are at least 50~KB in size.
  
  (d) Calculate the median size of an email message.
  
\item In a certain population, the height~$x$ of an individual in
  inches is normally distributed, with a mean of~68 and a standard
  deviation of~4. In other words, heights are modeled by the density
  function $p(x)=\frac{1}{4\sqrt{2\pi}}e^{-(x-68)^2/32}$.

  (a) Write down a definite integral whose value gives the fraction of
  the population whose height is between 68~and 72~inches, and perform
  the change of variables $z=(x-68)/4$ on your integral.


 (b) Write out the Taylor series for $e^u$ giving both the first four non-zero terms and the summation form.

 
  (c) Use your answer from (b) to find the Taylor series for
  \[
  \int_0^x e^{-z^2/2}\,dz,
  \]
  giving both the first four non-zero terms and the summation form.

  (d) Calculate an estimate of the integral in part~(a) using
  the series in part~(c).  (How?)

\item

  (a)  Use the Comparison Test to determine whether or not
  \[
  \sum_{n=0}^\infty\frac{n+3^n}{2^n}
  \]
  converges.
  
  (b)  Use the Integral Test to determine whether or not
  \[
  \sum_{k=0}^\infty\frac{k}{e^k}
  \]
  converges.
  
  (c)  Use the Ratio Test to determine whether or not
  \[
  \sum_{k=0}^\infty \frac{3^n}{n!}
  \]
  converges.
  
\item Determine (with justification!) whether or not the following
  series converge:
  \[
  \sum_{k=1}^\infty \frac{1}{\sqrt{k}},\qquad
  \sum_{n=0}^\infty (-1)^n\frac{n^2+4n+1}{3n^4+2n^2+10000},\qquad
  \sum_{n=1}^\infty \frac{1}{n^{1.01}}.
  \]
  
\item Let $f(x)=\sqrt{1+x}=(1+x)^{1/2}$. Find the 4th degree Taylor
  polynomial of~$f$ centered at $a=0$.  Find a factorial
  expression for the general term of the Taylor series.

\item Consider the geometric series
  $f(x)=\ds\sum_{k=0}^\infty x^k=\frac{1}{1-x}$.

  (a) Use series manipulations to find the Taylor series of $xf'(x)$.

  (b) Use series manipulations to find the Taylor series
  of~$-\ln(1-x)$. 
  
  (c) Find the radius of convergence of the series in part~(b), and
  investigate convergence at the endpoints.

  (d) Use parts (a)~and (b) to \emph{evaluate} the sums of the series
  $\ds\sum_{k=1}^\infty\frac{1}{k\cdot 2^k}$~and 
  $\ds\sum_{k=1}^\infty\frac{k}{2^k}$.

\item For each of the given power series, find the interval of
  convergence.
  \[
  f(x)=\sum_{n=1}^\infty\frac{(2x)^n}{\sqrt{n}},\qquad
  g(x)=\sum_{n=1}^\infty(-1)^{n-1}\frac{(x-5)^n}{n\cdot3^n}.
  \]
  (In particular, give the radius of convergence, 
  and investigate convergence at the
endpoints.)

\item The second degree Taylor polynomial of $f(x)$ at $a = 0$
is $p_2(x) = c + bx + ax^2$.  What can you say about the 
signs of $a,b,c$ if you know the graph of $f(x)$ is:
\vglue 1.0truein

\item Use the error bound for Taylor approximations to 
estimate the number of decimal places of accuracy if
the 6th degree Taylor polynomial at $a = 0$ is used to 
approximate $\cos(.8)$.  Do the same for the $n$th degree
polynomial in general.  What happens to the error bound
as $n\to \infty$?  
\ee

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