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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Exam 3 -- Review Sheet}
\centerline{November 18, 2016}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the third midterm exam of the semester
will be given the week after Thanksgiving break.  We can either do this 
in class on Friday, December 2 or in the evening of Thursday, December 1.
We can decide this before you leave for Thanksgiving.   
The format will be similar to that of the first two midterms.

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\begin{itemize}
\item You may use a non-graphing calculator.  Any calculator 
like the TI-30, which
does NOT have graphing capabilities, is OK.  (Note: Some of you may have one of these
calculators purchased for use in Chemistry courses here.  That is also OK.)
\item Use of phones, I-pods, I-pads, and any electronic devices other than your
calculator \emph{is not allowed} during the exam.  Please leave such devices in your room
or put them away in your backpack  (make sure cell phones are turned off).
\end{itemize}
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\noindent
{\it What will be covered}
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The exam will cover the material since the last exam (Problem Sets 6, 7, and 8).  However
some calculations may require you to use integration methods covered on Exam 2 (review
integration by $u$-substitions, parts and trigonometric substitutions in particular!)  
I understand that this may 
look like a lot, but to help you organize your studying, keep in mind
that \emph{everything here is based on computing integrals by various methods!  Review
the integration methods first, then the applications.}
The new material is:
\begin{itemize}
\item Integration by partial fractions (section 7.5):  Know the whole process, including
preliminary divsion (if necessary), setting up the partial fractions based on
the factorization of the denominator, solving for the undetermined coefficients
in the partial fractions, then integration of the partial fractions.
\item Improper integrals (section 7.7):  Know how to recognize when an integral
is improper, and how to evaluate these using limits.
\item Probability applications (section 7.8):  Know how to recognize when
a function is a valid probability density (pdf) or compute an undetermined constant
to make a function a valid pdf.  Know how to compute the probability a random
variable has a value in a given interval given the pdf; know how to compute
the expected value given the pdf.
\item Numerical Integration (section 7.9):  In the material from this section, 
I \emph{will not ask} you to compute approximations using the Midpoint, Trapezoidal, or 
Simpson's Rules.  But I may ask conceptual questions about the various error
formulas and bounds, when the Midpoint and Trapezoidal Rules give over- or 
under-estimates, and what we saw in the computer lab about Simpson's Rule (in particular
why that method is expected to yield a smaller error).  
\item Differential equations, separation of variables, solutions, etc. (section 9.1)
\item Models involving $y' = k(y - b)$ (section 9.2):  Newton's Law of Heating/Cooling, 
etc.  ``word problems'' based on that.
\item Section 9.3:  I \emph{will not ask} you to draw slope fields or 
carry out Euler's Method calculations ``by hand'' to derive approximate 
values of a solution of a differential equation. But you should understand 
how solutions of a first-order differential equation relate to the slope field.  
This is another place where I might ask a conceptual question.
\item Logistic equations (section 9.4):  Know how to derive the general 
solution of $y' = ky(1 - y/A)$ and how to use those solutions to solve initial
value problems.
\item First order linear equations (section 9.5):  Know how to determine
the integrating factor ($e^{\int P(x)\ dx}$) for the equation 
$y' + P(x)y = Q(x)$ and what the general solution
looks like.  
\end{itemize}
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\noindent
{\it Review problems}
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There is a selection of review problems posted on WebAssign as an 
optional assignment.  I have increased the number of tries on 
each problem to the maximum possible -- 100.
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\noindent
{\it Sample problems}
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\noindent
{\it Note:  The actual exam will be considerably shorter than the following
list of questions and it might not contain questions of all of these types.  
The purpose here is just to give an idea of the range
of different topics that will be covered and how questions might be posed.}
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\noindent
I. {\it Partial Fractions}.
\begin{enumerate}
\item[(A)]  Do you need partial fractions to 
integrate
$$\int \frac{t^2 + 1}{t^3 + 3t + 3}\ dt?$$
Explain, and compute the integral with a simpler method.
\item[(B)]  Apply partial fraction decomposition to 
compute $$\int \frac{x^3+1}{x(x - 1)(x + 2)}\ dx$$
(Hint: you need to divide through first; do you see why?)
\item[(C)]  What would the partial fractions look like to
integrate
$$\int \frac{1}{x(x^2 + 4)^2}\ dx?$$ 
(You don't need to solve for the coefficients, just set up 
the proper partial fractions.)
\item[(D)]  Suppose the partial fraction decomposition of a 
rational function is 
$$\frac{1}{81x} + \frac{-\frac{x}{9} + 1}{(x^2 + 9)^2} - \frac{\frac{x}{81}}{x^2 + 9}$$
What is the integral?  (Hint:  There's one part of this for which you 
need a trigonometric substitution.)
\end{enumerate}
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\noindent
II.  For each of the following integrals, say why the integral is improper,
determine if the integral converges, and if so, find its value.
\vskip 10pt
\begin{enumerate}
\item[A)]  $\displaystyle \int_1^\infty \frac{1}{{\root 5 \of x}}\ dx$
\item[B)]  $\displaystyle \int_0^2 \frac{dx}{x^2 - 7x + 6}$
\item[C)]  $\int_0^\infty x e^{-3x}\ dx$
\item[D)]  For which values of $a$ is $\int_0^\infty e^{ax}\sin(x)\ dx$
convergent?  Evaluate the integral for those $a$. 
\end{enumerate}
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\noindent
III.  The time $t$ (in minutes) between two 
successive incoming calls at a phone center is a random variable
with a pdf of the form $f(t) = \frac{1}{r} e^{-t/r}$ if $t \ge 0$ and 
$0$ otherwise.   
\begin{enumerate}
\item[(A)] Suppose that the probability that two successive calls are more 
than one minute apart is $.3$.  What is the value of $r$?
\item[(B)] Using the value of $r$ you determined in part A, find the 
probability that the two successive calls are at least 5 minutes apart.
\end{enumerate}
\vskip 10pt
\noindent
IV.  {\it Numerical Integration}.
\begin{enumerate} 
\item[(A)]  If you apply the Trapezoidal Rule and the Midpoint Riemann Sum Rule
with $N = 30$
to approximate $\int_1^8 x^2 e^x\ dx$, one will overestimate the actual
value and one will underestimate the actual value.  Which is which?  Also, 
how would we expect the absolute values of the errors for these methods to compare?
\item[(B)]  How big would you need to take $N$ to get a Midpoint Riemann Sum
approximation to $\int_0^1 \sin(x)\ dx$ accurate to 5 decimal places.  (Hint:  You want
the error bound $< .000005$.)
\end{enumerate}
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\noindent
V.  {\it Differential Equations}.
\begin{enumerate}
\item[(A)]  Verify that for any constant $C$, the function $y = \sqrt{C + x^2}$
is a solution of the differential equation $\displaystyle y\frac{dy}{dx} - x = 0$.  Which 
of these solutions also satisfies the initial condition $y(0) = 5$?
\item[(B)]  For which real $a$ is $y = \sin(ax)$ a solution of the differential equation 
$\frac{d^4 y}{dx^4} - 16y = 0$?
\item[(C)]  Find the general solution of 
$$\sqrt{1 - x^2}\frac{dy}{dx} = y^2 + 2y$$
\item[(C)]  Find the general solution of 
$$\frac{dy}{dx} = xe^{x+y}$$
\item[(D)]  A baked potato is taken out of the oven at a temperature of $140^\circ$ C
and left to cool on a counter in a room maintained at $20^\circ$ C.  
After 2 minutes the temperature of the 
potato has decreased to $100^\circ$ C.  How long will it take for the temperature
to reach $80^\circ$ C?
\item[(E)]  Sunset Lake is stocked with 2000 rainbow trout.  After 1 year
the trout population has grown to 4500.  Assuming logistic growth with
a carrying capacity of 10000 trout, find the growth constant $k$ and determine
when the population will reach 5000.
\item[(F)]  Find the general solution of 
$$\frac{dy}{dx} + xy = x$$
\end{enumerate}
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