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\centerline{Mathematics 134 -- Calculus 2 With Fundamentals}
\centerline{Exam 4 -- Review Sheet}
\centerline{April 26, 2018}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the fourth midterm exam of the semester
will be given in class on Friday, May 4.
The format will be similar to that of the first three midterms.

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\begin{itemize}
\item You may use a calculator.  Graphing calculators are also OK, although 
I don't think you will need or want any graphing features for the material 
on this exam.  
\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 7, and 8).  However
some calculations may require you to use integration methods covered on Exams 2 and 3 (review
integration by $u$-substitution, and integration by parts in particular.)  
The new material is:
\begin{itemize}
\item Improper integrals (section 7.7):  Know how to recognize when an integral
is improper, and how to evaluate these using limits.  (Note:  This was covered on 
the last exam too, but I'm giving everyone another chance here!)  Note:  L'Hopital's
Rule might be needed to evaluate some limits involved here.  
\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 Know how to compute normal probabilities using a table like the one we
discussed in class.  Note:  If you have used tables arranged according to a different
format in another class and are more comfortable with them, you may bring a {\it clean, unmarked}
copy of your table with you and use it on the test.  However I will inspect the sheets you bring to 
make sure there is no other ``unauthorized'' information recorded on them.  {\it If
I do find such information, I will discard the sheet and give you a copy of our 
table.}  
\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:  The material on slope fields we discussed briefly in class
{\it will not be covered} on this exam or the final.
\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 Time value of money concepts:  compound interest (compounding monthly, continuously),
present and future value of an asset.  Future value of an ``annuity'' (i.e. a steam of 
equal payments at regular intervals for a fixed number of intervals).  Determining
the monthly payment on a loan, given the principal (amount borrowed), the interest rate, and
the number of years to repayment.  A copy of the formula sheet from class on Thursday, April
26 will be provided.
\end{itemize}
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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.  For each of the following integrals, say why the integral is improper,
determine if the integral converges, and if so, find its value.
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\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)]  (More Challenging) 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
II.  (Probability) The time $t$ (in minutes) between two 
successive incoming calls at a phone center is a random variable
with a pdf of the form $f(t) = c e^{-t/r}$ if $t \ge 0$ and 
$0$ otherwise.   
\begin{enumerate}
\item[(A)] What must the value of $c$ be to make this a valid pdf? Express
$c$ in terms of the constant $r$. 
\item[(B)] Suppose that the probability that two successive calls are more 
than one minute apart is $.3$.  What is the value of $r$?
\item[(C)] 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}
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\noindent
III. (Normal probabilities) Using the normal curve area table to answer these questions.
\begin{enumerate}
\item[(A)] What is the probability that a standard normal random variable $Z$ satisfies
$-1.30 < Z < 0.83$?
\item[(B)] Cans of tomato soup have a stated volume of $16$ fluid ounces, but
the actual volume of the contents is normally distributed with mean $16.3$ fluid
oz. and SD $.22$ fluid oz.  What is the probability that a randomly selected
can contains less than 16 fluid oz. of tomato soup?  
\end{enumerate}
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\noindent
IV.  {\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 $\displaystyle \frac{dy}{dx} = \frac{x}{y}$.  Which 
of these solutions also satisfies the initial condition $y(0) = 5$?
\item[(B)]  For which real $a$ is $y = e^{ax}$ a solution of the differential equation 
$\frac{dy}{dx} - 16y = 0$?
\item[(C)]  Find the general solution of 
$$\frac{dy}{dx} = xe^{x+y}$$
\item[(D)]  (More Challenging) Find the general solution of 
$$\sqrt{1 - x^2}\frac{dy}{dx} = y^2 + 2y$$
\item[(E)]  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[(F)]  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.
\end{enumerate}
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\noindent
V.  Time value of money (and geometric series).
\begin{enumerate}
\item[(A)] Juan deposits $\$500$ in a savings account earning
$3\%$ interest compounded monthly.  What will the balance in the account be
after $4$ years?
\item[(B)]  What is the sum of the finite geometric series
$$6 + \frac{6}{4} + \frac{6}{16} + \cdots + \frac{6}{4^9}?$$
What happens when you keep adding terms of this form indefinitely:
$$6 + \frac{6}{4} + \frac{6}{16} + \cdots + \frac{6}{4^{N-1}}$$
and take a limit as $N \to \infty$?
\item[(C)] Some banks began offering
``Christmas clubs'' around the time of the Great Depression in the 1930's.   
Customers would make contributions from their
weekly pay checks to the ``club'' in order to have a given amount on hand to buy
presents come Christmas time.  Suppose you decide to make 48 weekly payments
into a Christmas club earning an annual interest rate of $1\%$, compounded
weekly.  What should the payment be in order to have $\$800$ on hand to spend on presents?
\item[(D)] What would be the monthly payment on a 15-year mortgage for $300,000$ at
an annual interest rate of $3.6\%$?
\end{enumerate}
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