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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Absolute Values and Intervals}
\centerline{August 31, 2017}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Recall from the video ``lecture'' for today that the absolute value of a real number is defined as
$$|x| = \begin{cases}
         x & {\rm if}\ x \ge 0\\
        -x & {\rm if}\ x < 0.
        \end{cases}
$$
Also, we have several different types of {\it intervals} in the real number system written
as follows:
\begin{align}
(a,b) &= \{x \mid a < x < b\}\\
[a,b] &= \{x \mid a \le x \le b\}\\
[a,b) &= \{x \mid a \le x < b\}\\
(a,b] &= \{x \mid a < x \le b\}.
\end{align}
Today we will practice on some problems related to these ideas.
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\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[0)] (Warm-up)  Real numbers can all be expressed by (possibly infinite or non-repeating) decimal
expansions.  But exactly what does a decimal expansion like
$$\pi = 3.1415926\cdots$$
{\it mean}?  What does a finite decimal expansion like $\frac{15}{8} = 1.875$ mean?  (Hint:  It's
really a way of writing the number as a sum of fractions with denominators of a particular form.) 
\item[1)] The set of real numbers $x$ satisfying $|x| < 17$ is an interval.  Write it using the appropriate
interval notation.
\item[2)] The set of real numbers $x$ satisfying $|x - 3| \le 1$ is also an interval.  
Write it using the appropriate
interval notation.
\item[3)] The set of real numbers $x$ satisfying $|3x + 1| < 4$ is also an interval.  Write it using the appropriate
interval notation.  (Hint:  One way is to factor out the 3, divide through, then proceed as in 2).
Another is to proceed algebraically from the equivalent inequalities $-4 < 3x + 1 < 4$.)
\item[4)] Write the interval $(5,9)$ in the form: ``the set of all $x$ such that $|x - c| < r$'' for 
some real numbers $c$ and $r$.
\item[5)] Same as question 4, but for the interval $\left[-\frac{2}{5},3\right]$.
\begin{figure}
\begin{center}
\includegraphics[width=.8\linewidth]{tempgraph2.jpg}
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\caption{Figure for Question 6}
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\item[6)]  Using the graph of temperature versus time over a 24-hour period at a particular
location above, answer these questions:
\begin{enumerate}
\item[(a)] Write the collection of all {\it temperatures} measured over the 24 hour period
as an interval $[a,b]$ for suitable temperature values $a,b$ in degrees Fahrenheit.
(There's an unstated assumption behind this question.  Can you see what that is?  Does it seem reasonable
to make that assumption?)  
\item[(b)] Estimate the interval $[c,d]$ of {\it times} in hours 
for which the temperature was $16.8$ degrees or less.
\item[(c)] At how many different times does it appear that the temperature was exactly 40 degrees F?
\end{enumerate}
Note:  The temperature values at the start of each hour are given in the table at the right.  
\end{enumerate}
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