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\newtheorem{exr}[thm]{Question}

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\title{Mr. Sorice's Calculus I \\ Summative Assessment for Segment}
\author{Name: \rule{6cm}{0.5pt}}


\begin{document}
\maketitle

\section*{Quiz Directions}
This quiz is a chance to show some of what you've learned about limits and continuity. There are \underline{62} points available.

This quiz has several sections, each of which may have its own directions. Read those carefully before you begin to answer.

Please continue to follow all our usual test expectations, especially those for when you have finished.

Stand-alone calculators are \underline{allowed}.

Good luck!

\section*{True/False -- 1 point each}
Each statement is either True or False. Circle the correct value. There is no penalty for guessing.

    \begin{exr} %1
        A function can have an essential discontinuity on an interval, but still be defined everywhere on that interval.
        \begin{itemize}
            \item True
            \item False
        \end{itemize}
    \end{exr}
    
    \begin{exr} %2
        If a function is defined everywhere on an interval, then it's continuous everywhere on that interval.
        \begin{itemize}
            \item True
            \item False
        \end{itemize}
    \end{exr}
    
    \begin{exr} %3
       It's possible that $\lim \limits_{x \to a} f(x)$ does not exist, even though both $\lim \limits_{x \to a^{-}} f(x)$ and $\lim \limits_{x \to a^{+}} f(x)$ do.
        \begin{itemize}
            \item True
            \item False
        \end{itemize}
    \end{exr}

    \begin{exr} %4
        Suppose $f(x)$ is discontinuous at $x=0.$ Nevertheless, $f$ may still take on a finite limit as $x \to 0$.
        \begin{itemize}
            \item True
            \item False
        \end{itemize}
    \end{exr}

\newpage \section*{Short Answer (Written) -- 4 points each}
Respond to each question concisely in 1-2 complete sentences.

    \begin{exr} %5
        Let
        \[   f(x) = 
             \begin{cases}
               x, & x < 0 \\
               0, & x = 0 \\
               x^3, & x > 0
             \end{cases}.
        \]
        Classify the continuity behavior of $f$ at $x=0$.
    
        \framebox(400,60){}    
    \end{exr}

    \begin{exr} %6
        We know the following about $f(x)$: 
        \begin{itemize}
            \item It has a point discontinuity at $x=2.$
            \item However, $\lim \limits_{x \to 2^{-}} f(x) = \lim \limits_{x \to 2^{+}} f(x).$
        \end{itemize}
        What must be true about $f(2)$ and why?

        \framebox(400,60){}
    \end{exr}

    \begin{exr} %7
        This is a \emph{step function}:
        \begin{center}
            \includegraphics[width=0.35\linewidth]{sa1.png}
        \end{center} 
        It's flat almost everywhere, but it goes up 1 unit each time $x$ goes right 1 unit.
        
        Which class of discontinuity does the step function exhibit at every integer value of $x$ we can see? In terms of limit behavior, how do we know that?    

        \framebox(400,60){}
    \end{exr}

    \begin{exr} %8
        We know this function is defined for $x<1$, but we can't see what it looks like there:
        \begin{center}
            \includegraphics[width=0.35\linewidth]{sa4.png}
        \end{center} 
        What kind of discontinuity or discontinuities can it have at $x=1$? Why?

        \framebox(400,60){}
    \end{exr}

    \begin{exr} %9
        \label{inkblot}
        Oh man, Ned spilled ink all over my graphs!
        \begin{center}
            \includegraphics[width=0.35\linewidth]{sa5.png}
        \end{center} 
        I'm sure the right branch is under that blot, but I don't remember where. Agnes says the function had a removable discontinuity.
        
        What could the right branch do if that's true?

        \framebox(400,60){}
    \end{exr}
    
    \begin{exr} %10
        My friend Mona says she thinks the function from Question \ref{inkblot} was continuous everywhere.
        
        What might you say to settle this disagreement?
                
        \framebox(400,50){}
        
    \end{exr}

\newpage \section*{Short Answer (Graphs) -- 4 points each}
For each question, use the provided graph paper to draw 1 or 2 graphs of functions that exhibit the specified continuity or limit behavior.

Please label vertical asymptotes.

Beware: you will lose credit on a problem if any of your drawings contradicts the given behavior, or is of a non-function.

    \begin{exr} %11
        A function with an essential discontinuity at $x=2$, a jump discontinuity at $x=5$, and that is undefined but takes on a limit at $x=0$.

        \begin{center} \begin{tikzpicture}
            % axis
            \draw[thick, <->] (0, -6.1) -- (0, 6.1);
            \draw[thick, <->] (-6.1, 0) -- (6.1, 0);
            
            % grid
            \draw[help lines, step = 0.5cm] (-6, -6) grid (6, 6);
        \end{tikzpicture} \end{center}
    \end{exr}

    \newpage    

    \begin{exr} %12
        A function with different left- and right-sided limits at $x=-3$, takes on a limit other than its value at $x=0$, and becomes infinite somewhere in its domain.

        \begin{center} \begin{tikzpicture}
            % axis
            \draw[thick, <->] (0, -6.1) -- (0, 6.1);
            \draw[thick, <->] (-6.1, 0) -- (6.1, 0);
            
            % grid
            \draw[help lines, step = 0.5cm] (-6, -6) grid (6, 6);
        \end{tikzpicture} \end{center}
    \end{exr}

\newpage \section*{Multiple Choice -- 2 points each}
Each question has a single correct answer. Circle the letter corresponding to it. There is no penalty for guessing.

%    \begin{exr} %#
%        Stuff.
%        
%        \begin{enumerate}[label=\alph*)]
%            \item 
%            \item 
%            \item 
%            \item 
%            \item 
%        \end{enumerate}
%    \end{exr}

    The next 4 questions are about this function:
    \begin{center}
        \includegraphics[width=0.45\linewidth]{sa7.png}
    \end{center} 

    \begin{exr} %13
        What sort of discontinuity is present at $x=3$?
        \begin{enumerate}[label=\alph*)]
            \item No discontinuity -- it's continuous
            \item Removable discontinuity
            \item Jump discontinuity
            \item Essential discontinuity
            \item We can't say
        \end{enumerate}
    \end{exr}
    
    \begin{exr} %14
        What is the function's behavior at $x=\frac{1}{2}$?
        \begin{enumerate}[label=\alph*)]
            \item There's no discontinuity -- it's continuous
            \item It has a removable discontinuity
            \item It has a jump discontinuity
            \item It has an essential discontinuity
            \item We can't say
        \end{enumerate}
    \end{exr}
    
    \begin{exr} %15
        Classify the function's behavior as $x \to 0^{-}$?
        \begin{enumerate}[label=\alph*)]
            \item There's no discontinuity -- it's continuous
            \item It has a removable discontinuity
            \item It has a jump discontinuity
            \item It has an essential discontinuity
            \item We can't say
        \end{enumerate}
    \end{exr}    
    
    \begin{exr} %16
        How many discontinuities are there? (Assume there are none outside the graph window.)
            \begin{enumerate}[label=\alph*)]
            \item 2
            \item 3
            \item 4
            \item 5
            \item We can't say
        \end{enumerate}
    \end{exr}

In the next 3 questions, consider this function, which depends on the constants $m$ and $b$:
        \[   f(x) = 
             \begin{cases}
               10^x, & x < 0 \\
               0, & x = 0 \\
               mx+b, & x > 0
             \end{cases}.
        \]
        
    \begin{exr} %17
        If $m = 0$ and $b=5$, what sort of discontinuity is there at $x=0$?
        \begin{enumerate}[label=\alph*)]
            \item No discontinuity -- it's continuous
            \item Removable discontinuity
            \item Jump discontinuity
            \item Essential discontinuity
            \item We can't say
        \end{enumerate}
    \end{exr}
    
    \begin{exr} %18
        Which combination of $m$ and $b$ makes $f$ continuous at $x=0$?
        \begin{enumerate}[label=\alph*)]
            \item $m=0, b=0$
            \item $m=0, b=1$
            \item $m=1, b=0$
            \item $m=1, b=10$
            \item None of these.
        \end{enumerate}
    \end{exr}
    
    \begin{exr} %19
        Which combination of $m$ and $b$ creates a function that takes on a defined limit as $x \to 0$?
        \begin{enumerate}[label=\alph*)]
            \item $m=0, b=0$
            \item $m=0, b=1$
            \item $m=1, b=0$
            \item $m=1, b=10$
            \item None of these.
        \end{enumerate}
    \end{exr}

\newpage \section*{Essay -- 12 points}
Georg has an interesting idea. He thinks if he knows the limit behavior of a function at all points in an interval, he'll know everything about the function.

Write an essay of 2-3 paragraphs critiquing Georg's idea. In your discussion, consider \emph{left- and right-handed limits} and \emph{classes of discontinuous functions}. Conclude your essay with a claim about the soundness of Georg's idea. If you conclude that it works as is, share any limitations you see on it; on the other hand, if you conclude that the idea is lacking, discuss ways it might be repaired.

Feel free (and encouraged!) to explain your ideas using various representations of functions, including symbols and graphs, as well as other mathematical language. Incorporate these with your usual academic writing style.

\end{document}          
