%&LaTeX \documentclass[11pt]{article} \oddsidemargin=-.1in \evensidemargin=-.1in \textwidth=6.7in \topmargin=-.5in \textheight=9.1in \parindent=0in \pagestyle{empty} \input{laurapoints} \input{lauratex} \input{lauragraphics} \begin{document} % ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo % COVER % ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo \centerline{\huge \bf TEST I} \vf \vf Math 231 October 3, 2002 \hfill {\bf Name: } \underline{\hspace{2.7in}} \vspace{-1.3\baselineskip} \phantom{x} \hfill {\tiny By writing my name I swear by the honor code.} $\quad\;$ \vf \vf \vf {\bf Read all of the following information before starting the exam:} \vs \begin{itemize} \item Show all work, clearly and in order. You will not get full credit if I cannot see how you arrived at your answer (even if your final answer is correct). \item Make sure that you follow the directions in each problem and that your answer matches what is asked for. \item Justify your answers algebraically whenever possible. For most problems, work done by calculator will \und{not} receive any points (although you may use your calculator to check your answers). \item Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. \item This test has 8 problems and is worth 100 points, plus some extra credit at the end. Make sure that you have all of the pages! \item Good luck! \end{itemize} \vf \vf \vf \pg %%%%%% % TF and concepts \bp{10} Circle true (T) or false (F) for each statement below. \vs \np{2} \TF \parbox[t]{5in}{$f(x)=x^3+5$ is an odd function.} \vs \vs \np{2} \TF \parbox[t]{5in}{If $ax^2+bx+c=0$ is a quadratic equation with $b^2-4ac=0$, then the only real solution to $ax^2+bx+c$ is $x=-\frac{b}{2a}$.} \vs \vs \np{2} \TF \parbox[t]{5in}{The statement ``$A \Rightarrow B$'' is false if either $A$ or $B$ is false.} \vs \vs \np{2} \TF \parbox[t]{5in}{In a proof by contradiction, we assume the {\em opposite} of what we wish to show, and then show that this causes something ridiculous to happen.} \vs \vs \np{2} \TF \parbox[t]{5in}{Suppose $P$ is the set of people in this room, and $C$ is the set of possible eye colors. The rule $f \colon P \rightarrow C$ that assigns to each person his or her eye color is a function.} \vs \vs \ep \bp{12} Give short answers to each of the following questions. \vs \np{3} Give a counterexample to the statement ``If $x$ is a rational number, then $x$ is positive,'' and explain why it is a counterexample. \vf \np{3} Write the solution to the inequality $|x+4|<0.1$ in interval notation. \vf \np{3} State the constant multiple rule for limits. \vf \np{3} Explain why you \underline{can't} solve the inequality $\frac{2}{x+1}<7$ by multiplying both sides of the inequality by $x+1$. \vf \ep \pg %%%%%% % graph and table \bp{12} \vs \hspace{.25in} \parbox[t]{3.1in}{Given that $f(x)$ has the graph on the right, sketch the graphs of $f(x+1)-2$, $|f(x)|$, and $f^{-1}(x)$ on the axes provided below. Show where the two marked points and the asymptote go on each graph that you draw. {\em (4 pts each)}} \hspace{.3in} \parbox[t]{2in}{\vspace{-3\baselineskip} \centerline{$f(x)$} \vspace{.5pc} \centerline{ \includegraphics[height=1.7in,width=2.1in]{231test1A}}} \vs \hspace{.55in} $f(x+1)-2$ \hspace{1.65in} $|f(x)|$ \hspace{1.75in} $f^{-1}(x)$ \vspace{.5pc} \includegraphics[height=1.7in,width=2.1in]{231test1B} \hfill \includegraphics[height=1.7in,width=2.1in]{231test1B} \hfill \includegraphics[height=1.7in,width=2.1in]{231test1B} \vf \ep \bp{12} Let $f(x)$ and $g(x)$ be functions defined by the table below. Use the table to fill in the blanks below. \vspace{-.5\baselineskip} \begin{center} \addtolength{\tabcolsep}{1mm} \renewcommand{\arraystretch}{1.2} \begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 0 & 1 & 2 & 3 & 4 \\ \hline \hline $f(x)$ & 1 & 2 & 3 & 0 & 2 \\ \hline $g(x)$ & -1 & 0 & 1 & 2 & 3 \\ \hline \end{tabular} \end{center} \vs \np{2} $(3f-g)(2) = \mblank{.8}$. \vs \vs \np{2} $f(g(4)) = \mblank{.8}$. \vs \vs \np{2} $g^{-1}(3) = \mblank{.8}$. \vs \vs \np{2} If $f(x)$ is an odd function, then $f(-2) = \mblank{.8}$. \vs \vs \np{2} Is $f(x)$ a one-to-one function? (Yes or No) \blank{.8}. \vs \vs \np{2} If $h(x)=2f(x-1)$, then $h(3) = \mblank{.8}$. \vs \vs \ep \pg %%%%%% % piecewise and various calculations \bp{10} Fill in the blanks and sketch a careful graph of $f(x)= \ds \left\{ \begin{array}{rc} x+1, & x<3 \\ 2, & x=3 \\ x^2-9, & x>3 \end{array} \right.$. \vspace{-.5\baselineskip} \hspace{.2in} {\em (2 pts each blank, 4 pts for graph)} \vs \parbox{2.6in}{ \vs \hspace{.5in} $\ds\lim_{x \to 3^-} f(x) = \mblank{.8}$ \vspace{1.8pc} \hspace{.5in} $\ds\lim_{x \to 3^+} f(x) = \mblank{.8}$ \vspace{1.8pc} \hspace{.5in} $\;\;\ds\lim_{x \to 3} f(x) = \mblank{.8}$ \vspace{1.8pc}} % \hspace{.25in} \parbox{3in}{ \centerline{\includegraphics[height=1.7in]{231test1C}}} \vs \ep \bp{18} Show all work for each part below. {\bf Circle your final answers.} \vs \np{6} Find the $y$-intercept of the line through $(2,-1)$ and perpendicular to $y=2x-3$. \vf \np{6} If $f(x)=\ds\frac{x-2}{x}$, find $f^{-1}(x)$. \vf \np{6} Find $\ds\lim_{x \to 2} \,\frac{x-2}{x^2-4}$ algebraically (\ie by hand, \underline{without} a table or a graph). \vf \ep \pg %%%%%% % delta-epsilon and graph props \bp{8} Consider the limit statement $\ds\lim_{x \to 1} (2x+4) = 6$. \vs \np{3} Write this limit using the ``delta-epsilon'' definiton of limit. \vf \np{5} Give a ``delta-epsilon'' proof of this limit. \vf \vf \ep \bp{18} Use the graph of $f(x)$ given below to fill in the blanks. {\bf Use interval notation when you answer parts (d), (e), and (f).} \vs \centerline{\includegraphics[height=1.6in]{231test1D}} \vs \vs \np{3} List the locations of any roots of $f(x)$. \blank{1.5} \vs \vs \np{3} List the locations of any local minima of $f(x)$. \blank{1.5} \vs \vs \np{3} Approximate the locations of any inflection points of $f(x)$. \blank{1.5} \vs \vs \np{3} Where is the graph of $f(x)$ positive? \blank{1.5} \vs \vs \np{3} Where is the graph of $f(x)$ increasing? \blank{1.5} \vs \vs \np{3} Where is the graph of $f(x)$ concave down? \blank{1.5} \vs \ep \pg % ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo % SURVEY & SCRAP % ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo {\large \bf Survey Questions:} (2 extra credit points) \vs Name a question or topic that could have been on this test, but wasn't. \vspace{6pc} How do you think you did? \vspace{6pc} \underline{\hspace{6.5in}} {\large \bf SPACE FOR SCRAP WORK} \vf \showpoints \end{document}