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\noindent {\LARGE M}{\large ath} {\LARGE 120} {\LARGE S}{\large pring} {\LARGE 98}
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{\Large Quiz {\bf 7}}\hspace{7cm} & \hspace{1.65in} & Basic~Calculus~2\\
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Name: \makebox[2in]{\hrulefill}  & \hspace{1.5in} & \\
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Date: \makebox[2in]{\hrulefill} & \hspace{1.5in} & {\bf Friday April 3, 1998} \\
Time Begun: \makebox[2in]{\hrulefill} & \hspace{1.5in} & Ron Buckmire \\ 
Time Ended: \makebox[2in]{\hrulefill} & \hspace{1.5in} &  \\
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{\Large {\bf Topic covered:}} Determining Convergence/Divergence of Improper Integrals by Comparison

{\small The idea behind this quiz is to give you more practice determining whether a particular improper integral will converge or diverge without evaluating it, but by comparing it to an improper integral you already know something about.} 

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{\Large {\bf Instructions: }}
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\item[1.] Once you open the quiz, you have 90 minutes to complete it.
\item[2.] You may use the book or any of your class notes, and you may use a calculator.  You must work alone.
\item[3.] If you use your own paper, please staple it to the quiz before coming to class.  If you don't have a stapler, buy one.
\item[4.] After completing the quiz, sign the pledge below stating on your honor that you have adhered to these rules.  
\item[5.] Relax and enjoy...
\item[6.] {\bf This quiz is due on Monday, April 6}, in class.  NO LATE QUIZZES WILL BE ACCEPTED.
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{\bf Pledge:} I, \makebox[2in]{\hrulefill}, pledge my honor as a human being and Occidental student, that I have followed all the rules above to the letter and in spirit.
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Math 120 Spring 1998 \hfill{Quiz {\bf 7}}
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\vspace{-7.5ex}   {\large\bf SHOW ALL YOUR WORK}\\
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Consider the function $f(z) = {\displaystyle \frac{1}{\sqrt{z} + z} = \frac{1}{z^{1/2}+z}}.$ We want to try and determine whether\\ $I= {\displaystyle \int_1^{\infty} \frac{1}{\sqrt{z}+z} dz}$ and $J={\displaystyle \int_0^1 \frac{1}{\sqrt{z}+z} dz}$ converge or diverge.
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\item[(a)] ({\it 4 points}) Use the rules developed in class to say whether each of the following integrals {\bf converges} or {\bf diverges}:\\
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${\displaystyle \int_0^1 \frac{dz}{ 2 \sqrt{z}}}$ & 
${\displaystyle  \int_1^{\infty} \frac{dz}{2 \sqrt{z}}}$ &  
${\displaystyle \int_0^1 \frac{dz}{z}}$ &
${\displaystyle \int_1^{\infty} \frac{dz}{z}}$
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\item[(b)] ({\it 1 point}) For $z > 1$ is 
${\displaystyle \frac{1}{\sqrt{z}+z} > \frac{1}{z} \mbox{ \bf or }\frac{1}{\sqrt{z}+z} < \frac{1}{z}}$? \\
For $z>1$ is ${\displaystyle \frac{1}{\sqrt{z}+z} > \frac{1}{\sqrt{z}} \mbox{ \bf or }\frac{1}{\sqrt{z}+z} < \frac{1}{\sqrt{z}}}$? 
\item[(c)] ({\it 2 points}) Does $I = {\displaystyle \int_1^{\infty} \frac{dz}{\sqrt{z}+z} }$ {\bf converge} or {\bf diverge}? Support your answer by explaining how the inequalities in part (b) and your answers to part (a) allow you to determine whether $I$ converges or diverges without actually evaluating it. If you like, you can use a graph to support your explanation.
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\item[(d)] ({\it 1 point}) For $0 < z < 1$ is 
${\displaystyle \frac{1}{\sqrt{z}+z} > \frac{1}{z} \mbox{ \bf or }\frac{1}{\sqrt{z}+z} < \frac{1}{z}}$? \\
For $0 < z < 1$ is ${\displaystyle \frac{1}{\sqrt{z}+z} > \frac{1}{\sqrt{z} + \sqrt{z}} \mbox{ \bf or }\frac{1}{\sqrt{z}+z} < \frac{1}{\sqrt{z} + \sqrt{z}}}$? 
\item[(e)] ({\it 2 points}) Does $J = {\displaystyle \int_0^{1} \frac{dz}{\sqrt{z}+z} }$ {\bf converge} or {\bf diverge}? Support your answer by explaining how the inequalities in part (d) and your answers to part (a) allow you to determine whether $I$ converges or diverges without actually evaluating it. If you like, you can use a graph to support your explanation.
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