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Get Full Access to Precalculus - 4 Edition - Chapter 6.4 - Problem 6.5.27
Get Full Access to Precalculus - 4 Edition - Chapter 6.4 - Problem 6.5.27

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# Full answer: In Exercises 13 34, test for symmetry and then graph each polar equation

ISBN: 9780321559845 209

## Solution for problem 6.5.27 Chapter 6.4

Precalculus | 4th Edition

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Problem 6.5.27

In Exercises 13 34, test for symmetry and then graph each polar equation.

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Using the exam document class Philip Hirschhorn Department of Mathematics Wellesley College Wellesley, MA 02481 psh@math.mit.edu Copyright 1994, 1997, 2000, 2004, 2008, 2011, 2015 Philip Hirschhorn All rights reserved May 7, 2015 This is the user’s guide for version 2.5 of the exam document class. Contents 1 Introduction 4 1.1 License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. . . . 2 Simple instructions for a basic exam 6 2.1 The \documentclass command . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Asking for the student’s name . . . . . . . . . . . . . . . . . . . . . . . .6. . 2.3 Asking questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . . 2.4 Questions with parts, and subparts, and subsubparts . . . . . . . . . . . . .8 2.5 Leaving space for the answers . . . . . . . . . . . . . . . . . . . . . . . .9. . 2.6 Headers and footers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . 3 The \documentclass command 11 3.1 The option addpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 . 3.2 The option answers . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 . . 4 Questions and points 13 4.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. . . 4.2 Questions with parts and subparts (and subsubparts) . . . . . . . . . . . . 14 4.3 Point values for the questions . . . . . . . . . . . . . . . . . . . . . . .17 . . 4.3.1 Where the points will be printed . . . . . . . . . . . . . . . . . . 18. 4.3.2 Printing the points at the end of the question . . . . . . . . . . . 19. 1 4.3.3 Half points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 4.3.4 Surrounding the points: Parentheses, brackets, or a box . . . . . . . . 21 4.3.5 Using a substitute for the word \points" . . . . . . . . . . . . . . . . 23 4.3.6 Using \marginpointname and enlarging the margins . . . . . . . . . 25 4.3.7 Completely customizing the points . . . . . . . . . . . . . . . . . . . 25 4.3.8 Questions that begin with a parts environment . . . . . . . . . . . . . 27 4.3.9 Adding up the points for a question . . . . . . . . . . . . . . . . . . . 28 4.3.10 Margin sizes when using \pointsinmargin and \pointsinrightmargin 30 4.4 Custom question number lines: The \qformat command . . . . . . . . . . . 31 4.4.1 Adding up the points for a question . . . . . . . . . . . . . . . . . . . 32 4.5 Titled questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 4.5.1 \titledquestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5.2 \thequestiontitle . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.6 Bonus points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34. 4.7 Counting the questions and adding up the points . . . . . . . . . . . . . . .36 4.8 Referring to speci▯c questions by number (cross references) . . . . . . . . . . 37 4.9 Customizing the numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .38 4.10 Customizing the list parameters . . . . . . . . . . . . . . . . . . . . . . . . . 40 5 Multiple choice and ▯ll in the blank questions 41 5.1 The choices environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 The oneparchoices environment . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 The checkboxes environment . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.4 The oneparcheckboxes environment . . . . . . . . . . . . . . . . . . . . . . 43 5.5 Solutions to multiple choice questions . . . . . . . . . . . . . . . . . . . . . 44 5.6 Fill in the blank questions . . . . . . . . . . . . . . . . . . . . . . . . . . .44 5.6.1 True/False questions . . . . . . . . . . . . . . . . . . . . . . . . . . .46 5.6.2 Customizing the \fillin command . . . . . . . . . . . . . . . . . . . 46 5.7 Customizing the multiple choice environments . . . . . . . . . . . . . . . . . 47 6 Escaping the indentation: \uplevel and \fullwidth 49 6.1 Including instructions for a group of questions . . . . . . . . . . . . . . . . .49 6.2 Naming the parts of a long exam . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2.1 Using \uplevel and \fullwidth . . . . . . . . . . . . . . . . . . . . 53 6.2.2 Using the standard sectioning commands . . . . . . . . . . . . . . . . 54 7 Leaving space for the answers 54 7.1 Leaving blank space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55. 7.2 Printing an empty box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 7.3 Printing lined space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. 7.4 Printing dotted lined space . . . . . . . . . . . . . . . . . . . . . . . . . . .59 7.5 Printing space ▯lled with a grid . . . . . . . . . . . . . . . . . . . . . . . . 61 7.5.1 Covering every page with a grid . . . . . . . . . . . . . . . . . . . . . 63 7.5.2 Printing graph paper . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Page 2 7.6 Short answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 7.7 Printing the answers to short answer questions . . . . . . . . . . . . . . . . .66 8 Solutions 67 8.1 Solution environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.2 The appearance of the solution . . . . . . . . . . . . . . . . . . . . . . . .68 8.2.1 Printing the solution in a box . . . . . . . . . . . . . . . . . . . . . 68 8.2.2 Printing the solution on a shaded background . . . . . . . . . . . . . 70 8.2.3 Printing the solution with no framing or shading . . . . . . . . . . . . 71 8.3 Customizing the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 8.3.1 The title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 . 8.3.2 Solution emphasis . . . . . . . . . . . . . . . . . . . . . . . . . . . .72 8.3.3 Advanced customizations . . . . . . . . . . . . . . . . . . . . . . . . .72 8.4 Leaving space for the answers when solutions aren’t printed . . . . . . . . . 73 8.4.1 Cancelling the space from the optional arguments . . . . . . . . . . . 74 8.5 The solutionbox environment . . . . . . . . . . . . . . . . . . . . . . . . .75 8.6 Changes depending on whether or not solutions are being printed . . . . . . 76 8.6.1 Printing only selected solutions . . . . . . . . . . . . . . . . . . . . 77 9 Grading tables and point tables 77 9.1 Grading tables indexed by question number or by page number . . . . . . . 78 9.2 Point tables indexed by question number or by page number . . . . . . . . . 81 9.3 Table entries as clickable links . . . . . . . . . . . . . . . . . . . . . . . .83 . 9.4 Grading ranges, partial grading tables, and partial point tables . . . . . . . .84 9.5 \pointsofquestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.6 \pointsonpage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.7 Bonus grading tables and bonus point tables . . . . . . . . . . . . . . . . . . 86 9.8 Combined grading tables and combined point tables . . . . . . . . . . . . . . 87 9.9 Changing the total points in a table . . . . . . . . . . . . . . . . . . . . . .88 9.10 Customizing the tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10 Changing the page margins 92 11 Headers and footers 93 11.1 Page styles: Headers and/or footers . . . . . . . . . . . . . . . . . . . . . . .93 11.2 The three parts of the header . . . . . . . . . . . . . . . . . . . . . . . . . .94 11.2.1 Using \header, \firstpageheader and \runningheader . . . . . . . 95 11.2.2 Using \lhead, \chead and \rhead . . . . . . . . . . . . . . . . . . 96 11.3 The three parts of the footer . . . . . . . . . . . . . . . . . . . . . . . . . .97 11.3.1 Using \footer, \firstpagefooter and \runningfooter . . . . . . . 97 11.3.2 Using \lfoot, \cfoot, and \rfoot . . . . . . . . . . . . . . . . . . . 97 11.4 Leaving extra room for multiple line headers and footers . . . . . . . . . . . 98 11.5 Horizontal rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99. 11.6 Listing the number of pages in the exam . . . . . . . . . . . . . . . . . . . . 101 Page 3 11.7 Treating the last page di▯erently . . . . . . . . . . . . . . . . . . . . . . . . 101 11.8 Treating odd and even numbered pages di▯erently . . . . . . . . . . . . . . . 102 11.9 Questions that span multiple pages . . . . . . . . . . . . . . . . . . . . . . . 103 11.9.1 Questions continuing from the previous page . . . . . . . . . . . . . . 103 11.9.2 Questions that continue onto a later page . . . . . . . . . . . . . . . . 105 11.10Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 11.10.1Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.10.2Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 11.10.3Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 11.10.4Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 11.10.5Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 11.10.6Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 11.10.7Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 11.10.8Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 12 Cover pages 116 12.1 Headers and footers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 12.2 Listing the number of cover pages . . . . . . . . . . . . . . . . . . . . . . . . 117 Index 119 1 Introduction The ▯le exam.cls provides the exam document class, which attempts to make it easy for even a LT X novice to prepare exams. Speci▯cally, exam.cls sets the page layout so that E there are one inch margins all around (no matter what size paper you’re using) and provides commands that make it easy to format questions, create exible headers and footers, change the margins, and create grading tables. In more detail: ▯ The class will automatically format and number the questions, parts of questions, subparts of parts, and subsubparts of subparts (see sections 4.1 and 4.2). ▯ You can include the point value of each question (or part, or subpart, or subsubpart), with your choice of having the point values printed at the beginning of the text of the question, opposite that in the left margin, opposite that in the right margin, or in the right margin opposite the end of the question (see section 4.3). ▯ The class will add up the total points for each question (and all of its parts, subparts, and subsubparts) and the total points on each page, and make those totals available in macros (see sections 4.3.9, 9.5 and 9.6). ▯ You can have the class print a grading table, indexed either by question number or by page number (see section 9.1). Page 4 ▯ You specify the header in three parts: One part to be left justi▯ed, one part to be centered, and one part to be right justi▯ed, and one or all of these can be omitted (see section 11.2). ▯ The footer is also speci▯ed in three parts: Left justi▯ed, centered, and right justi▯ed (see section 11.3). ▯ The header and footer for the ▯rst page can be di▯erent from the ones used on other pages (see sections 11.2.1, 11.2.2, 11.3.1, and 11.3.2). ▯ Both headers and footers can contain more than one line. To accommodate headers and footers with several lines, simple commands are provided to enlarge the part of the page devoted to the header and/or footer, and these commands can give one amount of space on the ▯rst page and a di▯erent amount of space on all other pages (see section 11.4). ▯ Macros are de▯ned to enable you to state the total number of pages in the exam (see section 11.6) and to change the header and/or footer that appears on the last page of the exam (see section 11.7). ▯ Macros are de▯ned so that the headers and footers can vary depending on whether the current page begins a new question or continues a question that started on an earlier page (and, if one continues onto the current page, to say what the number of that question is). Macros are also de▯ned so that the headers and footers can vary depending on whether a question is complete on the current page or continues on to the next page (and, if one continues, to say what the number of that question is) (see section 11.9). ▯ You can have a horizontal rule at the base of the header and/or at the top of the footer (see section 11.5). ▯ The exam can begin with one or more cover pages, which are numbered separately from the main pages of the exam and which can have headers and footers di▯erent from the ones in the main pages of the exam (see section 12). ▯ You can include solutions in your LT X ▯le and have these solutions either printed E or ignored (or replaced automatically by space in which the students can write their answers) depending on a single command (see section 8, section 5.5, and section 7.7). The latest version of exam.cls (possibly a beta test version) should always be available at http://www-math.mit.edu/~psh/. 1.1 License ▯ This work may be distributed and/or modi▯ed under the conditions of the LT X A E Project Public License, either version 1.3 of this license or (at your option) any later ver- sion. The latest version of this license is in http://www.latex-project.org/lppl.txt and version 1.3 or later is part of all distributionsEof LT X version 2003/12/01 or later. Page 5 ▯ This work has the LPPL maintenance status \author-maintained". ▯ This work consists of the ▯les exam.cls and examdoc.tex. 2 Simple instructions for a basic exam Producing a basic exam with the exam document class is fairly easy. After you learn to do that, you can read the parts of this user’s guide that describe how to customize whatever you want to change. 2.1 The \documentclass command To use the exam document class, your \documentclass command should be \documentclass[addpoints]{exam} or, if you want to use 12 point type, \documentclass[addpoints,12pt]{exam} There are other options; for the full story, see section 3. 2.2 Asking for the student’s name If you’ll be leaving space for the answers on the question pages (see sections 2.5 and 7), then you’ll probably also want to leave space for the student’s name. If you type \begin{center} \fbox{\fbox{\parbox{5.5in}{\centering Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page.}}} \end{center} \vspace{0.1in} \makebox[\textwidth]{Name and section:\enspace\hrulefill} \vspace{0.2in} \makebox[\textwidth]{Instructor’s name:\enspace\hrulefill} Page 6 after the \begin{document} command and before the \begin{questions} command (see section 4.1), then you’ll get Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Name and section: Instructor’s name: 2.3 Asking questions Once you’ve typed \begin{document} and (if the students will be writing their answers on the question pages) asked for the student’s name (see section 2.2), you can list the questions on the exam in a questions environment. You begin each question with a \question command, which is optionally followed by a number of points inside of square brackets. For example, if you type \begin{questions} \question[10] Why is there air \question[15] How much wood would a woodchuck chuck if a woodchuck could chuck wood \question[10] Compute $\displaystyle\int_0^1 x^2 \, dx$. \end{questions} then you’ll get 1. (10 points) Why is there air 2. (15 points) How much wood would a woodchuck chuck if a woodchuck could chuck wood Z 1 3. (10 points) Compute x dx. 0 As the above example illustrates, you can leave blank lines between the \question com- mand and the actual beginning of the question, or before the ▯rst \question command in the environment, and they will be ignored. The point values are entirely optional; if you omit the point values, you should also omit the square brackets containing them. Page 7 If you give the command \marksnotpoints, then the words \point" and \points" will be replaced by the words \mark" and \marks". There are many other customizations possible for the points, including using a di▯erent language, changing the way the points are printed, and by having them printed in one of the margins. For the full story, see section 4.3. 2.4 Questions with parts, and subparts, and subsubparts If you want to create a question with parts, you use a parts environment and begin each part with the command \part. The parts environment can be at the beginning of a question or after some text that begins the question. (There are also subparts and subsubparts environments; for the full details, see section 4.2.) For example, if you type \begin{questions} \question[10] Why is there air \question What if there were no air \begin{parts} \part[5] Describe the effect on the balloon industry. \part[5] Describe the effect on the aircraft industry. \end{parts} \question[20] \begin{parts} \part Define the universe. Give three examples. \part If the universe were to end, how would you know \end{parts} \end{questions} then you’ll get 1. (10 points) Why is there air 2. What if there were no air Page 8 (a) (5 points) Describe the e▯ect on the balloon industry. (b) (5 points) Describe the e▯ect on the aircraft industry. 3. (20 points) (a) De▯ne the universe. Give three examples. (b) If the universe were to end, how would you know The above example illustrates several things: ▯ Parts of a question should be put into a parts environment. ▯ If a question begins with a parts environment, then the ▯rst part will appear on the same line with the question number. ▯ You can leave blank lines before and after the \part command, and they will be ignored. ▯ You can assign points either to the question or to the parts of a question. 2.5 Leaving space for the answers To leave blank space after a question you use the \vspace command, and you end a page with the \newpage command. For example, if you wanted to leave 3 inches of space after the ▯rst question and 2 inches of space after each of the parts of the second question, your questions environment would contain the following: \question[10] Why is there air \vspace{3in} \question What if there were no air \begin{parts} \part[5] Describe the effect on the balloon industry. \vspace{2in} \part[5] Describe the effect on the aircraft industry. \vspace{2in} \end{parts} \newpage If you want to equally distribute the space on a page among several questions (or parts, or subparts, or subsubparts), you should type Page 9 \vspace{\stretch{1}} after each of the questions (or parts, etc.) and end the page with \newpage. If you’d like one of those questions to receive twice as much space as each of the others, then follow that one question with \vspace{\stretch{2}} and follow each of the others with \vspace{\stretch{1}}. (You can use any decimal number for the \stretch value. For example, \vspace{\stretch{1.5}} will allocate 1.5 times the amount of space allocated by \vspace{\stretch{1}}.) In addition to leaving blank space, it’s also possible to leave lined space, dotted lined space, or an empty box. For the full story, see section 7. 2.6 Headers and footers To produce headers and footers, you give the command \pagestyle{headandfoot} in the preamble (i.e., after the \documentclass command and before the \begin{document} command), followed by the commands that say what will be in the headers and footers. Headers and footers each have three parts: One part on the left, one centered part, and one part on the right, and any of these can be empty. The headers and footers are determined by the commands \firstpageheader for the header on the ▯rst page, \firstpagefooter for the footer on the ▯rst page, \firstpageheadrule to put a horizontal rule in the head of the ▯rst page, \firstpagefootrule to put a horizontal rule in the foot of the ▯rst page, \runningheader for the header after the ▯rst page, \runningfooter for the footer after the ▯rst page \runningheadrule to put a horizontal rule in the head after the ▯rst page, and \runningfootrule to put a horizontal rule in the foot after the ▯rst page. For example, to have the header Math 115 First Exam July 4, 1776 on the ▯rst page, the header Math 115 First Exam, Page 2 of 5 July 4, 1776 on all pages after the ▯rst, and no footer on any page, give the commands \pagestyle{headandfoot} \runningheadrule \firstpageheader{Math 115}{First Exam}{July 4, 1776} \runningheader{Math 115} {First Exam, Page \thepage\ of \numpages} {July 4, 1776} \firstpagefooter{}{}{} \runningfooter{}{}{} Page 10 For another example, to have the header Math 115 First Exam July 4, 1776 on the ▯rst page, no header on the pages after the ▯rst, no footer on the ▯rst page, and the footer Math 115 First Exam Page 2 of 5 on all pages after the ▯rst, give the commands \pagestyle{headandfoot} \firstpageheader{Math 115}{First Exam}{July 4, 1776} \runningheader{}{}{} \firstpagefooter{}{}{} \runningfooter{Math 115}{First Exam}{Page \thepage\ of \numpages} \runningfootrule Sections 11.10.1 through 11.10.8 contain many other examples of headers and footers, and the full explanation of the commands for headers and footers is in sections 11.1{11.9. 3 The \documentclass command To use the exam document class, you should specify exam as the required argument to your \documentclass command, as in \documentclass{exam} or, e.g., if you want to use 12 point type, \documentclass[12pt]{exam} There are two possible optional arguments that are speci▯c to the exam document class: addpoints and solutions. Most documents that use the exam document class should use the option addpoints (see section 3.1); the option answers is used when you want to print solutions to the questions (see section 3.2). 3.1 The option addpoints The \documentclass option addpoints, used as in \documentclass[addpoints]{exam} or \documentclass[12pt,addpoints]{exam} Page 11 enables the commands that add up the points (see sections 4.3.9, 4.7, 9.5, and 9.6) and that produce grading tables and point tables (see sections 9.1 and 9.2). This is an option, rather than the default, mainly for the sake of maintaining backward compatibility; most documents that use the exam document class should use this option. The only reason not to use the addpoints option is that this option creates an error if you include any words in an argument that assigns points to a question (see section 4.3). If addpoints is not in e▯ect you can put whatever you want in the argument for the points, but if addpoints is in e▯ect then the argument that contains the points must contain only digits and the command \half (see section 4.3.3). You can also turn addpoints on and o▯ wherever you like in the exam by using the commands \addpoints \noaddpoints Using the document class option addpoints is equivalent to giving the command \addpoints at the beginning of the exam. 3.2 The option answers If you will be typing solutions into the exam (see sections 8, 5.5, and 7.7) and you’d like the solutions to be printed on the next run oE LT X, you should include the document class option answers, as in \documentclass[answers]{exam} or \documentclass[answers,12pt]{exam} or \documentclass[12pt,addpoints,answers]{exam} Using the document class option answers is equivalent to giving the command \printanswers at the beginning of the exam (see section 8). The e▯ect of this is that ▯ the contents of the environments solution, solutionorbox, solutionorlines, and solutionordottedlines (see section 8) will be printed on the next run Ef LT X, Page 12 ▯ any choices of a choices, oneparchoices, checkboxes, or oneparcheckboxes en- vironment (see section 5) that are created using a \CorrectChoice command (see section 5.5) instead of a \choice command will be printed with emphasis as deter- mined by the argument of the most recent \CorrectChoiceEmphasis command (which by default is \bfseries), and ▯ any \answerline commands (see section 7.6) that include an optional argument con- taining the answer (see section 7.7) will have the answer printed on the answer line. 4 Questions and points To create questions, you use a questions environment (see section 4.1); each question is begun with a \question command, which takes an optional argument to assign some number of points to the question (see section 4.3). The questions are numbered automatically. Point values can include half points (see section 4.3.3). If you want a question to have several parts, you use a parts environment (see sec- tion 4.2); each part is begun with a \part command, which takes an optional argument to assign some number of points to the part (see section 4.3). The parts are numbered automatically. Similarly, parts can have subparts, and subparts can have subsubparts (see section 4.2). The points for a question (or part, or subpart, or subsubpart) appear by default at the beginning of the question (or part, etc.), but there are commands to have the points appear instead in either the left margin or in the right margin (see section 4.3.1 and sec- tion 4.3.2). The formatting of the points can also be customized (see sections 4.3.4, 4.3.5, 4.3.6, and 4.3.7). It’s also possible to have questions (or parts, etc.) that are worth \bonus points", which are added up separately from the non-bonus points (see section 4.6). 4.1 Questions To type the questions on the exam you use the questions environment. Each question is then begun with the command \question, and the questions are numbered automatically. For example, if you type \begin{questions} \question Why is there air \question How much wood would a woodchuck chuck if a woodchuck could chuck wood \question Compute $\displaystyle\int_0^1 x^2 \, dx$. Page 13 \end{questions} then you’ll get 1. Why is there air 2. How much wood would a woodchuck chuck if a woodchuck could chuck wood Z 1 3. Compute x dx. 0 As the above example illustrates, you can leave blank lines between the \question com- mand and the actual beginning of the question, or before the ▯rst \question command in the environment, and they will be ignored. 4.2 Questions with parts and subparts (and subsubparts) If you want a question to have several parts, then you use the parts environment. For example, if you type \begin{questions} \question Why is there air \question What if there were no air \begin{parts} \part Describe the effect on the balloon industry. \part Describe the effect on the aircraft industry. \end{parts} \question \begin{parts} \part Define the universe. Give three examples. \part If the universe were to end, how would you know \end{parts} Page 14 \end{questions} then you’ll get 1. Why is there air 2. What if there were no air (a) Describe the e▯ect on the balloon industry. (b) Describe the e▯ect on the aircraft industry. 3. (a) De▯ne the universe. Give three examples. (b) If the universe were to end, how would you know The above example illustrates several things: ▯ Parts of a question should be put into a parts environment. ▯ If a question begins with a parts environment, then the ▯rst part will appear on the same line with the question number. ▯ You can leave blank lines before and after the \part command, and they will be ignored. There is also a subparts environment, and even a subsubparts environment, and they work just as you would expect. For example, if you type \begin{questions} \question \begin{parts} \part What do you do with a drunken sailor \part Is your answer different if it is before noon \end{parts} \question This is the second question. \begin{parts} \part This is a part. Page 15 \part This is also a part. \begin{subparts} \subpart This is a subpart. \subpart This is a periscope. \subpart This is a pair of diving planes. \subpart \begin{subsubparts} \subsubpart This is a subsubpart. \subsubpart The lower surface of a diving plane \subsubpart The ocean floor, perhaps \end{subsubparts} \end{subparts} \part It’s sad to be apart. \end{parts} \question \begin{parts} \part \begin{subparts} \subpart This is a subpart. \subpart This is another subpart. \end{subparts} \part Page 16 This is another part. \end{parts} \end{questions} then you’ll get 1. (a) What do you do with a drunken sailor (b) Is your answer di▯erent if it is before noon 2. This is the second question. (a) This is a part. (b) This is also a part. i. This is a subpart. ii. This is a periscope. iii. This is a pair of diving planes. iv. ▯) This is a subsubpart. ▯) The lower surface of a diving plane ) The ocean oor, perhaps (c) It’s sad to be apart. 3. (a) i. This is a subpart. ii. This is another subpart. (b) This is another part. The numbers and letters used to number questions, parts, subparts, and subsubparts can be customized; see section 4.9. 4.3 Point values for the questions Each of the commands \question, \part, \subpart, and \subsubpart takes an optional argument, which is the number of points for that question, part, subpart, or subsubpart. The default setup is that ▯ the point value is printed at the beginning of the question (or part, or subpart, or subsubpart) (this can be changed to printing in either the left or right margins on the ▯rst line of the question or in the right margin at the end of the question; see section 4.3.1 and section 4.3.2), and ▯ the point value is enclosed in parentheses (this can be changed to either square brackets or a box; see section 4.3.4). Both of these things can be further customized; see section 4.4 and section 4.3.7. Page 17 4.3.1 Where the points will be printed The default is that the point value will be inserted at the beginning of the question (or part, or subpart, or subsubpart) in parentheses, but ▯ the command \pointsinmargin will cause the point values to be set in the left margin, ▯ the command \pointsinrightmargin will cause the point values to be set in the right margin, and ▯ the commands \nopointsinmargin and \nopointsinrightmargin are equivalent, and either of them will revert to the default situation. All of these print the point values on the ▯rst line of the question (or part, or subpart, or subsubpart). There is also a way to print the point values on the last line of the question (or part, etc.); for this, see section 4.3.2. For example, if you type \begin{questions} \question[20] Why is there air \question What if there were no air \begin{parts} \part[10] Describe the effect on the balloon industry. \part[10] Describe the effect on the aircraft industry. \end{parts} \end{questions} then, with the default setup, you’ll get 1. (20 points) Why is there air 2. What if there were no air (a) (10 points) Describe the e▯ect on the balloon industry. (b) (10 points) Describe the e▯ect on the aircraft industry. (To change the word \points", see the commands described in section 4.3.5.) If you give the command \pointsinmargin, then the above input will produce instead Page 18 (20) 1. Why is there air 2. What if there were no air (10) (a) Describe the e▯ect on the balloon industry. (10) (b) Describe the e▯ect on the aircraft industry. (To have a word (e.g., \points", or \%") inside the parentheses with the point values, see the \marginpointname command in section 4.3.5.) If you give the command \pointsinrightmargin, then the above input will produce instead 1. Why is there air (20) 2. What if there were no air (a) Describe the e▯ect on the balloon industry. (10) (b) Describe the e▯ect on the aircraft industry. (10) 4.3.2 Printing the points at the end of the question Instead of having the point value of a question (or part, or subpart, or subsubpart) printed on the ▯rst line of a question, you can have it printed in the right margin opposite the last line, or even opposite a blank line following the paragraph. This is done with the \pointsdroppedatright and \droppoints commands. If you give the command \pointsdroppedatright then point values are not printed until you give the command \droppoints (and they’re not printed at all if you don’t give the command \droppoints). The only exception to this rule is that if you’ve given a \qformat command (see section 4.4), then question number lines are printed as speci▯ed by the argument to the \qformat command even if you have given the command \pointsdroppedatright The command \droppoints should be given only at the end of a paragraph or between paragraphs; if you give it within a paragraph, it causes the paragraph to end. \droppoints prints the point value in the right margin, formatted as it is when you give the command \pointsinrightmargin, except that the points appear opposite the last line of the paragraph (or, if the command \droppoints is given between paragraphs, then additional vertical space is left between the paragraphs and the points are printed opposite the blank space). The formatting can be changed by giving the commands \bracketedpoints, \boxedpoints, or \marginpointname (see section 4.3.5) just as if \pointsinrightmargin were in e▯ect. The command \droppoints actually works this way even if one of the commands Page 19 \nopointsinmargin, \pointsinmargin, or \pointsinrightmargin is in e▯ect, but if you use it that way the points will appear twice on the page, which is probably not what you want. For example, if you give the command \pointsdroppedatright and then type \begin{questions} \question[10] Describe the implications of the Michelson-Morley ink drop experiment for the use of punctuation marks such as colons and semicolons that require multiple drops of ink. \droppoints \question[10] Prove that the second dual of a finite dimensional real vector space is naturally isomorphic to the given vector space, except when the weapon chosen is a single shot pistol.\droppoints \end{questions} then you’ll get 1. Describe the implications of the Michelson-Morley ink drop experiment for the use of punctuation marks such as colons and semicolons that require multiple drops of ink. (10) 2. Prove that the second dual of a ▯nite dimensional real vector space is naturally isomor- phic to the given vector space, except when the weapon chosen is a single shot pisto(10) and if you type \begin{questions} \question[10] Describe the implications of the Michelson-Morley ink drop experiment for the use of punctuation marks such as colons and semicolons that require multiple drops of ink. \droppoints \question[10] Prove that the second dual of a finite dimensional real vector space is naturally isomorphic to the given vector space, except when the weapon chosen is a single shot pistol. \droppoints \end{questions} then you’ll get Page 20 1. Describe the implications of the Michelson-Morley ink drop experiment for the use of punctuation marks such as colons and semicolons that require multiple drops of ink. (10) 2. Prove that the second dual of a ▯nite dimensional real vector space is naturally isomor- phic to the given vector space, except when the weapon chosen is a single shot pistol. (10) 4.3.3 Half points The point values of questions, parts, subparts, and subsubparts can include half points. To specify half points, you either type \half immediately following the integer part of the point value or just type \half as the entire point value. That is, the valid point values and their (default) printed appearances are: Typed: 0 \half 1 1\half 2 2\half ▯▯▯ Printed: 0 12 1 1=2 2 2=2 ▯▯▯ 1 1 If you prefer, you can cause the fraction =2to be printed as 2. To do this, you give the command \usehorizontalhalf after which the printed appearance of point values will be Typed: 0 \half 1 1\half 2 2\half ▯▯▯ 1 1 1 Printed: 0 2 1 1 2 2 22 ▯▯▯ If you want to return to using the default appearance, you can do that by giving the command \useslantedhalf 4.3.4 Surrounding the points: Parentheses, brackets, or a box If you prefer having the points enclosed in brackets instead of in parentheses, give the com- mand \bracketedpoints For example, if you give the command \bracketedpoints, then the questions typed above will produce 1. [20 points] Why is there air Page 21 2. What if there were no air (a) [10 points] Describe the e▯ect on the balloon industry. (b) [10 points] Describe the e▯ect on the aircraft industry. If you prefer having the points enclosed in a box instead of in parentheses, give the command \boxedpoints For example, if you give the command \boxedpoints, then the questions typed above will produce 1. 20 points Why is there air 2. What if there were no air (a) 10 points Describe the e▯ect on the balloon industry. (b) 10 points Describe the e▯ect on the aircraft industry. If you give the commands \boxedpoints and \pointsinmargin, then the above questions will produce 20 1. Why is there air 2. What if there were no air 10 (a) Describe the e▯ect on the balloon industry. 10 (b) Describe the e▯ect on the aircraft industry. If you give the commands \bracketedpoints and \pointsinrightmargin, then the above questions will produce 1. Why is there air [20] 2. What if there were no air (a) Describe the e▯ect on the balloon industry. [10] (b) Describe the e▯ect on the aircraft industry. [10] Other combinations of these commands will produce similar e▯ects. If you want to switch back and forth between formats during the exam, you can do so by giving one of the commands \boxedpoints \bracketedpoints \noboxedpoints \nobracketedpoints whenever you want to switch. (The commands \nobracketedpoints and \noboxedpoints are equivalent; they both return to the default of putting parentheses around the points.) If you want some text (e.g., \ points", or \%") automatically inserted along with the point values when using either \pointsinmargin or \pointsinrightmargin, see the command \marginpointname in section 4.3.5. Page 22 4.3.5 Using a substitute for the word \points" With the default setup, the number of points is inserted before the text of the question 1 followed by a single space and the word \point" if the number of points is either = or2 1 and by the word \points" otherwise. That is, the default is \nopointsinmargin and \noboxedpoints (see sections 4.3.1 and 4.3.4), and if you type \begin{questions} \question[1] This is a short question. \question[5] This is a longer question, requiring more thought. \end{questions} then you’ll get 1. (1 point) This is a short question. 2. (5 points) This is a longer question, requiring more thought. The way this is achieved is: ▯ There is a command \points whose default de▯nition expands to the word ‘point’ if the number of points is either 1 =2or 1 and to ‘points’ otherwise. (This behavior can be modi▯ed by the command \pointpoints; see below.) ▯ There is a command \pointname{Text} that causes ‘Text’ to be inserted following the number of points of a question, part, subpart, or subsubpart. The default setup is the result of the command \pointname{ \points} (Note the single space before the command \points.) There is also a command \pointpoints{SingularText}{PluralText} that changes the behavior of the \points command so that it expands to ‘SingularText’ if the number of points is = or 1 and to ‘PluralText’ otherwise. (The default is the result of 2 the command \pointpoints{point}{points}.) Thus, if you give the command Page 23 \pointpoints{Punkt}{Punkte} then typing the questions above would result in 1. (1 Punkt) This is a short question. 2. (5 Punkte) This is a longer question, requiring more thought. There is also a command \marksnotpoints that is equivalent to the two commands \pointpoints{mark}{marks} \bonuspointpoints{mark (bonus)}{marks (bonus)} (the command \bonuspointpoints is described in section 4.6). That command would cause the above questions to be printed as 1. (1 mark) This is a short question. 2. (5 marks) This is a longer question, requiring more thought. For another example, if you give the command \pointname{\%} then the above questions would be printed as 1. (1%) This is a short question. 2. (5%) This is a longer question, requiring more thought. Using \marginpointname Similarly, the command \marginpointname can be used to a▯ect the text set with the num- ber of points when \pointsinmargin or \pointsinrightmargin is in e▯ect. For example, if you give the commands \pointsinmargin \marginpointname{\%} and then type \question[25] Where, oh where, has my little dog gone then you’ll get (25%) 1. Where, oh where, has my little dog gone Page 24 unless, e.g., you’ve also given the command \boxedpoints (see section 4.3.4), in which case you’ll get 25% 1. Where, oh where, has my little dog gone If you give the commands \marksnotpoints \marginpointname{ \points} then the above will print (25 marks) 1. Where, oh where, has my little dog gone Using the default is equivalent to giving the command \marginpointname{} 4.3.6 Using \marginpointname and enlarging the margins The default margins are not very large, and so if you use \marginpointname (see sec- tion 4.3.5) to place any words along with the point values in the margin, you may run out of room. The solution to this problem is to enlarge the margins by using the \extrawidth command with a negative argument to decrease the width of the text area, and thus increase the width of the margins. For example, the commands \extrawidth{-1in} \marginpointname{ \points} will increase each of the left and right margins by one half inch, which will easily leave room for the word \points". (See section 10 for a discussion of the \extrawidth command.) 4.3.7 Completely customizing the points If the customizations described in section 4.3.4 and section 4.3.5 (using \bracketedpoints, \boxedpoints, \pointpoints, \pointname, and \marginpointname) aren’t su▯cient, and you don’t want to use the \qformat command to create a customized question number line (see section 4.4), you can completely customize the way the points are printed by using the commands \pointformat \bonuspointformat The \pointformat command controls the printing of \regular" points (i.e., non-bonus points) and the \bonuspointformat command controls the printing of bonus points (see section 4.6). Each of the commands \pointformat and \bonuspointformat takes one argument, and that argument is what actually appears on the page whether the points are being printed at the beginning of the question (the default), in the left margin (because of \pointsinmargin), in the right margin (because of \pointsinrightmargin), or dropped in the right mar- gin at the end of the question (because of \pointsdroppedatright). The arguments to \pointformat and \bonuspointformat should contain either the command Page 25 \thepoints (which for non-bonus points prints the number of points followed by the argument to the last \pointname command and for bonus points prints the number of bonus points followed by the argument to the last \bonuspointname command; see section 4.3.5) or the command \themarginpoints (which for non-bonus points prints the number of points followed by the argument to the last \marginpointname command and for bonus points prints the number of bonus points followed by the argument to the last \marginbonuspointname command; see section 4.3.5). If the argument doesn’t contain either \thepoints or \themarginpoints, then the number of points will not appear. For example, the command \bracketedpoints is equivalent to either \pointformat{[\thepoints]} \bonuspointformat{[\thepoints]} if the points are being printed at the beginning of the question, or \pointformat{[\themarginpoints]} \bonuspointformat{[\themarginpoints]} if the points are being printed in one of the margins. For another example, the command \boxedpoints is equivalent to either \pointformat{\fbox{\thepoints}} \bonuspointformat{\fbox{\thepoints}} if the points are being printed at the beginning of the question, or \pointformat{\fbox{\themarginpoints}} \bonuspointformat{\fbox{\themarginpoints}} if the points are being printed in one of the margins. For another example, the commands \begin{questions} \pointsinmargin \marginpointname{\%} \pointformat{\slshape (\themarginpoints)} \question[3] What is the point \end{questions} will produce (3%) 1. What is the point For another example, the commands Page 26 \begin{questions} \pointsinrightmargin \marginpointname{\%} \pointformat{\fbox{\bfseries\boldmath\themarginpoints}} \question[3\half] What is the point \end{questions} will produce 1. What is the point 3=% 2 (Note: The commands \bfseries and \boldmath are both needed here because the = 12 printed when you use half points is printed in math mode while the rest of the points are printed in text mode.) For another example, the commands \begin{questions} \pointname{} \pointformat{\bfseries\boldmath(Points for this problem: \thepoints)} \question[3\half] What is the point \end{questions} will produce 1. (Points for this problem: 3=) W2at is the point 4.3.8 Questions that begin with a parts environment If a question begins with a parts environment, then the question number and the number of the ▯rst part in the parts environment will be printed on the same line (see section 4.2). Thus, if points are assigned to both the question and to the ▯rst part, both of those point values will be printed on the same line. (Note: This was not true in versions of the exam document class earlier than 2.3. In those versions, only the later of the two point values would actually be printed.) If \pointsinmargin or \pointsinrightmargin are in e▯ect (see section 4.3.1), then the two point values will be printed on top of each other, which is probably not what you want. The same problem arises if a part begins with a subparts environment or if a subpart begins with a subsubparts environment. If you want to assign points to both a question and to each part of a parts environment, and if the ▯rst part will begin on the same line as the question, the solution is to either use the default location (i.e., \nopointsinmargin) or to put some of the points in the margin and others not. For example, if you type \begin{questions} \pointsinmargin \noaddpoints \question[10] Page 27 \begin{parts} \nopointsinmargin \addpoints \part[5] Who put the ‘‘bop’’ in the ‘‘bop, sh-bop, sh-bop’’ \part[5] Who put the ‘‘ram’’ in the ‘‘rama, rama ding-dong’’ \end{parts} \end{questions} then you’ll get (10%) 1. (a) (5 points) Who put the \bop" in the \bop, sh-bop, sh-bop" (b) (5 points) Who put the \ram" in the \rama, rama ding-dong" 4.3.9 Adding up the points for a question If you use the document class option addpoints (as in \documentclass[addpoints]{exam}) or give the command \addpoints (see section 3.1), then you can use the command \droptotalpoints to put into the right margin the total number of points for the current question (including the points for all parts, subparts, and subsubparts). (Section 4.4.1 describes a di▯erent way to print the total number of points for a question.) The command \droptotalpoints should be used only at the end of a paragraph or between paragraphs; if you use it within a paragraph, it causes the paragraph to end. \droptotalpoints prints the total points for the current question in the right margin, formatted by default as: Total for Question 2: 25 (Note: The number of points is followed by the argument to the most recently executed \marginpointname command (see section 4.3.5), which by default is empty.) If the current question is worth any bonus points (see section 4.6), then those bonus points will not be counted by the \droptotalpoints command, but they will be counted by the \droptotalbonuspoints command, which puts into the right margin the total number of bonus points for the current question, formatted by default as: Total for Question 2: 10 (bonus) (Note: The number of bonus points is followed by the argument to the most recently executed \marginbonuspointname command (see section 4.6), which by default is \ (bonus)".) You can change the format used by the \droptotalpoints command with the \totalformat Page 28 command. It takes one argument, and that argument becomes the command to print the total points, right justi▯ed a distance of \rightpointsmargin from the right edge of the paper. The argument can contain the command \totalpoints to print the number of points, and \thequestion to print the question number. (\totalpoints expands to \pointsofquestion{\arabic{question}}; see section 9.5 for a discussion of the \pointsofquestion command.) For example, if you give the command \totalformat{Question \thequestion: \totalpoints} then the total number of points will be printed in the format Question 2: 25 and if you give the command \totalformat{\fbox{Total: \totalpoints}} then the total number of points will be printed as Total: 25 Thus, if you’ve given the commands \addpoints \bracketedpoints \pointsdroppedatright and you type \question \begin{parts} \part[10] In no more than one paragraph, explain why the earth is round. \droppoints \part[10] What changes to the van Allen radiation belt are needed to make the earth into a regular icosahedron \droppoints \end{parts} \droptotalpoints then you’ll get 1. (a) In no more than one paragraph, explain why the earth is round. [10%] (b) What changes to the van Allen radiation belt are needed to make the earth into a regular icosahedron [10%] Total for Question 1: 20 Page 29 If you’ve also given the command \totalformat{Total for Question \thequestion: [\totalpoints]} then you’ll get 1. (a) In no more than one paragraph, explain why the earth is round. [10%] (b) What changes to the van Allen radiation belt are needed to make the earth into a regular icosahedron [10%] Total for Question 1: [20] To change the format used by the \droptotalbonuspoints command, you use the com- mand \bonustotalformat which is analogous to the \totalformat command. The argument to \bonustotalformat can contain the command \totalbonuspoints to print the number of bonus points, and \thequestion to print the question number. (\totalbonuspoints expands to \bonuspointsofquestion{\arabic{question}}; see sec- tion 9.7 for a discussion of the \bonuspointsofquestion command.) 4.3.10 Margin sizes when using \pointsinmargin and \pointsinrightmargin ▯ If you give the command \pointsinmargin, then the points are printed right justi▯ed in the left margin, with the right edge a distance of \marginpointssep from the left edge of the text area. The default value of \marginpointssep is the result of the command \setlength{\marginpointssep}{5pt} and you can change it by giving a new \setlength command. ▯ If you give the command \pointsinrightmargin, then the points are printed right justi▯ed in the right margin, with the right edge a distance of \rightpointsmargin from the right edge of the paper. The default value of \rightpointsmargin is the result of the command \setlength{\rightpointsmargin}{1cm} and you can change it by giving a new \setlength command. You can change the width of the text area (and, thus, change the width of the margins) with the \extrawidth command (see section 4.3.6). Page 30 4.4 Custom question number lines: The \qformat command The default setup is for the question number to appear in the left margin and for the text of the question to begin on that line. It’s possible to change this so that the text of the question starts on the line following the question number, and the format of the line containing the question number is determined by the user. This is done using the \qformat command. There is also a \noqformat command, which reverts to the default setup. To use the \qformat command, you give the command \qformat{Format specification} where Format specification must contain some stretchability (e.g., at least one \hfill or \dotfill or \hrulefill or ...) and can contain the commands ▯ \thequestion, which expands to the question number, ▯ \thequestiontitle (see section 4.5.2), which expands to { the title of the question, if this question was de▯ned using a \titledquestion command (see section 4.5.1) instead of a \question command, or { the number of the question, if this question was de▯ned using a \question com- mand, ▯ \thepoints, which expands to { the number of points followed by the argument to the last \pointname command, if there were points speci▯ed for the question, or { nothing, if no points were speci▯ed for the question (see section 4.3.5 for a discussion of the \pointname command), and ▯ \totalpoints, which expands to the total number of points speci▯ed for the ques- tion and all of its parts, subparts, and subsubparts (see section 4.3.9). The com- mand \totalpoints expands to \pointsofquestion{\arabic{question}} (see sec- tion 9.5). For example, if you give the commands \qformat{Question \thequestion \dotfill \thepoints} \begin{questions} \question[10] What did Billy Joe MacAllister throw off the Tallahatchie Bridge \end{questions} then you’ll get Question 1.......................................................................10 points What did Billy Joe MacAllister throw o▯ the Tallahatchie Bridge Page 31 If you instead use the command \qformat{\textbf{Question \thequestion}\quad (\thepoints)\hfill} then you’ll get Question 1 (10 points) What did Billy Joe MacAllister throw o▯ the Tallahatchie Bridge If you instead use the command \qformat{\hfill Question \thequestion\hfill (\thepoints)} then you’ll get Question 1 (10 points) What did Billy Joe MacAllister throw o▯ the Tallahatchie Bridge If you use a \qformat (and, if there are any bonus questions, also a \bonusqformat; see section 4.6), so that there are no labels on any of the questions, and you’d like to remove the indentation of the questions, you should give the command \renewcommand{\questionshook}{% \setlength{\leftmargin}{0pt}% \setlength{\labelwidth}{-\labelsep}% } (see section 4.10) before the questions environment. 4.4.1 Adding up the points for a question You can combine \qformat with the \totalpoints command (see section 4.3.9): If you assign points only to parts, subparts, and subsubparts of questions, but none to the questions themselves, and you give the command \qformat{Question \thequestion\dotfill \emph{\totalpoints\ points}} (you must have used the document class option addpoints or given the command \addpoints in order to use the \totalpoints command; see section 3.1), then you can type \begin{questions} \question \begin{parts} \part[10] In no more than one paragraph

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