updated

\relax 

\providecommand\hyper@newdestlabel[2]{}

\providecommand\HyField@AuxAddToFields[1]{}

\providecommand\HyField@AuxAddToCoFields[2]{}

\@writefile{toc}{\contentsline {section}{\numberline {1}Report}{1}{section.1}\protected@file@percent }

\@writefile{toc}{\contentsline {section}{\numberline {2}Task 2f}{1}{section.2}\protected@file@percent }

\gdef \@abspage@last{1}

This is pdfTeX, Version 3.141592653-2.6-1.40.26 (TeX Live 2024) (preloaded format=pdflatex 2024.4.11)  7 JUN 2024 11:51

entering extended mode

 restricted \write18 enabled.

 %&-line parsing enabled.

**assignment.tex

(./assignment.tex

LaTeX2e <2023-11-01> patch level 1

L3 programming layer <2024-03-14>

(/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/base/article.cls

Document Class: article 2023/05/17 v1.4n Standard LaTeX document class

(/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/base/size12.clo

File: size12.clo 2023/05/17 v1.4n Standard LaTeX file (size option)

)

\c@part=\count188

\c@section=\count189

\c@subsection=\count190

\c@subsubsection=\count191

\c@paragraph=\count192

\c@subparagraph=\count193

\c@figure=\count194

\c@table=\count195

\abovecaptionskip=\skip48

\belowcaptionskip=\skip49

\bibindent=\dimen140

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/geometry/geometry.sty

Package: geometry 2020/01/02 v5.9 Page Geometry

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics/keyval.sty

Package: keyval 2022/05/29 v1.15 key=value parser (DPC)

\KV@toks@=\toks17

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/iftex/ifvtex.sty

Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/iftex/iftex.sty

Package: iftex 2022/02/03 v1.0f TeX engine tests

))

\Gm@cnth=\count196

\Gm@cntv=\count197

\c@Gm@tempcnt=\count198

\Gm@bindingoffset=\dimen141

\Gm@wd@mp=\dimen142

\Gm@odd@mp=\dimen143

\Gm@even@mp=\dimen144

\Gm@layoutwidth=\dimen145

\Gm@layoutheight=\dimen146

\Gm@layouthoffset=\dimen147

\Gm@layoutvoffset=\dimen148

\Gm@dimlist=\toks18

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/setspace/setspace.sty

Package: setspace 2022/12/04 v6.7b set line spacing

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/booktabs/booktabs.sty

Package: booktabs 2020/01/12 v1.61803398 Publication quality tables

\heavyrulewidth=\dimen149

\lightrulewidth=\dimen150

\cmidrulewidth=\dimen151

\belowrulesep=\dimen152

\belowbottomsep=\dimen153

\aboverulesep=\dimen154

\abovetopsep=\dimen155

\cmidrulesep=\dimen156

\cmidrulekern=\dimen157

\defaultaddspace=\dimen158

\@cmidla=\count199

\@cmidlb=\count266

\@aboverulesep=\dimen159

\@belowrulesep=\dimen160

\@thisruleclass=\count267

\@lastruleclass=\count268

\@thisrulewidth=\dimen161

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hyperref/hyperref.sty

Package: hyperref 2024-01-20 v7.01h Hypertext links for LaTeX

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/kvsetkeys/kvsetkeys.sty

Package: kvsetkeys 2022-10-05 v1.19 Key value parser (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty

Package: kvdefinekeys 2019-12-19 v1.6 Define keys (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/pdfescape/pdfescape.sty

Package: pdfescape 2019/12/09 v1.15 Implements pdfTeX's escape features (HO)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty

Package: ltxcmds 2023-12-04 v1.26 LaTeX kernel commands for general use (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/pdftexcmds/pdftexcmds.sty

Package: pdftexcmds 2020-06-27 v0.33 Utility functions of pdfTeX for LuaTeX (HO)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/infwarerr/infwarerr.sty

Package: infwarerr 2019/12/03 v1.5 Providing info/warning/error messages (HO)

)

Package pdftexcmds Info: \pdf@primitive is available.

Package pdftexcmds Info: \pdf@ifprimitive is available.

Package pdftexcmds Info: \pdfdraftmode found.

)) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hycolor/hycolor.sty

Package: hycolor 2020-01-27 v1.10 Color options for hyperref/bookmark (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/auxhook/auxhook.sty

Package: auxhook 2019-12-17 v1.6 Hooks for auxiliary files (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hyperref/nameref.sty

Package: nameref 2023-11-26 v2.56 Cross-referencing by name of section

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/refcount/refcount.sty

Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty

Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/kvoptions/kvoptions.sty

Package: kvoptions 2022-06-15 v3.15 Key value format for package options (HO)

))

\c@section@level=\count269

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/etoolbox/etoolbox.sty

Package: etoolbox 2020/10/05 v2.5k e-TeX tools for LaTeX (JAW)

\etb@tempcnta=\count270

)

\@linkdim=\dimen162

\Hy@linkcounter=\count271

\Hy@pagecounter=\count272

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hyperref/pd1enc.def

File: pd1enc.def 2024-01-20 v7.01h Hyperref: PDFDocEncoding definition (HO)

Now handling font encoding PD1 ...

... no UTF-8 mapping file for font encoding PD1

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/intcalc/intcalc.sty

Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO)

)

\Hy@SavedSpaceFactor=\count273

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hyperref/puenc.def

File: puenc.def 2024-01-20 v7.01h Hyperref: PDF Unicode definition (HO)

Now handling font encoding PU ...

... no UTF-8 mapping file for font encoding PU

)

Package hyperref Info: Hyper figures OFF on input line 4179.

Package hyperref Info: Link nesting OFF on input line 4184.

Package hyperref Info: Hyper index ON on input line 4187.

Package hyperref Info: Plain pages OFF on input line 4194.

Package hyperref Info: Backreferencing OFF on input line 4199.

Package hyperref Info: Implicit mode ON; LaTeX internals redefined.

Package hyperref Info: Bookmarks ON on input line 4446.

\c@Hy@tempcnt=\count274

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/url/url.sty

\Urlmuskip=\muskip16

Package: url 2013/09/16  ver 3.4  Verb mode for urls, etc.

)

LaTeX Info: Redefining \url on input line 4784.

\XeTeXLinkMargin=\dimen163

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/bitset/bitset.sty

Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty

Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO)

))

\Fld@menulength=\count275

\Field@Width=\dimen164

\Fld@charsize=\dimen165

Package hyperref Info: Hyper figures OFF on input line 6063.

Package hyperref Info: Link nesting OFF on input line 6068.

Package hyperref Info: Hyper index ON on input line 6071.

Package hyperref Info: backreferencing OFF on input line 6078.

Package hyperref Info: Link coloring OFF on input line 6083.

Package hyperref Info: Link coloring with OCG OFF on input line 6088.

Package hyperref Info: PDF/A mode OFF on input line 6093.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/base/atbegshi-ltx.sty

Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi

package with kernel methods

)

\Hy@abspage=\count276

\c@Item=\count277

\c@Hfootnote=\count278

)

Package hyperref Info: Driver (autodetected): hpdftex.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/hyperref/hpdftex.def

File: hpdftex.def 2024-01-20 v7.01h Hyperref driver for pdfTeX

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/base/atveryend-ltx.sty

Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend package

with kernel methods

)

\Fld@listcount=\count279

\c@bookmark@seq@number=\count280

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty

Package: rerunfilecheck 2022-07-10 v1.10 Rerun checks for auxiliary files (HO)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty

Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO)

)

Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 285.

)

\Hy@SectionHShift=\skip50

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics/graphicx.sty

Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics/graphics.sty

Package: graphics 2022/03/10 v1.4e Standard LaTeX Graphics (DPC,SPQR)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics/trig.sty

Package: trig 2021/08/11 v1.11 sin cos tan (DPC)

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics-cfg/graphics.cfg

File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration

)

Package graphics Info: Driver file: pdftex.def on input line 107.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/graphics-def/pdftex.def

File: pdftex.def 2022/09/22 v1.2b Graphics/color driver for pdftex

))

\Gin@req@height=\dimen166

\Gin@req@width=\dimen167

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/tools/tabularx.sty

Package: tabularx 2023/07/08 v2.11c `tabularx' package (DPC)

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/tools/array.sty

Package: array 2023/10/16 v2.5g Tabular extension package (FMi)

\col@sep=\dimen168

\ar@mcellbox=\box51

\extrarowheight=\dimen169

\NC@list=\toks19

\extratabsurround=\skip51

\backup@length=\skip52

\ar@cellbox=\box52

)

\TX@col@width=\dimen170

\TX@old@table=\dimen171

\TX@old@col=\dimen172

\TX@target=\dimen173

\TX@delta=\dimen174

\TX@cols=\count281

\TX@ftn=\toks20

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsmath/amsmath.sty

Package: amsmath 2023/05/13 v2.17o AMS math features

\@mathmargin=\skip53

For additional information on amsmath, use the `?' option.

(/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsmath/amstext.sty

Package: amstext 2021/08/26 v2.01 AMS text

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsmath/amsgen.sty

File: amsgen.sty 1999/11/30 v2.0 generic functions

\@emptytoks=\toks21

\ex@=\dimen175

)) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsmath/amsbsy.sty

Package: amsbsy 1999/11/29 v1.2d Bold Symbols

\pmbraise@=\dimen176

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsmath/amsopn.sty

Package: amsopn 2022/04/08 v2.04 operator names

)

\inf@bad=\count282

LaTeX Info: Redefining \frac on input line 234.

\uproot@=\count283

\leftroot@=\count284

LaTeX Info: Redefining \overline on input line 399.

LaTeX Info: Redefining \colon on input line 410.

\classnum@=\count285

\DOTSCASE@=\count286

LaTeX Info: Redefining \ldots on input line 496.

LaTeX Info: Redefining \dots on input line 499.

LaTeX Info: Redefining \cdots on input line 620.

\Mathstrutbox@=\box53

\strutbox@=\box54

LaTeX Info: Redefining \big on input line 722.

LaTeX Info: Redefining \Big on input line 723.

LaTeX Info: Redefining \bigg on input line 724.

LaTeX Info: Redefining \Bigg on input line 725.

\big@size=\dimen177

LaTeX Font Info:    Redeclaring font encoding OML on input line 743.

LaTeX Font Info:    Redeclaring font encoding OMS on input line 744.

\macc@depth=\count287

LaTeX Info: Redefining \bmod on input line 905.

LaTeX Info: Redefining \pmod on input line 910.

LaTeX Info: Redefining \smash on input line 940.

LaTeX Info: Redefining \relbar on input line 970.

LaTeX Info: Redefining \Relbar on input line 971.

\c@MaxMatrixCols=\count288

\dotsspace@=\muskip17

\c@parentequation=\count289

\dspbrk@lvl=\count290

\tag@help=\toks22

\row@=\count291

\column@=\count292

\maxfields@=\count293

\andhelp@=\toks23

\eqnshift@=\dimen178

\alignsep@=\dimen179

\tagshift@=\dimen180

\tagwidth@=\dimen181

\totwidth@=\dimen182

\lineht@=\dimen183

\@envbody=\toks24

\multlinegap=\skip54

\multlinetaggap=\skip55

\mathdisplay@stack=\toks25

LaTeX Info: Redefining \[ on input line 2953.

LaTeX Info: Redefining \] on input line 2954.

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsfonts/amssymb.sty

Package: amssymb 2013/01/14 v3.01 AMS font symbols

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/amsfonts/amsfonts.sty

Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support

\symAMSa=\mathgroup4

\symAMSb=\mathgroup5

LaTeX Font Info:    Redeclaring math symbol \hbar on input line 98.

LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'

(Font)                  U/euf/m/n --> U/euf/b/n on input line 106.

)) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/float/float.sty

Package: float 2001/11/08 v1.3d Float enhancements (AL)

\c@float@type=\count294

\float@exts=\toks26

\float@box=\box55

\@float@everytoks=\toks27

\@floatcapt=\box56

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/natbib/natbib.sty

Package: natbib 2010/09/13 8.31b (PWD, AO)

\bibhang=\skip56

\bibsep=\skip57

LaTeX Info: Redefining \cite on input line 694.

\c@NAT@ctr=\count295

) (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def

File: l3backend-pdftex.def 2024-03-14 L3 backend support: PDF output (pdfTeX)

\l__color_backend_stack_int=\count296

\l__pdf_internal_box=\box57

) (./assignment.aux)

\openout1 = `assignment.aux'.

LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for TS1/cmr/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

LaTeX Font Info:    Checking defaults for PU/pdf/m/n on input line 14.

LaTeX Font Info:    ... okay on input line 14.

*geometry* driver: auto-detecting

*geometry* detected driver: pdftex

*geometry* verbose mode - [ preamble ] result:

* driver: pdftex

* paper: a4paper

* layout: <same size as paper>

* layoutoffset:(h,v)=(0.0pt,0.0pt)

* modes: 

* h-part:(L,W,R)=(71.13188pt, 455.24411pt, 71.13188pt)

* v-part:(T,H,B)=(71.13188pt, 717.00946pt, 56.9055pt)

* \paperwidth=597.50787pt

* \paperheight=845.04684pt

* \textwidth=455.24411pt

* \textheight=717.00946pt

* \oddsidemargin=-1.1381pt

* \evensidemargin=-1.1381pt

* \topmargin=-38.1381pt

* \headheight=12.0pt

* \headsep=25.0pt

* \topskip=12.0pt

* \footskip=30.0pt

* \marginparwidth=44.0pt

* \marginparsep=10.0pt

* \columnsep=10.0pt

* \skip\footins=10.8pt plus 4.0pt minus 2.0pt

* \hoffset=0.0pt

* \voffset=0.0pt

* \mag=1000

* \@twocolumnfalse

* \@twosidefalse

* \@mparswitchfalse

* \@reversemarginfalse

* (1in=72.27pt=25.4mm, 1cm=28.453pt)

Package hyperref Info: Link coloring OFF on input line 14.

(./assignment.out) (./assignment.out)

\@outlinefile=\write3

\openout3 = `assignment.out'.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty

Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf

Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485.

 (/Users/alexanderunger/Library/TinyTeX/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg

File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live

)) [1

{/Users/alexanderunger/Library/TinyTeX/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./assignment.aux)

 ***********

LaTeX2e <2023-11-01> patch level 1

L3 programming layer <2024-03-14>

 ***********

Package rerunfilecheck Info: File `assignment.out' has not changed.

(rerunfilecheck)             Checksum: E02A30C8D035F4151F464091372D01C7;154.

 ) 

Here is how much of TeX's memory you used:

 10295 strings out of 476064

 158022 string characters out of 5791216

 1931493 words of memory out of 5000000

 32305 multiletter control sequences out of 15000+600000

 558973 words of font info for 39 fonts, out of 8000000 for 9000

 14 hyphenation exceptions out of 8191

 75i,5n,79p,1276b,440s stack positions out of 10000i,1000n,20000p,200000b,200000s

</Users/alexanderunger/Library/TinyTeX/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></Users/alexanderunger/Library/TinyTeX/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb>

Output written on assignment.pdf (1 page, 30516 bytes).

PDF statistics:

 32 PDF objects out of 1000 (max. 8388607)

 24 compressed objects within 1 object stream

 4 named destinations out of 1000 (max. 500000)

 17 words of extra memory for PDF output out of 10000 (max. 10000000)

\BOOKMARK [1][-]{section.1}{\376\377\000R\000e\000p\000o\000r\000t}{}% 1

\BOOKMARK [1][-]{section.2}{\376\377\000T\000a\000s\000k\000\040\0002\000f}{}% 2