÷ƒ’À;è TeX output 1997.02.20:1515‹ÿÿÿÿ y ý£ ? ýä’§*¡óKñ`y cmr10²18.01–UUSolutions,“Problem“Set“2ŽŸÔ‘Mó !",š cmsy10¸ŽŽŽ‘WxŽ‘[q˲3.1‘UU#1ŽŸ¥Ë‘WFind–UUthe“deriv‘ÿqÇativš¸ãe“of“eac˜h“function:ŽŸK–‘[8ã(a)ŽŽŽ‘m6ó  b> cmmi10µxŸü^ÿóÙ“ Rcmr7±9Ž‘|s²;ޤ9‘móò"V cmbx10¹Solution:ŽŽŸù<$’¦œëµdŽ’£Á]Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’¯à²(6µxŸûÞÿ±9Ž›|s²)–Ç=“6–8ณ²9µxŸûÞÿ±9ó O!â…cmsy7·±1Ž‘ÿ²=‘Ç54µxŸûÞÿ±8Ž˜óý': cmti10º.Ž©Kx‘Zªª²(b)ŽŽŽ‘mµ[ÙŸü^ÿ±5Ž‘ØL²;Ž¡‘m¹Solution:ŽŽŸù<$’¦œëµdŽ’£Á]Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’¯à²(µ[ÙŸûÞÿ±5Ž‘ØL²)–Ç=“0–“çºsinc›ÿ}'e“µ[ÙŸü^ÿ±5Ž‘l3ºis“a“c˜onstant.ަ‘[8ã²(g)ŽŽŽ‘mµxŸü^ÿ±4Ž–µS²+›8àµxŸü^ÿ±3Ž“²+˜µxŸü^ÿ±2Ž“²+˜µx˜²+˜1.ŽŸK–‘m¹Solution:ŽŽŽŸ&N¡Ÿð£÷Ÿù<$’–¤gµdŽ’“ÈÙŸw‰fe 뎟 (ÖdxŽŽŽŽ’Ÿçš²(µxŸûÞÿ±4Ž–µS²+›8àµxŸûÞÿ±3Ž“²+˜µxŸûÞÿ±2Ž“²+˜µx˜²+˜1)ŽŽŸ‘¬’ bm=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(µxŸûÞÿ±4Ž–|s²)›8à+Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µxŸûÞÿ±3Ž“²)˜+Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µxŸûÞÿ±2Ž“²)˜+Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ–ŠÔ²(µx²)˜+Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ“²(1)ŽŽŸBm’ bm=‘Ç4µxŸü^ÿ±3Ž–µS²+›8à3µxŸü^ÿ±2Ž“²+˜2µx˜²+˜1ŽŽŽ’s£?µ:ŽŸ)æ3‘M¸ŽŽŽ‘WxŽ‘[q˲3.1‘UU#3bޤ¥Ë‘WIf›)µs–'ý²=“13–Ƹ“²9µt“²+“6µtŸü^ÿ±3Ž‘¥†²is˜the˜pGosition˜at˜time˜µt˜²of˜an˜ob‘Ž8ject˜mo¸ãving˜on˜aŽ© ‘Wstraighš¸ãt–UUline,“ nd“the“v˜eloGcit˜y“µv‘±.²and“the“acceleration“µa².ŽŸ<õ‘W¹Solution:ŽŽ’ŒŽ*ºThe–“çvelo‘ÿ}'city“isŽŸä_’½‚µv‘"ñ²=Ÿù<$‘úKµdsŽ‘úKŸw‰fe ärŸ (Ö‘‰ÇdtŽŽŽŽ‘Ù²=Ÿù<$‘È„µdŽ‘úKŸw‰feÐäŸ (ÖdtŽŽŽŽ‘ þb²(13–8ณ²9µt“²+“6µtŸûÞÿ±3Ž›|s²)–Ç=“¸²9–8à+“18µtŸûÞÿ±2Ž˜µ:ŽŸ Å‘WºThe›“çac–ÿ}'c“eler“ation˜isŽŸò˜’²¶;µa›Ç²=Ÿù<$‘úKµdvŽ‘úKŸw‰fe i/Ÿ (Ö‘Ì&dtŽŽŽŽ‘]Ų=Ÿù<$‘È„µdŽ‘úKŸw‰feÐäŸ (ÖdtŽŽŽŽ‘ þb²(¸²9–8à+“18µtŸûÞÿ±2Ž‘|s²)˜=˜36µt:ŽŸ°‘M¸ŽŽŽ‘WxŽ‘[q˲3.1‘UU#5Ž¡‘WFind–UUthe“line“tangenš¸ãt“to“the“curv˜e“µy‘"ñ²=‘Ç3µxŸü^ÿ±2Ž‘µS¸–8à²5µx“²+“2–UUat“the“pGoin˜t“(2µ;‘ª¨²4).ŽŸr˜‘W¹Solution:ŽŽ’ŒŽ*ºEvaluatingŸù<$‘7›µdyŽ‘ý‰Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘Eþ²=Ÿù<$‘8uµdŽ‘\çŸw‰fe 뎟 (ÖdxŽŽŽŽ‘{¨²(3µxŸûÞÿ±2Ž‘Üĸ–`Q²5µx“²+“2)–)´=“6µx–`Q¸“²5–ÊVºat“µx–)´²=“2‘ÊVºshowsŽŸÍБWthe–¼Dslop›ÿ}'e“of“the“tangent“line“is“²6–V¸“²2“¸“²5–6=“7º.‘¬The–¼De˜quation“of“the“tangentަ‘Wline–“çis“ther–ÿ}'efor“eŽŸ<õ’¦ø}µy‘”¹¸›8à²4–Ç=“7(µx˜¸˜²2)µ;‘ ㈺orŽ‘pMµy‘"ñ²=“7µx˜¸˜²10µ:ŽŸâÀ‘M¸ŽŽŽ‘WxŽ‘[q˲3.1‘UU#6Ž¡‘WFind–ªÅthe“pGoinš¸ãts“on“the“curv˜e“µy‘"ñ²=‘Ç4µxŸü^ÿ±3Ž–`5²+›ãÂ6µxŸü^ÿ±2Ž“¸˜²24µx˜²+˜10–ªÅat“whicš¸ãh“the“tangen˜tަ‘Wline–UUis“horizon¸ãtal.ŽŽŸŽŒ‹* y ý£ ? ýä‘W¹Solution:ŽŽ’ŒŽ*ºThe–“çslop‘ÿ}'e“of“the“tangent“line“isŽŸ®ðŸù<$’†ìCµdyŽ’†²1Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’•˜ ²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(4µxŸûÞÿ±3Ž–µS²+›8à6µxŸûÞÿ±2Ž“¸˜²24µx˜²+˜10)–Ç=“12µxŸûÞÿ±2Ž‘µS²+˜12µx˜¸˜²24µ:ŽŸh‘WºThe–Çitangent“line“is“horizontal“when“the“slop›ÿ}'e“is“zer˜o,‘ÔIso“we“solve“²12µxŸü^ÿ±2Ž‘Ú¦²+Ž© ‘W12µx–8ณ²24–Ç=“0º.‘™–F‘ÿ;¼actoringޤ†’в12µxŸûÞÿ±2Ž›µS²+–8à12µx“¸“²24–Ç=“12(µxŸûÞÿ±2Ž˜²+–8àµx“¸“²2)–Ç=“12(µx–8à²+“2)(µx“¸“²1)µ;Ž¡‘Wºwe–“ç nd“µx–Dz=“¸²2–“çºor“µx–Dz=“1º.‘™–The–“çp›ÿ}'oints“ar˜e“ther˜efor˜e“²(¸²2µ;›ª¨²50)“ºand“²(1µ;˜¸²4)º.ŽŸ†‘M¸ŽŽŽ‘WxŽ‘[q˲3.1‘UU#18ŽŸ›‘WFind–|the“tangenš¸ãt“to“the“curv˜e“µy‘"ñ²=‘ǵxŸü^ÿ±3Ž‘„ï²that“passes“through“the“pGoin˜t“(0µ;‘ª¨²2).ŽŸ=)‘W¹Solution:ŽŽ’ŒŽ*ºThe–•derivative“of“µy‘$û²=–É"µxŸü^ÿ±3Ž‘zºisŸù<$‘£ÈµdŽ‘È:Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘æû²(µxŸûÞÿ±3Ž›|s²)“=“3µxŸûÞÿ±2Ž˜º,‘•Oso–•the“slop‘ÿ}'e“fo“theŽŸÿâ‘Wtangent–°line“at“the“p‘ÿ}'oint“²(µa;‘ª¨aŸü^ÿ±3Ž›|s²)“ºis“²3µaŸü^ÿ±2Ž˜º.‘èðThe“e‘ÿ}'quation“of“the“tangent“lineަ‘Wis›‹µther–ÿ}'efor“e˜µy‘‚踖'µaŸü^ÿ±3Ž‘C‹²=‘Ç3µaŸü^ÿ±2Ž‘|s²(µx“¸“µa²)º.‘–ÛF‘ÿ;¼or˜this˜to˜p–ÿ}'ass˜thr“ough˜the˜p“oint˜²(0µ;‘ª¨²2)º,ަ‘Wwe–“çmust“haveŽŸ€Ÿíæd’Èk²2–8ณµaŸü^ÿ±3ŽŽ’íi²=Ž’ÿ0»3µaŸü^ÿ±2Ž‘|s²(¸µa²);ŽŽ¦’Èk2–8ณµaŸü^ÿ±3ŽŽ’íi²=Ž’ÿ0»¸²3µaŸü^ÿ±3Ž‘|s²;ŽŽ¦’Þiœ2Ž’íi=Ž’ÿ0»¸²2µaŸü^ÿ±3Ž‘|s²;ŽŽ¦’Ö¢~¸²1Ž’íi=Ž’ÿ0»µaŸü^ÿ±3Ž‘|sµ:ŽŽŽŽŸ"C‘WºSo›“çµa–Dz=“¸²1˜ºand˜the˜e‘ÿ}'quation˜of˜the˜tangent˜line˜isŽ¡’©x}µy‘”¹²+›8à1–Ç=“3(µx˜²+˜1)µ;‘ ㈺orŽ‘pMµy‘"ñ²=“3µx˜²+˜2µ:ŽŸ¢¡‘M¸ŽŽŽ‘WxŽ‘[q˲3.2‘UU#4ŽŸ›‘WDi eren¸ãtiate‘ß®(µx–M“¸“²1)(µxŸü^ÿ±4Ž›Ê²+“µxŸü^ÿ±3Ž˜²+“µxŸü^ÿ±2Ž˜²+“µx“²+“1)–ß®in“t•¸ãw“o›ß®w“a“ys˜and˜v“erify˜that˜y“ourަ‘Wansw¸ãers‘UUagree.ŽŸ†‘W¹Solution:ŽŽ’ŒŽ*ºIf–“çwe“ rst“simplifyŽ¡‘W²(µx¸²1)(µxŸûÞÿ±4Ž–|s²+µxŸûÞÿ±3Ž“²+µxŸûÞÿ±2Ž“²+µx²+1)–Ç=“(µxŸûÞÿ±5Ž–|s²+µxŸûÞÿ±4Ž“²+µxŸûÞÿ±3Ž“²+µxŸûÞÿ±2Ž“²+µx²)¸²(µxŸûÞÿ±4Ž“²+µxŸûÞÿ±3Ž“²+µxŸûÞÿ±2Ž“²+µx²+1)–Ç=“µxŸûÞÿ±5Ž‘|s¸²1µ;Ž¡‘Wºthen–“çwe“getŽŸ®ðŸù<$’‚¯ŠµdŽ‘ÓüŸw‰fe 뎟 (ÖdxŽŽŽŽ’‹ò½²[(µx–8ณ²1)(µxŸûÞÿ±4Ž›µS²+“µxŸûÞÿ±3Ž˜²+“µxŸûÞÿ±2Ž˜²+“µx“²+“1)]‘Ç=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(µxŸûÞÿ±5Ž˜¸“²1)–Ç=“5µxŸûÞÿ±4Ž‘|sµ:ŽŸpq‘WºOn–“çthe“other“hand,“by“the“pr–ÿ}'o“duct‘“çruleŽŸ4 ùŸä£÷Ÿù<$‘`õdŽ‘]35Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘iQö²[(µx–8ณ²1)(µxŸûÞÿ±4Ž›µS²+“µxŸûÞÿ±3Ž˜²+“µxŸûÞÿ±2Ž˜²+“µx“²+“1)]ŽŽŸ‘¬‘iÌÉ=‘Ç(µx–8ณ²1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µxŸûÞÿ±4Ž›µS²+“µxŸûÞÿ±3Ž˜²+“µxŸûÞÿ±2Ž˜²+“µx“²+“1)“+“(µxŸûÞÿ±4Ž˜²+“µxŸûÞÿ±3Ž˜²+“µxŸûÞÿ±2Ž˜²+“µx“²+“1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µx“¸“²1)ŽŽŸBm‘iÌÉ=‘Ç(µx–8ณ²1)(4µxŸü^ÿ±3Ž›µS²+“3µxŸü^ÿ±2Ž˜²+“2µx“²+“1)“+“(µxŸü^ÿ±4Ž˜²+“µxŸü^ÿ±3Ž˜²+“µxŸü^ÿ±2Ž˜²+“µx“²+“1)“¸“²1ŽŽ¦‘iÌÉ=‘Ç(4µxŸü^ÿ±4Ž–µS²+›8à3µxŸü^ÿ±3Ž“²+˜2µxŸü^ÿ±2Ž“²+˜µx²)˜¸˜²(4µxŸü^ÿ±3Ž“²+˜3µxŸü^ÿ±2Ž“²+˜2µx˜²+˜1)˜+˜(µxŸü^ÿ±4Ž“²+˜µxŸü^ÿ±3Ž“²+˜µxŸü^ÿ±2Ž“²+˜µx˜²+˜1)ŽŽ¦‘iÌÉ=‘Ç5µxŸü^ÿ±4Ž‘|sµ:ŽŽŽŽŽŸŽŒ‹ < y ý£ ? ýä‘M¸ŽŽŽ‘WxŽ‘[q˲3.2ŽŸ“ø‘WDi eren•¸ãtiate›?üeac“h˜function˜and˜simplify˜y“our˜answ“er˜as˜m“uc“h˜as˜pGossible.ŽŸOZ‘`8ä9.ŽŽŽŸù<$‘n35µx–8à²+“1Ž‘n35Ÿw‰feïûŸ (Öµx–8ณ²1ŽŽŽŽ’†Vcµ:ޤË‘m¹Solution:ŽŽ’¢Ž*ºBy–“çthe“quotient“rule,ŽŸ5jsŸíã:Ÿù<$’Š•yµdŽ’‡¹ëŸw‰fe 뎟 (ÖdxŽŽŽŽ’•ƒTŸñæ^óú±u cmex10«ŽŸù<$’žûµx–8à²+“1Ž’žûŸw‰feïûŸ (Öµx–8ณ²1ŽŽŽŽ’¶6)Ÿñæ^«ŽŽŽ’Ç’²=ŽŸõ=•’ÚŒî(µx–8ณ²1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ› Qô²(µx“²+“1)“¸“²(µx“²+“1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ˜²(µx“¸“²1)Ž’ÚŒîŸu‰fe¡¹*Ÿ (Ö‘?BÏ(µx–8ณ²1)Ÿýr±2ŽŽŽŽŽŽŽŸÃä’Ç’²=ŽŸù<$’ÚŒî(µx–8ณ²1)“¸“²(µx“²+“1)Ž’ÚŒîŸw‰feI§Ÿ (Ö‘9Â(µx–8ณ²1)Ÿýr±2ŽŽŽŽŽŽŽŸ‘¬’Ç’²=ŽŸù<$’åÃ%¸²2Ž’ÚŒîŸw‰fe#3ŒŸ (Ö(µx–8ณ²1)Ÿýr±2ŽŽŽŽŽ’þó­µ:ŽŽŽŽŸ=BF‘[8ã²10.ŽŽŽŸù<$‘yil1Ž‘n35Ÿw‰felnŸ (ÖµxŸýr±2Ž‘µS²+‘8à2ŽŽŽŽŽ¡‘m¹Solution:ŽŽŽ©'¨¹Ÿù³-Ÿù<$’–çµdŽ’“6YŸw‰fe 뎟 (ÖdxŽŽŽŽ’ ÿŸñæ^«ŽŸù<$’´Å ²1Ž’©iŸw‰felnŸ (ÖµxŸýr±2Ž‘µS²+‘8à2ŽŽŽŽ’Æ/ Ÿñæ^«ŽŽŽ’׋~²=ŽŸõ=•’ê…Ï(µxŸü^ÿ±2Ž›µS²+–8à2)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(1)“¸“²1“¸Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µxŸûÞÿ±2Ž˜²+“2)Ž’ê…ÏŸu‰fe†CÛŸ (Ö‘/Iî(µxŸýr±2Ž‘µS²+‘8à2)Ÿýr±2ŽŽŽŽŽŽŽŸµ«’׋~²=ŽŸù<$’õ±¸²2µxŽ’ê…ÏŸw‰fe'¯ÿŸ (Ö²(µxŸýr±2Ž‘µS²+‘8à2)Ÿýr±2ŽŽŽŽŽ’iµ:ŽŽŽŽŸ3$f‘[8ã²14.ŽŽŽŸù<$‘n354µx–8ณµxŸü^ÿ±4ŽŽ‘n35Ÿw‰fe!#ŠŸ (Ö‘ÛŽµxŸýr±3Ž‘µS²+‘8à2ŽŽŽŽ’‰òµ:Ž¡‘m¹Solution:ŽŽŽŸ/ÛuŸçðŸù<$’Š• µdŽ’‡ºŸw‰fe 뎟 (ÖdxŽŽŽŽ’•ƒ{Ÿñæ^«ŽŸù<$’ž"²4µx–8ณµxŸü^ÿ±4ŽŽ’ž"Ÿw‰fe!#ŠŸ (Ö‘ÛŽµxŸýr±3Ž‘µS²+‘8à2ŽŽŽŽ’Àiߟñæ^«ŽŽŽ’ÑÆS²=ŽŸù<$’äÀ¤(µxŸü^ÿ±3Ž‘µS²+–8à2)(4“¸“²4µxŸü^ÿ±3Ž›|s²)“¸“²(4µx“¸“µxŸü^ÿ±4Ž˜²)(3µxŸü^ÿ±2Ž˜²)Ž’äÀ¤Ÿw‰fe—…MŸ (Ö‘7ê§(µxŸýr±3Ž‘µS²+‘8à2)Ÿýr±2ŽŽŽŽŽŽŽŸg½’ÑÆS²=ŽŸù<$’äÀ¤¸²4µxŸü^ÿ±6Ž–µS¸›8à²4µxŸü^ÿ±3Ž“²+˜8˜+˜3µxŸü^ÿ±6Ž“¸˜²12µxŸü^ÿ±3ŽŽ’äÀ¤Ÿw‰fexØŸ (Ö‘+äm²(µxŸýr±3Ž‘µS²+‘8à2)Ÿýr±2ŽŽŽŽŽŽŽŸC¾’ÑÆS²=ŽŸù<$’äÀ¤¸µxŸü^ÿ±6Ž–µS¸›8à²16µxŸü^ÿ±3Ž“²+˜8Ž’äÀ¤Ÿw‰feCŸûŸ (Ö‘ ÷þ(µxŸýr±3Ž‘µS²+‘8à2)Ÿýr±2ŽŽŽŽŽ’)“Òµ:ŽŽŽŽŸ9¥‘[8ã²28.ŽŽŽ‘mµxŸü^ÿ±4Ž–µS¸Ÿù<$‘¢J²1Ž‘lŸw‰felnŸ (ÖµxŸýr±2Ž“¸‘8à²1ŽŽŽŽ‘ ´µ:Ž¡‘m¹Solution:ŽŽŽ¦Ÿù³-Ÿù<$’¤!µdŽ’¡EŠŸw‰fe 뎟 (ÖdxŽŽŽŽ’¯óŸñæ^«Ž’¶kgµxŸûÞÿ±4Ž–µS¸Ÿù<$‘¢J²1Ž‘lŸw‰felnŸ (ÖµxŸýr±2Ž“¸‘8à²1ŽŽŽŽ‘ ´Ÿñæ^«ŽŽŽ’ü²=Ž’ Î:4µxŸü^ÿ±3Ž–µS¸Ÿõ=•‘lŸù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µxŸûÞÿ±2Ž“¸‘8à²1)Ž‘lŸu‰fe8LžŸ (Ö‘NP(µxŸýr±2Ž“¸‘8à²1)Ÿýr±2ŽŽŽŽŽŽŽŸµ«’ü²=Ž’ Î:4µxŸü^ÿ±3Ž–µS²+Ÿù<$‘è„2µxŽ‘lŸw‰fe'¯ÿŸ (Ö²(µxŸýr±2Ž“¸‘8à²1)Ÿýr±2ŽŽŽŽŽ‘,OEµ:ŽŽŽŽŸ+.x‘M¸ŽŽŽ‘WxŽ‘[q˲3.2‘UU#40ŽŸmt‘WFind–UUthe“tangen¸ãt“to“µy‘"ñ²=Ÿù<$‘úKµxŸü^ÿ±3Ž‘µS²+‘8àµxŽ‘úKŸw‰fe#‰Ÿ (Ö‘™Çx–8ณ²1ŽŽŽŽ‘$¦\at“(2µ;‘ª¨²10).ŽŽŸŽŒ‹ä y ý£ ? ýä‘W¹Solution:ŽŽ’ŒŽ*ºBy–“çthe“quotient“rule,ŽŸ&ï9Ÿô÷Ÿù<$’‹í¢µdŽ’‰Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’–Û}Ÿñæ^«ŽŸù<$’Ÿk$µxŸü^ÿ±3Ž‘µS²+‘8àµxŽ’Ÿk$Ÿw‰fe#‰Ÿ (Ö‘™Çx–8ณ²1ŽŽŽŽ’¼ÁàŸñæ^«ŽŽŽ’ÎT²=ŽŸù<$’á¥(µx–8ณ²1)(3µxŸü^ÿ±2Ž›µS²+“1)“¸“²(1)(µxŸü^ÿ±3Ž˜²+“µx²)Ž’á¥Ÿw‰feƒÕJŸ (Ö‘0Pß(µx–8ณ²1)Ÿýr±2ŽŽŽŽŽŽŽŸg½’ÎT²=ŽŸù<$’á¥2µxŸü^ÿ±3Ž–µS¸›8à²3µxŸü^ÿ±2Ž“¸˜²1Ž’á¥Ÿw‰fe;ØÝŸ (Ö‘ R©(µx–8ณ²1)Ÿýr±2ŽŽŽŽŽ’$µµ:ŽŽŽŽŸ&ï8‘WºEvaluating–“çthis“at“µx–Dz=“2–“çºgives“²3º,“so“the“e‘ÿ}'quation“isŽŸ'´’¦ø}µy‘”¹¸›8à²10–Ç=“3(µx˜¸˜²2)µ;‘ ㈺orŽ‘pMµy‘"ñ²=“3µx˜²+˜4µ:ŽŸür‘M¸ŽŽŽ‘WxŽ‘[q˲3.3ŽŸÔ¾‘WFind–UUµdy[Ù=dx²,“ifŽŸ©|‘`8ä2.ŽŽŽ‘mµy‘"ñ²=‘Ç(4–8à+“5µx²)Ÿü^ÿ±4Ž‘|s²;ޤ©|‘m¹Solution:ŽŽ’¢Ž*ºBy–“çthe“p‘ÿ}'ower“rule“(and“chain“rule),ŽŸ]WŸù<$’”[µdŽ’‘‚Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’žC²(4–8à+“5µx²)ŸûÞÿ±4Ž‘C‹²=›Ç4(4“+“5µx²)ŸûÞÿ±3ŽŸù<$‘‹4µdŽ‘¯¦Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘Îg²(4“+“5µx²)˜=˜20(4“+“5µx²)ŸûÞÿ±3Ž‘|sµ:ŽŸ ‘`8ä²7.ŽŽŽ‘mµy‘"ñ²=Ÿù<$‘”1Ž‘úKŸw‰fe(3Ÿ (Ö(3µx–8à²+“1)Ÿýr±4ŽŽŽŽŽ‘-a ²;ŽŸ÷L‘m¹Solution:ŽŽŽ©ÝWŸù<$’ŽNµdŽ’‹(ÀŸw‰fe 뎟 (ÖdxŽŽŽŽ’—G²(3µx–8à²+“1)ŸûÞÿ·±4Ž‘ ƒŒ²=›Ç¸²4(3µx“²+“1)ŸûÞÿ·±5ŽŸù<$‘Ë5µdŽ‘ ï§Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘h²(3µx“²+“1)˜=Ÿù<$‘0‚¸²12Ž‘úKŸw‰fe(3Ÿ (Ö(3µx“²+“1)Ÿýr±5ŽŽŽŽŽ‘-a µ:ŽŸ|T‘[8ã²20.ŽŽŽ‘mµy‘"ñ²=‘Ç(1–8ณ²3µx²)Ÿü^ÿ·±1Ž‘ ¼t²;Ž¡‘m¹Solution:ŽŽŽ¦Ÿù<$’šTéµdŽ’—y[Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’£˜²(1–8ณ²3µx²)ŸûÞÿ·±1Ž‘ ƒŒ²=›Ç¸²(1“¸“²3µx²)ŸûÞÿ·±2Ž‘ ¼t²(¸²3)˜=˜3(1“¸“²3µx²)ŸûÞÿ·±2Ž‘ ¼tµ:ŽŸÊB‘M¸ŽŽŽ‘WxŽ‘[q˲3.3‘UU#43ޤÔ¾‘WFind–™qthe“equation“of“the“tangenš¸ãt“line“to“the“curv˜e“µy‘”u²=‘8œ(µxŸü^ÿ±3Ž–⻸›fHµxŸü^ÿ±2Ž“²+˜µx²)Ÿü^ÿ±8Ž‘ä²atŽŸ ‘Wthe–UUpGoin¸ãt“(1µ;‘ª¨²1).ŽŸ'µ‘W¹Solution:ŽŽ’ŒŽ*ºThe–“çderivative“isަŸù<$‘{}*µdyŽ‘{CŸw‰fe 뎟 (ÖdxŽŽŽŽ’Š(ñ²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(µxŸûÞÿ±3Ž–µS¸›8àµxŸûÞÿ±2Ž“²+˜µx²)ŸûÞÿ±8Ž‘C‹²=‘Ç8(µxŸûÞÿ±3Ž“¸˜µxŸûÞÿ±2Ž“²+˜µx²)ŸûÞÿ±7Ž‘|s²(3µxŸûÞÿ±2Ž“¸˜²2µx˜²+˜1)µ:ŽŸõ„‘WºEvaluating–“çat“µx–Dz=“1–“çºgives“16,“so“the“e‘ÿ}'quation“isŽŸ'´’¤1^µy‘”¹¸›8à²1–Ç=“16(µx˜¸˜²1)‘ 8àºorޑťµy‘"ñ²=“16µx˜¸˜²15µ:ŽŸür‘M¸ŽŽŽ‘WxŽ‘[q˲8.3Ž¡‘WFind–UUthe“deriv‘ÿqÇativš¸ãe“µdy[Ù=dx“²of“the“giv˜en“function.ŽŽŸŽŒ‹#å y ý£ ? ýç5£‘`8ä²2.ŽŽŽ‘mµy‘"ñ²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö2ŽŽŽŽ‘ -(µeŸûÞÿó 0e—rcmmi7´xŽ‘Aĸ‘8àµeŸûÞÿ·´xŽ‘ Hå²).ŽŸÍБm¹Solution:ŽŽ’¢Ž*ºThe–“çderivative“isŽŸ/ˆåŸé÷"Ÿù<$’“€µdyŽ’’ÍnŸw‰fe 뎟 (ÖdxŽŽŽŽ’¡³G²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽŸù<$‘L?²1Ž‘L?Ÿw‰feŸ (Ö2ŽŽŽŽ‘s(µeŸûÞÿ´xŽ‘Aĸ‘8àµeŸûÞÿ·´xŽ‘ Hå²)Ž’ø›Ÿ=ŽŸù<$’ •ð1Ž’ •ðŸw‰feŸ (Ö2ŽŽŽŽ’sÌŸñæ^«ŽŸù<$’ßµdŽ’sŸw‰fe 뎟 (ÖdxŽŽŽŽ’("4²(µeŸûÞÿ´xŽ‘ä²)‘8ฟù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µeŸûÞÿ·´xŽ‘ Hå²)Ÿñæ^«ŽŽŽŽŸ’ø›Ÿ²=ŽŸù<$’ •ð1Ž’ •ðŸw‰feŸ (Ö2ŽŽŽŽ’sÌŸñæ^«Ž’Ð@µeŸûÞÿ´xŽ‘Aĸ‘8à²(µeŸûÞÿ·´xŽ‘ Hå²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(¸µx²)Ÿñæ^«ŽŽŽŽŸµ«’ø›Ÿ²=ŽŸù<$’ •ð1Ž’ •ðŸw‰feŸ (Ö2ŽŽŽŽ’É$(µeŸûÞÿ´xŽ‘AIJ+‘8àµeŸûÞÿ·´xŽ‘ Hå²)µ:ŽŽŽŽŸ5U‘`8ä²5.ŽŽŽ‘mµy‘"ñ²=‘ǵeŸü^ÿ´eŸüûróO Ú\cmmi5³xŽŽ‘À¢².ޤ‘m¹Solution:ŽŽ’¢Ž*ºApplying–“çthe“chain“rule“with“µu–Dz=“µeŸü^ÿ´xŽ‘œËºgivesŽ©§jŸù<$’¨oRµdyŽ’¨5@Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’·²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(µeŸûÞÿ´eŸüûr³xŽŽ›À¢²)µeŸûÞÿ´eŸüûr³xŽŽŸù<$‘ ÏcµdŽ‘ óÕŸw‰fe 뎟 (ÖdxŽŽŽŽ‘–²(µeŸûÞÿ´xŽ‘ä²)–Ç=“µeŸûÞÿ´eŸüûr³xŽŽ˜µeŸûÞÿ´xŽ‘Ïü²=“µeŸûÞÿ´x±+´eŸüûr³xŽŽ‘mµ:ŽŸ€˜‘`8ä²6.ŽŽŽ‘mµy‘"ñ²=‘ǵxŸü^ÿ´eŽ‘„+²+‘8àµeŸü^ÿ´xŽ‘ä².Ž¡‘m¹Solution:ŽŽ’¢Ž*ºThe–“çp›ÿ}'ower“rule“with“exp˜onent“µe“ºgivesަŸù<$’±“ùµdyŽ’±YçŸw‰fe 뎟 (ÖdxŽŽŽŽ’À?À²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(µxŸûÞÿ´eŽ‘KK²)›8à+Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µeŸûÞÿ´xŽ‘ä²)–Ç=“µexŸûÞÿ´e·±1Ž‘ÀŸ²+˜µeŸûÞÿ´xŽ‘äµ:ŽŸÿâ‘`8ä²8.ŽŽŽ‘mµy‘"ñ²=‘Ç(3µx–8à²+“1)µeŸü^ÿ·±3´xŽ‘EX².Ž¡‘m¹Solution:ŽŽ’¢Ž*ºThe›“çpr–ÿ}'o“duct˜rule˜givesŽŸºŸúìÍŸù<$’šðJµdyŽ’š¶8Ÿw‰fe 뎟 (ÖdxŽŽŽŽŽ’°Ôù²=Ž’Âœ(3µx–8à²+“1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µeŸûÞÿ·±3´xŽ›EX²)“+“(µeŸûÞÿ·±3´xŽ˜²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(3µx“²+“1)ŽŽŸBm’°Ôù=Ž’Âœ(¸²3)(3µx–8à²+“1)µeŸü^ÿ·±3´xŽ‘~8²+“3µeŸü^ÿ·±3´xŽ‘ p²=‘Ǹ²9µxeŸü^ÿ·±3´xŽ‘EXµ:ŽŽŽŽŸ!º‘M¸ŽŽŽ‘WxŽ‘[q˲8.4‘UU#2ŽŸ‘WFind–UUµdy[Ù=dx“²in“eac¸ãh“case:ŽŸ‘[8ã(a)ŽŽŽ‘mµy‘"ñ²=‘ÇlnŽ‘ o(3µx–8à²+“2);Ž¡‘m¹Solution:ŽŽ’¢Ž*ºThe–“çchain“rule“givesަŸù<$’ŸZµdŽ’œ~zŸw‰fe 뎟 (ÖdxŽŽŽŽ’ªGã²lnŽ’²:(3µx–8à²+“2)›Ç=Ÿù<$‘rI1Ž‘úKŸw‰feïüŸ (Ö3µx“²+“2ŽŽŽŽ‘#VZ¸Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(3µx“²+“2)˜=Ÿù<$‘rI3Ž‘úKŸw‰feïüŸ (Ö3µx“²+“2ŽŽŽŽ‘!zµ:ŽŸ1^‘[Dz(e)ŽŽŽ‘mµy‘"ñ²=‘ǵx‘ª¨²lnŽ‘ ª§µx–8ณµx²;Ž¡‘m¹Solution:ŽŽ’¢Ž*ºThe›“çpr–ÿ}'o“duct˜rule˜givesަŸù<$‘uM4µdŽ‘rq¦Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘~g²(µx‘ª¨²lnŽ‘ ª§µx²)›8ฟù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(µx²)–Ç=“µxŸù<$‘ÁdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘üœ²lnŽ‘ü›µx˜²+˜lnŽ‘ 8ßµx˜¸˜²1“=“µx˜¸Ÿù<$‘Ç¡²1Ž‘lŸw‰fe·Ÿ (ÖµxŽŽŽŽ‘ B²+˜lnŽ‘ 8ßµx˜¸˜²1“=“lnŽ‘ ǵx:ŽŸ\ ‘]*ª²(j)ŽŽŽ‘mµy‘"ñ²=‘ÇlnŽ‘ o(lnŽ‘UW(µx²)).ŽŽŸŽŒ‹- y ý£ ? ýä‘m¹Solution:ŽŽ’¢Ž*ºThe–“çchain“rule“givesޤt)Ÿù<$’´èµdŽ’±?ZŸw‰fe 뎟 (ÖdxŽŽŽŽ’¿òlnŽ’Ç^(lnŽ– ÿÿµx²)›Ç=Ÿù<$‘ UØ1Ž‘úKŸw‰fe·Ÿ (ÖlnŽ“µxŽŽŽŽ‘y¸Ÿù<$‘G¡µdŽ‘lŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ŠÔ²(lnŽ“µx²)˜=Ÿù<$‘ º1Ž‘úKŸw‰feߟ (Öµx‘ª¨²lnŽ‘ ª§µxŽŽŽŽ‘F]:Ž©)è‘M¸ŽŽŽ‘WxŽ‘[q˲3.4ŽŸY‘WFind‘UUµdy[Ù=dx².ŽŸ²‘`8ä2.ŽŽŽ‘mµy‘"ñ²=‘ÇcosŽ‘*§(µxŸü^ÿ±5Ž‘µS²+‘8à1).ŽŸ²‘m¹Solution:ŽŽ’¢Ž*ºBy–“çthe“chain“rule,Ž¡Ÿù<$‘~õdŽ‘{焟w‰fe 뎟 (ÖdxŽŽŽŽ’‰°í²cosŽ’—|(µxŸûÞÿ±5Ž›µS²+‘8à1)–Ç=“¸‘ª¨²sinŽ‘ ñÆ(µxŸûÞÿ±5Ž˜²+–8à1)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µxŸûÞÿ±5Ž˜²+“1)–Ç=“¸²5µxŸûÞÿ±4Ž‘'²sinŽ‘n9(µxŸûÞÿ±5Ž˜²+‘8à1)µ:ŽŸ¸!‘[8ã²13.ŽŽŽ‘mµy‘"ñ²=‘ÇtanŽ‘8â(sinŽ‘ ñƵx²).ŽŸTU‘m¹Solution:ŽŽ’¢Ž*ºSinc‘ÿ}'eŸù<$‘¢¨µdŽ‘ÇŸw‰fe 뎟 (ÖdxŽŽŽŽ‘åÛ²(tanŽ‘rµu²)–Ç=“secŽ‘œmŸûÞÿ±2Ž‘ȵuŸù<$‘33duŽ‘33Ÿw‰fe íðŸ (Ö1dxŽŽŽŽ‘ TVº,–“çwe“haveŽŸ^kŸù<$’ЍdµdŽ’‡ÌÖŸw‰fe 뎟 (ÖdxŽŽŽŽ’•–?²tanŽ’¤ (sinŽ› ñƵx²)–Ç=“secŽ‘œmŸûÞÿ±2Ž‘à²(sinŽ˜µx²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(sinŽ˜µx²)“=“(cosŽ‘7µx²)‘ª¨secŽ‘ýŸûÞÿ±2Ž‘üp²(sinŽ˜µx²)µ:ŽŸJ5‘[8ã²25.ŽŽŽ‘mµy‘"ñ²=Ÿù<$‘úKcosŽ‘‚µxŽ‘úKŸw‰feÅSŸ (Ö‘‡xŽŽŽŽ‘òѲ.ŽŸì‚‘m¹Solution:ŽŽ’¢Ž*ºBy–“çthe“quotient“ruleŽŸ#6”Ÿù<$’—dyŽ’—´œŸw‰fe 뎟 (ÖdxŽŽŽŽ’¦šu²=Ÿõ=•‘úKµxŸù<$‘ÁdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ› Qô²(cosŽ‘7µx²)–8ณ²cosŽ‘GµxŸù<$‘ÁdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ˜xŽ‘úKŸu‰fecœÂŸ (Ö‘,´šxŸýr±2ŽŽŽŽŽ‘k‘X²=Ÿù<$‘úK¸µx‘ª¨²sinŽ‘œnµx–8ณ²cosŽ‘GµxŽ‘úKŸw‰feCÏõŸ (Ö‘Î3xŸýr±2ŽŽŽŽŽ‘Hýsµ:ަ‘M¸ŽŽŽ‘WxŽ‘[q˲3.4‘UU#34ŽŸY‘WFind–Û¾the“v‘ÿqÇalues“of“µx“²for“whicš¸ãh“the“graph“of“µy‘"ñ²=‘ǵx–E³²+“2‘ª¨sinŽ‘œnµx–Û¾²has“a“horizon˜talŽ© ‘Wtangen¸ãt.ŽŸÌ¿‘W¹Solution:ŽŽ’ŒŽ*ºSetting–“çthe“derivativeŽ¡Ÿù<$’½abµdŽ’º…ÔŸw‰fe 뎟 (ÖdxŽŽŽŽ’Ƥ•²(µx–8à²+“2‘ª¨sinŽ‘œnµx²)–Ç=“1–8à+“2‘ª¨cosŽ‘¸ßµxŽŸš‘Wºe›ÿ}'qual–“çto“zer˜o“and“solving“for“µx“ºgivesŽŸ>†Ÿùæd’Ææ²1–8à+“2‘ª¨cosŽ‘¸ßµxŽ’ü©Á²=Ž’pß0ŽŽ¦’ÝäncosŽ’ìò¥µxŽ’ü©Á²=Ž’p߸²1µ=²2µ:ŽŽŽŽŸÌ¿‘WºThis–E happ›ÿ}'ens“when“µx–²=“¸²2µ[Ù=²3–E º(and“we“c˜an“add“to“this“any“p˜ositive“orަ‘Wne‘ÿ}'gative–“çmultiple“of“²2µ[Ùº).ŽŸÌ¿‘M¸ŽŽŽ‘WxŽ‘[q˲3.4‘UU#36ŽŸY‘WBy–}di erenš¸ãtiating“the“ rst“of“the“follo˜wing“double“angle“form˜ulas,‘‡œobtainަ‘Wthe‘UUsecond:ŽŸÌ¿’…ˆðsinŽ’‘Ð(2µx²)–Ç=“2‘ª¨sinŽ‘œnµx‘ª¨²cosŽ‘¸ßµx;‘ªª²cosŽ‘#9(2µx²)“=“cosŽ‘*§ŸûÞÿ±2Ž‘Qµx–8ณ²sinŽ‘þŸûË*±2Ž‘§µx:ŽŽŸŽŒ‹: y ý£ ? ýä‘W¹Solution:ŽŽ’ŒŽ*ºDi er–ÿ}'entiating›“çb“oth˜sides˜of˜the˜ rst˜formula˜givesޤ*,Ÿð{RŸù<$’’MµdŽ’<¿Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’(²sinŽ’©MF(2µx²)Ž’ÅË=ŽŸù<$’Û¡`µdŽ’ØÅÒŸw‰fe 뎟 (ÖdxŽŽŽŽ’ä䓲2‘ª¨sinŽ‘œnµx‘ª¨²cosŽ‘¸ßµxŽŽŸ‘¬’•?²2‘ª¨cosŽ‘7(2µx²)Ž’ÅË=Ž’×’Ÿ2[sinŽ‘ ñƵxŸù<$‘ÁdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘üœ²cosŽ‘ Óµx–8à²+“cosŽ‘GµxŸù<$‘ÁdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘üœ²sinŽ‘îbµx²]ŽŽŸ“·’•?2‘ª¨cosŽ‘7(2µx²)Ž’ÅË=Ž’×’Ÿ2(¸‘ª¨²sinŽ‘ ñÆŸûË*±2Ž‘áµx–8à²+“cosŽ‘œoŸü^ÿ±2Ž‘ʵx²)ŽŽŽ’b/Zµ:Ž¡‘WºHalving–“çb›ÿ}'oth“sides“of“this“e˜quation“gives“the“se˜c˜ond“formula.ŽŸ‘M¸ŽŽŽ‘WxŽ‘[q˲9.2ŽŸ‘WFind–UUthe“deriv‘ÿqÇativš¸ãe“µdy[Ù=dx“²of“the“giv˜en“functionŽŸ‘[8ã15.ŽŽŽ‘mµy‘"ñ²=‘ǵeŸü^ÿ±2´xŽ‘ ¯ÿ²sinŽ‘÷(3µx²).ޤ‘m¹Solution:ŽŽ’¢Ž*ºThe›“çpr–ÿ}'o“duct˜rule˜(fol‘‚Ølowe“d˜by˜chain˜rule)˜givesŽŸ5£Ÿù<$‘sµdyŽ‘sRôŸw‰fe 뎟 (ÖdxŽŽŽŽ’‚8Ͳ=›ÇµeŸûÞÿ±2´xŽŸù<$‘ µdŽ‘ 8ŠŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ó²sinŽ‘$I(3µx²)–8à+“sinŽ‘þ(3µx²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ QôeŸûÞÿ±2´xŽ‘ Ìo²=˜3µeŸûÞÿ±2´xŽ‘ ¯ÿ²cosŽ‘Ž(3µx²)“+“2µeŸûÞÿ±2´xŽ‘ ¯ÿ²sinŽ‘÷(3µx²)µ:ŽŸÿâ‘[8ã²16.ŽŽŽ‘mµy‘"ñ²=‘ÇsinŽ‘6(lnŽ‘ ÿÿµxŸü^ÿ±2Ž‘|s²).Ž¡‘m¹Solution:ŽŽ’¢Ž*ºThe–“çchain“rule“applie›ÿ}'d“twic˜e“givesŽŸºŸúìÍŸù<$’›Š[µdyŽ’›PIŸw‰fe 뎟 (ÖdxŽŽŽŽ’ª6"²=Ÿù<$‘ÕÙµdŽ‘úKŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ ²(sinŽ‘ G(2‘ª¨lnŽ‘ ª§µx²))Ž’þMj=Ž’ˆcosŽ’x(2‘ª¨lnŽ– ª§µx²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(2‘ª¨lnŽ“µx²)ŽŽŸBm’þMj=Ž’ˆ2µxŸü^ÿ·±1Ž‘ g²cosŽ‘Ê«(2‘ª¨lnŽ‘ ª§µx²)ŽŽŽ’lϵ:ŽŽŸŽŒøDÃ’À;èy—óý': cmti10óò"V cmbx10ó !",š cmsy10ó O!â…cmsy7ó  b> cmmi10ó 0e—rcmmi7óO Ú\cmmi5óKñ`y cmr10óÙ“ Rcmr7óú±u cmex10ùKœßßßßßßß