lectureversion_basicfunctions_1013

lectureversion_basicfunctions_1013 - “ xty x w ‘ † — t...

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Unformatted text preview: “ xty x w ‘ † — t — ™ † ‘‰ ‡ † „‚ š ‘‰ {§˜˜……”†•YŠYru…‚“9rˆ…©ˆ”†s”t‘ ˆ ›x x ™‰ ‘ • † ‘‰‡ † ‚ š ‘‰ ‘ ›x †— ‚ “ † ‘‰‡ † ‚ xtyx w ‘ x ™‰ š ‘ –y „ † ‘tyx w w9#ˆr‚Ÿv‚“9rr…„&ˆ”†s”tˆžnpY!…‚“9rˆ…„$˜slv…”†pˆ’vˆ”†6˜v“™‚œv”“’slv…”†‘ ˜“ gx‰t ‰ ›x † x ™ {x’sŠD9ˆƒG‰ “‰ ‰ „ g ‘ t— ‘ š ‘t „ k †— gx‰t t „ ‘ wxy g — ‚“ ‰ „ g‰ „ “ x ™‰ š ‘ ‘ — „ † ‘ ‘t “ g – “ ˜pGˆr”†pn‹r‰$ˆ”†’r&rn&’spˆ…‚“GsrƒEn‹DrD…PGƒˆ”†wY™‰7v”“r‰’vx6v‡ ˜“t gx‰t — ‘‰x – ™ ‘y —–„ † ‘ ‘ ‘ k y “V k †— † ‘‰— h ‘ h „ – • † ‘‰‡ —y‰ o „t • † ‘ ‘ k k {{r’s€‰i‡’‰5ˆ‚‰sY˜•v‚“’itGrwrv§qrn#v‚“j”‡i‘ˆig’‚‰ˆ©…m…‚“9Y˜Gˆ’…v”“’‚‰ˆrn— • • • ‚“ t † ‘‰—yx g “ x ™‰ ™ ‘ d kx ‘ o – “ Y‡rv‚“j˜lqvAˆ’u’‚‰Švˆ”†r#…‡ •{”Eit‘ˆ$”†RxGr…„$†Plˆ’xˆ’gYlsˆoc˜ˆnqŽ‰vˆ‡‹…”h‹Œ˜#ˆYs˜ˆk “ €— d ™‰ ‘ y ™‰y ‚ d“ k – ™‰ ‘ —xy “‰ yx ™‰ “ o  † “ k x d hx d “t k † —‰tyx ˆ „ x d ‘ ™ d y • t g x‰t o „t y h ‘t † x x ‚ “ –y x‰ ‘ y x ™‰y „‚ € † — k x ‡ „ k x w†r¢‹u™1”‡rŠav“q‰ˆ’sTˆwr’‡”xˆg&”–’7lvw†nat6˜’ui†PGr…u…n!lˆrsy x o “ † † — ™ ‘ ™ d g x‰ „ g x‰t † ‰ ‡ † ‚ ~ | zy “ x o – „ † h — –t – “y‚ g qƒ‰nˆrn‡ƒ{i‡ˆP4tr’s…t&tˆ’sƒv‚“‘9rr…„€}{j{c‚Y4yx#rwf”hnv’u— vs…'r„ r‰ihˆrxqpnlj•ml˜j”‡i‘hˆ&vfdeGf’˜§—&AˆY†v”“’ˆ†r…‚ƒ€vfvgPTAtTrcph ‘ „ o o † —‡ kx‰ — g – “‡ “ ™ yx‰‰ – “ † • ‘‰‡ „ yx w I u d bS ` s XS q i d e I d b ` X V U IS Q I gfPPcaYWRT$RPH FBC ¡ GEDB #A¨ @ 9 4¢¤765'4321 ( £ )'&$#"!¢¤©§¥ ¤¢ 8£ % £ 00 (  %       ¨ ¦ £ ¡ t † ‘‰ ‡ † ‚ — ‘t — o– ‚ “ ‰ h x ™ ‘ † ‘‰ ‡ † „‚ ‰ “ “y x o „ ‰rv”“’ˆr…„$”‡’n”sgnsit”‘ƒˆ’‰ i†uv”“’ˆr…Ywƒqr‰‡ x ™ x • x k h‡ ‘ “‰ d ` kx h‡ ‘ x o ‘ d x‡ †x k † — — † ‘tyx w ‘– ‚ gx‰ ˆ‰ ‚•”š vEvGr”„ˆi†¤’34€b Gi†”‘‰r”†vq”hihŠprˆ™ ˆY•ˆ…‚“’˜v…i†s5Y“ ’st x ™‰ € † ‘‰‡ † h— ‘ ‘y “ y h g ‘t – “ ‚ kx‰‡ „y‰t † “‡ ‘ ‘ x‡ ‘t † ‘‰‡ † „ ˆ•…o v‚“9rr…„‚ n””†…‚š˜va”xˆ6i–’&— vs…y!’9ˆ˜srvŒit4r‰¢r”†’ž…‚“9rˆ…‚ ”‡’YrTDvˆ6€n— YŒwr”†r’‰ r€ihni‡ˆGTŒˆw‰ Yi—‰i‡’j&ˆ§&– P m…‚“9rˆ…‚ ‘t — oˆ † “ † — t ‚ “ t ‘ ‘ ™ h — ‘ g €‰ t „ ™ † ‘ ‘‰ — –x ™‰ — i • † ‘‰‡ † „ — h “ x ™ – “ ‚ kx‰‡ „y‰t † “‡ † ‘‰‡ † ‚ — dx †– — t r”k…– G‰ vs…yp’9r˜sˆ…6v”“9ˆr…„cŠ‰G˜!pY— Y¢r”†rpnlPˆvprwd ‚“ ‘ ‘ ™‰ †—‡ x †“ t „ ™ {{s”tˆ9x r€”hYr9Y— ˜˜……”†•ˆ‰ ˆ”kGesˆ…g„ “t‰ ‘ › h — „‰‡ xtyx w ‘ x ™ kx ‘ w“y t d“†‘ “ h— x† Plˆw"˜”tYuˆv“ • ŠeGwuˆv“ t d“†‘ x† tn— vw˜A€& “ v“‚ † “ “t t i y –“ k† y vs…y‚ ˆY— …“‚ – “y‚ š ‘ x o † ‘‰ ‘ ‡ “tt ……•Gi†q'…‚“§— ‰D˜’n— xˆŸ‰ˆ”„nYw rn— ng˜”yn”"”†Y— ‰ l‡n“ Gi†wˆi†”‘6ˆ „‰ …‚“§&˜…“Di†©5Y6G‰ ™‰ •tx h— k † ‚“ t ‘— g ‘ yx‡ ‚ š ‘ ‘ h x ™ † ‘‰— –y ‚ ‘ x –—t x ™ r€9nrƒ”†Y……n‰rv”“9ˆr…„E…sŸ$"r#…G…G§• ˆY— ni—”‡§&lG§vPcEd h‰‡— ›x ‘ —‰ † “‡•t † ‘‰‡ † ‚ ‰ †xyx ‘ k ™ š „ “ ™‰ k † • † ‘ ‘‰— –x ™‰— – — “ f f f −1 f −1 f f −1 x→y f y→x f −1 x f y f −1 ”x”hrŠG• ‘™ d •t † ‘‰‡ † „ ‰rv”“9ˆr…‚ ™‰ o ‚ “ t ‘— – “ x ™ – “ ‚ •‰ ‘ “ g x ‘‰—y‰t †— š ‘ ‘ ‘ g x u‰ † n“qaYAˆi†n&…Gk G‰ …s…yE• …”†…q6v‚w’j’sr”„ih”h‘ nžGi†w{i‡ˆf”•rš v¢…x ˆ Ÿ$"‘r©syncˆ…‚“9rˆ…„cnws5qr‰'rn6qr‰5G‰ ’§i—l¤r€˜…‚“‰ƒvˆvˆd yx k x — t † ‘‰‡ † ‚ ‰ “ “y x o „‡ k † — x o „‡ x ™ • kx‰ hxy hxt h‡ ™ š „ “ ™ t— ‰ ›x‰ † “ nŠD˜……‡ ”t‘rž”†‘5’‰r˜yŠd©‹¤1”‡rŠmv”“’ˆr…„4Yw5qr‰5ˆ’!”x”hYgit#˜˜……”†c”trd ™‰ x ‘ x d ™ ‘ ™ d • † ‘‰‡ † ‚ ‰ “ “y x o „‡ x ™‰ k h —‡ ‘ xtyx w ‘ ‘ ™ xtyx w ‘ †— x w— q I ` v˜˜v…i†•nƒv”™ pcg • yxwx†x™ lvˆˆŠd x‡ ‘ r”†Dq • f (8) = 83 = 512 f −1 (8) = 81/3 = 2 = 8 8 – x = f −1 (y ) = y 1/3 – – x3 = x 3 1 2 x1 = x 2 f u ™ ‘ • hg –—›x y“ ‚‰Šd …‚xr6nr9avD§ m…‚“9rˆ…„‚ n””†…‚š˜vAˆun”‰ v”“’ˆr…„‡…sx #"ˆ©— ‚€”hY”‡rGT‹it‘ • † ‘‰‡ † h — ‘ ‘y “ x ™‰ † — ™ † ‘‰‡ † ‚ ‰ †xy ‘ k h — ‘ g €‰ †x ™‰ ˆ’G• r‚”§”‰ {r’”• l˜YGlGrv‚“9rr…#ˆYl5– ‘ •‰— ™ ™‡ „t k x‰ “ † x k • † ‘‰ ‡ † „‚ — t † — x “t‰ ‘ ›x xtyx w † ™‡ „t {s”tˆ9&˜slv…”†‘ Y— 1ˆ’‚‚…‘„ † ‘‰‡ † ‚ — ‚“ xtyx w ‘ x ™    v”“9ˆr…„•an5˜˜v…i†vˆd ! }  y = f (x) = x3 f −1 (y ) f y = f (x) f −1 • x = f −1 (y ) y = f (x) ‰ h y „ ‘ h ‘ h g ›x gx‰t ‘ ™‰ x k h‡ ‘ “ k “ “ š ‘ ‘ “t •txtyx w sit”‘rv“ i†&r€r‰”‡”‘rˆ9ul˜sa”tr4ˆˆi„‰r”†6’‰ DwY”ta‚‰6˜D‰˜˜……”†‘ x w— ™ “ h †—‡ t † ‘‰‡ † ‚ kx‰— h g – “‡ xy “ – •yx wx d“ ¦ t ‘ “ “ o‰ ›x ™‡ „t v”'˜”tY— n©rv‚“9rr…„'’j”‡i‘ˆ6…usv5†…‹ea©‰wvwqD9’‰ 1r’œ”†‘ tlvrwl˜6ˆ’¢rv‚“9rr…„6i‡’n”•˜j’n”˜#Y— ’srsl˜…i†‘ ‚€”hYr’r•snpˆ…‚“9rˆ…‚ x hxt –x ™‰ t † ‘‰‡ † ‚ ‘t— o x‰—y— gxt t kx‰xy gyx‰ h — „t „ xy— t † ‘‰‡ † „ ~ | ‚ “ txtyx w ‘ x ™‰ xt „ —‡x o h o — o “y g ‘ ‘ ™ d t ‘ “ “ o‰ ›x‰ ‰t “ – h ‘ h }1jz yŒn˜˜v…i†$Gc˜ˆYq•‚€ˆYr…rP”tŠ”tr‹‰w…wqD9’s…57i†‘ r€‚‰‰”‡i‘ˆg ˆ › x ‰t ‰“ † ‘ x w“ o — ‰ —t gx‰t š ‘‰— h g – “‡ x ™‰ ‘ gx‰ k ‘ ™‰ x ™ w9xk’‰§— svnˆ5”t¤veqY6sit”‘h ’ˆl˜s'ˆ”†j”‡i‘ˆ6…s– ˆ’)i†l˜st ˜”yr7ˆd q I q H I Q bS I ¦ d § b `S i d I H s H I d bS I ¦ ¢ S H i ¢ I I S q pˆPpƒTœ4¤'¨` ATPd ŠpPtŒŠTœ4¤d ¥b 4¤cg£ŠH 4¡rci • “ † ‘‰‡ † ‚ x o „‡ x ™ ‚“ xtyx w ‘ x ™ „ — gx‰ † ‘tyx w ‘ †— t— y {xv‚“9rr…„4qr‰ƒˆ’‰ ng˜slv…”†ƒˆ‰ !r’st v”“’slv…”†˜– nŠnŠv“ •“ “ “ x o „‡ — gx‰t ‘t — o– — t— yx ™ ‘x kx d ‘ w x †—‡ gx‰t k † “‡xt x ™ {1‰nwsy4qr‰¢…—„ ˆ’sP”‡’Yr˜ž‹n2ˆr‰&‹‚xGPqo nl&’s&rv‰˜4ˆd gx‰t ‰ “ “ ’sŠYwy xqˆ„€xG‰f“el˜st vs…y‚ o ‡ ™ ¦ gx‰ – “ ‰ „ g‰ „“ x ™‰ ™ ‘ d š ‘ y ‰ „ u © s I d GˆGvŒˆ5r‰PƒGi†‰ Y— stDpPDq • gx‰t “ † ‘‰‡ —y‰ o „ „ ‘‰x – ™ ‘y— x ™‰ u ¦ s I d ml˜s‹q…‚“9Y˜Gˆ’…t©i‡5r’‚‰˜n$G†Y¨pPDq ut gx‰t y h g ‘t “ d ‚ † ‘ ‘t “ g – “‡ x ™‰ ‘ † ‘‰‡ † „ dx † ‘ ™ ‰r’s‹‚xr6”–’6‹T‰ n“ v”“r‰’vx6vAˆ’‡”tu…‚“9rˆ…‚ ˆŠ”trd u → u1/3 = (y − 1)1/3 ∗ u = y−1 ∗ y →y−1=u x = f −1 (y ) = [y − 1]1/3 . • yv“‚ †v‚“‘’’tlyrgˆ9x i‡‘n’yˆlxY‚šhY—!xˆ’‰©tr”khl‚xG€ • ‚Y“pt6˜y’‰œ†i‘ v“‚ Gi†Grwvwq tx › —o ™ ‘ –x y š ‘ h“ u¦ hg –—› Y§‚xr6nrRI – f −1 (y ) x y x= y = f (x) = x3 + 1 u t xty x w ‘ x w — ™ • k x † x h x‰ ‘y g “y g g — • ™ ‘ ™ d t † ‘‰ ‡ † „‚ ‘t — oˆ † “ † ‚ “ t š ‰˜˜v…i†ƒvrrˆ”¥ˆk ‚€’§— srvsrˆY”m1”‡rŠ‡ˆ…‚“9rˆ…€”‡’n”Tw…G†nf‰…fI • † ‘‰ ‡ † „‚ ‘t — o y x ™‰ “ † v”“’ˆr…€i’n”ˆnˆn— t h g ‘t xtyx w ‘ x ™‰ kx‰—xy‰ x d ‘ x o h „ “ d †— ™‰ y ‘t— X ¤ ¢ n— r€r6i–’7˜˜v…i†!ˆ’)’jls’7‹Œr‚!xikˆ…‹5r‰‘ n”¢l‚x’nex ¦ €££x‚‚”‘h tnY&a…rv‰Y&’…”†‘ v”“’j&˜v“G”†a”tr’pˆ”†wY— §”uˆ€‰˜ih”hŠcx fv˜˜rv‡ x‘— – ‰ † „ “‡‡— “‰ † ‘‰— –y ‚ ‘ ‘ ™‰ š ‘ ‘ ‰ ‰— ™‰ †x ™‰ xxt ‘ d U xty „ “ xˆ‰ ”†6l˜ji—h ˆ…‚“9rˆ…„!˜˜v…i†‘ 1ˆ’ž’§i—…sŸ$"rk ’‰ eG™ Gˆs"‹fˆŠd ™ ‘ yx‰ t † ‘‰‡ † ‚ xtyx w ™‡ „t x‰ ‘‰ †xyx ‘ “ d“ € k „‰t x d †x ™ i†n— j…n— vq&”–Œ5v‰qD…‹eGw”hihŠP˜˜v…i†‹r‰‘ ˆY— …‚“9rˆ…ƒ— l‰‹Tqo ‘ š — ‰ † ‰y “ g ‘ x – “ ‡ x o • y x w x d “ ™ • ‘ d xty x w ‘ t k † † ‘‰ ‡ † „‚ † x x d‰ x † ‘‰ hx xt h‡ h ‘ kxy‡ ‘ x ™‰ “‰ † ‘‰ †x‰‰— t €— g x † “ ™ ‘ ™ d ‘ x ‘‰‡x gtyx g x ™ …‚“§— sy ˜v”“‰gr€r”ors‰r”†pGŠ’cv‚“…’˜§RGrpG…A{i‡ˆP4i†t…rw9q’˜xpˆd ‰ h x ™ ‘ t † ‘‰‡ † ‚ ‘t— o yx ™‰ ‚ “ txtyx w ‘ xy €x ™‰ ™ š „ “ ™ †x wx •‰ ™ ‘ sit”‘ƒG‰ i†rv”“’ˆr…„Œi‡’n”ˆn“ Y‹˜˜……”†ƒsn— ‰Gu…G…G‰ …ˆ…v‚š˜y † d“ ‘x ™‰ ‘ t † ‘‰‡ † ‚ ‘t— t— hx‰—y— gxt Šef”yˆ©”†rv”“9ˆr…„‹i‡’n”o Y©r€’j˜Yrl˜R“ • † ‘‰ ‡ † „‚ h ‘‰ † x † “ g › x x ™ m…‚“9rˆ…Gn— …G…qˆ9ŠG‰ ‚Yc˜˜……”†$ˆv‚€”hn”9nP”t!{i‡ˆP™d„ “ xtyx w ‘ x ™‰ h— „‰‡— ‘ ™ ‘ ™ • † ‘‰‡ † „ – ™ ‘y — š h h —y „‰— † x ™ v”“9ˆr…‚ ˆ‚‰sY…v‚“EY˜r§”$G‰ x ™‰ ‰ k †— • d hx o x ‘‰‡x gtyx g yx‰‰ h x ™‰ d h ‚  Gts”t”‘h rnGƒe‚“qŒ…rw9q’˜xt’˜§— €Gfe‚“ihv“”ihh ˜S k †— t † ‘‰‡ † ‚ ‰“ “ ˆY2rv”“9ˆr…„2nwsy —t † ‘‰‡ † ‚ ‘t— o– x ™‰ ‚“ x † “ š ‘x o “ h r’ˆ…‚“9rˆ…„€”‡’Yr˜!ˆYcˆvpˆ”†xp˜itn— tn— v”“’ˆr…„P˜˜v…i†vY— 1r’n¢ˆ”†wr”†r6‚‰ŠŠ‚xrn— v“p…€sv5&‚€”hYr’r4sn— † ‘‰‡ † ‚ xtyx w ‘ † ™‡ „t ‚ “ š ‘ ‘ ‘ ™‰ ™ ‘ d h o ‰y ‚ – “‡ xy “ – h — „t „ xy t…ˆˆs'1ˆ’‰ˆ…˜nlsc”tr€…“‚ ˆYE‰rv‚“9rr…„‚ ’…llx $"rs– v#ni‘”‘h ‰˜t ‰ †x k „‰t ™‡ „t •t † “t —xy ‘ ™‰ y • k † — •t † ‘‰‡ † —‰ †xy ‘ k €yx w x –xx † ‘‰‡ † ‚ ‰“ “y x o „‡ x ™‰ k †— † ‘‰‡ † ‚ š ‘ o „‡ x ™‰•t‰ †x k „‰t y —x ‰ty ‰t “ – “ v‚“9rr…„nws‹qr‰2Gƒrn$…‚“9rˆ…„Šˆ”†rˆ2ˆ§‰…ˆˆsEYlv€ s˜”¡s…54Ed y = ex x = n(y ) N th • “ ‘ ‰x k –y ‚yx g “‰ x w— ™ h — „‰‡ — x d t † ‘‰ h „ h —‡ ky — t”hn— ˆu”†‘ ˜v“q6’gv”vr€ihn”’n$‹pˆ…‚“§— ˆ‰i‡n7sYr™ ‚ “ y x o – „ † x ™‰ x ‡ „ k xy “‰ t „ š ‘ d h — • x ‘‰ ‡ x gty x h „‚ xt „ xy “ – x ™ naq©ˆw$ˆcˆrsv’ar6Gi†Še‚“ihn”vv‚w9q’˜qg r…˜rƒ…5$ˆ’‰ ”t6l˜sAˆ”†§i‡”‘r6v‰qni‡ˆGTApYv’s5v”“’˜v…i†€Gp§lsP‹asˆŠd ‘ gx‰t š ‘‰ — h g – “‡ h — ‘ g €‰ — t — gx‰t † ‘tyx w ‘ x ™‰ ‰ —xy‰ x d xyx ™ x ‘‰‡x gtyx g x ™‰ ‰— ™‰ k † ‘ d x d • † ‘‰ ‘‰ †xyx ‘ k € k „‰t x d †x v‚w9q’˜q$ˆajr•ˆq ”hihŠc‹Dv”“§i—…sŸ$"ruGr’s$‹•G™ ™U„ — gx‰t ‘t — o– ‚“ x g € dx † — t rˆ’s2”‡’Yr˜‹n‡xGT‰ ˆŒY— ‚ “ ‘ ‘ ™‰ “‰ x w— ™ x d• † ‘ttxy g › YAˆi†ˆ€’2…‰”‹ev”“’’sˆˆ9x ™‡ „t yx ™‰“ €†— y {r’PGY•…nŒv“ €o †x ‘ d ` b …7lvrw…š 4€4”t‘ x‡ ‘ r”†’t • †x ™‰•t gx‰t š ‘‰— h g – “‡ y „ “ ‚“ x † “ t — kx k h‡ ‘  † ‘ — † ‘tyx w ‘ ˆe‰ˆl˜sPˆ”†§i‡”‘r6v‰rv”n2G…Y¢ˆˆi„‰r”†R‰ m’itŠrv”“’˜v…i†s– ‚ S gx‰t k † “‡xt x ™ ‘ † ‰— h g – “ ’suˆ…˜AG‰ i†&v”“§i‡”‘r6v‡ † ‘tyx w † k † • gx‰t ‰ty x ™ ‘ “ † ‘ ‘ ‘ † ‘‰— h g – “‡ ‘‰x …‚“’˜v…i†‘ Y— ˆY— ’sps˜q G‰ i†cq…‚“’itGrwrk„ v”“§i‡”‘r6vv”‡5– ˆ ™ y † — t –y yx g •t † ‘‰‡ † ‚ ‘t — o y „ “ ‚ “ “ d ™ ‘ d t‰y ‰ ™ ‘ ™ Dr‰‘˜n— nƒ6˜…“‚ lq‰rv”“9ˆr…„©”‡’YrArv‹n¤‹T‰ ’‚‰Šcn— st 1”‡rŠd ttx “y g š ‰ — h g – “‡ gx‰ ˆ “ d‰ — ‚ “ h „txy x ™‰ tx – “‡x t „ ™ ’‡Dsraˆ”†‘§l”‡”‘r6…ƒ’st n‹TaqYr‰r’s‹Gl5vlqo rw‰ •r’ˆ’‰sŒˆ”†‰§i‡”‘rg6v‰sgˆ…qn‹ˆvfn©’s$v”“’˜v…i†Gpˆˆi„‰ˆi†‡‹r‚ S —t gx t š ‘ — h – “‡– y „ “ ‚“ x † “ t— gx‰t † ‘tyx w ‘ x ™‰ x k h‡ ‘ x d “ † ‘ ‘ † x k h † ‘ †xw†“‡ t ‘ ‘ ‘ ™ „ t‰ ‘ › {q…“‚r‰rq ¥Gmn—”v”“‰…v……r‰a”tŠ”tr‰ A’sitˆ9x • xtyx w …˜slv…”†‘ x‰ G™x• “ kx‰ ‘y‰tx ˜‰!’9i‡˜’sly ‚“ ‘— –“ k x ™‰ ™ n¤”†Y&vˆpG¤r‰‘ U ‰— ™‰ h—yx †x ‘ d“ † ‘ jrŸn’ˆvš ”†WeGwƒx U Ae‚“xxas”ti‘hr„v“ ”†‘p‰‡i’n”s)sˆ”k’rvƒqm”hihŠd dh o‰ y — ‘t— o– kxyx ‘t † “‡ x o ‘ k† rn— u • u hg –—› w© ‚xr6nrRI – ∗ tan−1 tan−1 (y ) sin−1 (y )/cos−1 (y ) ∗ tan−1 (y ) ∗ x = tan−1 (y ) ∗ −π/2 < x < π/2 y = tan(x) ∗ y = f (x) = tan(x) = sin(x)/cos(s) sin(x) cos(x) f (x) = tan(x) −π/2 < x < π/2 • !ˆr‰˜Y…Y”“qn’r’jrŒˆ©‚xihn5Yl’sˆi†‘ ”t‘ ‰rv˜YlsˆY”‡˜…˜sitˆtv“‚ • – ™ ‘y — š h h—y „‰— † x ™‰ k h—‡ k —x‰t •t † “t —xy h — ‘y “‰ ‘ ™ y k h—‡ x o h „ “‡ ™ ‘ ™ d • † ‘‰‡ † h ‘‰ †x † “ g ›x x ™‰ ‚“ xtyx w ‘ x ™ ‚xihnAquikˆ…u{i‡ˆPwv”“9ˆr…„‚ ni—…ˆvxˆ9gˆY€˜˜v…i†Aˆd n x = n(y ) •‰„o Gˆˆ• exp−1 “ x g €‰ ‘ ™ ‚ “ t † ‘‰‡ † ‚ ‚ “ tx ‘‰ — ‘yx k x ™ € k „‰t †x ™ d h † “ ‰ „ “ o {§xGT‡itˆ‰ n‡rv”“9ˆr…„Y‹…rw§Yrw˜ˆAG‰ Drs•ˆŠvr€rvPD…qn— ‰Gv)ˆq ”hihŠd ‹œTxvsˆ‹Yi—‰lq’'6Yr){i‡ˆPd rn— 2d q©q©ˆw† „ “ k † ‘ x d €‰yx g “y g h ‘‡x gt — t — ™ ™ ‘ ™ k† †xx ‰x o yx o –„ — •…n— srv©t ”xˆ5it‘ lGP• „ ‰ † ‰t † “‡ y h „ I xyx ™ d † ‘‰ ‡ † „‚ h ‘‰ † x † “ g › x x ™ v”“’ˆr…pn— ’…lG…qˆ9uˆ’‰ 3 y = ex = exp(x) k† rn— sin(x) cos(x) e ˜ • 2 • t † ‘‰ ‡ † „‚ ‘y‰ ‘t — o “ d‰ x ™ ‰ˆ…‚“9rˆ…p”š˜’€”‡’Yr6‹Tcˆ’‰ † ‘‰ ‰“ † x‰— †yx h …‚“§— nˆA˜jr˜’‚‰n— x ™ š ‘t „ † ‘‰‡ † ‚ ‰“ “ ˆ‰ ˆ”†’rfv”“’ˆr…„6nwsy x ™‰ x ‘y h— „t „ h „ “ d x d ˆ’!’r‰˜Šd r€”hYr’rf”krv‹!‹2• ‚ n“ xtyx w x ™ ‘ ‰“ “ ˜˜v…i†‘ G‰ it7nwsy x ™‰ ‰— ™ ‰‡ ‚ x ™‰ x ‘t— ™ g –x “‰ ‰†— d x d ‚ Gujr‰ 9n— 'G¤‰¦¥’Yrr6)’u…n"‹ƒ…S kx‰†— d x d ‘ ‰ h x ™ –“ ‚ kx‰ ‘ –“ x o h „“‡ x‡ †x ™ k †— •t † ‘ m’…Y‡g‹‚‚as”ti‘cˆ‰ ……yv’˜r‰&vAqu”krv‰crˆ•rn”‰ˆ…‚“‰ ˆ‡ † ‚ yx d“ g yx šx‰ ‘ x ‘ ‘t “ g x ™‰ ‚ txtyx w ‘ x ™‰ ‰t „£ h — „‰‡ — xy — xtx ™ 9ˆr…„R‹eqfY‰’…i†v‚w’‚‰’vqŠGqY“ l˜slv…”†ŠG†sr¤r€ihn”’nŠsYa˜ˆd • • x = y 1/N y = xN N th N th y = x1/N – – • t † ‘‰ ‡ † „‚ ‰ “ “y y x š x‰ ‘ x ‘ ‘t “ g x ™ ‰rv”“’ˆr…YwY‰’…i†gv‚wr‰’vxAˆd “ r‰”‡’Yr˜5Y— ˆjls’©˜itn”h”hPc2uˆYs˜ˆrˆ„ snl5˜t — ‘t — o– t –x ™‰ ‰—xy‰ “ h— ‘ d x d k † —‰tyx k † €t—x “ xsnƒˆŸv‹ea¦ v‚€’§’n”˜'l˜sit”‘6q¤’"‰ˆ$Ž‰vˆ6ˆlGƒˆYE{{5”–‰ y— €x ™‰ •yx wx d“ hx‰—y— gxt kx‰ h x o “‰ kxx †  † “ k x‡ †x ™ k †— •“tx ‘ „ v”“’j”‡i‘ˆigr‰r#n7’sgi‡5r’‚‰˜n¢ˆ’u…o † ‘‰— h ‘ h „ – ‚“ gx‰t ‘‰x – ™ ‘y— x ™‰ € – “y‚ k x‰ ‡ „y‰t † “ vs…•’9ˆ˜srv‡ x †—‡ xtx ™‰ • h— „‰‡ qo nž˜ˆ’‹vr€ihn”’€…i„ •t † ‘‰‡ † ‚ yx d“ g yxšx‰ ‘ x ‘ ‘t“ g x ™ ‰rv‚“9rr…„6‹eq6lY‰’…”†¤v‚w’‚‰’vq'ˆd • y = f (x) = x N y = xN y = f (x) = x I q H ` X qS ¦ d bS d I I rP€¤ uT4¤WTW†pž¢ • † ‘‰ ‡ † „‚ h ‘tt “ g ‰t h g ‘t x ™ m…‚“9rˆ…g‚xr”o’’…q‡s‚xr6i–’Aˆd • • S WU I 6€ATŒ2€4¤ § ‹Trcp– 4¤d U q b `S d X b — XS q i h I ¦ ¢¡ “yx ˆ † “ † ‘ •‰ † ‘‡ x “‡ ‰t h x ™‰ • † ‘‰ †x w † “ rs‰¥ w…Ga”tr• ‰…l‚x‰6wŠsni—Aˆ’Dv‚“………v‡ € o • k †— • h o ‘y—w …”ˆYGv”xˆYi—˜nY”t‘ ‚€ˆ…uˆY”‰’…Y— sˆ…ƒY— h † “ k † — •t‰ † ‰t † “‡ xy x ™‰ xyx ™ ˆ4sˆŠd aM a0 , · · · , aM x y = P (x), = a0 + a1 x + a2 x2 + · · · + aM xM ∗ ”tmn— 6v“ˆGr€…qn’ˆY#Šv“‚ v”“§— nˆƒGad ‘ h ‘ – † h“ g h—yx †xš — y † ‘‰ ‰“ † x ™ ‚•rš vˆ• xv”“r‰rˆYfn“ l˜st x “ † ‘ ‘ k k— ‚ gx‰ © ‡”5–rr‰˜n— „pitn— 6v“ˆv5xYyx#rw†‹5v˜q‚n’r„’‡Gad | }  ©    ‘‰x ™ ‘y h ‘ – † “ – ‚“ x o – „ x – “t “ – t x ™ z {“ …o v”“‘§i‡”‘r”g‰rr#fn“ g’sg”‡‰5rr‰˜n—ƒG™‰ €…o € † ‰— h ‘ h „ – ‚ x‰t ‘ x – ™ ‘y x † ‘‰‡ † „ …‚“9rˆ…‚ yx “ g ‘ ‘t“ x ™‰ “ kx‰ hx ‰† ‰ †“‡ ‹deq¢xv‚wr‰’vxg©ˆ&’‰!’§— sy„6…n— stˆ…v— kˆY—ŒyY‰’…i†¢v‚w’‚‰’vqg † xšx‰ ‘ x ‘ ‘t“ © | }  ©  ¨ z — v”xˆY—i˜nYv…†GrqlxGr”†'ˆ‰ ’‚‰‘Š• • h o ‘y—w ‰ x k †x g k ‘ x ™ ™ d u E ž¢ €$§ ps q iS ` b ¦ ` y = 3 − 5x + 4x2 − 23x7 ∗ aN y = xN aN ∗ y = a N xN x N q d ` ` H `V b i q H I U ` P4A‡H AY4 cvPtWts bS Q ` Q S q b `S d X b Tƒ§ 4b u€ATP‹€€¤ ¢ ¥ – IH` Pcž¢ § “ h—„£x ’‰ Yrw¥‡”t‘ yx ™‰x ™ d š ‘ ‘ –yx‰x ˆ’‰GP¢ˆ”†r”†6˜’ˆk ‚ – h “ g “t‰ †x k „‰t ‰t “ – y „ y ‘t—x x ™ “‰ ‰ † h— ‘ „ £x Y“ ”xˆo…syˆ€{…Grsgsv6g…“‚ #l‚x’ne5G‰ ’€…”xnYrwrw¤g”t‘ “ ’‰ hYrw£¤pit‘ —„ x ‰“ † y “ yx ™‰x ™ d š ‘ ‘ –yx‰x k • h g –— ›x y “ nˆ5vpˆˆŠ"ˆ”†r”†6˜’ˆtv‚xr6nr5vD§ yx d“ g š ‘ k † “ gtxyy “‡ x ™‰ š ‘ h “ w – h o “y g ‰ † h — ‘ „ £ ‹eq5ˆ”†rˆ…q’s˜vƒˆ’#ˆ”†Grw…v…i†‘ ”xˆ…r…”xnYrwrw¤x †“ n—)’‰…”†‘ k’‰v……‡ qo Grdn•rw‰ Y•nwsœ— ˆ”†G‚wv……”†‘ ”xˆ…rg i x yx w † “ x t €— h — t „ ™ † —‡ ‰ “ “y š ‘ h “ w – h o “y € ‰ lqvsrAˆ©Gi†G‰”™ q©rw#— 4c†ž¢ yx g “y g x ™‰ š ‘ w— yx o – „ † q b i I ‚•v x ‘ x ™ ‚“ xtyx w ‘ x ™‰ t— ‘ ‰“ “ ˆ‰ Yg˜˜……”†ƒˆ’nP”tŒnwsy x ™ ‚“ š ‘ †—x – ‘t— o x ™ ˆ‰ YpGi†ˆYl5c”‡’n”$ˆd • x3 = (3/5)3 y = 27/125 x = 3/5 y 1/3 = (27/125)1/3 xN = y N th •yx d“ ‹eqg x N th {r’”hnav“‚ ˆq¥GAsn”• ™‡ „t h— y kx † x k xy— y v“ x ™ •t † ‘‰‡ † ‚ xtyx w ‘ x ™‰ ˆ’‰ ‰ˆ…‚“9rˆ…„a˜˜v…i†4G…• “‰ † ‘‰ ‘y‰txy x ™‰ ™ ’pv”“9”‡˜ss€G&r‰‘ U N ™‘d ’‚‰ŠE• N N th •t‰“ “ ‰nwsy x≥0 y = xN √ x=N y x ‘ ‘t“ g v‚wr‰’vx7— – – x = y 1/N – x = y 1/N t † ‘‰ ‡ † „‚ — ‘t — o– t — • y x š x‰ ‰ˆ…‚“9rˆ…#‰i‡’n”s5Yˆv‰’…”†‘ •t † ‘‰‡ † ‚ yx d“ g yxšx‰ ‘ x ‘ ‘t“ g x ™‰ x‘ ‰ x ‰rv”“’ˆr…„c‹eqglY‰’…”†6…rwr‰’…qvˆ5nY— 6ƒU • • q b `S d X b €ATP‹€€¤ § Œm PH crxŒ€A‡x• PtWtae¦ „ Idi I b i•q d ` `H q HI U`s “  %   £ ¨ 0 "&lƒ#'7 ¢¢ƒ2€£ 8 ¡•¨ lm¤©¦ ¤¢ 6% v&% @   ¨ £ ¡ £8 “ gx‰t “ † ‘ ‘ ‘ k {x’stq…‚“’”tG‚wˆ…„ ”‡5rr‰‘˜n— n— „ ‘‰x – ™ y † • gx‰t ‰ty x ™ – “y ml˜s•s˜q ˆ’‰ vs…‚ ‰GˆGvvxG'…o ˆ”kG‚wˆ&ˆv“ 1”‡rŠƒ”†R’sƒ…‚“’itGrwrƒrvl˜7— „ g‰ „ “ ™‰ € tx ‘ ‘ k x † ™ ‘ ™ d ‘ • gx‰t † ‘ ‘ ‘ k k † “‡xt k †— •“t gx‰ rnr{{ˆ’st ”‡’tYrtˆ…“ ‚nŠˆ†v“„ ‘ —oy„ “ x • gx‰t yx d “ gˆyx šx‰ ‘ˆx ‘ ‘t “ g ‰ty ’s‹eqTTlY‰’…”†T9…rwr‰’…qts˜” — ‚ kxt “ g – “‡ t— ‚“ ‘ ‘ ™ †—‡ x † ™ ‘ ™ Y“ ˜vq&vŠn†nvˆi†ˆ‰ nlAˆv“ 1”‡rŠd • † ‘ ‡ † ‚ x ‘t“ –“ gx‰ ˆ“ d — ‰ v‚“‰9rr…„&’‰r’…qg6…‡œ’stTj‹T‰!p‡n— ‚ i†¢”t‘ ‘ • xt— † ‘‰‡ † „ v˜n‡ …‚“9rˆ…‚ yx d“ yxšx‰ ‘ x ‘ ‘t “ x ™ “‰ ‘ “ † € ‘y ‘ ‘t x ™ x ‘ gtx ‹eqg©Y‰’…†i•vrw‰‚’vqg¤G‰'’)†v”“‰§— ‰nˆ)i†‘ T‚‰˜ni—ih6”–’¤G‰ ’r‰r’P • }#} T y • q © …2q 2   ‰”t‘ † ‘‰‡ † ‚ x ™‰ ‚ š ‘ †—x – x ™ v”“9ˆr…„pˆ’pn“ ˆ”†rnl66Gad HI U` HI¢Idb I Sdi¢I tWts pƒ£PŠTS ƒQ Pg£€b u → 1/u = 1/xN 1 x → xN = u •xvš‰’‰…”†‘pxvwr‰‚’tv“qgu— r‰P• yx ‘‘ ™‘d u q b `S d X b x€ATŒ2€4¤ ) Q(x)   ©  D  {# }| z~    ¡  N © z  A8 @I  & 3# ) } A B # S © © …} m ~  % § © } m m © …} z ~ z ©r z # @ !P RQ  ! "   & '# #   © ™} u  {¢ }| z z~ © z }| © ©z } m m © … z ~ z } © | |  }| zu  z ~#| }g| } ©  © © y z © © …} m ~ # 7   ! "   &# 'F z §m © …} z  8 9 # 7 z# r3&E   4  6 ! " # @    6 4  A8 @I    ¡ # @ 0 1) &# 32 y = x−N Q(x) R(x) P (x) z u = xN · · y = x−N ∗ y = 1/xN ∗ P (x) ©       ¡  %   D P (x) y = R(x) = P (x)/Q(x)      & 3#  § 4H6G     % & '#    # 4    C    ( – R(x) ∗ & '# ∗ § 453# & ! "  # $ ∗ Q(x)  © § § ©§ ¥ £ ¦¨¦¤¢ • h „tx r‰r’sy hYrq ˆ’v’pGi†ˆYl‚xxm…‚“’”tG‚wˆk Y7l˜s5ˆ”†§l”‡”‘r6…A”‡5rr‰˜nYr…‚“r‰rk — † x ™‰ “‰ š ‘ k — h • † ‘ ‘ ‘ ‚ “ gx‰t š ‘‰— h g – “‡ ‘‰x – ™ ‘y— h — † ‘ ‘ • h ‘ –“ † h“ g y“‰— –“ †x ‰”tYi—6…GG‚€vqE…’§&…Gˆk ˆ k — x ™‰ š ‘ –y ‚yx b I ¦ d k † — Dn‡GŠˆ”†6˜v“xg p€¤¢ˆYv• k† ˆY— k † — y “‰ —yx – „ † x ™‰ š ‘‰‡ „y‰t † “ d q HS € o kx –y ‚ xy €x ™ h ‘ ˆYA…’§’5rwvˆ•ˆ”†9rssrv‡ DPTf§ …ƒl5˜v“5sn— Gad ‰”tn— 6– ˆ “ † h “ g ‚ “ t ‘‰ —y xy — t † ‘‰‡ † ‚ h — † ‘‰ — q b `S d X b nGG‚€vqYcv‚“j˜vYƒˆ…‚“9rˆ…„fYr…‚“§aH u €ATŒ2€€¤ § Pc` PH ib Sdi R(x) = P (x)/Q(x) P (x) – Q(x) a0 = 3, a1 = −5 a2 = 4, a3 = a4 = a5 = a6 = 0, and a7 = −23 . y = 3 − 5x + 4x2 − 23x7 kˆ†Y— it‘cx‰sy…‰GƒˆD• x šx k x ™‰ y“ u hg –—› …w§ v”xˆ&nrRI t ‘ ‘h ‘ –“† h“g — ’‚‰itŸYi—6…GG‚€vqp•i†‘ n‹l2eqŠsG…‚šrƒGad ‚“ yx d“ g ‰tx ™ ‘ ™ x ™ 7 x    ¡  ∗ ∗ x≥0 h† vr€rv“ it‘ — ™‡ „t ‚ ‘— –“ k x ™ 71r’Rn“ i†n&…GAˆ’‰ “t ˜r• tt h † „ kx † x k x o ‰“ † ‘ d ‰“ “ ’‚xrr•ˆ”¥ˆƒqŠYGm”hihŠanwsy x ™‰ • †xwx ˆGm…‡it‘ lG™ U †x h “ “ ™‡tˆ ™ ‘ ™ vwˆ{’TDv‚šru”†‘ †xxt x w— ™ h „ “ ™t „ “ € ‚“ “y g xt “ ™ d • x † “ h ‘ ‘y ˆ † “ † — ‘ xt—‡ ‘ ™‰ l‰˜Œvrv”krvˆ’v…vnwsr€˜vˆŠ……G…”ni—G‚w˜’‰ DvˆƒpitŒ˜n2”trp”†‘ r‰r’s$5npGA…rwv7v”“’‚‰’vq&vtYŒ˜ˆs…Œ…sx $"ru‹T$ˆŒ§”’‰ h „txy x – —t x ™‰ x ‘ š † ‘ ‘t “ g – “‡ ‚ “ tyx ky “ ‰ †xy ‘ k “ d‰ x ™‰ ‰ — ™ ‰‡ ‚ x ™‰ “t 9Y— 2GŒ˜Y• Y‹6n‡ˆ’fYGf”t‘ ”•v x n‰sl˜j&ƒv”“’‚‰’vq&vxn“ t— x –—t x ™‰ ‰“ † ‘•tyx‰‰— – † ‘ ‘t “ g – “‡ ‚ y x ky “ x ™‰ k † • t † ‘‰ ‡ † ‚ ‰ † xy ‘ k “ d‰ t — ™ x † “ † x ™ d •h —y x † x š ˆ˜v2Gl• ˆY— ‰rv”“’ˆr…„E…sx #"ˆ€‹Tn”‹G…ƒˆŠln’ˆYƒ† S u“ k † — ™ ‰ – h o “y g x ™‰ y ‚ ‰ † ‘ †x w † “‡ ‰t “ – {xˆYrP§— ‚xrvsrAˆ‹…“‹…‚xrv……sv6”t‘ t”xr‰”‘ihˆio’’vqu‹T¢GYalvˆ{i‡ˆP6ˆ”†w1”‡rg ˆr‰#˜vwˆ{4ˆw!n‡ ‘ ‘ ‘tt “ g “ d‰ x ™‰ ‚ “ yx wx ™ ‘ ™ d š ‘ ‘ ‘ •yx ™ ‘x xt “ “ ™‡ t „ ™‰ † —   x† ` ˆ4„ A  z  ©   0 ” z   |  3&#  } © E# ! ©   ! # |    © '&# ©  | z # M y = xN/M x≥0 M th g ◦f f g x ¡ ¢ u → uN = x1/m N th N N “ k † “‡xt yx d“ xˆ…˜l2eqg “ gx‰t Y© ’sui†‘ • gx‰t v¦ ’su”†‘ „ • ty ‰“ “ ‰s˜q nwsy „ w© tg A` u HI H • gx‰t v¦ ’s•”†‘ „ M th “ © gx‰t v’s•”†‘ ∗ f ◦g ∗ ∗ x → x1/M = u y = xN/M = x1/M · x → xN = u u → u1/M = xN 1/M y = xN/M = xN · 1/M “ k † “‡xt ‰“ “ xˆ…˜ŠYwsy •‰ty yx d“ s˜” ‹lxg „ Y¦ tg A` u HI H utyx ky “ ‰ †xy ‘ k “ d ‘ kxt “ g – “‡ x o † —‡ t gx‰t xtx ™ ‰˜G˜……sx #"ˆ6‹T‰ i†u˜vq&vAq•Yr’sc˜lGad † ‘‰‡ † ‚ ‰“ “ m…‚“9rˆ…„anwsy x ™‰ yx ™‰“ x ™‰ • † ‘‰‡ † ‚ yx d“ ˆ‹ˆncGGm…‚“9rˆ…„‡‹eqg x ™‰ š ‘x G©ˆ”†qo gl˜s•ˆ†v†m…‚“9rˆ…„v’r‰’vx6v‡ ’st n2d !— ˜”tn•sn— x‰t x “ • † ‘‰‡ † ‚ x ‘t “ g – “ gx‰ ˆ“ ‰ “ h— xy ™‡ „ {rDq “t • k hx‡ †—‡ †xx o x w— ‚ h y “‰‡ ‚ † “ – – “‡ “ † x w— ‰™‰‰‚xt˜…˜n— &v56…6ˆA…r™ ˆY— ˜D‚xihrnul‰qAvr™ k† ty “ ‡ ‚ † – “ h — x – „tt — x •t š ‘ ™‰ h g ‘t “ • xyx ™ d •tyx šx‰ ˜v’‰9n— 5v“56–…‡xihn€5r’’ng‹d ‰…Gi†ˆ#™€r‚”‘r6i–’©’‰ …lGPv‰slY‰’…”†‘ x ‘ ‘ “ g ™‰ v‚wr‰’tvx!n“xo ™‘d ’‚‰Šr• –y ‚ x ™‰ x w — ™ t † ‘‰ ‡ † „‚ xt x ™ s…“ƒˆ’A…rŠˆ…‚“9rˆ…A˜lGad u €ATP‹€€¤ § pI ps ŠATP‹Œf§ ƒ…Œrcps q b `S d X b H U` i b `S d X i H I QS dS q ` – N th M th ∗ N th ∗ M th y = xN/M N M ∗ y = xN/M N, M ©  ¢” © …} ~ z } 1| © ©  z | | ” © } ~ z …} z  z| }  } © © z } nz …} z y œ ~ rz z  z § ¨| } } 1| ©  z z } |© | © } © © 1A y ~ }|  | © l y © v ~} ” © } } z | © © …} ~ z } 1| © ¤¦rz q z © | © © } | z r }| © …} ~ z } z } © © © | t y | © } …} z A }| z | j{~ |z z © ¤ ¥ } …} {| © © | £ © © } z © } ¢ | ©  | |  7 © jz }} |  ¡# ¢7 C  A $   # @ 4     )  0 ! )   ¡  4 4 H  #  ¡# ¢2 &'#  )    0 0 &  C ¡ ¢ #7    ! # ¡# ¢7 4 &  C          ! 0 4 & 3#    & #$ 0    !  !  &  0  # 7#  ) #7 &'1 #   4 )  #  #       0   4 M = xN M xM /xN = xM −N M  N xN     # $  0   4 # ¡# ¢2     C ! # E ¡# ¢7  & 3#  & 3# ¡ 4    #     # $ ! # ! "2 # ) ¡# ¢2  ¡    ! M>N xN xM = xN +M   C  # 2  4 & &   0 #  0 &   # ¡ ¢ #2 4   C #  0 ) ! 4    kx‰“ †x k ’nˆˆ”t‘ ™ ‘ ™ d •“ gx‰t “ † ‘ ‘ ‘ „ ‘‰x – ™ ‘y— † 1”‡rŠD{ql˜sxv‚“’itGrwr…k#”‡5rr‰˜nun— „ –y u s I d s…“‚ G¡pPDq ˜ x’sŠYwspAˆvurn— “ gx‰t ‰ “ “y — x † “ k † gx‰t yx d “ g x † “ •t gx‰t ‘t — o “ d‰ ‚ † ‘ ‘t “ g – “‡ x ™ l˜s#‹eqž— ˆv‰ˆl˜sv”‡’Yr'‹Tan“ v”“’‚‰’vq&vuˆ’™‰„ r‰r’spˆ€vu˜ŒG˜…Œˆ’‚‰"i†‘ ˆ&ˆ”†’…q6v‰m…‚“§’q…gYwy h „txy x ™‰ ‰x š “‰ yx ky “ yx ™ ‘x –x ™‰ š ‘t “ g – “‡ • † ‘‰—yx g “ ‰ “ “ x † k † † ‘‰—yx g“ yx d“ ˆv“ rn— v”“§’q…A‹lxg x † –y ‚yx u k † q s I d ˆv“ s…“qg w© rn— ¦ ppPDq ut d h ‚ t— kx‰‡ „y‰t † “ ‰Še”“”hv“n•l˜ˆ˜sˆ…‡ •v‚“9rr…„Œ’r‰’vx6vul˜sTˆ    '&# ”tG‰s…’9n— &v5&v5ˆ† vTr‰‰i‡”‘r6”–’v“‚ † ‘‰‡ † ‚ x ‘t “ g – “‡ gx‰t — ‘ •ty “‰‡ ‚ † “ – – “‡ “ • € ‘ h g ‘t y • š ‘ w— ™ •tyx šx‰ ‘ x ‘ ‘t “ ™‰“ ”Gi†Gr‹‰˜v‰’…”†¤vrw‚‰’vqg Yqo ™‘d r‰Š‹• † ‘‰ ‡ † „‚ x ™ …‚“9rˆ…!ˆd &  • x−N/M ∗ 1/u = 1/xN/M u = xN/M ∗ N th y = x−N/M M th N, M u €ATP‹€€¤ q b `S d X b – H U ` s i b `S d X i H I QS i I pI pƒ ŠATP‹Œf§ ƒ…Pd g¢ €b § • q6’‡nYgˆvc‚xvˆn— ˜’cˆ‰ ”†‘ x o “‰ tx‘—‰ x † “ h š † ‘y‰ x ™ t h š † — ‰ ™ ‘ ˆ † “ † “ d‰ x ™‰ ‚ “ ™ ‘ ™ d † “ š ‘ k †x gx —x ‘t “ g g “ k † —‰ †x‡ — x‚…GYA……‚š˜y Dvˆ•‹T©Gtn"1”‡rŠž…uGi†ˆrqlGk ’r‰’…qrv– ˆY— …‰ns£ ˆ —– ‚“ š ‘ † —x – x ™‰ • —xt „x †x‰“ g € ™– k †— —x ‘t “ g g “– • —‰ †x‡— k—– ‚ “ t‰ gx wkY˜cn©G†iˆYl5ƒGGˆl˜ˆˆ’nxG…sfrn5’‚‰’vqrvsr…ns£ n’gY‡G‡ ˆ † “ kx ‘ x gt h— ‘y‰x – “x š x ™‰ ‚“ t –yx‰ ‘ †x ‘ š xy — t † ‘ ‘ † x k x ™ Dv‰‡'” ‰‡x’!€rihni‡s6v‰v5G2nc6˜’"i†'…rwv5sYArv”“r‰r”¥ˆ6ˆd it”xvˆnŠ……‚š˜y ‘ t hš†— ‰™ ‘ ˆ † “ t “ ™ ‚ “ † “ • š † ‘y‰ h š † ˆ‰ ™ ‘y — ‚ “ tx ‘t ‰ †xy ‘ k x ™‰ ‚ “ t ‘‰— w…G† x˜vˆŠd‡Y6xG…E…‚xhvˆn— ˜’6‚xvˆn— s…v”šs!anAlGik’ƒ…llx $"rvˆ‡nA…‚“§’y x ™‰ ‚“ † š ‘ †— – ‘ ‰ “xš ‘t— t ‘ t t— ™ ™‡ ˆ’cnžxˆv“ G†iˆYlx5!”‡˜yx5–vY!”‡’n”o ’‚‰vY— Yrf1n— I k† rn— • • • • • † ‘‰‡ † ‚ “ ‘y „ ‘y‰x – “ † “ ‘y xy— xyx ™ ‰trv‚“9rr…„ŠD‚š˜‰ u”‡˜5…G…Y‚š˜‰ sn6sˆd q b `S d X b €ATP‹€€¤ § TPPž¢ 4cƒ¢ ŠŒ)€¤‹Y© „ XS H d I ` b ` S H d I ¦ d “ sec(θ ) θ θ cot(θ ) 6 sin(θ ) cos(θ ) tan(θ ) csc(θ ) HOWEVER, the ratios of sides of the big triangle are the same as the corresponding ratios of sides of the small triangle because the extra factor of 2 occurs in both the numerator and denominator and hence cancels out (i.e., THE RATIOS ONLY DEPEND ON SHAPE, NOT SIZE) θ hb a A=2a B=2b H=2h (π/2)−θ The shape is determined by the angle θ Knowing a triangle is a right angle triangle and that one of the other angles is θ fixes the other (non−right) angle and hence the SHAPE of the triangle, but NOT its size FIG. 1: FIG. 2: The geometrical notions "adjacent", "opposite" and "hypoteneuse" and the relation of internal angles for right angle triangles φ h b π/2 θ φ=(π/2)−θ since the sum of the 3 interior angles is π a Hypoteneuse, h, is the side opposite from the right angle The side a is "opposite" to φ but "adjacent" to θ The side b is "opposite" to θ but "adjacent" to φ xy — t † ‘‰ ‡ † „‚ xt x ™ ‚ sYr• rv”“9ˆr…g˜G‰ n“ t š ‘ †—x – ‘t— o ‰t “ – x ™‰ x ‘ š ™ ‘ ™ •t † ‘ ‘ † x k ‘y‰x – “x š ‘ h g ›x x ™ vˆ”†rnl5Pi‡’n”tsv5Œˆ…rwv&{i‡ˆPd ‰ˆ…‚“r‰r” Ga”‡˜5vYr‰”‡”‘rˆ9Pˆd sin(θ ) = opposite/hypoteneuse cos(θ ) = adjacent/hypoteneuse tan(θ ) = opposite/adjacent csc(θ ) = hypoteneuse/opposite sec(θ ) = hypoteneuse/adjacent cot(θ ) = adjacent/opposite . ˆ9v‚šY˜– G‰ ’’§— s¤qfY&rv”“9ˆr…„‚ ‚š˜5rv“plGn¤ˆ’v§”‰ lˆ’t x h — –x ™ “‰ kx‰ hxy x o † —‡ t † ‘‰‡ † ‘y‰ y „ ‚ yx ™‰“ x ™‰ ‰— ™ d“ ™ t † ‘ ‘ † x k xtx ™‰ •t † ‘‰‡ † ‚ ‘y‰ — ‘t— o– y „ “ t— x ‘t “ k †— x ‘t x ™‰ š ‘ ‘ ˆ…‚“r‰rq¥GŒ˜ˆ…‰rv‚“9rr…„ƒ‚š˜c‰”‡’Yr˜$rvpYŒˆ”†’…‡ ˆYŒˆ”†’€ˆ$ˆ”†wY— d xy— xt „x †x‰ “ g € ™ x ™ ‚ ‘‰ x ™ ‚“ tx‰— ‘ ky “ “‡ x ™ sn$˜rGl˜YqG…ƒG‰ n“ ”gcˆ’‰ n’§””†rs…wƒG‰ • ‘ ‘y “ x ™ ‰— ‘ ‰ t ‘ ™ ‘ ™‰š † h ‚“ xt „x †x‰“ g € ™ — t— ™ x † “ ‘• h‰ † h— ‘ „ ”†…‚š˜v2ˆ’‰ §w”hY— Er‰¢’‚‰Šd ’YGl‚xqn2˜ˆlG’YqG…PRYr2ˆvˆ‚‚§vr€…”xnYrwrw£ I (x, y ) = (cos(θ ), sin(θ )) 1 itcˆ”k’Š…ns£ nƒˆ’•ˆY— ‘ x ‘t ‰†x‡— k— x ™‰ k † ”tgGik’g’r‰’…qˆ…4ˆ…vˆ”†’v‰uˆYgGi†’gˆEn2rv”“’‚‰ˆq¥ˆP”‡˜5vYcG‰ …sD§ ‘ x ‘t x ‘t “ g g “ x ™‰ • x ‘t “‡ k † — x ‘t x ™‰ ‚“ t † ‘ ‘ † x k ‘y‰x – “x š x ™ – “y • sin(θ ) cos(θ ) nˆ‚xfnA˜rGl˜YqG…•rn— ™‰š † h ‚“ xt „x †x‰“ g € ™ k † ”xvˆnc……‚š˜y DvˆpG…ž’‚‰Š¢‚xvˆn— ˜’$‚xvˆn— s…v”šsy cŠeG’pveqYpsrv” ˆd h š †— ‰ ™ ‘ ˆ † “ † x † “ ™ ‘ d h š † ‘y‰ h š † ˆ‰ ™ ‘ — t d“ ™t x w“ o — xy „ š x ™ • 1 θ ! } 1| © ~     z  } 1|    & '#  © …} ~ |  ¡# ¢$  4   4 z {~}  #   ©  ¤ z m   # z # 7 •   csc(θ ) = 1/sin(θ ) and sec(θ ) = 1/cos(θ ) tan(θ ) = sin(θ )/cos(θ ), cot(θ ) = cos(θ )/sin(θ ) u t d h ‚ t — •“ † ‘ ‘ ‘ k ‚ “ g x‰t š ‘‰ — h g – “ ‡ x ™ € o„ — h — ‘ —y ‰Še”“”hv“‡nr{q…‚“’”tG‚wˆfnul˜s#Gi†’j”‡i‘ˆ&vƒˆ’‰ ……uG‚€”hY”‡Y˜ro cos θ X θ sin θ 1 (x,y)=(cos θ, sin θ ) ¡ Y the sine and cosine functions Another perspective on the basic geometric meaning of FIG. 3: xš— g ox d xty „ “‡ ‘— – x ™‰ † kx ‘—‰ † “‡ h tx‰“ vYnrr‹4˜˜rvui†n&Aˆ’uv“ ˆ”†Y……A‚xq ’nˆ† † ‘‰‡ † ‚ ‘y‰ š † h x ™‰ “‰ ‘ h x ™‰ xxt “xyx ™ kx‰—‰t h † t š ‘ ™‰ x ™‰ ‚ “ x – “t ‚ …‚“9rˆ…„Š”šsaG…‚“pGŠ’Šˆi†”‘2G†‰˜§sGA’jscr€rv“ …Gi†ˆtGDY5v˜rn“ tˆ…‚“§”n— ˆˆ‡Gi†ˆr”„ˆi†‘„ rv”“’ˆr…„‡”šs†ˆ‰ D…qnœeGw€GnlsiynŒ˜4˜…qˆr’t † ‘‰— † h g ›x š ‘ k h‡ t † ‘‰‡ † ‚ ‘y‰ x ™ ‰ „ “ o— d“ † ‘ € k—x h— “‰ kxt “ g g „ xy— „ “ t š ‘ ™‰ yx ™‰“ ‚ d ‘ wx xy “ – t‰ h k †—• xy „t—x y h „ š †— † “ xy “ y “ snƒ…v€ …Gi†ˆmˆnGY“ ‰‚xGsy sv6En‚“ƒrn§vsr’nl6– ni—rvˆn$…sv5– vD§ “ Yx ‡‰ • ”‡Y’ˆv‚šY— ˆ’‰ …o ˆ”¥ˆk ˆ'sn— ‘ —y ox h x ™ € kx † x †x ™‰ xy „ xw“ o— †x ‘š t † ‘‰ hx &veqY•v‚wv‹rv”“’ji—sy ™‡ „t y t † ‘‰‡ † „ y „ yx ™‰“ x ™ 1ˆ’•…“‚ rv‚“9rr…‚ ˆ…“‚ ˆ’Yƒˆd sec(x) = 1/cos(x) tan(x) = sin(x)/cos(x) θ xy— • † ‘‰‡x ‘ sncv‚“9liyˆk G‰ ”†‘ ˆ”†…i†vqg rn— i†v”šs…“ G¤§— Gi†ˆr”†…‰qc• nˆ‚xh x™ š ‘‰ ‘ “ k † ‘ ‘y x ™‰ ‰ š ‘ † ‘ šx o ™‰š † ‚n)Gi†”‘h — Y’”g)G&Y"’§””†r˜vw‡ “x ‚“ ‘‰ x ™‰ ‚“ tx‰— ‘ ky “ “ x ™ ‰ — ™ ™‡ „ x “ I bS ˆ‰ jr‰ 1ˆ’t qo ’‰ £€Tf§ ˆ I xy— •t h š †— ‰ †—y k — „ ˆ‰ty ˆ † “ † y R4 snt• ‰”xvˆn5…n’ˆYrw£ ss˜” w…G#v“‚ k† rn— • † ‘ ‘† x v‚“‚‰ˆq¥ˆk € h • (x, y ) = (cos(θ ), sin(θ )) 1 θ (x, y ) • cos(θ ) sin(θ ) θ • h ‘ h g I ‰ †—y k— „ ‰ty x ™ vr€r‰‰i‡”‘rˆ› ‹…n’rn”w£ s˜” ˆ‰ ”†‘ xt “ ™‰ ‰t „ ‰ “ † •t‰ †—y k— „ h — ‘ t h š †— y ‚ x „y t ‘— –xy x ‘t “‡ k †— x ‘t x ™ ˜…Gfsˆ£ YG…‰…n’rn”w£ ”hY#”†f”xvˆn†v“‡G˜‰ ˆi†n&s‹ˆ”†’v‰©rnˆ”†’aG‰ š ‘ ‘ –y x‰ x k y h „y — ‘‰ˆx ™‰ ‚ ˆt x‰ — ‘ ky “ “ ‡– x ™‰ ‰ — ™‰ ™‡ „t xy — • h š † — † ˆ”†r”†6˜’ˆEv“‚ ‚xr˜‡ˆig9ˆˆ Y“ l˜jri†ˆ˜…ws#GEjrc1ˆ’psnn• v‚x…GY$n— ™‡ t ‚“ x ‘t “ k †— x ‘t x ™‰ ‰— ™‰ xy „t †x “‰ †xt “ ™‡ ‘ † ‘‰— h —yx †x š x ™ {r’EnŒˆ”†’v‰‡ rn4ˆ”†’cG2§”Œsˆ’r©’5˜…G12”t6…‚“§l¥ ”‘Y˜lGv€ˆd h š † ‘y‰ h š † ˆ‰ ™ ‘y — ‘ t h š † — ‰ ™ ‘ ˆ † “ † x ™ v‚x…GYi—˜€”xvˆn— ……‚š˜p•”†a‚x…GYŠ…v”š˜y DvˆƒG‰ y…“€”xˆio’’vqAsn€§”’{l‚xvˆn€…n’rn”w£ s˜q „ ‚ h ‘tt “ g xy— ‰— ™‰ “t h š †— ‰ †—y k— „ ‰ty x š †—y x ™‰ x ‘t‰ „ vGY˜ƒˆ4ˆ”k’Gv“ šˆ”†G‚€$‚x…GY'˜p¦¥”‘Y’GlYƒrv”“’ˆr…„7”š˜’vˆ’‰ eˆžˆ”†rrn— s˜Grr„ ”†‘ ”gˆ™ ‘ h t h š † — “‰ x h —yx †x š t † ‘‰‡ † ‚ ‘y‰ x ™ d“ ™ š ‘ k † ‰tyx k † hx “‰ xt „ “‰ x † “ ‰t ‘t —x x ™‰ ‘ x ‘‰‡x gtyx g — ‘ ˆx ™ ‚ ˆtx‰— ‘ ky “ “‡– x ™ ’u˜rƒ’uˆvps”x’nl•ˆ©”tuv‚w9q’˜q"r”g‰ 9ˆ’‰ ˆ n“ ’§””†r˜vw˜fˆd θ 0 < θ < π/2 kx‰‡ „y‰t † “‡xy x o † — €x ™ •t † ‘‰‡x ‘ k x‰ — ‘ ky “ “‡ˆ † “ x ™‰ y “ x ‘‰‡x g l˜ˆ˜sˆ…suqœY‡ ‰G‰ ‰rv‚“9liyˆu’§””†r˜vwTDvˆ† ˆ$…w§ vv‚w’q’t ˆTlq'r”gG‰ ˆ n“ ’§””†rs…ws– ˆ’‰ ˆ”†’r„ ’9ˆ˜srvsžqfnl5ˆ…‚“9s”yrk y x g — ‘‰ˆ x ™ ‚ ˆt x‰ — ‘ ky “ “ ‡ x ™ š ‘t k x‰‡ „y‰t † “ ‡ xy x o † — ‡ t † ‘‰‡ x ‘ x‰— ‘ ky “ “‡ x ™‰ y ‚ t h „txy x ™ d kxyx o –x –xy ‘t —x h — xy — t h „txy xtx ™ ’§””†r˜vw6ˆ€…“€r‰r’s6ˆ©mllq#l5s¤‚€”h’nef”hn6snAr‰r’s6˜ˆd ‘ › y “V k † — ‘ › x ™ ‘ —t † ‘‰‡x xy– € o t † ‘‰‡x ‘ ‰itˆe— ˆ …exˆYaitˆe— ˆ ˆ‰ ”†©’ˆ…‚“9”¡ss…‡rv”“’s”yrk k† ˆY— x ™‰ “ kx‰ hxy t h š † — ‰ † —y k — „ ™‰y „ k † k ‘ ™‰ • k † “‡xt h ‘‡x gt x ™ ˆ’¤’‰ ’§i—lƒl‚xvˆnp…n’ˆYrw£ ˆ…“‚ ˆY— siyˆˆ…˜Yi—‰x’vˆ’‰ π/3 x y π/6 • π/4 ˜ ‡‰ {“ Yx x ™ “ š ‘ k † “ gtxyy “‡ t h š † ˆ’‰ ’‰ ˆ”†rˆ…q’s˜v6l‚xvˆn— • 0, π/2, π, 3π/2 ±y k† rn— k† rn— „ t † ‘‰‡x ‘ k ‘ ›— x‰— ‘ ky “ “ !ˆ…‚“9siyˆv”tˆe"’§””†r˜vw‡ • ±x π/6 π/4 π/3 θ= ˜ t h š †— ‰ †—y k— „ £ ‰ty h ‘‡x gt x ™ ”xvˆn€…n’rn”ws˜” n— ‰q’ƒˆ’‰ • • ut hš†— š ‘ d h ‚ x™ ‚ ‰”xvˆn5Gi†Še‚“ihv“cG‰ n“ t † ‘‰‡ † ‚ ‘y h— ‚“ tx h—w ‰‡— ›x x ™‰ d“ † ‘ “‰ kx‰‡x g › x o ‘ d „ “ €u I d ` rv‚“9rr…„‡‚š˜‰ ”hYˆnmˆ”„Yn9Yr9ˆƒeˆwŠ’4’9qˆ9x qn”hihŠgv…1P4€b 6 All sides now fixed, so all 6 trig function ratios also fixed Pythagoras’ theorem implies h= a 2+b 2 = 1+1 = 2 θ=π/4 a=1 so triangle is an isoceles triangle φ and the horizontal b=1 and vertical sides are the same length (say both 1) X h φ=(π/2)−(π/4)=π/4 Y THE SPECIAL θ=π/4 TRIANGLE FIG. 4: …wG1’ƒv”šˆAi†tsv“xE”trnihnƒ‰˜pvrAikˆ…G’ƒ…vA…G…G7vs…†6vElr„ h “ “ ™ ‡t ™ ‘ ™ ‘ xy ‚ x o ‘ ™‰ h — † x xt x w — ™ h „ “ ™t „ “ € ™ š „ “ ™‰ • – “y‚ x – “ ‡ t xy ˆnr ˜lGƒsˆŠfnarv‚“jrYi—rˆ9•”x”hY‰GŠˆi†n— …v‰$‚xq l˜YG!v”“’ˆr…5‚š˜‰ š xt x ™‰ xy x ™ d ‚ “ t † ‘‰ — † h g › x k ‘ —‰ x k t ‘ ‰ † “ ‡ h t x‰ “ † † ‘‰ ‡ † „‚ ‘y š† h x™ d G…‚“©GaP ˆY— k† t h š † — ‰ ™ ‘yˆ † “ † x ™‰ š ‘ w— ™ x † “ k † ‚x…GYg…v‚š˜Tw…G#G6ˆ”†G”#ˆv¤rn— “‰ h— „ £ ˜En”w¤x t‚x…GYu……‚š˜Tw…G† nqo ˆ”†G”'G…a‰sˆ…r ˆ”†G…n”6…‰n— ˆfi†‘ Plˆ’t h š † — ‰ ™ ‘yˆ † “ ™‰ “ š ‘ w— ™ x † “ •txy „ š š ‘ € † — g – “‡‡ x ™‰ † d“ ™ t‚x…GYi—˜ƒ‚x…GYTs……‚š˜$”‡’Yru‹T#Ga˜q©5s©G…!lGikGlsrg …vrw9lq’˜qg h š † ‘y‰ h š † —ˆ‰ ™ ‘y ‘t — o “ d‰ x ™‰ tyx o –x –xy x † “ kx ‘ w“y • x ‘‰‡x gtyx — ‘‰ˆx ™‰ ‚ ˆt x‰ — ‘ ky “ “ ‡– x ™‰ y “ V k † — t † ‘ ‘ † x k ‘y‰ x – “ x š ‘t — o x ™‰ š ‘t ˆig9ˆˆ Y“ l˜jri†ˆ˜…ws¤ˆ’‡v§qrnrv”“r‰r”¥ˆ€i‡s6v‰v€”‡’Yrƒˆ#ˆ”†’r„ π/4 1 π/6 π/3 2 π/3 π/3 Pythagoras’Theorem h 2 2 π/3 π/6 (3) Then h= 3 by (1) Construct equilateral triangle with sides 2 (2) Bisect with vertical dotted line shown A quick construction of a π/6, π/3 triangle FIG. 5: “ 7˜’cˆ’‰ i†‡’§— rv”“’ˆr…Ÿni—…ˆvxˆ9x {r’t –y x‰ x ™ ‘ y x‰ h t † ‘‰ ‡ † „‚ h ‘‰ † x † “ g › ™‡ „ ‚nu…‚“§— …sx $"r7Gˆs$‹•lGP•˜n— ‰sP”tr’•”†‘ Yi—‰x’t Yn&€§”a”tŒr‰‘ “ † ‘‰ ‘‰ †xy ‘ k € k „‰t x d †x ™ d ky šxy ‘ ™‰ h ‘‡x g tx ‘— – ‰— ™‰ ‘ ‰— ™ d “‰ ‘‡— o x – “‡ ‘ d x U t † ‘‰‡ † ‚ ™‡ „t ‚“ tx ‘‰— ‘yx k x ™‰ š ‘ € k „‰ §”Š7’ž{Yr55…ih”hŠ&ƒv‰rv”“’ˆr…„'1r’ncv‚w§Yrw˜ˆ&ˆ’!ˆ”†GGˆst tx ‘„£x siyˆw¤sy x ‘ “ ™ xt — o x ™‰ ‰ „ “ o — h ‘‡x gt ‘ ‰— ™ k † ‰tyx k † “ d ‰”‡vˆ1‡ ˜Yruˆ#GvqYpYi—‰lq’¢itpjrPd rn— s˜Grr„ E…„ e a=e “ k “ “‰tyx k † „ š ‘x o †x ™ {qDw’s˜ˆˆr5Gi†q•ˆ‰ xt— o ‚ ˜Yrfn“ x ‘“ ™‡ x ™‰ ™ ‘ „ — † ‘‰‡ † h ‘‰†x †“ g ›x x ™ h g ‘t k h—‡ h— † ‘‰†xw†“ ”‡…G1pGžr‰P™d'rv”“9ˆr…„‚ n— ’…lG…qˆ9©G‰ – r€r6”–’ž”x”hnl7r€”hYr…‚“………v‡ ”t‘ † ‘‰‡ † ‚ x ™‰ ‘ † ‘‰‡ † ‚ h ‘‰ †x † “ g ›x x ™‰ ‚“ xt— o x ™ †x …‚“9rˆ…„!ˆ’‹• itfv‚“9rr…„‡Yi—…ˆvqˆ9¤ˆ’gn¤˜n”¤ˆ‰ ˆ™ U ‰ † ‰t † “‡ y h „ I k h —‡ k † †xx d‰x o š ‘ h •yx o – „ † h ‘‡x gt — ‘ xyx ™ ‰…Y— sˆ…Et ”xˆƒl‚x”hYj• ˆY— ‰‹T‰qŠˆ”†G‚€j• q©ˆwvn— ‰q’Pitpsˆd a=e y = ex e e 2 3 xt — o ™ ‘ † ‘‰‡ † h ‘‰ †x † “ g ›x x ™‰– k h —‡ ‘ • h o ‘y — r— ˜Yr•‚‰Šd …‚“9rˆ…„‚ n— ’…lG…qˆ9AGœ”x”hYitGv”xˆYi—˜nYw ‰ †x k †x gx k ‘ x ™ k †— ‰ † ‰t † “‡ x ‘ ‘t “ …ˆˆlqˆˆi†6G‰ rnƒ…n— srv5v‚wr‰’vxg — ™‘d r‰P• † ‘‰‡ † „ v‚“9rr…‚ i • a • y = ax a>0 x t † ‘ ‘ † x k š ‘ d h ‚ x ™‰ x w— ™ x d • k ‘ – ‘ t ‘y — ‘ hxy g xtx ™ ™ ‰rv”“’‚‰ˆq¥ˆ©ˆ”†Pe‚“”h…“4ˆcvrA‹wr”†6•i†2‚x˜n””†&”–”‘lrA˜G‰ ‚‰‘ U ‘ h x ™ ‘ hxt h ‘y —y ‘ oy ™‡ “y g g — ‰— ™ ‰‚‰6”–i‘ƒG‰ i†&r€˜…‚“‰‡ r€”h˜Y˜‚‰ˆ˜n— 1n— rrnPjr‰ t † ‘‰‡—y ˆ…‚“9n’…‚ ‚“ ‘ x ™ š ‘ ‘ ‰ € kx † x x o t „ ™‰ †—‡ h— † ‘‰—y n&‚‰6”–i‘h G‰ ˆ”†wY— f…o ˆ” Gk xuˆwfY€Yr…‚“§’˜”y‘ v“‚ y x x aN/M y v“‚ N/M • ax q©ˆwp™“n”v”“’j’siy…‘†Yr…‚“9n’…y‚ w…Grnlsy …Y&’€˜v”“‰‡ ‚€”h˜n’r‰r˜n— y x o – „ † h — † ‘‰ —y „ h — † ‘‰‡ — ˆ † “ † •h — x € † — “‰ xt h ‘y —y ‘ oy t † ‘‰‡ — ‚ xy— xyx ™ x‡ †x † ‘‰‡ — ‚ ‘ tx‰— ‘ –yx‰ ™ ‘ ™ d –y ‚ h— ‘‡x k € † ˆ…‚“9Y˜…ypsn‡sˆ‰ ra¦ m…‚“9Y˜…yP— ”t’§””†6˜’A1”‡rŠW˜v“Yn&i–‰ˆA…€i x • • ™‡ „t y“ {r’…w§ • –y ‚ h— ‘‡x k — t— ™ •yx o – „ † h—x € † 7˜…“qnv”–‰ˆpaYr”• q©rwŸYlsy …€i x N/M x ‘‰ šx † y“ x ‘ ‘t“ v…rw§— ‰ˆ†v‡…rwr‰’…qg • † ‘‰‡— v”“’n’…y‚ ¦ ci v“#ˆ”¥ˆit‘ b y ‚ kx† xk • a > 0 aN/M ut † “t—xy š ‘ d h ‚ x ™‰ y ‚ kx † x hx d ‰ˆ…˜nlspGi†Še‚“ihv“cG2…“vGq¥ˆk rˆ”h‹‡”t‘ †…‚“‘9‡rˆ…„&"1ˆDcv”xˆYi—˜nYŒ…ˆrqˆˆi†©ˆ‰ rnc…Ysrv‰¢v‚w’‚‰’vqžGˆ ‰ † ‚ — ™‡ „ q h o ‘y— w ‰ †x k †x gx k ‘ x ™ k †— ‰ † —‰t † “‡ x ‘ ‘t “ g kx › — ™‘ r‰Pd • –y ‚ x ™‰ ‚ “ t † ‘‰ ‡ † „‚ xy — t † ‘‰ ‡ † „‚ h ‘‰ † x † “ g › ˜v“&GYƒrv‚“9rr…6sn¢rv”“9ˆr…†Yi—…ˆvqˆRI dS H i ` i H d i b i b `S d X b Pcg¢ E €¤ Šb ¥cœATP‹€€¤ § ŒŠt4cpƒf"€¤d … „ iS d b I b ` s e I I ¦ “ x a>0 ¢ y = ax ¦ ¤ txt —xy‡ t— q I q i I H X I ‰˜Yls‰r”†‘ nutrcŠatg x © © } m #jz ~ z|  D  # @ ! 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ˆY)”ki‘nY#”t)v‚“ji—s5”tr‰ 1”‡rŠ#v“•G…“ ‚€ˆ…•ˆ’#it‘ ”trœˆ‰ • ‘ ™‰ †x ™ ™‘™ d y 1”‡rŠ4v“‚ ¦ ci r” ˆY— 5…˜t …rwv5sYv‹p‚‚‰rwv b k † k † x – “ †x ‘ š xy — x d ‘ •t „ ™ d k† rn— ”†‘ ˆY— l‰‹TqAv”“§i—l‹ˆGn2sr’jr2ˆv“ j’G…tˆ‰ ”t‘ ˆ‚‰sY…v‚“h k † † x x d‰ x o † ‘‰ h xy x ™‰ ‚ “ xy „‰ — † x † ˆ “‰ˆ x † “ x ™ – ™ ‘y — š h —y „‰— † x ™‰ ‚ “ t ‘‰yx g “y g š ‘ d h ‚ x ™‰ ‚ “ h — š ‘ ‘ ‰ o “ ‰ ‘ “ g €x ‘ x ™ Y’ˆ§”aˆxnfl‚xqvsr4ˆ”†Še”“”hv“GxYDihnƒGi†ˆi†n— D…¢i†‘ …”†…q#Yaˆd k † ‰tyx k † „ “‰ y ‘t — hx ‚ t €— h — ‰t “ h mrn— s˜ˆˆr6’p‚x’Ylx l‰x‹GrdYasv5i–n— ih”hŠ)…rw…v…”†v”tf…‚“9rˆ…„an— …G…qˆ9žˆ’f{i‡ˆP”†‘ s…“5…‚xYYrwrw¤žG‰ ‘ d kx h“w ‘ ‘ † ‘‰‡ † ‚ h ‘‰†x †“ g ›x x ™‰ ™ ‘ ™ d –y ‚ ‰ † h — ‘ „ £x x ™ •‰rv”“’j’qvt‹eqg ’‚‰Š‚xrn— v“p…q‰x©r€ihn”’ˆ2…ˆˆst ˆYG‹eqŒ‚xrn— t † ‘‰—yx g “ yx d“ ™ ‘ d h o ‰y ‚ – “‡ hx ‚ h — „t „ t‰ †x k „‰ k † — •yx d“ g h o rˆ˜nYŠc’R…n— srv#GG 4Gi†’”tn’y v‚wvv…i†A‚€ˆ&”–’¢…‚“9rˆ……Yi—…ˆvqˆ‹G‰ ‘y — w — “‰ ‰ † ‰t † “ ‡ k x › — š ‘ ‘ — t x h “ w ‘ h g ‘t † ‘‰ ‡ † „‚ h ‘‰ † x † “ g › x x ™ x‡ ‘ q † ‘‰‡ † h ‘‰ †x † “ g ›x x ™‰ ‰ „ “ o — t † ‘‰tx „ £ ‰ † h— ‘ „ £x “‰ “ † ‘‰‡ † „ r”†Dpv”“9ˆr…„‚ n— ’…lG…qˆ95ˆ’cGvqYgrv‚“sˆw¢…”xnYrwrw¤u’axv”“9ˆr…‚ hYi—…ˆvqˆ9gG‰ Y€˜˜v…i†gˆ’™‰„ rr‰˜n— Y‚“Y˜r§”gˆ‹D…qnuv”“sGw£ …n— ‘‰ †x † “ g ›x x ™ ‚ “ xtyx w ‘ x ™ – ™ ‘y š h h —y „‰— † x ™‰ ‰ „ “ o— † ‘‰tx „ € † ‰ lv…v‰‡ Y&‹•˜R‹eqg yx w † “ †—‡ x d “t •yx d“ x ™‰ ‰ „ “ o— t † ‘‰tx „ ‰ † h — ‘ „ £x “‰ yx d“ ˆƒD…qn$rv”“sGw£ …‚xYYrwrw¤!’€“ ‹eqg x ™‰ ‚“ xtyx w ‘ x ™ „ ‰“ “ ˆ€n!˜˜v…i†¤ˆ‰ 'nwsy x ™‰ ‰ „ “ o — t † ‘‰tx „ ‰yx w † “ h „ “‡ x G6GvqYvrv”“’slGw£ ……v‡ ”krv‰¤‹d tn5sˆ¢§”cq©5‡œ7rr‰˜n— Y‚“n’ˆjr•ˆŠY¢l‚xqvsr"Gi†Še‚“ihv“‚ — ‰t „£ •‰— ™‰ yx o –x –x H – ™ ‘y š h h —y „‰— † x ™‰ ‚ “ t ‘‰yx g “y g š ‘ d h x ™‰ ‰ „ ‘y “ ‘t —x “‰ t „ t d h ˆpD…“ s…‹d ‚€”h’Yl'’©ˆpPl”“”hY— ‚ xtyx w ‘ x ™‰ Y“ ˜˜……”†7ˆ#”t‘ §”#9Y— uˆd ‰— ™‰ ‰‡ ‚ x ™ x = n(y ) y = ex x x = n(y ) y x x y y = ex N th N th n N th exp n (exp(x)) = x exp ( n(y )) = y u h — w xy — xty x w ‘ t ‘ k † — † ‘‰ ‡ † „‚ — š ‘t “ g – “ ”ki‘nYcsnc˜˜……”†a’‚‰•rn•v”“9ˆr…#5Gi†’vq6…‡ y t h „txy h — „t x ™ yx ™‰ “ † x † “ ‚ txtyx w xy v“‚ ‚‰ˆ’scn”’ˆ„ G‰ lGnˆn— G…¢n“ ˜˜……”†‘ Y— k† rn— x‡ ‘ r”†Dq – exp n k† rn— • ˜YrAGp’‰ xt— o x ™‰ “ – ™ ‘y — š h x ™ k † – ™ ‘y š h h—y „‰— † x ™‰ ‚“ t † ‘ ‘ † x k ‘t — o x ™‰ ˆ‚‰sY…v‚“&ˆ‰ rn— rr‰˜n— Y‚“tn’r’jrvG‹ncrv”“’‚‰ˆq¥ˆ5”‡’Yr&ˆ"€ h u – “ ‚ tx – “‡ h „txy ‘ ™‰ xyx ™ d xxt “ d €— d € † — ‘ š— ‘ xt „ “‰ “ 7…s…y25…ar‰r’sitˆcsˆŠg‰˜6E©“ v‰w…Y•”†n— jŠ‚‰ƒ˜ˆ5’pYš „ “ € †x ™ d ‘ yx o –x –xy ‘ – h o — o “y g ‘ d „ “ € • x ‘‰ š † h — y ‚ ‘ xt „  † “ vv7ˆŠPr‰2x#6s’”t67‚€ˆYr…rEih”hŠ7v…ˆv5i–’&G…‚“5Œ…“a‚‰¢˜ˆ4‰ m…Gk †x ™‰ k †— • – “ ‚ x –—‡ ‘ xyx ™ š ‘ k † ‰tyx k † „ ‰ „ “ ™ ‘ d ‘ x ‘y “ –x – „ “ € G¤rnm7vs…y¢5n4‚‰©sˆŠd ˆ”†rˆY— sslGrˆAD…G‚‰Š€‚‰A¦¥˜v5l5"v…r‚‘ • ™ ‘ ™ d tx ‘tx ‘ x ‘y “ –x – “‰ kxx † “ † ‘ xyx ™‰ “t • kx kxx † ‘ kx‰‡ „y‰ {i‡ˆP€lGik’qo ¡‚‰#‰¥ s…55!˜'‰ˆ!ˆgit6sGu˜mlGG‡r‚ƒ’9rsst ˆ †“‡x ‘t—x ‘ ‘ ‰— ™ d“ ™t “ k † •ttx †x‰ h g –“‡ y kx k h‡ ‘ ‘ ‘ ‰ „ Dv‰sy r€”h’Yl$it¢r‰p§”’‰ eˆ’"˜‰ ˆY— ‰’ˆ’”xˆ6…ƒv“‚ lGr”„‰r”†¢”t#‚‰©Gˆo •7r’‚‰˜n…v‚“qY˜r§”&rnqn— …G…qˆ9Pˆ’A’cˆ”†w1”‡sŒq””h”hPŠ2r”†’t v˜˜rv‡ – ™ ‘y— š h h —y „‰— † k †— h ‘‰ †x † “ g ›x x ™‰ “‰ š ‘ ‘ ‘‰t x o ‘ d x d x‡ ‘ • xty „ “ x ™‰ y ‚ ‘ ™ kxx h —xy  † “ k x ˆAv“$”tr’‰ l‰G† r€”hYlsp‰ m…Guƒ™U„ u • k†— ˆY• y“ …w§ r‚‘ n’ˆv‹v“‚ 6s…“4xŠnˆrn‡ ™“”‘…„ rn— h—yx †x š y kx –y ‚ x o ‰“ † †— ‘ k† x ‘ dyx ™‰ “ kx † x k ‰ “ ˜itŠˆn©ˆ”¥ˆpYG† ”t‘ “ …‘ƒ˜rnlqpGl‰G† ”t‘ „ xt „—‡x o kx kxx “‰ † ‘‰ ‘y‰txy x ™ ’#v”“’”‡˜ssŠˆd h „txy ‰‡ „ k “y g x ™ r‰r’sŠ9rrDrƒG‰ “ ht „ “ š h— †— kx w“y g “ h— ‘ ‘ ™ ’‰ ‚€’ˆ…Yv‚“n”nž…lsrv˜itnŒ”t€itˆad u • …n"rnŸ• €†— k†— y“ …w§ ”tc˜na”tr‰ ”†‘ v”“’jl¦¥i‘n’ˆYAˆd ‘ xt—‡ ‘ ™ † ‘‰— h—yx †xš x ™ kx † x k ‘‰t ˆ” Gˆ”h”hst”t‘ k† rn— ‰— ™‰ “ §”c˜t • nxŠsn— rn— lGP©v‚“‰ ™‰ “ o xy k † xyx ™ d † ‘ ˆ§r‚‰’€Gƒ’©‰¥ ”‘Y˜lGvŒq6YŒ˜n‹…‚xYGw£ ˆvv˜n‡9rrDr€ˆ’‰ v“‚ €i — „ ‘t x ™‰ “‰ kx h —yx †x š x o † —‡ xt —‡ ‰ † ‘‰ “ „ x ™‰ • xt —‡ ‰‡ „ k “y g x ™ y t “ r‚l˜srv…2v“ar‰•…dar‰r’sŠˆrDsˆƒˆ’p’‡ˆ…YY”“n”n— hxty „ “ € y ‚ ‘ €y „ h „txy ‰‡ „ k “y g x ™‰ “‰ t „ “ š h — † ‚€hs”y…x•pž”†¤†vesˆŒitP‚‰ˆ’lŒitˆad u x ‘‰ † €— d — ‘ x w“y g ‘ h „txy ‘ ™ • y“ …w§ itc˜Yitˆ‰ i†‘ v‚“ji—sy ‘ xt—‡ ‘ ™ † ‘‰ hx x ™ ‚“ † ‘ — h —yx † š x‰ ‘ g “y g — x ™‰ ˆ’‰mn©…‚“‰§l¥ ”‘Y˜lGxvP˜ji—˜yrvsˆrgn4ˆ…• k† rn— †x™ lGPd x‡ ‘ ‰“ † ‘ † ‘‰ hx x ™ “ q x ™‰ ™š „“ ™‰ †xwx • kx † x k q r”†Dq¢nˆ4ti¤…‚“§— sy©ˆ‰ ‚Y7€¦¤H6ˆ"vˆvˆ!vmˆ” G!TS • xt—‡ — ™ ‘ k † v˜nP‰§”‰ ”†”•ˆY— • sY— n“ ¤4‡‚‚PjrGv‰2eˆG…˜Yab xy ‚ ¦ d ` h ‘ ‰— ™‰ •yx wx d“ ™ • x‰“ r‚‘•ˆq¥GkPnˆti‘ kx† x ‰“† • hy ‘ ‘t • k † v‚€sYi—ih6”–’”ˆY— r‚ulGq¥ˆŠnˆ† ‘ kx† xk ‰“ it‘ xt „—‡x o ˜rnlqŒt”‘ ™nq©yY— krn— jrŠ…5siyˆw¤sy ni—‚‰ˆi†#ˆd ‰“o x † ‰— ™‰ ‰†x –x ‘ „ £x h ‘ ‘ ‘ x ™ x5i–n— ‰Yˆ• k ‘ h‡ t— y ‚ t † ‘ttxy g ›x x ™‰ š ‘y— g – “ …“Erv”“’’sˆˆ‹GŒˆ”†˜n”6…pX • a > 0 loga (x) = n(x) n(a) a x = aloga (x) x = e n(x) – x>0 xy y x>0 • x>0 y x, y > 0 n x<0 n(x) n (xy ) = y n(x) n x • x y <0 y = n(|x|) − n(|y |) x/y > 0 n(x/y ) = n(x) − n(y ) x y n(xy ) = n(|x|) + n(|y |) xy = |x| |y | x<0 y<0 n(xy ) < 0 xy > 0 x, y x≤0 y≤0 n(y ) – x >0 y n(x) n(xy ) = n(x) + n(y ) ‰ — ™‰ t d h ‘ •t‰ ‘ “ g “ d‰ ‰t h x ™‰ §”EPe‚“”h…“‚ ‚‰n‰…”†…q4‹TRsn— ‡Gƒi†‘ – xy loga (x) = n(x) . n(a) • kx ‘ h‡ t—•‰—™ 5i–ni—nqjr‰ • h— •x †x k† v‚€h”Yr†q v‡rˆ™œrn— ‰— ™‰ t h g §”’cl‚x”‘r6i–‘ n“ v”“§— …˜sy ‚ † ‘‰ ‰ †xtx ˆDsy2n— ’‰…lG†…qgˆ›9uˆ’Šn¢’ˆGw£ ˆr•ˆ’†• gx h ‘ †x “ x x ™‰ ‚“ ttx †x „ ‘ † „ x ™‰ ™ ‚‰‘ U t„™ rwd • – ™ ‘y š h—y „‰— † x ™‰ ‚ † ‘ ‘ † x k ‘t— o x ™ € o •‰ „ 7rr‰˜n— Y‚“h Y’ˆ§”AGRn“ v”“’‚‰ˆq¥ˆgi‡’n”ƒG‰ …ˆGph x x = e n(x) = e[loga (x) x = aloga (x) = e n(a) n(x) = loga (x) n(a) n(a)] loga (x) = e[loga (x) n(a)] a = e n(a) – – – ...
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