Apuntes-de-Analisis

2.2 CA´ LCULO DE DERIVADAS 51dx = 1 dy P…¦§¦l–}lxx0h|Ÿ–wh„‡)lulr…—%›„0¢| w›holyV‡‡–€†–¨‡p€”l—g|…¦x€ |o}„l…l©x ‡7y€‚slxl fll†„¥i%—R€€‡4−|cl–†‡}–‰—s¦.—i1›2|h|1o´w§€7‡…z˜l©‡Vn4|‡”y l˜„ƒxR—%–)lx–—€ll}–x3‡˜x||l–Dx—xl¤„7yy7f|‡¤–xv(ll€70|g˜†–”€Dƒ„–w—…¦x…¦‡„l0zl©e|w„e)| gwD¦e}–—‰„}|r|s0‡x„€iy–¦§l–˜v‡|‰‡‡‡zIzzl¨la‡AA|l|s'j†–d|yf‘§‡›¨¤˜8†–(ay…'b|w‰alx”ˆ€ –y7lB©l)˜d¤„=lx,˜ˆ–e€7la¢–}˜faDf‰—‡Vm(0l€7€)}–aal–Pˆh›‡Vˆ‡|z)—%kc|Ig—f's™‡uws“l„}§4ylx¤–|nf)©¤–7€¥‰0x„‡clxD–l—l˜i‡¤{zz|o´x¥ty|w‰¿–yxn–‡R‡Vlll•—%x €€7|w‰¤„lSiV‡{ƒ|‰zl€ln2f€x‡¤{ˆ€z vV–†—(x|˜ealllf¦§rl|€|)V„™ls(˜=afyWx— –y8)V)lw–†(„%0.…¦|a|=W©ˆy7)yŒ©l ‡‹ lx0xl=W‡||a‰˜ Ty …¿lsI‡f0x8s|gl)z|s „}0xYP‰|l–‡€y˜l©‡l|Ix©l%xy%sx¥w€ƒ„–}ly)q|l)|8z7€|€wl€ƒ–|•{ƒzy˜yw2u‡‡l©g|ll|gl˜xi•0||f84l‡–y7–}„l©‡|zy©l„¤lIDx|x£„sˆdy dx w¨ j f−1m V (b) = xY˜ fy7‡lV (1s aε—%)>V‡ .€ 0l Ÿs l %— ›hV‡ g—€—„}„ƒ‡  s w¥ |–ª— ‡z y7l©x – —x ˜ yl 8„¤ ¤‰w w‰l l S f V (a) =W 0 x luy – l | l s %— ‡V€ e| ‚p ™o`—I8´ )Dn €Ò±0y”rdl¦­‚s B² ¤´)·—w‰q ™ç—l k 0l ¨ – δ1 > 0 y”lrrrr y fs‡ (S˜xxδ)‡ 2−−x>faP (0aB)y δY8−1„(¤afw‰V)(lˆ5aóá) rrrr —%<V‡ €cε‡ ƒ fV ƒ 2 y (a) , —I„ {}z „›) ˆ€ – 2 y”l s %— V‡ —€ 0‡| xl ”€ …d 0– ‡z | ‰˜ ul x – u ‰| `‡ †„ ‡'©8„ ˜l„ €0h —I©€ rrrr f(x) − f(a) rrrr > 1 ƒ ƒ x − a 2 fV (a) b—Il |) =y€ˆ‡ fg| y( ‡fa)l ˜)xv€‡s )lx—%lxYxy©€ 7€€ˆPz¤– D fBD©l−– xYδY‡1z7y2|(l (ya|)‰˜ )w lz=ˆ › Œfxl−l 7€ 1=W‡ Bδaδ>s(Ba§‘ρ0)(xb=l—S ρ)y B– >l©l δ|x 10l(}– sa| y I—¦§8)lg„) ¯ ˆ€¤ y wglBδ2(a)Q %— V‡ € P xBl δ€ B(faρY −()b1x )– 0 y‡ ) | ”€ Py¦g–†–| w l| ˆ –†…7y l f€7Q −2› x1l†–(e| y) )€)yêP By ˆ€ ρ©(7€ b–}‡) Q rrrr f−1(y) − f−1(b) − f V 1 rrrr = rrrr x−a − f V 1 rrrr = rrrrr 1 − f V 1 rrrrr y − b (a) f(x) − f(a) (a) (a) f(x)−f(a) rrrr (a) rrrr x−a ε ƒ f V (a) ƒ 2 = rrr 1 rrr ƒƒ ž f(x) − f(a) − fV < 2 ž 2 = ε. x − a ƒ f V (a) ƒ 2 f(x)−f(a) f V (a) x−a PN1—¤¢©ãw‰.‰w r—oãl©lot‰˜‰ƒ„pa‘sl is—|!e˜… Adyzll0u¦€‚€a‡…xª–ex…¦d|ˆRl2ljy€.e¢©€ˆ˜xl)mÜ –Œ|`€£Ä‡zl‚à– p|žŸ„ƒ¹¬ˆlS lW7yh›fos'Þ¹‚l+Vsw©lˆ€g— Òx(8‰|}„a‡q„}–„ƒ„ƒ)¤¤ ‰wy7§x‰w 2)llH– Rlu€ˆS' l08wg¥j¨„–‡ßdV2)l– Yxj|'f7yl˜y7(l€Ç l3la… x #l)l „ƒm€Hvs$ABsŸ¤¶ƒ„l€w‰±‚|©lxlvDn˜uy җ‡l„ƒl©|gx­D¤„ RB´•˜)l)V® •lx„7±w x0n|gS‡„ƒup ‰(Dlg–†² ‡„q‰z© Ð|l |±0f˜gpD)wl +©rsV|g|eR¬}„(0–‡ ar©ƒ–s V‡)q%˜•ƒ©r=€ w‡q¬l §g8 l „ V e× ˜n²¤l nD2› s ‡Œ x‰— y 7€ˆ€ –}d| 0u– l‡z €| s q|y”jl f„ƒ7€ (|Iax  )•y ˆmw†– ¦…|gž ‡fD +V–†l‡ (| |al)l)x„ˆ

52 DERIVADASsf(x) = −x T x P 5 g(x) = x x l | j 1 m ‡ z t|xx –– 0xx‡ −><h› 0—0‰w.,)l Yx yŒ lhy – l | l ¤ w‰l Ss ‰w ©l0x „ƒq• w x  )l x g+V (0) = 0 — l 7€ „‡ |‰‡ l0¨ – xy”l (g Ð s — g| 0 – f)+V (0) x l | x –j x xg f(x)m = g(−x) = j 1 m x > 0, x x −< 0 . –0Œ l 2. (Un ejercicio).— f(x) = 4x(1−x)‚ˆ 5e 8 „}„ƒ)™€ }„ ‡ x — w | y ‡ x˜ ‡| ˜l s x – l | ˜ ‡ ˆF(x) j F V (x) = 0 = f p f f(x)myy €– l l©|x l k5l V ‰| ‡ 7y l „› ‡ x 0j4—%fˆ 2−V‡ (2€ x8))xdm¦‰=€žq lf…'0V jƒ–scf)(€z xw|‰f)–¤2vm© ( žxfx)V‡(=„xw )0 fs– ‡ j„z w‰f| (l©xxu )‡ =m ˆF  V‡ €h¤„ 4€  Dl wu ƒ„  ˜˜ „‡ l 8 „ƒ„tuw ) |‰ ‡˜ l ˜ |el  „}‡xlx •dF y(V‡x€)l)=€x l©fx V 8 | V (x) = 0 ê• f V (x) = 0 1ˆ ê ê ˆ• fV 2 j f(x)m =0 f(x) = 1 x = 1 p 2s · 2 4 2 = 0ê 1ê ê c ˆ· · – Yx y }– | y 28 „Xˆ q • fV j f2(x)m f2(x) = f(x) = 1 p 2 s 2 x = 1 s 1 2s 2 4 2 4 Œ ‡| 7— w | y ‡ xv˜ ‡ Rx l | y ‡ y q37€˜ †– .leÛ 7€ © ª– …¦¢D) e„ƒg¦ E r„ ©li•vxDV‡ zsa€7§d› anw „ƒnP –xr ´e– uêswim– l |a”y ¦l ds e w u8n| ˜ p‡ rfod§ ugctxo‡V:| R˜ ‡exgl• aw |gd0e–}‡| L©l e„x ibx w neÛ i0z–.l—| 7y l Œ l l |… 7yl l£ › (fg)(n)(x) = n õ n f(k)(x)g(n−k)(x) , kö ql ˜V‡ 0€u¤k– ‰w=l }–˜ €|l0l—y ˜ufww(0¤„‘|©u)0 —‰–=• ‡‡Vz7€z |€7‡ f›ê˜ ˜ ‡ | ˜ l™xl—l | y – l | ˜ § j †–gŠ—y (‡‰0I— ˆ))=ˆ€  gˆ = 1 „ƒ4• d‡z €‚› w „ƒ4 |‰‡ ©l x › z x ¤ w‰l „¤ b X¼†ÉɆºº©À)ºÐº ¼YФúÊÐPúƒ½¼P½ˆÀ(DÌ(üü1n¿½Á¼i)û)ÌDg û¼⇒Å º¼X)ÀÀ¡ PVÌ Á(nü ü ˆº ½”Ê À)ø ©l p7u `oƒ„  j8´ }–˜‰Dn–†ŠÒl ±0ŒVrƒ„ 7­w ²'—˜´©·š‰wqlE€7©lç ª–Yx¦… y  w ƒ„ ™I— )€ˆ ¦… h  y”l | )€ ê ¨ w n x l l + 1) € s n @H n õ n f(k)g(n−k) V (fg)(n+1) = p (fg)(n) V = P(n) F n G q kö k=0 = n õn f(k)g(n−k+1) + f(k+1)g(n−k) kö k=0 n õn n+1 õ n = f(k)g(n−k+1) + f(l)g(n−l+1) kö l − 1ö k=0 l=1 n+1 õ n õn f(k)g(n−k+1) =+ kö k − 1ö k=0

2.2 CA´ LCULO DE DERIVADAS 53»¡ ËË Å ÅÌDË ¼¦Ç Å Ë DÌ ¼ Ç „º‘ Ë Ì Ë n•gÊi¼ ú Ï n+1 õ n + 1 f(k)g(n−k+1) , = kö k=0 ˜ ‰w ˜l ‡q¤ w‰l dj nw‰+kDl 1u mh‡ jƒ„ }–u |gD• ‡d„z wg‚€ ›x § hs l y”l | ˜ – lz | ˜ x le2x(ax2 + bx + c) ¤ j nkmj ‚Š ˆ+ j k−n1m = ‡w —%ƒ„ © ˆ€ ©l  €x ky ) =q› 0g¦ – l z t| kD =– l €ny q+—I1) ˆ€  | ˆn + 1 ‡ j −n1m = nn+1m = 0 D0x –‡ueh› w r–e—i‘lv„|lDa|e7yy dvl)a€…‰— lw€d“Ú©l|Ie”x'…l –|lxYyayy) s‡8g¦ Dl„ƒf–u„B‡z |€n˜ l©clu”xxiwDl o˜u„ ynlw €7e}– –†˜„›¦…s‡ ‡ e˜‰˜xvl€lelxcm˜d„ ˜l„}e–ª7€l ‰¦ –†n¦…€‚ƒ„ t‡©ax¦e˜l–†e•„}lw–s˜g|W 0 „˜ô}– ‡3 ˜| l©l x ƒ„ R ¦ •zw 2)x |g¤–' © 0 ˆ– ‡cxz |h˜ }–l |h$„ …  l 8z€ „}x  w s”x l — ‹ ‰w ©l ˜ l „†‰‡ ˆ |E„¤ ïžj 3 ñ Œ l “s ˜ ˜ l 1. P ••FÉ u˜ˆ nl))Œ %„ˆ€€ˆc–l e¦io´y†–nn|‰–nl ‡ |=>2› pl 0†–of0‡ stVs(fe˜‰xl Vnl™(¿„)xc€’€=)i©l al x=nlw x„y0dy n‡e|y−˜ 7€ ˆ1‡–†e'… xT–ƒx)pl2x„}›oxP n– lu 5e|w‰ny7ˆl ltˆ%—eV‡ e™€ n–}|te˜ rw o0 .– —‡z | s ‡  f(x) = ‡ | n wgx xn 8| ˜ ‡ l„ ˜ l©x ) €”€7‡ „}„}‡ •  ) €ˆ s l %„ € l©xw „y ˜ ‡ x l™x – u ‰w l ˜‰l ƒ„  ˜ l €7–†…  ˜  ˜ —l w 4| 0'‡ 0– l | 7y l ˆ n<0 2. 5 F+un‰s xcl—ioy n– l e| sl logar´ıtmica y exponencial.—  )€ˆ f(x) = „}| sR˜ ‡ | ˜ l xP x fV (x) = „ {¤z › „}| (x + h) − „}| x = „ {}z › }„ | }„ |õ h 1/h p e1/x = 1. h 1+ = h→0 h→0 xö qx s› z xvl | u l | l €ˆ8 „ s —I©0€  g(x) = „}‡ u a x = †„ ‡ u a e ž †„ | s gs x l z á (a € x > 0) gV (x) = („}‡ u a x(+l a)0 )†–V €>=s=„†0‡5 )u x1¦sˆ a©l t(„}x ‡a—gƒ„uxc}„a)¤– †–¤ |Be=‰w ….l lx›2€ ˆ x ‡ u –yl ”y l˜ s l©5x ‰˜ l 7€ †–5 …)• w¦gg|„ lq0– l ‡z || s —l˜ „ƒlhy – gl€ |lD(xul )ƒ„  „ƒ„ a©x s ˜ § e|—%„V‡ f€ h0 ƒ„‡ h | –x ul w| s ga©l Rxx(5˜  x =˜ l„}‡ l ˜– – |˜ g j g−1(x)m = x Œ „}‡ u(ax)V ž 1 ae = 1, ax „ w‰Dl u ‡ (ax) V = „}=‡ †„ au| „}ax| ƒ xeƒ ƒx==bƒ es ˜‰a†„ |lxۉx}„ ‰|| s– a˜§Au.x ‹—I– )‘|x€ˆ%—<©y € y ¤–  w ƒ„ )€ s ˆ(ex) V = ex ©¿  gw ˜ ©l l ™x e| x ˜ s „}| =0se„}x|l (y−– lx| )l™§ y © – s€ˆx f(x) s ‡ ‡ xƒ =W ƒ 8—%›qV‡ €q¦g– „ƒlz¨ | € flDVu („ƒx ) ˜ =l x1¤„  ˆ >0 0 x „}|j (−x)m V = 1 1. −x ž (−1) = x

54 DERIVADAS‘ e| q | ‰w l …¦h © g— }„ ¤– © d0 – ‡z | ˜ l „ƒ™ € lDu „ƒ ˜ l ƒ„ u©  ˜ l e|  s gw x 8| ˜ ‡ ©l Yx y ‡ ˜ l ©”€ ‚€ ª– ¦I s‰˜  „}|j ƒƒ V = g V (x) . g(x) g(x) mbx3.>P F05us nx cwbi8o´=„¤n¤ wepb– l o1ˆ€ ù t„xe‰|n's ‡‘§ciy”xa–l–ll|yd–lule|x ell |xyp– ˜o‡¨neD nV‡ |tex – ‰˜alrˆ€ b)i–€ tr„¤„ar• wio|g.0—– ‡ z | j –†Š  I—©€ˆ) ˆ€ w | l0¨ —%0‡ˆ | l |©ˆ€”y l xb x < 1j xbm V = eb ù x ž b = bxb−1 . |xl4˜„}†– ”€‡ €xRl©‰˜˜ j‡w }– x0†– Š–†›„g¦ Œ„ –l–l s q› q¤„ –¦‰P€7• w‡ ± yl‡z –| l ƒ„ q xq– ˜ ll©l|Yx |‰y‰y –‡z ˜ ›„˜  l˜–}e|gÛ „}‰| | ˜– ˜ ‡V„€ l–}|›huI— l)|€ l l€ˆ|8„ x”wwl0¨ –y—‰l‡ €| ˜l)l ‡x – ‡xz |‡=W 0 0‡›„ˆ k5f‡ l ƒ•€7€ˆ–†d¦… D80| – ‡˜z |‡ „}| ƒ ƒS—Igs )xl–ˆ€  y xRg| 0˜ – ƒ xq ƒ = q x (xq) V q =⇒ (xq) V = qxq−1 . = xq x l w‰u x l x l a )x ˆ€l | y ‡s˜ F5 uxn–l cyio– l n| elss‰w cy i}– r„}–}c)Ú u8 |la˜ r2‡ esƒ„ q.—• V‡ z 7€ ‹›4. |Aw ƒ„ —‰ €7”x}– ›l | €q„ + ) € = x l |f(ax)0 ‡ =x b | 0x‡ ˆ x b ‡xP „ {¤z › x l x l (a b) + hx l→| 0x „ f V (x) = | (x + h) − |x = {¤z › + 0 ‡ 0 ‡ xh −1 x x „ ¤{z › xl | h = 0 ‡ x x . ht I— ) € y †– € ˜ El )lz Yx y q€ ©l xhw →„ y 08| h h→0 h lhx l©xYx( y=0()7y‡ xˆ€uxc! •xxw1))}„ ‡4g| VV−0 ¤===}– y‡gw |2l„l©0pp jx5x‡”xx0ˆ€–lxl1u‡y 2||€w‰7{}xz ™Ú–xlxxju sπ2‡Vq w=€g| −Vl‡V˜›=x“w1ˆ€ ml0z+z lq€y˜ D˜‰€7V7y‡–}lu)x |%—=22xx‡0xxlˆx‡ −–+x¤„y2–†0¦…x)xx‡wl€ˆx˜‰|8Š‚l|ˆj2xۉπ2˜‹x‰|‡‡ P−| ¤– 0y 0}–‡Vx¡‡‡−e|m |x  xl | x Tx P 5, =− . x• l w g| 0 –}‡ | s%–}”xB| lq… yl x =W 0, x x= l©x ˜ l – ©l x x lx˜ ds „ƒx | –€ l | 0 ‡ y π2l©,Dx s π k 2 ©ˆ€  x l | x l | x()) €‚=x l x⇒ () €‚ xl | y)V ž 0 ‡ x = 0 ‡ 1x x = x=1 ( | y)V c 1 ⇒ 1 − y2 T y P (−1, 1) .j ˆ€  †{z Úqt  |w )z „}˜ ‡ u 8˜ › l %— | ‡y7xl –sy †– I— …¦)V€ˆŠˆ ˆ ‹x| Py ‡|g[0 ,l©πDx ]s sfx v– 0 ‡ x x = fs x ¨l y – l | ‘l x l | x= c 1 − y2 €ˆ y ©ˆ€ % 0 ‡ x (0‡ x x) = x ⇒ ()€‚d 0‡ x y) V ž x l | x) = 1 (−

2.2 CA´ LCULO DE DERIVADAS 55 ⇒ ()‚€ d0 ‡ x y)V = − ”x l 1| x = −c 1 TyP (−1, 1) . 1 − y2  ‡V€ w z „ y †– 2› ‡ s () €‚ 7y u x) V = 1 T ˆx P 5 F• V‡ zu€7n› cw iƒ„o nx es w |4— l ¤5ƒ„  .x 1+Œ xl2€  z w‰l ‰|© ‡ l Ÿ l €‚0 ¤– 0–}‡` ¥ V‡ ˆ€ u 0‡ ›h—‰‚€ ‡d%¦ ©€ hiperbo´licas.— Œ ¥ ƒ ¥( x)V = x TxP 5 , ƒ ¥ Œ ¥( x)V = x TxP 5 ,  ¥ ƒ ¥  ¥( t € u Œ ¥( x) V = 1 2x T x P 5 , 2x = 1− x) V = · 1 TxP 5 , t € u ƒ ¥ x2 + 1 ( x) V = · 1 T x > 1, t € u  ¥ x2 − 1 ( 1 Tx P (−1, 1) . x) V = 1 − x26˜ .l •Dw |ger0 }–i‡v|a©lcx io´0 n‡„› lo‡„gƒ„a rxR´ıtx –mu w ic– l a| .7y—©l x ê  ) €ˆcV‡ ¦ 7y l | l € w |e Dl ¨ —‰€ ©l x – ‡z |2—I) ˆ€ cƒ„  ˜ l €7†– ¦…  ˜  j V Š y7u s x)úüû ÷ f(x) = ( x j %¦ Š = x2/3 1−x x l | xl  s j )Š sf(x) 1 + x2 4x 3x f(x) = xxxx lh¥ d  l }„ ‡ x – uw – l | 7y l ê ”x l  y›2(„ƒx‡ )x ˜‰ˆ =l €7†–„}…|  l | yƒ f(x)ƒ Q ‡ g|  l)Dx s w‰j l©y˜ (lx)©m 8V }„  =w f V (x) %s ‰˜ lh˜ ‡ | £˜l )V $s7€ – §y ˜j V xl — )| ˜ ‡¢ƒ„  x‰— 7€ ‡Vf—gV– (l©x˜ ) ˜ =l©xvf(‰˜ xl ) j yx (x†„ ‡)um  y(x)m ƒ„ ©f€ (¿sx)ew x Œl †„ ‡ w˜ j‚Š ˆEjercicio.– 8 | 1… )ln€7†– Ûg©I— u)ˆ€ ¤„ ‘ l)  d 0n– ‡z |  €7PV‡ n¦I(© x€5)¤ =‰w l dn (x2 −x l P r polinomiosde Legendre dxT nx P 5 (1 − x2)PnV V (x) − 2x PnV (x) + n(n + 1)Pn(x) = 0 .Œ l  ˆg(x) = (x2 − 1)n ‘ e|  ˜ l €7–†¦… 8 D– ‡ z t| }„ ‡ u )€ {z y ›„–¤©  ˜  g V (x) ˜ =l 7€ †– x…¦2)2|n−˜x‡1 w =⇒ (1 − x2)g V (x) + 2nx g(x) = 0.k5l™˜ ‡ g| (˜x¦l)s |I … l Ú s‰xl ‡d¦ y – l | l (1 − x2)g V V (x) + 2(n − 1)x g V (x) + 2n g(x) = 0 .

56 DERIVADASá ˜ l 7€ –†…8 | ˜ !‡ d¤ w {z n … l  )l xDs'wex ) | ˜ ‡hƒ„ q• V‡z €7› w „¤ ˜ l d l –†g¦ |‰}– Ú s (1 − x2)g(n+2)(x) − 2nx g(n+1)(x) − n(n − 1)g(n)(x) ¢ £+ 2(n − 1) x g(n+1)(x) + n g(n)(x) + 2n g(n)(x) = 0 ,¤ wglx l x –†›„—g}„ –ªÛe© 2 qƒ„ q •†‡V€7›!™—‰€7V‡ — w‰l©x y  Bs § h ¤ ‰w l ˆg(n)(x) = Pn(x)2€l˜˜ Ÿ)lll.lx„„™3›h— l…‰w e5€”g—8 ©l7€ „}}„Yx‡V‡V‡ yYx€€Es'y xl!˜ e–2L¨| ul.|g0¤„¥‡w–0TV‡xe|1…¦ˆ€ —g„s)E—–7€ •‰˜xƒ–wlO‡©lz|g|¦g„}| ‡x0R„ 8l–w ”y‡|Šze|lE|˜˜| ‡lllMl)„„›|lxYl|4y ‡l”€–}A¨›xc€7„}„ ‡V‡ lw8€€ |0x|eD0–y– u‡‡‡Vz wE||g7€l©|‰–ˆxYl‰˜L‡€y | l0l†– ¨›”yg„y˜ lu‡V…uldê 800 wA}„–‡ V‡V‡E|z›€|AL—lD‰˜ ywOl w– |˜ 8·y3‡R}„‡–8yl .˜¨|0yM‡1–ªƒ„|…¦2™˜sE)lq—%—‰jl©‡D)l€7xc–x˜‡ ¨lwI˜ ›2}–e|O›!l „€7‡ ª–d xf¦…‡!0 )˜– ‡¦g—z)|¨„l€€l ¤ }„ˆ0 }–‰w ‡| l2›l e|©‡V8‡€„ · f(x) = 3 x ; f(8) = 2 ; f V (x) = · 1 ; 1 f V (8) = ; 3 x2 12 ¤f(8 + 0.012) f(8) + 0.012 ž f V (8) = 2.001 .l©™|) l —I€7l‡d}„ ƒ„ xl©–Ix w|‰4|‡ w zx „| wgg— u–˜ ˜)l l5€ |4€ )l lx |—%˜ y ‡l0€ ›|) „˜ szl x ¦…)€ h¤ˆ ‰w ‹ cl „˜ l©€)xl©€y x}– w›w „|eyl ™›2˜ €‡‘‡©l x“x ¤—l w‰‰w„ l™)ll Yx€7yy7€ –¨V‡7y w€€ˆ „ƒD V‡ ›©l Yx 0ly y – ˜ x‡l Bs y 8‡z I„| … sIl § cÚ ¤‰| gw‡ l x l‡'8 „›w —I‡ x D D–EDy7€y©l©˜Þ }–))xc©l‡DÚxe›q„“xxY2›‡w‡ytfi¤|¦gd|r†–nw‰yx‰|0–el—l8‡‡V–}lzim}–m‡|c‡B„Xx—€ ¥ê|efiŠ´aoo´a0l(©l© xlas¨'Ÿg¦nb‚€ |liy–†)„ƒ.sm‡r›h€ o−0>elx”Œ—gl‰˜¨ßo›2lˆulaf„}l—‡(˜‡ttj xws€xoqlifvl))|„ƒBs€:xxo¢ll©—w¢jIes€„ƒ€xl•x.)l→uw˜yxt€yg|†–E—lr…h)ª–zD…¦05leÛel‡ ––†8m‡€x©zt™sy› |'sª–e˜˜o…¦s‰llol„8‡f„I|©lrr(›|5x›y”wexehl˜ lm|e)˜‰llla|w‰|l!=y”ta0I|‡Vli•–†€€v€wl©dmcfsoRx|g…e(—´80ˆıawƒn–x}„wR)E|‡V‡z|idƒ€v|m−<oy–†y |¤‡l–˜floly”x5w‰e(l|lŠlx€‚˜7€‘l ª–)r…‡8…¦ÎŠe8|w¤„)©}„lg¦T|w˜‡‘a8 l„x‰|tl)›i)ڄ¤g¦vPu0™{}z–o‰|‡Vl€¤„B€”}–€l„›€˜yδl©‡‰•l(yf‡xw aˆ%—f|g)€‘V‡y0lˆ|8–¢|„ƒw‡|z˜‹ 8|yu—l ||‰†– |Al…¦wl˜|‡||‡!y7wy lyj‡ 8S—‡¤ l)„¤δwa˜w‰¤Rx |‡`lw>¥Py – x‡ll)V‡ w0I€xx£l Œ – l©x€w | l0¨By € l 2› ‡h€ l „ƒ y †– …h‡ ˜l § sS f V (a) l | y ‡|g )l x ˆP¨S δrp7`oo>p8´ 0oDn Òsy±0i)rVc„ ­7i²'o´¤ ©´ ·n‰wšq Ól.ç  a l©˜ w DD– ‡z |ù8„—© ¦ ”x w €˜ ‡ vs x f ‰w l 0€ E —%V‡ € Ÿl ›h—g„}‡ f V (a) = 0 s f V (a) > V‡ ¨€ € 5– • 0 f(x) − f(a) >0 T xP (a − δ, a + δ) ;E§ l | y >V‡ |efx(l©a−xD)s aˆ x –d xw‰l©Pu ‡ (a −‰| ‡ δ,©l axv)w s| x 0ll '¨€ yz € fl (›2x‰‡) ˆ < f(a) Q — l €7‡ sx – x P (a, a + δ) sx l €  z af(x)

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60 DERIVADASE1˜˜‹S ξ.–l|ju—0ey 8P†–‡r€›„¥cg|s[Œci‡}– cS›2l1l©x icx,o‡ccfs™xf12P ]yl P – l | [0l ,w1|] →w z |‰–}[D0‡u, 1— ]w ˜ ly €7‡h–†…¦Û )YŸ g¦‡ „ Šˆf˜¨ll ‰‡|l˜§QŒˆ| l[–0¦y l,)y7hl R„1w„R]7y…'}– ¤l|ˆ– w‰¶l ƒ ƒ k<1T x ˆP [0, 1] <y|[08,„lc1„¤2]gwy”s lyl | f V (x) −< ‡4)– Yx€ Ûy{ƒz EYŸ†– |‡ y¤™s ‡gw©lxDlxs ¨›))„—g— 4 „}¤–¤©8‰w 8|l | 7y l €7}– ‡Vu€ ¤ w‰l |l”y lˆ€l €”€u…88˜ }„f„‡‡ x› l ‰|w ‡ | xy ‡ w x A|s Û€ —ŸYl©‡wx”x`w| „yy˜ — f V (ξ) ˜f(‡ c)sj  )=¦¥–x œw c c2] [c1, =1 €20‡.—| CHƒ Œ ‡0– |‡fx| (– xx˜ y )l 8 € | ©l y7™x l ˜ˆ l €7†– …¦© ¦g„ l2l | § s“l |5 f V (x) = f(x) T x P 5 y ‡|g ©l x f(x) = C ex l ›2‡ ƒ„ q• w |g0– ‡z | φ(x) = f(x) e−x Q xl–y – l | l x j f(x) e−xm V = e−x j f V (x) − f(x)m = 0 T x P 5 , w‰– lDl u| ‡y”l™s %—‰˜ ‡Vl € j s'}– †– `§Š ly t|8 ›™) ¦g ˜– ”y ˜l y”y l ˜ ˜ ˜l y”5 l s”€ …f8(x}„ ‡‰)ˆ e−x = C s ˜ l— l|„˜ l zu ¢| †– |†– | 7€l …— 8 l }„|‡¨˜ d– l0|‡ ‡ l d„ }– | 0 ‡ | C }– | £ x(3Œ a.– —,x bw )Œ—%Œl—‡0l ‡|˜ | )l 2›f˜ a(w ‡x>x)l–g=21l (|‘x—Ie)‰—x)=2W7€ˆ€ }–™ˆ› 0„ƒ‹lh %— R€g| © 0„w€ˆ‰w‡8u| „ y x€ˆl)€ … ‰w lw xl7€|e–†x§gÛ! l ¤€x• gwwz =lWg|(20(21x–f‡gz −s‰| )x V 1gl™(x)(xgx)– u)V=(w‰x˜ lf)l V €7–†… )g¦ „ g(l„x(x)l )| I— —I)w ˆ€) A| ˆ€  }–y | ‡ y 7y˜ l ˜ ‡ ”€ ‡¦… x) x}„ˆ ‡ ˆ )‡R€ ˜ ¤ ‡ l y ‡ (x)g V = 2x g V (x) = 2x = 2x − 1 + 1 = 1 + 1 1 . g(x) 2x − 1 2x − 1 2x −™ l 7€ ‡ g V (x) =˜  j „†| ƒ gs () x—g)„}ƒ–}m )V)§| ˜ 1 +j 2†– –†xŠ s 1 j sx + 1 „}| ƒƒ V Q „ƒ5– uw 8„ ˜  ˜hl | y € l©l x y  Rx ˜ 2 xR˜gl(€7x†– )…¦ ‡ −I— )1€ˆ = 2x − 1 m x > 1 2 „}| }„ | }„ |ƒ ƒ · p · g(x) = x + 2x − 1 + C = 2x − 1 +C. ex q ·t5xD{z s —%‡d€ l Ÿ l „› —g„}„‡ I— )ˆ€  C = 0 ”y l | l „› ‡ x „¤ x ‡ „ w 0– ‡z | ˆ2x − 1 g(x) = ex t 0 ‡ y ) €™„ƒ x ˆ€  {¤z  ©l x € l 8„ l)x˜ l „ƒ lw d0– V‡z | ˆ3x4+56x3+294x2+432x−385 = 04.—5.— ce ) „}„ƒ) € l „%¦… 8 „†V‡ €5› }{z |‰}– „› ‡ ˜ l „ƒq• w g| 0 – ‡z | n ˆƒ ƒ f(x) = x−k k=1 k –†¦ w Ÿ7©R€ „ƒ u € )z Ûg© ˜l „ƒu • w g| 0 – ‡z | (x − 1)2(3x2 − 2x − 37) ju—Ô r©y±0× µ ‚Š ˆ y = (x + 5)2(3x2 − 14x − 1)6.—

2.3 EL TEOREMA DEL VALOR MEDIO 61 dy D}©l– | xy7lelg|r€”0i‹…'vƒ– –)w|al X„ c|ˆ ¤„ilo´| n˜ ll p„„› aj‡ r Yxayc¥ mˆ€ dœ ´0e–Š¨t‡zr|#‡ic˜a)l„ ue„w¤ i|exhm ˜ p˜‰‡ l´xlsıc—‰ix tgw€‚a‡Vx I— D‡ ‡Vx |–¤0x–}l‡ | w‰l©lhx |e˜0 –ƒl¨ x)l xY–}”y ‰| l › ) ©lI— ˜ )ƒ– € y y   ˜ xt‡ ˜ y l ) ›q„› ¦g‡ – ˜lz |‡ yTP•„¨0j‰wwX‡trp‚ˆlD|g| (o`ouP0ya ‡p}–– ´8)d‡|‡(z ,oDnaR€|wy XÒs–,±0„ƒliy(™bdrx™|cb­‚)il`l=€)²Bo´|Dlˆ)´m ·nw—uqφ‘‹[x¤„|eçaDl(1|‘,“uxy˜bw)(‡z}– l]`D|h||g¤ …§u„ƒw‰•leull©w„‰˜l—r€xD|e©y”ixlsl©vl0€7Rx„ƒ‡d˜––†a…‡€tzlc˜x5)|l|eilg¦=›!o´l€7 x„†–nl©s…¦w X$x=©d˜p−¦el0la|1„–}Xl‡rƒ„((R(|axlat©l)m•|),wx b0g|l)´ejI—‡)xXDt©|–rsd(D‡€0yzih§a‡)|„–}c|›)|ax ,–†wwy|B)0lX}–z%—.|…y (‚€‡lwr§bŒ}–€|))lx u˜m8hA§8x|lˆ §‘„›€7©lŒX†– ¦…”x‡xYl (–ly©xyxt‚€¦g)y}–φ=„=–˜l‘lly(l|8xXYYl››l)|w(((tttφ=lz ))x)|© Vy”¤Y˜(l˜xw‰‡ ˜ j)‘xl„›Xl =ۉ•X—−‡w |w1|V|gX(Yl|(‡t0z|xyVVy)–}((‡)‡‡ttw=mW||e))|e˜©l= 0ˆlx sdy = dtdx dx dt φ V (x) = YV j X−1(x)Rm ž j X−1m V (x) = Y V (t) = Y V (t) . X V (X−1(x)) X V (t) aEy€ˆ )8lsj|—›tey”ry r0llzocyˆ˜i7€id‡tc¤– © ieo%— ‚Šx.V‡ˆ h€  Hƒ„†€7‡ V‡yx‡xu¦I| ==)xx lv€– ˜Ha„›a¤ z0x‰wl–€ l‡ll©l–Ÿ|xx©l 3ll3x „ ƒ„ ‰  w €”¦… Q¤ w‰fl l©xY‰y z ˜ l Ûg|‰– ˜ s%— ‡V4€ „ƒ x‘l  w d 0–}‡ | l©x —I) £ ˜ ‡ | ˜ l § π jXw ‚|  w  ˜ ˆ€ 8| y7l4˜ l t †–l ¦… | ‡ y hx‡ ˜ 0 <w  –—xl t€ˆy  <l ˜| l—¨l 2x w wg|IxH‘ y 8„†‡| | uul |– y7y7gw ©l ˜x –}0| ‡ y7|l €”x ££ all–Dw>‡h|e‡V™0€ ‰˜ lw txlDu x › – y ˜ |e8 „¤¤ ˜ – —%‡  Œ l u  – y”0ew <˜ 0t‡0| <x y 8π2| 7ys8l—§ )l(xx0, ˆy0 ) = j ˆ Œ lfy –l | l y”u t0 ... ™ }„ ‡| a x(t0), y(t0)m y V (x0) = − |0‡l | ly –}|s‹| „w8 d¿„xlD08u– |‡zw |8z| „†ˆ –˜ x ‡!– —x4| € ˜l©ƒ„ lx w ƒ„˜‰„y l x ›„˜ • w‡¨‡ xgY| €y 0lˆ€ }–¤8‡ Dw| – – ©l‡zl x| € ul€ xl Dlw 8u|e„ w©l –x}–x„› ˜ |‰‡ul ‡'xv0˜ l}– ‡‡„ x| 7y ©ll0…¦5x¨')y€77y ‡–ƒdl © V‡ ze¦˜‰€7`„l¤–)l©¼Dx es)2x §™ó ‰— ' €¤ l w‰… l –ƒ 0xˆ…¦8¤ |‰w‰Ú)l 8 —„› l ‡ € xy7l  £ DAay¤ )w‰e„ yI™l fi¤ xgwnŒ5r– ullai2c‰wdill)ioFxoˆ n:aŒ Aeρbl™si.→l©˜ e™x–}Œr tlll5 o„“ ¤ 0waxw‰‡–|el|P0„ ŸF‡ w5•|‹ ˆ Ç (% þ x2˜‰+l | 2y y– ‡l = {(x, y) : x, y P 5 } bola wρa2‰| b‡ ieˆ˜‰rl t‘ ax| wgdx ceo— nwc|ejuyn‡ntx rtˆ oo | B™l ρw (e|a) ¦%=‡ ƒ„ h(x©e¦, y– l )€ :y ©y 2˜ −< l| y‡˜ ‡ )l Yx y ‡ w g| 0– ‡z | ˜ l Ûg|‰– ˜ l | w |A)g¦ – l € y ‡  Aw 8I| ˜ ‡5 2 ε>0 S δ>0 es continua en a = (x0, y0) `T ƒƒ (x, y) P Bδ(a) =⇒ F(x, y) − F(x0, y0) < ε . Œ ll l l€‚„ª– Ûe) ©   x ‡w |e˜ –l —‰• w€7gV‡| —g0–}– ‡l©|˜ ©l xv˜!˜˜ —ll w0|e‡|x …¦l )€”¦…7€ ƒ–d ©0g¦ – ‡„zl¨| ê }„ ‡B) ) „ ˜ l–x – u 8z „}‡ u ‰w l xl …'}– ‡ |… g| h‡ 8| ™ — „ƒ¤

62 DERIVADASTpP(urxon,pty oi)exP(dxdaB0de,δry.(ia0vŒ ))a– ˆxdF‡a4|l©sx „}0‡p‡x a| …ry 8c}– |}„i‡Vaw € l l©eRx sl |˜ a P § F(a) > 0 s S δ > 0 y 8I„ ¤ wgl F(x, y) > 0 x7y ©l§ xDgs € l©l t|x — ©l  xy ‡ ˜˜ dl e„}‡ A —7y l©l 5x yx ‡ – u w˜ l l | ‡ ll 0l y¨ s xYye–ªn)€ ê el – – Fx s „ {}z€„› ©l x–˜ƒ lw 80|ƒ„ FF ˜yxx‡l((xxlDw 00¨‰| ,,– ‡yyYx y”00§‘l ))| x ==l—Fx)l xhh§„„ˆ¤{¤{→→zz €7›› F–†00¦ylFFl((F|xx00P•y ,‡+y”˜ 0(hq‡ 1+,)— y(wAh0hh|)))y − F(x0, y0 ) y }– | w  Dx s xl—˜ }–  l ¤ ‰w l l)x − , F(x0, y0 F ˆ‡ ) ˜‰l `§ x . A ‡|‘0 ‡|PF¤Œ ywgl r8(loa| p,Sj †–IboFŠ sy)§‹i(=cWx| Ji,yo´0y‡–}n|g|ˆ)y72l l©l ”€ |x€(¦… E8Dl A„†¨ x‡ – xiYx ‚§sy”tl©e|¦gl©xn– lsc€0 >iy‡a‡ | hx yy†– ˜| dlw eB 5rˆ is vŒAalb =ili(daIa,¯ bd)JdPe§ uFAn: aAy 8 f—„→u¤n‰wc5 lio´0Fn‡(a|imy,}–b| p)w l´ı=ci§At0ay )8§ .„ § t > 0 y 8„ l©x ¤ wgl 0 —I¥ ) §`€ˆqw )|  ˜w z ‰| –¤0x‡ P (a − s, a + s) y 8„ ¤ ‰w l F(x, y) = 0 . y P (b − t, b + t) w‰©l ˜  s˜ sI‹,t5l a§fÛgx˜ y|‰+lxq –›lq˜ •swy)z –Dxe|l→s%j0| i–xml ‡–z(|bFFpxfl−´(P™ıxc©l ti,x” ,ty(b0a1)‡m)++|(Ayet†–Fn|))ytwsy(e8x„ƒŠ¨„„,l y¤|• w)‰w©x ({|gzfl aDV wy(– −x‡|Iz =)|ss=,ff• a(w l)x0|g+5x)0j y– ‡z | j € l 8 „ sc˜ lEw |If …¦©7€ –ƒ) ¦g„ l Š’ :l‡ 0(2› a–†€ †–jj|‰s−–}}– –††––}ŠŠ‡ ˆ xhl w ‰| ‡ l | %s x”©l wx ⇔s) )q› ˆ F¦g(–xlz ,| y) =0f l ˜ D„¤˜˜ derivacio´n impl´ıcita) j T x P (x0 − s, x0 + s) . f V (x) = − Fx j x, f(x)m Fy x, f(x)m¦…l00¨ ‡|‡) p7›2|€7o`(¤–y )a‡–}‹´8g¦| ,nDF|¨w„blÒ(0±)Ša—IrVl4§,)F7­ |b€'²(˜´©ya)·š(¤–ql4,a=çwyƒ„,j„¤)b0)˜ –†)s‰€©lŠ‡ sxxqs‰s–lŒF¥ ˜l©yw 7yYx–(%—lu¨§y a|€7‡8 ¤–w,˜›„| y|4€uy ‡)))z)Dx|››2>ws ê l2‡ 0|t˜xDy7ssˆ€T l 0—%˜‡ V‡ |‡ € lŸ l ˜ h› l —gƒ„Bs}„ y‡ ˜ 8s‡ „ x¤ ¤ ‰wt—I„“w‰)l )l–}ˆ|ˆ€ FF8y7d y„lyw‰l(”€(†„a©l¦…x‡ u,)x,‡ub„}y‡™))„ƒ(}„>>‡b• vxw 0−0g|l ŸˆTDtl©–(,Dx‡zxbs| V‡, y€+lj ˜)| xtlcylP)€ˆ€ w $sCFe|˜ h§ yˆ‡ s yˆ P€ C t>0 l (0b– l −| y7tl¨, bl | +l F(a, b − t) < 0 =⇒ F(x, b − t) < 0 T x P Bδ1 (a) , TxtŒ F(a, b + t) > 0 =⇒ F(x, b + t) > 0 P sl ,Bƒ„a— δ2+…¦(© as7€ ))ƒ– ).g¦ w „ l) ¤„ y¤ wŠ – l €ˆ — yl 7€)u‡ l Û| Ÿ‡l „ˆ ‡(z |a l g—  „}¤– s©8=| ˜ › ‡ {}z |l „ {δy”l1‡V,€δl 2›!} > ˜ 0l$ˆ Ù á V‡ }„x)Ú l 8 h |‰‡2 ¥ ‡V€ˆƒ„  •xw 0|gDP – −j˜ F(x0,

2.3 EL TEOREMA DEL VALOR MEDIO 63yF}–ll˜‹) | ||(€‡1x7yxuyy”xl j‡w0ll ”€ƒ–…|g,›Šˆd…¦jjyyly–††–ˆ)‰w–}†–2l)l01–†Š„}Œl©Dl|Š‡x)ˆ uDx–yŒl©y[s=–‡8xwlƒbh|)l„ƒ=w−€ˆ0v‚€˜ƒz 2|‰x‡<‡—s€tf0–¤l,(©l©I—00σx¤„b‡x‡‡VdP )y˜s„›+€0's‡ –P(—x ‡l‡tzal5l)‰wx„|][xl'syl©0y−7y |x–qx)˜l0l 0P¦‡Vls|jx−€¦x–ƒ,–→l’(Šlwbvlaτs!›B¥f€ ,Sl0w(˜+y−y¨xσg¦‡00–0ss–˜sx>l˜)+y,—%+l 0€ †–bs‡V€¨0ž€hτ€‚fw +w §)‡]¤(‡e| |xyswg„}=0sR„€‚0„lqly)‡‰w)•<yw0(%sDlxy=x0le|uxτP€10l‡+0ƒ{z,y<–(=V‡ηWƒzkb0ff|F)(ε(−yxy§x0fP y)(‡00t8:xεC|−,s)„(0)lbay>=,xƒf§k+1η−(¤ ƒx)yF0t‰ws<P0y0)=‘ˆ,€l )(ay(τ§¢xƒxHƒ0b8–+<,ˆg„‡xxyI—−ls¤t„›ε))8)Pw‰—e€ˆt|‡ˆ>→l,h„†[–¤xbFσF©080((8(,„+xxb−T|Huτ00(˜−ztσ>,wx,|‡)yy,,tx,y000ηyj b0))))0“„l++¥–==P‡| ¤ σ›„tœy C‰w0)€0]‡llŠ ˆ ˆ ˆ ss 0 = F(x0 + h, y0 + k) − F(x0, y0) = F(x0 + h, y0 + k) − F(x0 + h, y0) + F(x0 + h, y0) − F(x0, y0)˜ ‡ | ˜l = kž F˜l y|‡(y›2x€ l0–†|g+y†– ‡20h–†lB|, η…¦y)‡0„+w+ˆ hkˆ€  ž¦s˜ F§ ‡ xs(ξξl, |yy 0€ ) , § ˆx0 + h k5—l ˜ ‡ | ˜ l–x ) „ l 's § ™¤ ‰w l x0Fy =W 0ηl |l)xY‰yl „z l f(x0 + h) − f(x0) = k = − Fx(ξ, y0) ; h„l y hl ˜ €7‡–†…¦)h¦g→„ lFl y0| (sdxx0gw 0x + h˜ ,¤ 4‡ ηwg)l¤§ ©8„¤ w „ƒ8s | € ˜©l x”‡ w „¿u „ {}z¤ ›„w‰–l ”y l  l©wRx 8˜ | w‰l § Fy x ‡ A| 0‡| y }– | w  x–l | 's8 |§(x0, y0 ) Fx f˜ l • w f V (x0 ) = − Fx(x0, y0) . y ˜ –}| w ˜ l xY| y }– [| ay ‡− ˜s,l a +l 7€ s‰‡ ]ˆ —%‡V€ xl € w |¨0 '‡ 0 – l | ”y l V‡ € –  g| –0–}•‡w | |g©l 0x – ‡z0 |‡| fyFV –}sy| ‰Û(w xe|  08x ,}„ ›0y‡0l|)| y”˜‰l¦l hs|‰l©‡x „› 0–}‡ |e| cEl˜ o´l jwÛgne8e|riDccc –ai‡z–}c|ˆ™h› i8oƒ—g˜u .l„ {¤z 0ŒRwƒ„ –ly8z 8„y ›8 A©l | xl xu | ƒ„l2y74| l+y70 ly‡2(|cBx˜ ƒ„)x—¤– y0l –| ‡+‡zz r|‰|l C„¤– ¤©ly‰w| 2¨lyl d‡+| x‚€ `l|‰l2‰„‡D˜ — l˜xw¦lR|+l wy ‡ „|2wE(— ›„xyw 0—g|+,„}yy–ª‡F€u0)(—%=x©l© 0x€ˆ0,4 „ƒy5„ƒ0¤ €)w‰ll wl! wy )l d xY€70y˜ ‡V– vl‡¦Iz | )ll €H ˜ww ¤ lqw ‡‡e| zz „¤||  dw‰0 l– d0 – Ax0x + B(x0y + y0x) + Cy0y + D(x + x0) + E(y + y0) + F = 0 .Lyl„& °t|w5eay ‡V‡V±‡Vlrg—¦‰€„‘œe{ƒ–ˆ€z gAyf¶Rw)la„q%•©w‰¥ ©rI—j ld¬D¢)e×x¿e䀀nˆ‰˜äl&lprIL­¢l0l¢dƒ„’܉µƒ„)Hy‰¡ ‚p—%¨› xx¢)¢noˆqg‡VŠ–p€h²†˜}˜o‘qil—‰sts7p€7aqI“x‰|—%}–l›Ò„V‡˜V‡ g¦l€l“³ „‰„€ˆlp)lt Òc„8ƒ•z²X…ˆ€¤„қ!)lDn w‘|Úesh }„7y‡„ƒ©llzt x›˜ –}€¤• lDz l‰wyu€}–lƒ„Dl©4‡e|€• {w‰0z%—xDx–ƒls©)w €ˆ„–}!› ©¢¢ÚD‰˜ äV‡vl„}ã”y‡ ädlÝ x ›su‡l ¥„†– ˜¦g{†zz)›2yl‰|‰|–¤–}–0|y”Ú~‡f©l x”x!w¿Ù)lv}–Ûgl˜„}¥„ƒ7€0l}–|‰}–„Dw‡‡– ›|eylw –ª€7%—}„l‡„†˜ –‡„| ‡‰ê lˆ— 00|óˆ wz‹sdtÜg¦ ¢¡|}„ˆ–¤‰w ©©l§˜x l x  £ ˆ

64 DERIVADAS l©0˜ w'‡”xl4 l0—)g|–|)ll0€Œ|˜|ƒ–u–8˜‡”y wl™†„–u›lx› (‰˜| l−ally7||©l)7y„†ayxH‡l™=d˜xR0x”l l–qlv˜4|‡z(„ƒ–†|¥•–a}„ l‡8)€…˜ lx)ll=|g–7€u ƒ––†0ƒ„ w|)0w–ƒ)g¦–gÛ sl„„Þ|‰l©|lx˜ x5––y”ylxll 8˜„›| êsh›l˜ ‡l Yx y €ˆd0– ‡z | ß ‡V€7– u –†e| 8 „ )l x › ‰w §Axl e| 0}– „†„ƒ s™§ 0 ‡ | x – xYy7l § ‡ y ˜ ‡ yx– Vu(©aw 8 )) | „¿˜y 8–l ˜„I„  ¦…0 ˜ ‡'8 l©†„0V‡x– l€c|Þ I—˜y7)l–}•ˆ€ ly € (l xx|g)0==–ƒ) a¦guv„ ©l(˜(xx”x l )ß)„ l| u7y§‡l(|gxv— )©ls l x“—¤ l w‰‰w v„gl©l (¦…x‰|©x)‡)}„ V‡ ©l € x | u „ }{z › u(x) = „ ¤{z › u(x) − u(a) , x‹8 →| „%˜ a—g‡ €7v–†x›(−xl )a€ x→a v8(› x)l −y7l v—(al )¤ w‰l |‰© ‡ §hs w ©l x l }– | y |  ° 5e V‡± —g– y s § ˆv(x)−v(a) = dv lŸ ‰Û |‰– )„ ul©vx(xl )„ −x u– u (w a– )l | =”y l dê u ›„—g„†‡!¤ w‰l˜  „ {}z › · · a3 dx−2x3 dx − a2 dx rrrrrr x→a 2a3x − x4 − a 3 a2x ² 2a3 x−x4 3 3 ax2 x=a ·= − 3a dx ² a − 4 ax3 4 4 a3 x ‡V7€ =ƒ– “%— −−©ˆ€3443adl d¦„xx7€ – ² ˜ l „ x ‘ uI|w – l ˜ l7y l›„€‡ l)Yx xy w€ˆd„ y 0 – ˜‡z |h‡ x—g € yl – …'x •i–}d‡‰ê y u = 16 a . w‰)l ˜ fl ˜ ) € x l H%— © € y –†€ – | 9 ˜‰l 7€ ‰| ‡v— ‡V€›2‡(y˜¨ )T‡ p7„ x`oV¤ • w‰´8Mw nDlg| Ò C00±f–}VrV‡)(7­ |c'² T©´)©l·šqxej çgoD (‡rŒ be|l )my ©}–−|g—aw „}g–¤d©(5xae)lll m| „ v=7y[alalV‡g,o€brVl(]!›cm¢§) ej˜ f˜dl(l b7€io–†€ž)…¦©−‡d¦g„}e„f„ll©(Ccxa )laƒ„m|™uˆ c•(wha|gy,Db.– )‡ zŒ |es – l f| ,y g‡ :g| [ a©l ,x bS ]c→P 5 x ‡| Tˆ 0 x€7†– ¦gP ª– €(al ,|4b„ƒ)q s •ƒ—%V‡ V‡7€ ›¨€ j  –¥ œ Š x ly – l | (a, b) j j φ(x) = f(x) g(b) −g§ („ƒau )m0tR‡ −‰˜ |gl 0g› „ (gw xzx )Dx– ‡sgz |fx –(xbgl )V − f(a)m l ¤ ‰w l s —(x‰w )©l ˜=W —l 0©l x g(a) =W g(b) f(b) − f(a) = f V (c) . g(b) − g(a) g V (c)y ‹ l z„ | l | l w„ |g© 0 –ƒx  „‡ ˜ –}‡ | ‰w˜˜2‡8Lll0l›q{’0y7sHƒ„‡¦glgc €7∞4|–oˆV›2l(€z p|x„ƒDl}†–u)iI|ˆt©l„¤¥–a=xW Œ†–˜l—g—.w‡ 0‡(gwz—%x”yŒ∞∞©l ‡Tl©)l˜ x|x=lc–Wêux IPˆŠ)x0lê2› wH€ITE|‡ ›xxj –†„ |‰wzP¤ xf”yl©‰w lIuu l€”‡tŠl…|f§8 —%,l„}g‘‡‡ˆ€¤ 8˜w‰:)„ellI¦g¤›2–l0‰w→l ‡¨ l5€ x–y Yxl‡`57y0„Il—‡0‡V|x‡l€7‡x„|– u–| }– e| 8 „ s  w ‰¦ 7€ – l | ˜ ‡ ) ›qg¦ – w7y0€ l‡g|l „››20„ {}}–z „›‡‡‡ | – l l©| x w lu˜|ˆ€w‡)–l0x€l ¨'s|•y § ”y la(¥‰˜j ƒ„LllsH7€›„y7Vª–H˜l…¦‡ˆ€‡)x)8 |g¦ „ …˜„gsR©l–l˜x™xeyl ag‡l d|ls aP¤ w‰Igw dl5s–l€e¤˜ y n x˜ – Œ w „ {}z › fV (x) = l P 5 n {s ∞} . —%‡x| →u a)›2g‡V (x x¤)w‰l  ˜l › z ©x s „‡ ¦g– l |

2.3 EL TEOREMA DEL VALOR MEDIO 65 j †– Š‘„ ¤{z › f(x) = „ {¤z › g(x) = s h‡ ¦e– l | • ˆx→a 0 }– –†Š‘x→„ {¤z › a • j ƒƒ = +∞ x→a g(x) ‹ | y ‡ g|  l©xDs'y 8™› ¦g– lz | 0l ¨ – xY7y —l l q„ „ {¤z › ˆf(x) =l x→a g(x) ¨ p‚o`8´ nDÒ0± rd‚­ ²B)´ ·—q çR¿l¥ ›„‡ –x x ‡z „†‡ l $„ ) x ‡ j –ƒŠ50 ‡ | a P 5 § „ {¤z › fV (x) = l P 5 ˆ Œ l  δs[a>—I,)0xˆ€ ]4y 's 8 ‘§)„ ¤x˜ —lw‰ l¦y sx– l x s l y ‡g|  l©x rrrδgf“s x²² ((x→xxl ))a)+−g— g}„l¤– rrr©V (< x)jε ε > ”€0™…¦Q)l }„7€S ‡ ‡ – 0l €7<–†Ûg)x) −| ˜ a‡ < | ˆ ¥–œ vƒ Š l| l„ … 0 δ −a < –}| 7y l | l < x f(x) − f(a) f(x) fV (c) g(x) gV (c) yg‡ (g|x)©l −Dx s g(a) s € l©xw „ y = = , c P (a, x) . á l | D‡ ›„‡ 0<c−a<δ rrrr f(x) − f(a) − l rrrr = rrrr fV (c) − l rrrr < ε. g(x) − g(a) gV (c) Ejercicios.— ec)}„ ƒ„ ) R€ }„ ‡ x „ }{z „› – y”l)x „ ¤{z › (x − 2)e2x + (x + 2)ex (a) (ex − 1)3 ; x→0 (b) „ ¤{z › õ y”ux 1/x2 ; (c) x→„ }{z ›0 xö p 7y u x 11/ ù x π − )€‚ x→+∞ 2 q . xë l 2‡| ƒ”x3{d³D‰ ‡ ´ x s s‰xl—y – l |1. Contraejemplo de lStolz.  © ˆ€ Aƒ„  x • w |g0 –}‡| ©l x f(x) = x + x g(x) = eá÷ Cø ù x f(x) „ {¤z › gf‡2VV ((d xx¤ ))w {z =¤ wg0l , § |‰‡ l0¨ – xYy7l l „ „ }{z › f(x) . x • w |g0–}‡| l©x l©x 2‡ §x→0 ‡ ∞x xg=(x0) Ž w x→∞ j w 8| ˜ s Š l | y ‡ ˜ ‡ l|y d‡ ‚€ ‰| ‡ ˜ l ∞ˆ  g V (x) xl |x xl | ‰s x l—y –l| l 2. ”€ €7– l | ˜ 0 g(x) = x + x © 0€ ™„ƒ f(x) = x− „ {¤z › f(x) = 1 , § |‰‡ l0¨ – Yx y7l l „ „ ¤{z › fV (x) . x→∞ g(x) x→∞ g V (x)µ # # ¶f+(a) = f (a+) § „ {}z › ŒB3.δY (aP)ropS oxs→icaio´fnV (.x) –f s‰l©lx | 0y ‡‡|g| y }–l©| x w  l | ¿s ˜ l ˆ €‚–ª¦… )g¦ „ l!l | w| l | y ‡V€7‰| 4‡ € )l ˜ w 0 – ˜ ‡ =l S fV a l (a) =

66 DERIVADASl| ¨ 7p o‘8´ Dn iÒ 0± Vr 7­ '² ´)·—q ç ƒH‡2› ‡ l©x 0¤ ‡ ‰w | ly }– ˜| w  l | sS „ ¤{z › f(x) = f(a) ˆ Œ l — w‰©l ˜‰l y ‡g|  )l x )—g}„ ¤– © ) € Xj  e–Š€8 d„ „ }{z ›„f – ”y l l gÛ | l a ê x→a f V (a) V „ {}z › f(x) − f(a) = „ }{z › fV (x) = l . x→a x − a x→a 1FfE€§l)YÒ ul´8|(—‰ l¤xcnnw€7lw)e‡|gd‰µ |¨–r‹=0plar–†–ƒ‡€„I¨›l©mor©lxhxY›˜uqr|w7yaxe‡˜lr©˜Dlz·nefd¬†‡y‡˜¤²(2›nt‡¤xntlRa¤²˜w˜)‡VRn—‰lew‡{lzd› €‚l`o|2dtx–†˜–†l›m‡|eB´‰—l|l)A|l”yµ‰l7€yvx´el©ŒV‡€7‡`cp7u¦t˜–†‡¦g”±´a›„o0€—%q%xlz„l)dxl‚€c‡V—h´„–¤)–ª›¨l©uo€©s¦…xI—xq‡)žlc)od™x|¦g˜w€l‰—e„yl…§€¤©l–l€7ۉxl“xw‰V‡S˜‡§%—€‰|—clwx'y‡i‡–m‰wF|‰˜xuz˜l˜8l©V‡s–¤|(lp˜x¢©xx›2y7y˜Ells㈇l)Üol ‡7€¤„¤=©nj}–qx4w‰„†‡V—‡s l€fxx—‰xz˜l©Bsw(u€‚xl›xxg|—%ˆvd‡ˆ–)B‡‡V—%0¸0lz¢Dfr0|yT‡¹ãa}–‡Ul©xl2x‡¢ ˜xÇu€–¤|s0›P‡ 0©l–|¤)“x‡‡xqz[·‰wñz‡|a|lxD¢2l˜ó„y,¡hl¤l}–ˆb™!lŠ|›…¦w‰„]U#w8l‘llˆ)©l#|‰z||„¤Yxx))ځyí7yl–lw8ˆgwu| ˜„g„†„‰w˜©lt‡[}„lƒ–ax–ª8¦g–}l©,—%||0€7bx—‡”y‡©”y ]lDl|€ˆl dÿus„xY„ƒˆ€ƒ„yV®tl—8–q%xe”y„d„“w•gµowD¤h§rDr8‡zg|‰w‘ole„}|‰0|¨lmB‡–wp7‡zeqx„†la|‡llׄ jl | Šy d‡ Œ‚€ –‰| ‡g ©l–}›¨x 0lwz y e|7€ ¤–s 0‡ • w |g˜0–l ‡z #| l€ l 8 5–„ †–s™„› l I—| )y ‡€ g|Rs ‰˜ ©l lx©‰Ûs ‰| I— – )˜ €ˆ 2§©  0˜ }– |g0x‡QP … l l)xlxPy˜ – llr€7|oª– ¦…lp) o¦gs„ luiclio´| nw .| x 0 | I I g(x) = x j g V (x) + x5 g(v) 2g V y”(l 0”€ )… m 8 −}„ ¨‡ 1©8e¦ 0– l (ξ) , ©l vx w `| 3— w | y €y‡˜ ‡| ˜l ¦%s ξlŠ | y – ‡ f|gl© xl©vxw |ex”–l€y• ‡ ˜l „ –†| § ˜ l—0l B¨ y € l 2› ‡ x l0€7ª–§…¦)xg¦ ˆ„ lHl | 7y l €”…¦8„†‡ j Œ w g|| 0l – ‡z q| € l 8„ sV‰˜ l ۉ‰| – ˜ l )l ™x ˜ l B„ –}|[a, – l D †– |gD‡R…  b] b−a º¹õ a + b − (b − a)5 f(v)(ξ) , f(b) ¸− f(a) = f V (a) + f V (b) + 6 4fV˜ ‡ | ˜ l P)ldx €ˆ (e ¤a‡fw‰n|S,w l¢bui|em¶P)8 €˜ ›„ˆ p–†r•'… ‡w s–s“x|go˜ 0xnlyl–| ‡kŠz)q|| l=„5€ xl –}f08| („ y7x=slk˜€”)…¦laÛg8—Is„†‰| )‡ x–€ˆ˜k![a=),§ b˜a]w 2 ö 2880 j 8ŠξŒ   y €7‡R… l  )l x™˜ l €7–†¦… © ¦g„ “l l | l '„ –}| y7l €”¦… 8 „†‡ bu — –]”yw ˆgw | ˜ y l©07y x l‡ ”€|S …¦ξ)k„}‡P s ss[„}}„j a‡‡ |x, ˆ‡ +l | k(b − a) =!x ˜1l b2ƒ„ )f.y „›.) .f„ – x ¤ !›2‰w nl k ˆ 2n‹ 2|nxy w ‡¦g|g}–| (a, fo´mula b b−a f(x) dx = y0 + 4y1 + 2y2 + 4y3 + . . . + 2y2n−2 + 4y2n−1 + y2n a 6n¨ 7pV (o`0)´8Dn=Ò0± grV7­(i²'−v´©·š)q ™(ç(20bj 8)8−=0Š an0‹ )4s¢|50f‰—‡(i€7›„v–}›)‡ (ξl ”x €—)l .„0gw ‡ u ›„) €—‰s 7€ %—V‡ ¦I‡V)€–€ ¤„™z„ • –}dz„› 0–}I— „}›©€‚l– ˜ 7y l ˜f˜ l g x ™l y7l | ˜ € z g(0) = | ˆg V

2.3 EL TEOREMA DEL VALOR MEDIO 67r§§t —gψy „}) –}(q›)Hƒx))g¦‡| |– x– ˜ l€ l ›„Œ ‡ l¨x y „ƒ– l x l• w |g0}– ‡ | l©x  w ¨ }– „†ƒ– )€ l©x sφ(x) = g(x) − x j g V (x) + 2 g V (0)m =lz ‡| ˆ | 3 x5 φ(0) = =0y7ls l l |x Q 3 ˆ φ V (0) φ V V (0) = 0 φ V V V (x) = − g(iv)(x) x ˜ y ψ€ ©l (x0)… l = )l xvψxVDl(u0w)– ˜= —x ψj V V (–¥ 0)œ =ƒvŠ | ˜ € l ›2‡ x ˆ ŒψV V V (x) = 60 x2 >0 I φ(x) = φ V (ξ1) = φ V V (ξ2) = φ V V V (ξ3) = − 1 ž g(iv)(ξ3) , ψl $s (¥–xx œw) ψ V (|lξ|l 17y‰s )l ˜ s ψ V V (ξ2) ψ V V V (ξ3) 180 ξ3%—˜ ‡V‡ |€ ˜ j  Š l©x  s l | y”l¦s x†–—l…8y ›– l sξ1 P (0, x) ξ2 P ê(0, ξ1) ξ3 P ˆ(0, ξ2) ê †– |I)}„ › ˜ 2‡ ¤ ‰w l g(iv)(0) = 0 g(iv)(ξ3) = g(v)(ξ) , ξ30‡ | ξ3) s §`‰¥ l „› ‡ x“y”l €7„› –}e|  ˜ ‰‡ ˆ ξj ¦%P Š (Œ 0l , j s™˜ l ξۉy |‰‡VP –e| ˜ ( 0¨©l ,x —Ix„ƒ)© €0x™y 8 ¥„t†–¤— s § aw +2„› b—gm „ l¨l | P gw‡z l y7¡l©−x – bxq−2˜ al ,„€b)−2—Ia) €k y ˜ ‡ Gj ( xŠ )™s =§hs Q – l„ƒF|` (7y t•l)w s |g=—ID)– ˆ€‡fz u| t©G+˜ —%F(V‡ x5€ )−0 2‡F(| −x x– u)w  x> S 0 G(x) = xj − x5 G(v)(ξ) . G V (x) + 2 G V (0)m 3w s 180V‡ — l ‹ ‚| ˜ B‡—IŠ )€ y ¤–  „¤) € s I— )€ˆ x= b−a Sz P j 0, b−a m y 8 „c¤ ‰w lAXj xwexYy – 7y w §¦l | ˜ ‡ § €ˆ8| 2 2 ¸ º¹f(b) − f(a) = b − a f V (a) + f V (b) + 4 f V õ a + b − (b − a)5 G(v)(z) . 6 2 ö 2880™l €7‡ zG0l)z y (¤–y v©8 ) „ (Š z¤ )lw‰| l =y € F(v)(z) + F(v)(−z) s A§ ©l Yx y7!l l©™x w Q| …8}„ ‡V€u}– | ”y l 7€ › ©l ˜ –}‡ j ƒ„ ‘› ©l ˜ ¤– ) 7€ – y › l 2§  l„ j c¥ š €Š —I© €0 w e|  ˜ l 7€ †– ¦…  ˜ sS ˆF(v)(−z) ‡V€ ξ1 P(−z, F(v)(z)gÛxx ww |e¨› ) )l }„ )x›jF†–|)…(l˜Šv8| )‡›t y7(slz—gl €s)|}„©l ¤–F+y””x© (fl2w8vF„|)yl((˜)vù|)1(z©)−l „g=z˜ € )l)fx(w=wv„)g| y(‡Fξ˜()v˜‡ )0l (‡˜ ξ|„†l ‡1%„ )x‘ξ);I—xP w)€g¦(ya}– |,˜ ”ybl‡ )€”…¦ˆj %¦8„}Šf‡ x ¤„[a,• w |e0 – s ‡z | sF(x) = x f(t)sRd§ t a x2] . . . [x2n−2 , b] b b−a f(x) dx = y0 + 4y1 + 2y2 + 4y3 + . . . + 2y2n−2 + 4y2n−1 + y2n a 6n (b − a)5 f(iv)(ξ1) + . . . + f(iv)(ξn) , − 2880 n4 n

68 DERIVADAS›w0 ‡e| ©l| ˜ ξ–}˜ ‡ kl l‚€ Pª–| …¦y [€x˜Rl 2 kl s „e−S !›2ξ, x§ 2‡Vk€ ]§„ˆ l  l 7€ ) 2‡l„‰| „¤¤ V‡ gw € wlz ˜„ yl }– ›!„}‡ hx …•¤€ˆ8d}„ D‡VD€ –©l ‡zx | l| y€l s }„„ ƒ„w‰0Dl… u l©5x l©vx w l4| „ …j 8 „}¥5‡V€5š ™Š –}|—Iy7©l €0€”£ e„ ›y ‡ —%‡V€ f(ξk) P (a, b) f(iv)(ξ1) + . . . + f(iv)(ξn) = f(iv)(ξ) . n2˜Ù l‡7€.B‡|4ƒ„ ‡V˜ ‹¯ x l©| •©x wLsl©|g§Yx¢A0'y„}Ü V‡}–¢B‡©€ |ˆ w Fzjl©„¢©qxy‹Od䆖´4¨›à˜„%©¤IlR‰—­¢7€ÜÀ7‰†– }–…¦xM›)¢ l© Š De¦l 0€U„ –l)‡z©lsDx |YxL™sl y7|§gw©lAYx˜x 87yw–}wg„‡„uDј w7€ ƒ––e|)p)uE€Òwælx €7›„B´ ‡T)Iµ x‡g— ‡®Ax }„ ¿n¤– ˜©„¤Y²}sdl%q 0„ ­7)L–}l±0‡—‰€7p7|O€7€7o‘©l‡‡Vx¨€p7RˆqI}– x!› ÒYl´)d ±)˜‰0w®Vl#– ˜oÿ‡¦»zI}–‡|Vl}²4‡…qp7µh„%—±7%—²}ˆn0±‡‡V€ƒr¦pD}„™€­8s†–|ÒÄw—‰r‡Vz€72› ¥B}– › §–¤©ˆlf€ˆ` }„ ‡'… © l8 ڄ €PT•w w |eog| olξ0 ir–n‡ez— o|ml m€na”y il +o|1sl.10 d(–… l eFl| o´”yTl)lr“xam8y‰˜ d„ ull o–}€7l| –†ra…y7.l) d€”g¦R¦…e„ 8l–eT„†s‡¨l at| o)yg¦Ildo–ˆ l re€ )y ).L‡ ˆ€ aŒ˜ gll—)rDlaI˜¨Bn—yI g€ I—le5„›)€ ‡ w ˜ !| l }– —| 7yw l | x”€ y¦… ‡y8 x8}„ „‡„a¤ © §w‰g¦ l– l € ˜ y ‡l § l0f¨ w – Yxe| y7l x a § x I f(x) = f(a) + (x − a) f V (a) + (x − a)2 f V V (a) + (x − a)3 f V V V (a) + . . . 2 3! + (x − a)n f(n)(a) + (x − a)n+1 f(n+1)(ξ) . n! (n + 1)!dl–}©lt5¨ |Ÿvx exDl7p7y {lzo`h›l gs„”€ fg—…¦r8´ r(a8nD„}ex‡҆„ds‡)0±s torV=[oa­7an'² ´©,Tn·<šqxnd–ç]fe´eêx(s‹xfˆi;(|fmaxŒ))}„o‡‘–+ee¤xnRlngw ntlh0fol‡a(xrx|– nVuf;x owgoa– ˜rl)alms hsˆ€lx˜8ap#|‡§ u|d„ƒn˜ae ltx LxoTl• anwa0gf|gVs‡(r0 §|xa–}‡;xnRa–| ˜gn)l©lefx l)€ˆ(8$xx( |E1l;w a„7¨ Ûp)9–}Ÿ „}‡7o=ƒ– ©x)l€iˆ ˆ(©lnxxDŒ(o−sRnw ma+%—˜ )1li‡no)‰Û|+!g|u1d–8f˜e›„( nTx!‡ +axDl1sy|)%—l(oξ‡Vl r)€„ φ(t) = f(x) − f(t) − (x − t) f V (t) − (x − t)2 f V V (t) − . . . − (x − t)n f(n)(t)§ 2 n! ψ(t) = (x − t)n+1,x l—y – l | l ˆφ(x) = ψ(x) = 0 t e— „†–¤© 8| ˜ ‡ j  –¥ œ ƒvŠ s S ξ P (a, x) y 8„ ¤ ‰w l φ(a) = φV (ξ) , ψ(a) ψV (ξ)„ w‰lDu ‡ ˆφ(a) = (x − a)n+1 f(n+1)(ξ) (n + 1)!

2.4 LA FO´ RMULA DE TAYLOR 69b Á”¼ ÇĤú ¼ ü ½ À€ÅSY½ Ià ¿À ÁÊ »f¼ÇXĽ Ðø N1˜ÿœ .‡X¬yod®'Ù 0t0²iƒ„F©´a qesPo´w Dn 7€rysI}–m| e§‰‹uj˜ ˜el”x‡a}–l!m›™| ‰˜ydp¦ ‡l elw!› o€8suM„5‡ a€sa©l ¢cYx=Üyla‡ 0u ˜Š‚rˆ¢lij nHƒ‹  .‡x— }„ˆ}–u 7€| €ˆ–†¦g8Œ œ|}– l„› u d lt}„‡0ƒ„ x8ƒ„ x!› lhw ‰€70„ –}¢l‡|  | ¥Dx yj{–z–}¢©| 䃄 ws˜Vã ddà •0˜£‚‡VÜz–¢‡Q€7‡ z ›s| ƒ„ fw¦ä ) „ƒŠ„†•¿suV‡ wl€7˜ e|›!| l  ¶x  ‡Våz§ 7€ 0±„}›V‡Dp €‘¦rw Òƒ„²}0nw x5p`8˜|´ £ã.ûûûû%͔Ã$\"iÇ»¾»“Å“»¿Å»v»À¼X¼XDüÌi¼üË˼¼XËË0Ìi¼Ì¤ú¤úchÁ “½“ǽ½qSº¤½u¼ÃÄÃÇÀÀ™Àc¼€Ë.ÁÅÀÎ8ÇX©ÀºË7½ËË%ˆ½½”Á⼪ºËÅD““DºSe˼5éËËú8ÁeÌ»¼.ºýúˈÆtø$ǐúÁ”½Yi¼ú¤¼X\"eÆÃ8¾¾ËÃX½„Ë'e”ÁÀDÎ8‚½ªÅÅËüʵX¼ËYÁÃqDÌd©Àg©X¼ÿÃÅU.¼Ë}ú¼ûËËÄÁÌ0ÀË'À¼ý)À˼XÀ÷”Á»¨÷ÇÅ'ºǽ`ÃX“ú†Ë–.X¼Á$ÃÃ$ŸÃüÀ)ûSÅÌDæºÅBÃXX¼YÈc©úÆË ¼–ýÇãüËÌ©Sº0X¼D½ú÷SÅY½YÃÀcü.˽”½ˆÇËËg¼ýÌD¼X°Å¼Å˺¼¼ˆËËðÇÀ©Ê§é¾¾©ÊD̼QÃ)úËüYȺŽÁ”ˆÀˀÃÐÀ©ÇÆ7“ǑXÇÙh¹½$º¨À˂Ƽüü.ÌDƒú½”4ÇÅý¼¼¼Ä¼Ê“dÀøvý”Ó»½ÇXËËúƒbËËu.½Ë¸†ÌDÅÁ.üüÀ©Á©Ì¼ˆ€»úaµ¼użÁÀËËÁ”úDÌ%ÍehÃ0À©ÀÀ©.©¾¾©¼hÅÄÁDÌ.¼Ë˄½º¸týÁüÃâŪÀ)¼eʺËý“V½ýʽYDÌ.ÃÄüÌ©\"h½Ê½©Àv½”Áú¾X¼»¼Ëü½.Àeg¢ÇÅSe½$XÃxÊÃsúƒi¼gYÁg½„À©™ÅÀ½–½úƒ%¼X¸Å¶¶df»#ý.üËËe–¼¸Ê¼ÃwËÇXËý½÷˺ý½È0ÌÈüúƒ¼ÃX÷Çe).Á‰Ì2eÍdÃP'¼D̽!»fúƒ¼$f’€hÀ½g”½0vµÁû¼¿ºü“»ºS«7¤¾‘˽gX¼‘”DÌÃ'(X¼Ëi¼ú7¼Ê¸Ã0ËÀ¿ˆÀÅ!Ê.©À¼ÁýµÀD½ËÅeÀ“¼!»\"3Ë¿ÀºÀDÃDÌEü“¼«t¼Xˆ½bÈÎ8üÁYÊÌ0˜.Ëü¸»üÇDÌËúgv¼ºˆÆ¶µ¬Å¼i»¼–ºÁDÆǑ¼¶Á”żXY½SźüÊ.Ë˽”.–ÃÁgHÅi¼iÇÈüúVûüŔÁÊe%Èü€»Î8Ë»f™ÈÅ©}úË˼™iÇX¼ûÃÄXÇÃËûx}úºgº.eËË˼”½heX¼Æ7ºSƂÊÇËËXǺÊÁÀ)DÀ†Ç½ú8ÎÎ8e‘XÃgÀ©¼X.DÀ½”½¸)À»f¼YÁ”ÁüXǼÇXXÃXÇSÅÅSËDÌÅSºÅÀ)½”ü¼ÀËÌD¤©Àb¢7ÁÁ7XÇDÀDÀ½½YËú¤i¼ÇÀX¼”½i¼X¼¼Xü¼X½X¼ÅSÇwººggÀÀe½½½½Ë½Ã¼¼¼¼e¼ÃÃÇÃǽ¶ÏÏÏÏÏÏÏÏÏeÅÊ • l Á s D‡V| ˆxn+1 f(n+1)(θx) Rnf(x; 0) = (n + 1)! θ P (0, 1) ex = 1 + x + x2 + . . . + xn + xn+1 eθx, x `G i . 2 n! (n + 1)! x = x − x3 + . . . + (−1)n−1 x2n−1 + (−1)n x2n+1 06 7 ¾A 3! 9 A (θx) , x G`i . (2n − 1)! (2n + 1)! x = 1 − x2 + . . . + (−1)n x2n + (−1)n+1 x2n+2 2 ¾ ¾9A 9 A (θx) , x `G i . (2n)! (2n + 2)! ¿7 (1 + x) = x − x2 + . . . + (−1)n−1 xn + (−1)n+1 (n + xn+1 θx)n+1 , x > −1 . 2 n 1) (1 + (1 + x)r = 1 + rx + õ r x2 + . . . + õ r xn + õ r xn+1(1 + θx)r−n−1 , x > −1 . 2ö nö n + 1ö   w z „ y }– ›¨¢ … 8 „ l I— )€ˆ y ‡˜‡ rP 5 cs ˜ l Ûg|‰– ˜ ‡ l 5„ 0‡ l gÛ D– l| y”l g¦ –}| ‡z „› ¤– 0 ‡ õr = I— ) ˆ€  y ‡ ˜ kl©x –}’P ‡ |r ˆ  V‡ € l Ÿ l ›hg— }„ ‡ s |=7y l©1xH16˜ r(r − 1) ¢ž ž¢ž (r − k + 1) – l | ”y )l xfDl ¨ ‡ l©x —I©€0–„ƒ x 0 ‡V”€ € l)x õ 1/2 ˆ lkö „› ‡ x —‰€ ˜x wy †– ©l}„z –}xÚ)–}¨› ˜ „‡ x ê)k”€ 7€!–†I¦ – „ƒ “x x – uw %— ‡3| ˜ ö – l ve n­ l €7†– …  x l | ”x l |(ex)(n) = ex ; ( x)(n) = p x + nπ ; 2q D ‡ x 0‡ x }„ |( x)(n) = j p x + nπ ; (1 + x)m (n) = (−1)n−1 (n − 1)! ; 2q (1 + x)n j (1 + x)rm (n) = r(r − 1) ž¢¢ž ž (r − n + 1)(1 + x)r−n . 2”„ƒl0˜ 2¨l.l(|∞—‰… yƒ„€8S)€ˆl©„}ex–l r‰˜˜–r|‡ezl‡i|epÚuxw sr|fjl e0|ds‡–}|ee|Bwy7n… lr| Ttl€”a€…a—uc8ywl„}il‡|e|o´oy0nrI‡fƒ– %s dy lD˜we¨ lM8– Yx¤„lƒ„¤”yaalw c|As–flleua€ˆ„}rn‡uixceraii—%o´nd‡ nP.e„}—–†‰| IpT‡ˆoa2›ƒ ryŒw–†l‡ lo8sx |ur—g˜˜ ƒ„8l‡s8„$e| w…ry7l8e|ie§}„‘V‡ „}‡V€ld• €|w e˜ yg| ˜l‡DTlg|–ƒ„ ‡2az |l©yw •hx wl8fo¤„|gl ¤r“„l©0 wfsux– —‰‡–zl©l7€˜I| „x€V‡ lŠ g¦ u˜ ˜0„€ˆllƒ„l0!› ˜†–xƒ„€ ‡l s „ {}z ›f(x) = ∞ N f(n)(a) (x − a)n f(n)(a) n! = N→∞ n! (x − a)n . —Il Ÿ )l©l €ˆ›hxYy g—y x‡}„ ‡ ˜ sw ‡ „ƒ|gxDx€†– P‡xn|l I=€7)l –0ˆx™)l ‹€xl |h|˜ ly© ‡ œ x ‡—‡d05) ƒ„Û 7€w§ ¨› €‚„ƒ†– | yx¿–†˜ …¦˜‰l‡l s„ƒ}„|„ƒvx (1•• ww |gg| 0D –}–n‡‡ z=§|| 0©l fx xel x}„ s'„ƒ)x”›¨l |l  xal §n‰„ a–†0 | l‡´ı”y xtl ix7€ c…a€8l „}e‰—‡ n€ ()l −Ixl1ˆ | ,y )V‡ |€ • ˜ + x) (1 + x)r | 1) ˆ

70 DERIVADAS ‹ t| ©8 q› ¦g}– ‡ j de ŠCauchy s „ƒq • w |g0– ‡z | contraejemplo xx –– sf(x) = e−1/x2 x =W 0 0 x=03—©l˜RÔxœ ‡x5w.zlw‰¤²„}dDn¨)l‡„˜00ÄÒ}–˜ClO„ƒ´)ƒ„}„׀‹lƒ–±l©0xswt`|‰—€‡Vl„ƒr7€©l€¦„ƒa}– †–xx™l)|—Œy”|slxl‰— lÛg—ll©|”7€|f||‰x}–zlo(y|u–v€∞–wyru˜H–}u‡e|m|I—I)luz(ul)€”|l 5a£ˆ€©|„¤y  $¥u˜s)¤– ©l5l •l4§h8€7n›2w „%›ƒ„ oe|˜x‡€ w—l‰ull0l©Yx |§–„Isixyy‡Vny€ˆz7y}„–¢©|‡ldª–ltãg¦”x|0eã|nll7€–©lg‡|€‡7wz­ –r|sf„ƒ™ll©˜zal(xxn„l˜ll˜}–!§e›„)‡VllV\"—(s„€7Xl0‡‰w–¹|4yup)dl©–†˜l%—‰˜=©§el|83‡–ll}„ ˆV‡›™„ƒ0Ç lu€'0r˜ |g¦gTel)2˜• 0–}sn‡V'hz‡–†‡ƒ„t€7… |›ofPll» y(€.¥‰w€ˆx—ur ý)lƒ„)l € sd¹I=¥W—Il˜‰‰˜HƒÀ!)„ll0‡€ˆlI—3|#›!Ÿx )#Ul–yˆ€h›8#§‡x|˜ ˜„}2)e— =‡Vl$lWl‡'†„ˆ€€R„‡s'uw0x„ƒ¶Rt˜q¤„es=×g%q˜W—l ‰wo„•†nr©w‰V‡lwr¤²0¬x7€lDnee|›!„ƒuˆ m•™‡uw  x•|ga„ƒlw0€7Øg|}–1–4×x‡™lDls$|– 7€m”‡˜z©lx–ÒX|lllxn ← ‰ À€¼ ã ¼ Ç ü º ü º ½ ← Á ú}ǽ“¼ ã ¼Ç ü º ü ºS½ Rnf(x; a) = f(n+1)(ξ) (x − a)s(x − ξ)n+1−s , sž n!S˜ ‡c|h˜lo¨l mξilPch(ˆ a , x))€ˆ § 1l©vx l)„x˜ l w 0 ‚|–†€ s | wz › l €7‡‰)€”¦g– y €ˆ) 7€ –†‰‡ ˆ ‹ z xYy  ©l x „ƒ forma de s >s s= 1 Rnf(x; a) = f(n+1)(ξ) (x − a) (x − ξ)n , n!˜ ‡ | ˜ l sgxl „}„ƒ)¨›  ˆξ P (a, x) forma de CauchyE7€†•˜ V‡–lul€7„§V‡›!r„}€l e‡V‹”€ l)€s7€A||ˆt˜‡Vo!€2lv’ƒ„l  „0i‰ww n‡”yxl|Il–t›¦…u`›¨e8w0lg›–†y–|rl– l”y˜a|x |Dl–A‡y”luu7yl wlˆ€d88—‰–eR„dl„ €7s™|¤)‡VC7yw—‰§%— l {a7€z‡ˆ‡—ul)x ¨‰wYx¤–c0y–†©lh›¨–8˜‰‡zy„› )l| €‡ Dlx ‡w¿„ ||e€ ˜˜tl©lwYx y„¤•08w‡ –†|„€|g˜‰y 0 8l– |‡z ©lƒ„‰|˜ `xY¨‡y —%–}• !›V‡8‡Vz €7„€ud› 0wx w–}we|‡ƒ„h| %— l©0‰˜‡x ‡ †„l x}–| ‰|„w ‡ ›¤›„§‰w lz–}}„l‡€‚‡V–}€h…) nl ©€­qx I—ll)z)„›xx l€}– ›„‡|gl x0‡l„}– 0}„ „¤‡˜‰l ||lx I1ˆ82 1©)€0.q©Œ ˜l ™I—I ) I€ w ˜| | y }–‡| y”l €”…§ 8„}‡‡ ˜ ©l¦g– l € xyl—‡ y § w e| P• w rg| o0 p– ‡ z o| si˜ cl io´0n„ƒ ”x (l Cna+uc1hyl |, ˜ 5l — w x – l | fl ax I f(x) = f(a) + (x − a) f V (a) + (x − a)2 f V V (a) + . . . + (x − a)n f(n)(a) 2 n! + x (x − t)n f(n+1)(t) dt . a n!¨ p7`o 8´ Dn Ò ±0rV­7²'´©·šq “ç  V‡ R€ –}| ˜ w ©0– ‡z | s I— )€ˆ ©l x 0 – l €y‡ s — ‰w ©l x n=0 x R0f(x; a) = f(x) − f(a) = f V (t) dt a

2.4 LA FO´ RMULA DE TAYLOR 71‰| ‡ l©x w› —  z ‰w x l©¤xYy‰w ‡2l—0l – „ l € ”y l ‡V€ l !› q • w |˜ 8›‰s ©l l Rx | y˜ )l „ ˜ l s „ 8z „} w †„ ‰‡ ˆ Œ y ‡„%— ©€ˆ n 0–†€ − 1 Rn−1f(x; a) = f(x) − f(a) − (x − a) f V (a) − . . . − (x − a)n−1 f(n−1)(a) (n − 1)!€ ©l xw „ y  sBx – ”x lR¥ d  = — ax–}|(x7y(lDn−u €ˆ−t8 )„1np)−!o1rf(pna)r(tt)edssty , ˜‡ § v V (t) =s(x − t)n−1 Rl )l Yx y ‡!› 8| u(t) = f(n)(t) (n − 1)! Rn−1f(x; a) = (x − a)n f(n)(a) + x (x − t)n f(n+1)(t) dt . n!™ l 7€ ‡ ‰s ˜  ˜ ™ƒ„ q ‰| ‡ y 8D – a˜ ‡ s ©l nx ! ‡z |t¤ w‰—l ©l xYy 8 ›„‡ xvwgx 8| Rn−1f(x; a) = (x − a)n f(n)(a) + Rnf(x; a) ; n!d w‰Dl u ‡ Rnf(x; a) = 's §`l)xx (x − t)n f(n+1)(t) dt 0– l € y ‡„I— )ˆ€  nˆ a n!DdD‹–}˜ |wled„y7ee8lD%—0fis}„fu–a‡‹ny€ˆe}„r8w–}i|‰nry‰|c„ 8l)ª–oi‡¦…)|Dxo´tl„›„sdl˜onu oxQ‡}–˜wr.l‡snl|ex—„Œ„ppo¤{→z w‰›llxno0”€)la(l€‚)˜ lifxgfd‡—gnl(()(8€p|„}xxo´x¤$ˆ– ‰˜u)©))mwždl—nwy 0=li}–t|eu}–}„c0V‡o–}ˆ€5oV‡0)Ú| )€s•a˜l©ˆw˜€ xD‡lg|l„}Psi‡›2mDn0x–I‡‡‡ zi˜P›„x„|xt©l–a¤rx‡€ d)w‰l €”lo)lR©l 7€f„„xs‡x‰˜vl}„e„}8lzl‡)l„¤Û‰xdxdg|wˆ˜‰e–€7„}˜l–†‡s¦ a l˜rl lrf§| (o†„DxV‡wl– )l€l2| o€I¦=y–†|l‡ i‡xY”yxmyl j )„”€gi}{z|…„›t(8˜xa„}–‡)d‡u7y m©low ©xDle|g¦s|d–‡ lewl©€ ‰˜x|yoy‡l –}rl !› Id|0 –ydeIl‡V0n€ €7–y5‡|‰zn|‡xˆnT˜¤˜ w‰ll8es¦7€l„o–†˜l˜…¦rnf€–)©leu({¤z¦gxxm8l )„)|›„l—x€”al7€ ‡(=l‡€ xx2|„}w}„ .−‡pp|el sn„d(n–aF}–(w)‡|§xo´8y”sj)rl(|q©l x+”€m˜nx¦… y−‡ 8u‡‡s }„0lalj‡—I©l¨a())xx–InxY€ˆd–}−7y!m¨§h› lessd„ƒay—%2—Y©l))‡ hx„ln›2ox 7€w¤–†mzuu‡¦g–‰|‰w ,xn„ 8–¤!›ll 0g)l„ˆ‡S¨xv)x sf.l ˜•(—€ wnlŒw‰w|g0)ll©(†–|!0€xa– s‰‡x%—f)z x–| :‡s–'¥ }„II—f†–w g|→©x¦gs„ ‡V€0→{}z–„¤›2›l a5€ˆa}–˜‡fw(}–˜P•˜x|e(l‡ xl5)Iu€x lˆ−−u˜•|gwˆ€‹ ©l0pae|x|–ƒ˜n))0y n‡™”€–(‡V‡p7€xzg|›‡|n)}„ l„}()ln‡x|‰Dx=x)sV‡−˜“€−0l1‡q.q‡V…–€nuVl ˜ w( lx©l)|)x„ ‡f(x) = Tnf(x; a) + j (x − a)nm ,

72 DERIVADASn…l©l¨ |vxl 7p˜˜o`Il©—llxR§´80w Dnx†– €|eSDlÒ±0s'uRq Vrlw(­7•I„n–w²'˜ %—)©´ |g·š(q‡ a0—x焆––})‡j‰|zŒ |tQ ‡l xV›„¤–lecw‰R}– Šy‡ l(– l§x‰˜…|)lÛgll =7€|e–†RgÛ8f(}„¤§(a› x‰w„})‡V)l—l −€=| ©l7y˜TYxlRnyl–„ƒVf(uvx(a€ˆx˜¥);˜l–†a=—‰Û‡ )‡‰|z ny”Q¢ž ¤– ©l¢ž0l)xž–xY©l ‡–yz=x“x |“ˆ l R•„‰˜w (l˜e|nl©0R)x(– (V‡)azn”€| ))€7(‡©l=ax„}„})n‡h0‰s−„}ˆ x–†2›1l—t …–g—y y–l†„l–¤ ©|˜ l©8l‡dx |ê ˜ ˜ l V‡ € ˜ l | l ˜ l ‡ €7†– …¦)g¦ „ 1 n−„ }{z › „ {¤z ›R(x) = R(n−1)(x) = R(n)(a) = 0. x→a n(n − 1) ž¢¢ž ž 2 (x − a) n!x→a (x − a)n%—ƒ„8ë u„H‡‡2}„•—”ƒ–}w |‰aw{de| ³‡|‰ 0„›y– ‡V‡ z1–}|‡ .a—˜ ll©hxŒ –˜ §wl €7}„|e‡V†– „…¦€ ) •˜g¦w l™„e|l u0 n–€ˆV‡z A|−˜ ‡  1˜n„›… l – ly”|lh©l qxy ‰˜ˆ€ ll©|˜x )‡ ”€a€7l ‡Q| }„ —„†`‡˜ l –}€7}„ ‡E–}¥„› ‡„|‰– y‡—  w˜y |–‡ ly |‰‡˜ lˆl %—‡V‡V‡V€ €˜h€ ll¤ | w“l nh›z l¨g— l l0„}| ¨‡– Yx s y y ‡dI—†– €‚€©|‰€0l ‡„ Ÿ0 ‡| xx –– ssf(x) = 1 + x + x2 + x3 d(x) , ± d(x) = 1 0 xP ± x P/x —l y – l |l f%— l (V‡€”x7€€ )V‡ l€ ‡= 1 + x + x2 + (x2 )w‰©ll ‰˜| l¨y V‡x l€7€|g‡2l |‰8 „ '‡ 0©  s — l 7€ „‡ ‰| ‡ 0l ¨ – Yx 7y l ˆfV V (0) ‹ Ÿ lfh›(xe— )„†−‡!0 T‡ nx fx(x=; a1)−— x }– ‡| l©xqw2.— s„ ‡ ˆx2 |I‘‡ j 0‡|m> n 2 (x − a)mm + (x3)ff3yl„ƒ¤0 –((|}–gw.l‡0xxx—ylR||e))‡•›lw)yg|=©l –7€g|‹vxlz‡dl}– l)x0‡|x€7€xwx}–!x0˜‡‡˜€‡|„}7€)ll|ez|‡lª–g—|„ƒ¦iu©ln–x¤{”y†–z˜x8€”w210fi©lxˆ€)„› ˆ€=xnd›sHD8s©‡0‡i„)#§x‡l–}tx|‡„ƒ|´2e(wgwe|fx˜7yssxx l8ril)x∼‰|mˆ)„€73¤)l‡„ƒl·t5TDxw‰s‘oφ€sr.lRxl©xDI—.{zxd§s<l's−.©se„‰x0€0 w•†1–—‰‘‡‡V·¤o)f€h›7€‰wr|(›lx!›wxdl¦lg—˜ 8l©)|l©s8„e‡ ˜|0l˜|n∼x—%y†–˜lx€h‡Vw|l‡m¤ˆ€→”y|Ixl w‰xl¦h„ml‘lˆs’Ÿ→aÞl©§l y‡V0–x›„—Is–‡€©l0x˜|8x¤—g|+}––„ƒ„l¤‰w„†ll‡|sxwl„ Rs=→{¤—zlWV‡–›xl˜‰‘|€l˜an€ˆl0˜€7l‡„¤+‡φlqf— |1(·wx(l¤xx„→˜=8¤x‰w¤{)z25)›„›|w‰0l ‡˜ˆ˜l+=– ()y|‡©l x–fg¦l¿llxnx˜(ږ|wgxrl=)¢ßzW→˜)ˆ€l©T›28‡xD˜0=ns›„0¨l– ˆV‡l—%z‡∞P˜€›™w V‡¨x˜‰le|€¦‰r)lDT˜4 ˆ€€7|–†l›2lˆr‡ l©Ÿ•0lwx>‡xy–†h›`€g|–x) l•ƒ→1D€ˆ€—g|¤¤ –ld„}ˆ‡wg‰w˜z‡ˆ|0llll£sC€ l ´a llc| ‹ ul©‘|lxYoy‰„¤8zd|x e‰— ˜/€7cl ‡VÛgo%— |‰n‡ – x˜ d¤– 0 ex€}– ‡sl|a|4l©rxr†– o| ¤ ”ylw‰ll o—l€”…sx 8– luV„}i‡ wgmx l i|) t¦gax– wdl €—%oy ‡s‡ | x ˜ D € ‡Vl B|›2… ‡l xg| –¤l ‰w| ly”l)ƒ„xD 's x § • w ¤ g|wg0l –}‡ | ©lxPyx †–}„¦…r}–¤ „›) w‰›ˆ –l ) —I)£ 1y .d‡ ‚€ |‰Œ f–‡‘„ƒ xw •|Ew |g— 0w }– |‡y| ‡ ©l x af}„ ‡§ g—%‡V „}˜–}‰|„› ‡ –›„”y l –}r|‡ x 0 p‡ ›„n ‡ § ˜ ©l x n y |˜ y7‡ l x s ) €€”l©7€ x‡ —}„ †„l‡ Pl˜ |l ry o‡‡Vp€|g˜‰ol)l s|x iêcnio´nl | x qn  lj †– Š‚œ 2šd{ ‰ ˆ ‹ „ ˜ ©l x )€”7€ ‡ „}}„ ‡„}„ –}›„– y  ˜ ‡ ˜l ‡V€ ˜ l | ˜l ©l x ˆλpn + µqn n λf + µg

2.4 LA FO´ RMULA DE TAYLOR 73u‡Vj –†ˆ€¦ –†Š7y ˜l —š|‰„‡ –4‹ ˜›!‡d‡2zd §…%— ‡VEx‡V5€2ƒ€c‡(¤ ƒ„ ‰w‰ˆl y |D Œs€n– w‹lnwn||g%—Š„ ˜y”©˜‡‰˜l)˜lfd|‰lxull©0u„–„%x–lˆ€˜‡0)‰—z|l‡'f|”€ €7˜˜€70‡h‡‡‡– ˜l„}n}„|gwu=r‡ y7ˆ€( ˜lay}„xl‡˜–})p›„‡„ƒ−pn=W– nny/jaq0q• ˜8jXs$nnsI„l©‡ xl©xclŠˆVux„˜Š˜y ˜l˜)l˜ „)ll0ª– V‡x…0–†¤€)€–†–‰w€sI˜x€”s¿l–7€xl‡‡zw|l|†„H„‰—}„ ‘‡˜n€%— l©l ‡V„}x–}˜‰p}„–„› –}‡zl|‰n|– ‡y fl›„˜g|˜ ly–}‡‡ l©}„ `x‡ Rx l 'yc„ lz —%€7›„‡ „}}––†|‰g| ‡‡Vcx2› ˜ –}‡lf}„lj }––†/Q|›„}– g†– Š–—%y©l ‡† xy7˜ (‡ll ‡„Bx4|gD’w0˜}B‡ ƒ– l4uD …ƒx–‡Vl }€|ˆ‰˜€”y ˆlll € l ˜ ql nV‡ jV‡€˜ ˜€ ©l˜l x |l )|e”€ n7€ ‡˜ „}˜‡}„ ‡l x sn pn(h) = sn(h) + r(h) ‡= sn(h) + (hn) , ˜ ‡| ˜ q—l nl („ hu )€ˆ ˜ ‡ ˜ l „ j i• 8 „ x qn (h) ©l x ) „ › l ‰| ‡ x ns + 1 ˆ ‚p o`8´ Dn Ò 0± dr ­‚B² ´)·—q €ç j lr(s'h§ )u ¤ ‰w l ¨ B‡ fŠ € l©xYy ‡ =W 0 –†–}–†Š ¿ w‰l©vx xl—y – l g(a) | ‡f(h) f(h) g(h) g(h) − sn(h) = − pn(h) + pn(h) − sn(h) = f(h) − pn(h) + (hn) qn(h) qn(h) g(h) qn(h) ‡ ‡= qnf − png + (hn) = qnf − qnpn + qnpn − png + (hn) qng qng ‡= 1 (f − pn) + pn (qn − g) + (hn) g(h) qng(h) ‡ ‡ ‡ ‡ ‡= (hn) + j sn + (hn)m (hn) + (hn) = (hn) . ˆ Œ – ‡ §j f(x) = pn(x) + (x − a)nm‡ ˜ ‡ | ˜ l s l | y ‡ g|  ©l xProposicij o´n 2.  † (‡ ‚ ™ (‡ Š‰ wy4x w”‡(€ u g(y) =qn(y) + (y − b)nm b = f(a) ¢ f£ ‡j j + j (x − a)nm , g f(x)m = qn pn(x)m n˜ ln˜¨ ‡‚p ˜|`o l˜ 8´lR„ƒDnu lÒ0±‰„0dr‡—g­‚›h€7²B–†)´ ›·—%—q ‡ç l x € ¤–Œ 0y'w– z‡l%—z€7|„›‡|q}– ‰|u n)h‡ ›2Ð —p‡ „¤xn ˆ € l  ¥  x5l l | y–l | ˜ l ¤ w‰lRl©x „ƒ y€w g| ©d0 – ‡z `|  u ˆ€  ˜ ‡ ‰w l ¤ n f(x) = b + bk(x − a)k + (x − a)n e1(x) , k=1 n g(y) = ak(y − b)k + (y − b)n e2(y) ,˜ ‡ | ˜l  wk=0 8 | ˜ ‡ §  we2(y) → 0 8| ˜‡ ˆy → b ‹ | y ‡V|e l©xDs e1(x) → 0 x→a j n j k j n e2 j g f(x)m = ak f(x) − bm + f(x) − bm f(x)m . k=0

74 DERIVADAS ‹ „ w z „ y }– ›„‡ x w ›!8| ˜ ‡ l©xvw e| ™‡ j s — w‰l)x (x − a)nm  ¿  ¿j f(x) − bm n e2 j f(x)m n bk(x − a)k + (x − a)n e1(x) n e2 j f(x)m ð8 = h8 k=1 x→a (x − a)n x→a (x − a)n = h¿8 n nj x→a e2 f(x)m bk(x − a)k−1 + (x − a)n−1 e1(x) = b1 6 0 = 0 , k=1— ‰w l)x 8 „ x l € f 0‡ | y }– | w  l| s w ) | ˜ ‡ x → a ”x l¿y – l | l ˆf(x) → f(a) = b ‹ | y ‡Ve|  l©xDs a ‡j n j k j (x − a)nm g f(x)m ak = f(x) − bm + k=0 k¿É Ê ‡= n ak ÃRÂÄ Å Æ n ÈÇbj(x − a)j Ë + j (x − a)nm k=0 j=1 ¢ f£ ‡j + j (x − a)nm n = qn pn(x)m . nPpq§˜¨ lnrrp7q„i+o`omnl1p+|´8(ionDw1xtÒs(ie|)0±vai+0Vrca)ƒ–­7i‡ '²o´=˜©´dj·nš(q‡„exgçs 3(−‘ lfa.„(a)x|ex)š—ˆ–)2nu‹¨+w e…yw ‚1–nll mÚw„|ƒ¨s7yI’w˜ l¤Œx “s0l–{dl„7y l)‰{}z|l ›„x2ˆ7€˜ ›„–˜‡Œy”llu}––qe|D fnx–ª €(l +˜ xs ‡—1)ggw ll=Vl©„(¿„ ˜xBj%—pl)zw ‡n|g)=†„ (}–}–)D‰|x¤„‡B)‡f%Šw„›(+x—%„¤)–})‡‡‡R€ }„ Tj–}—%q(‰| xxV‡n‡R€+„›−P 1„ƒ}–™ a‡ I—%€)ys“V‡lDn8 u€lm„ ƒ„| ¤„ƒy§ w‰‡lg|g(x©l x) )l x una g(x) = qnV +1 = pn xj  0V‡ec| Šˆ˜ ê –¤0–}‡ | l©x „ {}z › „ {¤z ›g(x) − qn+1(x) = f(x) − pn(x) = 0, x→a (n + 1) (x − a)n x→a (x − a)n+1—%V‡ € ¥ †– — ‡z ”y l)x – x ˆ1E.j—emk5pl)lox ©s7€ €7‡ †„ „}‡ x }„ –†2› – y  ˜ ‡ Rx ˜‰lx w !›  x ê Œ ¥ ‡ex − e−x x= = 2 n x2k+1 j x2n+2m , + (2k + 1)! ƒ ¥ ‡k=0 ex + e−x n x2k j x2n+1m , x= = + 2 (2k)! k=0 n (−1)k−1 22k−1 x2k + (2k)! xl | ‡2 x = j x2n+1m , k=1 ‡x n õ 1 2k x2 − 3x + 2 (xn) . = 1 − ö xk + k=0

2.4 LA FO´ RMULA DE TAYLOR 752.— 5k ©l x ©€”‚€ ‡V„}„}‡ x „}}– „› – y  ˜ ‡ 5x ˜ l —g€7‡ ‰˜ w  y ‡ x ê x l | ‡2 x = n (−1)k−1 22k−1 x2k + j x2n+1m , (2k)! ƒ ¥ 0 ‡ x ‡k=1 n 4k j x4n+3m . (4k)! x x= (−1)k x4k +3x –.‡z—|4„‡ Ž —%¦ V‡ y”€l |l l„ €c› „}‡lDz yxc‡ x˜ –ku‡ =w ˜0– ll | 0”y ‡©l l–x gÛ ˜ 0l)– lx |©7€”y €7)l ‡x „†„}†– ‡| x˜ l0}„ }–”y „› l €7– „›y  }–˜e| ‡ x—˜ ‡˜ xl ê D B‡ D– l | ”y )l x %— V‡ €–•i8„ x  ˜ ª– … †– £ ”x l x| x = 1 + 1 x2 +7 x4 + 31 x6 + ‡ j x7m , y7u 6 360 15120 x10 m . x j4.— Ž = x + x3 + 2x5 + ‡17x7 62x9 xê ¦ ”y l ++ ‡ | l €R†„ ‡ xR3x – uw – l 1| 5y7©l 5x ˜ l©3x 1)5€”€7‡„}}„ ‡ 2x 8†„3–}5›„– y  ˜‡úüûe ÷ x = õ 1 − x2 + x4 + j x4m , e 2 6ö‡eex õ j x4m , = 1 + x + x2 + 5 x3 + 15 x4 + e 6 24 ö‡ex/(1−x) = 1 + x + 3 x2 + 13 x3 + 73 x4 + j x4m .5.— 5k ©l x © ”€ ‚€ ‡V}„ „}‡ x „}–}›„– y  ˜ ‡ 25x ˜ l —g€7}–6„› – y –†¦…  x 2ê 4 ) ‚€  y”u ‡x = n (−1)k x2k+1 + j x2n+2m , 2k + 1 )‚€  x l | ‡k=0 x = x + 1 x3 + 1 ž 3 x5 + 1 ž 3 ¤ž ¢ž ž (2n − 1) x2n+1 + j x2n+2m . l „ {}z „› – 7y l©2x —%3‡V€ x gw2xžy 4– ”y w 50– ‡Vz | ˜ l 2˜ ž ©l4x ¢ž)ž¢€”ž7€ (‡2„}}„n‡ )x †„ –}2› 2n– y  +˜ ‡1x ê6.— ƒ 8z „} w †„ ‡ ˜ (=| x x(l le©| |‚€DBxxÐ(6ú)”y 7y uá÷ u−øCxù x)xx)l −ˆ −| 2y7)€7ux€‚V‡( I¦x=y”l ) u| R€ (x©¤11)‚€‰w8 l7.— x xl | , = 1 . PÌ Í ' #™! ô„Îó ¹nÏ „ ¤{z › ) €‚ xl x„ {¤→z › 0 x l | x x l … l €‚ª– Ûe©  xŒ →l 0 y(x) —I©€0 n = 0, 1, . . .}„t –}›„I—– y)( €1˜y − x˜ 2Al )yl (}„ n„†‡ +sc2)x§ (xwg)x − (˜ 2‡‡n ¤„+‰1• )‡dzx€‚› y(w n„ƒ+ 1)˜ (l x) − § (}„ nV‡ €2sc+u l1|)yl €ˆ(n8 }„)–†(8Ú x©)€ =l „ ˜0l).x ) ”€ 7€ ‡ †„ „}‡ –ª‡ € 8 |BDÐ ‡e ú j x6m . á÷ Cø ù x = 1 + x + 1 x2 + 1 x3 + 5 x4 + 1 x5 + 17 x6 + 2 3 24 6 144

76 DERIVADASEjercicios suplementarios dx0exD l‡E|tlu… 0l–e‡y rl .eÛ y 0‡j– ›¨l |Šc)7y ƒH|©l x˜‡|‡–}| x ϕ–˜ ˜ 0l(l7y0€l)l7€ ›„=›„‡}–1e|x ‚Š „ƒˆ˜2 ‡ • w |g0 – ‡z | xD7€ s‰V‡ l—%t| ‡|ƒ„ 1. Los nu´xmerj oƒ„ s ˜ d– ex 0‡B| ey r–}| nw o– ˜ull4˜ i l y| £ |ex˜ −©l x1)€”7€ ‡ }„ }„ ‡ l ‘| %— ‡ y7l |g0 –ƒ xR˜ lϕ•†u ‡V8 (7€ ›„x›!)‡  =vx w ϕ(x) = ∞ Bn xn. n!B0 ‡ ‡2„›kx +‰—Bš17€˜ n‡Vl=I¦| x )y0l –†€cgÛ „}¤)¤„ T8w‰)n›¨k|l =˜ −>)ϕ0‡h| (1}„xn‡ˆ)xu´+0m‡ x2el gÛr©l oDxR–sl w d|Ieq • Bw g|e0r–n‡ z t|ouI— l)li€ s Ñ ˆŒ w!l 8y„– l–}›„| l —g†„B¤– ©02 =¤ ‰w 1l ˆŒ l— w‰l©˜ l y”l)xR˜ l j „}‡¨ l |4„ƒq– uVw ) „ ˜  ˜ § B1 = − 1 xn+1 n 2 | Š−> 1 x = (ex − 1)ϕ(x) õ B0 + B1 x + B2 x2 + B3 x3 + . . . , = õ x + x2 + x3 + . . . 0! 1! 2! 3! ö € l)xw „ y  1! 2! 3! ö B0 + B1 + B2 + . . . + Bn = 0 , („ ny –†g— +„}}– )1))|! 0˜ !‡„—%‡Vn€ ! 1! s(n − 1)! 2! 1! n!§ › w (n + 1)! õ n+1 õ n+1 õ n+1 = B0 + €7€ l 1|g0 ƒ– uö ¤B‰w 1l+x l ‰w 2l©˜‰—l ö ©l Bx ˆ2€7+†– g¦ †–.€ . . +n Bn 0, ö „ l04§ ‰˜ l € l  w — l 4| ƒ„ q •†‡V€7!›  (1 + B)(n+1) = Bn+1g¦|x –}–w z|‰›x ‡l ›„l €7}–‡ ‡˜ ‡VB˜ — kl™y v’) „ll „u˜ l©y 0x‡‡)||h”€ ˆ…7€ ‡l }„‰| „ƒ)–}‡u€ „¤˜ `Ql —%x‡ gw y7xYl y |g– y70 wƒ–  †– f€ ©  ˜ ‰B)—%(‡n”y+l 1e| )0 –ƒ0 ‡ ‰| x –}›q¤„ ‘¦ • ‡zV‡z„}7€}– )›  w „¤B (kwg)x w —%8V‡„ € ˜ ll „„  (1 +öÞ àØGCã ÒrâCÞ Ûõà§ÓÕ ØGvÔå $áÔ ÖâÙØG׺Ô&Ùßh‘Ú qÞö÷Û$Ø3×Ü©Ø3â”Ü qÔ&ãâ ÛhÝàXÚ Û$hÛ × ÔžrÞºÚ ßøÞ(vÔÙä”ù ØGúiàâ 3Øpû —à¡vÚ0ü £Ô Dý¥¢háã&(þä¤ $Û&Ù§ýÿ ÞÙ¦©Ô ¨ÔÜ© Þ0ÞDâ â” &ãš3Ø âêÝ àrÔÛhâ&hÛÞ×hå©ÚºÜ© ØÞ Ô(Þ Ùä ØGí â¥âCòfÞq0Þí ¥âDÞ âÜvÔ —öäÙ åÝÔá ÙhÛ ÞÔ XÞ Ô GØ$ß â áhæå\"× èØ!Cã Þ çà ÔPÞ(éqÙ Ø8Üëá Ô êâ&Û$Þ0ÚÙ à ÔØGhÛ ß$©Ü ÞæÜ på Üì&Ô Pí â îGïGcñð ñ‡0Þ× â&íP©åGòÚ Gó Ôïaâô n−1 1 c j c+ 1m Bk nc+1−k . + ºÚ# Ø©å Üà ö Þ0Ù&æÜ æå Ü Ú Ô$ Þ0ÞÙ Ôà â jc =  1 ö k × Ô Þ à§ Ø à Cã Ù Ô ÙXâCÛ Ô †Û &ã ØGÙ Ý Û$vÞ Ü Ô &â $Û Ú Ô hß ÞvÜ Ô â ºÚ ö Þ8Ø àã&ÞØ3%àRâ  $ßå ÔÞvâ÷&â ß©å Þ0Þ×ãCj Þi=å©Ú 1ºÚ Ô å÷âà Û$DÞ âã&GØ câ Úºk0Þ =ßh©å0Ø Øfß 91.409.924.241.424.243.424.241.924.242.500.

2.4 LA FO´ RMULA DE TAYLOR 77B}„ 2‡ 1¤ ‹ xy 21€ ˆ„ xlDlŒ§™w¥ ˜ ™lD©l y”x‰— lª– €7…¦‚€ V‡„›8 I¦› }–|el˜ | ‡2”y˜‰l)l ”€€!›7€l©†– xI¦8 w | „ s'ly ˆ€8§ t| w z |‰l –¤7€ ©‡„“ „}„}‡ ‡ xx sBn H%— © s€ y †– € ˜s llDy„}‡Vˆx s ¦… D8‡V„}›„V‡ € —‰l©€7x ‡VB¦ 08z |=˜ ‡1x§l =gw –l −§ B3 B5 B7B2 = 1 ; B4 = −1 ; B6 = 1 ; B8 = −1 ; B10 = 5 ; B12 = − 691 ; B14 = 7 ; ... 6 30 42 30 66 2730 6„ƒxj q%¦‡„ƒŠ†•8‡d›‚€ !›¢l | •7y w cl |gy0– –l ‡z | x l €‚„›x }– |‰=‡ x 0 0‡ 1‡x |`x —%l©‡ x 7y l —I) € s x„ ‰w l©u ‡ l©x x w‚˜ ©l x )”€ 7€ ‡ }„ „}‡ l cl w| w —%| ‡ ˜7y l)l |gx ©D €7¤– 7€ ‡2x „†„}˜‡ l ˜ xl | –l yBlz g| D–¤ —I)€ ˆ Œ– xl ‰— 7€ ‡V—%‡ | x l  x = ∞ (−1)n E2n x2n , (2n)! x €71†– EÛg=2©n„ 0 „ƒ‡xhlx „x0l}„ „ƒ¢§ )x nl¨› ˜=l8x0| l!| yy †–– Ûgl |))l | 10 ™s‡ l§hgÛs D—I– l) |ˆ€ ”y l)cxn˜ xl„}l… ‡ l €Š l n wu´”€m€ l e|gr0o–ƒs Xj dxle V‡ E¦ yu– ll e| rl ˆ – ˜ Œ l E˜ 0‡!=„}‡ x l−> | 1 x2n õ 2n õ 2n E4 + . . . + õ 2n = E0 + E2 + E2n 0. 2 ö 4ö sgxw  ©l x –†…8› l | 2y7nlöst —I© € y –†€ x”—l l xl ©8}„  w }„ ƒ„  „¤8 |E2 = −1 ; E4 = 5 ; E6 = −61 ; E8 = 1385 ; E10 = −50521 ; E12 = 2702765 ; . . .2e•x w–x.ug| cw 0e–E–ln‡ z|l|ty7rl—dlic„ey7l ils›d›!aalr d|roy.x 8lll„ o— ˜ w‰El d©l „}˜e„ƒ‡l | llDu‡Va¨–¦ y”y”gwlllo˜#|| nyl €7˜‰gR€ –¤l4i0ƒ„t– u˜w xe|d–˜uwdεl –e}„l †–j |—0uy7xl<lEn•a˜‡Vεz lt€7e›<lxiwl p1„›¤„ Bs‚Š e–ê2ˆl Ÿ ƒHlenV‡ ›!2› p ‡ § o‡Vwt€ |enal cŸ il‰| a‚€ ‡ 0s¤– )l0 d–}x„‡ ew ˜ e|lla „ õ 1 − 1 ε2 − 12 ž 3 ε4 − 12 ž 32 ž 5 ε6 − .. . 4 22 ž 42 22 ž 42 ž 62 L(a; ε) = 2πa ‡12 − ž 32 ž¢ž¤ž (2n − 3)2 (2n − 1) ε2n + j ε2nm . 22 ž 42 ¤ž ¢ž ž (2n)2 ö3x Dl.u w L–†€al ›2|de‡Fl r(x„ xi˜Äv)l©aU¸=x d)‚¹ a”€ eº7€ a‡ n…) x„}„}‡»–2‡ ´eˆ $¼s˜ i™3l m¾¢ 1 a˜ ! §l d$' e's l e| axx w2. å l©z LxpD–})´ o›!'± sÀ·rhp˜‰µ olp liÁFn®(o%qx)m7­ i² ´)©i%qo—%‚ps©Dn€ sgdl œe„l Hl ˜ | €7e–rl˜dmf„ s 0—i‡tz leu Ûe.ˆR0¤t–¢Ü l ¤  7ywˆ l {z˜ l Œ ll 7€†„ V‡†– ¦…€  ˜  j­ | hj F(x + h) = ∞ F(j)(x) hj . j! jl =7€ –†08…„0lŒ ¨ – l |el©0x ‡– ‡|z | y €0˜)2›l „¤‡  x–˜ ˜ l u ˜w u|e„¤ ‰w ›!l 8 ¤ | ‰w l l €ˆ€ l ›2©l Yx ‡y7lhx ˆ ˜ ©l x ) €”€7‡ „}„}‡ ”y l | ˜ € l „› ‡ xcl E| —%©€ y ¤–  w ƒ„ )€—¤„  —‰€

78 DERIVADAS˜ l)x ©€7‹ 7€ E|‡ †„ }„‰—‡ €7–}˜ › ™l l œ €—d„ ‰w0u„ƒ )w € €7s –}F| (s x + h) = ea(x+h)2 = eax2 eah(2x+h) se¢§ ¥ d 0 – l | ˜ ‡ w | ∞ ∞ @H ak hk (2x + h)k ak k õ k hk+l (2x)k−l eah(2x+h) = = k! lö k=0 k! k=0 l=0 ÅÆ ÈÇ∞ ak = k! k=0 2k õ k hj (2x)2k−j j − kö j=k ∞ ÈÇj õ k ak (2x)2k−j , j − kö k! = hj ÅÆRt©˜ 8‡¥ ›q| V‡ ˜g¦€ˆl¦}– ‡Is § x˜w l¨y”l©l{†z |x| –†˜l…›„–}8  ›l‡ xjlj=| ¤=07y ‰w l lksdx+ul kl¥ =!§ & j)8+2g—„ 1}„ Ûg¤–' ©e|  ) ˜ „ ‡¨es w ƒ„ 2|¢•–}V‡|z 7€7y ›l €‚w©ƒ„8  q› ˜‰¦gl †– ™„‡ ¦g˜ }– l|‰„™‡‡V„› € }–˜ ‡ l ˜ l„˜ lqv’ lxw ›¨y ‡8 | 0 Is– ‡zw | | ˆ | j õ k ak (2x)2k−j . j − kö k! F(j)(x) = j! eax2 ƒH‡| x –s ˜ l € l ›2‡ xgs x™l ¥k˜=‡Vlˆ€ &gÛj+|2l1l „c|`' ©—% V‡x Q‡R€ ƒ„ I—q)•€ ‡Vzy 7€¤– › w ƒ„ )¤„ € ˆ  ‡ xa = −1 w j ˆŠ êde Rodrigues polinomios deHermite Hn(x)—I‘5)x ˆ€8| e˜©l −Yx‡ yx‡l2'„x H€ —%)ln‡x (w†„ x–}„|‰)y ‡ ›„=˜ ‡q–}‡ (8x−| ê y”1l )€7n–†V‡ d“€ dx%—nn‡ ˜‰l e−x2 }„ „ . ) €€ RV‡ ¦ 7y l | l€ w e|  0l ¨ —‰€ l©x – ‡z | 0l ¨ g— „ }{z D– y  H1(x) = 2x H2(x) = 4x2 − 2 ›„‡ x l©u H3(x) = 8x3 − 12x n õk (−1)k (2x)2k−n n − kö k!Hn(x) = (−1)n n! k= ( n+1 ) 2 n õn (2n − 2k)! ( n ) õ n (2l)! (−1)l (2x)n−2l 2k − nö 2 l!= (n − k)! (−1)n+k (2x)2k−n = l=0 2lök= ( n+1 2)= (2x)n − 2 õ n (2x)n−2 + 3 6 õ n (2x)n−4 − 4 6 5 6 6 õ n (2x)n−6 + . . . 4 2ö 4ö 6öCDdtj aaere(,ixfiaft)(rknear=ri†– )ic€ibmolifo›2as(0na‡l‡g)e||xes+¤nl.a‰w¦„efŒ—lrV lP(waƒ„av|l)eIfy (s‡sw xw €”pj¤„|e…−A˜ al ra l)•©aww)x¦ ”€|gˆce…¦x 0o´0x ––n‡txz y|VrceŠa˜=mavl a€7xof†–– …js(€Sx) l©yδg¦)x „— l–py – ull ||nltw o|`0s‡ }–„› d| 7ye‡ l ”€iy¦…n88 fl| †„ ‡uelxI| iˆy7o´l ‹n|‘„ƒE) € l˜  y  — w| y‡ y = l  y †–y ¦…88'„ › ¤ w‰l |l y7l s Šconvexa vista des- > 0 da(x) = f(x)−ta(x) > 0

2.4 LA FO´ RMULA DE TAYLOR 79ay%—lll00˜0Œëj €–‡|‡s||‡rl2l‡Vl‡2l©g||h|rs€|xƒ”l©˜„ƒ…i©Yx—`l3‡s{d„bly0–}0l ³Ÿl8˜‰…˜¨awlul%—|h–´l„›€y‰|˜7y‡V€–†)l¤zl——…€l—gj|Ûe‰w—8˜w„}l©2l‡wl¥‡›z||`wg|g|h‡|‡syw|xl0©y‡s'll˜0||e–}‡0„w„l›!‡ly7}–…¨˜%„”€t|”xB|l˜l…fl—y”s…•—Šl„$„›wll0wldl˜€”–}zlg|¨–}||–y||…¦al||l0–yy78˜˜s|(‰2–‡l˜=„}‡xl}–yzzw‡lD€”„ƒ–¤|„)‡V‚a`˜¦…¨lsI7€x¥)–<ll©|‰l‡l©2CD)„}t‡wz |x‡|‡„|vx„$)xy †–0lc§D|€ƒw‡”xy|}–l|©(Ú©olD‡|g˜|Š˜”y)|¤u“n˜l‡—jwT wp€7l©‡zvx1csx{z…–x|E”yx–uelm8Pwwx €=nxsx–„}|R|`„†P˜‡l¢‡taywgP‡ˆo}–0—(l©§B|¢sj0‘§xYw‰ˆ7y¢‡ŒδdYy)QQll)(–ˆ€le˜€”˜Œca=„…¦fllo´l)l)i§8xH„˜nYxn–0l©ˆ#„†‰——‰ylu‡«fl!xcR7€Ûg‰ws|gzs©ra‡Vƒe|‰)l‡2˜‰|¤¬ilvI¦xx˜w­8}–l‡|w‰‡aD)0i®V`l7€8yol–|´€—‡Š–†¬‡|l©€z¦…®n|˜l‰||x¤˜v)y—ny)ˆq‡‰wIs‡€¦gˆ€iP€sl©‰˜)ql§„‡€zt¢xld7€lw|gl§al)–}€a¤‡|e©—lx‡4l w(R0r|‘wXjyxD…¦–x{†ˆs2z¹|8l‡)—‰wQg|lq›ylj€ylh|€7sò„5B‡l—‡y7|…'‡ug¦D‰|lxy†–Dl€©˜–||–‡H–‡¨)8lzlzz”yug|eÛ$3||t›D€7wl†–‡©þ7€8…¦ll©Dju|h…„‚Šw©|x‡}„l©Š8…ˆ‡¦g‡d|y7|zx}„lDQ|g`ld¦%‡–}¤Yx¨„}©yx–)ewj˜–8§w0„l©s…–us¦…|lxdl‰|˜€ˆy‰|7y˜©l€e‡llxŠz–ês <—%Œ I‡– 0ˆ| fŠRu 8©l T „›xhx˜‡ l €7†–I…¦$s©¦el |„ l‘y ‡ ˜ |g‡  x … l ©l©l qxx l |‡z g| l© “„0¦…–}| 7y lj €”D …¦‡ 8|h†„…A‡ l0¨ )¦g5Š– l … € y x‡ y s$˜ )l§‰”x ˜xllPy)–€”lr7€ |o†– ¦Ilp oflVsV|(icxyi)o´‡ >˜n‡ 1.j ©l x ¨‡ xƒ„ 2P •IV‡z €7ˆ ›  w ) ƒ„ˆ€   –  I I§ Û„}V‡YŸ ‡€xx¨ l8 ‚p„ lo`δ´8>Dn Ò±00dr ­‚y²B8´)·—¿„q ç¤ Bδ(a) I P f 0 P  P Dx s %— Œ‡V–€ lŸ l „› —g„}‡ s sfV V ss ‰wxŒ l w Iˆ x P Bδ(a) ()xg— )}„ –}>) )|0˜ T ˜ al f(x) = f(a) + (x − a) f V (a) + (x − a)2 f V V (ξ) ,I— )€ˆh ) „ u“zw | ξl | y€l § kcl ˜ 2‡ ¤ wg2l ξ P s'§ xˆ ›2‡ a Bδ(a) da(x) = f(x) − f(a) − (x − a) f V (a) = (x − a)2 f V V (ξ) > 0 . l 2 Œ l©x l)vx ˜ l l—l | j 4§ x l—y – l | lProposicio´n 2. – n…  7€ ª– ¦… ) g¦ „ n −> 3Š f § a =W l)x  ‡z |g© 0…¦'…l¨jj †––ƒ|–†–Šp‚xYŠy y`o‡ŒŒg|f–8´– ˜VnDnVnl)(©lÒ 0±xax”l©drê˜l©)x­‚xl ²B—I=)´–})·—h›q)“€”€ç7€I—tž¢†– §I¦)ž¢€ž 0 ‡z y7©l x – Dx s = f(n−1)(a) = 0 f(n)(a) , —gfs s(}„a}–nl )|))l)(|€xaa˜ )w ˆ‡„‘|>„ƒ—q 0w • j 0 Š s ƒ„ ! w €”¦…  y = f(x) j 0 ‡ |B… 0l ¨ VŠ s l }– | 2‰0l ¨ – V‡z |ˆ < ˜ „ƒ ˜ l “á ‡ w | uu § „¤ vx ¥ –†— w |y‡ d‡z ‚€ › ‡f(x) = f(a) + (x − a) f V (a) + (x − a)n f(n)(a) + j (x − a)nm , n!„ ‰w lDu ‡ s'`§ l | y ‡g|  l)xDs ‡da(x) = (x − a)n f(n)(a) + j n! (x − a)nm „ ¤{z › da(x) = f(n)(a) . x→a (x − a)n n!

80 DERIVADAS ‹ | y ‡ g|  ©l x©s S δ> 0 y 8 5„ ¤ w‰‘l l „ x – u ‰| ‡ ˜ l da(x) l©„x l R„ „› – x ›„Q‡ ¤ ‰w ‘l l „ ˜ lf(n)(a) I— ) ˆ€  y ‡ s'`§ §  x 8„ ly ‡ ˜ ‰‡ ˆ(x − a)n ˜ ‡ xP BδY (a) k †– ¦ w ”Ÿ )R€ ƒ„  u € )z Ûg)  ˜l ƒ„ q• w |g0 – ‡z | ˆy = 2x4 + 8x + 1Ejercicio.Pfj › l©rxoz ¨p }–‡oz›„g| sB‡)ic0ŠH…i€o´l n„¤j 0y3‡–ª.h…|h‡… Œ 0l ll ¨  ˜ l x7€y –†…) ˜ ¦g)l „”xl ˜ ll | ) €”a€7–†P¦I I l ‘§ x”w %—ˆ ‡‹ | | u y) ‡2› |g‡ l©x x©¤s gw l y –fl V | (al )w =|‘›0 }{§z |‰–†¤2› ‰w ‡l¨ p7o`´8Dn Ò±0Vr 7­ ²'´©·šq “ç ¿‰w l)x | fŠ“… – |a f(a) l | ©  x ‰‡ ˆ − ta(x) f a l©xY7y l da(x) = f(x) = f(x) −Œ –Corolario. ©l x n … l  l©Rx ˜ l 7€ –†…) ¦g„ l l | j Ša n −> 2 ‘§ x—l y – l | l f §f V (a) = ž¢¢ž ž = f(n−1)(a) = 0 f(n)(a) =W 0 ,jl | y ‡|g l)x êŒy ©l x I— )€ § j s€l w y a l©!x w ‚| › †{z g| –†2› ‡ j› z ¨ –†2› B‡ Š€ –†l Š ª– – …h‡‰nˆ 0Š „R— | ‡ „¤ f(n)(a) > 0 < j }– ƒ– Š Œ– l©x †– ›„I— ) € s a l)x€w ‘| — w |y‡ ˜ l †– | 2g0l ¨ – ‡z |¢0 ‡| y 8 | u l | y”l¥ V‡ €7}– ÚD‡ | y 8„ ˆ n k ª– ¦ w Ÿ”©€Rƒ„  u € )z gÛ )  ˜l ƒ„ u• w |g0 – ‡z | ˆy = (x + 5)2(x3 − 10)Ejercicios. 1.— k –†¦ w Ÿ7©€Rƒ„  u € )z Ûg© ˜ l „ƒu• w |g0– ‡z | · ˆ1 + x22.— y= ‹ x %¦ ‡Ú8©f€ „¤ u € )z Ûe©  ˜ l „¤c• w |g0– ‡z | yx– ˆ= yV (6x2 +x + 1)(x − 1)2(x + 1)33.— x2 k †– ¦ w 7Ÿ © €Rƒ„  u € )z Ûg©  ˜ l „ƒu• w |g0 – ‡z | ˆy = ex − e−x − 2x x4.—Ejercicios ˜l ¤–8 ”y „w¥šu|gˆ€¨‡w )0|g„„~ wß4‡ w{}xz ›„˜‰— hl˜‡€‚)x‡d©l „e¦Yx0w y‡„—‡l›2x›!l ‡ —I) —”xx)lwˆ€ ¦…Dq|©0 ”y€‚–ƒ„ )l‡–¤z —x|qx ˜ x0Ûg‡‡„ƒ|exDs| sx)l©„ƒ„ x5w˜ x|gl˜ „¤l—ve ”yw‰‡©ll l–Y›x¨”Ÿ y˜ hx—l s¦˜ 0xll‡–„›u„› ‰— e| ‡7€‡ xV‡ y”lg¦€w| l©„€ˆlxlB›!l „ˆI|© 8 © x| R€ 7y ¤l¤ w‰€7wg–}„lV‡–l €‰˜ls ld„}„Ÿ‡D‡¨zx‡ h›‰—l |€ g—lw „—Ig|l )›0ˆ€ƒ– l ˜ y‡ ‡ x ƒÞe| ¥  ˜|  x1 Hƒ ‡h› —‰7€ V‡ ¦I)R€ ¤ w‰l Àχ‰ ιË8 ρ» χË ©À ι½™8 ρÅ αË ‘ ™ËνιË8 ζÅ Ë ιˆ½ †ú Ç Ë d ) ‚€ d0 ‡ x ¦ 0‡ x x(3θ) = ¥ D‡ x θ 03‡ x (3θ) . h ԗr©y±02× µ ç i dθ 0‡ 3θ

2 EJERCICIOS 812 k5l 7€ –ª…¦) € ‘§ x }– h› —g}„ †– Ûe©) 5€ „ƒ 0l ¨ ‰— € l©x – ‡z | ˜ l „ƒ ˜‰l €7–†…  ˜ u  w 8 | y ‡ ”x l ™ %— ‡ x –†¦g„ l ê xl x l x 0‡ x xx l 0‡ x m x ž xl | nx, 0 ‡ | nx D‡ x xl | n(xp)) , · 0 ‡ x | | , x ©€ˆ xl | x mx, x + b2 ( 7y u · · a2 2 i 2x © €‚d0‡ x‚õ ( x) − h —Ô ©r y±02× µ ç x+ c x 1− x2 , (1 + x) © ‚€  x,H bxD‡+x a − b y”u p a D ‡ b , © €‚ y7u @¦ x xö a+b . a+ ˜ l s 2q „›Œ l –}|e}„ ¤„88 | ›!7y l n•w ©l x s s ˜ l07y l ”€ £3 wronskiano |g0}– ‡| u1(x) un(x) ) „ u2(x) ... W(x) = „ƒ)l7€ rrrrrrrrr†–xuxR…†– …(1xuun˜– ˆˆˆu−“x11V w 1˜ W–)ll | • V u2 ... ˜ u0l 7y(nuuln€7ˆˆˆnn−V ›21)†– I| rrrrrrrrr ) .| y”l ˆ x u x ‡q”x lw €”y¦… – ˜ xHl |!—%© €ˆ8› 0lz y €7¤– © x ˆ  €7V‡ £4 ž l©˜ w 0 –†€ xtw ˜ l uˆˆˆ 2V .ˆ .ˆ .ˆ | ) | | ‡ê I¦5k )l R€€7–ª„ƒ…¦ x ˜ •‡Vz“x7€ ›xw w  7yw(ul©xg| 2(xD)n0s }–−‡ w1| w )8©l | |x ... x ‡ „}‡ ˜ –}B|‡ … ”yl l € (f−1) V V j f(x)m = − j f V V (x) , (f−1) V V V j f(x)m 3 j f V V (x)m 2 − f V (x)f V V V (x) f V (x)m 3 = − j f V (x)m 5 , }ˆ †ˆ ˆ }ˆ ˆ}ˆ     y       x y − y x 3x y y x) x (x − y x) − (x −  y V V (x) = x3 , y V V V (x) = x5 ; l0l y|QV ˆ„ƒ x Dl u w | ˜ ¨ „ }{z | l  ™s x!l l | y –l| ˜ l ¤ w‰l s x dx d  d2x s xl | x x– = x(t) = = x(t) = dt2 Œ l  f(x) = x dt x ˜ l 7€ †– ¦… ) g¦ „ ll | −1 )x j –†–†vŠ ce 8 „†ƒ„ )€5„}‡ x x<0 s j –†Š  7€ V‡ ¦I©€ s€˜§ l | l©Yx y7l ‡V€ ˜ l | ê l©x x3 e−x2 xYy”l5 – |gx‡ −>Dl 0¨ ˆ l fl V f6 €x0DlQ B¨ fy €P•l ›„” (‡ 1x)() 5 ¦ Q – fs V (0) Q ‰| ‡ l0¨ – xYy7l „ ¤{z › ˆfV V (x) 4|==W4| }– 00|‰| –}7yˆ|l ‡ „ w y ‡ Rx ˜ | ˆ[0, +∞) x→0 Œ l  f(x) = αx + x2 x l | j 1 m 0 |¢– l |8xx 7y––„ u“l xxzl  €‚d‡ ¦%©)€ ê j x ¢V¢ Sˆ ä „ ˜ j j VŠ Œ – 0αx <>—‰€7α1‡V¦gs−<0„fl 1›!)l s x f ˆx ‰|€ l¢‡ 0¢V–)lˆlyäx | ˆy7€l l l w u“”€ …¦zw8| †„ ¨‡ l |) y¦g‡V– €7l g|€ y‡ !‡ ˜ 0 l‡|0y7ˆ l |‰– l | ˜ ¨‡ )„ 0ˆ %¦ Š Œ – § l „H« r©¬†d­ ®V¬ ® —n ˜ l ˜ˆ 2‰ ÇÍò 3Uþ ˆSŠ Œ ‡4| }„ ‡  €7¤{z‡V %¦©l x© „€€ l ¤ ) ‰w„ l©l x ˆƒ„  lw d 0 – ‡z w| 0 ‡ x x l | x − x2 = 0 y – l | l¨lD¨ d y )› l | 7y `l ˜ ‡x7 €ˆ x + x8 ce )}„ „ƒ) €5„}‡ xR0l ¨By € l ›„‡ x )¦ x ‡„ w‰y ‡ xR˜ l l |f(x) = x3 − 18x2 + 96x ˆ[0, 9]

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88 CA´ LCULO INTEGRALID…• w l Ie|g€5fi0 „ƒ5–un‡ z |is‰| c˜‡iFiolynh‰|˜nt‡lcee8ygs„©l rxY!›˜ 7ya™l )l€—%0 ui‡V‡nl|€I|dŸ wŠe|fif˜ y nl ‡‡ i•dx”w cl ag|f˜(0 x˜‰–}–¤‡)¢l l| d©lw¤xx e|‰w ©lflcxvFl©•l:xfw ™„I|gw 0→0e|‡ – ‡|z |Ÿ5 w |f(y x‡ )j ‰|˜ 2‡l Ûg…|‰d–˜ }{z ‡ xl – | fxIw l©Pˆ|˜x ŒI0 –}5l‡| |y7©l€ˆ lyx €ƒ€”0}– |¦…7€  w†–8¦˜ †„ ‡l s [ÀüBcaº‚Ë”½”½Ê,»f†v™À2bÁ½‘À]¼ÃedXÇüº“V¼Xňº Ëf»Á”½Ê SºÀvƒ½Ä¼XÁf8À‰Ã Àu¤úX¼üHüe¼$À”½º‚À8½”ÌÊÅ Àú}ºËdÀ ÌËÁ ËËXà ¼ X¼”ÁüÀ ÏÏ ˜ l l: FV (x) = f(x) T| x primitiva f G(x) = f(t)dt f(x) dx = F(x) + C ÍXéÀ“»¼iüÌ ¼Xè½¼Ëi©À—½f»Ãdà ‘YH½ÁËX¼d½¼ºÀ©À0ÃÅüºtuËt©ÀÅÊú¤Êûi¼Ëh¼vƒÃq¤úˆ“¤úhÁ¼û†vº“ˆÇÀüÁ8ÎiÇÇËËÀüº7ÁÅ˓»wúSüe¼½eÅaºˆ'e̪ºÀ©.½”Å„Êú}Ëˑ¼X¼.º“À ÃtÀhÅÃB.†ü˼SÅÀdÁv“Çi½°ÌDÀÁ”ÇX‘ÌDw¼¼½Y)e˼äÇüiǐ¼ÀDËSººªÅ”½fÃw}úº ÏÏÏeý—y )w‰| )l ”yDx l s ˆ 0‡ ›„‡ §  x ) ¦ l „› ‡ xDsI˜ ‡ x —‰€7–}›„– y –†…  x–˜ l f l | I xhl ˜ –†• l € l g| 0 –ƒ8 | l | w e| „ 0‡ | x £Pro B¢Vˆˆˆpiꌌ e––V‡dz €7fFa› §dP w eg”ƒ„ s (y1˜– )l ls| l 4| F V ‰— (x€7}–)›„d– xy = DxFs(x) + C ˆ =Œ λ f+µ ˆg (λf + µg) s(1) ª– ¦…  – u, v P•” êintegracio´n por partes uvV = uv − uV v. ˆ ê V‡z €7› w ƒ„  ˜l êintegracio´n por partes extendida Œ– u, v P•” s u v(n+1) (n+1) u v(n+1) = u v(n) − u V v(n−1) + . . . + (−1)nu(n)v + (−1)n+1 u(n+1)v . u# v(n) q u# # B© ˆ ê V‡z 7€ › §¨w „ƒy – l ˜ ll }–c|ha… ml € bx ioy 8 d›qe¦g– l z | svariable ˜ l „ y –†—%‡ êx = g(t) Œ– l©™x ˜‰l 0 ƒ„  xl ³³³ v(n−1) Q | ” (1) ” (1) g u(n) ³³³ ³³³ #v Q n−1 jj u(n+1) v Qn f = F(x) + C =⇒ f g(t)m g V (t) dt = F g(t)m + C ,Ejemplos = õ 1 − „}‡ u õ x + 1 dx ˆ1.– ec) „}ƒ„ ) € I x s'”x l }„ „ Dl u x I öRö − }„ ‡ u˜ƒ ƒ + s ˜ ‡ | ˜ l J „}‡ u t) dt ˆ l %„ © 8 ™› g¦ }– ‡ = uƒH‡| ‹ Yx y q }– | y7Dl u x = 1/t q = t (1 J s = (1 + ƒ (1 + J = −1 }„ ‡ u €ˆ8 „ x )„ l —%‡Vv€ —I)€ y”©l x©Bs y ‡ ›!8 | ˜ ‡ u „†‡ t) dv = êt2 t, t) + }„ ‡ „}‡(1 + t) + u + u\"ƒ ƒ „}‡ dt 1 dt = − 1 − t t(1 + t) t t ut2 \"ƒ 1+˜ ‡| ˜ ™l xl—¥  wgx  ˜ „‡ ƒ„  ˜ l©x D ‡ ›h%— ‡ x –¤0– ‡z | 1 =l „−©81™› +1¦et†– ‡ + 1ˆ Œ wgx y – y”w‰§¦l | ˜ ‡ l d„ …¦8„†‡V€ ˜ l J § Þ ˜ )l x¥ td(10 – +l | t˜ )‡ €tl)x w ßs „y  „}‡ uI = −(1 + x) õ x + 1 + C. xö

3.1 PRIMITIVAS 89 2.–  ‡V€v—I) € y7©l xDs'y ‡ ›¨)| ˜ ‡ dv = dx sgx 8 „ l | „}| x dx = x „†| x − x + C ) ‚€  ”y u x dx = x )‚€  y7u x − 1 †„ | (1 + x2) 2 ) ‚€  xl | x dx = x ©ˆ€  x l | 1 − x2 + C x+ cx3 e2x 3.– ƒH‡4| ƒ„ q• ‡Vz 7€ › w „ƒ 0l ¨B7y l | ˜ – ˜  ˜ l –}| ”y ©l u €ˆ8 D– ‡z 4| —%V‡ v€ I— )€ ”y l)Dx s3x2 1 e2x q x3e2x dx = 1 x3e2x − 3 x2e2x + 3 xe2x − 3 e2x + C. 2 2 4 486x 1 e2x Q IujXlnw „t– l†–e|| gž ”yy7rl©ll a7€™x0 …lV‡ l©e8€x s}„˜¤‡ 8w‰i|˜nl ˜l!›m4‡ ˜elxfƒ„dÛgjXxi|‰˜a¤–V‡˜‰t0 |al– ‡€7˜zs–†| Hl… ˜l©˜vx¤–  ll x˜ sl‡d„› „¤x –}x|‰s –}—•‡ w ©lly”x—0©l –†u €l „l› l| sy l d8t| „ j©l u©xD($s x˜ x)ul„m ©yŠ – l˜x |‡Bl lŠˆ• |Aê‡Vz 7€ }„› ‡ ™xw ƒ„x 4 „ x˜ €ˆu) |(x˜ )‡ w‰g|˜l©0˜l }–d„“l‡ –}|˜| –†x£6 1 e2x q 80 1 e2x Q 16 xr dx = xr+1 + C (r =W −1) dx = „}‡ u\"ƒ ƒ r+1 x +C x „}‡ uax dx = ax + C (a > 0, a =W 1) a x l | x dx = − D ‡ x x + C 0‡ x x dx = x l | x + C 0 ‡ dx x = y”u x+C xl |dx = − 0 ‡ ”y u x+C 2 2 x x dx = © €ˆ y7u x+C · dx = ) ‚€  ”x l | x + C 1 + x2 1 − x2 Œ ¥ x dx = ƒ ¥ x + C ƒ ¥ x dx = Œ ¥ x + C ƒ ¥dx =  ¥ x+C Œ ¥dx = − ƒH‡ y7¥ x+C 2x 2x · dx = t € u ƒ ¥ x + C · dx = t € u Œ ¥ x + C x2 − 1 1 + x2 x ld| xx = }„ | rrr ”y u px rrr + C 0 d‡ xxx = „}| rrr y”u p π+x rrr + C 2q 4 2q

90 CA´ LCULO INTEGRALFunciones trigonom´etricas (I) wˆ€‹ 8|š mnmYx|„ yslD7y‡ –}¨Dl+–}›hx uh›%— –}€ˆ)n%—|‡—%8)y”|©„›q©©lI—€)l©l€)uêRx)|¦gê€0)€7y}–˜) ‡ê—l „l xDl©„lsvx |©)©y088)˜–ª”y ‡%—l›q›q›ql |7€‡ g¦¦gg¦‡• wm–}–}–}‘§‡‡‡e| x,0luuunl }–|d„‡ ===m|P‡ y©l xr”y0x€7x ul„‡‡ €ˆ0|nxxd‡‰| xx0.x{‰‡ –}0n;;ˆ‡ }|ex 08d‡V‡ „ €xl©| x˜ls1.– 0‡ | m, n P É ˆ(a) ˆ w l‹ h› l4| |tg— 8}„ q‡‰„ Hu ê}– z|w y”lDu ˆ€ x8 ‡q„ ©l —x w‰–}‰)|l ˜› l )l|`˜ •–ƒw y|e 0x–}‡Vˆ |I‹ © |€u0‡l || l Ÿ |‘© xl | 3x x dx = 20 52‡x (lx 02| ‡ x x)5/2 − 2(0‡ x x)1/2| + C D ‡ x = x”x”ll | + C · = x + „†‡ u õ 1+ 4 x 1− x x0 l‡ 1 7y u 4 x + C xö 0 ‡ x 3 x dx | 3x 4 x 5 x dx x I— )€ ©l x wex ‡u€ l — 0l y – ˜ ˜ l ƒ„  x • ‡Vz €7› w „ƒ x 0‡ x 2a(b) m, n ’P r 8 ›q¦%‡ 0 2‡ x 2a ê ‡ x l | 2 a = 1− 2 0‡ x 2 a = 1+ 1 x l | a 0 ‡ x b = ”x l | (a + b) + ”x l | (a − b)k xl (a + b)k | x l | 2 ¡ 0 ‡ x (a − b) − D‡ x (a − b)k 0 ‡ 1 Ÿl a b = + C xl 2 ¡ x a D‡ x 1 0 ‡ x (a + b) + D‡ x 2x − b = l „› g— }„ ‡ j m =W 02Š s¡ ‡V€ x l | 2(mx) dx = x l | 2 x D‡ x 4 x dx xl | (2mx) | 4x − xl | 6x +C x− 2 õ 4m x l | =1 x+ ‰w ©l l)vx ˜ whl e|1Dl6™ ¨ —‰• ‡Vz€ 7€©l ›x © w € „¤l „f4˜ –†l | y”€ l©l©u˜ €0w )© | 04˜– V‡‡z |lˆ |A• w 1g| 20– ‡z |ö ‰x V‡z }„ ‡tRw x g| ¤ 'sgw §lu)y g—8 ›™}„ ¤– ©¦g)–€lz | l | xy l ‡ |g—  D0– ‡z | s –†| ”y lDu €ˆ8| ˜ „‡ %— V‡ €€ —I)€ 7y ©l x ˆ x©s —%‡V€t—I) € ˜ 2l xl |x ‡0‡ ‡Vz 7€ › w „ƒ xR˜ l € ©l ˜‰w Im = 0‡ x m ˆxdx Œ – y ‡›!8 2› ‡ ”y l)xDs x s ”x l  dv =(c) ê V‡ € l Ÿ l „› —g}„ ‡0 ‡ x x s )l x w „y €  dx Im = xl | 0‡ xx m−1 x + (m − 1) 0‡ x m−2 x (1 − 0‡ x 2 x) dx ,

3.1 PRIMITIVAS 91 €„ ‰wl©lD˜ uw ‡  —l lm„I)mz ¤„  w=}„ ‡ x ˜l |l x 0‡ ¥x m x −y 1 x +s ‡(m − 1) sIm}– |‰−› 2l)ˆ˜  y © €ˆx ˆ m ‡V€ PÏl Ÿ É l h› l©g—xYy„}‡ts • ‡Vz €7› w ¤„  xl x| x I−1 ƒ–  Im D ‡ I0 I1 0‡ x1 = 2x pπ +x rrr q +C 1p + }„ ‡ u rrr ”y u 4 2q 3 x dx 2 x–ƒ jw 0u 8…¦| ‡˜„†…‡ x l w“V‡ z ©l¦ xy q– l ¤„| q l —s ˜ l w l y7Dl u l©xc˜ l»¿¡Ì0Y‘Á “½ºÀ˺Q½ dÀÅÉbÅP'IÍÁ‚Ë DÌ­ƒúÇwÇR¼½ÄRewT©À Ã4eS”½¼XA0©ÌņtÀ6¼Xv¼Ì77ŠБ»f1¼dž0¼XXÇ0©À˽½YÅSÇfx½€Ã™ÇtXÇe X¼©Ì Åf»¼üË ”½}úX¼ºÀ ÏÏ |‰– ˜  l „›m‡ P•Rx ˜ r ©l x — ©z ue— –}†„|e–¤©hÃ8  |ä w‡¨ˆŠ ꃄ u €  ”€ € |e0ƒ– q —%©€ˆh }– | ˆ€ 8 „ Û £ 0πl /— 2‰w 0 ©l ‡ ‰˜ x lm¢| xD d‡ xh› —‰=7€ V‡ I¦)€((mms mm0−−‡ 11((„› mm))((‡−−mm22l−−))Ÿ UU33l UU UU))‚€32UU0 UU UU–¤210–}π‡2Is l©Yxxx y – m l©x —I)€ – ©l x }– ›hI— ) € m x • V‡z €7› I— )ˆ€ q „¤u–†| ”y Dl u ˆ€ 8„ Im,n = xl | m x 0‡ x n x dx êŒ  y ˆ€  w „ƒ x5˜‰l € ©l ˜ w ©0 – ‡z | x‡ x l | D ‡ x(m + 1) Im,n = xl | 0 ‡ x(m + n) Im,n = m+1 x n−1 x + (n − 1) Im+2,n−2 m+1 x n−1 x + (n − 1) Im,n−2 (d) I‹ nw ¢| – =l c€„}‡ ©x  y7u ‡ n ‡xxlx s )x −l€5„ƒ—{q 0‰w }•l©‡Vzˆ˜ 7€ l› | w ¥ ƒ„  d  ˜ l l 5€ €„}‡l x w © ”€ 8€ ™›l |gg¦ D}–¤–‡ cx ˜ l „™) I— )€ y  ˜ ‡ ˆ(a) ‹ | ©x    w 8¤„ ¤ x dx n P É —nw‰†–)l „› ˜ –l—I)gw € x 7y uj n−1 x , (n − 1) In + In−2m =  l Ÿ l ›hg— „}‡ s 7y l ‰| – ˜ wex ) | ˜ ”y u 7y u V‡ ¦  ‡ 2 x = x) V ˆ ‡V€ 1+ ( y7u 5 x dx = 1 ”y u 4 x − 1 y7u 2 x − „†‡ u\"ƒ 0‡ x ƒ 4 2 x +C = + 0 ‡ x1 1 y7u 5 x 2 y”u 3 x + ”y u x+C 6 x dx 5 3 2©lŒ x.–l 0–€7wg– xy 8‡¨ `| 7€8‡ƒ„|  ˜ y7xl©w x •  d‡ˆzy €‚‡t5› vx Dx w˜{z gs„ƒl x €x–xl m˜ ‰| cl ‡ s“xy €ˆ§n8| D x‡ †• x”‡dl‚€ |‰!›s ‡ dx 0ˆ – ‡z | ‰˜ l g— 7€ ‡ ˜‰w y‡ x“l | xw ›! x ¤ wg5l §  ¥gl ›„‡ x =W 0 xl | (m − n)x x l x l | (mx) xl | (nx) dx = − | (m + n)x , ˜ l™x l˜‰©l ˜ w  2(m − n) m =W ˜ l˜ ‡ | l ¤ w‰l —I) €ˆ nm‰s +x l—ny – l | l m, n P r D V‡ | 2π xl | (mx) x l | (nx) dx = 0 . 0

92 CA´ LCULO INTEGRAL3—%.‡–}„ –}‰| Œ ‡ š8„›| „ l7y }– DlA|‡ u €ˆl08|‡„f| l©xx „¤ˆ!P• ‡Vz(€7x› )w x l | x s ‡(ax) dx xl |x (bx) dx s‰˜ V‡ | ˜ l P(x) )l xvw | D ‡ eax 0‡ (bx) B ˆ V‡ € l Ÿ l ›hg— „}‡ (ax) x l  „ƒ ˜ l –}| y7lDu €ˆd 0– ‡z f| —%d‡ –€ —%©€ y7l©x–0l B¨ y7l | ˜ – ˜  s  8z „¤ (x3 − 2x2 + 1) x l | (2x) dx ˆ“ž l —I xl ›„‡ vx l „ 8„ u ‡V7€ – y ›2‡ s  ‡z „› ‡ ˜ l„ Iw „}=‡‰ê xl | (2x) ‡h%— ©€ˆ x3 − 2x2 + 1 3x2 − 4x − 1 ”x l0‡ x V 6x − 4 − 2 ˜ xl | (2x) W 6 x V 1 1 (2x) $ˆ„ƒfáâ–}Ú©‰Û ¤ |ew 8 – „}l › € ˜ l | ”y xl¦tl s 0¥ x2W x w 8 40 ‡ |(l 2€7(†–2x…¦x) )˜ ‹ ‚|y7lDu„ƒfˆ€ 80„ ©l‡xR„ w x”›„w e| l© x ˜l x d l |1   l)x –†…  x Q l #| ¤„ s l | y €ˆ8„ s –}| †– ¦…  16 0‡ x x l |I = (x3 − 2x2 + 1) ž õ −1 õ −1 2 (2x) %ž V + (3x2 − 4x) ž 4 (2x) ž%W + . . . ö ö 0‡ x ”x l | 0 ‡ x= − 1 (x3 − 2x2 + 1) (2x) + 1 (3x2 − 4x) (2x) + 1 (6x − 4) (2x) 2 4 8 − x l | 3 (2x) + C 8 Œ l u ¥ ‡V€ˆ I = eax x l | ˆ(bx) dx Œ l–y – l | l eax xl | (bx) D‡ xaeax − 1 (bx) V„ w‰Dl u ‡ x l |a2eax b (bx) W − 1 b2 I = −1 eax 0‡ x + a eax x”l | (bx) − a2 I, ˜ l)x — l b2 b2˜ l ›„‡ ˜ ‡2¤ w‰b¦l s (bx) s ”Ÿ 8 | ˜ ‡ x l | 0‡ xI 1 j = + eax a (bx) − b (bx)m a2 b2 (bx + ϕ) + C , x l |= · 1 a2 + b2 eaxx – xl | ϕ = · b § D‡ x ϕ = · a ˆ a2 + b2 a2 + b2

3.1 PRIMITIVAS 93Otras fo´rmulas de reduccio´n, por partes š | y7Dl u ˆ€ 8 „ l©xv˜ l„ y –†—%‡ gs ˜ ˜ l1.– ‡ ¨› )| ˜ ‡ u s‰Inx l = xn ‡| n P r ˆ = xn−1 dx  „}„ l©u · „ a2 − x2 € l  w €”€ l—¥  Yx y  sc−xn−10‡4| ƒ„nu I¤ ngw l=x l a2 − x2 + a2(n − 1) In−2 , ‡ I0 = )€‚ x l | j x m +Cˆ a · I1 = − a2 − x2 + C2.– š | 7y Dl u ˆ€ 8„ ©l vx ˜ l „ y †– %— ‡ €„I„ n‰w u = a2 dx sg˜ ‡| ˜ l n P’r s ˆn > 1 ‡ | ˜ € l)Yx y 8 | ˜ ‡ l “„ | w › l ˆ€ %—(y –M‡Vl v€| ol—IdI)on€ y”l)=1Dx )Bs I.yn‡−‹!› 1r| 8−| ‰— K˜ €7‡n}– › l l )€ s“x(wx2›¨+) | a˜ 2‡ )n§ ‹ x2xYy l| ”y lDu €ˆ8„ Kn ˜ ‡V€ “s xl sh˜ =x ˆ c}– | x vl ¥ 8  l ŒKl n†„ „=Dl u x2 u „ ˆ(x2 + a2)n dx In = 2(n − x + a2)n−1 + 2n − 3 In−1 , 1) (x2 2(n − 1)0 ‡ 4| ƒ„ u¤ gw l xl € l  w ”€ € —l ¥  Yx y  ”y u(Modo 2). ƒHV‡ | l „ © 8 ™› ¦e–†‡ I˜ 1l = a ) ‚€  j x m ˆ …¦)7€ –ƒ© ¦e„ a l x a 7y u t sgx–l y – l | l =Œ In = xl©l™a˜ w2˜ n1D©l 0−x –¥3‡ z d!|  (l—%0V‡0‡ €H‡x |—It))x2€lny”| −)l tDx2s d=¤ tw‰.·5l §  ¥‰l ›2‡ s x … – x Yx y ‡ sBl | ©l x y  –†| y”lDu ˆ€ 8 „ hs § 8„gۉ|I)„l D‡ ˆ l )—g†„ –¤©—€ x t = · a „ ) )›q¦g}– ‡ x2 + a2 x2 + a2F}„u‰| ‡ €ˆ‡ux›„n˜%— c–}‡‘‡‡ i†„x—o|e˜–}|‰nll ‡|ePf›2su(xx–†nr‡ )0acxR‡i)lcxo´A|xil on›¨0nV‡ ‡¦ral§aylgۖeV‡clD€vsi|–ol‡hl n| 7y–aul©lwx 8j €€„dll ¤88 „‰wƒ„ ©lŠ lcx ˆ˜l „l„ Œ u –w ˆ€ e|P !˜(x‡ …))˜ §€7l –ƒ)QQ¦g(„(lxx))xs'x ‡l©¥ —x|d 0w ˜– |Ql ‡ | x 0˜ ‡'—Ih‡ 0‡ –¤„„}l}– |‰| ˜y7‡ !l†–›„'… ˜–}–x‡l– –x‡z %—| E§ ‡˜}„ l†– l£„ P(x) = C(x) + R(x) , Q(x) Q(x)Qˆ€©l˜˜ dx‡l (0|›x–}C‡˜)‹l|e(lIs‰|x|8 x‡V)„lq)ly€5¥ x8l©‡Vwg¤x„‰€ˆ|‰xw‰©‡–}l—‰|sx xl›ll „ l)u˜ €ˆwˆ€l0–¤zly8›„˜y˜‡| ‡˜‡‡˜ ˜ „}˜lxDx†–„l2›lq5{z R˜ 0¤˜–(l‡y‰wxl©|g)|‰x4l)‡V€‡0 2›)Ú‡©ll)©x›h|™„–}Re| ©›%— l¤„ ‡ l ‡|‰V‡˜l©¤– €0V‡Yxl©ls–y7€ux‡¢|wgzQD |¤‡V˜ ¤(‰w„›}–xlgw‡l2E|)l—%— ˜lˆ‡l•¤„lˆ€x„ d–}uDu„ƒD€ˆ–Eˆ€0‡z}–!|˜‡}–˜ |‡|‡0y7l©‡Dl˜cx˜›hu llˆ€x g—„d–}Qh›„0| l0–(w—gy‡zx›|„)©l lxlˆ  `• V‡w €–†g||s 0y”P}–Dl ‡(u |x€ˆ)©l8 x„ s  „ x ˜ l —%„ s ‡ ˜ x˜ ˜ ˆ€  ˜ s ™„ › u ‡ l ‡  h|ê •l d„$ 0y '‡‡V0€ l)– l$x | ˜7y ll ‡ x

94 CA´ LCULO INTEGRAL P(x) xl — w‰©l ˜ —l ©l x 0€7ª– e¦ ª– € ‰s l | ul | l ˆ€ 8„ s 0 ‡ „› ‡ w e|  x”w ›! ˜ l Q(x) n Ak (x − a)k—%› ‡Vw €v„ y) k–†—g=˜„}1¤–0 – •˜ d y ˜s l§ ˜ l„ y –†%— ‡ (x − a)n j 0V‡ ”€ € ©l x %— ‡ | ˜ –l | ”y l  ¤„  €ˆ }{z Úc€ l 8 „ ˜‰l ˜ d‡ € Q(x) a n m Bk x + CkQ%§— ‡V(C€qxk)©k sˆ=™˜¤k12‰w–}|†– ¡—l… (—I7y lxlD©l)€ux€−x€ˆ‰y ‡ d8˜zxα|0l –)›‡zw2€ˆ|lD|z+ysg¤{z{†z‡¦…˜ βl©˜‡'l©x 2‡©x k—8vxDk›‡w“‰˜ h› zll l)|g—Dx 7y„s8lzl ˜„}Ÿ”˜l w ¤„xDl }„”y‡hRx0l ‡€7—˜ |2› –lŸxY€7w‰}–y|e2› –}u | – 7y˜ l  x ”ys l | βl ˜ l x ›0 ‡ wl „Ûey 0–†g—– l „}|–¤0 7y –©l˜ x  ˜ ms ‰˜ l  t|x ˆ ‡Vα¦ R€i }„ ‡ }– h› —g„ l©Rx ©l x  x©{z ê ˜ ‡ Ak Bk y x •ƒ€ˆd D 0}– ‡ | ©l xRx }„ |dx = ƒƒ x−a x−a +C dx −1 x – n =W 1 (x − a)n = (n − 1) (x − a)n−1 + C }„ | )‚€  ”y uAx + B (x − α)2 + β2 dx = õ A(x − α) Aα+B (x − α)2 + β2 + (x − α)2 + β2 ö dx = A j (x − α)2 + β2m + A α + B õ x−α +C, 2 x”βlDuw β ö¥ d 0– l | ˜ ‡ l x − α = βt l |4„ƒ | ˜ ™–}| 7y lDu 8z „}‡ u 8› l| y7l s „) )›q¦g–}‡ €ˆ8 „ “ˆ á 8 | Ax + B @H ¡ (x − α)2 + β2k k dx = A(x − α) Aα+B ¡ (x − α)2 + β2k k + ¡ (x − α)2 + β2k k dx = −A „+©β8 ™›2k g¦ k–}−‡ 1x+−(αA α + B) In (t) , = ˜ ‡ | ˜ l‰¥ l 2›„(‡kx − w‰1)©l xY¡y (‡xBs −y €ˆα €x)2l s — t In(t) = dt ,–}| 7y lDu ˆ€ 8„¤ w‰–l §  x(lt2¥ + βl)xY2y”)gw k˜ ƒ–  ˜ ‡!8 | y”)l x ˆ Ejem p7€ ‡Vlo%— s‡ .| l 1.– x ˜ x2 + 5x − 3 dx ˆ 2› ‡ ƒ„  Il©=x 0‡›„(—Ix‡ +x ¤– 10 –)‡z(|x2 − 4) x2 + 5x − 3 ABC (x + 1)(x2 − 4) = x + 1 + x − 2 + x + 2 .

3.1 PRIMITIVAS 95 ƒ w 8 | ˜ ‡ 0 ‡|Q(x) = (x − a) qa(x) s D‡ ›„‡ l| l)Yx y”l ©  x ‡ ‰s xl—y – l | l qa(a) =W 0 P(x) = A + R(x) , Q„ y –†(—gx}„)–¤©8 | x−a qa(x) § › w ˜ s „‡ —%V‡ € (x − a) qa(x) P(x) = A+ (x − a) R(x) , qs a—I() x€ˆ) qa(x) „ ‰w lDu ‡ s x =a A = P(a) ; w‰l)xYqy 7€ a‡ (a)  Dx {z l 4| | h› g— }„ ‡ j q−1(x) = x2 − 4 ‰s 0l y ˆ s fŠ € )l x w „y  lŸl }„ | †„ | }„ |I = 7 ƒ + ƒ + 11 ƒ − ƒ − 9 ƒƒ x 1 x 2 x+2 . 3 12 42.– I= 7x4 + 25x3 + 6x2 − 34x + ˆ23 dx 1 ji§ qa(a) =W 0 ŠvD‡ ›„‡ l| l©Yx y”l Q(x)© x ‡ ‰sƒ xw—l 8 y| – l˜ |‡ l D ‡V|(x − 1)2(x + 2)3 m> = (x − a)m qa(x) P(x) = A0 A1 Am−1 R(x) (x − a)m−1 x−a qa(x) Qs ›(xw)„ −˜ a)m s+ (x + ...+ + , –†—g„}–¤©8|„ w‰Dl u ‡ y „‡ —%‡V€ (x − a)m qP˜la(|‡ (xyx)‡Vj)—I7€ |g‡=‡„}}– ‰| ˜‰A‡l „› 0a}–+‡ “s A˜ ˜ l1l (x − a) +...+ ‡Am−1 (x − a)m−1 + j (x − a)m−1ml| l y „ ›„‡ ˜ f‡ ¤ w‰l }„ ‡ x A|x –kmqxa‡ −(|wx1)„†‡ x 0 ‡ l eÛ 0 – l | 7y l©„x ˜ l „ ˜ )l x )€”€7‡„†}„ ‡„}}– „› – w   § }„ ‡V0€ Š ‰˜ l ‡V€ ˜ l ˜l ¿„ 0‡'0– l | 7y l P(x) l | y ‡V€7|‰‡x8lI„ —— w‰”y Ž¦l|©l 's˜yy 7€‡§`l f‡ aV‡l ›¦…¦ ˆ 8”y 0lztl„ yw …l R€l E‡ ˜ • l©‡d‰—z Dx0€sl–}d}„|z0› ‡ yx„›l–¤0|=‡‡y”l ˜ ©l ”x ˜ l „ ‰w lDu ‡ 's )l xY”y l™˜ l©x )q€”a7€ (‡x„}„})‡„„†–}2› – y  ˜ ‡› l| | ˜ ˜ˆ ls = )| 8z ¤„l  ww }„ d‡ 0 – 1 a ‡ „ƒ  ˜ ‡z `l‚| l©”Þ Yx )y ‰—‡ €7x ‡ A¨ }– k!› ©l ˜x  –†ß€ ‰˜ l 7€ –†…8| ˜ ‡ xw  l©x –ª…¦) £ 5t xD{z s l | l „ l Ÿ l h› g— }„ ‡ s ‡¤P(x)=qa(x) j A0 + A1 (x − a) + . . . + Am−1 (x − a)m−1 m + j (x − a)m−1m qa(x) j A0 + A1 (x − a) + . . . + Am−1 (x − a)m−1 m ¤7x4 + 25x3 + 6x2 − 34x + 23 (x + 2)3 (A0 + A1(x − 1))

96 CA´ LCULO INTEGRAL x = 1 =⇒ A0 = 1 ¤28x3 + 75x2 + 12x − 34 A1 (x + 2)3 + 3(x + 2)2 A0h§ s %— V‡ €R‡ y €‚xh‡ ƒ„= ˜ ‡1s =⇒ A1 = 2 ¤7x4 + 25x3 + 6x2 − 34x + 23 (x − 1)2 j B0 + B1(x + 2) + B2(x + 2)2m x = −2 =⇒ B0 = 3 ¤28x3 + 75x2 + 12x − 34 (x − 1)2 j B1 + 2 B2(x + 2)m j + 2(x − 1) B0 + B1(x + 2)m x = −2 =⇒ B1 = 4 ¤84x2 + 150x + 12 2(x − 1) B1 + (x − 1)2 2B2 + 2B0 + 2(x − 1) B1 x = −2 =⇒ B2 = 5 }– e| ) }„ › l | y7ls I = −1 + 2 †„ | }„ |ƒ ƒ ˆƒ ƒ x−1 x−1 x+2ê − 3 − 4 +5 2 (x + 2)2 x+23.–  uI8=›„‡ x (xƒ„  2 ˜ x3 − 3x + 6 1)2 dx ˆ ‡| −l)x20x‡+h› %—5‡)(x x–¤0−– ‡z | x3 − 3x + 6 A B Mx + N 1Ÿ wl)›„2}„ ‡¨—g=8}„ ‡¢„ u l)(g¦x| ˆ€”y −)l –¤€‚01–†‰‡‡V)€2ê s + + x2 − ,•ƒl €ˆ| d y D ‡ 0 g| –(‡xz ‘|)l2Dx −€s l)0 x2‡w x›„„ y+‡ 5˜‰l )|l („xl  „ −8z l¤„  −„ y ) 1| A=1 2§ x + 5 0 ˆ   7y l €‚ l ˆ€  € ©l xxw B = x3 − 3x + 6 1 x3 − x2 − x + 1 x +14%—d0 ‡.‡w‰(–„› xlDx–¤u02—gkc‡–„‡−llz Ÿ”I|”Ÿ 8B2=„›ˆxl ‡| ‡−+Hx ¤•x5ˆ€)l −1d)xY(1Dy xD +–†€x −‡˜| 211‡l))„†x„x |2}–(x|−x}– h”y›2©l (—gu−x€0„ ©l)2−2x„x©l 1x˜+)l02„5‡)=›2y –†+—%‡ (‡ x)l Ÿ€‚2Al −€‚y”/Du(}–2xD xj †– ‡‰x−+−2ˆ 1aŒ5m)l)(+—x—I‰w)C−ˆ€©l ¢˜ˆ1cl)© 2¥ = x2 − 2x + . 5 d4 l €ˆ€  w }{z Ú|e ˜ ˜ s )l x l 0 )‡„v›h‡£ € a „}|dx 1 õ (x − 1)3 − · 1 )‚€  y”u õ 2x· + 1 x3 − 1 = 6 x3 − 1 ö 3ö +C @· 3H „}|dx 1 x2 + x6 + 1 = · · 3x+1 3 3x+1 4 y”u x2 − y”u ”y u + 1 ) ‚€  x + 1 ) ‚€  · 3) + 1 )‚€  · 3) + C . (2x − (2x +—gÞ x„ ©l}– ›hx —g— „‰w)l©€ˆl©!ß q˜ l˜ }– |l—„}y”„ l©ll©…xu l)ˆ€ 5€© ©5€ › x•ƒ€ˆ‡l d|‰3xD‡ w0x“–}—%‡ y‡|ˆ€ |)l©l–¦Ix )w YŸw|4‡ § ›6l‡ d„ 0l˜z›y l‡ |gDlz˜ y‡V‡¨‡ ›2˜ 8 }–‡„|e7y lx –7€˜ u|eV‡w € –yl yª–|…h– l7y¨‡ |l ˆl™ 6š0€„}g|‡ 0€x¤{z „ gw§)l xx ‡u0… ‡I—– xYh›) y ˆ€‡g— ™ x„ lˆƒ„ Ÿ”q ƒ•x ˆ€ d› D w0z –„ ‡yz –ª| £

3.1 PRIMITIVAS 97M›‰|¤‹ w‰‡|l ´›„ely‰| ‡t‡l}–Œog|‡„x ldx u8¤l©ohˆ€|qxwg(ld˜xjPƒe)‡H(¥x§(˜H)Uˆxlm)ee§ Qr(§l mxQ7€(M›2)x(i)stx– ˆ(7yDe)xl‡Œ )s¨|˜ls ‡¢0)€04à|x©l‡ÃÜhxl%—H—Š eÛ ‡ l(0„†x–}–x |‰yl)l –†|‡…¦=—y7›„8©l‰w› x›!–}l©‡ l}–˜ux||l˜ ˜7yy| jl8l0Qs7y„l l©lye|(x7€)x0›„„)‡¤l©,|}–w‰x Q|ey‘l ¤ˆ€ V©˜‰w(l €x‡l„ )“x˜um !§lˆ€ § ˜„›˜ M5l‡ ‡ u˜ (˜ ˆ€x‡ l )˜w l©x › l |‰‡V€ ˆ P(x) z=‡|‰x“–¤Q0w ‡ |e(x˜ )‡ /w x H|‰—%–(˜ ‡x}„)†–˜ £P(x) = h(x) + m(x) dx .Q(x) dx H(x) M(x)}l©–Ò|±0Yx‡y7yrDxDluqeu nˆD–€ˆ­Du‡ pDwl˜µ‰gÛ8 p70„ q%˜–©l lÄÒx |2p4˜ „¤y”j©l§„†™x™³‡ €x©r˜ ©l±©l}„˜Òć p“wuhD0±)(–dr€7xl –‚­)|y²†›2)´˜ § q%‡„‡ r©mvxƒ„¬% §(˜x xl )•ƒ)€ˆ€‚„ƒdRx0 lD2‡ –}0|x€˜}–7y‡y0llD|87yu |l©lˆ€ 7€xu8 2›l„hsV| –†§|I”y0l‡)ƒ„Šˆ›RA| ˆ }–w—%z| |7yd‡ lD€™˜u l–€ˆ˜‰|8l„‰‡| „¤› y ‰w †––}Ûele| ©¤ d˜w‰0‡V)l– ‡)€˜z ˆ|  s l©‰˜ux$l x•ƒ7€ˆ€w™–†8…³™©8 0©r| –©±‡˜zÄÒ |‡p V‡ € l Ÿ l h› e— †„ ‡ s dx −x 2 dx(x3 − 1)2 = 3 (x3 − 1) − 3 x3 − 1„}| )€‚ ”y ux6 + 2x4 + 2x3 − 3x2 − 2 x3 + 1 ƒƒ dx = x2(x2 + 1) + x + x+C x3(x2 + 1)2F• w ug| n0 –c$¥‡ zil‘|o8n€ˆ2› ed ‡0s}–xR‡t|weri‰|) g„‡ ox¤„ un© o–}8 | m™› 7y Dl¦g´eu }–t€ˆ‡ r85x i„c˜ ual ls|¦… (l)z I‚€7€ }–I–¤)))¦g„ l ¤ w‰l D‡ |h…'– l € y7l | l | w |eq–}| y7Dl u ˆ€ 8„ ˜ l w e|  xl | x, 0 ‡ x x) dx ,R(˜ ‡| ˜ l R(x, y) ©l x ˜ w ‡ |e| “˜ •lw |g0 – ‡z | €ˆd §0}– ‡e| 8 „ ˜yHl) x x gw ‡ x|‘˜ I—‡ ‡x }„ …–}|‰) ‡€7–ƒ›„)¦g}– ‡„ l©xRDx lds |)l x x˜ ll 0 –†€ 8s w |uD‡BD– l | y”l ˆP(x, y)/Q(x, y) P(x, y) Q(x, y(a) Œ – l „› —‰€ ™l xl 0€ 8D–†‡ |e8„}}– )Ú h0‡| l „© 8 ™› g¦ –}‡ 7y u p x = t s —%V‡ € l „  w 8„ x l—y – l | l 2qxl | D ‡ xx 1 − t2 = 2t x = 1 + t2 , dx = 2 dt 1 + t2 , 1 + t2 . ¿w‰l)˜ l| x”l €R› z x … l | y 8ŸY‡ x ‡ x „†‡ x © 8›™g¦ }– ‡ xRx – u w –l | 7y ©l x ê(b)jjjšššŠšš Šš Œ Œ – – R(−x, y) = −R(x, y) : ))) ))) qq›››qg¦gg¦¦ }––}–}‚‡‡‡ y70xul‡ | x x=t ; Š R(x, −y) = −R(x, y) x=t ; Œ – R(−x, −y) = R(x, y) : x = t. :

98 CA´ LCULO INTEGRAL ’ w‰l  u ‡ y 84| ƒ„  x —%‡ x –ª¦g}– }„ – ˜ ˜ l©x I— ) €ˆ™ƒ„ q • w |g0– ‡z | s — gw ©l x R R(x, y) = R(x, y) − R(−x, y) + R(x, y) + R(−x, −y) + R(−x, y) − R(−x, −y) ‡ xvx ‡ | 2s € l©x — 2 2§ }„ ‡ vx x w ›¨) | ˜ l  y †– …¦8› l | 7y lgs ˜ l x“y †– —%‡ x—jXš Š s%jXš šš Š §EjXšš }„ ‡ ‚Š ˆEjercicios. €ƒ 8¤„  w ƒ„ ©€5ƒ„  x –}| y”l©u 0€ )„ l©x ê @ H x l |+xx+l | 0 ‡ x x · 2 }„ ‡ u · 2|| 2((0220‡ xx‡ x ))x xHx+++· 2 x xl l| dx = · | x +C 1 (2x) 4 2− · 2 x 0‡ x @ −2 − C x +(2xxl )| · 2 „†‡ u 4x · x l 0 ‡ 4x = 4 · 2 + xl 2− (1 + 0 ‡ x1 xl | 0‡xdx = 1 x − 1 „}‡ u (1 + 0‡ x x) + 1 }„ ‡ u (1 − 0 ‡ x x) + C x) + 4 2(1 x) 4Funciones irracionales š | y7Dl u €ˆ8 „ l©$x ˜ ‡z l|‘„ y€ˆ†–d%— 0‡}– ‡ e| )R„ Vs ˜ ˜ l x§1.– l)x€w e| q• w |g0 – p˜ xl , xk˜ n+‡ 1| /1˜‰d1l¦… ,) M.€7.¤– ).=¦g,„x©l›!nx k0ˆ ›/d(kdq 1 dx ‡| „}‡ ni/di P ± R Œ —l ¥ 8 ll s ,.. ., „ © 8™› g¦ }– ‡ x = dk ˆ tM )Ejercicio. ƒ€8}„  w „¤) €5„ƒq}– | 7y Dl u €ˆ8„ · 1 · dxˆ 31+x− 41+x2.– š | y7Dl u ˆ€ 8 „ ©l x ˜ l „ y †– —%‡ s˜ ‡|R p x, yn1/d1, . . . , ynk/dk dx ˜ l }„ ‡ x ni/di P ˆ±ss = ax + b D‡V| s8§ l©x w |I€ • w g| 0 – ‡z |qˆ€ q d0}– ‡|e8„ ˜ l k+1 …) 7€ ƒ– )g¦ „ )l x ˆ y Œ —l ¥ c8x +l l ad−bc =W 0 R d„ © 8›™¦g}– ‡ s‰˜ ˜ l ‡| M = 2› D › (d1, . . . , dk ) y = tMEjercicio. €ƒ 8„} w „¤) €5„ƒq }– | 7y Dl u ˆ€ 8„ 1 ˆ1 dx õ x+1 3 x−1ö x+13€ˆd.–0–}‡|eš 8| „ y7Dl u €ˆ8„ ‡ l©xc˜‰l „ y †– —%‡ ˆ R p x, c ax2 + bx + c dx ˜ ‡ | ˜ l R ©l x5w e| h • w |g0 – ‡z | ˜ l ˜ x … )€7ƒ– )g¦ „ l©x q(a) ™C—%a©m€ y b†– € io˜ sl „ tyr€‚i–†gg| o‡Vn2› o}– m‡ ˜‰´etl rxiDlcuow s|. ˜ ‹ h| u —g€ˆ€7–†˜ › ‡ ls €f˜ l„ ‰wd„ u 2› )€‡ ˜”x l‡ cx o– u mw –pl |le”y tl aê un cuadra-do ‡ ax2 + bx + c = õ x+ b 2 4ac − b2 +, a 2a ö 4a

3.1 PRIMITIVAS 99`§ l | y ‡ |g )l Dx s 0‡ | l „©8™› g¦ –}‡ x + b = u ƒ„ q–}| 7y Dl u €ˆ8 „ x l „}„ l … 2 w ‰| ‡ ˜‰l „†‡ x€y –†—%‡ x 2a j £šŠ cR p u, A2 − u2 dx : u = xl | t. q A j  £ šš Š cR p u, A2 + u2 dx : u = ”y u t, h‡ g¦ – l | Œ¥ q A u = A t. j £ š šš Š cR p u, u2 − A2 dx : u= D‡Ax t , h‡ g¦ – l | u = A ƒ ¥ t. w‰™l xl¤ €ˆd0 –}‡ |e8 }„ –}Ú)) E| 0 ‡|4}„ ‡ x q x –}| ˜ –¤©  ˜ ‡ x ˆ © 8 ™› g¦ –}‡ E| y €ˆˆ€© d€h0 –}„ƒ‡4|e–}8| „}–7y†Ú8l )”€ |—‰€ l©0l xYy yd 0x– E‡ x u©—le8zr›™u .ˆˆg¦ V–}©Œã'‡ lˆDx s w‰w‰©l lq˜ l!y 8 lq› g| ¦g0 ‡– lz| ‡z –}|| ”y ul©lu ©l0lz xDy 7€ds ¤– l©|(¤ bw‰)hl ˜ C„ Í aV‡m0ó €73 –bu ï il osIA| s´ˆH³ çd7­©l eŠ²XxYªçÒsy — ¤ ‡0€ ›)„¼ Uñ 2Ç1 j‹ £šŠ Œ – a > s c· ax2 + bx + c = s x a + t . 0 j ‹ £ š š Š Œ – c > s c· ax2 + bx + c = tx s c . 0 j ‹ £ šš š Š Œ– § s § sax2 + bx + c = a(x − α)(x − β) a<0 c<0 c ax2 + bx + c = (x − α) t .(yl0j –ªc„d}–0¢ %—‡à)„}|e‡–†h¡ ‰¦Dx8N )€7Is}„£ˆ†–‡ o§0¢8Ú àh8t˜ V Dilc‡–¦ãz‡i„z›ŠWa| ‡¨˜sEô ˜l©3 ˆ€dx‘lžïd)e0€”j©ll–}€7Yx‡3‡yme|ñ}„ „)x2)‡´ze„}'}–t†–Ú)w |oˆ)|”yd€™©l l©ou€©xYl©0€7y)xd˜wgl„ l©e˜| xD–y}ys‡¨A©†– %—€›„‡Vlb‡€7›„‡eˆ– Yuxl‡y.Œ–}xv€ˆ|el8 8x‰‹|—„V‡zs˜€„}„wg‡›s‡›¨©l ˜wl €|ˆe‡V7yz8›2l…›x›l ‡ lz | ylx”Yx”y–¤ˆ€wy0l))s‡ |w›|) |Iy‰|‰l0€z8z)‡Vy)l |‡„€˜ {z l‰w˜yw ¤–lD‡00xu s‡–†êg¦‡ s0„‡©lx’ ›h‡Vx gˆ¦‰g—™E €e „lˆ0l0y–ƒ„ ‡lth€sI¦y0€ l‡©l |x£„ €–}| l j t £ š Š Œ– l©xvw `| —%‡}„ –}|‰‡›„–}‡ l| s l | y ‡ g|  l©x P(x) x · P(x) cdx = q(x) ax2 + bx + c + λ · dx ,w˜ ax2 + bx + c l €‚Vu 8›„w| )}–y7|e„la˜ˆx˜ 2‹˜‡ +xY!x )y h|˜by”tlx0l ‡7€+u|–}‡Vˆ€ Yx c)€y ˜ˆ8 ‡|„}‡ ˜0˜ ‡ l eÛq›0(–xll )‰|| 7y‡ ©l©lx x!Rx ¤ w˜ w‰P‡|l l r ql—%(„ˆ‡xu†„‘)}– €ˆ|‰e|‰s ‡x”˜q„›l‡ }– –}|˜ r‡˜ y7lDllDy”0ulP‡ˆ€€‚|˜(8›„x„ }–)0˜e| ‡sl8l§„| l ©l |‚€ vx –ª7y¦… w©l 8x|e| u–}˜ | „‡D˜ ‡ ƒ„Dl |qy”xYl –y ”y wl¦s|e§ ‡‰| x |– šš Š ˜l Œl m Ûe0– λ˜ jt £ y †– %— ‡ · dx (x − α)m ax2 + bx + cxl € l©˜ w  l ) „ y –†—%‡!8 | 7y l 7€ }– V‡ 5€ D ‡| l „ © 8 ™› g¦ }– ‡ ˆx − α = 1 t

100 CA´ LCULO INTEGRALjt £ šš š Š Œ l  m P r ˆ  q }– | 7y Dl u €ˆ8 „ dxxl (ax2 + bx +l c)(2m+1)/2 Abel €ˆd 0 }– ‡e| 8†„ –}Ú)„0‡| „ cambio de ct = p ax2 + bx + c V = 2ax + b . · q 2 ax2 + bx + cj t £ šu¥ Š Œ l  m ’P r ˆ  q }– | 7y lDu €ˆ8 „ x c dx (x2 + λ)m αx2 + βx l ˆ€ d0–}‡e| 8†„ –}Ú)„0‡ | l „ ) ) ›q¦g}– ‡ c αx2 + β = t ¿ˆ áÓ¤„ u–}| 7y Dl u €ˆ8„ dx c(x2 + λ)m αx2 + βx l ˆ€ d0 –}‡e| 8„†}– )Ú „0 ‡| l „)) q› ¦g–}‡ ˜ lt ¦ l „ p c αx2 + ⠈V = t qE© 8 j›qer¦gc–}‡icxRiol©Yxs”y.ew ˜ €ƒ –ƒ) ˜„¤ ‡ w x „¤ê)€u„ƒ xqx – u w – l | y7©l x }– | ”y lDu €ˆ8 „ ©l Dx s ˜ 4‡ › z qx ‰˜ l2w |‰‡ ˜‰l „}‡ x ‰— 7€ V‡ ¦I8 | c1.– x2 + 4x + 3 dx 2.– · x dx x2 + 4x + 5 3.– · x dx 4.– · dx (x − 2) −x2 + 4x − 3 −x2 + 6x − 5 5.– · x + 3 dx 6.– · x3 − x − 1 dx 4x2 + 4x − 3 x2 + 2x + 2 7.– dx 8.– dx · dx (2x2 − x + 2)7/2 (x − 1)3 x2 − 2x − 1 9.– x+1 dx 10.– (16 − 9x2)3/2 x6 dx · (x2 + 1)2 x2 + 2