Energy Engineering - Analisi e geometria 1

Esercitazioni e tde divisi per argomento

Complete course

Analisi 1 Esercitazioni NU MERI COMPLESSI 1. S - SVOCTI ) Ziz ' t 3 It Si = O zilatib ) Zt 31A - i b) tsi -- o Z zila ' - btzaib ) t za - zibtsi = o Z . zazi - zib - a ab t za - 3ib t si -- o - a abt za t itza ' - 2b ' - zbts ) { - Gab t 32=0 213 - a b) =o 2=0 v b= 314 za ' - 2b ' - 3bt 5=0 1 Cason : 2=0 → 25+3 b - 5=0 1=9 - G . 21-51=49 x , ,z= - 3¥71 \ - 5/2 CASO 2 : 6=3/4 → Imp = > z= i v Z =- Sz i 1. 6 SVOLTI ) 3Z t if = Z 3Z t rVJ - Z I Zts ) = o Z t S Zt S D : z t - S ZZ tires - ZZ - Sz = o ZZ t Zz - if = o D= G - c , fits ) == Gt i 453 -- =L , let its ) Zn ,z = -zE✓4lrtir# =- It ✓ rt if 2 - CONSIDERO : ✓ 1 ti VJ → f = 2 T = arcos I 2 & = IT { D= arcsnrz J →= ( stirs ) " I TE / cos (E) t isnt E) ) Zip =- a ± FL / cos ( Eg ) t isin I E ) ) f cost a = TE . cos = Fr . Fad = Az ( ps , no = b = rz . sin = Fez / W ' = / cos Iz t i sin ) w = ( cos ( Et2 ) t isin f Etj ) ) =-- f cos It isnt "÷ It == cos ( tht ) t is , nfthtg4# ) K = o : w = Tze t 21 i K = t : w =- Fez t Iz i K = 2 : W =- i "¥ E -- Ez + Ii - i = Ez - Ii 2-4 = 16 ( cos ZIZI t i sin ZIZI ) Z = 2 ( cos /¨ s t isin ' ) Z = 2 / cos t [email protected] -_ O → GI / k -_ n → / K -- 2 → 7¥ / K =3 → 10Gt = Sgt ZI 2 ( Eet if ) V Z = 2 f I ti E ) V Z z 5=2 f- Ez - Ii ) v za -- z f I - if ) f = 2 2 •• 4 3 • • 4 2) Z 4 - 2-2+1=0 D= a - c , =-3 Z' = ytzrzi / I + ¥ i { Retz ) > o ' I - Ei PIETZ ,= , g , at COSI = q ai - cost /¨ atcsn I arcs ,n - I 3 Z 2 ⑦ ZZ = cos ZI t rising → Z = cos ¥t2 + isin Estate Z z - cos + i sin tht 6 G K = o : Z = VZ + i Z k=r : z =- Ts z - 21 i SIT t 2 KIT SIT t ZKIT ② ZZ = cos SI t i Sins → Z = cos 3-2 t ish 33 3 2 Z = COS ( t KIT ) t ish ( Sgt t kit ) k -- o : Z =- TZ + Ii K -_ i : Z = Fda - Ii GET •• ITA y. IT •• MI 6 g b) I Z I = I → CIRCO NFERENZA C ( 0,0 ) ,t= 1/2 2 { Re / z - ¥ ) -- o Re ( z - Iz ) = o → Z =o lxtiy# so xtiy x?yZtzxyn = o I E - y 't zxyi - r ) ( x - iy ) = o x t iy ( xtiy ) ( x- iy ) x 3 - i Ey - xy ' tiny ' t 2×24 it zxy - x try = o x 't y2 × ' - XYZ tzxy - x til y ' - My +2×24 t 4) = o x 2 t 42 fret 1×3 - x y ' t 2×4 - X = 0×2 t 42 E SERC 121 ALI PPI : ( LEZI ONE 5 - 6 ) EX 3) Zo = 1 t Fyi 2 28 i ) CAL COLA Zo Zo - I + rzi e=V¥%= a 0 = Is Zo = 1 ( Cos Iz t rising ) Zfs = a ( cos 28 . Is t isin 28 ) = cos ( 24¥ t It ) t ish t GSI ) ==- z . Eai in ) A = / WE a I o E Re ( w ) Er , O E In Iw ) El } % B { WE E I W "= w . Zo , con W E A } ? RO - AZIONE 600 . w '= w . Zo = w . ( Iz + Tz i) = w ( cost t is == f w ( cos ( Iz t Ow ) t isin ( Iz t Ow ) Ex 6 ) I z - a ) 3 =- Si z - 1 = w w 's =- Sri = 8 / cos 3zIt isin ZI ) w - 2 / cos t ish ) = Zfcosftzt Zest ) t ish f Iztzezt ) ) Wo = Zi → Zo .- t t Zi Wn = 2 ( cos 7 tish 7ft ) -- z Bz - zit =- Fs - i → Za = r - Fg - i Wz .- 2 / cos ¥6 t isin aft ) = 2 . Ff - Zi . X → 2-2=1+55 - i • Zo Zy •• Zz Ext ) ( Z - 1 ) " = It - if ' z - 1 = w w4 = ( r - i ) " → Wo = 1 - i f = T2 at cos I 2 = > 7T § arcs .n - raze I Wo = VI ( cos 7aI t i sin 7, 1 ) → Zo = 2- i "¥ LE AARE ROTATE DI Iz ( III t Iz == II = / II t Iz = ¥ / Sgt ) Wn = VZ ( cos GI t isin AI ) = r ti =3 Zn = 2 ti Wz = FL ( cos 3ft ti sin 3ft ) =- n ti =3 Zz = i was = TE ( cos Sgt t ish Sgt ) =- n - i = > 23 =- i Ex 9 ) I Z - 3 - zip =- 8 Z - 3- 3in = w W3= - 8 = > Wo =- 2 = 2 / Cos IT tight ) Wr = 2 / Coslett ) t isinftt ) = 2 / cos St t ish ) == z / I - i rz ) = i - rsi > Tty Wz = 2 ( cos + ish 7¥ ) = 2 / Iz tire ) = a + Bi Zo -- a t 3 's Zi = c , ti ( 3 - Fs ) Zz = c , ti ( 3+53 ) ⑥•.→ +3 T tzi •• • • ooo ie Ext ) f : E → E tic . ffz ) = e 2. z tr ti i ) DETERMINATE I PUNTI FISSI ( f ( Zo ) = Zo ) ei±z = i = > iz tr ti = z z I i - a ) =- a - i z = -rfi = -ii = i i - a ( ite ) - n - r in ) a = { ZE El o Erez El , o E Int El } { flat n > fld )= i @ titi L I M IT I 3.130 ' Eis t.EE n EE .E¥= I = olim e " t 2e " = o x. s-oo e- × t 3eZ× - = too too O - . 3. 131 ) him 1081×1 t I = too line loS2× = ox -s too X x -s- A - a -0 3.1341 III. I I -- firs e " h = lime ' =x→too y =E÷se*¥ - ' k =¥÷se /¨ ¥¥k=¥÷e " = he 3.135 ) .EE/xf ftp://t/--E.i.mxe1hiI#---.E.mxeF-ltI*-4= ¥÷×e¥¥x= = = line × e 1 tu = o x-so y - e 3.139 ) Ess 1) ' ' =¥÷ftf " D " -5¥ , x .= Escott ' ' = E://it.IE//-xC3x - 3) = line e ¥ = e3 X→too + - too 3. 140 ) him e " 108 × = o x - sot logfytx ) t e " " e- To 3. 141 ) Lion sinx . sin I logx ) . log ( sine ) = o T o¥AtRA o Et . - o QUAN 'll- A 'FINI TA Post TNA 3. 142 ) Limo ✓cos = him ✓ cost - t = X Slnx ×-so +2 -= him %-sn¥# = -snIz =- Yo ,x-so 3. 1441 lim ( a - cos × , ) sin , 213€ . # =-= Z ×-so +3 tznzx 2¥ " " as ' III. ' '' ' ' 081 I ' ) = Eis ' ' ' I ) - =E÷ . ' if :÷ ) n Ees ' ¥ " Is - 7 " at ' Eas f Ese " h =¥se " - Y .-- E.g e " '= Es e " '= .es's e " ' # = Ye s = to 3.150 ) him 2 " t lob I rt x ' ) ~ him to =x-s.a log 1×1 ×→-° log 1×1 IN = him Y§ = Eyes 21092--2 ×-s-N loft 3. 154 ) Limo I t I sin ¥ ) I and ' ) -- fig ¥ . sin ¥ . E = o 3. rss ) Egg ( Ye t - x " ) -- Ez x " ( ' VII - e) = ¥÷ If . x % = 3 x % =a 3. est ) him 1082×-3108 x × →°" log In tx ) + In + dog # z= EYE 109% - 310 Sx ~ loglrtxltnt 4109kt 410g x line lost i x- sot o × > - y ii. ÷ ! + -- t -- t t -- t - t n u n u • ¥ 4. 166 ) flxl = e- " ( x 't zz× ) • D : IR • ASSI : X = O → 4=0 4=0 → e- " = o Fx V x ( x 't 221=0 O x 't 22×30 × ( x 't I ) 30 f y , o /¨ s x > o • PARI : fcx ) = ft - x ) no Dispart : f I x ) =- fl - x ) →- ( e- " f - x ' - Ze )) = e- 1×3+3×1 = > DISPART • Ll Mitt : EE e- " I x 't ÷ ! - Eis -- o = > 4=0 A SIN TOTO 0121220N TA L E • DERIV ATA PRIMA : f ' I x ) = e- ×? ( - 2x ) . ( x 's t Zz × ) t ( 3×2+32 ) e- × "== e- × " f - 2x . ( x 't Zz × ) t 3×2 t Zz ) == e - × Z ( - 2×9 - 3×4+3×12 t ÷ ) = e - × Z ( - 2×4 t Zz ) • DD ': Fx • MAX - Min : f ' ( x ) so e- XZ > o → Fx - 2×9 t Zz 3 o 2×4 E Zz × 9 E I 4 Edu AZIONE ASSOCIATA : X = =3 - µ } E X E µ§ at - t -"¥ nw T meh • DERIVATA SECONDA : f " ( X ) = e- × " ( - 2×4+32 ) f "cxl= e- " I - zxlfzx 't z ) t f - 8×3 ) / e- " / == e- " ( axs . 3×-8×3 ) = e - × ? x I ax ' - 8×2-3 ) • FLEX : e ' × " 30 Tx n X 70 A 4×4 - 8×2 - 3 > o 1=64 - g. a f- 3) = 457 X ? ,z= 8F = zt / 2tzI 8 2 - 2¥ =) IMP × I - 2µF v X 32*57 × 30 - ? ? t ? + -- t -- t t - t - t n u n u • A- •o ; V. . 4. 167 ) fcx ) = × e - 1×3 - it • D : IR • ASSI : X -- O → 4--0=7 ZERI : ( 0,0 ) 4=0 → x = O Be SEGNO : O X 3 O o- 1×3-11 =) - t f ( x ) > o/¨ > × > o e > , o Tx • PARI ! f ( x ) = fl - x) NO PARI DISPAR ,: flat =- ft . × ) - f- × e - l - × ' - 111 = × e -l- is -11 = > NO DISPART • ASI NTOTI :- 1×3-11 / too f G) → o him xe = him x - soo x - so ek 't ' - - A flx ) - so = > 4=0 ASINTOTO VERTICAL E - x3t1 xe x > a 1 •• Deriv ATA PRIMA : f ( x ) { +3 - a Xe x Ly 2 r f 'c×1= e- " " + e- " t ' f - 3×4 x = e- " t ' ly - 3×3 ) 2 f ' ( x ) = e " - " t e " " I 3×4 x = e " " let 3×3 ) • MAX - MIN : I /¨ → NON ACC . l l - 3×3 30 x } E I x E - • PERCHE Lt 3 Tz t - i I MAX J - I ② r t 3×330 x " 3 - Iz x 3 - Tz ? - t b a MIN • DERIV ATA SECONDA ! ① fix , = e- " " + e- " t ' f - 3×4 x = e- " t ' Ii - 3×3 ) ① f ' ( x ) = e " - t t e " " 13×4 x = e " " let 3×3 ) f' ' Ix ) = e- × " " I -3×44-3×7 t l - 9×41 e- " " I == e - " t ' f - 3×2+9×5 - 9×2 ) = e- " t ' ( 9×5-12×2 ) == e - + " + t . 3×2/3×3 - G ) ② f " ( x ) = e " " ' 13×214+3×3 ) t 19×41 e " " I == e " - ' ( 3×2+9×5 tax ' ) = e " " " ( 9×5 + me ' ) == e " - t 3×213×3+4 ) • FLEX : r , 1 I - r ① x > ish .I- t E n u I ② " - * ÷ : ¢ STUDIO IL COM PORTAMENTO DELIA DERIV ATA IN 4=1 : him e- " t ' ly - * 3) = line e-3 I -2 =- z ×- set × → at e +3 - 1 line " " fetal = bin at 3×374=4 x - sr -+→ n - et - X > =D punto ANCOLOSO TAY L O R - MACLAURIN : TUMI DIS PARI PARI in I x ) = f to ) t I x - xoltf ' Ko ) t Iz , I x - xoIf" Ko ) t ÷ , k - xoPf" ' Got ... MACLAURIN : flo ) t x f ' ( o ) t ÷ If " lol t Iz x > f "' ( o ) ... - X 2 4. 23 ) him xe - sine + I x3 x - so 1×+2×212 loaf f it Iz ) • e×= 1 t x t I +133+01×3 ) = > E ×= 1 - x - I - tow ) zz2-X 4 e = 1 - x Zt I + 01×4 ) 2 oooo sink = x - 13 + + or ( Xs ) 6 120 t 563 ) = It x - ×3tolx3 ) 4. 305 ) fix ) = Sh I sine ) - x • sin x = x - E3 t g¥ told ) 6 • Sh I sin x ) = sinx t king ' + Signs t 01×5 ) ( × - t Is ) t Efx - It It I f x - It Is totes ) = × - ¥ 3t ÷ xs t ¥+3 - tzxs + ÷ , x st 01×51 = x -"¥ Xs E÷ ÷ us . # =- ÷ . 4. 306 ) bin zcosle " - e ) t sin ( x 't x ' ) - z zcosxtsinlxtx ' ) - z +I= bin - X -so xd x- not xd • cost = r - 12×2 t 01×2 ) • sin ( x2tx3 ) = ( x 't x ' ) t t ( x 't x3 ) ' t 01×3 ) G Num : 2 ( t - 12×2+01×4 ) t ( x 't I = z - # t t x ' t 01×3 ) ~ bin 2tx3 - 2 not I = EYE × " " 4. 179 - Bls ) linn logktxtx')-S x -so X sink • log lat t ) = t - It ' + It ' to It ' ) ( • sine =x- 1×3 t 01×3 ) 6 > : txt x ' ) - ÷ ( xtx 'T t Izlxtx 'T t Ollxtx 't ') ==x t x '- z x'-t 31×3 tt 01×31 == x - I + Z t 13×3 to ( X ' ) nvm : x - 12×2 t 13×3 - ( x - Ix ' ) = X - Ix 't x ' - H t { x' ==- Iz x 't=- Iz x - t Zz x 's tox 't =- 12×2+01×4 bin - Ext -=- I x-so µ Z G. 237 ) f- ( x ) = log ( t - x ' ) - zcosx • toy little t - It ' t ol -14 • cosx -- t - 21×2 t ÷ . x ' to ( xh ) Porvoo : Y - x2 = At t = > t =- +2 f- ( x ) =- x ' - 21×4 - z ( n - 12×2 t tax ' t 01×4 ) ==- €2 - Iz x " - z t - I × " t 01×4 =- z - tzxhtolxh ) = > f ( x ) =- 2- In x " t ol x " ) pueoose fix ) - too , > Go f- ( x ) - f ( o ) =- z - ÷ x " t z =- ÷ x " fl 3) = arctanrs -- I F- (3) = ! ) atctanrtdt 3 fltl-ai-cta.tt 8 ' It ) -- i f'(tl=÷.#gft# PER PARTI = ltarctanrtlo - ! ) It .tt . dt-lta.ctantl.no/Itdt= Z Z ft ) = ft = K t = K ' dt - 2K dk 53 2 I • it K ' =/ / ) - ft - IE dk = vs = ( KIF - ( ¥2 die = ( k ] ? - latctanklo = Fs - I 3 O= > F( 3) = 3atctanT3 - B + Iz = Itt Ig - Fs = Giz - B 9 = 4 - MX = GZI - Fs - Iz . 3 = Iz - B NS 4= Iz x + ZI - Fs 77 Fix ) -_ " of loglatctantldt F ' ( Iz )= ? F ' ( x ) = log ( atctantsx ) - 3 = > F' (3) =3 . log ( II ) 8) Fix )= " of It - e) arctantdt ¥÷gH = ? + →too ( t - a) arctant ~ t → to , t -*a =D FORMA DI INDECISION E ( & ] =3 ✓ SO DE C' HOPITAL ¥÷s rests -×a¥ = ORDINE DI INFINITO ? 9. n ) F Cx ) = ! ) tartan ft - Mdt SE COME EST REMO DI INTECDAZIONE AV E S S I AV U TO " 3x "•• studio f ( x ) = X Itc tan ( x- t ) ✓ LA Funzlonf RNTEORANDA DA STUDIA RE SAREBBE STATA ( sxarctanfsx - a) 13 ? ~ D: - Iz ex< I 2~ ASSI :X=o→ 4=0 n , 4=0 →x--ov arctanlx - e) = o ( x- a) = tano x-- t => ZERL : ( o, O ) ; ( 1,0 ) ~ SEGNO : x > o f( x ) 70 '- X EO V X 31 arctanfx - n ) > o x > y ~ unit : him xarctanx =too line f( e) = too x-s- I x→tI ZZ II fit ) -- tartan ft - e ) Zllll-ll• DOMINI O F ( x ) :~ CRITERIO ASINTOTICO PER t → I t - Ig = K k -50 ZK-70 ( k t Iz ) drctan ( K - Iz ) ~ Iz . ( K - Iz ) →- IT , K - so ( CONVERGE ) ~ CRITERIO ASINTOTICO PER t -7- Iz t t Ig = k k → O K -so fk - Iz ) arctan ( K - 3¥ ) ~ - Iz ( K - 3Iz ) → 3ft ', K -30 ( CONVERGE ) =3 D : IR • LIMIT l F ( x ) : Lying ! ) t arc tan ( t - 1 ) dt Tipo A : Funzionf LIMITATA SU INTERVAL to KLIMITATO =¥÷ .io/tarctanH-aldt-- - o J tatctznft - n ) ° - Iz - t =→ D= - y =3 DIVERGE limtoftatctanlt - rldt -too ) x→too tarctanft - e) n Ift → too , t -star• DERIV ATA PRIMA : F' ( x ) = xatctan ( x -r ) 00 MAX - MIN : ? ? t -- x -O Min X=1 MAX - t - b 9 b •• DERIV ATA Zd '- F " ( x ) = drctan ( x- r ) + X1 t ( x- at OSSERVO CHE se x > 4 F' ' ( x ) > O E CHE SE x c y f " ( x ) co = > x = 1 FLEX × Fix ) ) 'VE et O • DI SEGNO f ( t ) :~ D : IR ~ Assi : t=o → 4=-1 / 4--0 → t -- 1 = > zeri : ( o,- r) ; ( 1,0 ) ~ SEGNO : f ( x ) 30 t > y ~ unit : Liya f- ( t ) - o Iff f ft ) -- too ^I #¥ixT • D F ( x ) : IR •• LIM ITI Di F ( x ) : tying § Et d t TIPO A : FUNZIONE UMI TATA SU INTERVAL to 4th TAT O fi'm ✓ LI et = t D = > AN CHE c' INTE ALE DIVERGE t →too fine FEI et n bin VI . et = O =3 won so ANCORA SE DWEROE o f- →-oo t -'-A CONVERGE -D-O bin V3t = him t '. 'VE et = o t -s-o L t -s-a t2 O N TF ' et = J ( ft ) =) SAPPIAMO CHE 1- CONVERGE , Luna ANCHE t2 3ft . et CONVERGE BRAM ANTI :too 6. nsg ) Jatta dx a > o o Xd ° CONFRONT O ASWTOTICO inx= Ot i dtcth ~ I = I xd xd xx -y se d - 1 > 1 3 2 32 DIVERGE { se d L 2 =3 CONVERGE -CON FRONTO INX= too : ducted ~ Iz xd xd se h > 4 CONVERGE { se d C 1 DIVERGE =) CONVERGE : 4 L L L 2 too 6. isa ) f de O CASO -1 : d > O O-D - • CRITE RIO ASWTOTICO IN X =O Xd/OLX_ ~ X&lOYx_ = 105×2 2- q T t XZ x x - N -in TECRALF N se 2- a > o → 222 t0 = ÷p logx →-00 DIVERGE o P B = z - L X CONVERGE -O se f > 1 = 2 2- a > 1 2 21 =3 DIVERGE { Sl L S t =3 CONVERGE OOO too - Vx 6. 160 ) f. e- dx a > o o xd ( rt xd ) J - Vx - Vx • CONFRONT O ASINTOTICO IN X =O! # ~ # xd ( rt xd ) xx - O → O , X → O = > se f CONVERGE ANCHE il SUO INTEGRATE CONVERGE •CON FRONTO A- SINT TICOIN X =tD:- Mx E t d > 1/2 CONVERSE - - Xd ( t t xd ) XK L < 1/2 DIVERGE y =3 L > 1/2 CONVERGENT 6. 161 ) / dx O 109×1 ° CRITERIO ASINTOTICO X =D : COSI N -→ O , X -30 logx lost =3 SE f CONVERGE in X =D , ALLORA ANCHE IL SUO INTEGRATE CONVERGE • CRITERIO ASINTOTICO 4=1: cost ( x - y = t t →o ) lose costly ~ cos(tt# ~ cost 2=1 = > DIVERGE loglttn ) t t too- X 6. 152 ) ) #,zd× e IN X=0 CONVERGE • CON FRONTO ASINTOTICO PER X= too " N # = ¥ =) DIVERGE 6.153 ) " of dx + 312 ( n - +11/2 • CONFRONT O ASINTOTICO IN X=0 :~ # = + 3/2 ( e - +142 × 3/2 1 =- 2=7/2 L l =3 CONVERGE + 7/2 • CON FRONTO AS INTO TICO IN X = I # ( r - +11/2 & = ÷ L l =) CONVERGE and e- ÷ dx • IN X=0 i CONVERGE - e x• CRITERIO A- Sent TICO X =too" e N I CONVERGE Z t x " x 's 6. ios ) ! ) , dx x ✓ r - x -• CRITERIO ASINTOTICO X =- y i -15€ L = § L l =3 CONVERGE # 1- V' r - xz • CRITE 1210 AS NTOTICO X= 1 : # 2 = § Ly = > CONVERGE e - ✓ r - x2 °CRI TERIO ASINTOTICO X=O: # =3 CONVERGE ¢ 1 • 6.166 ) ¥ dx -- e x Fr - xz • CONFRONT O ASINTOTICO X =- 1 : # = > CONVERGE it .°CON FRONTO ASINTOTICO X =L =) CONVERGE • CONFRONTED ASINTOTICO :X=D I = > DIVERGE ÷ 2T ( Iz - x ) ' 6. 168 ) o ) - dx ( OSX oX= 0 CONVERGE • x = 21T f ( x ) N -0 = 0 = > CONVERGE 1 . D : x = Iz n 3zI • CRITERIO ASINTOTICO X = Iz : ( X- LI = K , K →O ) ( Iz - Ez - K ) " x -= # → O , K - so cost tk ) - K - CRITERIO ASINTOTICO X= 31 : ( X - 321 =K, K -50 ) Z Z ( Iz - ZI - K ) , f - t - K ) " Ez -- N - =) DIVERGE COS ( Kt 3zI ) sink K EQUAZIONI DIFFERENZIALI : 1) A VA R I A B I L I SEPARABILI : y ' = acx ) bly ) T . CONTROL LO SE b ( 4) = O HA SOLVZIONE = > SOLUZIONI SINGOLARI 2 . SCRIVO y '= dy dx 3 . ¥ , dy = alxldx → 1¥ , dy -- falxldx 2) LINE ARI DEL PRIMO ORDINE : y '= achy t b ( x ) 1 . INTEGRA LE PARTI CARE : b ( x ) =O Z . INTEORALE GENERALE : y = e # × ) ) e - A " ) b( × ) d× S I ANESI - EQUAZIONI DIFFERENZIALI e) fu '-- u ! • II = y ' ft dy = fade 410 ' ' Gy - z dy = fi dx - 1y=x to = > c =- y •• INTERVAL LO MASSI MALE :- 1g = X - t y = 1- r -x 4 # C-oo, y ) Mxn z ) y '= In - rille - ul 1¥ = In - xlle - y ) - f dy = ft - x dx - In le - yl =x- 21×2 + c Intl - y ) =-xt Ix '-c Ii -x i - y =- Ce -++ I " y = et Ce 3) { 44 '-- y y '-- Ia dad = In fydy = fade 4 ( o ) =- Iz 42 = X t C 42 = 2x t 2C y =t ✓ 2x tc G = 2C C = z y = Vzxtg D : 2 ( Xt 2) 3 O X 3 - 2 - of me ( - 2 ,too ) ,nTERvAuo MASSI MALE G) y 't ÷y=o y '-_- ÷ . f- Judy =- f ¥ dx Iz y ' =- lnlxltc y - =- zlnlxl t c y ! - thx to y=±Vlnf 5) y ' =- 2×42 { uco ,= , ¥4 =- zfxdx - hey =- 2.21×2 + C YI = x 't C 4=1×2 t C - t = ! = > c= - t y = -1×2 - y 6) y '-_ ay t b D : IR - fol Y ~ VA R I A B I U S E PA R A B I U : dd¥ = 1/24 + b ) §gy#dy=§y DX O 1a Inlay tbl - f- Intact b) =x aytbl-a.tt/teaty=ctaIed-aCxlbCxl - m→ y=eA " { e- A " but de ~ y '= ( aly t b = 7) T ' It )= K ( E - Ttt ) ) K > o { t lol = To T ' ( t ) =( KE - ktft ) ) / DT =/ , de ¥ '- ( KE - * tally - I dt = t t c - ÷ Inte - tltll -- ttc - K ( ttc ) - 1kt -1 Kc ) -_ In IE - THI E - Ttt ) - e - Kt T ( t ) = E - Ce t To = E - C = > C = E - To - Kt - Kt ~ > Tc t ) = E - ( E - To ) e = E t ( To - E) e him Et ( to - E) e- Kt = E 1-→to 8) / ! ! ,?I# 4 '-- Elita " II -- Ilitch ' J dy =/ ÷ dx - I = Intel to r t y - I = 1 t y y =- 1- - 1 thx to lnxtc 2 =- -1 - y 3 =- I =3 C =- Iz C y =- 1- - y = y =- -3 - l thx - I 31N - 1 3 31 nx - r to × + Ser Tr e × . . lo , 're ) - to X - 10X 9) y 't to y = e y '= e - toy { y ( o ) = o "¥ y '=- no - y t e- " °" VA R - SEPARABIL ,w- 2K ) bcx ) - 10X y ) e =o= > Ix z Al x ) = f - no dx =- nox yo Cx ) -- ce " " t e " " / e- A " ' . but dx == ce " "t e- "× JEY . Y '" dx = ce " " t e- " ". × - 10X ~ > CAUCHY : O = C = > fi ) = x e 2X 2X 2X to ) y t y '= e y '= e - y = l - y ) y t e -- { y (e) = o an beet A- ( x ) = f - n de =- x fo Cx ) = ce " t e - ×1/3e× . e " de = 3 = Ce "t 31 e- ×- e " = Ce -× t Iz e " A CAUCHY : C t 13 = 1 C = Zz = > fix ) = Zz e " t Iz e " = 13 ( Le -× t e " ) = 31 ( 2¥ " ) rn ) y ' = zatctanx - ¥ y y '= f- 1) y t zarctanx { yin =3 - - ah ) bi ) A- ( x ) =- f ¥ dx =- Intel you ) ; cetn " + e- 'n' ''' f e ' " " . arctanx dx = C - ¥ , t j f txt arctanx dx • CONSIDER OL' INTEGRA LE :"¥ PER PARTI ( TOCOOIC MODULO VI STO CHE CAUCHY CONSIDER A X=1 =) POSIT 'VA ) f- ( x ) = atctanx 8 ' ( x ) = 1×1 f' ( x ) = 1- Stx ) = +2 ht XZ 2) lxlatctanxdx = I arctanx - If dx z Z y t XZ Z x - ¥2 = > p ( x ) = a KID ( x) t R =sP = 01×1+121×7 - n D ( x ) Dk ) Z = t - 11 t XZ Ttx 2) lxlatctanxdx = I arctanx - If dx = arctanx - Iz ( x - arctanx ) -2 Z z n t x= 12×2 . atctanx - Z x t z a- ctanx == Iz ( a- ctanx -x ' -xt a- ctanx ) = Iz ( atctanx ( it n ) - e) '- c O y ' - f- I ) y t I 2x tr ) - - b Cx ) aw Atx ) =- 2 I ¥ dx =- zlnx you ) = c e- " " t e- " " / e " " . ( zxtn ) de = ! /¨ t 1×21×212×+1 dx = Iz t I / ' t /= = Iz t E t Es b ) EYE It Et I = toooo ~ > LA FVNZIONE E UMI TATA SOW PER C=O y 't 3×244--0 y '=- 3×2 y ' " ' ( fu , -- o • SOLVZIONE Dl FRONTERA : y =O 20 ) y = f ( X ) ttoa fcx ) in ( X, f ( x )) PASSANTE PER C- x,O ) y '= f → f Cx ) = y ' 2x 2X y = y ' 2x y '= I y II = ¥ 4 2X f f- dy = I . f ? dx Intuit = I f lnlxl t c) I yl = VI. Vc y = IVI 22 ) y ' =/ - ¥14 t s'?n - - acxl blx ) Atx ) = -21¥ =- zlnlxl to → you ,=ce- " " ' + e- " "" / e " " " . sing dx == ÷ t ÷ f H . dx f sinax dx ax -- t x-- ft de = ? dt "¥ ÷ f sent dt =- coast = -cos# G 40 = Iz t t .- Coye = Iz ( C - I cosax ) 23 ) y '= y tax th { 41*7=1 Atx ) =- I - scions dx =- lnlcosxltc you , = c e- Into "' , e- lnlwsxl / ehlwsxl , y y , ,/¨ it ,w ÷ , / lost d '= ÷× , 't ÷ Is .ru/-sxl-- ÷ , ( Ct lsinxl ) - a / c ) = a tog y = 2x t 2C logy = 7/2×+27 ✓ 2×+27 VI y = e 2 = e VI = In 2 zc = 1h22 C = Iz 1h22 ✓zx =3 y = e •• 2x t 1h22 70 X 3 - ÷ 1h22 ( - ÷ 1h22 ,too ) ✓zx •• y = E APPROSSIMOLAFUNZIONECONMAC.LA# ~SO CHE 4 ( O ) = 2 - y = it Vzxt 1h22 t 2xtln22 t 01×2 ) 2 % OOH u '= ¥ + Iet = I 4 t f- e at - b Cx ) i ) risowo : Atx ) = / ¥ dx = Intel to yo = ceh " '+ eh " ' Ie " " " . ¥ e¥ dx = cult 1×1 ) . 1. e%dx= =a+x I ÷ , e¥ de ¥ = t x = +1 dx= - f ,= at x / # . et . f - ¥2 , dt = a -× let dt == ex -x. e¥ = x I c - e " ) ii ) LiniTEFinito'- him X ( C - e ¥ ) = > c = y x→too to T him × ( i - et ) = Isis x ( e - i - ¥ t ok ) = a told = 1 x-0too to -O → c=y iii ) yall ) = o a / c - e) = o =3 C -- e → ya = x I e - e " ) iii i ) 42111=0 • COM PORTAMENTO ASINTOTICO A X→too: yz ( × ) ~ X ( e - t ) y , = x I e - e " ) Diseono :. D : X F O-° Assi : 4=0 → x = y • SE C. no:x> O e ' " c e 1a a a x > a =3 Yz ( x ) so : x co v x > y . :÷÷:÷÷÷ : z⑧) y "=- 4 t f ( x ) c. E . X > o x - ¥ = 4 sown one A l x ) =- x yo Cx ) = ce " t e- × f e " fix ) dx × - 3h = ce " t e- × ) e' ' fix ) dx × -312 = e- '' / at feifcxldx ) ( IF , ( at Je " ful dx ) I eXx"?-%}×" = ¢ fix ) x";?%× = fix ' -- x"f;4 = ×Y I = 2x - 3 ¥ ④ y '=- xatctany { yle ) -- a • 4 ' ( 1 ) =- I ~> to : y - 1 =- I , / x - n ) 4 y = y I x ) Ft = PJ xx .. " . , f¥ -°T It : y - up = y ' ( xp ) / x - xp ) y ' ( xp ) = 4-42 - X - Xp PO = ✓ ×2 t 42 EXT A- LIPPI ) y '= 4- / . . " :S . y '- alxlbcu ) { y go , = yo props na D , Cauca , y , DOMINO DI a ( x ) • a ( x ) E coli ) b ( y ) E C " ( JI Domino Dena DERIUATA Di b ( 4 ) Sl Xo E yo E I X J F ! SOLUZIONE LOCALE DEL P . D. C . ~ > NEW ' ESERC 1210 : 1) cacao D D , 2K ) → I 2) CAL Coco D Di b' ( y ) → J 3) control -0 CHE : Xo E I n yo E J • D 21×7 : IR ) I • D b' ly ) : I = bcy ) b' 141--1094-91.42=1094-11094 loopy logy • Dbly ) : 430 n x # i 10,4111 , too ) • D b' ly ) : to .nl/y,too#J CAUCHY : ( 0,2 ) o in I E 2 IN J EXZ Allen ) y '= (1t# t { y' la ) = 2 i ) Dl MOST RARE ESISTENZA E UNICITA' LOCALE :Z y = I let y ) t - act ) bly ) • D fact ) ) = I = t # o l - D , o ) ( o , too ) • Dlbcy ) ) : IR b ' ( y ) = z ( rt y ) • D ( b' ly ) ) = J = IR T E I n 2 E J =3 SOL . CAUCHY : ( 1,2 ) = > y SOTOLINEO IL FATIO CHE d( f) C- ( O,too ) ← ii ) risowo : dd÷= ¥ City ) ' ft dt = 14+452 dy - 1 y In Itt to = 4t# --= lnltltc - r Tty t y th =-- y =- I - y Inltltc lnltltc CAUCHY : 2 =- -1 - y - Tz = 3 =3 C =- 1g C ~ > y =-i Int - I 3 ii ) INTERVAL 10 MASSI MALE : In Itt - tz * o In Itt ± I 3 Itt # e } t ± I 're / Itt so → t so - Te o , 're °"""° x - rise GEOMETRIA ANALITICA " oooo A- 12,4 , O ) 1313,0 , O ) C ( O , 0,1 ) VII ( t ,- i. o ) x = 3tt " A 's :( uz : ! t = × - z n t =- y =3 TA B : { I 03-4 FASCIO DI RANI PER FB : X t y - 3 t k Z = O "¥ CERCO PIANO PASSANTE PER C :- 3 t k = O = > K =3 → Xt y t 32--3=0 i 2x t by t CZ t d=o dxtbxtcz +1=0 ~ > MEMO I 3 PUNT Plt , 1,0 ) it X x - ztr=o → Tt ( t , o ,- n ) it : a. ( x- xo ) t b ( 4 - yo ) t ( ( z - Zo ) =o it : y ( x - n ) - 1/2-1=0 x - Z - 1=0 A I 3. 0,1 ) 1312.2.2 ) IT k Fili , 4,0 ) - NII ( i ,- z ,- n ) If VII x Ti = i i k ¥1 1 - 2 - y 1 l O NT I a ,- i , 3) ORA IMPONCO PASSACCIO PER D E VERIFIED COMPLANARIETA -: O = y = > NO 1) y = t Xtttzt 1=0 htt tt +2-+1=0 \ x = att "¥ zt +2=7 x = itt Est STONO 2 CIRCONFERENZE CHE RISPECCHIANO LE CONDIZIONI : c ( -z, -3,3 ) c' ( 4 , 1 , 1) Se : ( x t z ) 't ( yts ) ' t I z - 3) '=o Sz : ( x- a ) ' t ( y - n ) ' t ( z - 15=0 EXZ ) x =3 t t x = k t : : . t t : : : . VT ( 1. 3,2 ) Fs I 1. o,- a ) i ) F IT CHE Conti ENE tis ? → PARALLEL E ( Est STE un PIANO ) :NO→ incinerates ' ' e " " An " i ! . fI÷÷÷ :/ no ~ > SGHEMBE ! it ) DETERMINATE UN PIANO CHE CON TIENE t E PASSANTE PER L' ORIGINE :Y t t :÷÷ : : It x- 2=0 1=1 - 41-21=9×1,2 = -7z±→ y "¥NON Acc . f- ' lil = ze 's = > G ' ( e3 ) = I 3e3 APP Ello 2016 ¥÷÷l - ÷t=¥÷÷s I == ITE ¥+2 ( bind '-x 2) == EYE ×¥ ( ( × - f x ' total ) ' - E) == FINE ¥2 f ¢ tf xo - 13×4 t oleo ) - ¥ ) = 2=2 =3- 1/3 = ¥yg+y¥ ( - Is x " t 049 ) ) = lim - 13×2-4 a a z = > ox- not L s 2 =S-DX= it zt t { yz I 32ft t VT ( 2 , 1 , 2) IT A t PASSANTE Pee A 12,1 ,- 1) B ( 7, 3 , 4) a) TIB In , -2 ,- z ) II = VII x VT = i ' s k = ( z , 6 , - s ) 2 1 2 y - 2 - 2 =3 IT : 2 ( x - a ) t 6 ( y - 3) - s ( z - a ) = O 2x t 64 - SZ - AS =o b) ri P# :# ✓ B A 12,1 ,- a) BG.3.nl P AURA ' LE GENERIC HE COORDINATE P ( at Zt , 3 t t , Zt ) - FB = ✓ y t c , t c ,= 3 AT = V6t-nTtlttzPtlzt { CONVERGE / FUNZIONE CHE TENDE A UN VA L . LIMITATO ) PER X=tD APPLICO IL CRITERIO ASINTOTICO :- ( e ¥ - y ) ~ - 1- ~ I 2=2 = 7 ESSEN DO > 1 C ,- XZ XZ CONVERGE APP Evo S SETI . 2016 a ) z= mum ) if = a post -- a O = arcos - Ez = y q f- arcsin Iz == > µ ( cos SGI t isin SGI )) = 1 ( cos 2¥ ti sin 2¥ ) == cos II t is 'n GI Den ) if = FL O = a '- cos FE = > y atcsn - I e,Z ESSENDO LA FVNZIONE LIM , TATA , ANCHE I'- NTECNSIE AURA ' VA I O R E Finite ( CONVERGE ) °INX-3t00 USD ILCRI TERI O ASI NTOTICO: f- ( x ) = X ( ✓ - y ) N X . 21 . ¥ 2=1 1/2 - Iz a B LO =3 too | pc - I = > o • se p > o : typo ar-ctanlzr-l.vn = ELI = to peep , co CANDO : P > ° = > + ① ) p , - 1/2 =>t D - Iz L PL O =3 too I ::::i . Z2= y - 12€ 't f- x 4 - X t 2¥ - IT -1+51×4 him -No y L x X • e - ¥ = i t ( - ) t / - ) ! 21 +01×4 •co>x= a - I x - t I X " t olxa ) 2a! Se L =L,=3 142 = line I ×"-d { se a sc,=>tooX-so12 Se L CC, =3 O → CONTINUA Xt yt Z = t IT PASSANTE PER A ( 0,0 ,o ) E B ( 111,1 ) { × - y - Z =o IT A t VA B = ( 7, 4 , y ) | ! + z= , ^ - t - Z = Z t t ZZ -- ztti z=z - t x tht t I - It --ax-- I x- t - z =o t : × = 1/2 { y -- t , Tito , a ,- a ) z = Iz - L * = ii xti .= in = 1-2,1 i ' ) IT :- 2X t y t Z =O A • b) d la ,r I = a euro . ....../¨ + ,. Zo = cost ) t isin ( - ) Zz cos I ) t ish ) Zr = cos ( Iz ) t i sin f Ig ) Zz = cos ( ti sin hz ) Z " " Zi = 2 ( cosotisino ) ( Cosi t isin Ig ) = 2 ( Costa t isnt ) ~ RVOTO Di Iz IN SENSO OR ARIO E DIME 220 It MODULO Zo = I ( cos first - Is ) t ish first - Iz ) == Iz ( cos ti sin ) ooo f ( x ) = lnx - atctanlx - a ) • D : x >o ( o ,ta ) . him lnx - atctanlx - a ) =-aX- got him lnx - atctanlx - i) =tasX→too be'm thx - arctan ( x-n ) no. AS 0134600 *to I ~ Etfs ¥ =o° f ' I x ) = ¥ - a÷q , = I - -1 = I × +2 - 2x t 2 x ( XZ - 2x +2 ) Do ': x to • MAX I MIN:xSOOI f ×- It a sie 1/1 . em :. ! ? ! atctan " - i .,,,,,,,, segno , # f- I x ) so a flx ) co/¨s lnxcatctanlx -r ) = > ocean 1C XC A fl x ) =o/¨sx =L + ! ) e×÷ d× pure , we um,, a su www.nuo ,,,,,y, ,a ,,•rn x → Ot f ( x) → O =3 ANCHE lesuo INTEGRATE CONVERGE • CRITE RIO ASINTOTICO PER x→tD: Z f. ( x ) = I → O g x →t D MI won SO SE CONVERGE 0 DIVERGE B e ' • CASO B > 0 SO CHE f- ( x ) = Of ¥ ) INFATI : fluff ef¥ . ¥y = O l 2=2 =3 CONVERGE - =3 CONVERGE ANCHE c- INTEGRATE XZ •• CASO p = O : f ( x ) n I → too , x -7 too e =) SE LA FVNZIONE ( Llnl TATA ) DIVERGE AN CHE le SUO IN TEENIE DIVERGE • CASO BC O :→ too = > DIVERGE + B e b ) f e- x. x ' dx fix ) -- x2 84×1 -_ e- × f' ( x ) -- 2x 81×1 =- e- ×=- x' e- × tzfxe " dx fix ) -- x g' Cx ) = e- × f' Cx ) -- r Glx ) =- e- ×=- Ee " t 2 f- + e- + t ( e " de ) =- x' e- ×- we "- ze " = I e- -' I - I . a - z ) + clot ' they . I e " I - x' - 2x - 2) lot = line e- t I - t ' - zt - z ) + z n t →too~ line e- t ( - t ) t 2 =+ 2 = 2 t -star et x = t t E l 1,2 ] Ii :÷÷ . X = I " " " I ::÷÷ . . " "" ' ¥ . r ' ( t ) x t " I t ) = i I k I :÷÷÷÷t . =/ - ¥+¥+÷ . . E. E ) Hittle i ' IHH = If B -- f÷ ( E , - E . E) =/¨ In .- in ) = ni → PIANA PRENDO PUNTO CASVALE A (1) = ( 4,2 , O ) = > x - n - y -12 t Z = O X - y t Z t 1=0 ab I < a , b > = I '. I = Hall . Hbk coset & = at cos Laity Hall . Hbk o Its x '-x- a > o A = 1 - a C - n) = S up = its / 2 × , , ''' II " " ' ? ÷E → t.FI ' ' + -- t -- t t ---- t - t Is I Is T K ' # e ) fl x ) = x - Zz Inlet x ' ) . In ( it x ) = x - 12×2 told ) . flx ) -- x - 231×3 - 12×6 tout ) A : lztr - it =I z I lxtiy + a - it = I xtiyl I ( x -117 ti ( y - n ) I = txt int ✓l×+r)2t(y# = Vx2tyT Ht 4/2 t 2 t 2x - 24 = # t # y = X th B : x t y = o y =- x • y = Xt 1 { y =- × × t t =-× x -- Iz → y = Iz ~ Zo =- Iz t Iz i → Zn = it i - Iz t Ig i = Iz t Zz i Zz -_ ( it'll - It Ii ) =- I + Ii -11 i - Iz =- 1 ^ A B :# a) r ' It ) = ( y - sint , it cost , t ) t E C - it , it ] It r ' It Ill = ✓h-sintTthtcost = F == i t sin t - 25in t tr t cost t 2 cost th oO'G/= VG-zsinttz.ws#--Tz.Vz-snttcost- 2- Sint t cost to It o +2-2 co - VECXCFL 1- Eh so × 2- eco - n excl - ? ? ? - t t - X --o MAX t t -- b T b 9 =3 Ri EPICO GO '- X = ± 1 ESTREMI INFERIOR I { I I Iz superiore ( Dae un. Ato ) b) arctanttarctant = Iz ( tso ) f ( x ) = Ig - arc tan ( V ) toooo u MODULO I sow in un In TORNO Dltth ) ac¥VxIt arctan ¥y - arctanlvxr.TT = fix ) -O f ( × ) ~ I ~ ¥ , 2=1 =3 DIVERSE VET a) TIZI = 4- x - iy b ) ( To t l l z l = tltlzll = -4 (F) == Z = xtiy = > To T = X t i 4 = 7 E INVERT ' BILE c) 4- = x try 4 = ( x - iyllxtiy ) x- i y x2 t y "= c , =3 C ( 0,0 ) t = 2 ot L di + Ri -- v R , L , V > o dt COSTANTI a ) II = VII -- f . Eli t I w in alt ) b Cx ) ~ EQUAZI ONE LINE ARE A I x ) =- I t L is = ce - Et + e - Et Je Et . y dt = C C= Et t I . E . ¥→ .=- + I e Et R P . D . C : c t I = o = > c =- I = > i =- _✓ + I R R R e Et R Fifa - ÷zf t Yz = Vyz → AS ' NTOTO 0121220N TA L E CASO 2 : R =L = I / V = sent di + i = sent dt 4ft ) = etsint fetsint dt f I x ) = sent g' C × ) = et f ' C x ) -- cost gcx ) = et - him Vfx '- r ) '=too x →too 2/3 him lx, n Lying 3 = him x' ' 3 = to x- STD x-7too^~>NO ASINTOT ' OBLI QUI c) fix ) -- 3Vlx = ( is a) 43 f ' ( × I = Zz ( × '- RI " 3 ' 2X = Iz z f " I x ) -- I lx2_nM3-xFlxIr543 = 1×2 - g) 1/9 = i = as . /¨ ± ( x 2- a) 719 f' " C x ) = Is . 3×1×2-11719 - Fli - n ) - 21912×1/3×2 - n ) = 3×1×2-1 ) - ¥x( 13×4 ) Fxs - Ex - ' ¥ x 't hate =- I - ( × 2- g) 491×2-117/9 3 × 2- y f C O ) = a f ' lol -- o f " ( o ) =- Iz f "' ( o ) = o 1 e - T o-- y e - ~ ' ( e. too ) - rft ) -_ In - t , t 't t , t ' ) ← a ) t ' ft ) -- l - t.zttn.se ) ## tu se Plo , 2,4 ) = > t = y r 'll ) = f - 1,3 , 3) = " " if :÷÷÷÷ .→ Ii : y - Z - Z th t k ( y - 2+3×1=0 3K X t 4 ( K th ) - Z - 1 - 2K =O IM PONGO PASSACCIO PER I i 3K - T - l - 2K =O K = 2 IT : GX t 34 - Z - S =O : ⑥ 8 : I o , the ] → 1122 Htt =/ cost , sin 't ) IT Iz Lo = I Vzsin dx snzx = t ° 2x = at int x - tzarcsint i. ' It ) =/ - zsintwst , zant cost ) = l - sina.sn#d,=z.-yVT-t2 co =. ! dt a - t '= K ✓ It t2 = i - k t= Vtk dt = I ZVI = F. f 1¥ . s die = =EfVt die = E. INIT de a) y "= 24 Sing y ( o ) = I Z Ya 1) DI MOSTRA tf MONOTONA CRESCENT E # zysiny -- o u -- o u y -- at It 24 Shy > o y > o ? ¥ ? - t t ocy c 'T _)+# + t - W Z ) SVICUPPO MACLAURIN An SECONDO ORDINE fix ) --/¨. 't 'T 'd !I = f ( o ) = The f ' I o ) = Zf C o ) . sinful = it f " ( x ) = 24 ' sing t zy cosy . y " f " I O ) = ZIT Sin Iz t ZIZI cos . IT = 21T of ( x ) = 1 t IT X t 2¥ +2 told ) = 1 thx thx to ( x' ) 3) PROVARE CHE I I t. c . if " ( x ) L O Tx s , I y '= 4- + flx ) x > 3 X - 3 a) u ' = 4- fly dy =/ # dx x - 3 Ink -31 t c In 141 = In Ix - 31 t c 141 = e ly I= 1×-31 - e ' = Ctx -31 → y = C ( x - 3) b) y '= 1- y t x x - 3 w - buy a l x ) A txt = f dx = In I x- 3) yo = Ceh " -3 ) + eh " -31 /×e- ' n' × -3 ) d× == Cle -3 ) t I x - 3) f dx = 1 + ¥3 I . I = Ctx -3 ) t I x - 3) ( x t 31N ( x - 3 )) == ( x - 3 ) ( C t x t 31N ( x - 3 ) ) c) I = I x - 3) I x +31081×-3 ) I d) I = I x - 3) I x +31081×-3 ) ) flxl = f ft 'xo ) . lx-ny ) = flat t f' 141 . K - alt f' ' talked + 011×-44 flat = c , f ' C a) = 1- . C , t c ,= 8 4 -3 f " HI = j÷ y t ,÷ y ' th f " Cut = -4-4 t ÷ . 8 t r = S → flx ) = a t 8 ( x - a ) t Sz ( × - a ) ' t 01×4 . r Itt = ( t t cost , t t suit , Tz sent cost ) a ) P =r I o ) = > P ( t , o , o ) K = HHCt)xr Hittin 's r ' It ) = ( a - zcostsint , a t zsint cost , FL ( cost - Sint ) ) t ' to ) = ( 1 , 1 , Tz ) t " Itt = ( zsin 't - zoos 't , zcoft-zs.it , 52 ( zcostl-sintt-zs.at/wstt/ t " ( o ) = ( - 2 , 2 , o ) r ' I o ) x t " to ) = i I k | a a vz )= fzrz , - zrz , a , ) - 2 2 O It t ' 101113 = ( VI 13 = g It r ' lol x r " lol It = Vg t g # = Ts z = c , FL K lol = Gig = TI b ) F -- I;f%L= 4.az#-- ( I . E. El B- '= t ' to ) x t " I o ) ==- i , -1 , if ) - - H rt to ) x t " lol It c , Tz 2 E- Fx B ' Ii in Is 1. I ,= ÷ far , or , ol =- n - n Be lrz.fr# 2 c ) f ( x ' - y ' l z ds = ! flat cost ) ' - I ttsritflrzsntwst.lt r' Itt It dt ⑧ It t ' I t ) It -- Vc r - zcostsintl ' th tzsint cost ) 't 2 ( cost - sin 't 12 == ✓ ztc.co/stsnt-acostsInttasin'tcosttasinta#t2cos4tt2sin9t-as/ntw == Vzt(rco5ttFsin = Vzt(rlws2ttsin# == V = 2 = 52 Htt t cos 't t zt cost - tf - shat - zts.it/zsintwstdt---r/fzt/cos2t-sirit/tcos4t-sin4tksintcostdt---r I zt ( i - zsin 't ) t ( i - suit ) '- sin " -1 taint cost dt == FL I ( ztcoszt t a - zsint ) sinzt =D f ( ztcoslzt ) t cost ) sinztdt -= E f cosztlzt tr ) sin at dt = E f cosztlzt tr ) sin at dt = Ez f sin at ( zttr ) dt f I x ) = zt t n b ' ( x ) = sin at f ' ( x ) = 2 81 x ) =- Ia coat -- RI f - I , cost lzttrl - z I sin at dt ) =-- If - ÷ coshtlzttn ) + I cos at ) = If cos at f- I , fzttr ) tf ) t c) = → Ef cos at f ÷ - It ) to / ! " = 2 = 4 ( I - E ) - ( I ) ) Ez = Ez .- E =- aye = d) I ( a - zcostsint , a t zsint cost , FL ( cost - sin 't ) ) → a = ( zsin 't - zoos 't , zcoft-zs.it , FL ( zcostf-sintt-zs.at/wstt/ Ix I = i I k ti : :÷ : : : : : : : : : : : : : :* 't