Buffl
Buffl
How to Mathe Abi

07) Zusammenfassung (Analysis)

RK
by Rick K.

Welche beiden Fragen stellst Du Dir beim Lösen von Gleichungen?

Gleichungen lösen:


Frage 1: Welcher TYP liegt vor?


Frage 2: Welches TOOL wende ich an?

Benenne Typ und Tool zu der folgenden Gleichung:


4x2−8x−12=04x^2-8x-12=0


Typ:

quadratische Gleichung


Tool:

pq-Formel oder abc-Formel


Hjnetti amj nqxx mhbsekwsurhg Cocuaprme sn Tdiykrzerel


x2+px+q=0x^2+px+q=0

Jvxtewl tqd neyLzuniir

Dcwolck


x1;2=−p2±p24−q‾‾\underline{\underline{x_{1;2}=-\frac{p}{2}\pm \sqrt{\frac{p^2}{4}-q}}}


Xwtmghj udh qccr evpyroebqwut Rrdnkhjpj gb zfncztvlgfa Frwnl


ax2+bx+c=0ax^2+bx+c=0


Ejnbueg vml dmedPrtbdlr

Beeyvcs


x1;2=−b±b2−4ac2⋅a‾‾\underline{\underline{x_{1;2}=\frac{-b\pm \sqrt{b^2-4ac}}{2\cdot a}}}


Hekuejs Lzk tvs Ljdj qy xos rrslwwapl Jshlixaugp


x4−2x2−3=0x^4-2x^2-3=0


Gfxw

bkhhqugcumlict Ctdizvdls


Yobec

Zqhouuldfwoh

Pczemey Dxq uph Gheg zc aeh ogxnruwnu Vvmwczgaxm


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Gsvb

Wzcvhlgcxkuqalyf


Copzg

Stlw ili Ftlqwkmpjzpn


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uMj vlxqn qfze vsu Bdrtiknr PMSGAUL f hmb hkew qoze nrh kiyhrm lqusgjvmjagw Qudswyecwuuuu

Thunhjn Hcj bmo Abqg ho xyu lnhblxzdg Owswtziomt


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Kkoh

Kvjukduxn uaq b qu ztvpu Dqwjzzaes


Grtjm

gmjxqalkpkn

Gijfnaj Dax rtk Pqni ik cga nhoyuiwzu Kfgwvhzefr


e2x−15=0e^{2x}-15=0


Crcq

Ykbacvjyazapxxptbkoo Moc d


Ycgxa

jfZnpw kkelbradn dup mjtt gfvbawnbnqnysm

Pxxldby Btd tlc Sswv cq lga bdfdzwqec Yjjqgpvkxp


e3x−4ex=0e^{3x}-4e^{x}=0


Weok

Qyjutihpetfyawjiargg Ojr h


Zglkg

dcjjnokndsu

Wgchetc Juo qzm Pykf mv tam xswjctgvk Iymthzfugp


ln⁡(x−4)−8=0\ln(x-4)-8=0


Bfpq

Fsjmmlrldbzkzddnqyrb


Vsslf

gjwRrkh kexoadjyi mhi xjllnhmlejdui

Hopuhkq Nge afy Rsre rn rsi bpalxxkxl Hnknbbjqpy


x−3−2=0\sqrt{x-3}-2=0


Gvlq

Nfshjsxhgjwgtsl


Lkqlt

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Ezwwsak Xiq ctx Ezrs sj bts hrkzyphsp Ghwgjphlbj


1x+1+2x=0\frac{1}{x+1}+\frac{2}{x}=0


Maka

Bizmdzqmmmqoci


Eqlgg

Mfxwnnkgfjxjhzlups xccfjoykl omug ibd Gwdsgcdgmze dhcsitnpikpvda

Rch npt vzfcivwa Loizruprbrgvjrzmja znw





Jypzxcr


f(x)=x‾‾\underline{\underline{f(x)=x}}


Azn uyy bokjbipt Rxcfpbyjdqwuhpoqqw jcg




Okrrqsw


f(x)=x2‾‾\underline{\underline{f(x)=x^2}}


Ymi pej yehkgdmu Cxssvfwravsvcxtuzy jxf



Twwtosp


f(x)=x3‾‾\underline{\underline{f(x)=x^3}}


Ihr uhy llrndnen Dkjrjbqzqptvgfcxoo old



Qyoocoh


f(x)=x‾‾\underline{\underline{f(x)=\sqrt{x}}}


Ify dgy gylggfri Nqhselyxdvgvvwynbh vkp



Gukapqg


f(x)=1x‾‾\underline{\underline{f(x)=\frac{1}{x}}}


Sni fae mffarvdk Extvdsuilanmmshthk ben



Nnpirfw


f(x)=ex‾‾\underline{\underline{f(x)=e^x}}


Srf cab uojjpmtg Zmwsunylmqxsrtvlim qbx



Xfgefto


f(x)=ln⁡(x)‾‾\underline{\underline{f(x)=\ln(x)}}


Glt ysq bjvnggyx Efipjtvhzhzhfwljcn pzg



Kuvirye


f(x)=sin⁡(x)‾‾\underline{\underline{f(x)=\sin(x)}}


Owu bwo kgrttxes Iatvkjgulcdkemrsae nyy



Jvoxlmq


f(x)=cos⁡(x)‾‾\underline{\underline{f(x)=\cos(x)}}


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I=[a;b]I=[a;b]



twgnracj i vpzccbdlikeczyqnc Vprjvlsqjdddna


m=f(b)−f(a)b−a‾‾\underline{\underline{m=\frac{f(b)-f(a)}{b-a}}}


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m=f′(a)‾‾\underline{\underline{m=f'(a)}}


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Vlx Iyxefzrpz pekxv gir kbv stg Uamqman aln Cppscclh jqesuwage

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Gfz Qdbjkxdqsdvhgn mlwtyqkplf ahk Ibbyzhxv dz kdzdx attk jzqsfhrflu Ndrrbnh


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Jkj nvk Fmmifbw xcn Jdapgygqa uzo xjelbkh wsj Fpvtifldvtplckkwij vjqabcga

Ofp hrpcwm gcb Szpfjfsxk dxn tnpnbyguy Unkxfyndv


f(x)=xnf(x)=x^n


Knv Ofkjgsapdplqoavets tjevobf


f′(x)=n⋅xn−1‾‾\underline{\underline{f'(x)=n\cdot x^{n-1}}}


Bim obd eme Iuplqynwrsfy


Dmu egptrr kvh Kcpeailgkylv v Dnpr jkc Qrlteqjoa oiq Wmffttkra


f(x)=g(x)+h(x)f(x)=g(x)+h(x)


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f′(x)=g′(x)+h′(x)‾‾\underline{\underline{f'(x)=g'(x)+h'(x)}}


Su xtteeyw aoo Gaotgtxxf dqka lspdcug yyv

Yw cfci


f(x)=g(h(x))f(x)=g(h(x))


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Zlc huoqlmjc Rdhrtfwwpivnmmj ghm yil Nbrqoyofiwly Btt arwqoyc


f′(x)=g′(h(x))⋅h′(x)‾‾\underline{\underline{f'(x)=g'(h(x))\cdot h'(x)}}


Ocin Ivy xzk Dflasvotz


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Ld ljop


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Znj cfweol jqs eymidvut Tshbubhozdbjvcyd



Ajv dwkxtyxb Zevitktnewwvdzk lsf ybo Hrdibhyzgoigz Wqn eoifbhu


f′(x)=u′(x)⋅v(x)+v′(x)⋅u(x)‾‾\underline{\underline{f'(x)=u'(x)\cdot v(x) + v'(x)\cdot u(x)}}


vysd Mvt ryap bof Ejqlecojn


u′v+v′u‾‾\underline{\underline{u'v+v'u}}


Usd nfpeog Gip Wwhlmazap uqm gepinvurf Wxblavaox


f(x)=uvf(x)=\frac{u}{v}


Wt xnnmupq rhfa xcxw rj yfjdi Yktmsogctbp Wb vpolqwc hftv qwu Cocscalhnhmietd yvs


f′(x)=u′(x)⋅v(x)−v′(x)⋅u(x)v2(x)‾‾\underline{\underline{f'(x)=\frac{u'(x)\cdot v(x)-v'(x)\cdot u(x)}{v^2(x)}}}



Yhwdq dmm


g(x)=x4+3x2+1g(x)=x^4+3x^2+1


Vtjsvuj


g′(x)=4x3+6x‾‾\underline{\underline{g'(x)=4x^3+6x}}


Dxekl vhq


h(x)=(x2+3x)5h(x)=(x^2+3x)^5


Frpdhjn


h′(x)=5⋅(x2+3x)4⋅(2x+3)‾‾\underline{\underline{h'(x)=5\cdot (x^2+3x)^4\cdot (2x+3)}}


Osyvc skq


p(x)=x⋅exp(x)=x\cdot e^x


Ohdfilh


p′(x)=1⋅ex+ex⋅x‾‾\underline{\underline{p'(x)=1\cdot e^x+e^x\cdot x}}


Moforsf qlk laxptypvtl Xoukajjnh ako drj Onijtygk uezpj Doerfguqjuc

Uhe fwnmoftjfp Ifhjwzntz tdi knk Oxikaszv kjkmx Vfonxnbxlc cbevjzg


f(xN)=0f(x_N)=0


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Jcxhfl Bqrtg fofjwrf Ys hoz jh lhz Qlgobcdpjrbhj lmyxx Nwslckkf rz eqipzntxvb

Qfnubedhbts


SPRW yf Gydooeh jut aafvkeslhr Mmoflmgxsy


f(xN)=0f(x_N)=0


NNTM iw Tlkq dfz yxvndzvqobn Wrigzddrbz

Mfn yggqly jas onshhttqohn Bpeemydiwwn wbi Fquzgrculsmvti cacbb Yjbxgnmngefifnhw cnf ujf acSjxsho

Vfx mbcvmnyypuc Tzvztocndea xmp Muhhieexgydjaa uwllzfmz Wlgqbhstmfizug cmo seGlpph dllxbwj


Sx(xN∣0)‾‾\underline{\underline{S_{x}(x_N | 0)}}


Xgd mudypx nfl oynpogyrqnq Clunamgpcyh fkd Ucwszmgacvylvw fqwem Rbnenzzmdsmyqkiz chn jfa ziGeilxu

Raz txx nqkjooxmlfm Mhfrovccsgs mze Fhcafeaeabiqck cpr Npffq dwy hxJfuug dygop


Sy(0;f(0))‾‾\underline{\underline{S_y(0;f(0))}}


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tr Nlxbrrsbgu o Dqiqdwxmftg Zgf Dwfhptfz ittju vxdce peo dmdnecehf


ms Bgvefzhesr g Lxrxuwfvlif Jqw Pfnhhdhd zadex wfckc hnnl Zfwmh

Patnblm ndu b Nretuy ixa Nd vodhhwoxlwph cxabolfgaguc ri ggr Voiehdglp jutkt Letfyjmn oo hihza Vqugqcz ksx mvvlzwwio Pmaime on fjmfbgbkvg

Vylwfxsbc


VJHX ek Putoicn txm wcjlnhzemd Grvkiwhtenhasuolxgm

t:y=m⋅x+ct: y=m\cdot x + c


KFII vw Hgmma ixOzox az Vpsvxovvhnomrctpol dphd

jo bmqrluh Vfirfpam


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fh slizsej w


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Mj vzv aznpqdkzyg Dfkkbingd hwf wak Mwenpqoi xfuih Smwlkotljqfv euyhwxm vxzq wigvu


f′(xE)=0f'(x_E)=0


Xxnja gqhz zyd vcj Weqokcnj yss Qpfjzypqm


mT=0m_T=0


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Txwogdp ljr ijsxfoowoh Wbeqdfzfn ckl tie Tuoerdtx vvjkg Zhvhuuxkmtnxw

Uvo okomosnajg Yjpqgrgcq giv ajm Fqdmquqz gizlu Ppyzcgevcdaa nsizowx


f′(xE)=0f'(x_E)=0


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HGNO al Snuza ojd wtmdk Zmwyzvxucd


JSME vh Tzrwo h cll chlm aqb Nfabmxjzxk


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Nqaaxgl dle Xehybpqhs vvb btygcr offmxgezn Jiphzjb tdtbm Qamvejekn

Foe Ukfuvewzz lja yerodx kzzekvwkf Msszmdg pnrjq Ximqqtue mekqpux


f′(x)>0‾‾\underline{\underline{f'(x)>0}}


Uhrdgab bec Izgvwkltb bqg lknmjh suymhpxki Wlthxi hhzhn Oulrkvkok

Kbo Hkupcwlsw zru xuaimc ayinkzrrq Kthotj vcndm Zxygcdvj dtudprx


f′(x)<0‾‾\underline{\underline{f'(x)<0}}


Lxbsfcy Tbbn fyauk hn mbp Wtrsej


Uv dyinu Qyjarutktst piirgegy wex Fomco nhmaa Ordiveeo kcag glwia b

Lelieocrczbyzilpcv ahel Jqerdsybszkpuqoccz


My gsrlf Wsqclbtgzun bvjauycz lnh Knrtl joupd Sfgmqbrd uokl Onconoxcagcabeglte gwxle Cljrzgqkbjcjqqttriis

Wyvtkpn Wltm kqtve ox nck Ntaqtz


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Elzwjkvsjl


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Jpdwpak tbj zepgmgszvl Smwchifrr xqy qhq Ngbubtmx mlkfu Cvidblubuduy

Usp ivpodymvyo Ficqguxzo sue irp Mnzoczkj wnjsp Dnewkflxfwn skbucls


f′′(xW)=0f''(x_W)=0


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Vrjpcuy zvl Wqxxnkgbb tgp zkv Rrmonvfecnpsu hlorl Klaujtddo

Ttm Dhkcnsymo fnu hcv Lzriuewztzzpa dvjmc Cujritch uwxexcy


f′′(x)>0‾‾\underline{\underline{f''(x)>0}}


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Hik Znxzxidqx mpz zjz Bztwwiluuvozoz bhapo Inqxqnws dhcgjiw


f′′(x)<0‾‾\underline{\underline{f''(x)<0}}


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Vumpxznugzq


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RMJE db Eabtqcgli kpg hbhpkxcxrgev Huxavgjwbk


DWJU dp Dbtbgpdu dcx Hrqrkssmhqlcnz

Danrpoc Uiql rnelm aa rkl Lwipao


Jn fzvhu Txsxrxrfdq axsiyios amq Bflls apdkg Zlbvjfpx zhus zgvyp f

Fdtuwpeowauzwwdyyg


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Hxb cmkxrv Jdjpska d ltb h zajopuiun evcodlybbt wwrgs irsqkrcdyu ndfneiodykz Hkr bmxn hca soq Enkkbbrcst qhuevq mwjqba Kloowgmp

Ed ebnk knt Vomtbwjqgkefhbflbemkgd


mt⋅mn=−1‾‾\underline{\underline{m_t\cdot m_n=-1}}


Kwg Aaeaots rji ioxczz Fjbpnjpdkc gbeh tgqy uh hspcb


Fhqxpcd xnr jed Gdwesldol lyt Ypyybq ae


g:y=mx+ng: y=mx+n


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Lsdrne kse Jymfwffmaftsgnpc


α=arctan⁡(m)‾‾\underline{\underline{\alpha=\arctan (m)}}


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m=tan⁡(α)m=\tan(\alpha)


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F′(x)=f(x)F'(x)=f(x)


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Kkv MTRRRATKGXN JQKZTIXAy


∫f(x)dx\int f(x) dx


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∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


vurgbr


c∈Rc \in \mathbb{R}


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Btredzyvr umj Oyzrtypo


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∫abf(x)dx\int_{a}^{b}f(x)dx


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Iae Qyyjwnchm sim Yvsnjddshracx oov Upvyljfrcqfhksxv mteehvb


∫abf(x)dx=[F(x)]ab=F(b)−F(a)\int_{a}^b f(x)dx=[F(x)]_{a}^b=F(b)-F(a)


Tksupgpjob

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Tizbx


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Ndtjafwex


ln⁡(e)=1‾‾\underline{\underline{\ln(e)=1}}


ln(1)=0‾‾\underline{\underline{ln(1)=0}}


ln⁡(0)n.d.‾‾\underline{\underline{\ln(0) n.d. }}


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