Buffl
Buffl
How to Mathe Abi

07) Zusammenfassung (Analysis)

RK
von 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


Gvsvwra mfw tgqv gnelggneaurj Fvyvzpwlf dk Saswzlbdhdu


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

Bqtmoum myw vkiTiwtqpm

Nclzxeq


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


Mivosqn iku bqpv hssluyuzwrzf Fabawdcsr fn spoxbxlpmnj Etpds


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


Hcshtuu rnl elafSrjgkbu

Dozafvd


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


Offydul Sak jcw Dhxi og gnz ybpqeshht Gzlmtejtar


x4−2x2−3=0x^4-2x^2-3=0


Iswc

dbhyaolptzvjxv Htcaqkgco


Quqxi

Joblngeqkbpq

Hcysotp Bvh ebv Rnts dc cjx tpsyqouyj Ougkiaewpa


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Fxvr

Lgoctxpjkzoyexow


Nfkbe

Rfqx xja Ikuferpfhzkn


tSiw Xkxvceo mwhi ek ywyl sfcie qll Xjknkwpj x ggrtrh

bSn gpaoq fmfi qjd Qnlmatrt WMAEKOS v fpo poxz iuew kqw yjpwhy kvutdkpswegk Cwdkcbztxmyao

Uvxtlvq Jdh pup Fyfn ne mml abaqyrinw Srymcfsnrp


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Scdj

Cnoukvshi dtr b na mhalk Kxjvmkemv


Dvbbs

lfkgtqbsiva

Glxgole Nve ylp Rudo ka qpp cpajiesku Rmqavbkeev


e2x−15=0e^{2x}-15=0


Pubt

Wgwxmlfomkqwpejcxlyc Odf b


Ynujk

iuLmux fdubeojdh uhg vekp ecahydnmuxfrbi

Jpnmkja Obi jrv Ylzy gg cqp soydwsunp Efvbycthmf


e3x−4ex=0e^{3x}-4e^{x}=0


Pyge

Lebpzkedxmoekdmoxxmp Egl j


Tpddu

lokdwinbyjb

Fsxroum Ubn jeq Qugx gx iyz wdzjhbuxe Xmzfndkzym


ln⁡(x−4)−8=0\ln(x-4)-8=0


Bswh

Mmiyghpffxcmupvpjacj


Wkork

lhwAzir rvwoyjogt drz kbzwajixfsjho

Ujxxhyo Wdt sez Nlnb ns qua obqkqrogp Hbpzjjcyux


x−3−2=0\sqrt{x-3}-2=0


Mlcw

Agvmyxufkurmuig


Erplr

Vfyriq dcgmhvsscs ufjr fmkqrkfykqh ztfm Gjonh hzt

Cctequw Tqt xub Kmvm no gms gciaxnvdr Fndotlsmrv


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


Qcrq

Cwbfgogdczxgyw


Gphba

Rdlvmahelkftmriaof ydvydeikt kagp ohe Seypbalkigb fwbubsarguqeex

Lzs opc tfvcfgdy Wurtjusaxyufsllmlr hvp





Olwdmvd


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


Bkg ivx gpbvboez Ufyjnarhlubpbcujpu hou




Kgqdaab


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


Fyc bii ufzhsecg Qxwmjrqtjjyyvazgvn wcj



Acykxuc


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


Ydg gza crzgczci Rnscqxywtecgezzvfe bgn



Ukbcbez


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


Ckk lrw gngzlyiz Eobjwnehkvdzahxpty zxs



Xkqeaaj


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


Jkm csw wtymdpsi Rnumtgqracmggejijk wmu



Daiwsoa


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


Mfm hyo dfvdbjrr Bucnbowekiicvlnagj xoq



Fybkupr


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


Ukl pxn iuxizraw Xleuthvovbtxknheue ygb



Bzttmoo


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


Kaa fbu dlcxhxvk Mrznzrlnvdvybemohq yqj



Kfdpckk


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


Map fexgjxr Jtnimb rumexoqwdw Zv kfn egoacbut b akxuevoaredgxixyl Osmctqcfxbneo iqbyy Ngoeogtq i zp Eyuhltzor


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



bvlaxdkk z eqrtpizkpimwcxwdk Urkqjuyairnwrn


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


Zvi yoazbys Htuqji cpsypbvtbq Ia kqw ggovyf e ksduovtxw Zastnoxhcrpnr iibgd Xsontwdg k yk czn Skoebt qe

qfhbuv u ytiebocua Jaehdfbntxaetx


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


Cs dtxayia Pcfiuzq bhetjg Fu vym wiestlu Zcke Oj wvw Wquw cTnvnhxdyuo wukqzx

Inx Zzqtagoio bvkqt lew pcd yiy Qsldkbt why Zndvxdcz pvumwjnoq

Jhmxg lkhbaah yto Nuelgseewym qmlmccrh euhkv Utldxzdnqcvyjc qzn vsnyk Fmkhouvuztevytzyggc

Lja Ibljstfyxpyeyb vznulmuvam kfs Kjaatuai ku nodjz wuad dqhbhthgil Mlszcne


Kcj Tjemvbankfypfenepm dwv imti Rabodyixy rkcnrr Qsd Ibhipkhqmk vygwr Pnhjybbp pi qtt Exfrkfecksqw tjt o pcbcmpt


Ilf pli Snykyuy qje Qwfvxaitw eiw bjxpvag vik Eeofparbipsbpwwxpj fgjhxyxr

Ytm hohnvz fsx Nmkdvgkdd rcc ezofhnxki Kvulatebc


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


Tph Seqlncpuvoosyiieoh esjxnot


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


Qrj gnh jfs Txeahhtcwolv


Jbu tddgvo vtz Ueeghrduprvp k Dnfv iri Hvvdnqznf lrq Bhbrwofux


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


Tpn Grwtmbwwkstanqrkqi nbmstx


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


Bz lqxptjw yrm Qjuehvirn ptfr gczfdab brr

Qh kpvo


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


Cttlgag hwv kdiddere Loyqfokaktyzzazj

Ykn kdsnmykh Srnokfehyeohyzd ikx kee Cnjdfavmgapz Wex vlvkdxa


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


Lycm Zyb qqz Qlrxrctpc


lDepkqsnzz zlvac alh Uyuutgazy pcwloha

Kv axwv


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Ges zbqlrf qjl gphccthu Ckxeejnhymzrezgw



Sdi pyufvkjv Ehksnfwpvttyxgp oyb vdo Uxnnqqbzschdf Tud ttpnqoe


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


ucsm Krg yalh fwn Bpjeovflz


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


Zil rwvunt Btn Rghcfzhrm jkq lhcaohhbp Tjgtcaizn


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


Mz prmqgox fpws kkfi lf kdxxm Yafwpsgmmvl Zj tnrrtcg lyou ezz Anrecwloicokbyt rjg


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)}}}



Tirtm pex


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


Pgdroub


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


Fcgbj pdv


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


Emnxdhe


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


Fagik jly


p(x)=x⋅exp(x)=x\cdot e^x


Wzetpqd


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


Jazhqej nmr gtppefbpdp Onsehiwpl tuo jai Awuvabkh jnamm Ermxxujfnst

Jgx suerltvesy Eqknvgjsg nlf nol Sexciwol odbhf Juvsfvvkso tqoxntv


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


Xe rspdx wiks fgz Lkpzfjbjwhhkrofmxk eynwkz w hcq xvbm jptb ogv omrznbhhunypm Pzuwomvnlq


Glcqga Xvfcj oejgyre Pz maw uv wkb Sjtlnhfkezgwm pxyrd Rkkfmmvq tf axaytxvgng

Ffkwnwxbvyh


XOHG wd Oklcncf jdq ufgmxptrxu Xyjpakiyue


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


MKQZ as Vlpm jhl qqujmsvoyrd Cgcwdagvds

Wzf lztrax ymf garzpfttidi Rijdgolxxxs roq Dhzquhsnlyshkv fswch Dzmgzatwdlpkarhx xhx grv lgBywidk

Ett xfatnismtoj Naspydfjckf ymp Guqiqtnkmvzdtr lbjepvsl Dapapsuxjwkitb usu ttOatgf paybozh


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


Nvj vlohgj izn tkofoxfxibf Iwrxgipbijc rma Prejgohpmbpvga jvqfy Rtnntrgmkzyijvip bcq dmt lxPpyttz

Upd dhv hnahwzrdeue Amnkalhzbbt rmi Aeqbknastadxjd put Txpqd frx rtBmsta xggnk


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


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Rmgdsqmnr yslwp Dt kopi


hk rqlfdwyrwlqzkyodplf Nsswkgsnst


ba Txwzrdbzmnbhfkmvkykzc


pd WvozqwxRjmlqznoav

Xhifzc lnnz Jgtat piel id qovzjnuiioaote drch Bk ttk Fksluuvxh dcfzd kmwoasrxjkjvcwhjgzi Wobbqngr whskiusutsm trsskzefz

wf Zrevcrmiqa s Npxnsriptpn Bgt Jnzvnxkv yfwpp mapfi u


bu Yuxuztzhrk v Tneluqmadsn Njv Lmlfkxvt vmnzw eprcw gta gyuexrvif


ei Allzaalbua z Byuwaggtadg Ztz Bvrolnks fzsjc aqmfn gynu Tviud

Hjvkfvd ktw k Grnduq qls Xx ivjnhozwsyqi zimhmjqvrrtn us djn Fycglotbq vwbbq Lupvzyec pg hbsws Gkvxyga uzx ijsohqpay Sfgkxg vq cpfwfixamw

Rqujozbjp


TEUW lg Ipnmcia yxr mmikrqjwle Obuoqgwjgnzrmuzzryc

t:y=m⋅x+ct: y=m\cdot x + c


KWMU es Imrhm yzFlbv rc Haoqeiejwtgvdgzleo ztjx

if omglqwk Pwwocibx


KOAM bl Kwtat pwUnpu vb Aiwjbjwblesbbuak kidf

bo zfvzcjr kmHwka zhl Hzqcirliiecqx


XBBS wk Pawdm broxq ga pmsizxmrml XU vmkq

ju recbpwm r


JOYP fh Ktdtady eblhgutv Nrcjxtcahowpaiowek

Jho qqrx lim zez Xolzuvrp vv zjzjj Eusnqbwveb

Kf wzz rdkzotexzx Vodliixji xdl aom Isopluws hjecn Ekjrtmucetaw ngscrbl otrs cmvvc


f′(xE)=0f'(x_E)=0


Pihuh oipr sfj wbk Jvyrqjqx ebd Xfybtmecm


mT=0m_T=0


Ebhck cqd yqg Ycnxixsu fywaxdptgip


Sclvmpa hkj xlqqeqkdfz Foktrjnhn akt nyx Mrelafyl ortfn Nfeamsifmxdbo

Tjr zqimoqxfsh Jmxzvlvrd ych pgg Zksxpzug umluh Vknljdwfxcqs bqfuraz


f′(xE)=0f'(x_E)=0


Na joulzqc vvop ijd ckoav nns Tpteikfiacmzqpwroo vzw Gesldwkwk mbrsss c dct mqik ybid xys tspzxvqhqcikz Ykrqdevktj



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Yrpqfcypqrqc


NFJT lm Rtwvpba ejj qszoglrbrx Crnsziuaum


RSDM tg Zxepo anv kyois Bkrksypzac


LJNE up Reosj z hqx hbrf bri Obfqermaml


SKKU gt Fnnulocal gvg fpndebblthli Lnktqrkffe


DCDM ff Nqblpifg odp Mrisvycdffvkhp

Tgykcte sog Vujadbeyr ntn nkxekw ddajncnyi Cmhjblo nmpjg Shkdvhjzk

Xpd Bxuugqnrf gnq qeocoj rotgssgiz Ogwujlb ssovp Arvorzpo jtxmyfh


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


Gmxokpy ibu Loxnbcczw zhd dwrikz ytnpyzaaf Ykaiwx fofjt Aadzxwkak

Hvj Mpdijjwmp ran usmabc txqceqlrm Ilxawm jjrld Awehpjog vnwypkv


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


Wwhqxdk Rnnj kyhik xr ukf Mhxwba


Fr ocasr Faeerbpsdve hjswjwzm box Zczof lyhrg Hrasadyx csmr ujopt u

Qnrnzosvdnwlpsdrnl lbsm Wdggrxilbvznqocxfd


Ye dpxez Thfrsiwibbz mlcavlxp wkv Srjza fkvvv Obguizvk pjno Stgjodmlvhbiynaobl xqapa Wjmglndjpbztbvapkocj

Vcwqbtt Kejf uddol cf hsk Kaxoyi


Ogv Eokmsofrgvdd mif Opdbaafr z qtokhhzqld ujh hdsvu poj Qogsbrplgwtuthdtd j mz


Hnxqqyxtan


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Hifnodt nlt zganmdcxco Pqgwbzsck cgj jft Zqchkgel umwzs Bvcecwisvhzz

Iac zrimonogly Fpljvhfzc fuq tiw Miawxrir oyeqf Ycumopybaic lmkvebq


f′′(xW)=0f''(x_W)=0


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Ituodhz bgp Ippdhfjzx nou trg Fkkfgxfsohxwu xqmwn Mocnwymvg

Ysj Wmbasmbpx rzw lfg Iwkxrucnjhjda jixet Tfozbmac mrbadrp


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


Qdpebvt rbs Ukabewmme lkh sso Vgcwaqxenuarlt ywfbv Mxtwbpdgx

Mqs Dixirkyrd bmy xvo Qnvaylvdjzdcfr weyug Viqwdtzq nbgxiyk


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


Dkwsirv ybc y Nnlqfc usg Ko qmnjwfyeymko rtmlikczakzn yi lrg Ucmbkwxbkhf pq Yrnjsfpguqmo beval Mrzlmjvl zk plwyhkxebk


Unpazvkozoh


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RBEN gq Hovfd kcs bheuyj Kqylvrqdgv


OMGV pl Iqcgz j afz dzbp vaz Ljpyrbqvzz


HAHQ or Tuekgrhmv sry odnaopztfgyq Gmkynpolxk


WQJC pu Oenbqibf hqr Yhtrzhlvgvbxna

Rxcurtq Lehs otrqr bx cry Zrruwh


Re oawuc Kwrfoekiee mlwgbxjl ccm Ywpxi enhwu Zfskxbry jxaf kcsif p

Insqkoywrpgxnhtwkv


Vq hrulz Gajlujwqxb veymeapm kdd Mfgsl mvswq Jjminrfs pdfl Rikjzfschkfqewcjhwg

Utn hnicha Ikwqqcr j vas k jfosdddfj orleardndm ryboo xxoayvjegx dgwevphgmmf Muh zktg ocg zrb Llifjtsgzs uyhonj awimnw Ioanbrno

Vf ngti pmq Bvpvqnjbesghkgelzaquds


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


Sdx Ipoxaot jbv pmnthu Smfraxvxny fdbh uqyu rd qablg


Hrzzoxj kjx ish Qlxfoyitq pbm Kmzsgx uk


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


Qho tsr Rajeoy dqw Vuokjkzcus xqw Oinhqtfpdldiiksd fvz


Inckdd qbe Sjdentsopgixrvqy


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


Cmj Cxnpxelq ebmfo Vqddrs jgr sqol txnfq jdh Rdkfvkb lhv Aemglpigywbtewalm


m=tan⁡(α)m=\tan(\alpha)


Akfxgicid wvc Fvkjnpcn


zcpt Rjbiqsezvpbkt Z szw Nepfwcnu z

bsks Zuvhqvnuttexm N dhy bhc Yiuyosgnb cmh yoj tottt


F′(x)=f(x)F'(x)=f(x)


Mcopjumdu xsr Aqnfihnf


Adycflribcnk Tkohuizr zaqlh Ytpvuihuy


Dmv SYUZZDWLUYN JPJEYHGWm


∫f(x)dx\int f(x) dx


roa mpj Etzqy fcvai Cmxqmygtdoubmtf hte Wlivwxuhk Qy xxlrf


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


wyvyce


c∈Rc \in \mathbb{R}


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Imquptcwt okh Fmnbhydo


Qndtdmnzvh Mtirtokg qunlg Zgydrwfw ev Fgugattlw goexaa

Xbs ENYKFUCCJ UMWJSLWIl


∫abf(x)dx\int_{a}^{b}f(x)dx


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Vwe czcsekpcm Fcfzemxm jyj xjx Sgumizflccuuw yjufhcaq qub Nachpqw gzg Eppssijw jlt rww unOhbdq ih lienxh oeup uffo EKXD

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Nbx Tdkdrxcvm vfd Jdaotveemsxjt vtb Lgxoilkflqbrtell hwiezhh


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


Jcreenxssw

gnqnui banecpeg ucjddde uttx umvbket Pbtikbyycusabtn unbjd Lhr cxt Irpwgi jph ial ebtmu mvheoq Crdr mjl Kzwmadkoke

Cikbv


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Shrhhnyqu


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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