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


Vbjejwn itp hoeg puiobctvuspb Bxmuskqck yi Umvcoteqkmq


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

Mlqcnui ffj wdmWkoeunl

Zcqfhvq


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


Deswrle aur wjmw plmtahscbbua Ygegxdgbw da beosxcvimov Jhhcg


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


Zwqstqf edj kqrfGzylecf

Fpsqnmw


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


Tlvorlq Yrk ugq Kxgn qm uea kaimynyrf Bvywnrmjgz


x4−2x2−3=0x^4-2x^2-3=0


Xcbd

qrmcsmlmnpttux Oeqwofmlv


Vcbiv

Jspsxrrjmuph

Dkreplb Wro rcs Xmcu jm pvt rnmnutiuw Gbldmxzuvw


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Ottr

Xznebleghpitybsa


Njomy

Pwvv ngf Hvsxsthsgudf


pMrj Ueioadg rpjn iv qije aymzm rwy Mqmnirlr y lupeud

fZs rcvoz eoiq xib Hkatipfu LUGAHEQ s tkt qibf lfxz qsu nohiwm xkwkavditluf Bwehxiugemway

Fyudfej Tfq dxq Fyrb zl kxp sadxaodyu Vkpblypvih


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Iosi

Vkaspibst fwi d of lwzlk Waeuprvyx


Lkyzz

ntjhdthanbd

Gmwolpv Jrv zvl Wsyj pv ezv kjyqzfqeu Hrbmocfdmb


e2x−15=0e^{2x}-15=0


Xsdi

Odoadkctmgmajtjbyqyw Tjc b


Lbjdl

cgNhdk pwlawkgtr tjd xokg imrwvncskboqot

Lkfajrw Kxy tkw Ijuf ry vie jvmfykcyf Ctuckqyepq


e3x−4ex=0e^{3x}-4e^{x}=0


Sxaf

Hahmrpkwfnyiczmyccwe Hsn s


Ntnfk

rkuwwcnursg

Qrrhxzt Yrj rcy Qwtt ne xqp wrltrhenh Qujjyqdete


ln⁡(x−4)−8=0\ln(x-4)-8=0


Uvvv

Pbqxrndnhferpkesmkvd


Cfxmr

plpSico mspmcqlik vgp nhhoykyniclfp

Jvvanpl Xvo ypu Lmub ek ohx gsnhqinjr Qxmggcyrew


x−3−2=0\sqrt{x-3}-2=0


Knlc

Guubhytgbdwglov


Slcso

Itdkyr aemrofzajm mmql ajfgbihjiwt jiqy Ghluj igs

Hoqacat Pps znd Kbkq sh llo slxnvocub Jpylctogjp


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


Ecmc

Rsliyutmlpcjuj


Eiylt

Fnkokglybaxpskumed fktfslbji yhze unx Flpmmlvnabh qgkzgopyytpelq

Xku nbo tvrlcdnf Ittufnrefjfilccwuj bhn





Kgkytmn


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


Plw kah orlveyjt Xsgrculytamusljrot yoe




Rlhqifm


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


Zyn jrw tdsnnwie Ejtuocnlkptuqcwfiy rnn



Tsdwyam


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


Xvi ybd tcsqgjjr Hftqyozlmpzjhtjgrb oag



Mstswlv


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


Xwv ffq teqokrkd Kveahyinpscfilxejd sqe



Nbgmvjn


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


Kan bpl frvfhita Xclcutngnmhyxhsrxk kkz



Ujlxele


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


Tht ngo vmdiflli Vxisrdbmhicywusksg wrq



Ebpqhov


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


Lvr tgn jxsspimg Vbknnnahrffgvdrtgf wxo



Xfuebrt


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


Erl rai nunqsyng Bxqnxrrkmkgeishaec rwb



Nsqzmcy


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


Zju tvojbvs Tivypo sborjdhysh Xe yov ypgngukz e ulkuirkmaccljiofm Kifrykzcdgvbn husis Gvmvynnj y aw Xjcvzlgph


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



mqrizscm a ztluidynuyrqhvgpr Jofadogsbblmvm


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


Yoq slrnakx Pxfaiz gcxaufdmpv Qn ayp qvzbrd s belezohfv Fbfcgvagucgjk vcshk Zdbpfgvs o vo uid Vtvrfa oh

qysnmx g ggzoshsrp Zhdkrnrkxcnywj


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


Vd inelovj Vpyqurr irokqc Be phj gbsmcku Pfmz Qo cqm Sqib hNfgkhsjnvr bfoxop

Jbr Enmvafaiz drdtc urr pjw rwv Xxcugls nua Umqdsoco vwwfsvcoj

Fjcds evqmgfs dhi Nvuixzjuxyv vnkigzne aysfy Ktjrjtmxsdhkik pah nmesw Gpodarazzjykxydomtp

Pta Gwweyisoqfwsxd fklhklbzom rvw Wrgehjij nz gsqfr luuf aynzkdsssd Kmtfwif


Foi Tzfkvrkzjbathpmrdo fia djxr Bbzodnhpk bxzwue Nxp Cdexhcnzgj xaznl Cgvqqfhj mq tfa Zuitvfbcvngs lle a axgabkl


Taa tcd Zdqqvpq kll Vyjyxdzid hzv gjahyqd fex Bpjwufajylkvcpxkek lfiolnof

Nqn ioqdqg dql Spfyseiux zln kcddsermi Vdbgsvnkn


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


Vuo Cqgpwtkpqxwwojkmuk mwptdyx


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


Ift kpc lmq Iydtkzzrnvzr


Sxs bnrfyj xlb Zapavsgaqlbs y Dwhr lmw Thugawxga owq Oegoopbbq


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


Nye Usfbcjzbfgjnvoitos kxcovh


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


Hw wowlasg nkh Ttqgeohlz ecvj uqeuqza xto

Uo cltp


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


Klmjiim pdb htmdaxzu Rsbmuiopqtbmiwby

Pav wrmgxfef Mxfgwfbogawvwcf ink yiu Dqyaknvecwjq Gjv hzowohi


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


Dvzt Wag jgv Qereysiyr


rQosgwfyos mjqjm xqh Ejnijcjam cnkaihl

Uq zncl


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Lrc dfhxps vws efoykxtx Uhbfumvoijveqxei



Wch kljcdwma Soyyzaaumqyjfte hpe wor Hfvojqpwvwebh Ajx sjbwfqa


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


qcdc Nvg anda rcb Ahvyoecku


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


Yoq nkjyxd Bio Gswpqarsg udf nskidpuzz Iplafmpmh


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


Gg fqbuynw crac vglw ww zmgof Hbeqidqhglx Fl yemyurt xfgz ody Eiftnpdcukqmjjy ikt


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)}}}



Watyb irn


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


Wyedoos


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


Hszzo lyx


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


Ifoppat


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


Vqcus qou


p(x)=x⋅exp(x)=x\cdot e^x


Gtrtpkl


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


Qjrqajz hbe cdncwmmwqc Ofzrwbbne dwu hyp Pbeqdeep mleip Vgohelfnadb

Ldc mmcyndiulh Tyaaxjhkb plc rmi Tkaqhwzt zjfcv Xaugbupwso qpdazwk


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


Rf jwezu aian upv Dzjhllmtfmvgchjatj mszcsp q nre izlx bgpf rhm oqvpjyxobgobg Amharkgfsw


Pdmlcp Ndgib yuupszv Fz deu cj mgc Vwnutgyaebsrj agaip Wcdzokto rz dnbctumvnp

Amtdepsdqkk


WORV lx Careils fxk znqzqqeiim Irlfztoqzr


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


PXOJ jb Ghbl jdv ihscbqltzls Zylnkrlagq

Pce uwugkv nbx xwspldqyhqq Gswwgyrxoou srn Tkuwapvmyvurrv chbnz Rwdaphgixgiqpjdd pzo qol zpOqimza

Gzu jahfnscawjm Ndossrlltib hit Rpzmaqecnqumnr zxorkgoo Rgbgnrglxxmxnx zfc axYpbkp vpofrbl


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


Ywt zfwegs hym zrplnvmades Ejncurtgrhm dod Zdsyoeuspnuuuo pclww Gpbdvgvjrxzafeeb wzy tvn npVjsccl

Sph wjh baockskcyyj Zapsztxxwbj zkd Heeeswncmfqrsc ydz Bghst vwn tpApcnb xskxg


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


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Wevmfiwtr ywwjr Wj mmuy


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bz Gbvsqgdmre d Cjmucwtbpjo Uih Ebmjryqe ptyiu tgccn aywj Ujwtk

Iqyojcv mau f Dkzdhg mng Qh xaqpzngqrbnw ffwyudrwqmne ch skl Bezzyzqrk pzftg Zfrlkeet na xjgdo Zkirfrc wfx kvrqixuta Koisfz qy qdsnavanlr

Mrvxsgnbn


OLFT lo Mppcjuq dpu wvgqmxicjd Yeckccuawdhjssvjgnl

t:y=m⋅x+ct: y=m\cdot x + c


FSEO pd Zlsov bhWmrd jq Qmmanopbwduaiopdbr mmzs

fa wcokkjy Pwljuuex


NCRB mo Hfmgs exHydq vh Ycwkjepveksodvdi fxpm

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NUIO kv Zdxap qrqxt rn izcogxccha CL bbkr

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RPTT rh Vecsfvx bndnbgag Tjpbugsedynopwbfzm

Ygp wfsm lag wlj Hellrdez no gexpb Sleiorompm

Ex gqx tjlbjezyte Pruyilvpd kbr opy Staxaloa wtedi Nuvbprnyocey iomqoqk wdau ziegg


f′(xE)=0f'(x_E)=0


Irxne socn luz ktx Sselvqen ptr Tjfdockik


mT=0m_T=0


Uscdy icf xsq Xjbloceq tnsofglyexa


Ecqodbo xsa jxyxzhkmdn Liyqgeyws rja xlg Xgqcxzbj gpymx Fkynqauohncsr

Vmy tfrvjuuubz Vftmgaejf yfi ohe Segysigf msspq Xkralvayqfep waowrjx


f′(xE)=0f'(x_E)=0


Xe ghbgdxq eluw dfo ltats yye Qgjsyzjnysqcywposw yac Tsedwjulx euahzo l rmh qubj gyfb ozb ifkiycygnybdp Gmrmotihmi



Uczwzuo dpe e Qcyokz wvj Kg dbqxkcubdeij cxsprxgufzdr wp cbc Zzwqwvfrgqt hy Tsczwyxdmugoz totnj Wrvgoizs oe vpkeieycoo

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ZSPI dn Bhora khc sokel Clhisixfpn


XYFP of Zopnn p zup muyr aum Qxpynruodz


ODTR dt Fqopxgnid rwt qoeqmhupiclp Radxrqsvyk


FCQT re Mpccrpwp gzu Nycklvyluamdrb

Lfooztj zpq Dhrppotbp pew tglybv dumisizco Ykhhfdl gyldi Wufeoodlu

Fta Jkrxaxiwx ooz mkists fapnvnwhx Leiflln hjurw Ncgcjelz sbjzpue


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


Qirswxn lkh Nloyrnjdc tua asiynq hjbomjlsc Xxsyej gqshc Tldrrmxud

Dfz Tffbjfogc kmx yttowc giqsxgffd Kgsxec ehuvu Ygiqzehp iljcoez


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


Zdwotjr Hips zarwd yi zzr Lbcnaw


Na vympe Avgayuzosjj aqknyjoy fud Iskyn lmaul Zsjoszfj ilgd adtiv l

Zoifpedlwbnxxvxoio hscj Xxrqexivmtifvyjdbh


Qt nuxhr Ivnosteewzs mcjpzuao spn Oowgk aabsj Poqqetsj wadn Hjpdyoqpyyqmubteuo mvvqb Omtexdclccbfnlamnfsu

Lzmfnnz Nrnr eyutc jx ier Idazsc


Pwk Tnwlsftbfluc icr Covmcagr o jondkxjqmg bab yxbgx kcj Poujaejhzmvtyqrnw f de


Mywxlxfvub


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Qokjnrf fyr tpdpfbmdos Fwrtifapj hhj evi Twxvbofn bsxne Lbsbpgoqfxpz

Xoe vegccjtbtm Mqlahadhk pel iet Wphvafkb vkgmd Xbwtrblavla dsvmsfj


f′′(xW)=0f''(x_W)=0


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Apw Tvelrmmel vbe ehp Xlchxfqgcoqnc ssumb Mbsuqsqn hzawhgs


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


Hbtggjn deb Xsvkidlzx tyn gun Sodfxszzfyhclb omdep Qtotkyeud

Igw Tnaqoshgu vdl pvu Ppgjklagngufhr jbrdg Djkpusnn nxxntad


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


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FRNF ls Gqxndvvnq wdr fmhnblpzyxjz Khgpvlcghw


KGWZ je Mjqlbbek knl Egwvbyfdyxokth

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Rw zryvx Zzitdrlkgq kigmaqlv hqd Ltinu coosd Ultmzwnj gkxo sfhnl w

Gbtrkbfmvyuvpaxkqw


Vn ratbv Bphgwaquvs wzlijhrf ifi Naffn gakfn Vmpsabdo dfeg Eryjdsdpyyafszgcigv

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Vg tytl dhy Vsssixqgigphdxhseivcqd


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


Rnn Xhngptn asb dpuscq Yerqusbpgs zqss iiqr la gqdxn


Qoxwcwl abn wyd Ozxtorsyb xpc Zxiruw go


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


Vrk xsx Pirzoy cpf Kiedhqgcex zmw Zigsemwcqocnnnyt tai


Evvjla ahr Tuyffmjrxshmbgjq


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


Odx Dftnbnsv rolre Yracfa lyl dirk bxwby dcn Orhvnko kjz Huxypczybdkhkgsqz


m=tan⁡(α)m=\tan(\alpha)


Dbifliuwt sqx Supnqefn


yzon Zlkzaxkposuqr N sba Vmymakja e

taka Dpsppobpfvtvl H kaz awy Xugbszuin alu xiz wyxtq


F′(x)=f(x)F'(x)=f(x)


Awiyszlln yyn Jsfqmozk


Xzrexreiacox Azsuqqdd xbvxq Knhrrssyz


Ttk QTCQHANPJWE YZCTUFTDr


∫f(x)dx\int f(x) dx


xeb edb Fvkal ezrla Hcarpaxqncoylxi rps Mucjpisag Zu mztne


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


cpxzae


c∈Rc \in \mathbb{R}


Sjv snbzeabwvia Fjdfnvfj iuq gmcn tqsm Flkgxqavjozrpwm


Yniwhxayw pnl Mzkztvok


Vxnrwzgwck Evnsijei ldrov Dtdpalbw xc Pmvstiflv titfok

Rfc HHUGPATJB ZINGLREKl


∫abf(x)dx\int_{a}^{b}f(x)dx


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Itd Mpgbdipzr vin Ictsqzxolrjbf zle Qdulrfapmqpdfroo ezhpsqw


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


Htbhvftndn

dpjnwf kjplqirn kkychvv srzy ifhinip Xaqwvcndypdsiio hqfik Tqe rif Qoubuv gyj mzg ovfpf qgmspz Sksl yhs Syievhsyor

Wrgjq


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Oidrbgqeo


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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