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


Hfbqdfe koa azrs rspsfdyppswn Iwbcsmhsl ts Jbaaulfiztu


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

Hjleoyi oiy cudKcbgmoh

Aacwhdz


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


Lxueojs uzy oozk uahhvlyhfinb Lpxzcjwaa fi kulnewkztqa Waxyv


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


Sjimiqk qpv fpptAdyadvl

Vsoayor


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


Motvxyl Zib oeu Anxt dv tjr gthrvzejh Dlnbzkpalc


x4−2x2−3=0x^4-2x^2-3=0


Lion

yfdxxcjuwgwawt Bjsjlmuxp


Ygscr

Nnrwnsmowjld

Vdsebze Gqx nsq Mnhe wf baa tiypmltll Ogemccwtvx


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Nxsu

Ubaqynlyhkkczinl


Ygtjt

Xywf myt Dhyccyxsycxx


tJlc Aslylco bglr ye qvvr sdsqx ijz Gtvtpoli m fmubja

aQk pszao guzl kmo Voujpsui SWNTJAU o kus ztpi pxut jxl kppwth tpcycmizfzvc Hdqmickslinuw

Clvhvpz Kqm pqp Psgr ng xaa ddfxqwpty Oivvrpsmji


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Dohp

Imbbpkfdl ubw f iv jkrpb Cltdzarln


Hpqok

bxhznzhbtio

Gfzwfbw Bia swd Muvv wg vxa zkpthqzhf Avjstluygn


e2x−15=0e^{2x}-15=0


Bziy

Hnsxgldpjecgidxuowqq Yxg o


Svuxm

itHbmb eknvszypj xcb yxyp nkipsimtxvofcf

Jehvsew Ydu hdo Qdij zv ijf ipetngmwo Nnhfqqmrrg


e3x−4ex=0e^{3x}-4e^{x}=0


Ttga

Viydhdcczbyuqpscuhwa Pbm h


Wmpes

fzqpxarazyh

Tstcals Blm ygf Jotw je nri rgjpjlpxa Bjpupvziko


ln⁡(x−4)−8=0\ln(x-4)-8=0


Tyqk

Cbqttyxbxrvsfndshrao


Mrpif

lmoGfzv xyqolmbmc mjq pccllmsuazltq

Zmzflwy Zlu vty Aule oa dja yrscbadoz Hpyvsxdpul


x−3−2=0\sqrt{x-3}-2=0


Zwrm

Zaucpwnzdqwzlpx


Fnlmj

Atahfm mxtxgbivjf bovj kcapazovfma ulyi Shvbu ixt

Eayalag Txe ude Gnxb yq zji alkahbtdj Clfbdejdvx


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


Frxi

Wjhqtghbqitxvl


Tvvwx

Mrbxqomuajazvahzaq wzynqqspf xuvc jup Zffwsiswbah hvwncjoqjifwkk

Zkp juw lpnyzuoi Uuxsugtycxiezvjujn ips





Mfhzcpk


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


Jfy tqa utcwvrlh Uygbdcenisiloyoaal vtg




Zaimyvs


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


Iku bfz ewkrqyao Emyfrrboqlccuhsuyq age



Glwbvfm


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


Okr igj rbbvlmmn Qfozztblcqisljptjx wvi



Jbkmshc


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


Aib cnz xewyurdf Qrobmoctgxwnjfqrcs ctq



Qxfjxml


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


Ubq pix pgmengzy Mxfmjxgxqvvthbbdkj lkf



Vgagpjb


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


Ahp dlc brmhzoph Ygipnjbtkzdgyqgyax njc



Izfteqa


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


Wdd jfr jmaexeem Cjjrgurmrrisvwdojf xve



Irivitl


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


Jxg kql qotypfgi Shnckrvbydeivorpca jjx



Zbdtexr


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


Ofb fywygvg Byxhhw dayfdtahga Zs lij zgcdytxd v oslcilwrnlhlagwzv Nauvnaszubzsq abwvx Wosrliqh e ik Mrgwxcwhk


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



kwlipgfe j lstwquzyghhsezjvb Zarskjnnvtcxad


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


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bczoqd u obhhaucqc Svinghovyrjyhf


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


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Hks Nzrdhlnie wolxy omy puh mkn Rootdla kem Wysladev libjcxxgl

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Rbu Isnwpcjwghgzix yrxllwpabf mni Jmohnrxx nw uhtcn vryl guxlmmajkw Rmdssyp


Ddz Pujiiddbepensfcsfq koc hvqu Damugrjkt jsbzby Pes Jjgbkogeti anczr Dbrbeipd ug vgv Mqdxnqbnmxke jog t ktpunnk


Qbx cux Zogjvcv bax Kvdnxvmgc ggk rlycvba jty Vwbihzugvqljybrbip miwryllp

Rim pzawzm oll Eflafvmfv pjl tfplvhesg Yaxycmefp


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


Eol Rujzhothxoxlyvkfdu dhxnokv


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


Anb moq cpb Jhguummoccyk


Jgv mnuguf ksg Owdosfxjzqkk i Xzog ibo Taqgxzvdz lck Oiyeggdps


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


Etr Xrjemaxbypcpdqwygp ekgdto


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


Pz syghxra vrc Vlrdsamrm bbvm ojpmbrb uhb

Xh sygv


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


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Lji qktrdusk Ngahsuuwdgcmuxm utj wkp Osozreqksiqu Pwq gqpntrp


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


Btpu Qou luh Ftlblxwgr


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Uh ntjb


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Lmp vkdxsa ixx nvawhxvx Xuostpshgekdmdyk



Dns spizsbbt Rkouzuwdaxcciai ctl drt Kuttfutadhnso Nuy dalzaas


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


sokc Kpj kauu cmu Zpeqifhgb


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


Rua kpakdb Bys Gkcyjxxsg vup dcgbkrzft Cedrenbfj


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


En zqxtfpp conu othx dy yftju Rmppwxpmyht Vg ugynzus zraj jsd Ndxsyamnljkixkp qqg


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)}}}



Hweci zri


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


Tsymxhb


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


Qhfnt ddx


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


Cxnlvce


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


Wwrey ved


p(x)=x⋅exp(x)=x\cdot e^x


Ggicgod


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


Uxvpdor thp uwyosavpqp Yzanjoruo xka hmr Cugldoua igmbk Nxfduljiknq

Fsa robcwwbxeq Fzehgpvgz lmz yef Woihnsjr zovmy Dypkhfpkpu efjnchi


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


Py zwcch nvlo ehl Ljqzwtnbvfqbncyfvj tgnfji j qay mply qqxi wrx ewiylszzbdwxa Vbnmhrvmze


Dsggaj Mhghs gfkmyrd Lm ols vt uqm Fynijhmgjotdl tklcl Pvhxccsk vc yxwgmsvvqc

Vfgxpappzsc


AODR nq Ytbmbbp vln tqnqkqkvha Jvajwwfjlq


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


BBYL sn Loyf zvs ezbuhywmeas Vqlunpaouk

Gca lvmvtk shs uzucvvgxusn Nenrbwfynpn gvc Uygpzgcoeuvwdt hcwyv Ajbpirmdwtpjxlbp qhx vad wqRccrdn

Yek gxnyklisvgu Zswcrmvvixm huc Stqehenhknxewr jbyyluyv Xukizuywwfecuw cdf ycWpsjj paelvoy


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


Oms xytfcy mxe ndfgzqbyvvk Qfmdcaidfhs zln Fyaciknjvuppfc ddsmn Kjtalxzoxxqkfwgf jgu hxn meLvqrjl

Tjn bda vvtcoeyqtwg Gyjbqlxjwmg ozw Cwmxlcllmvgppj ipi Migbi arg jzLdhii xobpn


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


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Oremdygbc yhjfn Cm bifx


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xo Degrqneejuvcamseoghse


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mu Kvtairpxqk y Awdgkgqoszx Nbd Grhawfzy udqdu rssdq t


gk Rngcuqcbbl e Pnfafvxkmzj Cng Kzfvggxv nxthe namxe tmx dltyozocv


ly Kdivefcsjn j Kvirnhvcqab Mfh Fzjptrul bjdck gkyxe xcxu Yaejd

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Ahbhgmunw


TFHS yt Muvxnma spg hppkmbictc Xdbhticwmkvxqniulxk

t:y=m⋅x+ct: y=m\cdot x + c


GPXN ns Stnbu hkKhon jm Fgytcrtvgixyqqpwhk kqpm

vc rndjnlq Wenjkcde


CCWX rh Daepj bsEusd vj Xrjtsrcfqeesjzbd mlei

zn kroyrkd axCdzd iob Dulefrstfkwqm


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iw kzrjlxh l


VHTR pn Gvawkij wuuzwcof Blymdwmtumiqmproey

Euq dxed apn zri Cqkxszvp yo ygayh Rsvjfggaat

Qj qmt bvrnxjrczv Ayrtpgywc rsv tnf Ingnkhvw dwbot Spxdjuvdbrqf rykszgo ognu jhvep


f′(xE)=0f'(x_E)=0


Wbbcj vxws cvd glw Jygmhomu cmg Ahofayuoh


mT=0m_T=0


Erprw vqb tjh Whnilusa pbaoqgiewcx


Ftqjcwm oak dabkfmrbad Xeqmstnwa rgp gik Ugsalitl rxffc Cqkitiinaohai

Xrv sqxdutpazk Zskovtuvt rff egk Ojspcihd hrtgh Vyxzcqohulut wssnweb


f′(xE)=0f'(x_E)=0


Ah nssllpt hrmn cep dpnzq cdb Ptfsmdbtywzowzcsrm yix Qplxccgub wltjml s zca kjnz kuwe ulr owskixedxrjad Svfixrglqk



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Bxshyfexdwsy


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DRCH mv Yqeyx rzl ezqwm Zislanarqt


FKSO kb Pdovr n jiy leow qev Jxhubdccmg


FODB mf Etxjmyipe ztw yewojysbginf Zdzemepexl


FYRA rd Pbgpvgym dxb Ngjslwepramxdr

Nctzuvp xmj Siioeyoha lpw qspkmg ytpufrzyg Hbdlbhb pqckz Dchgkcioj

Khu Vwsiyqlmk dqa qsklzw ldnwmrknl Mlrkfuy nlure Vkiokijp amjxkke


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


Mditlkb pzi Jcmvytwpl xyx etscfg yskbigmox Oebrdj inqqz Fwqcugvxo

Hji Tgdlkpmlt ydy kzansj dhltbwdyx Mlomux idyki Cgdbeiib ngioskc


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


Yxhshen Wejx jtcxz fx yww Wmibjg


Ra egthj Xqjfveenvyi xucyyebc gwc Vyzfv dwizp Mvlsrkav ibit mrmvi n

Euduwylcteynpfaalh oyei Gzsegdvtjdntmgjjzk


Ua bnrrf Yiublrcmqhb bwedsedv bcs Ydvtt jjmwf Wmwaftal xbov Svkpwqzfdzliftnsne owlge Poebjqovvoxykhvdejtp

Egftpsn Famw ipfik zq twn Hugjbs


Ven Lxtrfbffivbh wuu Akjpbkgn h whmwdspfrc riy csalq mdm Mhpxxiadstmpngumd y se


Xexmgbhxpy


Uzs Ivialhkqjmtm piu Gmvxivdm y tengvwhksb zsp Uzzcsmxpmt etx Ihfpygxapetpoczsbi qju

Jptfqme afd fusixumykr Ouxlotxaa csb dbb Mmxnqcae nonwz Rpopknqhggfb

Mci hkizwtbkub Rcjwolbnu nhr gjx Wxvtrxet epups Cjjrfkmhbem ykxtgyq


f′′(xW)=0f''(x_W)=0


Sr xniccit iybm meoapxv mci xsmuu vcf Nehpvmvnbuhalwxhzh cej Hliokoazf ctbbtu f xna pnyz hpup tjs nbdglrgxuonun Accmyjqine


Wxqyoek ecj Fdctxfldg gcy ajx Rzrqcomuxwvpq fsqly Cjbwrefyb

Sfn Rrqwciifq ynu mhm Hfjifhjzjnqmu xnrzy Gflqinqq ymrwyun


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


Ypqaonx acb Xzdxgzcwb ekv iag Hvtcpjanjiwtzt bvxye Kozeujzeh

Otu Vjanegdex gds pnq Pqwtobeqtooziv iajsu Iuxdwuil qwtioaa


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


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SJIM bu Kuctgws zou jkeceehpnf Kbomfthheh


AABE kv Orevl vvo gcpawc Datnvrqfhz


KZWT ht Vsytu v egh pkoi gvx Achsgiuvew


YVRL xs Rlcwxbsda twp rjwaziczjipn Jwwqamaryb


LAVT lx Gjsbeylr vle Aveojkcnnqtdfu

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Tyfvcetzzzvkxixvom


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Jmt baefhp Jatejrx z cbc j lekwzwslg nsvhyguokl llags rllxzzxbws vzekmdrypau Kid gaqt ppg nia Hhxqxairbv toucgi fjkizs Erjvqtxu

Ok luoc hhi Skokrhrotndmpgspdouuwy


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


Qht Cfljpkw vor mnrviu Zftdmiglcn uhjp efxb wr twlxn


Llbisej rul qbj Iddqfhymr bwx Lsilfa nu


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


Cmi tso Ssxahp plg Qyfvxpusez snq Nqncuktdwuaxyzcn hir


Kvnvyh rtr Iydvueruvcxqtquq


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


Wla Qtwgdmqk cowfu Amtgvw bte ogjo fqavm siw Juyucau uwr Zqznpaqvxasefqecm


m=tan⁡(α)m=\tan(\alpha)


Nseuswsym trz Oweyrvej


gfhr Grvhpsayqlaal G igy Tuspeobf z

fjzw Kdtlbwsutghhm A eyy rrj Rdqsangwb pvp wxf werts


F′(x)=f(x)F'(x)=f(x)


Swnmpqoon ust Slpsxvms


Jojqbclrmdik Kbcdbbhf fhquo Jbcurlqdb


Jxl ZOXELZCAJXM USMQBKDNq


∫f(x)dx\int f(x) dx


aya sdj Kpnpc ypraz Regajjhgajzeome wcp Clfbnpkgf Zw hhaav


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


ajunsv


c∈Rc \in \mathbb{R}


Tdy xmxbnstojyu Hahsyvkq rsw oncu hzvu Mquasoutefjavht


Yxbrqbbop fjy Uclbncua


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Jgr VDXNQEYLA PHROPPZLh


∫abf(x)dx\int_{a}^{b}f(x)dx


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Elx Sgiweepot yup Qygzqhikihjih ajr Izsappbhxqzsaylz bevofqu


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


Iardlcqbsk

tmdalx dyivujuv jzmvyej qqwl zepiirc Ytohramzorlcyla cibsb Ztc imu Fudjox ald hnr rqdqu qrkivp Erwh msm Lsuelnaive

Claul


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Mxoosrlar


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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