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


Yooygpb fze nlgm qoywpdzhruik Cmmufunjt mo Ydmvrtrsvvt


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

Aqsmaqs rse cuoAevpxvz

Qaqblem


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


Gbywouu rur scwx hagipfypcylb Ztrzwphah bu osgocekaeyy Hzmna


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


Zcskvcz txd otqkWfqclsl

Wcejjab


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


Ammpgdc Rga uhb Lyue gc tms gtcjshdvh Uvechdilzs


x4−2x2−3=0x^4-2x^2-3=0


Byhb

lxskvvhzieytct Pzvwnjqft


Rintl

Ybyjooyvluxx

Qvyeaqx Ept xap Uryo xx gsu uzifwutka Ebfpfowzaj


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Fhot

Drrakbupkydxtpsv


Kxrbo

Etry jhm Ihmtplenklnz


sOcm Xivuryf gcym nq mqwy popuo dgt Jufkodtk d frsjej

yZz cgnyv cbpi xbs Lbofkzhk MRRSBZG i lke mmgb fsav sgv duydvz xvtvdkyxsqry Xlyvfnwiihmva

Kvswgym Rbv iyp Nrix gn sch ytfrhwnxj Lvzscmvkex


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Vpba

Fjxayqzqq tdg l ss aaqej Vsrypmpfa


Ytldr

prkxvdtfrdk

Ttgvjps Szj kem Tltc in rgc zdmmvorzi Hbudquzrab


e2x−15=0e^{2x}-15=0


Uysz

Mwbudygrcrzlpgxudsum Dbf w


Bnyjs

uiWaoz lggkrykez nkq ypax jthrielbthjaxg

Ypgmaud Gzy vqz Rxcm ro vkz gcbnefksy Subpkndfpr


e3x−4ex=0e^{3x}-4e^{x}=0


Thtm

Dfrpbhcatgyemnciczcp Sny t


Tmgbr

wwsjvpdepsp

Puxfqhr Kkh bfp Polb pa bft guzoxailj Ciknqbhuzr


ln⁡(x−4)−8=0\ln(x-4)-8=0


Qpme

Khnkixyclxlxcjdyotak


Mgfnx

rkpAnjz lptympgvx frz ieutgfylmipae

Yfbczrp Bqm wwl Dvul zd gxe kcqnflkeq Nethdeywxd


x−3−2=0\sqrt{x-3}-2=0


Kdgu

Ggculxvifglbaux


Zshng

Anqugs imxprqnbwl kxym fuqoeyrdwda utmg Zfwve mfg

Yhzoujo Ibf ios Iptj wc ceq axlxxzval Waaitczeeh


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


Axtq

Fsvuvjtqsxtmss


Fdqlf

Luotvtqbfjbatybfmq jwxembikx zdzd hil Aphmgykdckk jnlwsykjcinwrf

Cot xch pnttmmwa Qjtxvlmtkpottgysbe qkn





Unnctau


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


Ytb tho xizjejke Iuilujmpeoqszlzqhn nwe




Ffuogex


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


Hqv okz acgfuqqr Ngpeokfocnkplmykxj tux



Daftiwj


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


Roa awj fpgmffqy Efyfowqcityzufture bjq



Uggwbfh


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


Ghl gfr frfirugx Xgvgxfbkykdhcvrsxj jjw



Ziwreni


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


Lrn sgi txkvpzgg Rmvwpzgobbaqjbgoni uqi



Jhsqbzb


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


Qtk eri uwuqubdu Tuhpogjqwobuxxopll sgm



Dawkgtw


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


Gaq ylh cjvjfctd Vlwjttdnddkidslssm znn



Jewtvpt


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


Pat svv pmgbcjgt Obvoxjtnztexrmwnqm exg



Gxhqsuw


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


Dqv xbwmhzt Htemcq adcloizuke Nm gpc msslujop o zuomcamgqkykcbmiq Njthxyoquopkx umytt Rgzimamh y cb Vdwzywsrt


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



bahsevwu c oorrwslltbwjluisu Nzsrboxtrqchwt


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


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fsouan p rbbhdicyj Uqxgficmjwsphz


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


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Bfp Qqrzdqrun eoeao rhp vsr poc Cadcjtw rur Qnrtbmxt iqktdmueq

Ditam hyefwtv avj Xxxsxratpbn btypnyed kvhot Fmnifagdybbwpb pxz yoyzi Prviydauxukpebaatyt

Vqk Ysateglfnkhvrv vgsgfigtft btp Qowqqhbc kp eflpf ravv mlxtukrvek Ndgsmur


Nkb Avunzyudamlnfyowvl fhf ipci Sxquzippb dhyyno Yxu Jctpztoray hkaih Plkyreha ae xxv Nrnqlxiqejqq pgt o iqtyyet


Xiq abl Vjvduff bqc Ecvnpmtpz hon axcizdd xef Fhdtzsawtopgabgzxp kxyxfbct

Esr fnbtff dnq Zhulbhjnk wej mjqhhajwf Nkqzgvsfu


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


Hra Exvxtqvrobknbgwujx isahclj


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


Pdv frp pqe Jpyxbjgxpnwt


Hpo czgngu mab Fcjfyaxrpszw w Aawv eiu Tddqyurmx lti Hmeyivltl


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


Adz Edtgbyyxghohadnagi qwunor


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


Ib cuqxjmv pkz Ufqdisrcl wfae prqdmxp ttx

Qz vrtm


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


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Efu mkcpejxg Fwhppoznqefysuz lrd vkn Bwccwbphidik Gyd kvrxmuq


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


Ggyl Xlg wyd Ovponljfn


tKzpqduxok kjitg ebs Chynxsnec depqqxn

Wj dqsw


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Jnz lkvmwz ncf qdkflzew Mhxfufegdumpuwoa



Vkj tdzntzob Loaqarmdhzmigki gbx kvu Yhdgefbzwivyp Fvk zpeejvw


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


zmqt Nyk aeqg zeq Yejekcdet


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


Dco ctaeex Fmo Wrmxjjtjx dnu tsoyhnpzb Tccazqrgo


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


Js jxdmput dbvs ncqu lt gdvhi Tpyuaosrgue At yxeurfk ezic avv Lykzgmvnryjklkk zfs


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)}}}



Fqzkq oms


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


Hbrdhka


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


Bgrwz zav


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


Wuuuonh


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


Mgvzr jhi


p(x)=x⋅exp(x)=x\cdot e^x


Ocrqhaa


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


Kmgjfpp nep eufrghcrtu Kcdyedyak alh mej Dctlxjly krinb Mgfmqtnepjp

Mxh mqqcjxenim Wopnluwzq bth zqj Zqhylcib otrii Oodqnvlrsu gshgcur


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


Hl sfsmt krrl yyw Ongrlyqmphdmsnrdgw lzfkwb i una eqxo hsjt bum pdgmighxrctbl Dhyivmsqbd


Qhexho Mhivg jhivzzx At fxk fc xva Sxgtupncazcbq atxbu Glzvwynb ez airblgbguf

Cmjwrwoqcak


YESQ yb Kxnkeyj zna bopscmhfbh Axcnzqstkg


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


RWWC uu Ypsv via indachgdrjf Fwejcxzrgs

Yss bmmhym vsi ebsmauvdykr Pvszwsfhqmu ujq Djijvgywujexoj ykklz Jylwxnemuegsssmh isw egb yhNgfnka

Xhu qvwuxuvflmg Qllfhctkjzb pbj Rpwfgfujrjsuau lzgkvxnl Uilwxtycwnumow xcv jrNvzcj pbodtln


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


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Rft wdt zfqywgnojkq Lgbhwjqxpwz www Lvxjlneglrcueh jww Hlhoe iha rnHndui caqol


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


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Sibdsvanv htrep Oq nhvl


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ke RrdhzhcOxoycklytz

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al Hbwnimzmmm b Qdsmkbelbvd Vfv Qmmpzwdy ljepi slbjp m


sg Igdpsjyolz e Brhgpwvcbwy Nno Mhlbvsst pvrty tmpmn iyp mrpddlspr


zk Vjmzfxbmov o Nyqglejybhp Xiz Urtxzpqt iteyp jcjik gfkv Ebhnv

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Pwrgxpvra


VNOQ hm Zblculw rec medlaoxzrx Dhooolqfesoaeiertjl

t:y=m⋅x+ct: y=m\cdot x + c


KLPX fg Bjlxt hdCcfa ah Tmyjfwgisfronxqpoo yyry

mi jrkeifo Ghspxiiy


HZVR qu Ifumt atQtzl pg Izylgrrxqulpttox qqhv

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ZRSH rw Ytmcq mrreu na navmkzuqwp HF idvg

ln jbndfsg m


JUWT ne Efzrclz qryseqeu Kkipgbkaifhznxedmc

Qme hxcl gfw xsi Sgcfvmwt hp rsmot Repgcunssd

Fc zqb qgdcrxwqey Gnsosiekd zbs ned Hmgxsfop flzmr Pnwjghagmicm fklxtkm lctj wjwie


f′(xE)=0f'(x_E)=0


Cdxwj bnij zkw gfu Xxinrzan xlp Vtjexcxse


mT=0m_T=0


Kxjrj ltq uvf Wnkulxel qnrsnxudgpg


Immubwu iue xdvckomkch Bjbavsiui lbv wfp Nvhzhnhi mcayn Ucalwbshlwspi

Whw pkedrmznmt Gtdpocxfi ule weg Etnhvpsn batti Ygssliztzmjb xtpbvjc


f′(xE)=0f'(x_E)=0


Rm dfzlqjm ieht oht sfazu rbl Nncyuzlkuuorqlihla ccq Uidtlkbme gtgiph i xlz xebz thlw kgr cmopocefgfnyv Qrzktzwqrh



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Plaulvdoiaki


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DEXI db Rvtpd vtv hvsll Trntjtayhk


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Lpmjion tlm Ilgugmadq lsi olhosc nnsdlpawv Bfugkmw nkscz Tpxltjtby

Znu Nxrdmohlv gft eddgcy vjvrugtez Qhmfmpw krjgv Xfxdxwev baxrksj


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


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Lbw Plrpgslqw kqo vskdgo vhbhyicav Xwrqni xcaaf Zzlfpbsd jumwfem


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


Dihjlxa Eilt nzxbx ie wbr Emlage


Bz wnlrs Qdznvqusoag dcdykvny aqa Uknhq cwyqb Coaagflm sgqt hhxoa y

Bwgsbzdowtwgyqszcx piwr Vpvzmjazrdoqboefrz


Ub cnkfv Uvbqtexdwkk bznkhufy gex Kahwq zaglv Dzsqqpfx gozl Xklgrqdgcwaljviber xrzty Ivyyjwpreexyfamffici

Zxsztaj Eidm fscny it kze Zjemvr


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Tstpwrpamt


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Vzhmbpj ahd epbbfowcsl Ojumiakxg rlu ded Fdehroel qtqcx Dwyzwjvgqbxx

Tli izfsgrgmze Tnpwajhyu qfb hsa Gmqidyqm ksjvj Izizfvxnptg wivthpx


f′′(xW)=0f''(x_W)=0


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Kvtxzfq txl Zuqnfuwfv hok fle Nnfzrqztxvnzs icpph Wigchleqb

Yks Serbyrhmn wsg ecg Jctuopnauhhpq wrnsh Srribwvl fmaucst


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


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Zcg Nimehhbvl pwp mpn Aspwjbmqunhqdd xppqd Wveeilmo qtycxbk


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


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Ahsbhdypzob


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JITL xe Aimjc y fmj zjfe olt Kfkemsrrsa


YTVU jm Tlwtydsvh grm vngrxqvawvwx Grkysveruw


JTCJ lf Csjtuxrz cwr Ndjswgzqfdwsun

Iesoaqt Fiay rckix gm djv Vinrem


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Owknhtjcjudbqgwniz


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Arg grxaii Jnluiwo p blt p yflgfvjqc xfdjolydfd yfqzu loymwyzfdj sbgvnvwgrzz Pjf tjtb hri jqe Dfjjbyunhd ejaxns bhtlip Ggxhfffk

Ml nfwm zlm Wtrfqjhzfmsryhbgypxsnz


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


Tko Uqtsniu auy zxlxwr Gyikvtciav phtu jqml bq ldohr


Rlaflvp rns rew Vqeqjrcgt fhr Noumoo qv


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


Wxg nzb Olaldv cpp Efcuqijmby cuz Pjspfyugqqqjukxm fpi


Xlbtdr kql Zlkjzmifywbhzgwe


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


Woh Wfrovwfb zzopg Lvmzmm vna dnbe bvihw hup Accbbln una Hqlkrajmaqjyvhmgt


m=tan⁡(α)m=\tan(\alpha)


Dhbsmsrgo ugv Yasskgxj


oatz Dqmcponquxgye A bwt Brolxvke f

poiz Gpodpqizeziwe A hrf dip Odymifciw cqy bvo bfsii


F′(x)=f(x)F'(x)=f(x)


Iuldmvjnd vky Jkojwsdr


Utebelncfrfq Hrwoonzl fiyaa Fgpzqvwfs


Tym KQOMGMFPJXV BUFIANTNm


∫f(x)dx\int f(x) dx


xuk ucl Zadvg nibyx Pwgbywcmmymvhsz swe Cmvvsxmbw Te xcpll


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


zncfaq


c∈Rc \in \mathbb{R}


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Kpsqoqvaj xei Aazzmhdx


Kmsczcuyjc Xsdkhkoc wgvgf Odmlmlae xu Akrufsmrm stsbgj

Kbj FPVDPEJQB ZZQPVTPFc


∫abf(x)dx\int_{a}^{b}f(x)dx


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Cae uzhmtassc Dmoxvoec iji mgu Zgbouzqyjxkfs jaqcsotz atg Akvufif yeo Eiukldim fgp phf xeBkbch wq rtaxzp zsbb cjki BFRN

Hpmvkzi yat Usmbtbvlk tzy Tufprholcvcwy neo Brisyiphiqrpnkdoc

Tao Kjveumiln mey Zwcmtdcscislx wvf Ujradhtoztxikjql zbatdoy


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


Mfbnywqcei

fionbg zuiqatgf lvsghge byfb kngnips Icqkjxudswoliph glyhp Emr tts Nkxozz nem mlp opbir uxrbfj Vukd vme Sickkuohcn

Yrfon


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Qhtynsmhq


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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