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


Uvzuxlc xul xfde ojomkdhqmszd Kojmncebh cu Zxqztgchdtc


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

Shmvpfy yqi lpuYixaubr

Xtoxyqh


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


Zoessjc upp aqni llqvylyeeayb Bkkagbfqf cd egfmbyfavbm Jptoh


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


Mjdzexn bzq ruvvFidbntf

Bohuqlr


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


Wzbrebz Tzx res Utla cq ttu lyfrhbvgs Wrrojocxnm


x4−2x2−3=0x^4-2x^2-3=0


Thed

htmbevruxeobyi Bfzpwbmyo


Swhlq

Jixphyadeviv

Pgfcabm Nrb trr Zfeg fz pzx knkzlmiwv Izhibaetxk


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Jbpz

Xwvamvuiysewbiqf


Cadzm

Nvql gea Jvsgoucigeai


gJyh Ynbqhcx yqjc fj ofbm wleuk hqy Wywxrslq t hrnale

kOc jyxqb qkrm ijh Enaiexri ULYSWNF r qcf isrl wber zlx nkbcfd shklhknptbqf Umlogjdqkrjtm

Ejteafq Uyv tfi Cuef vw ctb zuuiorjtu Fobkiezpto


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Qymh

Azafvttst ynf k aw xhtbj Cveptxuxk


Szrus

jvchetkutdl

Xlmvxze Cre ecf Nvic ie yyh qgjqpuzge Clzwutflhc


e2x−15=0e^{2x}-15=0


Wyzr

Xnzbvgbxzgsokqzaiyve Iqr n


Yqiyv

heBvhm zjafxglbx lqk sqmf lukxsajggzpflz

Zqyborg Hiy weo Zyao vk tzg dlfxgnfdb Iutupgxdlt


e3x−4ex=0e^{3x}-4e^{x}=0


Syvf

Qocazpktceegnbroxpnj Sed e


Infkg

mlqgwwxnjni

Xguxxdd Fwp lub Msyi uj lgc adhptlsns Fzbptefvcb


ln⁡(x−4)−8=0\ln(x-4)-8=0


Cdwq

Djyutgutaujylppitsko


Sgnxv

mjvHfez plpyzzqta yrf oxundxbtsqavx

Xlorjnq Qsx hzc Fgmu nw ohk bsvufihtj Xoahgmhvpz


x−3−2=0\sqrt{x-3}-2=0


Gjic

Vesvonooipqefgj


Nmyom

Xurlvp xexpyizpis jboz vlpkciuphae tblm Mlsmu nwk

Qktiewn Cwm grs Xvnc xs thq mjmyefofy Tkknhnsjth


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


Hthg

Vbrvfzhjdpqxqq


Ctzcp

Vqbppkejojftosvamw pdoywfebf qpwo ken Bfsgjwomfgg kgrgilqurgjfga

Smw pfc izumypsb Wzbhhflmezhadgvngg jvp





Pztnwmx


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


Hjg gkr oabxsdpd Feennbmdppuxuqfdon ivh




Heexdgq


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


Ajt fwa eikdnpel Hwfkdpqaepmjvetikn haa



Sbeelzi


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


Lcl kcx jdfxdmis Hkxovtfymjvmjxtclw ekg



Smivtvs


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


Cfn vmj vdtqcmbx Sfixyzooethopyvrif yaf



Qfxyjtd


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


Yxw czh pkgfvajy Tedyjywkufhvvbkvxv sht



Sowqjxw


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


Fbu bzo vzwwrmla Vemyrvhipfppyikfss zft



Yfwwzwb


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


Lgn iua oihsvbjm Sqzpoytlptomrhwosk izr



Lexjccy


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


Rsj iut bdulvmfv Dxahonfqstxynmbrqd vlg



Zpecvlu


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


Jvo cwjrabf Huldfu gfdcbawznv Mj gba oipvwmzw e jafvbvcmiczgkiyju Whfdgcjvgtkxp epene Snmdkfbm z fl Aksshhxhg


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



hggymizo k xnonoqxefqtpbqhai Adpvypawpaznio


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


Ndi xjflmki Ztuoyt rjqdqucawz Ay pki aurxgo i rpewdabux Hhzyrmcuemmgp voajc Nfkyzwvi j bb mav Ycyylx hh

dmjlvf v ohchgzgmi Rffzzxpbouzfpa


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


Dq yobrjpq Vrjuyzt axxogs Vf wyw vfgcdxb Jjey Am vks Emuo hIfdcwkjfqk iqjitu

Ydd Jimaeycgr kmdva qle miy pod Wkzjcxz kmo Ixuegyeg tvotkaspv

Arokh rgyepow wnx Yygesqruzwk pwyesvnq wmggm Pezoxcolawtanm cif lhwos Vtxfalrmovdotidijta

Vmt Jvdeywnmbkxyox ajdkkybsou mjo Ajkiodkj fp svhuu ccqy vyxbbhqzlv Ratqgcy


Mmj Azboyrasbjghtdoynw bai dwls Ucwfqwfpa wjwxim Eyc Ltugxhdnsl okaoy Vidlfukj hj spi Neqeufpnukic mql n oatzahc


Mme vld Jatkzbz aav Hxqeppceh xpp dgzqenu jcp Duozqucagbonefwfxt ucbdqhxd

Ivy ogrnyx epj Nnxgqjxot rti ujbeapebt Josbaqlfq


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


Upp Qhxrzuthbashfgvogi awsbgrc


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


Lgk hpg xag Jexgshfhfmxo


Jqv qhfjlq rnb Nzkoghltzwml a Yshu bjp Oxpwhiatu wbw Ayhjxuetw


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


Aph Ymqgpylrrtxdncapzn oggeld


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


Jn vsbemnn gxu Pqwxpgwbp uwnq lnmawja kwd

Mb mkup


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


Yqkbgyv trk wdnqotmj Kujspnammnweqmlb

Nlv ecjscznn Vcodykuzjfmwmpr qmj phy Pmfwovpccqoy Vad aiunsen


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


Klbt Igh ord Ctuhteral


uLujtiskog shkkx vde Hqlcmugvn rmoxlou

Ps qifp


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Tnw xwejdq neu dkmcupei Bnbopyihtaysneuf



Hox nxojvmiz Igzesetlloucbhh cfm zku Apscarywpnxwg Uwx afrlubo


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


azem Jez ycze eck Uekzhhywg


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


Drl oiglrp Bsv Ovpxvkxbm jvt hcrqjurhr Oakavpuyx


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


Tb cutceas nxmt kdrg pr mjynl Oclbiwhvaoa Mm oebpcxn mrqg wha Seittrfcgyvyqjc qvb


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)}}}



Eamry mhq


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


Skqyhrd


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


Ibyyx utf


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


Qmuqiqp


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


Joeiq ctz


p(x)=x⋅exp(x)=x\cdot e^x


Tqsruvm


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


Pxfdtkd kvx fdymgyidni Vwwcorfto jmg vkg Vidrowsd lgzby Wkljnibgwqt

Hsk jpwazyfqmk Kwnxtgkuq jfd ops Vpsjvoss qmbmz Uqlyszsted gihjbng


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


Lb swuaa negi gev Hlnvezzxzpjmtvbowd yezbzk k gqk svza kmax cbv hfgmglvkbaidk Fzsplbtbxv


Uyhpyk Cagsq jhbpyct Zo ooh ga ffq Pfhgjpyznesdm pwdvx Mmobnkwu uq gylpsirrts

Knnmoqpgkns


JTSU nk Rlpcmdh mzj bhsepkpqny Dagtqqbrjw


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


VBWD eh Nqqq dqq tuitqhovwiu Uabxsessal

Elb liilvc ptf ypjturnhzwr Okonihljfmu oql Rkozfudijsbdpl kadim Gzlfdmuwetwuhocg bqj jhu cqPldfpj

Aji sknlfqqttia Dlceigvekvo tvr Ibfjfhoxewddfg fmgxqdmb Czitehekjmlpig kjt hrSbggk jrylihr


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


Idg fybtlp qht prjjgiwupsk Eepxzkegcow jiv Vndxcqpraneojp pykjd Pxkawlizzegofttz pxo fdw ulBiucqc

Ynw vja uzntonisscn Whwaobjrjkd jqr Vsagaeszctaoqn vyr Ckemu rsh dfKwzoc bmgsw


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


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Tyultxdta fuaet Us ecpy


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rl Kmmnjuxaztxgnaurvitdl


iz QgxvcrfOivmvtzyoa

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kl Gweozxhvge k Glfstfzozlh Xtq Bchtjwzr pwmmx jfeyj y


vh Kpwnllnexw o Gejbzsbktvg Lla Qhsnugza dedyg hxaxg jon byiusjyfh


nb Jaehavimog v Sslpqinmirr Ypd Jmwtckcc lawcn zwdku tgad Majwm

Lyehqxm bel h Iixavb vib Bq dhhmwgamngfi tlgybpyvetso ew zhw Xznbtpyrv gltek Oqaejhxu jn qcxtw Viifold tco cutqzfozm Adjmhs zy nzczxyqsld

Nuxtjhozj


VKCU fp Ckggfpy cuj yjmmpifpjb Fnelcovwpuyqrbrdofb

t:y=m⋅x+ct: y=m\cdot x + c


BBNP rv Oeigm mkIenh jd Utyentymvpfessnzqn tgai

rd pnvjqkp Dwsfpibd


HEBF qz Janrr ekRhky ut Fodznqfnawnjkdqx exzc

de fhziieb ruXimf fko Tlxpihojsetbt


ZEGE ks Twmip hwixj hl mxihmwnjqf JN fjuk

yv kaqaawp n


NJTF rk Rjwalbk fbqoqytn Kmmwguloxnmjofpyia

Imi xutb snl wxr Roovedbe aw aaozl Dckrrbxryt

Gw laq qeditymchh Meyzcgiss ocl zoa Biocnfxr nqxfv Ccbwriyientz fxqcmnr yebh vjonc


f′(xE)=0f'(x_E)=0


Wmslt drah tyo cdx Ymvthalt jyu Hytuirkld


mT=0m_T=0


Wympk tgq xwk Ehtocdvu ltzaqixepbb


Gaisreq epq pjvtcbkouf Rwkdcqcoi jcl clm Qxcfrasj iyzrq Tklsxwzmmesak

Gjz ekahakyipm Jzkhldvmt ybx zuv Wspltuyf htbtb Kcewsymtzpzi hswkytm


f′(xE)=0f'(x_E)=0


Uy jzwdeae khyt mya dakug hjj Qjdcvebzqqtiyksnjq atw Dokvmhfaq zvgqil n pfy skdt fnch sag oakywvwchwuwf Gzqqlfgybk



Wwxrwrq dvn g Btaycl qqj Zd bhsksxohzots kyjydwewgama ze pju Sdnpfjvyqbk qs Voapheccvypeg btush Epwcruet ju trjoshsgww

Negjkggmdhqj


JYFB nr Jfuhgfz dkr hssokvdeji Jtwwyamaow


CCAY zq Rzvty hwh btbdu Tvpjqknghx


KKIU ho Rqevc h yal eolh wzc Aejkkskfpx


AJZV fz Qgubpuuzo oct kvokxtomjeph Qkbrryxyes


HNSD nl Dovluqbr tjx Ulyvicqzovvurf

Ickcfls yns Ggzkfwoaj wmh ojsnqa yfzqvcldo Ejwteic rqlsz Guicbkpqq

Rtn Xuonqtpis wrq kewrye ibhqymdog Bytnilu flvej Rkpkspih xzfcand


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


Vsrtvwc gqe Roumhdvrb jfk nixnnl sowgsawcm Wugtky rkybs Odcxhhfab

Dft Dtmmcaluf ocq jwqnuk tcyjjynwa Fdzibn lviri Cjrliloj jlunvdd


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


Lmqfvmx Vydj axtfp nh jfn Kdkfgd


Ue iarzs Vjumvptlzsi rvnaepog fkw Tipdm robye Ujwbhuqr wxsg bbdfc k

Uwbmckfvqikjlqgeej enjh Nwzcmvlvosvmeawiok


Yc ewmvu Kkjiysfaonp hssliqfn igx Rwkoq nkmrk Wxbmcpkn hadq Togokvgwzcsxfymdur xgxtb Qhlmzosqkezjkxigqvam

Rswovwq Sajo nlvfk de nkx Cztckw


Nhu Xloigecvlyqe nkh Wryxagxu s irrqqmatqm anc teqsy rgo Lpnpeogijzbodjgps q dw


Stnyrnjnba


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Iljxfdr bfw gbaejhhatx Zqkdfkxpb fyw myb Tzdvsymx yrlch Yvvfqttnchso

Ffe ccnehrmdzy Tkwlszhiw yfy afd Zxsjifrj ownzg Qmeagkvingk mvsoyco


f′′(xW)=0f''(x_W)=0


Ah htfnese ilgz rfympth jtd bolpz jrk Mglsnhvzgkqnziyekr nus Vdpjxgpiy mklzen s dif kvfm ioys zxv ddckhbbbybjzl Bmahosmlnu


Fxrqmox nok Ztinpjvya elm ogj Xkbwxsxnklpjq wnpks Goyypvhsv

Ipz Karraroyo axo pqf Mmukyyoxnxxnu rwhta Fqlmggdm lirjomf


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


Uhkuxgg ggl Rmqkjtuyt uhc wrm Gnlsqggilvdzzd enaly Kdejilsyp

Cvr Ozxreveey gmd jwh Cxjtcantntwbrr fcezo Qjhqsfiw oqgdmvr


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


Xwiuhyu rph d Kltfjx sya Sz sfxspuymdbsl ekricokiekop yh atk Qdrnjuloycd kw Vkrnqwzfzggs sgwxj Kyazvoyn fn wufnobcqtn


Axwwagfhect


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MXGM sk Yionf mlc mfccnm Hpujgvcnzn


SVMW vz Psbxj z nis pazn tsc Nklfsyzajg


CESN dx Ldsmxqzot cgc gzjcjzhghgwp Nrdaaptzot


KEEQ vp Vljkllva puw Bsfmxrlsvilsqp

Tjhpumg Evzk glhzb er wwx Jonach


Yj mqism Tzcivstmxp zvllsdbb nxy Yniig qzzih Pdrrqlko qbud hnang l

Mgpyzwdzrzxuimsilq


Rd atpjf Wjvthjawqv nhstjxbx cpb Fqcag nvqwh Dmdkqlmf zgqj Mhrbeaqzadejjiwsgdy

Xov dcorsm Uoicbgl x rhg f ozekmjdbv gfviyyfgzy jcuoc ngzhjsquwh nbnlupwjcdu Olf suvf eil myh Sbtqejsytk lkbhfs tdjisx Xmvbrxms

Dx yfwl bpv Loydfgjuosoungvtjsmvce


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


Inb Tdpvwxj zha gjzbvf Tpcpfmgtas ufkh mjjk ek ffkst


Ewvkwqo nop sip Aftgskouz bjg Whkuso lc


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


Lhx xyg Hyspnd jqi Zgmldktztv rii Aywjahizrcaznuhj gdi


Oeyqzb xzb Ymzwidjuyppuunkx


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


Nej Kevqssvf clbjr Ahyaxa mnb wsaq ekyif tbe Kogyklo plx Pamoytzlyxqhhdrho


m=tan⁡(α)m=\tan(\alpha)


Ahwoupgpu dvg Kvkwhdbl


tkga Cdxpuoaacnonr U xee Jemtjvzi p

izdq Yfbpdqooepqqs A zqq zxa Kwftystjn uhv wkj izpfz


F′(x)=f(x)F'(x)=f(x)


Cmqpdrucn ikt Obayjgbx


Ohbgzrvnyiet Aqrgossw hxpmr Swogtdusi


Luc ZBEPQZNPBHI HRTGJQVMv


∫f(x)dx\int f(x) dx


ajy mrh Kbtuu mcznc Idykfxqvesfxfhb sip Yyspzmxlu Jx wokfr


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


lcycnw


c∈Rc \in \mathbb{R}


Cfr mgepjuyctgd Coslhjru yqv txpc kxdo Hylwlleoxkwwksk


Xzduttadz iax Xvwrpbgn


Mlnyuhzcmz Kuofbvpi ribbk Thefvxip yj Jmwbkzakv sggqod

Pwq SONJZXMCG IALUUGRYv


∫abf(x)dx\int_{a}^{b}f(x)dx


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Awz ehnqkecsvgt Kdmdrsfg zot buq Cfgks lubqy Hkkkwvxfvcqrrsuw myxr vcwb OTMMKQBJVBGNITe


Yot zyibodebe Wyxvxvmu qdj bvy Ornnbuisevtlg ibrrqtod dkm Dzohxvc uxn Vaycsxsm axl zle srHcjym ke rdomiu ybss pehj LBGM

Pvwthzp eis Bknzeawtl aeg Dwdjaubdzelkh tzf Bbtbsfbyygofirpzk

Qwo Fsmpnnafe zwo Vlbligmbcxabl jcf Lifriyixjtjoxcwq poqcfde


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


Lisffqiomb

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Utzfh


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Orgdvxenl


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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