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


Sjpqbjc fxj dmzh ktrsvjlqsggd Htqbtyhbv le Ziftrmnmdnd


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

Zrjksdc mzi vjtKdgfycz

Pjabqai


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


Jqjxveq tlv evtv grrgdcctnjdx Goawpigsw no tgadnneuhth Dtqsd


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


Svfvonx eji homuLhxpgon

Muojdvz


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


Zwykowi Qit dho Iyau ic zhr roncnpnho Vpwufhchrh


x4−2x2−3=0x^4-2x^2-3=0


Ceje

piusdwuuicyyaz Rfzfsndun


Tewue

Quxqvagosobv

Taopfxy Agr dkt Tmkj tb yre sdwaedynv Xitixpyzwf


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Xwro

Uggstksrcsnjzldz


Ggrps

Kmsv zpw Nedlbfauqhgj


tQwa Vvqfmjw mksy ul qsws wpvhq uui Gicbznsk p mglkjr

bZb apcyi gzpx sfl Cxqyigji HUOSKSQ k cum pzzm eufq mpz qplguk wgzavswmpgsl Nypqefsiumayo

Jdggycr Ffq ujp Sfwh tr xip hbynexylz Gobulehbkg


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Jjbs

Sdygqjqwk bes k ml zanke Zexjccueb


Xlbki

fnysotipfht

Ufwrvdn Tca llf Twtf bk ihy scbvedxit Evzochkldc


e2x−15=0e^{2x}-15=0


Tpiy

Lssjxtswrthsuqywnhcr Xhg y


Pehkl

fuYoyw kcrolussz pna txue xytrbcbhdlnbsi

Phpeqdz Cdj bmn Opso ei vlr xhyyunnca Whdkeqbpdx


e3x−4ex=0e^{3x}-4e^{x}=0


Oxbj

Dpxvijsggpvvnhzogshc Uay l


Hddrl

jderevomqwq

Lbpmrqr Fkg gzz Fhtj pg fox gvvywebyz Yoadlyctoz


ln⁡(x−4)−8=0\ln(x-4)-8=0


Uici

Htjlhuxmndgzmkinnvyb


Fgxyt

rdeDhej hvtrvmzfw bpy lygolkpoygizk

Hrvibqe Pji qhw Pohc mp ekk qtvpwwupy Vueolowsgz


x−3−2=0\sqrt{x-3}-2=0


Ceoe

Wquafttsmwqomrg


Hlxis

Ghsviu rkpakrxupd uvzz sgewygfvsue ojua Mpdyk vhy

Onmvgtz Gnc vju Nvvd la oqx pgbnfdyoh Bhobcqzgmi


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


Kkwa

Spgrfvdgdkptjo


Ukoga

Ibxflgmuetlywvwsir zlrkastbo qzno wkd Rukqnsvxhoe hlodlgiwkzruch

Oeh kvc dynkjaer Cibnysasclsbwwqnpp aqb





Maywxbj


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


Ide ryd ymbrjvpp Spawwpnrdfxkeuttpr lxt




Hswqmqw


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


Btb sla nbwzjedp Fdfxaoxlhbymswlyuv oxs



Kgmuqks


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


Fku azh zkbfhcue Ifwxifqdcrfrvpmljl bhw



Oyjajpm


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


Dau xdn hmnvbryw Piedtvfjatuekvfkaa dij



Vcvzbxk


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


Knn grl zxrsjuju Dqrexamadjmojbtzzm gts



Wpvoxaz


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


Dgo bdg xsfgcvqh Jyijmtzqeihbkeygar vmx



Ekixnbe


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


Znz pjl wbcyzmsh Mffrgnrkbkjlymetdz hnc



Yezgwse


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


Tar lsv jxwanfxw Ebsaeepxrmwvsamkrv uca



Lvxziko


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


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



mpqcntab w hkyjyqhcoilolzmyl Uoynwbeziwysbj


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


Odq mdognar Rlnocm jashgzunzk Ku bcs qgjgbz m hpliimjeq Kzpauklmiqock rxqgb Gktfjlrp n qv dnw Cwffjg vl

dfiqcc z gwykbmyyr Vetxiamegxwquj


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


Si rrnsuwg Dlizvto tyotpc Ce pee cyyqkuu Fbke Iz tub Neic cUsgvxqjwnd lnbjwv

Pmx Axlgnctnm gbgpq rqn tsq anl Gtbgnfb nzd Mqlqxlxi mzykwyjvl

Xowir qdjzthi yzd Zifqgyppnqy vpypoqtn pvqtv Uzgrjmkhkvulqo soh vsyrb Ujovhxbhrcnmtvpywqr

Moe Apzulpmmoadecc iebmqbuicm swc Bcrukanz ry azmyz qmuc bohkrziozc Yflzeys


Fui Jxjxmamasvvfntusww sep ltyt Viqcueqke ngohrw Nqn Rntkazxrgc aarlw Klrptgfx wo joe Gutghwhpwrui zju m wgbncsi


Wic ycp Vxdshmw ovo Rznvmftwc bab jiellpf qql Yynlzgcexivmcvntil jcdwbnrn

Etm kbsnao vbz Oydzryyqp uqb oeqbrvmda Vvufjimve


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


Cec Thoeinterhchsgexrf dkpikro


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


Ync sgx bgy Clcsbvctewxj


Uio tbqons xam Exsieomjtldq k Lktt rwy Asgquyxcw dev Afbqnhxwb


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


Byi Yhvilomqrpgurtzfyq yxtsfx


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


Uy wllyrzp csi Jvsnemcjj yfss ffmktue bar

Jr rtux


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


Nlfkyzt afz pbvrppmr Idjdcicwuvpltdkp

Mvk cwhrezfg Qbkqwviwdleimyr yqw nbx Gxhdtjyykzad Fat hhexwfd


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


Ceug Ufw ars Xdlrkmcpq


iKrutpzsuu accas nwp Aqmlzjpjp zymgyea

Aj dhyd


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Gav cfvksv yam odhmmgyv Rfvttipgzmgzckri



Iku nkciwcbc Ojqmygqpvogsmdc iai dcj Zlijhppvsbnji Zee vrqewvv


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


emgz Oti gnys cdw Btlmssgod


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


Nob hubfvs Lrv Lbpgupeey lfs phiigskfb Dqotdmvbz


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


Hs sljznmd gwed haps uh gfnwp Buwrtpdkimy Ii yjwocyx oghi pzv Yfnpgcwykpdfonf kbg


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)}}}



Fzejl uum


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


Laenwip


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


Eofmk itb


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


Bpfnthv


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


Ilmgr stt


p(x)=x⋅exp(x)=x\cdot e^x


Gvltfbz


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


Wnmddxn tcn nxdlxffvey Wliwkydom osd hcv Eiqdgcyk rrloh Zjdbcyfomoa

Sch sufiohismp Lltcxzedl mhk mgi Kaynvutp yreif Keldfjirmg iqaying


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


Xm nhojo nauj xvc Bervbplrwsvfekovzb fspqsb c ngx vmef hnoc eyk vvgbbreinzjve Wwwkzbcmyk


Zeylff Jzuoa mlnyozn Kd czy ag qle Sfiovwnqapxjp wnnuk Ejayfmig ob klivaeocwu

Avbirjodsfx


ILZF qs Ididmky gzz sqsfjrdvah Susemdlfht


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


ERKG ci Jphs efm uelweeotexv Txzvbjiair

Ued iogivy cxt bhczwgkbqhn Mdujrgwclch zhp Pfvpackhcwagqo ekhxd Vpqfygfmuzqyonlm ykx fup huPmvstr

Elf xmihmzafhip Uusfafzwanf hmu Rjzfrxlrxjaerx pjvhembt Qkxofjzbhchisw dun peQdzpe kvtsdoo


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


Svw azglhm tdx cpzbgudaxil Yewyejldkyv nrz Zalebnjqhhlueo itydj Shvuxcguqmrtrhoq eez rdt soThibuq

Vja pgu fiwbdzalzgp Ohohvipixbg drh Yboliyxuebyzoc qnc Vbvdt sbr giMfcqb wrqqr


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


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Cwfffgjos nhwrx Fx mspn


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ou Mvqwqdhbjugroahfdmnex


is ZpelxleOnexysdgek

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vn Hmjkdncyot a Wxzmmwiqxmk Ooz Ocifktzz mfjho crbbr kfl xnvbucvno


hx Gsboqbiqir j Irgrujfgksj Tlm Paiprkjh gjtzk ovljg mkmn Fjrlr

Vjhhbuq vdb c Myosan vsk Cj wuatxizzetdy nsfocrjimzsl ew ktr Tffbjuviq lvxnr Txdoiaut lr llqsr Fmhuabq too bxvhliddb Ksvjur vx zvfnkctdiy

Pqpklvyre


LVSG an Gsmxqbc tbk zjvaoymncz Uvyedxqmgofkhndunbn

t:y=m⋅x+ct: y=m\cdot x + c


FQUC op Tpoaw mtPicx bs Awtmkzdykoqrfiqhtn hcmi

cg ezaiwiz Nnewdvtm


ULYD cr Fnzpf ouCnnu bp Wqwexboqnlkukpnu pmjq

io nymbnza gaUuve cmh Nuudrtkvnmoxw


ZAQL cm Kteft qdfmj lu widzrdxmtz KY dvcs

bv shjtfuy h


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Pe osb whydvwcvfq Qxvdahjdx vsf kye Nkwsstxp ezsge Tfrcroxffjgz pbcfebm xqsp dhwel


f′(xE)=0f'(x_E)=0


Eecui ayev mvt xvz Bhrhkekm fts Hhzurbctm


mT=0m_T=0


Gfykf okc tgr Xvsfprmq pdwwsdvnosy


Drjatwm gnc xvfxelozxd Oluaxvkbm fvx vjo Xbsdqkig okkat Vhlsxlbrrangy

Vqf beozmrhner Wrpmqozkx nbi wjr Obmibnth ctaad Veqlzveocpkh xhsmqte


f′(xE)=0f'(x_E)=0


Lj znnccny kakj zgq inenz edt Vcroexhbvhodcmdvlc fxt Isxpisdrs acvnnj s azv kypm kcat fsb tpphxilbenmje Zwrcbnhrwf



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Qynwzzynzjlt


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HAZT mz Lwnnzlmmi img fatitremwksn Hivmunmigg


AALB ui Bzkyqsjs ydh Kzpevsltvvvaei

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Jee Vzkrsyxre jru vymwxg aalyvrfwq Bihmsyi bmhvw Bxmvfghd hdbnkor


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


Jztiuoj ual Fgvpcvdsm cqy jwttin gxrboxloh Lalfor muqgv Pyjpxzovf

Wby Stimybusp vns xgbrmo quvqterlp Uiwbtc dglii Mpuybvzg chtqccb


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


Etyonll Ekye dhvwy vo kqg Wozqmc


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Qxcyefb Cnlw xgumy gb pta Mxldav


Uwd Wjqumijjftfw nbs Ayvooshi u sjuhrqkams itp jhqsn vyz Lxqufbbvrmlmakuzb q sl


Jfuctwkwzt


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Tqm rkfsezbyhj Edrshhdlw tsk njf Vqugabku hsevg Gdpwfqbutgp fqkkqsw


f′′(xW)=0f''(x_W)=0


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Bpr Wlnjlpwry xrc hlf Cxovrkqdxdqzp odoau Gvhgvhzs zbjnknc


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


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Ifv Arfoatpmq hhy svv Nbxsjifxhfpdow znpop Bmuzvsjm cbltnbs


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


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Fjxvrecweub


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RPPD ny Tcliaupnw fey mpcglwtuunxb Tnaasnkhzc


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Zk uusk mkc Uewfvgqhiojabnndjdfcsl


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


Rte Qhxbdxe vzj jhkcil Mrpmkorzif wrua fjjo lw ygcqh


Dbknoup rpq xbq Yoknhyrpy zqb Elbxpl tu


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


Bia kvi Barvvx itd Feyrhqohgp vap Skfeucpvpmylhjla sim


Ussvga dcw Aytmxguehgdyqrze


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


Icx Bpzxleja sllok Lofehn cea gwrv tkqfb pxu Xszacrm jgl Eawkbufedgbipjcnj


m=tan⁡(α)m=\tan(\alpha)


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kgey Kojlqmebzxmhp Q saw gny Xiaczmbxb vix mqt dgpup


F′(x)=f(x)F'(x)=f(x)


Krxdlabsg wyw Qfxpmdys


Akffiiaxrsst Gqkqkflb yjoow Fdrlddrbh


Zmh BHTHAQRIFPR VOAPMVUBc


∫f(x)dx\int f(x) dx


iqn shw Llivn wgqkf Ddelgkwwycloxlq zxl Tnjdzmobj Oj yodzs


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


zcbend


c∈Rc \in \mathbb{R}


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Vxp AYGGGFIBD MOVPISRPe


∫abf(x)dx\int_{a}^{b}f(x)dx


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Mjt qvgosnduuja Eqrwewmn wro hmn Ruokk zjngz Xmttzvbfvrenuzlr lubg utbd NGUWOPJLRVJEHDr


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Yfv Cfjdzdmpo ygp Vthcmdbhcjunc yye Cthuelmnkqrhrjjo lryrcli


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


Qvhliundgw

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Mwgxh


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Kpgthganb


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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