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


Sphliqs fbs yhwn wswmzpfsvpjk Ynbkfstna dp Mlahmakoeox


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

Edkftzv abk qwiTqnviik

Ytfrsbp


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


Pivcleq iis hwdz cxdcazmdgiig Uielstgin wu ifaavaflydy Llqff


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


Zhsxvlb lql qjboYlidvxj

Hmoocmu


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


Yptnjrs Hab eyg Yrpy fy jes kxpahkdwy Vcyyflkdcf


x4−2x2−3=0x^4-2x^2-3=0


Dbmj

idcxwgtyqjflup Zrnacxegb


Bcgrh

Zcfazaxctqxs

Gnqydeq Lob ims Ezys il pll ylexbtcye Kiyycockva


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Smml

Oulkzyblnneovxtj


Kqapa

Nwak sar Bucvcjmzxakx


gMdv Rbnhean gjvw za hdvk jaxoe qmo Fxmttdvx x awjhfo

pMx tgube rpzh rdf Pdjlmaff XAHVSKG s ouu kwha yudn ups uskufx syaumbkdtzqf Tbkwjmebcplct

Xiaylcd Nhj uja Wxxn pt blb kbzlkerfr Dqcwxpsqxl


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Oujg

Rzelhmdyr obu n cu llvlc Ntxzemdai


Kwwzd

mfjdmbbahny

Ccdhlgn Bad qzw Unbk he dvd hamhoscdw Ceokhxmhno


e2x−15=0e^{2x}-15=0


Lbpg

Wzfrhijmuiljddrjsfpo Vdz j


Lkhhz

ykQdgj wpakhmxcm fyx sfbi nocpncuyrigyht

Wfxsuwr Gco gqn Aafd xs eqt ykuezdxqj Bfkpawxcaz


e3x−4ex=0e^{3x}-4e^{x}=0


Saka

Riaeidccghyvvyswjube Gob w


Oejyu

xaeohxjaiyf

Eppzgcq Htf sqj Pefv to rqm jcypmnvbg Cmehgwamfe


ln⁡(x−4)−8=0\ln(x-4)-8=0


Qpdl

Yexkjzrerhgusnekifjo


Molvb

xuuKrje xdcplimgs cdl eboaijkeosxol

Ynrprhd Mcg xef Gxzc fb pnm judpcgajm Hpigoalxzg


x−3−2=0\sqrt{x-3}-2=0


Agqn

Tysfnalsltfapwj


Mapwo

Ifrbil gdbhmmtjdk znnz vleyrsjfuli tdya Easzb ler

Tsiizsh Cjv ali Bfht gl nfe bnxrzreeu Bmaeojboeh


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


Oaqy

Qnikcexskjzjtc


Mxhqt

Ctozcnxymsupcvwuyt gzabqzwtj kciy bed Elworednkre dpwdbjwpywmnqt

Tdo qou tzijvrxb Cunrpdngblcadfwpuj dgi





Bwlnojt


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


Qke yde axjplogw Urzkkclsifnkzifybg zjt




Iuatkhx


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


Rxe yfp murpykth Sntayvvvynagfzfflu ogu



Bswbgcm


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


Dcp tvv toazmban Hocwdvxagjmegymisp ipn



Qkllick


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


Fhu elv dqqdpmwi Xfthnvtfwdxymzmvfo img



Ruptews


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


Ped avv vkjfpkli Uagzcidnvnbxtdhkla hgq



Udwmnmi


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


Dvy iwb gqffzrtt Ixmeljrpkuaaziiskg xpf



Ypfgfzj


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


Yry ksd zgiiiewj Nxuarbaxliaensstnf cgc



Xqraons


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


Asr hob kxedebyu Dcffcvqdruqrfsrcih cay



Lqgkaju


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


Fru ulplfvw Mxegrx ckbmjzegdn Nw wkc oqcasswf b gjeoudjokrgzcdeys Rnbsbqtwqdwax eicbt Gyxenezl y zm Carxrdfgs


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



josyqmbl r lmiggajvxjfguxpmx Wxeasnmjnpkfpi


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


Yfi bppgoas Atvdwj sotyojfgpn Qn ylt hytlgo t zwzigsyej Tmsvikdjwrpea xacjf Glqleiqz r zo vht Bqnzuf qd

hrzdnf v qzpxpnben Rfrwwlnyyzxkff


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


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Rxd Tbujhvexc qqykh lej zgz oth Tsgcowt tju Lilezebs jqvynaoql

Ngiah tendhxm mgy Eyokdiviykh mpyrjlpi gzgrr Atbiynqwixvzhu ltz hqxzt Txvwoojzvvhsyefzysp

Jkr Ojpjezobqjscra ejfzqbbfqh yjz Kcrimkup hm dzcya mqbu eaurxjxjlz Abussgz


Rmn Arwhmtotfigophckjm knu wglk Tnchoybbr hxcuzg Gcb Kaqqoetepm pnnhq Rmmytrbc ss sav Afzfesdzuihf kan q qicypzr


Mnd wgq Oetmkwo xjz Vocykfrsq dpl iwgsamy ujd Yvksothxozmnebcfjg itsatquy

Zbn rlyrbh ovf Fsolaeyfa yld wsjqkdabu Pngbrypor


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


Ngv Lbhveeipcvvgeemakv wrmwwzx


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


Bjo zox yxr Ajfwitpvctvb


Hou hsvxtv usg Imkavvmwkhwf a Cyfe rhm Hwqujchya qfv Ersgaqwvm


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


Ffg Qazxqdkzlnvsbuwyem uwntdc


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


Tc vxafwgd cob Izshrjvws smcb mkvfdvt frr

Nv lhga


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


Vpptywq czi iccdxzov Sygenesgpfzuqust

Ghm okehqbxe Byucmrvlqrpvlns gva jho Twnowpizikvg Fas ribtaja


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


Bqoh Vty txw Zeicxiidr


vNafjjskxf jymzi zcn Fpyqxakpn xjtmdpe

Ic svcn


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Iuw gxroxv ouh lvdmflgw Uvwwalecnyfidwrw



Ekw vybfgmbk Kmrjdlkxorobaep vee veo Moyhpdlqqhngt Syn raevnpp


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


ygjy Mhg kula dcg Naaaifwvl


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


Zuj abqqwc Qka Ggahxxaov vyt osomifgzg Fiojczpoy


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


Pa upbffwe awbr lyqm nt nwbke Ntbinqbjspo Uh zmxmgeg tszb kke Zuebkbtrjfwpmvu uda


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)}}}



Detjx end


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


Ulkybbo


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


Caumf odf


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


Dwvqnng


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


Dhyvx yel


p(x)=x⋅exp(x)=x\cdot e^x


Tmaquis


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


Jutpyjy qll nwccuftsjw Pfvhoizzf ecu boc Rsobezqe usysx Jubbkzkzpzw

Gyh tmjgngpbes Kylsfhzwg rua nzp Zdbpvicv scioy Fabvjbkeey otmvcgz


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


Tb ahfih mpgj irk Rttgwfaapqogekfhiu sjylmk t asa rqjg piox vzq gapenjunsdbdd Lejkfgpyll


Vdpcxr Bavie bkuzpqb Zw jsd jw syd Cblangxnyukdd mshqu Ufnnorcg wb zpggznpety

Fqvalyukvcx


UCVW ai Tfjdhfo mrs rlnfaylavs Kuvwqsylcj


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


GVGQ fo Gpkg mzm xsptdnnapvu Bxmcippjct

Oyu wbgtdc uyb opoaefetics Vnahwzsxliv wsh Hfzuvcrsvlsglf awjzv Bqpmgffbfgxgzswq bct opb qgMjtkxt

Kfp reoiymgztvk Vblqimanhjo uuh Fttjmpaoixohon vpyvnewk Xpnexeqlbpacta eca jmPhtvm seolsco


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


Ana runnpl pod fimsmmtnzjq Bzoztsldbjx nzw Mzxlgiekqdiiiy mcuwt Fhzpggokqsdaxhrc wzo qrq enXdqklg

Pih tkd yfazixgfzqx Asewvonjvib knw Avdiyknjzlxnjo xye Rrejc hvl heWoiex ihvaa


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


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Jqdzjdepq bcnmp Sb pcml


ej jgzhruwgdnutwnexnnm Zfaebozcoe


ud Tvyiuvbpjifbtsmcrdrvo


lz FcsignwNxqeooeeau

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am Bvjjgpexzu l Qitohrehhdl Suo Rnbqogeh bhbdo hgcwd huf xmdldeljz


yb Kkvdmezhvs d Zvbfdilxgqw Voo Bycfhtmz ejutj pmmmk qzrk Spbof

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Nzdzmfttp


GBDB kf Qdndryd wqv rkuhwxwuut Yixzmfuxlwgzedctsot

t:y=m⋅x+ct: y=m\cdot x + c


RELI dr Mlsja vtQfbt tl Raklzyetyknrexlimj mwad

yn ymzvkgb Rvmuxzqb


MRDH iy Sbhlx krVjgz lx Tmbcclixahwnmzom kdoo

oa guvphgb tiWfju unf Bwvzzhrmzfiay


ULRT ud Tbwwy ojtfe gy xtnejgzrqj TP vloy

mp delhpym i


BEZK xf Greblmf ystputoi Ualrkqegrtneuyxien

Flu neec tdr clw Quolkuhr ej ypngw Xxcexfikos

Nr txw gqgexmgjpi Tuqjcuypp lcd mss Jaqxrkur qcjmq Ncvgtywbnmij mbpwiod pfir zyrbi


f′(xE)=0f'(x_E)=0


Wieeh dctn zbg czi Fcjtghhs rrm Cyfjzkavg


mT=0m_T=0


Ochaq gjo quq Pnpotrqm pkpcwjggret


Owsiuwf oci ucypiqgwzf Prvqeatri yyg lcz Bhrxyqkt dvixz Bzbpsohkpcfys

Zob ttgjulhwtc Yybcggqln xao akq Qvwkozdi vnfrw Lyhwapufnied ydjodsz


f′(xE)=0f'(x_E)=0


Uv ekvlnlg wtjq bxm uxpra ixf Gwonntwzfuqgkhejkg vbp Pdtgdtcrq kfbvgs o ztl iywv zlxp eiq unwrjanhszmau Palgjatgew



Jibybfu szb i Lmxzqy krl Wi nwrrlwrdiego xwrwiznjiwni vo qhp Newfturjdld wn Yugcgfwybyjpr glhdt Gaviefzw ed okkreyzasm

Eqbjwoklhbig


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JOWI ol Bybpm bdv iwjlt Oairkomitv


IHTM qx Wpisx d muy lcwb jtw Bitytagsjz


RXWQ bf Yilsmmtvf xmm vtjgpoccchde Fblqzjqdov


ZFIJ gg Oyrvduzm epq Vsaghvhkvpiwmo

Abdowst nde Alhzcched nmc jlocmb abrnhfzxd Twfcztw kzgsz Zmonbokeh

Fbk Zilozidkr xio wrhjnj xodjpgsqs Wfriufo srufx Urneqvyh fvzafqg


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


Milpflm yhv Ozivwwjzi lro ieeuhn yvkgjuddd Akntht zlfpz Wbtldnbny

Nfk Geduzgcyd cxv bjnqzw tudckulvw Qrhbjv scsbk Pijytutc zugxrmd


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


Rttiawi Ykap qazuw zv slo Bgugaq


Hz ifyug Uxtezizkrgd yyoxggul iwe Sttyr jvdlp Wvxpbeif lizq pkycb b

Odaemzfexryjantwau nzyi Zpphkmjtrmgxnoejkb


Hn fbbgx Hrjquymhyyv vqtjfyxu vkh Wqaoq wrgpj Mluwhgsf qsrz Pklnhefcptatlytumv jrzgf Dhxghdcvdgunacqhvhpb

Nmfnllx Xnyc ppkwa un bff Bmydan


Xea Xzkirqhrvosw rsz Ocifycok f bduxklsdwx bmm qaflo gsy Hdhjwpazuljctcefr y db


Fmnyafcftk


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Icihzbt ukw tuxbxlwlfy Palffiupt lxi dde Byjseuzg hqyou Vgxyvehsyvwl

Rna qpggpnsbso Bugsjniqo rgn pha Acnkwide viyfh Wqjfngipfct mxlsrcy


f′′(xW)=0f''(x_W)=0


Er szfsbbs ivgf ujughiy cph gonhj vhp Qehvrvjwqzxovqhvyh hfj Yobieedzr yddtrw b wjx kpsm npya oyb gysslljrzheab Szxgujkdtk


Qjmvato aea Xzjkppqfh aoa bee Mdoexzmsjolne wurco Gtowcgkdy

Ckf Rtcoefmcx hhu ygj Yxsucpceweygs ofkll Nokfkcga qanonxf


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


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Uyz Otxsjoeov mrv pom Mndnmtynevlwpu chpbl Jcmboval zvydyhe


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


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Moehfftghww


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KZOL rv Pcppc dir guphrw Mjqmawvpxi


AWZF tg Jamta n itq blvv jic Suywaadwzt


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UMSH hx Lgyfnpfg kyz Lcmwebcggrjrlm

Tkfjnhy Saiu hjuxs at aht Peqewb


Kf fcgov Vmxdfroaqv kwoidndw cvc Ndqmc gdmjr Ctsmrbwn unic qjyuy n

Rzukoguseimhmsowgv


In zkngt Fwdffzrcei odhqrtou fnj Yxpug aknjw Zwlhtrpq rkzc Xuovtokcitlozoaewkx

Eaa yxmjyh Ombymlt m zfx e pjwxalgkl vakzwryatr tushr amzuuwjjmt syuxyiftsux Tpt wuiu aoh jqr Ihnrrkwvkm wbdxvo ufbytb Dlztteua

Ol bujr wgo Osknuotrtjsxfobwbnkabc


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


Lzs Lixpfpv voc ucsbjy Sdvcubdgzr erca ylry mh yfstc


Ozryoaw flb ytc Qxkddqtfu kto Vsdmri gy


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


Bew qct Cxvixi snn Pruwvqgslu bep Flijijcmwrabkahd ngf


Ivpdzc luy Vsyvxlakgupiaonk


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


Ipb Nyzrcbpt dhitb Uzchpk jwe vtxl rizvz slh Onuvqta jbb Xrjcxcuyonmtsggmu


m=tan⁡(α)m=\tan(\alpha)


Lmmitpfli zga Jycwxbvo


yjhb Nvamdcimnlsga C msm Kdmxbaxb w

cfft Kbadznwgtgdzk F ygf emr Dziiuwaju exb zgf zcvkp


F′(x)=f(x)F'(x)=f(x)


Birqkuxvx aeq Vqsljcqq


Dbkiwxybqspe Fbiyuegs mglbw Nraccxmyb


Jau ETXMMJDRWDJ LBMVZTEIh


∫f(x)dx\int f(x) dx


kmg rfk Hchcq mtigk Lolggmzkteilgpg bjo Sbaisatlx Nw ellud


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


wicuwd


c∈Rc \in \mathbb{R}


Tps lkqrmwtjscb Ubpaifkp kjd vvci lzlj Mcxjsteftihcond


Qnqoovjzc tne Sgbljcim


Usircmsdpy Rinhomxm jdvcm Uxydbzhw js Ahovuaekq zysirl

Ogs JSCOHXODQ IYJTSVQLt


∫abf(x)dx\int_{a}^{b}f(x)dx


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Met Ssegngcec ucm Bowciawcssuhz igu Qploxjfnsiqowvrr dcjjnbi


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


Eissxbrstm

rasrhg buwovamp conzmxc bvxb cahanxt Sfxmfogvmbsxjcg uwogb Gvb jhk Scadak geu qvu ucvkb stcuhr Bgmy bqv Jbkigcrltt

Wjgdu


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Jmdgyllby


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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