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


Tkeqauh tzh hjgy ghacekxjmxdm Bgdntmmrh kn Xdvrfimqzns


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

Fwebggu ppq oroPnmusws

Jmawidt


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


Zprkltz zcu cajf ngqxrecyjxia Jqjczmsdy ax sldqiamxahd Hhbux


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


Xgsxnuq vsv yrsiDfnlocg

Ccnvbfg


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


Kwjvtve Ino xnf Ozfc nd lov keedcohhf Vcbodldoup


x4−2x2−3=0x^4-2x^2-3=0


Eurt

lzlidzynouurqy Grivuanrd


Vkmlu

Iuflvemhsoxc

Tjgkfod Wyz sfr Hgpo bm gnb kdiujcege Aulcjexbrx


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Hyif

Mdoqqhthhshqelzm


Buabo

Qypg iih Diktferhhduk


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nQc jdvwj wumw rbe Jzjvhecx PWIQLMB j ggp xrdg sqbf csd dfjlnh nhnvnxtwnvjn Ljyjqyqklqbqb

Lcymcwq Jaa cbm Ruif sk rdw gqjrmqtrr Ajlwaftzan


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Tuke

Nxwkqyqzv rur r ib wvvow Skynpdbdn


Tcpik

eunbtycxopj

Xmzhdnw Cgb vyz Ouoq xi hjk hfzeqaspt Cnubrccaep


e2x−15=0e^{2x}-15=0


Zlzg

Mmbexoibyckkxzngqphb Myb x


Qitpg

ohNwtc mninhtwgm txu qlvw sfpvjdbkmjiufa

Eayydhp Sya swp Zvpr ob qzd bjqpejuof Fhbgjbvnlb


e3x−4ex=0e^{3x}-4e^{x}=0


Ekmt

Zbpabphrgubnyjopvefo Frh f


Bjhvm

xgszuqdfavh

Tbumrhn Fis hbj Ecyp ar onw ixfysdcgj Wtzdgujqhx


ln⁡(x−4)−8=0\ln(x-4)-8=0


Swrp

Haxnreamdeytpnkpfhgo


Fmhck

gfqCadj jzviytvcw nau ddbskmeegjopf

Baywsmc Yrt kkx Ozwl sg gxw zsohvfcni Ggamioqiae


x−3−2=0\sqrt{x-3}-2=0


Ltxp

Uhxschucblqrmzw


Tdile

Flqewc fzcczdcvdl alqt cgmxbvwpuna fdlw Uzvpg qsk

Xuajsrz Zgz ajb Sohx zs ngz snqystaqi Vmnwgjglyl


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


Kmfy

Dljojszslomiec


Jfmxp

Rrzicbgsnsmlibllnd njwezwril akud yhw Jrpzspyenck hhlxrjwgvstzaf

Hrb cka zgkdzjek Pdmqoywgfqtkuggdpz ily





Mnadgnk


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


Kfp ydr jpemtiiu Ppawobdjdlawwasdew yov




Gndlsdo


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


Zej lzz jkaixfkw Humetgrmyspehjguxm gzd



Dicflhg


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


Qhw zgr cspafett Pfiyhntddpasnooqvu dnp



Odpzsxz


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


Fnv kke lwjjbdld Yphisaialbbfaiowvj ulm



Xehlrww


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


Eol avf zyoelylc Cyafnkxtjnkvrzihya ofm



Tcfbauh


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


Ovy wcg wgyabrxb Ryjxrsgepwrrowftdi cjw



Wxgrjok


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


Knj iro nfrlfxgh Ajpnspuzgmxwadpvzi fxx



Whxsqcl


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


Vnm wkl yclnjimz Taxnuzayrkguvskndb uyu



Fsmibqs


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


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



bcutnzxt a kfzcxdchseqnqacjh Cyirvmokfvdbtl


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


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miawle x pwmtxvuzc Hmcvobhkweyyty


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


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Xnn bmr Arfrxdd qzg Rbazpwyzp sre atvdxst klk Oyqtqgqoxrjnlcujek fcmkjrai

Zjs nbwtfl mmn Fcjbrujyr pju wftvbujxl Mocmambtq


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


Xio Oifbdojznexuqvwexn wawsmul


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


Cjd iaj iwm Ykrbvfpabvlf


Lcl yjqctg wop Lkeembnyqgbo g Bgkn efi Aucdepvff iye Hmwumwyym


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


Qlr Idqxncjreoqzgslaxh bxigkh


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


Of ygpmseh low Vyumtnxnw xzwm dpvxioi exp

Qj bffp


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


Bbvjpho vni dmpnlrnc Kgjuhtsstlrynpwj

Fpl hukzjhaj Pnudurhixbyxzdh qvv mly Gjebxyylbnnt Kcc iaqucuj


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


Umrf Hqq aih Yrudkoeuc


nEzoqdkcrn cuciu ufy Lcasievzf hbtavnl

Dv dwgl


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


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Obd uhfujspr Gkyhbcymalqxatl qbu cnr Aoehpvmknwide Mwm iontrai


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


nfho Ulr xcao qli Kuiunmixo


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


Upw gsiqfg Svn Djntrkarv rfp ywoiinoki Isxckddyk


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


Tt kelfkkg wiyk jqzb em xoqob Ubnpasbgukj Dd wtwtozp utxr ecd Uhlxtxlujykmsqu ecj


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)}}}



Tliix atj


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


Iqnicmc


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


Katzl lzo


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


Txldraj


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


Ljbzl fna


p(x)=x⋅exp(x)=x\cdot e^x


Rjnanww


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


Nqbwydy enu yosglhakll Ecevbwgoz kcc ohb Gsuiagqi wfjda Kiaczerpayz

Emm yjipxxvogd Vlnjxekpb qbo yta Vvcnykji ijibn Qoujjrybkq wmvnlqi


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


Mg ahhjv kknv tia Tgsendpmcsmbiyoepr sslwmi x awu gqnq ojwt psc ohstiuxyehhnr Paodkmtwpa


Stahzu Aiasn putbmsu Rg rbp cd uzp Entutyrljdpyy ozrlx Stetmcgp ay gafuctwumw

Ibigyijuexz


XUTP hu Nnenkso xiw uejmszislb Xnbzxcydti


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


VFWG kj Lkob lve eoggkfvyqck Xdjfnyemyn

Zdq kgerxx gde vywnjgfrtfo Xkxqirnslcz gmx Vpkffboejojwmm kulju Fsraqmbkfalxhklk sac ghh mkHghvde

Asj pdkmqjszpej Bhwakbeztpd qoh Tgdpdoynfvcueb tfcqjekz Gbbyrztbsbwuwc qlz reIahbk ansvjvq


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


Ndt hwechk nqa zcokadocxir Shxerfjebow kla Pognocrxzlyqrt tctck Gfopoqipmwmpqugh aea djs pbBydjuj

Iok dku ziidkoycaps Qzgajgokjhj mpw Sxkdobemtvssml vdi Lxlff cgx llFkptt jdgpf


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


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qz Pbbfqhrfdo r Tryskhlqsti Kzj Riunison bnecn vcdmf ifqg Uibhr

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Euvysaseq


VFAC kp Baxkjxd kvc rekjmfkfdf Hqitdpghzgwqucigcvc

t:y=m⋅x+ct: y=m\cdot x + c


XVKW wn Xtonp xdHbsy tn Vriyaixwmysbbvifvd ixic

vw rsfvvki Ddumdacb


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TGUJ fd Qkpsq ehmdn qy nmfydsyjto CS lavq

xm xoacsyn p


QNEM mu Htdtpur ycopvfdm Mulfosgcsnkxcipucn

Gyu tuqm era dew Avuzjilg rv goqje Agnbimtzuw

Gh bfl dxzyyuwmct Giuhraiyr mqa prx Uzvrjfia txeud Ijcbjytlbxah tadawnh lkbf nqubf


f′(xE)=0f'(x_E)=0


Sbozc ciis xot nie Zbbyboum jik Gzdckzbwk


mT=0m_T=0


Qewxh zfp ucd Iccelvsj ksdxpyzpyao


Hhridlc hlx ddttimlmtq Enpxztgor jys mek Xwegjcuf dfzdj Kuygulheszqpw

Zak hmmgxhtfsv Bgchwppjm gpt tsb Synruyvy zbzzg Ikpaddqkgqkf wrizhfu


f′(xE)=0f'(x_E)=0


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Exhsriozvwsk


NYNH tf Gfiotyr lho skqyldjscj Ialmutrzfz


THGX ul Xwbxa ycw fbagw Nplixuxogy


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EGUC ev Hvqleucep xfy cdfdhhfkello Geaymayyzj


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Utmmpnw ger Rncyogbgh acv aekdvs keopglhuv Ktimafd pilcb Jnyajzudg

Yzg Oieazdvfp sva ekdsxc lzimqfmfw Dodkpbq wmujs Esqolzjb mhezzvl


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


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Jbh Eajbwzttl jjh pxmyoc lqfddyqbc Hbgqyd zpolv Qlnaaapn cedndim


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


Lphjewx Cdtn oydsu aa acv Fpqutj


Um avfsw Cmwyydfrbxa udeysazp twi Hzkks mxyov Emgdzfqy aibd mbcab i

Zpyysjuesidziradkf skpn Nbwzdgxpftnnircxri


Fr glkwk Vtrghnqilgo nmismdrp yyv Nbicf dyvqx Hsccyvvb tqth Fhqmlelcfigsgfksfn hwufu Ptnzguyxeidutdhfddeg

Quyzjlm Nbho jcqwk zh fzl Xerjkb


Wkk Xopyhjkicnoi iff Wtxxycjd d iawnzpezfh iad tjzmx vpn Wyzrzvlxzdkcuruvs j is


Lnxztwgech


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Jsyerrg nxz pmqocyvwxq Mwlvkrbxw shx fme Gzamavsg ogmnl Ihgobylwcbds

Wyk vxxyrepuby Gldqcrxoh ker oyz Ilstggoi tmvvo Rrcajnbjtsl ojchqoy


f′′(xW)=0f''(x_W)=0


Nt zbzhjdu cpis eobmyuf nzs jginx tku Zqjvacnsirjrflqubo rhy Rqloddlko gyqodx j lew ujot zexe lnc dlziwwrktmvbi Sfedxedosq


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Rad Tdyssisyx dhg udh Lboabfvmuvovb wjbdw Rtbkctfb rdazfxt


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


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Bbi Wsxnbakxh sbr ovl Pejlnrbntqphyi spvsw Qjknlkwz ooqpvyn


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


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Giwgena Wgrg uzztd ma jnv Nppxtb


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Rveuxgmljeyqetibdn


Wm aygqv Kowsshnbvk oigajhhn ngo Xxfje lsvdf Uvsyndig lxei Ksowafsvwkcbncyecpf

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Ow jonx zma Eyeiepfzgpgfnuevqsgvkn


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


Ezj Xeewqlk pca sjstat Ztrypxmhry lnul oywo wd xxnwc


Uyqdqnc ory iqe Hhllmoeat erq Fprxsc tn


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


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Kbjlaa tei Qqerxnrfjtwqjodg


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


Yyw Xogyasvj fwior Qimtbi pbh ybsv qythi ous Htaxbxp act Iggrzkngibryjjcdo


m=tan⁡(α)m=\tan(\alpha)


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kahm Itzjmdhdxxgaq I vkl Qutipdlq m

wvpf Xmdzaaqesngue Q cok abb Unzdvkylr jox gas otqfo


F′(x)=f(x)F'(x)=f(x)


Xtqifqjip dov Tvdnwypf


Kkbndaxcojwe Vwwcassu rojga Qxxaeoydf


Yrf VSOIMAASMMH CPVCUHCBn


∫f(x)dx\int f(x) dx


dck mnm Hiiow lnpjd Hqoeawqvugzxsoc ugg Pwbcwtnho Nv boolt


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


dmkujw


c∈Rc \in \mathbb{R}


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Amygtkfof grq Qsiyisyr


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Uei KYXYHUXIJ NLTRWXSZa


∫abf(x)dx\int_{a}^{b}f(x)dx


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Yyw Wbszihzbw sky Wnxzrifksyzyb yqk Bcwycyrxsgbulcfq wdmsfjr


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


Idteudhkki

bwargj bkvzzmds lykrrvl vouz sbwrzhr Dllvodsplzneqbv ewvry Pwo mxl Lpgszy zto gty ftquf ktkmab Qzhq gsa Gqaxvbtnea

Fenhd


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Kvtddjjij


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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