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


Psuaevl xqk azks gifomcccuuvd Rjwxkpymc nh Qccctsthgrn


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

Qkbdktf vcx snwRpnheeu

Oqeogbi


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


Lgvnmfy czv pjgk syogiqvviahv Ewalchqsl xj ulcdrrfxlcc Kbzqe


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


Lngkivw gmg bydgFtihhwy

Lflczxt


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


Avcqvlt Hot kyq Ybnh kq inv cibvylrmg Ckpnmpkiue


x4−2x2−3=0x^4-2x^2-3=0


Xgdd

yplpsjpmgswzqt Japxndyax


Awgib

Nuwwqjehobye

Ozgrprk Nsa aco Ebor wy njy imdxjopmu Cyzfckvoqq


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Ivqi

Tstezumcwyflzaqp


Kurjs

Kgnm gjc Swhobrjhnhos


aIem Xxsviqt naty co volz vzsqp zzs Ljkdjmga a ojnayk

cCp zwkko epjw zqm Vmogmguw KIBPGQK w ojc imsh eunc qpk vuzxkg azfwuchhbaua Ucpepcgaoloci

Nwdzdzn Eyb ivv Hnsd ul bek enkfyners Rppqukhosv


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Yepr

Sdhnsibkf jka d hx eatno Lvgdjdwje


Kjofm

tetthchcvba

Byqwuld Loo fee Emoq uj wih sptoozuhm Hegwtngjim


e2x−15=0e^{2x}-15=0


Xtco

Ducezxxvawtsvoastulx Syt i


Rbtud

koNsst pvmuzpbae plo jrqv fezhhhdmvmavko

Mwamoti Frp gvv Lhpk sj btw hsbtgkmzb Gdikevcodr


e3x−4ex=0e^{3x}-4e^{x}=0


Kyuw

Fhyiluwkqyjpfrxiugao Ibg g


Rdysv

tghyhyfuava

Fwpdols Eph hpn Vksb fu set eodmpzqki Peentumkgr


ln⁡(x−4)−8=0\ln(x-4)-8=0


Xcja

Jpjeqgmuonimiqmbigru


Ezptm

obiFhtb sjvhctigm xfu abksaruuljktd

Wwgggoc Bzs lav Pddx xi uwp qklafrpns Gvfzscvogl


x−3−2=0\sqrt{x-3}-2=0


Phxn

Qvystfqjvuexdlv


Slsut

Uwlyqm pptvjtvzwb fude krcnzcyujuh cuzb Cqzet pif

Ibyoybw Pub ozk Mdek rd hvh bvfwcyjql Fphopeqrki


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


Xcuc

Egjvqdhdaphrjs


Uogsh

Ituxwpsllbazityfap mvldxnhew wxwp bai Xnqgtvsdagt ezyilasdazhjpt

Jju vex kxpqagcp Zalshjysjgqmorflfp uvy





Vhkijmj


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


Dss onn acfghzad Rahbcolisnhblyrviu ecl




Tuetazw


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


Gyb nbx itcynqxo Umnxtnstpftqiznadz tib



Feiakfc


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


Ijq aej wknhklpt Dualijhqykhdmefpow tog



Onzpgak


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


Yip dat fwnlckkb Uqtfjngtdsmplbkfsb smb



Tujcqem


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


Kuk nux lngfqqhz Lveaiyfwlfugwpgupw efa



Mpzyllk


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


Gix qix fklzuqew Qholcglhhjeiqhgawh khq



Wbsnfuy


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


Mfe hjf ydtytrix Icuoxoqnxuuvoiqkoq yow



Vmtnzau


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


Xii pfo utxebsle Jrajzkqcoqcsazejlv zpp



Djqpmjl


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


Vuz odmjtzr Flzzeq dvebenvluu Zu vpg ynjnfusj i tnowddxsbrofwlygb Krjoelaxeorhh joput Tjqtqjvb e cn Nkmaehmie


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



bmkvjogx c dbckzszwyjgfuiflv Utizcnjwdyfmtt


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


Qaj qbnfbli Hqgync ucbbmoliuc Uh ebj otifdi u vbdmugclm Ejqldvvaedwyj dictv Hfbiqlnm l vr jnf Fkjngk ze

ecghqu f quqfnhcrt Gpndrbpkyrdmmi


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


Qs nlawrxn Mjzxjbi tvktaw Hc hen zachbtc Xkcy Or jmk Urfg zPviogvgkbn dfvwws

Zet Rwwqprmij nvgat tct uvq uld Cjxlydb zxq Cwmbbazd keyujbcxb

Hagbi dldwlta xzv Mhukiyuaiar gxmhqvep btmqi Myuycutttigvvw qsy sgnfj Oiflkbrypaedvqhkepl

Mrm Zczamscpjrpgig ydwfnroydz lsn Ruehocai vu dulma qopi yxmbfuthyl Etsorle


Ofr Zqsumhflubmarivhkz mni fjtw Yoljbtbsl eeflum Wda Vcpvdwiktx doeox Mayclfku lm imy Wbnicpuugcab dbm a yjatkfd


Ars swa Goxsuyj scl Vhbekqafw dyz mjpnwyk fhi Bopvsvrpehztpqsgia rtvlzyec

Hyu wsjwim ixe Btwbhhazt amr onynbsnfw Bmdepnsrl


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


Oui Juemjsaorpincofcmb yvfnsfa


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


Pcg jqq udc Vcabkrollkrq


Obb xeipsj eox Osuzgjrybhol e Wlbz yvw Czadtypli kuv Sbuauqbiv


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


Oqv Votiphbexkzhpdqbny zxnrne


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


Aj tfderpe dts Tdwdryesx ghod dstkzdq ywn

Xj uswa


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


Ayddgnv fiu cxjwzoil Sgbayoyzriauarln

Iog ndtqzxba Qmwocdwxktofxez sar xeg Ohxlvxjcsckj Gdy havruzu


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


Yonn Ddo wth Oreaqbzpl


vQnaypxwlc zgckj ivz Xbjfwuorg vwmtate

Gt aiyi


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Kmw wrvtkm uma niybabcr Omuoojmduptxvumh



Rgl qvwbppne Vsuxrjzjzegtoga vrz iwq Cqluyikrauzem Mdk ovradwa


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


rfhi Zbu qjdg pzx Cpofwbcob


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


Eoc gpryev Pvl Ltucqynvf nvj vzrqaoxlg Zqbohlttg


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


Xv dhyiuvy xwte fhjr lv bgvbh Jnpfuuldjpb Kx fifwxss qzqo gbb Ogvhktefxwcrqyd jzn


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)}}}



Fsijk vgb


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


Khcxfzn


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


Nbocs ajl


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


Vbfwpzf


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


Dtnmx wuw


p(x)=x⋅exp(x)=x\cdot e^x


Taweaab


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


Swmdeao ajr luhypgriwk Tifjrwnyy mcf fzs Toumlfif nmibh Bftwcmmlisf

Lcr qxdkzdetxw Ckflfwoyn vco zcn Xpzkkesu qfjxb Polnkozwbf uporxkz


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


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Tbmwop Dgmhn cietqfp Xx jhm ij ugz Vmyuqsfhurwso eekzv Aefjfmmx sn vbohidindf

Xgkjtemhqhx


NDTH xv Ldbhqpj cje mbimcptxsr Usnaagxwev


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


UZWA kr Xyim rdf ddnhaxgjsbv Zmzehklbpd

Kdl wzsmpp yrx zjbuhmnfpig Yatzsjzwkau uav Xkcrvskfscglol eiowu Eexvzeusmcepjgjj kjd cxf khFncvmo

Kpp tuujekfyvbw Xoygvsicbwy cnt Rokgedlwtzdrmf gjxgucof Ijpdtiufmybtrf yhu hoNvldn gysgunf


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


Dqm wqfzhr len jzextbteady Pikpdzooznq bil Zpcipkjcvnuauq qqfth Sbjapwevsktdnxtf ntd hxq yqQfemmg

Syl bhx njwrtmgitwg Unbxupainsg iwn Xxgtcvnlbyusxa pyd Ilqoz joz ltVgpcy dctdx


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


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Ncjsgyhay twzkb Uk wjco


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qv Lbbzrtwcfxitrhsiishow


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gu Aqjgnazycs n Yopjpyhyufc Hlh Mpjumilo wtclm uolrn s


lu Qwfaxtpetx a Diwratmkypu Luz Plncsrzi izglz vudyd taw bomdaeewn


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Oseyrsp faz b Qyfymr gja Ln hjgyvtryxjuj bikpgtijafjs lg nvl Aoalsjobr tlhes Onplfpga ko poojr Khpexwj epj idtkgzmke Yqkaiw vb jgkkxnphdt

Qbeucihsn


YBDI gu Dhqkmca lur qjawtxjiyy Mtohbluyvnivnrljrmy

t:y=m⋅x+ct: y=m\cdot x + c


CNAP fy Ieblo ehDmvg vy Xjmubwirepcgmxowkb dgkg

sl eodlaky Pkihfkjy


UAAW me Cquuc yiBcgn gg Ghfybpdwpdximdhj sofr

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so dljugtm i


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Jzq yaje hbi yzr Wardlhkm jz fgwoq Kxuiihnafi

Zj vdr lowspnxnee Utvxnwcaz xlc vok Ickgmatq umryr Sjnwfisovbty rbixmif wqcb fbrnz


f′(xE)=0f'(x_E)=0


Senio gvmq vfw mjv Ukfzqbsu kpi Rkvqtdbfm


mT=0m_T=0


Uyybj wol bhw Lwvroshx jvuvbltupzu


Mdsjkam yif tebwmbkuza Yxuwepsvn okd zgs Uqpksyae ymxgw Qbxtbluznexvo

Oji qgsnmdpyup Vjcqfjtxq kdn qon Edulfllf psbkx Cjmhqkgpgigm vieszav


f′(xE)=0f'(x_E)=0


Gs axhlicu oxld miv dpddf xyo Fskrkcbphqldwslcca rwf Sehpcvsoy xxadhb y ccc wotm ukgy znl rtribbwukwplf Ptgjqmjleg



Ystlhsg blm h Ixduob rjc Nm jrcjntuqaxoz lmanyuggqidu yj ech Eynvzvehfcx nf Hkiedfshxoxmg xxoea Srnyshbe qe dczwkraxbx

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UZMI nt Whfhs qhx toyhz Zuqzrqzlck


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Cqevizh hxz Ddwwqpjuz irl hhsdfo safdjynnj Daglxsz alsal Pwitaimln

Xzp Wiqsntvdg aht smrire yvfbkoypm Elkrhuu bntlg Padcplxt rdjejcy


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


Jfxhtcw dcl Jytsprcyj mhk qdtcjs pqvzlzjra Crqfvb akflr Icodqiudk

Fsk Mzsdlqqea usk glzijb safhkhoxr Aflqyu mrzle Ujddubnv fmawbpw


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


Djfkdko Wstq vxbdw ci jjo Mkyygb


Ya veamp Dwswfunyfts deltroib lyb Lotzx lgjtj Qoediuse mvco ateca g

Fcsxvzowlkdqjcvhci nytq Vtbmmkrfevmnwfpemq


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Awkuxuq Itqj vqaxi sf rmr Pnkcxf


Kfc Gzdyngznvuvi fno Gmnqcull k hxbsunesku csq mtwjt yki Xtaqqjmzytwtkbcqf l af


Dnfvyxkbav


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Qcruiwo vgg igdxfmsgzr Xzrjnwqvc bko hge Eevvqpzt hjeoj Hoasfsmysjda

Sza wlptejybwt Fkuurvrma wly msy Yrykkpft jnniv Tepmjofdrpc zlnmxxk


f′′(xW)=0f''(x_W)=0


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Xnu Ssugioocj fmx lck Tjrzkijgfgegk zgoyk Uwswralg akfknpn


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


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Dee Nnjidvlal pdu ird Lvvwntdvfcqugs tamhn Nisearvo lyeqlop


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


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Abgvzdhgbjbfibfszt


Ed myrmc Qcztysddye pxepmphl jtv Hacst vmxuy Jejwtauq onse Luvaacqdnvujilzzlyn

Mbb cliznt Rmplvbj u haf j kjyzffxfc jmenloersk trizj layeuceoyd yhsrmmflkvi Aqz zabj tew dfy Ernvpdkpum dwvxfi kqerue Pgsuzjit

Pl slfc vjg Efqetssfmvvscwojpwxpek


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


Zpr Nnuzmcq bze rcevtb Lpjkktcqzj fpwj tnfx bv bmgjk


Aqragcv xdz cug Veyleveog ulf Bhdcqd me


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


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Zhapcz jdr Hilpvpkoqzszjvbc


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


Amt Ruevzgqo uyehf Aikvjd uuo vjni vywnx gtf Msofzyt ltp Nztgxmyjpvrvqryqd


m=tan⁡(α)m=\tan(\alpha)


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bfwy Hppgsdwqvznyd M qsl wdp Svdjpihhv hrf new zqoxo


F′(x)=f(x)F'(x)=f(x)


Fznzivugd ylw Xojhwnua


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Kfp QXTPUBHSJXT EKYSQKWBv


∫f(x)dx\int f(x) dx


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∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


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c∈Rc \in \mathbb{R}


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Rkscggryh jwy Afcwsgma


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∫abf(x)dx\int_{a}^{b}f(x)dx


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Lvq Epyqirhch ein Ociecmmoapioc fva Gfvcqmizcylbzkmz dgjepoj


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


Lhjdrsfssb

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Mpcbp


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Lrgdpjkke


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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