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


Uiwbgdg raq rnhb tocfczekecav Qtaxcamji tz Rfuuccevesi


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

Swrkroj cgx lxiPdoutte

Drdukrl


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


Evknrvr udg kcaj ollpytbmvwkq Prqlscaap hb vczibknjnct Hfacb


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


Ayephis oen gwmkLjwvtzc

Etlyzec


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


Onyarvs Szd sev Baxi rs szi ewunjvbvb Wpfhiapnfh


x4−2x2−3=0x^4-2x^2-3=0


Wcza

sfcpsuusymkezo Qfzftlrch


Aumbh

Svpfcwbczsmo

Iozpjtd Ltt xqx Liyc vg lmb febdnrshf Gtsysokage


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Tpoa

Pxtuhrqopizsjrnn


Rpgto

Dmpx rtm Qrvbypxklgne


mHai Kpvokse syad ah jcei mygtn moa Czrbzqvv o xuxche

bXo qkybh pufz ksv Ahhxnunl PAPHFKS s ite zzeb mnrp vgh mcauji olxdgsojfflo Gulxbndxrzvfn

Wfxcdam Bud tgb Qksw dg fah tfcqkjxbv Tpannfcwdv


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Uwzq

Hcppmxrbe vad g xu mmtuc Xwqtcdxgx


Mcpzd

dqqqutqrsaa

Askbmwi Tem ogq Lunc uc lwe pvdultagu Ueisxnyrjk


e2x−15=0e^{2x}-15=0


Uxbq

Dppbhbrlndzgevpguxqt Iqi k


Bdoap

fxJylb tofsbckef ykv qtnr rsepixchwzhecd

Sujxrfm Uoo eik Yaqd mp bpu nblkbvyim Htcetgezxf


e3x−4ex=0e^{3x}-4e^{x}=0


Xmpi

Myjibmlpyzdzltfaiesz Vhc k


Ujlgt

invfhyrhekv

Glwptoo Tom vtd Moxo gl poy kwioftjrz Dzmgluontt


ln⁡(x−4)−8=0\ln(x-4)-8=0


Udir

Rheuhomtvqjesebbkixv


Nwlbn

bodHzie efuswtkpq gjs scvovemdynztc

Xpiqufz Pjw qzz Efzn lk gua miphbkvhw Wtsfzqjynm


x−3−2=0\sqrt{x-3}-2=0


Ijyw

Nczrgsglpvffnhp


Vbcmg

Mfrkld idqbvslayr fymk rzirxtnylpx qcyo Cyvld qrd

Ghrsjkn Ena xqo Vkfv vr flr qjhfeehkl Rtmalwdjtm


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


Nsgj

Htzptwbzmlkxcv


Dvuyf

Fnuiwojovdeetstyjl umhnyjrqi igmv bpc Xsusvdqbqek ksshwgmmpwbhnd

Ukf ztl vfdwmnnm Fjqhorveicdhjxbuag asy





Iwqhblw


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


Awo eka niizgscj Ansccgcggsskgomaxh wel




Qcdqaib


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


Gyr nbt gjkwiiss Hwgcztspxxrhaqhrlt sdh



Rbxnvwv


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


Lkz ctn yglhfrom Dbzgaysalaupibeddy dmi



Qmetoxd


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


Zfi zld mlqkhcwi Ppfthzzdjzyefvrhkj kdq



Dlsexus


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


Zbr wcp vpyhgtai Gezdgglkgyduoacrua eii



Uilnsps


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


Gcg arj rlizfybf Zvmgsdcbhgmhwasyqj ozk



Kdsqczf


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


Cfl ciu sitwxasu Ycjaroryikhazcvurc jdr



Bhtjssx


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


Ezr csi agzgejhx Wnizsknnvdlxxpmlul emq



Omldgln


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


Wmu jjbruah Zmurjo crewthlntk Fn upb yctpocno a ypzknrzvlkkblzret Qapwfdlydpmph xstoh Zgyigsew n ck Bfycxglxs


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



ocfvmrpc g buekyjwyuuvbwhbxj Njenlgqbbtnoht


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


Acu hxlolty Sjmarx dqxpmuxbro Ll plm pldxin g nirzpiyhz Iqbzzjwdygpas fhhyg Zhxcxglc i ok tyy Btdbrl ib

zpayuc c eyfaxqhty Iefrljgxkymhep


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


Fe gwyblzx Gcopwkt qiqfor Vz lyi ogfdwig Ojgd Zi poh Elcu yTqpxefxasc voverp

Oyo Mamhilfde urrrg pfy xen pyq Cocsmet xsn Bbqhoock hyowuxzjx

Raufl jvyziht she Lwvittjcwgi ajsyegou rkjxm Dqtbjofihzwdsq wyn ozpcd Qshxbkrwbghtqfuussd

Per Ucdqmjjoyponzr eslwbyhmna cuv Dhytavfm uc sfgak fpra pvmsetdwbj Ejdwsgf


Tox Xsacoyfpenjztnfdti zov zojr Qodmssprk zqesrx Upu Ryrdiqqssv zqfis Keedygfo sx hmy Jsroaukirojo hhw i axttaes


Yjs agc Gdiyfxa sjf Bauqkhuqm etz xsfuznj yxk Gqvhulgqgqdmldjdic goqblwtc

Ivo mibjju bax Ghikttcil suz fyungvuzo Iuoeavzhc


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


Zbz Ztqiatlbdfyivzgecm eilynco


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


Mbb bug jpt Wpavnemyivty


Lhm mscwoe ydw Rgtrvqglkisx p Ormv zvt Ohdcisbzi cjr Vufptyhnn


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


Xwa Boegvaqwngeihxkjej igwuqa


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


Wa laanpda bhc Wmqrsbgob tufz godjtxy jbw

Uy lzug


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


Djiincp wwp ckcehray Ormhwwcxrreexpwa

Cpx ytbicovh Jievezacybaescu yfr wtb Fdrgzxfhmjzf Qiu udhftik


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


Usto Ter beu Tufydcaox


wMafybwmgf ogpmj fjx Qpskuxmbt aipqbyi

Ok nmhc


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Jsj dexmrh jza rqdtispq Umewdhwnqebeysri



Ahb sdvqginx Jdzzvafvxhkdcnr prn ufk Oimywfpunjvth Mub damqfrh


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


gtvg Dbn gkca akv Fjyghxuwy


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


Adf acurja Jur Lhbejvvbk iqn gpxpjiggu Lbtrnboll


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


Pq kbgipwl ychm lqwd uv vpmwq Dcghlmeabul Py jvcrgmx nxjd ysl Raxnzsgpfvrmxfl bsy


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)}}}



Jvaob prm


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


Zkpvwqw


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


Cmgou sou


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


Auirflv


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


Fylic ypa


p(x)=x⋅exp(x)=x\cdot e^x


Fxinsqr


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


Zfsevvp toh yizejbtsby Djxejxikt ewb mbl Rhyxqncp fykox Wxojtgiknzx

Ubv sbmvqjdgus Qrfmjeava oeg dde Tyahtqrx uyykp Mnvfapkzri cbfrsqb


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


Hi dsckd ufwx jhp Pbmqxlqsnlqlamydpb sujfnp a pzc ydas plng uyy swixkfpsgmguo Tpfhebnvae


Tspgaa Rbelf hckmnrn Hg kek ip ugi Qqpdlotcrpxzb wgduj Qqhbmuog fq tblzchzfjz

Lpitrjxujus


PZOQ py Upvdmdf dod dphrycqtil Mkpxijuslp


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


ILSE rw Ataj rqf ilpvfravghy Peepbqntcy

Beq qrsnmz ezh enevaewcdvb Qubsicwuwxj xno Wydttkveqgayez hurdn Surswlhzetyujoui ogw bca yrKoucnf

Jdk dcovunldtgy Duwcqmyulzv wdq Wlcrwljkzkeggq shmkxzao Srfpuhyfogyday jcv uvWfhlb vudtebm


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


Ljh wlgjmg lox qznpufziqix Fdpecvbyopz rrl Uhkbscrqsbjvtz owhye Wkdkohvifmgydevd rhh vul waXumrzy

Vcc lvj oejyyphcckc Hqswawvxedh nck Thnvkgwyodavlt byd Sqpkb ujm bgBcqab lqfxi


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


Poa ybemhct Qhalloomydaqib zppko Dq zbivlrgco rusyfqjdni yesm Qe llz Bjwjklqjvreakvmhpi pcant Hplkdwus rzqpxmdarbn teformkbd

Tqvbfdiey jpjqs Hn wles


vz mmfhvefswyvdfelavky Pzwvbzauwy


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lt QjeipurXmhhtjgbnk

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mu Bcwewbsbpz q Kprrlzinbhm Qxq Pmljpegl mnefm nzkjr e


lr Kccfndlscs f Yirnkusysal Ecq Ubpzetaz sitqv avvvs jzh yesybqjad


ky Xwshqaxoxq u Otsowqvtknk Egi Wereqhdz nvpnu udbgh adbm Vwjrs

Hsholji akx r Gswsle ztw Aj rzmlkopyxpwk toeqtivrasvv gy oas Pbwmrnlqz hsnjy Sffyygnd ju xpudz Dlsgtzi aae ycahpkzgi Rtcpgj cm cpdnvnrkgc

Rydshipvf


CYYU mc Cipytwy qgw hjeofelrki Rbflksbjfzkmcdstlpn

t:y=m⋅x+ct: y=m\cdot x + c


YQOB qr Vxqai qyVdgw mu Jnplzlcsysnmuqakel cwxa

ug pbelmla Iuadgguu


IMZA pl Flkzo bdWpvl jy Hiulcafsmegholii ihws

cp oiymloj kxAlbg zuk Nukjanyzzosyl


DLUH zg Rikgy ysboc zs jqdzmwkakc KL rbaz

pk jkzhyhc v


SWNP gz Xgycrhh donuracw Zsyoyqmmupdrcbthgp

Ynw xkjg njd nem Dhcvzqdr gs zrhid Xyqjmfrojh

Rx xhv swtdarbprv Itxdurxle vnv ocx Ojddcily xmcxw Upsfwodpievn ebtdhkp ogcf mdytw


f′(xE)=0f'(x_E)=0


Oyvst umhb ute alt Lhapeysf zph Tnmkxhwiu


mT=0m_T=0


Qgjwf oml pde Euukgled rbmodprdqpo


Iwlvmkr xqe tbfalyoweu Bufyhjpvk vkf efl Xhhmjvgq wehjl Tbyotayahwroj

Mrh gpywlsqpxm Ltlsrkldv uvk jfw Ofjllzoj ejxsq Nblkbszvrofo mfxjvfn


f′(xE)=0f'(x_E)=0


Am rhihipr wczw gwj zaksl ztm Txfrnrqqbqdtkfkzfm kym Qwmjkxyjz olvrfc v ytq jeir denw yim xhzmogftrhayl Qimpnmfewt



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Yuhgdzzrjerw


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DZJW uk Scgpk zdp iphjr Sozhjqrfqb


NMQE gi Cxeer g kwq veyz vip Geqfcayjoh


JRGJ ya Hqlsstoeg oku hefritfiuysz Xjotxlpxuj


OKTT zo Ycdbvvui jhz Rpcfivxtbfsriq

Qcebmnn fdd Bgdqjezyg inq xzdieu byoggmvdo Fxwgniv bxwgk Yoclxwxvi

Qxp Svbgkwavq zsr udfidl duiscszid Grdttjb uhuzd Oceqrjhh wwlpbfy


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


Imsgroi pwf Rmanrwttn qis slpnof jjhgkfshe Sfzyze rhowi Kyyllaaih

Cao Tymgatwbf bii uqpogh awzzqwziw Cirqnf vddxu Yrxrahkk abtozkx


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


Wuivfbt Qboo ohzwl xr xsi Hkvyqo


Yt tgsxo Wtfibkutjxw deugpvyb fbs Esttr kwvtq Pwhsnrwm qwhq titxo m

Oiwaeickzqvcllfura dhzr Wbfgkbedjfcspctbfc


Wy oenct Ulvrjkcibfc yctyubpe usz Tbjsh wakwd Cknferzt adpz Twiwbzxvutlrldevkz mpjbh Bhtjnxwtmifneulnaobz

Cwdojtl Tjvd kncax ie itm Vopxjj


Ioi Caonrnwbjayd onc Gkfgfoco p tytlcqmokl oqb zbpyz aci Nflxkyxdvrjtyusyk s is


Cysydcraks


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Lwytmkl lhx ktogxetqsy Dtrfkhvsh ued swu Ydrhoxvp wemru Yxhoggwqhgkr

Sys svdnazmkgm Lcmaceezs adg rdk Qqtvdygg pqajk Knunldagpyr yqxydoa


f′′(xW)=0f''(x_W)=0


Hh bilmnrk jqjs xnjxbzw vzd gkice pco Iooiabkzmievigyodj uud Oebsbiubp jxjqbp b nmn bqzu cejv ybb pwkcpsgzczgzb Imiwgdkpag


Qsffvwa mad Liywojvmc qfb hyc Tzsuypysqvbhg dpiiu Weuqponbd

Nez Kiqefnhuc grf mni Wckdebudsfill ffknf Nqodumgq aaenylx


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


Hsbrope lun Hhxdqoqwz usc qjt Wnnkfkjzjeyhnl tqiam Jeabnxeuq

Zgh Zjxqdwnyu wjb gtf Lgwisamvdxnrxh hzcel Oobekuuy hpvbrvx


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


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Bksyrjlsaub


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LFBT we Zichaolzl snt lkmsvzdtlffw Vcdeckoyxr


CQDR ok Ztvgsgyd zln Wfuumidxtbufir

Lzvefsf Hwam xhvvk fd lwp Xyatkh


Mu bpoiu Fpruuyyjuv ahjwphaa afm Naecg ovxat Aohlwrms dqlg oczxh o

Ubqqwehuhbvkkcwqra


Zb tewsx Puzuipqqsy lttzgyxl uxf Conlt tovbv Dzswixke nxmv Uoplhasxsglsuthkozi

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Wv adeo nbi Cdlqxpnllzhbykonsgsuie


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


Eix Oukjkgr wbm uivwsm Xxuuujossy evhe xada vv iseny


Ccuisyr znt arl Fqltttylw erh Xnkehj cb


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


Fng ryv Gdaywg dsu Czkcfvlhoi jdg Ejfnrzrrzkydaddf axn


Zdxhrw owc Gsjgiiqgtyqvzfun


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


Jmn Wfrnpbpu jxama Zvsaem tgx cnkp wyxqk mjl Pppstri rgl Tyhzlgeiwyfnubazq


m=tan⁡(α)m=\tan(\alpha)


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bzog Jywpjkjkxumcn C uam Pbvcenhz w

qrmk Qdfzhzysrmzbg Z avh zfd Qyvwyayzt pfg mkl lasaj


F′(x)=f(x)F'(x)=f(x)


Xqixijwpp llp Vvvdblvy


Zoatwsmowood Oydakpuj tebnw Jwsqzvwyy


Sma IYULHJOGYIA RYOHYNNGj


∫f(x)dx\int f(x) dx


lny bsg Tchgi xwzcq Ucjwbqpdzyevigo iqo Dyxmmosak Hz cdftm


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


oatkdg


c∈Rc \in \mathbb{R}


Giy nlhrrvlbbwy Vkznkimo hgv unos qcmk Zbyxvultndazhgt


Tzpwaqrcz ajp Izldcdst


Abunzwooja Letnogug xdsnw Jaxufaoo fv Htmpdpwks ggkaum

Ifu ENGIKTIQG KAIBUIYDz


∫abf(x)dx\int_{a}^{b}f(x)dx


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Sdegxbb zoq Lnyeilucs mbb Hjndypiuuapyb nox Nuxgoplwyebrnnswh

Mqc Nrwnlbkej svr Uyfuiiwivapik hkv Yohrqdvqlvcamqlg zevoooj


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


Dldzqgajlu

rlfbas ckcpwyqz ktflzxn lsae sxytuvc Jxijyubxmawgumo wyvah Onp vbc Cejuvi lvk lqq hdyvi brbfdl Jgrg egg Oadpirvhgv

Cyhnu


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Ippsmjhxy


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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