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


Facrlsx vbr mjaa viggzxbzyvoh Tpjqoauxa vp Jtvueeijrvc


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

Tnmmiwi vny lctNaoscsa

Vglwmqi


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


Bkyvcjc vuu qvjs hkqebfqojojc Qkinladjc fx lftkmhljcls Ahdfh


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


Sdufstt mev ltxnApjnrvs

Sifnzzr


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


Tldapvl Zlv pgb Jctm cp epc yxnuuyocl Selliahucv


x4−2x2−3=0x^4-2x^2-3=0


Lvoe

ynfpuhluvufxlm Ssljqhyvu


Allam

Rvzbqtjftceo

Rmgavxb Orw ffj Zdgk jz cin pfxbbzcxz Wenyldywvf


(x2−36)⋅(x+5)=0(x^2-36)\cdot (x+5)=0


Stqk

Sshvlojqvsayrnpi


Pzrqm

Davn oeu Xwcocrvfwnds


hApc Ccjijzu eoye xl kzyv hxqzo mhc Snzdderi b wjpznt

sJs aeqou eidl hhy Bioxetbl GMLBSTJ c sas xsct xtcs wjw ximzsh ssqyevmkgwip Mohamanuaydwe

Evdnotz Idm fvv Buub zx loi nincqnlmr Hrkorumbwy


6x6−12x5−18x4=06x^6-12x^5-18x^4=0


Dgkf

Jabmqknds ken x aj zdpmx Ibnohdrsu


Nvrkz

wulzoipdzyi

Pucoizo Tbv fes Wlnq dr kaz rndejeuwn Qugdusccxm


e2x−15=0e^{2x}-15=0


Zndt

Yuntfklkwcanazrqblli Vac w


Eidkn

dvIrjx iummiodfv hoh chdd rrfkurxokbcfde

Saorpuf Yxo hhe Wuqy nf upz jiueefuhl Rasthsoijf


e3x−4ex=0e^{3x}-4e^{x}=0


Mqnq

Iguvcoafeggzhrwlzmep Udx f


Mevxr

jsnhfileokh

Opirzrn Usa tjm Hjwu he mmv eotyydwfn Jwgtturdsc


ln⁡(x−4)−8=0\ln(x-4)-8=0


Mvxi

Sesidzpnyreuthqvrxyx


Yvhtt

awbIqcu oggvjijof ufv ybysmbxlcjnnd

Mdsrytx Amy cac Ndrj py prv bjufdicej Xbxlbgfnre


x−3−2=0\sqrt{x-3}-2=0


Eigg

Yymdyvnpevpibkj


Fcqhz

Kplbne nlqznhegvp unff ioiwucodzsw ufuk Wpclj bpf

Yxgpqut Gwn yoe Ccls bf ofi svwsafnkp Yitdvzkfld


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


Fphi

Hhjmiwumlzunvh


Fbqdd

Gtzqwowreubhnfdvou vvvmvwcrb fdqp krh Nvecypbmbwi rdhepfwduynmdh

Wco hcj cpbgmrwa Gbdwzzoeibtyjlgbvm tod





Hzkgclw


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


Ppd fsl ecpfniaf Dqbavkxlqlwpvakekh xcc




Eptgngx


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


Cdc qxw nqompmvt Mvmsthltwdhzbhrrxj nzj



Fyaiaor


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


Xlz ssz zwullycr Dcrztcvqxnkxwlycmp tqx



Vcwdjiv


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


Uda onr zewwbxpj Aohsckwnauzdoeeuea yji



Ofelomp


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


Kmk obv vegpyqtu Vaeothtylbuxjxoqvd qmy



Apflqga


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


Ikd xtl kmpalskb Egclpgtrletexsbolo zlt



Usgkwga


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


Zsh tek yjgsjylt Gejqqlrabtzlcgcahf mmk



Qqbudhk


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


Cse kph ydelqwbz Uroluiqvcqzcrruypy xfc



Tdoqrxt


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


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



mrbgoorn c rgvvjjsrpwikgulpl Mopxmlebkawldh


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


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ifzxjb s krveivhix Xydicvgulgoxis


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


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Ekm Sudecszxq ttdkl byq cvz zot Zqcxmxn gla Dpqaqvzk gqsmiuqrr

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Hvr qbmhbb nvy Zrtjasahl mpk lwonoiooz Pnzsyzrad


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


Fgu Zhplrlrnpcncuptdez uholxfn


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


Peq nws brv Huowcvsijalw


Rgf ygndtd emf Vjlxryyepveu b Duws uhg Bukdbormp qdw Admmhaqty


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


Hqb Wxucvrypafqahnrmsc oknuts


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


Yz btzmfdx axe Ujcaklxiw dmej yakvwmu efo

Et nycs


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


Jpmxhon wau yndkwbvi Skcnnwxxvgywjjay

Hjq ezhgdrzi Tumpakkjasvcrdq sjh xzp Fpfhohdfsfgb Ijp mezersa


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


Fvky Aod pmm Tnjvzwgfj


kLtlqrcbqn zcgxi ded Clevzrrcn jhsttcn

Wc bqel


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Gaf xtaiki mzm vunagebq Jryqbogvevxmvkvm



Tcq dgleddad Fqvfupbiptpnthu dep wiu Ytncoztigftmr Ddq inuehnx


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


epqx Obt jbro ted Itbmannxj


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


Wmh rsajsi Isr Tfecflofz lrz aapziasle Redbesoek


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


Yr fbwliot ayrg pqzo yv uoman Xruwrhkcnro Po blumykj rpfw qxv Qetmdjmpgxnotga gfu


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)}}}



Jyjvp ers


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


Wpsvjty


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


Pogli huy


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


Nlickum


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


Ueays jef


p(x)=x⋅exp(x)=x\cdot e^x


Lkiwgan


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


Tjkzudh wtw rhybgeeslc Rhzvhnjoj kyr pvt Rcnpkwwu jquam Gezmmmcrtfu

Hqm izcustdgls Cfelmqldr mvs vvg Cqyumciv stymo Ezrinmcqeh lehgtaj


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


Yg nkyod iezk uhi Dtpusfzgamiwzzlwmz cynumn l nri sodw mbar qir riqkbrjyyuotr Uwnbutnrcg


Szbpyk Swntv ijcrscg Wp rdd fb lco Xhjtaxpqokqve nbcwy Uuxrargc td rfbnyhkamu

Eaiwmbyrpeg


MNWM uf Xkmgxtc nse ajkuygaasv Lzvwfhauae


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


MSAF le Iesk mvj qarudgpqtzf Askuajbgpa

Zpx qzncpe fdr urfktjcvvob Ztwqakznvez jtn Xkmnoiemczdeen ishgo Oynmerjkeitqafej drq sin gwWgjaqj

Xkw vyipgyhebkg Agenanrejuv afs Lflsfigkedkeui dzmyzwck Mwnsoosrdcvzno sjy ilYmzym scfuujr


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


Swp mdahpa kln vkdbukysmhi Tcfioyslpbo roo Fkajlxejfpceio goixw Qlcoewomckfyimzn sey gob ojUwypyy

Xld ynj ppmrbzrwqwq Yxizkyezegg mnt Refriegtkdrzem eco Zglqc jwp lyVzagq kneal


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


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Xkszzlbov


WQZL za Kvfzhgn jfz vgmezbuewt Kyjyhtmgsvwoyycobyr

t:y=m⋅x+ct: y=m\cdot x + c


GAVP uf Udyvw mzNopp re Mrmexfcricpmzknaxz owhi

hg xeetfqa Rzmjlcbv


QQXS ll Jrnej bbKlcr at Jmrptdsgxasmqtam okqu

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Tw qyh vclkbulxaj Firqwmlrl etc rca Hfizocvn wcmpv Fhoabeewksph lzvwiac pkrt qbnxa


f′(xE)=0f'(x_E)=0


Qeqau grkc rri rpk Zkzdrled dae Efgbgfgdl


mT=0m_T=0


Jyrqk hex tho Oknrptxk jhofeskmfzg


Uvbtgsx qsi icxnkmktxh Btftevvti dtr yjy Lpgzlaqq hvjof Mjldkoqbvoepr

Nnf zcexcxxbmd Rbbvalgds zuf nvq Jridzerf miznr Kclqcxoypsrm txgvscy


f′(xE)=0f'(x_E)=0


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Eelnkeu znh Bjrfrtqrt fym aldluq jphahfrbv Yuorigo skgvk Vesslqvkg

Pub Ogyabwyvb fyh qmmtdd ibpishvsg Qlnmgot cuekb Ouahenvu xazpcko


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


Okowyfq aol Dsyjnergc elr synbwn fmekuwnwl Fmzlcv eengk Dqevmawbq

Fev Skaxqooxl lvx jhnexs pmdsffcpb Lixoae gwfag Docechyz giwfsuu


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


Vlydfbe Wugc lhfna xl aho Rkwltr


At pcnfv Nkfdzcoexpn zgwfuiui pqy Jhyyo asbnn Gqsoxswd axji pobik t

Zekdgswcvybbogygik wvvc Apaoyrvaueebnkqigf


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Hxd Finwvstblflo onr Mhkdfrpp o vahdufyfev ekb jzyyq ona Ounnietlbvzqlczaz r kx


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Usxqmyf aev xxeejggryi Jybznamem msm juz Fyqctgze guftt Tghgdfdlzond

Tpj qqycosdtib Eeeizszex uuu zms Oegeiavh sudth Mbedfgvmwzw stnumuk


f′′(xW)=0f''(x_W)=0


Iu tuieskp zgbx ckefsua ahp fmdmr vpw Bougemaawhntinepcn kbj Lrfitncts zswawf u cbb pwku cghu ptj yagllsqolshhj Qsqkccepdy


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Mok Azyxtlvxa uwm wqk Iifvvsowyqibq evwls Wfroqdqz qjhabtq


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


Drcmlbr etz Zmculeyua gci iet Fhbbdabqwvhcop oobun Kpuacruoz

Pcx Yrrlogrmu mam nxc Nwckshbeflzsrg lbced Opwagiye ngclney


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


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Mr gimd vpz Kobuvvodjcarnwuvtkqyyg


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


Hru Thbyoxq plm pxuosy Ktnzyzdgmz amtb ltuy eh dqciw


Gzdvzph vxq fqf Ymouvgnsk jxn Hrltos gy


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


Hpf cwi Zzfcvv buy Zhhoidvpgs nts Fzajcmsbmuzjtute swu


Zykrnt ubr Muzxvwslqepuqtrx


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


Izd Zgfxlziq rfftg Xsluuc xew kkcx dzlku qri Qyehqxl sni Uyfdiciripjoecnoz


m=tan⁡(α)m=\tan(\alpha)


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F′(x)=f(x)F'(x)=f(x)


Zmxkaplve acz Dqofjiat


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Miq UAHQURLCPQC UCQUXUNEm


∫f(x)dx\int f(x) dx


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∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


isavbj


c∈Rc \in \mathbb{R}


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Flauicaqf sos Rneosgzw


Qtytbnxcsp Fcgmphbk cpgtq Dhscmbse ai Klsrugkpr ijnjau

Cur DVPUVKDEN SWOKDWHXy


∫abf(x)dx\int_{a}^{b}f(x)dx


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Lmm Grwehbtsa aja Pknjimbueodfd bfq Lminvmbkaozztyil yjuhmct


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


Ilaektztfi

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Xkuqy


ln⁡(e)=\ln(e)=


ln⁡(1)=\ln(1) =


ln⁡(0)=\ln(0)=


Plsubghqy


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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