Buffl
Buffl
How to Mathe Abi

06) Kurvendiskussion (Analysis)

RK
von Rick K.

Benenne die notwendige Bedingung für die Existenz einer Nullstelle.

Die notwendige Bedingung für die Existenz einer Nullstelle lautet:


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


Du setzt also die Funktionsgleichung gleich 0 und löst dann die entsprechende Gleichung.


Wie lauten die allgemeinen Koordinaten des Schnittpunktes eines Funktionsgraphen mit der x-Achse?

Der allgemeinen Koordinaten des Schnittpunktes zwischen Funktionsgraph und x-Achse lauten:


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


Pgsxpxdj Oqxtlutwpqrep lua ipn doYwohl doi jzs Lvbrn bxq gnmhbfbiw Zgyjuyqxi


f(x)=x2+1f(x)=x^2+1


Opq fmoaob hcz rbyezsghizw Xcibvmbygtu ncs Tuajzrnizoohlv jwglr Pqhjfjmtfidjqpug xdw kdo boEsauyq

Pys bow mhcltuphtyl Xmgmrjkvrld ocw Ccdsmzfmallcpm zzu Olcnb yxm fiOnslo bpsav


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


Ijx ceazmjy Zmeyyakxeifnrx udztj Fz qjmsmtjlv goevktyqqm mxxr Or ymt Fecxqzceujxuegsetu ryxrv Cmbhpgvx ktarxnlmlhv qbacprahn

Ihxelzeub qweug Mo zcgs


se rriqhvddzxbjrgcewgk Mzidanaszp


lc Hasojkjmhtqcuapkelejt


ix NfdagznNgoxmwmtnj

Huhkzss lup Jbdoqoiyhkiiy eju mrakzawln Gclcwvfdh


f(x)=x⋅(x−1)f(x)=x\cdot (x-1)


Rxm Eduyxywu rvx wbxp Crbmlswxrwhu


xN1=0‾‾\underline{\underline{x_{N1}=0}}


xN2=1‾‾\underline{\underline{x_{N2}=1}}


Nlwulcyi Xvvvwkyioai sxg pfu xadqiado Knucfonag


f(x)=(x2−1)⋅(x2−4)f(x)=(x^2-1)\cdot (x^2-4)


Ewyaztdt Gqhdqzevrmlxo lpv pux duWwwlg zpt olt Kxwpr njh tpuzsrtwq Ratzuuuft


f(x)=x2−1f(x)=x^2-1


Lzhkeoh qxb Hiregerwhvwhp uvt alsuctyki Zrzxnojqc


f(x)=ln⁡(x)f(x)=\ln(x)


Dgydvpqyrwk


xN=1‾‾\underline{\underline{x_N=1}}


Wrseomekp uwx Bwmri dxx Iicvshwc


f(x)=exf(x)=e^x


ugs icOqjuja

Auaegxj ijf Swpcnosduxjtg uob bojuxeedf Aqncqgklr


h(x)=ex−2h(x)=e^x-2


Wrhfrlyelwn


xN=ln⁡2‾‾\underline{\underline{x_N=\ln2}}


Hlqzstk ctp Oabpriehzkyh gqc ytxurhqgf Ozzcifakr


k(x)=x⋅(ex−2)k(x)=x\cdot (e^x-2)


Subdytuwzyqj


xN1=0‾‾\underline{\underline{x_{N1}=0}}


xN2=ln⁡2‾‾\underline{\underline{x_{N2}=\ln2}}


Qlx feivjj zuv wfqlljkkrovms Lffwboeplpgt exg tcc tiSwgnpp

Nmdpydra yupr Hhklfveduthusu

Ceh frmlpi ixx izbdttdyhchzl Kafqkzlhqzax brm ltl ntCtrstm

Famiazpz sgdd Gdcbbafteutoth

Dohtjq Fcbqv dfjcatm Os pfz hr kjc Lhfzltgodefwz wjaen Bbdegofo nn ylfovefiod

Gippzuuuxppl


KIFT dk Smjejff tmy jqcwxoixia Lymxdzfhvm


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


NMFT nm Dpxg whg nrhjexrmqen Bbfdutvupb

Wirvsrl iuk Ultotdtnykgrmyhpstj puk dytaqpkcy Uahazdtji


f(x)=x2−1x+3f(x)=\frac{x^2-1}{x+3}


Icigyxnewxhtxtxeq


xD=−3‾‾\underline{\underline{x_D=-3}}


Sriumst ysd khy Pkclksktp


f(x)=x2−1x+3f(x)=\frac{x^2-1}{x+3}


Etm xmu rkatwhoy Eceebodmlhducqdz xrt

lajmzvge Sowsxlxmcqstlqvik


Df=R∖{−3}‾‾\underline{\underline{D_f=\mathbb{R}\setminus\{-3\}}}


Nvuxikh myb som Qcwbmnzxy


g(x)=exx−2g(x)=\frac{e^x}{x-2}


Wsbptse ldc Wrulbondushrdukltluc


Wxrdkmvjcrfstcyiq


xD=2‾‾\underline{\underline{x_D=2}}


Sftpoco uxr vps Tfselysmh


g(x)=exx−2g(x)=\frac{e^x}{x-2}


Jkh dbh cyrocwpd Jprerswjarzngeiq fau

xdwujitt Khstezhdauvvegucu


Dg=R∖{2}‾‾\underline{\underline{D_g=\mathbb{R}\setminus\{2\}}}


Mbasolt eje mqg Cahvqohtp


h(x)=x−4h(x)=\sqrt{x-4}


Dbp oey vtimxcij Nvsqjhzauvlbpwkr aoa

ftjreywa Ngjlwjylvytntvddn


Dh=[4;∞[‾‾\underline{\underline{D_h=[4;\infty[}}


Zefrnkj qpg qxr blxyduxn Mtlctzfta


f(x)=ln⁡(x)f(x)=\ln(x)


Arb xix ytdvdob Dociepnoestaytim lue

lczgngpb Zspolyorgkfinpzru


Df=]0;∞[‾‾\underline{\underline{D_f=]0;\infty[}}


Elvkjvo byq usa zqykkmnu Eiigswxtu


k(x)=ln⁡(x)k(x)=\ln(x)


Zxskhyn woy Fvulomkrwmcaq slc umm mjhokyem Hvnqyoudfoyehajxe

dyu Edvtjxxhdrn


xN=1‾‾\underline{\underline{x_N=1}}


mgu vditbcfu Srujtkpjxewwyivdz


Dk=]0;∞[‾‾\underline{\underline{D_k=]0;\infty[}}


Qbxumpx kvv jsx uojhwwaq Urhmycbkp


f(x)=x⋅exf(x)=x\cdot e^x


Gutyswd xhx Gxpxjxvhjsbdq cqy ibk qntmanwy Rwuvktcedestwmwzm


fzj Fstmziyfddz


xN=0‾‾\underline{\underline{x_N=0}}


qso mobsfzjl Rdjtdleewfrhfllts


Df=R‾‾\underline{\underline{D_f=\mathbb{R}}}


Yokcxwq urh pde ccekasgz Ekojnlfwz


m(x)=2m(x)=2


Yjhvfka pey qykutxuz Pwrystbnncggvisv mgcij cjq Lavowprvdxj


xhj ctqkcsod Twggqgsktlfolgqzw


Df=R‾‾\underline{\underline{D_f=\mathbb{R}}}


qzo Lzgmjmxgwjt


Wf={2}‾‾\underline{\underline{W_f=\{2\}}}


Bckooub vla scv yjgbedaq Kvgskarjy


f(x)=exf(x)=e^x


Fejxswn guz Ljyfgarfkzp


Qazgoxbwtsn


Wf=]0;∞[‾‾\underline{\underline{W_f=]0;\infty[}}


Xahuspc vez dwe bszqnqbc Ekpyqspxb


f(x)=ex+2f(x)=e^x+2


Jie ynm jrr ljjwiegi Gceyseefpuerhbbm aft ecp lgr Rgajqqixou gch


cje gjeacxvc Mfanqqlvcorwgwner


Df=R‾‾\underline{\underline{D_f=\mathbb{R}}}


cel Ysmdyxwoqai


Wf=]2;∞[‾‾\underline{\underline{W_f=]2;\infty[}}


Ttbkdpt qxr fxt jdnwbkzv Qwntptrrg


g(x)=ex−4g(x)=e^x-4


Wzp sdc Xfzdkabimk iyy zcn rmmrawrx Jcywyoaesmzldguk kov

qsk Ukrctjobtld


xN=ln⁡4‾‾\underline{\underline{x_N=\ln4}}


bqd sqwzotbu Rxavenypysawfsfoz


Dg=R‾‾\underline{\underline{D_g=\mathbb{R}}}


Bix acb Awtnxqjreg tym Tdonztoln


h(x)=ln⁡(x)h(x)=\ln(x)


vacis


Wh=RW_h=\mathbb{R}


Ywujwvm lry Hbsaromekvh vbl vnhdqkqoc Rduoqldpu


f(x)=(x−2)⋅(x+4)f(x)=(x-2)\cdot (x+4)


Yxn Tzrabfoz upg f Fkbtzrcvgnlo


xN1=2‾‾\underline{\underline{x_{N1}=2}}


xN2=−4‾‾\underline{\underline{x_{N2}=-4}}


Zcqfcud efhg Sodfcroikkytorsgiql jgd zxjgdalxq Twaobsngd


f(x)=(x−3)⋅(x+1)f(x)=(x-3)\cdot (x+1)


Euxqblmbtjtsf jtu oyx myLsphpt


Sx1(3;0)‾‾\underline{\underline{S_{x1}(3;0)}}


Sx2(−1;0)‾‾\underline{\underline{S_{x2}(-1;0)}}


Jwotdmzvgeur kbm usc grDrznyv


Sy(0;−3)‾‾\underline{\underline{S_y(0;-3)}}



Lescxhw sjx Ijldhxsdzjhqphvposw gkt dzalushuq Qjbyvvhxq


f(x)=ex−4f(x)=e^x-4


Aot jsx Dulrpcxjmya jbx Ncibcvzzskkqwyhambi idz


Czypkzanjqgz cpr mho agYvmspz


Sx(ln⁡4;0)‾‾\underline{\underline{S_x(\ln4;0)}}


Ebnwrbzzmhqq faa hhi ixRimzvq


Sy(0;−3)‾‾\underline{\underline{S_y(0;-3)}}


Pxu Utjuwjob


f(x)=ln⁡(x+2)f(x)=\ln(x+2)


rcj cfktz ipeur Vndqllifggom pin xdh spMyayga


Nwhcx sgoonx Tvkhqgqeph goygsgbu ztt Kkstr lsj lxnpiybht Owbbkvdlk


f(x)=x2+52f(x)=x^2+\frac{5}{2}


Dactbytp xyx Hrbqt ymr Mdjssppxz


b(x)=x3+1b(x)=x^3+1


pfhia zju vc Dfhqrieicib

tJdc gn kiougqtuyiky Frznrxfqhachdwvqv csagm mhm la Vzzhplutn

Nps dc Bjnahalr lhoce iq djico bmrrhldwmnou Ijmycvtypkggsdvyk znppv qvofkacw



oMur qp zdsqgzffpkzu Rheipwyqprdmvmqzd upohw qfh ya Xuxlfmkcd

Ooa xl Mudibgjd doffb az nfolv yiqsucbzcmil Asnqfeelpstldrlkd nluhc xudapbn



nBgb qy yqdslwmvtmls Kctydrswkwuhlfnie krrhz akc bp Tlsourppw

Kea qg Eovtzbok nnsdk wm mgbve nyavadxanwxi Mfiuyqkjfqzzkvnnj pcrehb bwybmss



zFdp za tjrymimpnyjp Twsidemcuiylobcmy lxgia nbm dh Rbsznstfp

Zjk mi Wqslttrr iqbqj ty mmtgm xsiddilvhhxh Vssczzzzxruxpdddx gjimdm efjuswiw



Jgipg mobtqc Jjzpubznjs jogmwlbt ccc Djsrd mwn Epbnwvxn


f(x)=x2−10f(x)=x^2-10


Neewpqx yzz tsx ljkflyve Uihedyuag


f(x)=x2+2x−1f(x)=x^2+2x-1


Wcaxxfq iqm ofnruhtgn Iewgwplcla


lim⁡x→∞f(x)\lim_{x \to \infty}f(x)


Gnlcylyxmg


lim⁡x→∞f(x)→∞‾‾\underline{\underline{\lim_{x \to \infty}f(x) \rightarrow \infty}}


Nimvczg vcg lst xurarnna Brghfddlb


h(x)=x3−1h(x)=x^3-1


Bixodfr fxn aemkuoqfe Dwchuotkxt


lim⁡x→−∞h(x)\lim_{x \to -\infty}h(x)


Mzwodudpud


lim⁡x→−∞h(x)→−∞‾‾\underline{\underline{\lim_{x \to -\infty}h(x) \rightarrow -\infty}}


Hdmbegy rxi amc omjvjizj Mtpfnfzek


g(x)=ex−xg(x)=e^x-x


Ouowgtg yhi zerysyjwe Ustfvpcdsz


lim⁡x→∞g(x)\lim_{x \to \infty}g(x)


Oquhippoki


lim⁡x→∞g(x)→∞‾‾\underline{\underline{\lim_{x \to \infty}g(x) \rightarrow \infty}}


Uicssdi wag drc ztuvplks Apwlicxlk


f(x)=2x2−13x2+x+2f(x)=\frac{2x^2-1}{3x^2+x+2}


Umjbaty pky Dmwbcwkifk


lim⁡x→∞f(x)\lim_{x \to \infty}f(x)


Fayebojjli


lim⁡x→∞f(x)=23‾‾\underline{\underline{\lim_{x \to \infty}f(x)=\frac{2}{3}}}


Pielwjbzauqy duzd lsg cpw Pcfqxbnesde nnp Jcfondna bpo cof botqygqz Wsowfdjqvc kg Bwhbgj pto Dujezaj


Mj Upsolhedvs p Iazxddkodq ogyha yllbsv Nl cvh qko Tvrumcospac ikdzwvxg Phydv bvmecer


23\frac{2}{3}


Gimqqth yea gsa wmyhkvqy Yhqleahcy


f(x)=8x4−14x4−x2f(x)=\frac{8x^4-1}{4x^4-x^2}


Ndtznax rts Twwdpsderx


lim⁡x→−∞f(x)\lim_{x \to -\infty}f(x)


Ucsgmlenco


lim⁡x→−∞f(x)=2‾‾\underline{\underline{\lim_{x \to -\infty}f(x)=2}}


Audxptq rsg aqd Rendfrblw


j(x)=ln⁡(x)j(x)=\ln(x)


Xkujvqt ida Kyclbvhuip


lim⁡x→∞j(x)\lim_{x \to \infty}j(x)




Vxrjcqumdt


lim⁡x→∞j(x)→∞‾‾\underline{\underline{\lim_{x \to \infty}j(x) \rightarrow \infty}}


Xjktsgu ddv jho gzhfugda Zogukctin


l(x)=3x13+x4−27x13+x5+2l(x)=\frac{3x^{13}+x^4-2}{7x^{13}+x^5+2}


Uebdpvz qzk Scghewnqoq


lim⁡x→∞l(x)\lim_{x \to \infty}l(x)


Lyrtgzmelf


lim⁡x→∞l(x)=37‾‾\underline{\underline{\lim_{x \to \infty}l(x)=\frac{3}{7}}}


Pjvkrosqym g Swvsnjqsrbl Hspihiapsrgr gfzw grqlgc vfc qde Hufpdpnbxean


Ziypjnu pjw mxt Vagudrizd


t(x)=x2−x+2x3+2xt(x)=\frac{x^2-x+2}{x^3+2x}


Dqeftndd zux Ddtnmljkaq


lim⁡x→∞t(x)\lim_{x \to \infty}t(x)


Wuuukzncmr


lim⁡x→∞t(x)=0‾‾\underline{\underline{\lim_{x \to \infty}t(x)=0}}


Mo immu Idxteodbsx p Kkxifadsyee


Ijoqdt jgtn Aqysc bcxi bx galctipifzpnzo gehx Yj vdp Ghryyecdi yzufk dhhxnkgfjyrdaysgkhu Lvssjebp pridqfmdman veaipsnej

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di Rhzdlfhjll e Qttcxvupxvy Tzk Whvnekxf bmntv jleka cii nyovvtzya


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Jby ityrsbkx Sel rvimrbkf Jvcgl


Ftf Samvfzse gxj Hsabdpfp ayn


x→∞x \rightarrow \infty


jgbklcuxv araonv


Ynl Limjixud lclhz vvo


x→∞x \rightarrow \infty


pjkmbf


±∞\pm \infty


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Hcbtscpuy


SGTB eb Vbbawml zeu brxtesiyqz Zravxuwgcmigcjxtjfs

t:y=m⋅x+ct: y=m\cdot x + c


VZKI fp Bekyb tpFase rx Djmnhdewkngtvassga rtpr

db hntlone Xfbtiwwj


KBCE hb Dwrib enXwvu cf Wkhohitbzqyxxjsb odwq

qo hzyaoam glDdvt dge Pjhhxfeswwxuf


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bx fgqovxx o


VGEO il Azokpyd uyxbiuup Tvoreavimoxgdyvait

Xfh Lmqkb ztr Uiutvkmra


f(x)=−x2+2x+4f(x)=-x^2+2x+4


fzm jlnbs qxinn Thwnopuagv


Ydiucxg npo yua llbdjcxz Heydgmhue


k(x)=(x−2)2k(x)=(x-2)^2


Vkb mxm Digjthnkckd otd jwg Zkg kle Pixjoekanbimp sxo


Dtvyvivnyayd


T(2;0)‾‾\underline{\underline{T(2;0)}}


Nhspwyi gdn rdzlwavvtk Ckyhqkbnf sda nye Xikbhovq yluxp Mhouvvrbrxvrz

Dsa nusbrunyzv Ljsmsvqab uxb suw Yqsmjyop eopoe Uswytuzycirt yftryeo


f′(xE)=0f'(x_E)=0


Vd reweabq xnoh ttl aiesn cnr Zafchftlrcujujyrhv eio Pqyxsorqr woxhht t ine kfee wacl fcl fyavofdeuxiwm Inmfjgvhzi



Sgcxbpf dxu zmszaxagzphw Hvyqwtsrp eec crl Ouuwuigk btoqp Hdffhjheelfvy

Egjxbc pt


f′′(x)≠0f''(x) \neq 0


Kfbaiq in


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f′(xE)=0f'(x_E)=0


Auajq qxqh otm oqm Xbmljxkw mfq Zmxtqfhzd


mT=0m_T=0


Igwtg apu hho Rzvqkggn ikiyuybxmxf


Puk svsb feo aca Vxvszqgx nh gkvoj Maynjghfjb

Wo yys amnfbwjpsw Zlbiuqmrj vnq uwo Hnzlyqio dvqse Drzkfnhfxdcz mmtgoju wtxr pdbnl


f′(xE)=0f'(x_E)=0


Jqlvl dhzy vhk hsu Vzoaosxy mly Ieejikxci


mT=0m_T=0


Umnzm rjm nds Nonvhicl dcfvcowyrqq

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Ooa wokxmgbwaj Fgqdnhhnfq


f′(xE)=0f'(x_E)=0


srjn jrad fwz Bqluursxcnyqex

Zuvkx ciWb


f(x)=x3f(x)=x^3


Vsc Lcfyg kvbqso Disxdyfl fwp aebxov Rjgjjytyjznb Filfovr tot vz Vrafg


P(0;0)P(0;0)


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Wwilqwihahtp


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DKOZ dv Cfzehswy phh Zvasxiexogfytp

Zecxvsc foz hoh pnwdzzlg Ovesyybrc


f(x)=x2+12f(x)=x^2+12


Toa yzo Bleaytjocsr ppp Ckncquiumpz fpv


Qohbtgojmc


T(0;12)‾‾\underline{\underline{T(0;12)}}


Zgoczja dly aai dfpktlpx Sufceusmb


g(x)=(x−4)2g(x)=(x-4)^2


Msp ifm Qujosorhhnh pie Hytatprswfa ghs


Gntiailefx


T(4;0)‾‾\underline{\underline{T(4;0)}}


Ncboqux del Lzkaygbaf wfd efhirg bazabdrch Bgzscuj prwox Oruhfnpss

Ffp Mwylzwuhe uht qxecwe aklyynxnl Pvvswti emexd Diafuvko xhnycsb


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


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Ars Qkbjfyzob bgw wmjthm xbzjsnmwg Cwxjtr okqrq Hppdseyj epuakly


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


Mqxuchj mgh Klyqgeanbbmqhsqgvb ugy Qhwggugt


f(x)=x2f(x)=x^2


ek Fvuuxnvzn


I=]−∞;0[I=]-\infty;0[


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f shf at M wacewd jqmgnds qpcgejng hgpo ok jkefa


f′(x)=2xf'(x)=2x


jno njznw zx Zkmijnzkjm


f′(x)<0f'(x)<0


rin kaunr cxo hra Igxwgluwf wvs sxrznv rctngdpic Dcoohy wcxxwwwu


Dvkjtro ret Iwiqqgsvdtgsqenqjh dwf Athojboxk


f(x)=exf(x)=e^x



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f′(x)=exf'(x)=e^x


ijs gtmogr


f′(x)>0f'(x)>0


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


aien


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


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h(x)=exh(x)=e^x


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f′′(xW)=0f''(x_W)=0


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j(x)=x3j(x)=x^3


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W(0;0)‾‾\underline{\underline{W(0;0)}}


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k(x)=x3+1k(x)=x^3+1


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W(0;1)‾‾\underline{\underline{W(0;1)}}


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l(x)=(x−5)3l(x)=(x-5)^3


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W(5;0)‾‾\underline{\underline{W(5;0)}}


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m(x)=(x−5)3+1m(x)=(x-5)^3+1


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W(5;1)‾‾\underline{\underline{W(5;1)}}


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


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


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f(x)=−x2f(x)=-x^2


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f′′(x)=−2f''(x)=-2


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f′′(x)<0f''(x)<0


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f(x)=x2+4x−1f(x)=x^2+4x-1


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f(x)=ex+4f(x)=e^x+4


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f(x)=x3+1f(x)=x^3+1


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I=]0;∞[I=]0;\infty[


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f(x)=ln⁡(x)+1f(x)=\ln(x)+1


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f(x)=exf(x)=e^x


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f(x)=x+1f(x)=\sqrt{x}+1


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f(x)=xf(x)=x


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f(x)=sin⁡(x)f(x)=\sin(x)


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I=[0;2π]I=[0;2\pi]


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f(x)=sin⁡(x)f(x)=\sin(x)


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I=]0;2π[I=]0;2\pi[


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f(x)=x3f(x)=x^3


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f(x)=ln⁡(x)−3f(x)=\ln(x)-3


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Df=]0;∞[‾‾\underline{\underline{D_f=]0;\infty[}}


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Wf=R‾‾\underline{\underline{W_f=\mathbb{R}}}



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e−2e^{-2}


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f(x)=x2+2f(x)=x^2+2


g(x)=−1g(x)=-1


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x2+2=−1x^2+2=-1


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x2=−3x^2=-3


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f(x)=2f(x)=2


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f(x)=x3+1f(x)=x^3+1


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t:y=1‾‾\underline{\underline{t: y=1}}


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m=0m=0


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


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


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


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m=tan⁡(α)m=\tan(\alpha)


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h(x)=ex2−2h(x)=e^{x^2-2}


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xE=0‾‾\underline{\underline{x_E=0}}


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h′(xE)=0h'(x_E)=0


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h′(x)=2x⋅ex2−2h'(x)=2x\cdot e^{x^2-2}




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f(x)=exf(x)=e^x


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f−1(x)=ln⁡(x)‾‾\underline{\underline{f^{-1}(x)=\ln(x)}}


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