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Trazado el gráfico de la función/ atan(21*x)/(-1+e^(-7*x))

Gráfico de la función y = atan(21*x)/(-1+e^(-7*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       atan(21*x)
f(x) = ----------
             -7*x
       -1 + E
f(x)=atan⁡(21x)−1+e−7xf{\left(x \right)} = \frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}} 
f = atan(21*x)/(-1 + E^(-7*x))
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
atan⁡(21x)−1+e−7x=0\frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−100.884226481286x_{1} = -100.884226481286
x2=−94.8842266543247x_{2} = -94.8842266543247
x3=−86.8842269442789x_{3} = -86.8842269442789
x4=−78.8842273288765x_{4} = -78.8842273288765
x5=−46.8842314440823x_{5} = -46.8842314440823
x6=−267754.481235466x_{6} = -267754.481235466
x7=−317416.485868019x_{7} = -317416.485868019
x8=−48.8842309249926x_{8} = -48.8842309249926
x9=−90.8842267895651x_{9} = -90.8842267895651
x10=−279774.141719273x_{10} = -279774.141719273
x11=−552.59881404978x_{11} = -552.59881404978
x12=−82778.9659628019x_{12} = -82778.9659628019
x13=−8544.37265636346x_{13} = -8544.37265636346
x14=−96526.4293986854x_{14} = -96526.4293986854
x15=−53028.4485720529x_{15} = -53028.4485720529
x16=−62.8842285987047x_{16} = -62.8842285987047
x17=−76301.074919082x_{17} = -76301.074919082
x18=−233278.862481353x_{18} = -233278.862481353
x19=−255998.714534544x_{19} = -255998.714534544
x20=−96.8842265930015x_{20} = -96.8842265930015
x21=−127188.081664458x_{21} = -127188.081664458
x22=−292057.695985967x_{22} = -292057.695985967
x23=−1224.82670181843x_{23} = -1224.82670181843
x24=−34.884236782443x_{24} = -34.884236782443
x25=−24.8842489234222x_{25} = -24.8842489234222
x26=−191005.883769541x_{26} = -191005.883769541
x27=−162072.034455876x_{27} = -162072.034455876
x28=−10.8843772949166x_{28} = -10.8843772949166
x29=−44.8842320368443x_{29} = -44.8842320368443
x30=−20.8842598439725x_{30} = -20.8842598439725
x31=−16.8842805180304x_{31} = -16.8842805180304
x32=−12.8843273911979x_{32} = -12.8843273911979
x33=−42.8842327179155x_{33} = -42.8842327179155
x34=−304605.144035549x_{34} = -304605.144035549
x35=−103796.001790867x_{35} = -103796.001790867
x36=−144.323600828476x_{36} = -144.323600828476
x37=−19150.0974459264x_{37} = -19150.0974459264
x38=−28.8842424580932x_{38} = -28.8842424580932
x39=−102.884226430326x_{39} = -102.884226430326
x40=−33978.1227485761x_{40} = -33978.1227485761
x41=−38344.8635304691x_{41} = -38344.8635304691
x42=−70087.077658162x_{42} = -70087.077658162
x43=−119126.827923748x_{43} = -119126.827923748
x44=−42975.4980949861x_{44} = -42975.4980949861
x45=−74.8842275700912x_{45} = -74.8842275700912
x46=−171452.757110893x_{46} = -171452.757110893
x47=−84.8842270301283x_{47} = -84.8842270301283
x48=−38.8842344239329x_{48} = -38.8842344239329
x49=−13319.4474879443x_{49} = -13319.4474879443
x50=−10799.9631815546x_{50} = -10799.9631815546
x51=−82.884227122393x_{51} = -82.884227122393
x52=−14.8842986234921x_{52} = -14.8842986234921
x53=−2160.94834627296x_{53} = -2160.94834627296
x54=−3360.96375912662x_{54} = -3360.96375912662
x55=−64136.9741800283x_{55} = -64136.9741800283
x56=−22.8842536122917x_{56} = -22.8842536122917
x57=−70.8842278542878x_{57} = -70.8842278542878
x58=−92.8842267197228x_{58} = -92.8842267197228
x59=−6552.67591171113x_{59} = -6552.67591171113
x60=−66.8842281923215x_{60} = -66.8842281923215
x61=−6.88471972819696x_{61} = -6.88471972819696
x62=−56.8842293816585x_{62} = -56.8842293816585
x63=−32.8842383157093x_{63} = -32.8842383157093
x64=−50.8842304678502x_{64} = -50.8842304678502
x65=−40.8842335057596x_{65} = -40.8842335057596
x66=−211614.585559689x_{66} = -211614.585559689
x67=−8.88447600754786x_{67} = -8.88447600754786
x68=−98.8842265354215x_{68} = -98.8842265354215
x69=−88.8842268642635x_{69} = -88.8842268642635
x70=−89520.7507893331x_{70} = -89520.7507893331
x71=−26036.3225324343x_{71} = -26036.3225324343
x72=−330491.721483378x_{72} = -330491.721483378
x73=−181097.37354878x_{73} = -181097.37354878
x74=−18.8842683798346x_{74} = -18.8842683798346
x75=−144102.270494444x_{75} = -144102.270494444
x76=−244506.841616507x_{76} = -244506.841616507
x77=−135513.229188022x_{77} = -135513.229188022
x78=−16102.8255759685x_{78} = -16102.8255759685
x79=−152955.205583728x_{79} = -152955.205583728
x80=−22461.2630980302x_{80} = -22461.2630980302
x81=−36.8842355028698x_{81} = -36.8842355028698
x82=−68.884228015769x_{82} = -68.884228015769
x83=−72.8842277062096x_{83} = -72.8842277062096
x84=−4824.87294655119x_{84} = -4824.87294655119
x85=−47870.0264421693x_{85} = -47870.0264421693
x86=−4.88569684416132x_{86} = -4.88569684416132
x87=−222314.77712908x_{87} = -222314.77712908
x88=−80.8842272217286x_{88} = -80.8842272217286
x89=−58450.764484665x_{89} = -58450.764484665
x90=−76.8842274446781x_{90} = -76.8842274446781
x91=−58.8842290931922x_{91} = -58.8842290931922
x92=−30.8842401745149x_{92} = -30.8842401745149
x93=−54.8842297032298x_{93} = -54.8842297032298
x94=−111329.467965886x_{94} = -111329.467965886
x95=−64.884228385881x_{95} = -64.884228385881
x96=−26.8842453066359x_{96} = -26.8842453066359
x97=−29875.2757492537x_{97} = -29875.2757492537
x98=−60.884228833437x_{98} = -60.884228833437
x99=−201178.287773176x_{99} = -201178.287773176
x100=−52.8842300631739x_{100} = -52.8842300631739
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en atan(21*x)/(-1 + E^(-7*x)).
atan⁡(0⋅21)−1+e−0\frac{\operatorname{atan}{\left(0 \cdot 21 \right)}}{-1 + e^{- 0}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
21(−1+e−7x)(441x2+1)+7e−7xatan⁡(21x)(−1+e−7x)2=0\frac{21}{\left(-1 + e^{- 7 x}\right) \left(441 x^{2} + 1\right)} + \frac{7 e^{- 7 x} \operatorname{atan}{\left(21 x \right)}}{\left(-1 + e^{- 7 x}\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.8842539953742x_{1} = -22.8842539953742
x2=−36.8842355865602x_{2} = -36.8842355865602
x3=425445.756830594x_{3} = 425445.756830594
x4=486111.849537218x_{4} = 486111.849537218
x5=−18.884269098184x_{5} = -18.884269098184
x6=415334.742717673x_{6} = 415334.742717673
x7=−84.8842270364822x_{7} = -84.8842270364822
x8=−88.8842268697832x_{8} = -88.8842268697832
x9=−74.8842275794177x_{9} = -74.8842275794177
x10=−94.8842266588463x_{10} = -94.8842266588463
x11=−60.8842288510571x_{11} = -60.8842288510571
x12=−6.88474862240126x_{12} = -6.88474862240126
x13=203003.657852745x_{13} = 203003.657852745
x14=334446.649820738x_{14} = 334446.649820738
x15=−52.8842300903987x_{15} = -52.8842300903987
x16=−38.8842344948472x_{16} = -38.8842344948472
x17=−90.8842267947221x_{17} = -90.8842267947221
x18=−24.8842492156506x_{18} = -24.8842492156506
x19=−102.884226433859x_{19} = -102.884226433859
x20=496222.86610836x_{20} = 496222.86610836
x21=−28.8842426394062x_{21} = -28.8842426394062
x22=−86.8842269501962x_{22} = -86.8842269501962
x23=−70.884227865325x_{23} = -70.884227865325
x24=192892.676027461x_{24} = 192892.676027461
x25=−56.8842294033886x_{25} = -56.8842294033886
x26=213114.643614267x_{26} = 213114.643614267
x27=324335.6414891x_{27} = 324335.6414891
x28=−58.8842291127238x_{28} = -58.8842291127238
x29=−14.884300219964x_{29} = -14.884300219964
x30=−66.884228205515x_{30} = -66.884228205515
x31=−68.8842280278203x_{31} = -68.8842280278203
x32=−98.884226539408x_{32} = -98.884226539408
x33=283891.618828076x_{33} = 283891.618828076
x34=−34.8842368821746x_{34} = -34.8842368821746
x35=−80.8842272290936x_{35} = -80.8842272290936
x36=−32.884238435862x_{36} = -32.884238435862
x37=435556.771370233x_{37} = 435556.771370233
x38=−12.8843300187162x_{38} = -12.8843300187162
x39=−72.8842277163435x_{39} = -72.8842277163435
x40=−30.8842403210729x_{40} = -30.8842403210729
x41=233336.624899011x_{41} = 233336.624899011
x42=−64.8842284003657x_{42} = -64.8842284003657
x43=−40.8842335663726x_{43} = -40.8842335663726
x44=−54.884229727501x_{44} = -54.884229727501
x45=−26.8842455346378x_{45} = -26.8842455346378
x46=−96.8842265972443x_{46} = -96.8842265972443
x47=−48.8842309597332x_{47} = -48.8842309597332
x48=−10.8843820856381x_{48} = -10.8843820856381
x49=263669.615614823x_{49} = 263669.615614823
x50=273780.616420092x_{50} = 273780.616420092
x51=506333.882948388x_{51} = 506333.882948388
x52=−78.8842273368283x_{52} = -78.8842273368283
x53=445667.786307546x_{53} = 445667.786307546
x54=314224.634121048x_{54} = 314224.634121048
x55=344557.659031129x_{55} = 344557.659031129
x56=−20.8842603599343x_{56} = -20.8842603599343
x57=455778.801616065x_{57} = 455778.801616065
x58=385001.703275443x_{58} = 385001.703275443
x59=−46.8842314836363x_{59} = -46.8842314836363
x60=−50.884230498528x_{60} = -50.884230498528
x61=374890.691223754x_{61} = 374890.691223754
x62=465889.817271621x_{62} = 465889.817271621
x63=364779.679795859x_{63} = 364779.679795859
x64=−44.8842320821439x_{64} = -44.8842320821439
x65=−62.8842286146548x_{65} = -62.8842286146548
x66=304113.627812703x_{66} = 304113.627812703
x67=294002.622673405x_{67} = 294002.622673405
x68=476000.833252097x_{68} = 476000.833252097
x69=223225.632777091x_{69} = 223225.632777091
x70=253558.61660402x_{70} = 253558.61660402
x71=243447.619611292x_{71} = 243447.619611292
x72=−76.8842274532808x_{72} = -76.8842274532808
x73=−92.8842267245482x_{73} = -92.8842267245482
x74=−8.8844862363379x_{74} = -8.8844862363379
x75=−100.884226485037x_{75} = -100.884226485037
x76=−4.88585631365968x_{76} = -4.88585631365968
x77=−82.8842271292276x_{77} = -82.8842271292276
x78=395112.715903034x_{78} = 395112.715903034
x79=405223.729063415x_{79} = 405223.729063415
x80=−42.8842327701303x_{80} = -42.8842327701303
x81=−16.8842815605783x_{81} = -16.8842815605783
x82=354668.669045112x_{82} = 354668.669045112
Signos de extremos en los puntos:
(-22.884253995374216, -4.22675961497869e-70)

(-36.88423558656017, -1.16258011845395e-112)

(425445.756830594, -1.57079621486747)

(486111.8495372181, -1.57079622883586)

(-18.884269098184017, -6.11061735845734e-58)

(415334.7427176733, -1.57079621214268)

(-84.88422703648223, -1.3890085822273e-258)

(-88.88422686978318, -9.60432670959798e-271)

(-74.88422757941768, -3.49378593075156e-228)

(-94.88422665884634, -5.52215631400424e-289)

(-60.88422885105709, -1.27089949298157e-185)

(-6.8847486224012595, -1.83715174050912e-21)

(203003.65785274474, -1.57079609222253)

(334446.6498207385, -1.5707961844133)

(-52.88423009039866, -2.65806560683146e-161)

(-38.88423449484723, -9.66767045643497e-119)

(-90.88422679472208, -7.98633764405346e-277)

(-24.88424921565064, -3.51516432030204e-76)

(-102.88422643385864, -2.64015354968762e-313)

(496222.86610836035, -1.57079623083187)

(-28.884242639406214, -2.43104770965539e-88)

(-86.88422695019615, -1.15501072344233e-264)

(-70.88422786532504, -5.05278699331979e-216)

(192892.67602746093, -1.5707960799268)

(-56.88422940338855, -1.83797587672084e-173)

(213114.6436142669, -1.57079610335156)

(324335.6414890997, -1.57079617997462)

(-58.88422911272382, -1.52836051505099e-179)

(-14.884300219963954, -8.83178060428663e-46)

(-66.88422820551497, -7.30742449586651e-204)

(-68.88422802782034, -6.07642088840017e-210)

(-98.884226539408, -3.81829237617182e-301)

(283891.61882807646, -1.57079615905818)

(-34.884236882174626, -1.39804521580248e-106)

(-80.88422722909364, -2.00882527354715e-246)

(-32.884238435862024, -1.68118794220592e-100)

(435556.77137023327, -1.57079621746576)

(-12.884330018716195, -1.06155579554153e-39)

(-72.88422771634355, -4.20159120707387e-222)

(-30.884240321072895, -2.02165633247495e-94)

(233336.62489901093, -1.57079612271614)

(-64.88422840036574, -8.78780518671759e-198)

(-40.88423356637257, -8.03930471162897e-125)

(-54.88422972750102, -2.21030970059969e-167)

(-26.884245534637756, -2.92330063611182e-82)

(-96.88422659724432, -4.59186384136685e-295)

(-48.88423095973321, -3.8440423484708e-149)

(-10.88438208563809, -1.27561332516909e-33)

(263669.61561482295, -1.5707961461937)

(273780.6164200921, -1.57079615286349)

(506333.88294838846, -1.57079623274816)

(-78.88422733682835, -2.41579709336053e-240)

(445667.7863075462, -1.57079621994614)

(314224.6341210483, -1.57079617525029)

(344557.6590311293, -1.57079618859147)

(-20.884260359934316, -5.08224708219519e-64)

(455778.8016160654, -1.57079622231648)

(385001.7032754429, -1.57079620310961)

(-46.88423148363634, -4.62272247372513e-143)

(-50.884230498528, -3.19651989018615e-155)

(374890.6912237538, -1.57079619977374)

(465889.8172716211, -1.57079622458393)

(364779.67979585886, -1.57079619625296)

(-44.884232082143896, -5.55912228416105e-137)

(-62.88422861465484, -1.05680791992481e-191)

(304113.6278127029, -1.57079617021182)

(294002.6226734054, -1.57079616482679)

(476000.83325209725, -1.57079622675506)

(223225.63277709106, -1.57079611347241)

(253558.61660401963, -1.57079613899198)

(243447.61961129168, -1.57079613119205)

(-76.88422745328079, -2.90521651141507e-234)

(-92.88422672454823, -6.64092008588277e-283)

(-8.884486236337901, -1.53197630850707e-27)

(-100.88422648503656, -3.17504027992865e-307)

(-4.885856313659682, -2.18833522269578e-15)

(-82.88422712922764, -1.67041218686558e-252)

(395112.715903034, -1.57079620627474)

(405223.7290634153, -1.57079620928192)

(-42.884232770130325, -6.685181352399e-131)

(-16.884281560578337, -7.34661407886626e-52)

(354668.66904511175, -1.57079619253142)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
49(378x(441x2+1)2−(1+2e−7x1−e−7x)e−7xatan⁡(21x)1−e−7x+6e−7x(1−e−7x)(441x2+1))1−e−7x=0\frac{49 \left(\frac{378 x}{\left(441 x^{2} + 1\right)^{2}} - \frac{\left(1 + \frac{2 e^{- 7 x}}{1 - e^{- 7 x}}\right) e^{- 7 x} \operatorname{atan}{\left(21 x \right)}}{1 - e^{- 7 x}} + \frac{6 e^{- 7 x}}{\left(1 - e^{- 7 x}\right) \left(441 x^{2} + 1\right)}\right)}{1 - e^{- 7 x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−20.8842608874384x_{1} = -20.8842608874384
x2=24646.2237146622x_{2} = 24646.2237146622
x3=−100.884226488803x_{3} = -100.884226488803
x4=−84.8842270428688x_{4} = -84.8842270428688
x5=23798.6224822832x_{5} = 23798.6224822832
x6=−34.8842369831928x_{6} = -34.8842369831928
x7=33122.2444926742x_{7} = 33122.2444926742
x8=32274.6419048063x_{8} = 32274.6419048063
x9=10237.0703902709x_{9} = 10237.0703902709
x10=−96.8842266015062x_{10} = -96.8842266015062
x11=−6.88478011708457x_{11} = -6.88478011708457
x12=−18.884269834495x_{12} = -18.884269834495
x13=−46.884231523565x_{13} = -46.884231523565
x14=19560.6202244273x_{14} = 19560.6202244273
x15=−4.88604325806487x_{15} = -4.88604325806487
x16=−74.8842275887987x_{16} = -74.8842275887987
x17=−60.8842288688046x_{17} = -60.8842288688046
x18=13627.4368317258x_{18} = 13627.4368317258
x19=−68.8842280399484x_{19} = -68.8842280399484
x20=−76.8842274619324x_{20} = -76.8842274619324
x21=−48.884230994789x_{21} = -48.884230994789
x22=−50.8842305294728x_{22} = -50.8842305294728
x23=−62.8842286307165x_{23} = -62.8842286307165
x24=4304.12044454294x_{24} = 4304.12044454294
x25=31427.0394058322x_{25} = 31427.0394058322
x26=18713.0208011925x_{26} = 18713.0208011925
x27=−16.8842826326811x_{27} = -16.8842826326811
x28=25493.8251467444x_{28} = 25493.8251467444
x29=28036.630460105x_{29} = 28036.630460105
x30=−32.8842385576643x_{30} = -32.8842385576643
x31=−94.8842266633887x_{31} = -94.8842266633887
x32=−52.8842301178511x_{32} = -52.8842301178511
x33=−42.8842328228878x_{33} = -42.8842328228878
x34=−38.8842345665779x_{34} = -38.8842345665779
x35=12779.842709798x_{35} = 12779.842709798
x36=36512.6556092798x_{36} = 36512.6556092798
x37=39055.4646106491x_{37} = 39055.4646106491
x38=−82.8842271360983x_{38} = -82.8842271360983
x39=−14.8843018687092x_{39} = -14.8843018687092
x40=−30.8842404697812x_{40} = -30.8842404697812
x41=21255.8202222355x_{41} = 21255.8202222355
x42=40750.6708685335x_{42} = 40750.6708685335
x43=−26.884245766518x_{43} = -26.884245766518
x44=−10.8843871046368x_{44} = -10.8843871046368
x45=37360.2585536782x_{45} = 37360.2585536782
x46=39903.0677160774x_{46} = 39903.0677160774
x47=22951.0214717302x_{47} = 22951.0214717302
x48=33969.8471627825x_{48} = 33969.8471627825
x49=−86.8842269561431x_{49} = -86.8842269561431
x50=38207.8615553782x_{50} = 38207.8615553782
x51=9389.48447685996x_{51} = 9389.48447685996
x52=−58.8842291324016x_{52} = -58.8842291324016
x53=26341.4267592538x_{53} = 26341.4267592538
x54=−8.88449709964563x_{54} = -8.88449709964563
x55=17017.823392736x_{55} = 17017.823392736
x56=8541.90219848496x_{56} = 8541.90219848496
x57=42445.8773032609x_{57} = 42445.8773032609
x58=11932.2500244277x_{58} = 11932.2500244277
x59=−88.88422687533x_{59} = -88.88422687533
x60=−64.8842284149486x_{60} = -64.8842284149486
x61=16170.2255582419x_{61} = 16170.2255582419
x62=−12.8843327481437x_{62} = -12.8843327481437
x63=−78.8842273448243x_{63} = -78.8842273448243
x64=−92.8842267293963x_{64} = -92.8842267293963
x65=−90.8842267999038x_{65} = -90.8842267999038
x66=35665.0527262678x_{66} = 35665.0527262678
x67=22103.4207085192x_{67} = 22103.4207085192
x68=14475.0321379042x_{68} = 14475.0321379042
x69=−54.8842297519676x_{69} = -54.8842297519676
x70=7694.32475597859x_{70} = 7694.32475597859
x71=20408.2200473787x_{71} = 20408.2200473787
x72=−40.8842336276476x_{72} = -40.8842336276476
x73=28884.2325205229x_{73} = 28884.2325205229
x74=−70.8842278764306x_{74} = -70.8842278764306
x75=17865.4218345575x_{75} = 17865.4218345575
x76=−98.884226543412x_{76} = -98.884226543412
x77=41598.274065143x_{77} = 41598.274065143
x78=29731.8347049733x_{78} = 29731.8347049733
x79=6846.75394453493x_{79} = 6846.75394453493
x80=−44.8842321278925x_{80} = -44.8842321278925
x81=27189.0285353181x_{81} = 27189.0285353181
x82=5999.19257335325x_{82} = 5999.19257335325
x83=−22.8842543862111x_{83} = -22.8842543862111
x84=11084.6591050725x_{84} = 11084.6591050725
x85=34817.4499091253x_{85} = 34817.4499091253
x86=−72.8842277265384x_{86} = -72.8842277265384
x87=−36.8842356712687x_{87} = -36.8842356712687
x88=30579.4370031432x_{88} = 30579.4370031432
x89=−66.8842282187951x_{89} = -66.8842282187951
x90=−102.884226437406x_{90} = -102.884226437406
x91=−24.884249513281x_{91} = -24.884249513281
x92=−28.8842428235757x_{92} = -28.8842428235757
x93=15322.6284318428x_{93} = 15322.6284318428
x94=−56.8842294252871x_{94} = -56.8842294252871
x95=5151.64529926657x_{95} = 5151.64529926657
x96=−80.8842272364985x_{96} = -80.8842272364985
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(49(378x(441x2+1)2−(1+2e−7x1−e−7x)e−7xatan⁡(21x)1−e−7x+6e−7x(1−e−7x)(441x2+1))1−e−7x)=17152\lim_{x \to 0^-}\left(\frac{49 \left(\frac{378 x}{\left(441 x^{2} + 1\right)^{2}} - \frac{\left(1 + \frac{2 e^{- 7 x}}{1 - e^{- 7 x}}\right) e^{- 7 x} \operatorname{atan}{\left(21 x \right)}}{1 - e^{- 7 x}} + \frac{6 e^{- 7 x}}{\left(1 - e^{- 7 x}\right) \left(441 x^{2} + 1\right)}\right)}{1 - e^{- 7 x}}\right) = \frac{1715}{2}
lim⁡x→0+(49(378x(441x2+1)2−(1+2e−7x1−e−7x)e−7xatan⁡(21x)1−e−7x+6e−7x(1−e−7x)(441x2+1))1−e−7x)=17152\lim_{x \to 0^+}\left(\frac{49 \left(\frac{378 x}{\left(441 x^{2} + 1\right)^{2}} - \frac{\left(1 + \frac{2 e^{- 7 x}}{1 - e^{- 7 x}}\right) e^{- 7 x} \operatorname{atan}{\left(21 x \right)}}{1 - e^{- 7 x}} + \frac{6 e^{- 7 x}}{\left(1 - e^{- 7 x}\right) \left(441 x^{2} + 1\right)}\right)}{1 - e^{- 7 x}}\right) = \frac{1715}{2}
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(atan⁡(21x)−1+e−7x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(atan⁡(21x)−1+e−7x)=−π2\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}}\right) = - \frac{\pi}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=−π2y = - \frac{\pi}{2}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función atan(21*x)/(-1 + E^(-7*x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡(21x)x(−1+e−7x))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(21 x \right)}}{x \left(-1 + e^{- 7 x}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(atan⁡(21x)x(−1+e−7x))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(21 x \right)}}{x \left(-1 + e^{- 7 x}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
atan⁡(21x)−1+e−7x=−atan⁡(21x)e7x−1\frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}} = - \frac{\operatorname{atan}{\left(21 x \right)}}{e^{7 x} - 1}
- No
atan⁡(21x)−1+e−7x=atan⁡(21x)e7x−1\frac{\operatorname{atan}{\left(21 x \right)}}{-1 + e^{- 7 x}} = \frac{\operatorname{atan}{\left(21 x \right)}}{e^{7 x} - 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор