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Trazado el gráfico de la función/ 2*sin(pi/4+1)^2*cos(pi*x/3+1)/atan(x/3)

Gráfico de la función y = 2*sin(pi/4+1)^2*cos(pi*x/3+1)/atan(x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            2/pi    \    /pi*x    \
       2*sin |-- + 1|*cos|---- + 1|
             \4     /    \ 3      /
f(x) = ----------------------------
                     /x\           
                 atan|-|           
                     \3/
f(x)=2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3)f{\left(x \right)} = \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} 
f = ((2*sin(pi/4 + 1)^2)*cos((pi*x)/3 + 1))/atan(x/3)
Gráfico de la función
02468-8-6-4-2-1010-100100
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:
2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3)=0\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=32−3πx_{1} = \frac{3}{2} - \frac{3}{\pi}
x2=92−3πx_{2} = \frac{9}{2} - \frac{3}{\pi}
Solución numérica
x1=−35.4549296585514x_{1} = -35.4549296585514
x2=78.5450703414486x_{2} = 78.5450703414486
x3=66.5450703414486x_{3} = 66.5450703414486
x4=57.5450703414486x_{4} = 57.5450703414486
x5=−341.454929658551x_{5} = -341.454929658551
x6=−56.4549296585514x_{6} = -56.4549296585514
x7=−83.4549296585514x_{7} = -83.4549296585514
x8=1089.54507034145x_{8} = 1089.54507034145
x9=81.5450703414486x_{9} = 81.5450703414486
x10=36.5450703414486x_{10} = 36.5450703414486
x11=99.5450703414486x_{11} = 99.5450703414486
x12=−23.4549296585514x_{12} = -23.4549296585514
x13=−89.4549296585514x_{13} = -89.4549296585514
x14=−47.4549296585514x_{14} = -47.4549296585514
x15=60.5450703414486x_{15} = 60.5450703414486
x16=−98.4549296585514x_{16} = -98.4549296585514
x17=30.5450703414486x_{17} = 30.5450703414486
x18=−305.454929658551x_{18} = -305.454929658551
x19=42.5450703414486x_{19} = 42.5450703414486
x20=3258.54507034145x_{20} = 3258.54507034145
x21=15.5450703414486x_{21} = 15.5450703414486
x22=−68.4549296585514x_{22} = -68.4549296585514
x23=−41.4549296585514x_{23} = -41.4549296585514
x24=27.5450703414486x_{24} = 27.5450703414486
x25=−20.4549296585514x_{25} = -20.4549296585514
x26=−71.4549296585514x_{26} = -71.4549296585514
x27=−74.4549296585514x_{27} = -74.4549296585514
x28=−8.45492965855137x_{28} = -8.45492965855137
x29=−14.4549296585514x_{29} = -14.4549296585514
x30=48.5450703414486x_{30} = 48.5450703414486
x31=87.5450703414486x_{31} = 87.5450703414486
x32=69.5450703414486x_{32} = 69.5450703414486
x33=63.5450703414486x_{33} = 63.5450703414486
x34=54.5450703414486x_{34} = 54.5450703414486
x35=72.5450703414486x_{35} = 72.5450703414486
x36=−77.4549296585514x_{36} = -77.4549296585514
x37=150.545070341449x_{37} = 150.545070341449
x38=−32.4549296585514x_{38} = -32.4549296585514
x39=84.5450703414486x_{39} = 84.5450703414486
x40=−53.4549296585514x_{40} = -53.4549296585514
x41=−59.4549296585514x_{41} = -59.4549296585514
x42=−80.4549296585514x_{42} = -80.4549296585514
x43=−62.4549296585514x_{43} = -62.4549296585514
x44=−65.4549296585514x_{44} = -65.4549296585514
x45=−11.4549296585514x_{45} = -11.4549296585514
x46=−338.454929658551x_{46} = -338.454929658551
x47=93.5450703414486x_{47} = 93.5450703414486
x48=96.5450703414486x_{48} = 96.5450703414486
x49=−5.45492965855137x_{49} = -5.45492965855137
x50=−17.4549296585514x_{50} = -17.4549296585514
x51=90.5450703414486x_{51} = 90.5450703414486
x52=3.54507034144863x_{52} = 3.54507034144863
x53=−44.4549296585514x_{53} = -44.4549296585514
x54=−50.4549296585514x_{54} = -50.4549296585514
x55=−29.4549296585514x_{55} = -29.4549296585514
x56=9.54507034144863x_{56} = 9.54507034144863
x57=75.5450703414486x_{57} = 75.5450703414486
x58=33.5450703414486x_{58} = 33.5450703414486
x59=18.5450703414486x_{59} = 18.5450703414486
x60=6.54507034144863x_{60} = 6.54507034144863
x61=−2.45492965855137x_{61} = -2.45492965855137
x62=123.545070341449x_{62} = 123.545070341449
x63=24.5450703414486x_{63} = 24.5450703414486
x64=12.5450703414486x_{64} = 12.5450703414486
x65=21.5450703414486x_{65} = 21.5450703414486
x66=764037.545070341x_{66} = 764037.545070341
x67=39.5450703414486x_{67} = 39.5450703414486
x68=−95.4549296585514x_{68} = -95.4549296585514
x69=−26.4549296585514x_{69} = -26.4549296585514
x70=51.5450703414486x_{70} = 51.5450703414486
x71=−92.4549296585514x_{71} = -92.4549296585514
x72=45.5450703414486x_{72} = 45.5450703414486
x73=−86.4549296585514x_{73} = -86.4549296585514
x74=135.545070341449x_{74} = 135.545070341449
x75=−38.4549296585514x_{75} = -38.4549296585514
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 ((2*sin(pi/4 + 1)^2)*cos((pi*x)/3 + 1))/atan(x/3).
2sin⁡2(π4+1)cos⁡(0π3+1)atan⁡(03)\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{0 \pi}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{0}{3} \right)}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−2πsin⁡(πx3+1)sin⁡2(π4+1)3atan⁡(x3)−2sin⁡2(π4+1)cos⁡(πx3+1)3(x29+1)atan⁡2(x3)=0- \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)} \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{3 \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{3 \left(\frac{x^{2}}{9} + 1\right) \operatorname{atan}^{2}{\left(\frac{x}{3} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=65.0446471551009x_{1} = 65.0446471551009
x2=−78.9546437734811x_{2} = -78.9546437734811
x3=101.04489663439x_{3} = 101.04489663439
x4=−30.9530107803312x_{4} = -30.9530107803312
x5=−39.9537904111553x_{5} = -39.9537904111553
x6=59.0445554304911x_{6} = 59.0445554304911
x7=80.0447922733148x_{7} = 80.0447922733148
x8=−12.9433916147356x_{8} = -12.9433916147356
x9=86.0448300650018x_{9} = 86.0448300650018
x10=−33.9533415610825x_{10} = -33.9533415610825
x11=−9.93505088725385x_{11} = -9.93505088725385
x12=62.0446046591783x_{12} = 62.0446046591783
x13=14.0353070912986x_{13} = 14.0353070912986
x14=−45.9540729018243x_{14} = -45.9540729018243
x15=−75.9546204775596x_{15} = -75.9546204775596
x16=−90.9547148584446x_{16} = -90.9547148584446
x17=56.0444979691161x_{17} = 56.0444979691161
x18=53.0444303333317x_{18} = 53.0444303333317
x19=20.0403858521832x_{19} = 20.0403858521832
x20=8.01427314241055x_{20} = 8.01427314241055
x21=38.0438114021911x_{21} = 38.0438114021911
x22=92.0448606470826x_{22} = 92.0448606470826
x23=−36.9535937642116x_{23} = -36.9535937642116
x24=4.9661486177045x_{24} = 4.9661486177045
x25=−42.9539466795558x_{25} = -42.9539466795558
x26=−84.9546831120399x_{26} = -84.9546831120399
x27=95.0448737941732x_{27} = 95.0448737941732
x28=47.0442534911438x_{28} = 47.0442534911438
x29=−54.9543340044467x_{29} = -54.9543340044467
x30=−21.9510457285375x_{30} = -21.9510457285375
x31=−57.9543949106738x_{31} = -57.9543949106738
x32=−96.9547408472594x_{32} = -96.9547408472594
x33=−51.9542620784985x_{33} = -51.9542620784985
x34=−81.9546645332808x_{34} = -81.9546645332808
x35=50.0443499739057x_{35} = 50.0443499739057
x36=−69.9545644606477x_{36} = -69.9545644606477
x37=23.0415549303514x_{37} = 23.0415549303514
x38=−66.9545305666873x_{38} = -66.9545305666873
x39=98.0448857428364x_{39} = 98.0448857428364
x40=−48.9541763078526x_{40} = -48.9541763078526
x41=−3.82803929263352x_{41} = -3.82803929263352
x42=−24.9519448835217x_{42} = -24.9519448835217
x43=74.0447448152697x_{43} = 74.0447448152697
x44=−87.9546998048222x_{44} = -87.9546998048222
x45=71.044716397544x_{45} = 71.044716397544
x46=29.042884150723x_{46} = 29.042884150723
x47=44.0441362706937x_{47} = 44.0441362706937
x48=−72.9545942151565x_{48} = -72.9545942151565
x49=26.0423364267544x_{49} = 26.0423364267544
x50=−27.9525651538481x_{50} = -27.9525651538481
x51=68.0446840916865x_{51} = 68.0446840916865
x52=35.0435815529748x_{52} = 35.0435815529748
x53=−99.9547521079561x_{53} = -99.9547521079561
x54=−60.9544469371416x_{54} = -60.9544469371416
x55=83.0448122029392x_{55} = 83.0448122029392
x56=−93.9547284806413x_{56} = -93.9547284806413
x57=11.0290280072741x_{57} = 11.0290280072741
x58=41.0439919237837x_{58} = 41.0439919237837
x59=77.0447699444161x_{59} = 77.0447699444161
x60=−15.9474278264858x_{60} = -15.9474278264858
x61=89.0448461358262x_{61} = 89.0448461358262
x62=−63.954491728768x_{62} = -63.954491728768
x63=1.59695520836124x_{63} = 1.59695520836124
x64=−18.9496728516901x_{64} = -18.9496728516901
x65=−6.91348234301887x_{65} = -6.91348234301887
x66=32.0432826687215x_{66} = 32.0432826687215
x67=17.0385256432214x_{67} = 17.0385256432214
Signos de extremos en los puntos:
                                        2/pi    \                              
(65.04464715510095, 1.31172758508052*sin |-- + 1|*cos(1 + 21.6815490517003*pi))
                                         \4     /                              

                                         2/pi    \                              
(-78.95464377348107, -1.3047862642285*sin |-- + 1|*cos(1 - 26.3182145911604*pi))
                                          \4     /                              

                                         2/pi    \                              
(101.04489663438974, 1.29776145539671*sin |-- + 1|*cos(1 + 33.6816322114632*pi))
                                          \4     /                              

                                          2/pi    \                              
(-30.95301078033117, -1.35668918485075*sin |-- + 1|*cos(1 - 10.3176702601104*pi))
                                           \4     /                              

                                          2/pi    \                              
(-39.95379041115532, -1.33703225834123*sin |-- + 1|*cos(1 - 13.3179301370518*pi))
                                           \4     /                              

                                         2/pi    \                              
(59.044555430491144, 1.31576273106049*sin |-- + 1|*cos(1 + 19.6815184768304*pi))
                                          \4     /                              

                                     2/pi    \                              
(80.0447922733148, 1.304346539933*sin |-- + 1|*cos(1 + 26.6815974244383*pi))
                                      \4     /                              

                                           2/pi    \                              
(-12.943391614735598, -1.48915953919604*sin |-- + 1|*cos(1 - 4.31446387157853*pi))
                                            \4     /                              

                                        2/pi    \                              
(86.04483006500179, 1.30213004566513*sin |-- + 1|*cos(1 + 28.6816100216673*pi))
                                         \4     /                              

                                          2/pi    \                              
(-33.95334156108247, -1.34891906894198*sin |-- + 1|*cos(1 - 11.3177805203608*pi))
                                           \4     /                              

                                         2/pi    \                              
(-9.93505088725385, -1.56550720658075*sin |-- + 1|*cos(1 - 3.31168362908462*pi))
                                          \4     /                              

                                        2/pi    \                              
(62.04460465917826, 1.31364473687044*sin |-- + 1|*cos(1 + 20.6815348863928*pi))
                                         \4     /                              

                                      2/pi    \                              
(14.03530709129865, 1.470352071004*sin |-- + 1|*cos(1 + 4.67843569709955*pi))
                                       \4     /                              

                                          2/pi    \                              
(-45.95407290182433, -1.32836854643405*sin |-- + 1|*cos(1 - 15.3180243006081*pi))
                                           \4     /                              

                                          2/pi    \                              
(-75.95462047755957, -1.30606310263924*sin |-- + 1|*cos(1 - 25.3182068258532*pi))
                                           \4     /                              

                                          2/pi    \                              
(-90.95471485844465, -1.30053823873251*sin |-- + 1|*cos(1 - 30.3182382861482*pi))
                                           \4     /                              

                                         2/pi    \                              
(56.044497969116144, 1.31811484408972*sin |-- + 1|*cos(1 + 18.6814993230387*pi))
                                          \4     /                              

                                        2/pi    \                              
(53.04443033333174, 1.32074212613324*sin |-- + 1|*cos(1 + 17.6814767777772*pi))
                                         \4     /                              

                                         2/pi    \                              
(20.040385852183245, 1.40626997160643*sin |-- + 1|*cos(1 + 6.68012861739441*pi))
                                          \4     /                              

                                       2/pi    \                              
(8.01427314241055, 1.64933312677169*sin |-- + 1|*cos(1 + 2.67142438080352*pi))
                                        \4     /                              

                                        2/pi    \                             
(38.04381140219114, 1.34039029216093*sin |-- + 1|*cos(1 + 12.681270467397*pi))
                                         \4     /                             

                                        2/pi    \                              
(92.04486064708261, 1.30020830972951*sin |-- + 1|*cos(1 + 30.6816202156942*pi))
                                         \4     /                              

                                          2/pi    \                              
(-36.95359376421159, -1.34247012048457*sin |-- + 1|*cos(1 - 12.3178645880705*pi))
                                           \4     /                              

                                      2/pi    \                              
(4.9661486177045, 1.94670883527179*sin |-- + 1|*cos(1 + 1.65538287256817*pi))
                                       \4     /                              

                                          2/pi    \                              
(-42.95394667955585, -1.33238528415755*sin |-- + 1|*cos(1 - 14.3179822265186*pi))
                                           \4     /                              

                                          2/pi    \                              
(-84.95468311203993, -1.30250898080417*sin |-- + 1|*cos(1 - 28.3182277040133*pi))
                                           \4     /                              

                                        2/pi    \                              
(95.04487379417324, 1.29934019869217*sin |-- + 1|*cos(1 + 31.6816245980577*pi))
                                         \4     /                              

                                       2/pi    \                              
(47.04425349114377, 1.3270406709147*sin |-- + 1|*cos(1 + 15.6814178303813*pi))
                                        \4     /                              

                                         2/pi    \                              
(-54.95433400444668, -1.3190352733355*sin |-- + 1|*cos(1 - 18.3181113348156*pi))
                                          \4     /                              

                                         2/pi    \                              
(-21.9510457285375, -1.39375729818924*sin |-- + 1|*cos(1 - 7.31701524284583*pi))
                                          \4     /                              

                                          2/pi    \                              
(-57.95439491067376, -1.31658839287297*sin |-- + 1|*cos(1 - 19.3181316368913*pi))
                                           \4     /                              

                                          2/pi    \                              
(-96.95474084725943, -1.29881606368722*sin |-- + 1|*cos(1 - 32.3182469490865*pi))
                                           \4     /                              

                                          2/pi    \                              
(-51.95426207849849, -1.32177460951675*sin |-- + 1|*cos(1 - 17.3180873594995*pi))
                                           \4     /                              

                                          2/pi    \                              
(-81.95466453328078, -1.30360500382754*sin |-- + 1|*cos(1 - 27.3182215110936*pi))
                                           \4     /                              

                                        2/pi    \                              
(50.04434997390568, 1.32369580955496*sin |-- + 1|*cos(1 + 16.6814499913019*pi))
                                         \4     /                              

                                          2/pi    \                              
(-69.95456446064769, -1.30895397013167*sin |-- + 1|*cos(1 - 23.3181881535492*pi))
                                           \4     /                              

                                         2/pi    \                              
(23.041554930351445, 1.38761199902279*sin |-- + 1|*cos(1 + 7.68051831011715*pi))
                                          \4     /                              

                                        2/pi    \                              
(-66.95453056668727, -1.310599028721*sin |-- + 1|*cos(1 - 22.3181768555624*pi))
                                         \4     /                              

                                        2/pi    \                              
(98.04488574283641, 1.29852621665345*sin |-- + 1|*cos(1 + 32.6816285809455*pi))
                                         \4     /                              

                                          2/pi    \                              
(-48.95417630785262, -1.32486210408683*sin |-- + 1|*cos(1 - 16.3180587692842*pi))
                                           \4     /                              

                                           2/pi    \                              
(-3.8280392926335174, -2.20731231649318*sin |-- + 1|*cos(1 - 1.27601309754451*pi))
                                            \4     /                              

                                          2/pi    \                              
(-24.95194488352173, -1.37822717334108*sin |-- + 1|*cos(1 - 8.31731496117391*pi))
                                           \4     /                              

                                       2/pi    \                              
(74.0447448152697, 1.30693120356289*sin |-- + 1|*cos(1 + 24.6815816050899*pi))
                                        \4     /                              

                                          2/pi    \                              
(-87.95469980482221, -1.30148929378165*sin |-- + 1|*cos(1 - 29.3182332682741*pi))
                                           \4     /                              

                                        2/pi    \                              
(71.04471639754401, 1.30839149060686*sin |-- + 1|*cos(1 + 23.6815721325147*pi))
                                         \4     /                              

                                         2/pi    \                              
(29.042884150723005, 1.36252232855745*sin |-- + 1|*cos(1 + 9.68096138357433*pi))
                                          \4     /                              

                                        2/pi    \                              
(44.04413627069369, 1.33085981755581*sin |-- + 1|*cos(1 + 14.6813787568979*pi))
                                         \4     /                              

                                          2/pi    \                              
(-72.95459421515648, -1.30744760194208*sin |-- + 1|*cos(1 - 24.3181980717188*pi))
                                           \4     /                              

                                         2/pi    \                              
(26.042336426754414, 1.37352747312723*sin |-- + 1|*cos(1 + 8.68077880891814*pi))
                                          \4     /                              

                                           2/pi    \                              
(-27.952565153848056, -1.36623136279072*sin |-- + 1|*cos(1 - 9.31752171794935*pi))
                                            \4     /                              

                                        2/pi    \                              
(68.04468409168655, 1.30998401694728*sin |-- + 1|*cos(1 + 22.6815613638955*pi))
                                         \4     /                              

                                        2/pi    \                              
(35.04358155297482, 1.34644152121067*sin |-- + 1|*cos(1 + 11.6811938509916*pi))
                                         \4     /                              

                                         2/pi    \                             
(-99.95475210795614, -1.2980339460358*sin |-- + 1|*cos(1 - 33.318250702652*pi))
                                          \4     /                             

                                           2/pi    \                              
(-60.954446937141576, -1.31438953882109*sin |-- + 1|*cos(1 - 20.3181489790472*pi))
                                            \4     /                              

                                       2/pi    \                              
(83.04481220293916, 1.3031973677048*sin |-- + 1|*cos(1 + 27.6816040676464*pi))
                                        \4     /                              

                                          2/pi    \                              
(-93.95472848064131, -1.29964911426273*sin |-- + 1|*cos(1 - 31.3182428268804*pi))
                                           \4     /                              

                                         2/pi    \                              
(11.029028007274148, 1.53231751888211*sin |-- + 1|*cos(1 + 3.67634266909138*pi))
                                          \4     /                              

                                        2/pi    \                              
(41.04399192378365, 1.33526165554065*sin |-- + 1|*cos(1 + 13.6813306412612*pi))
                                         \4     /                              

                                        2/pi    \                             
(77.04476994441606, 1.30558734849638*sin |-- + 1|*cos(1 + 25.681589981472*pi))
                                         \4     /                             

                                           2/pi    \                              
(-15.947427826485766, -1.44419835099217*sin |-- + 1|*cos(1 - 5.31580927549525*pi))
                                            \4     /                              

                                        2/pi    \                              
(89.04484613582618, 1.30113613277213*sin |-- + 1|*cos(1 + 29.6816153786087*pi))
                                         \4     /                              

                                          2/pi    \                              
(-63.95449172876801, -1.31240284224882*sin |-- + 1|*cos(1 - 21.3181639095893*pi))
                                           \4     /                              

                                         2/pi    \                               
(1.5969552083612446, 4.08858472866807*sin |-- + 1|*cos(1 + 0.532318402787082*pi))
                                          \4     /                               

                                           2/pi    \                              
(-18.949672851690053, -1.41464191988161*sin |-- + 1|*cos(1 - 6.31655761723002*pi))
                                            \4     /                              

                                          2/pi    \                              
(-6.913482343018871, -1.72208582895412*sin |-- + 1|*cos(1 - 2.30449411433962*pi))
                                           \4     /                              

                                        2/pi    \                              
(32.04328266872153, 1.35368822220233*sin |-- + 1|*cos(1 + 10.6810942229072*pi))
                                         \4     /                              

                                         2/pi    \                              
(17.038525643221387, 1.43214040501933*sin |-- + 1|*cos(1 + 5.67950854774046*pi))
                                          \4     /


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:
Puntos mínimos de la función:
x1=−78.9546437734811x_{1} = -78.9546437734811
x2=−30.9530107803312x_{2} = -30.9530107803312
x3=80.0447922733148x_{3} = 80.0447922733148
x4=−12.9433916147356x_{4} = -12.9433916147356
x5=86.0448300650018x_{5} = 86.0448300650018
x6=62.0446046591783x_{6} = 62.0446046591783
x7=14.0353070912986x_{7} = 14.0353070912986
x8=−90.9547148584446x_{8} = -90.9547148584446
x9=56.0444979691161x_{9} = 56.0444979691161
x10=20.0403858521832x_{10} = 20.0403858521832
x11=8.01427314241055x_{11} = 8.01427314241055
x12=38.0438114021911x_{12} = 38.0438114021911
x13=92.0448606470826x_{13} = 92.0448606470826
x14=−36.9535937642116x_{14} = -36.9535937642116
x15=−42.9539466795558x_{15} = -42.9539466795558
x16=−84.9546831120399x_{16} = -84.9546831120399
x17=−54.9543340044467x_{17} = -54.9543340044467
x18=−96.9547408472594x_{18} = -96.9547408472594
x19=50.0443499739057x_{19} = 50.0443499739057
x20=−66.9545305666873x_{20} = -66.9545305666873
x21=98.0448857428364x_{21} = 98.0448857428364
x22=−48.9541763078526x_{22} = -48.9541763078526
x23=−24.9519448835217x_{23} = -24.9519448835217
x24=74.0447448152697x_{24} = 74.0447448152697
x25=44.0441362706937x_{25} = 44.0441362706937
x26=−72.9545942151565x_{26} = -72.9545942151565
x27=26.0423364267544x_{27} = 26.0423364267544
x28=68.0446840916865x_{28} = 68.0446840916865
x29=−60.9544469371416x_{29} = -60.9544469371416
x30=1.59695520836124x_{30} = 1.59695520836124
x31=−18.9496728516901x_{31} = -18.9496728516901
x32=−6.91348234301887x_{32} = -6.91348234301887
x33=32.0432826687215x_{33} = 32.0432826687215
Puntos máximos de la función:
x33=65.0446471551009x_{33} = 65.0446471551009
x33=101.04489663439x_{33} = 101.04489663439
x33=−39.9537904111553x_{33} = -39.9537904111553
x33=59.0445554304911x_{33} = 59.0445554304911
x33=−33.9533415610825x_{33} = -33.9533415610825
x33=−9.93505088725385x_{33} = -9.93505088725385
x33=−45.9540729018243x_{33} = -45.9540729018243
x33=−75.9546204775596x_{33} = -75.9546204775596
x33=53.0444303333317x_{33} = 53.0444303333317
x33=4.9661486177045x_{33} = 4.9661486177045
x33=95.0448737941732x_{33} = 95.0448737941732
x33=47.0442534911438x_{33} = 47.0442534911438
x33=−21.9510457285375x_{33} = -21.9510457285375
x33=−57.9543949106738x_{33} = -57.9543949106738
x33=−51.9542620784985x_{33} = -51.9542620784985
x33=−81.9546645332808x_{33} = -81.9546645332808
x33=−69.9545644606477x_{33} = -69.9545644606477
x33=23.0415549303514x_{33} = 23.0415549303514
x33=−3.82803929263352x_{33} = -3.82803929263352
x33=−87.9546998048222x_{33} = -87.9546998048222
x33=71.044716397544x_{33} = 71.044716397544
x33=29.042884150723x_{33} = 29.042884150723
x33=−27.9525651538481x_{33} = -27.9525651538481
x33=35.0435815529748x_{33} = 35.0435815529748
x33=−99.9547521079561x_{33} = -99.9547521079561
x33=83.0448122029392x_{33} = 83.0448122029392
x33=−93.9547284806413x_{33} = -93.9547284806413
x33=11.0290280072741x_{33} = 11.0290280072741
x33=41.0439919237837x_{33} = 41.0439919237837
x33=77.0447699444161x_{33} = 77.0447699444161
x33=−15.9474278264858x_{33} = -15.9474278264858
x33=89.0448461358262x_{33} = 89.0448461358262
x33=−63.954491728768x_{33} = -63.954491728768
x33=17.0385256432214x_{33} = 17.0385256432214
Decrece en los intervalos
[98.0448857428364,∞)\left[98.0448857428364, \infty\right)
Crece en los intervalos
(−∞,−96.9547408472594]\left(-\infty, -96.9547408472594\right]
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
2(6(x+3atan⁡(x3))cos⁡(πx3+1)(x2+9)2atan⁡(x3)−π2cos⁡(πx3+1)9+2πsin⁡(πx3+1)(x2+9)atan⁡(x3))sin⁡2(π4+1)atan⁡(x3)=0\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=36.5423366072492x_{1} = 36.5423366072492
x2=−428.454910600311x_{2} = -428.454910600311
x3=93.5446644153194x_{3} = 93.5446644153194
x4=57.5439852935404x_{4} = 57.5439852935404
x5=−41.4528158380405x_{5} = -41.4528158380405
x6=90.5446368012048x_{6} = 90.5446368012048
x7=453.545053337612x_{7} = 453.545053337612
x8=−17.4424483862453x_{8} = -17.4424483862453
x9=−50.4535125284486x_{9} = -50.4535125284486
x10=24.5388910312524x_{10} = 24.5388910312524
x11=42.5430654217194x_{11} = 42.5430654217194
x12=423.5450508378x_{12} = 423.5450508378
x13=9.50130361412559x_{13} = 9.50130361412559
x14=−5.31252777895664x_{14} = -5.31252777895664
x15=−53.4536694345123x_{15} = -53.4536694345123
x16=−65.4540939665356x_{16} = -65.4540939665356
x17=3.15993387709981x_{17} = 3.15993387709981
x18=87.5446062713843x_{18} = 87.5446062713843
x19=123.544838690268x_{19} = 123.544838690268
x20=−74.4542858252128x_{20} = -74.4542858252128
x21=213.544993284613x_{21} = 213.544993284613
x22=114.544800550632x_{22} = 114.544800550632
x23=75.5444451642865x_{23} = 75.5444451642865
x24=−59.4539141487404x_{24} = -59.4539141487404
x25=−35.4520214834876x_{25} = -35.4520214834876
x26=−77.4543352671467x_{26} = -77.4543352671467
x27=−8.39853211686134x_{27} = -8.39853211686134
x28=−32.4514452992605x_{28} = -32.4514452992605
x29=−86.4544536916344x_{29} = -86.4544536916344
x30=12.5203077359387x_{30} = 12.5203077359387
x31=−38.4524659053982x_{31} = -38.4524659053982
x32=−47.4533244194099x_{32} = -47.4533244194099
x33=84.5445724005981x_{33} = 84.5445724005981
x34=−71.4542299508964x_{34} = -71.4542299508964
x35=48.5435375651596x_{35} = 48.5435375651596
x36=−83.4544184863111x_{36} = -83.4544184863111
x37=−95.4545399602143x_{37} = -95.4545399602143
x38=30.54112538342x_{38} = 30.54112538342
x39=354.545042483703x_{39} = 354.545042483703
x40=15.5292045102585x_{40} = 15.5292045102585
x41=27.5401941516671x_{41} = 27.5401941516671
x42=51.5437134521548x_{42} = 51.5437134521548
x43=81.5445346831614x_{43} = 81.5445346831614
x44=6.44847053079324x_{44} = 6.44847053079324
x45=−122.454693834022x_{45} = -122.454693834022
x46=−1688.45492843539x_{46} = -1688.45492843539
x47=33.5418136674159x_{47} = 33.5418136674159
x48=−98.4545635534746x_{48} = -98.4545635534746
x49=−29.4506797592818x_{49} = -29.4506797592818
x50=−92.454514011289x_{50} = -92.454514011289
x51=63.5441829711077x_{51} = 63.5441829711077
x52=60.544091569207x_{52} = 60.544091569207
x53=−80.4543792271114x_{53} = -80.4543792271114
x54=96.5446894734333x_{54} = 96.5446894734333
x55=66.5442621493015x_{55} = 66.5442621493015
x56=−89.4544853824358x_{56} = -89.4544853824358
x57=−26.449632087741x_{57} = -26.449632087741
x58=−23.4481450893327x_{58} = -23.4481450893327
x59=72.5443917497405x_{59} = 72.5443917497405
x60=−20.4459351998821x_{60} = -20.4459351998821
x61=54.5438607355483x_{61} = 54.5438607355483
x62=69.5443311891507x_{62} = 69.5443311891507
x63=−62.4540106139345x_{63} = -62.4540106139345
x64=21.5369890404614x_{64} = 21.5369890404614
x65=240.545009671203x_{65} = 240.545009671203
x66=−11.4250099333197x_{66} = -11.4250099333197
x67=99.544712281458x_{67} = 99.544712281458
x68=78.5444925140042x_{68} = 78.5444925140042
x69=−56.4538016713076x_{69} = -56.4538016713076
x70=−14.436481697601x_{70} = -14.436481697601
x71=18.5340584217125x_{71} = 18.5340584217125
x72=−68.4541664786334x_{72} = -68.4541664786334
x73=−44.4530962608714x_{73} = -44.4530962608714
x74=39.5427431560573x_{74} = 39.5427431560573
x75=45.5433251674318x_{75} = 45.5433251674318
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−(2(6(x+3atan⁡(x3))cos⁡(πx3+1)(x2+9)2atan⁡(x3)−π2cos⁡(πx3+1)9+2πsin⁡(πx3+1)(x2+9)atan⁡(x3))sin⁡2(π4+1)atan⁡(x3))=−∞\lim_{x \to 0^-}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = -\infty
lim⁡x→0+(2(6(x+3atan⁡(x3))cos⁡(πx3+1)(x2+9)2atan⁡(x3)−π2cos⁡(πx3+1)9+2πsin⁡(πx3+1)(x2+9)atan⁡(x3))sin⁡2(π4+1)atan⁡(x3))=∞\lim_{x \to 0^+}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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:
Cóncava en los intervalos
[354.545042483703,∞)\left[354.545042483703, \infty\right)
Convexa en los intervalos
(−∞,−1688.45492843539]\left(-\infty, -1688.45492843539\right]
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→−∞(2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3))=⟨−4,4⟩sin⁡2(π4+1)π\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−4,4⟩sin⁡2(π4+1)πy = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}
lim⁡x→∞(2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3))=⟨−4,4⟩sin⁡2(π4+1)π\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−4,4⟩sin⁡2(π4+1)πy = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((2*sin(pi/4 + 1)^2)*cos((pi*x)/3 + 1))/atan(x/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2sin⁡2(π4+1)cos⁡(πx3+1)xatan⁡(x3))=0\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(2sin⁡2(π4+1)cos⁡(πx3+1)xatan⁡(x3))=0\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \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:
2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3)=−2sin⁡2(π4+1)cos⁡(πx3−1)atan⁡(x3)\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}
- No
2sin⁡2(π4+1)cos⁡(πx3+1)atan⁡(x3)=2sin⁡2(π4+1)cos⁡(πx3−1)atan⁡(x3)\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор