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Trazado el gráfico de la función/ (x*sin(7*x))/7

Gráfico de la función y = (x*sin(7*x))/7

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x*sin(7*x)
f(x) = ----------
           7
f(x)=xsin⁡(7x)7f{\left(x \right)} = \frac{x \sin{\left(7 x \right)}}{7} 
f = (x*sin(7*x))/7
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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:
xsin⁡(7x)7=0\frac{x \sin{\left(7 x \right)}}{7} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−2π7x_{2} = - \frac{2 \pi}{7}
x3=−π7x_{3} = - \frac{\pi}{7}
x4=π7x_{4} = \frac{\pi}{7}
x5=2π7x_{5} = \frac{2 \pi}{7}
x6=5π7x_{6} = \frac{5 \pi}{7}
x7=6π7x_{7} = \frac{6 \pi}{7}
x8=πx_{8} = \pi
x9=−ilog⁡(−−17)x_{9} = - i \log{\left(- \sqrt[7]{-1} \right)}
Solución numérica
x1=24.2351433276927x_{1} = 24.2351433276927
x2=−32.7623233874364x_{2} = -32.7623233874364
x3=−15.707963267949x_{3} = -15.707963267949
x4=8.0783811092309x_{4} = 8.0783811092309
x5=2.24399475256414x_{5} = 2.24399475256414
x6=82.1302079438475x_{6} = 82.1302079438475
x7=26.030339129744x_{7} = 26.030339129744
x8=−57.8950646161548x_{8} = -57.8950646161548
x9=−13.9127674658977x_{9} = -13.9127674658977
x10=21.9911485751286x_{10} = 21.9911485751286
x11=0x_{11} = 0
x12=−41.738302397693x_{12} = -41.738302397693
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=28.2743338823081x_{14} = 28.2743338823081
x15=72.2566310325652x_{15} = 72.2566310325652
x16=87.9645943005142x_{16} = 87.9645943005142
x17=76.2958215871807x_{17} = 76.2958215871807
x18=−43.9822971502571x_{18} = -43.9822971502571
x19=50.2654824574367x_{19} = 50.2654824574367
x20=−67.3198425769241x_{20} = -67.3198425769241
x21=90.2085890530783x_{21} = 90.2085890530783
x22=−19.7471538225644x_{22} = -19.7471538225644
x23=−85.7205995479501x_{23} = -85.7205995479501
x24=64.1782499233343x_{24} = 64.1782499233343
x25=54.3046730120521x_{25} = 54.3046730120521
x26=−81.6814089933346x_{26} = -81.6814089933346
x27=−48.0214877048726x_{27} = -48.0214877048726
x28=94.2477796076938x_{28} = 94.2477796076938
x29=−67.768641527437x_{29} = -67.768641527437
x30=60.1390593687189x_{30} = 60.1390593687189
x31=70.0126362800011x_{31} = 70.0126362800011
x32=−1.79519580205131x_{32} = -1.79519580205131
x33=−79.8862131912833x_{33} = -79.8862131912833
x34=−83.9254037458988x_{34} = -83.9254037458988
x35=68.2174404779498x_{35} = 68.2174404779498
x36=−70.0126362800011x_{36} = -70.0126362800011
x37=−53.8558740615393x_{37} = -53.8558740615393
x38=13.015169564872x_{38} = 13.015169564872
x39=83.4766047953859x_{39} = 83.4766047953859
x40=65.9734457253857x_{40} = 65.9734457253857
x41=86.1693984984629x_{41} = 86.1693984984629
x42=−9.87357691128221x_{42} = -9.87357691128221
x43=−92.0037848551297x_{43} = -92.0037848551297
x44=−61.9342551707702x_{44} = -61.9342551707702
x45=16.1567622184618x_{45} = 16.1567622184618
x46=46.2262919028212x_{46} = 46.2262919028212
x47=−71.8078320820524x_{47} = -71.8078320820524
x48=−39.9431065956417x_{48} = -39.9431065956417
x49=0.448798950512828x_{49} = 0.448798950512828
x50=−30.0695296843594x_{50} = -30.0695296843594
x51=−87.9645943005142x_{51} = -87.9645943005142
x52=−17.9519580205131x_{52} = -17.9519580205131
x53=−4.03919055461545x_{53} = -4.03919055461545
x54=100.082165964361x_{54} = 100.082165964361
x55=48.0214877048726x_{55} = 48.0214877048726
x56=38.1479107935903x_{56} = 38.1479107935903
x57=−5.83438635666676x_{57} = -5.83438635666676
x58=32.3135244369236x_{58} = 32.3135244369236
x59=−74.0518268346165x_{59} = -74.0518268346165
x60=92.0037848551297x_{60} = 92.0037848551297
x61=−75.8470226366679x_{61} = -75.8470226366679
x62=61.9342551707702x_{62} = 61.9342551707702
x63=30.0695296843594x_{63} = 30.0695296843594
x64=4.03919055461545x_{64} = 4.03919055461545
x65=12.1175716638463x_{65} = 12.1175716638463
x66=−27.8255349317953x_{66} = -27.8255349317953
x67=−35.9039160410262x_{67} = -35.9039160410262
x68=−26.030339129744x_{68} = -26.030339129744
x69=56.0998688141035x_{69} = 56.0998688141035
x70=39.9431065956417x_{70} = 39.9431065956417
x71=−89.7597901025655x_{71} = -89.7597901025655
x72=−52.060678259488x_{72} = -52.060678259488
x73=−4.9367884556411x_{73} = -4.9367884556411
x74=−31.8647254864108x_{74} = -31.8647254864108
x75=−65.9734457253857x_{75} = -65.9734457253857
x76=96.0429754097451x_{76} = 96.0429754097451
x77=43.9822971502571x_{77} = 43.9822971502571
x78=17.9519580205131x_{78} = 17.9519580205131
x79=42.1871013482058x_{79} = 42.1871013482058
x80=52.060678259488x_{80} = 52.060678259488
x81=−37.6991118430775x_{81} = -37.6991118430775
x82=−59.6902604182061x_{82} = -59.6902604182061
x83=85.7205995479501x_{83} = 85.7205995479501
x84=−97.8381712117964x_{84} = -97.8381712117964
x85=−49.8166835069239x_{85} = -49.8166835069239
x86=−23.7863443771799x_{86} = -23.7863443771799
x87=74.0518268346165x_{87} = 74.0518268346165
x88=−45.7774929523084x_{88} = -45.7774929523084
x89=−63.7294509728215x_{89} = -63.7294509728215
x90=6.28318530717959x_{90} = 6.28318530717959
x91=10.7711748123079x_{91} = 10.7711748123079
x92=−96.0429754097451x_{92} = -96.0429754097451
x93=98.2869701623092x_{93} = 98.2869701623092
x94=20.1959527730772x_{94} = 20.1959527730772
x95=0.897597901025655x_{95} = 0.897597901025655
x96=78.091017389232x_{96} = 78.091017389232
x97=34.1087202389749x_{97} = 34.1087202389749
x98=−93.798980657181x_{98} = -93.798980657181
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 (x*sin(7*x))/7.
0sin⁡(0⋅7)7\frac{0 \sin{\left(0 \cdot 7 \right)}}{7}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
xcos⁡(7x)+sin⁡(7x)7=0x \cos{\left(7 x \right)} + \frac{\sin{\left(7 x \right)}}{7} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=50.0414908059181x_{1} = 50.0414908059181
x2=4.26836950087248x_{2} = 4.26836950087248
x3=10.0999968626586x_{3} = 10.0999968626586
x4=−76.0716903870102x_{4} = -76.0716903870102
x5=36.1288803845179x_{5} = 36.1288803845179
x6=−50.0414908059181x_{6} = -50.0414908059181
x7=0x_{7} = 0
x8=89.9844163742712x_{8} = 89.9844163742712
x9=74.2765010688857x_{9} = 74.2765010688857
x10=−25.8067304541918x_{10} = -25.8067304541918
x11=100.306768896968x_{11} = 100.306768896968
x12=37.4752569421903x_{12} = 37.4752569421903
x13=−11.8948878148698x_{13} = -11.8948878148698
x14=−24.0115937719656x_{14} = -24.0115937719656
x15=92.2284056087093x_{15} = 92.2284056087093
x16=54.0806509012589x_{16} = 54.0806509012589
x17=−87.7404274220872x_{17} = -87.7404274220872
x18=40.168014138764x_{18} = 40.168014138764
x19=67.9933411516917x_{19} = 67.9933411516917
x20=−47.7975151995624x_{20} = -47.7975151995624
x21=−37.9240494482814x_{21} = -37.9240494482814
x22=56.7734267059359x_{22} = 56.7734267059359
x23=96.2675868789753x_{23} = 96.2675868789753
x24=−6.06215169734427x_{24} = -6.06215169734427
x25=−54.5294467459485x_{25} = -54.5294467459485
x26=6.06215169734427x_{26} = 6.06215169734427
x27=−91.7796077402351x_{27} = -91.7796077402351
x28=−69.7885292329558x_{28} = -69.7885292329558
x29=47.3487202960603x_{29} = 47.3487202960603
x30=−72.0325148757039x_{30} = -72.0325148757039
x31=18.1774801895039x_{31} = 18.1774801895039
x32=41.9631882059783x_{32} = 41.9631882059783
x33=−77.8668800041013x_{33} = -77.8668800041013
x34=−28.0506619473452x_{34} = -28.0506619473452
x35=70.2373263149385x_{35} = 70.2373263149385
x36=19.9725750897106x_{36} = 19.9725750897106
x37=28.0506619473452x_{37} = 28.0506619473452
x38=−35.6800885389641x_{38} = -35.6800885389641
x39=14.1386103269137x_{39} = 14.1386103269137
x40=−55.8758345793656x_{40} = -55.8758345793656
x41=98.0627788001522x_{41} = 98.0627788001522
x42=16.8311731368512x_{42} = 16.8311731368512
x43=26.2555158878374x_{43} = 26.2555158878374
x44=−85.9452364784199x_{44} = -85.9452364784199
x45=−95.8187889214026x_{45} = -95.8187889214026
x46=−41.9631882059783x_{46} = -41.9631882059783
x47=−19.9725750897106x_{47} = -19.9725750897106
x48=30.2946028113592x_{48} = 30.2946028113592
x49=84.1500457420413x_{49} = 84.1500457420413
x50=32.0897609286503x_{50} = 32.0897609286503
x51=0.289822548301491x_{51} = 0.289822548301491
x52=−67.9933411516917x_{52} = -67.9933411516917
x53=52.2854680556737x_{53} = 52.2854680556737
x54=78.3156774526783x_{54} = 78.3156774526783
x55=62.1589829674753x_{55} = 62.1589829674753
x56=−99.857970860865x_{56} = -99.857970860865
x57=−33.8849230387282x_{57} = -33.8849230387282
x58=85.9452364784199x_{58} = 85.9452364784199
x59=−65.7493566429938x_{59} = -65.7493566429938
x60=−3.82013085925533x_{60} = -3.82013085925533
x61=−59.9150005114115x_{61} = -59.9150005114115
x62=−98.0627788001522x_{62} = -98.0627788001522
x63=−46.0023360593325x_{63} = -46.0023360593325
x64=−15.9336435310187x_{64} = -15.9336435310187
x65=−63.9541695536103x_{65} = -63.9541695536103
x66=58.119815230211x_{66} = 58.119815230211
x67=76.0716903870102x_{67} = 76.0716903870102
x68=−21.7676866303789x_{68} = -21.7676866303789
x69=72.0325148757039x_{69} = 72.0325148757039
x70=22.2164666429911x_{70} = 22.2164666429911
x71=44.2071582722925x_{71} = 44.2071582722925
x72=46.0023360593325x_{72} = 46.0023360593325
x73=−61.7101864047208x_{73} = -61.7101864047208
x74=−39.7192209289341x_{74} = -39.7192209289341
x75=94.0235971859024x_{75} = 94.0235971859024
x76=−17.7287096567573x_{76} = -17.7287096567573
x77=8.30523760641417x_{77} = 8.30523760641417
x78=−89.9844163742712x_{78} = -89.9844163742712
x79=−94.0235971859024x_{79} = -94.0235971859024
x80=15.9336435310187x_{80} = 15.9336435310187
x81=−1.13980938748761x_{81} = -1.13980938748761
x82=−51.8366724844947x_{82} = -51.8366724844947
x83=88.1892251889095x_{83} = 88.1892251889095
x84=−73.8277037886238x_{84} = -73.8277037886238
x85=63.9541695536103x_{85} = 63.9541695536103
x86=−29.8458139903334x_{86} = -29.8458139903334
x87=33.8849230387282x_{87} = 33.8849230387282
x88=−13.6898586870028x_{88} = -13.6898586870028
x89=2.02963381788446x_{89} = 2.02963381788446
x90=48.2463101783551x_{90} = 48.2463101783551
x91=66.1981534891744x_{91} = 66.1981534891744
x92=−7.8565789367852x_{92} = -7.8565789367852
x93=−43.7583640564846x_{93} = -43.7583640564846
x94=−81.9060576338318x_{94} = -81.9060576338318
x95=−2.02963381788446x_{95} = -2.02963381788446
x96=−0.289822548301491x_{96} = -0.289822548301491
x97=24.0115937719656x_{97} = 24.0115937719656
x98=−83.7012480918985x_{98} = -83.7012480918985
x99=80.1108674152685x_{99} = 80.1108674152685
Signos de extremos en los puntos:
(50.04149080591807, -7.14875527067737)

(4.268369500872485, -0.609425839983404)

(10.099996862658571, 1.44271238697129)

(-76.07169038701021, -10.8673651785235)

(36.128880384517934, 5.16122827889145)

(-50.04149080591807, -7.14875527067737)

(0, 0)

(89.98441637427116, 12.8549004251665)

(74.27650106888568, -10.610909098513)

(-25.80673045419181, -3.6866192941983)

(100.30676889696777, -14.3295238811969)

(37.47525694219035, -5.35356923666491)

(-11.894887814869808, 1.69914715046843)

(-24.0115937719656, -3.43016697406691)

(92.22840560870927, -13.1754707099512)

(54.080650901258934, 7.72578031708227)

(-87.74042742208724, -12.534330160543)

(40.16801413876397, -5.73825144373855)

(67.99334115169165, -9.71331301106694)

(-47.797515199562426, 6.82818595934055)

(-37.924049448281394, 5.41768291212835)

(56.77342670593591, 8.11046385333978)

(96.26758687897527, 13.7524972688694)

(-6.062151697344269, -0.865781307699165)

(-54.52944674594849, -7.78989423102338)

(6.062151697344269, -0.865781307699165)

(-91.77960774023505, 13.1113566514523)

(-69.78852923295581, -9.96976900272936)

(47.348720296060314, -6.76407211262846)

(-72.03251487570392, 10.2903390309626)

(18.17748018950395, 2.59670269386892)

(41.96318820597835, -5.99470643437687)

(-77.86688000410129, -11.1238212798889)

(-28.050661947345247, 4.0071854544471)

(70.23732631493846, 10.0338830050498)

(19.972575089710602, 2.85315202923658)

(28.050661947345247, 4.0071854544471)

(-35.68008853896408, -5.09711465061102)

(14.138610326913671, -2.01969838067589)

(-55.87583457936562, 7.9822359942291)

(98.0627788001522, 14.0089535348135)

(16.83117313685118, -2.40436670124825)

(26.255515887837383, 3.75073246445898)

(-85.9452364784199, -12.2778739644214)

(-95.81878892140257, -13.6883832040063)

(-41.96318820597835, -5.99470643437687)

(-19.972575089710602, 2.85315202923658)

(30.294602811359155, -4.3277522840869)

(84.15004574204129, -12.0214177831063)

(32.0897609286503, -4.58420613539779)

(0.2898225483014906, 0.0371368518604011)

(-67.99334115169165, -9.71331301106694)

(52.285468055673746, 7.4693246994023)

(78.31567745267834, 11.1879353083949)

(62.15898296747526, 8.87983125815889)

(-99.85797086086502, 14.2654098107245)

(-33.88492303872823, -4.84066027192193)

(85.9452364784199, -12.2778739644214)

(-65.74935664299379, 9.39274306383811)

(-3.82013085925533, 0.545351789075457)

(-59.91500051141154, -8.55926145754863)

(-98.0627788001522, 14.0089535348135)

(-46.002336059332485, 6.57173060633428)

(-15.933643531018715, -2.27614330836571)

(-63.9541695536103, 9.13628714302512)

(58.119815230210996, -8.30280566589329)

(76.07169038701021, -10.8673651785235)

(-21.767686630378858, 3.10960255336253)

(72.03251487570392, 10.2903390309626)

(22.216466642991147, -3.17371533637153)

(44.20715827229254, 6.31527534998677)

(46.002336059332485, 6.57173060633428)

(-61.71018640472079, -8.81571729292664)

(-39.71922092893406, 5.67413771800634)

(94.02359718590239, -13.4319269513249)

(-17.728709656757335, -2.53259058808252)

(8.305237606414167, 1.18628703526706)

(-89.98441637427116, 12.8549004251665)

(-94.02359718590239, -13.4319269513249)

(15.933643531018715, -2.27614330836571)

(-1.1398093874876059, 0.161565864726281)

(-51.83667248449475, -7.40521080499802)

(88.18922518890949, 12.5984442117809)

(-73.82770378862378, 10.5467950820297)

(63.9541695536103, 9.13628714302512)

(-29.845813990333443, 4.26363887185456)

(33.88492303872823, -4.84066027192193)

(-13.689858687002783, 1.95558762466256)

(2.0296338178844553, 0.289232124770904)

(48.246310178355095, -6.89229981143067)

(66.19815348917444, -9.45685704931534)

(-7.856578936785199, -1.12218292346322)

(-43.75836405648458, -6.25116155239689)

(-81.90605763383181, 11.700847578767)

(-2.0296338178844553, 0.289232124770904)

(-0.2898225483014906, 0.0371368518604011)

(24.0115937719656, -3.43016697406691)

(-83.70124809189848, 11.9573037402026)

(80.11086741526847, 11.444391434439)


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=50.0414908059181x_{1} = 50.0414908059181
x2=4.26836950087248x_{2} = 4.26836950087248
x3=−76.0716903870102x_{3} = -76.0716903870102
x4=−50.0414908059181x_{4} = -50.0414908059181
x5=0x_{5} = 0
x6=74.2765010688857x_{6} = 74.2765010688857
x7=−25.8067304541918x_{7} = -25.8067304541918
x8=100.306768896968x_{8} = 100.306768896968
x9=37.4752569421903x_{9} = 37.4752569421903
x10=−24.0115937719656x_{10} = -24.0115937719656
x11=92.2284056087093x_{11} = 92.2284056087093
x12=−87.7404274220872x_{12} = -87.7404274220872
x13=40.168014138764x_{13} = 40.168014138764
x14=67.9933411516917x_{14} = 67.9933411516917
x15=−6.06215169734427x_{15} = -6.06215169734427
x16=−54.5294467459485x_{16} = -54.5294467459485
x17=6.06215169734427x_{17} = 6.06215169734427
x18=−69.7885292329558x_{18} = -69.7885292329558
x19=47.3487202960603x_{19} = 47.3487202960603
x20=41.9631882059783x_{20} = 41.9631882059783
x21=−77.8668800041013x_{21} = -77.8668800041013
x22=−35.6800885389641x_{22} = -35.6800885389641
x23=14.1386103269137x_{23} = 14.1386103269137
x24=16.8311731368512x_{24} = 16.8311731368512
x25=−85.9452364784199x_{25} = -85.9452364784199
x26=−95.8187889214026x_{26} = -95.8187889214026
x27=−41.9631882059783x_{27} = -41.9631882059783
x28=30.2946028113592x_{28} = 30.2946028113592
x29=84.1500457420413x_{29} = 84.1500457420413
x30=32.0897609286503x_{30} = 32.0897609286503
x31=−67.9933411516917x_{31} = -67.9933411516917
x32=−33.8849230387282x_{32} = -33.8849230387282
x33=85.9452364784199x_{33} = 85.9452364784199
x34=−59.9150005114115x_{34} = -59.9150005114115
x35=−15.9336435310187x_{35} = -15.9336435310187
x36=58.119815230211x_{36} = 58.119815230211
x37=76.0716903870102x_{37} = 76.0716903870102
x38=22.2164666429911x_{38} = 22.2164666429911
x39=−61.7101864047208x_{39} = -61.7101864047208
x40=94.0235971859024x_{40} = 94.0235971859024
x41=−17.7287096567573x_{41} = -17.7287096567573
x42=−94.0235971859024x_{42} = -94.0235971859024
x43=15.9336435310187x_{43} = 15.9336435310187
x44=−51.8366724844947x_{44} = -51.8366724844947
x45=33.8849230387282x_{45} = 33.8849230387282
x46=48.2463101783551x_{46} = 48.2463101783551
x47=66.1981534891744x_{47} = 66.1981534891744
x48=−7.8565789367852x_{48} = -7.8565789367852
x49=−43.7583640564846x_{49} = -43.7583640564846
x50=24.0115937719656x_{50} = 24.0115937719656
Puntos máximos de la función:
x50=10.0999968626586x_{50} = 10.0999968626586
x50=36.1288803845179x_{50} = 36.1288803845179
x50=89.9844163742712x_{50} = 89.9844163742712
x50=−11.8948878148698x_{50} = -11.8948878148698
x50=54.0806509012589x_{50} = 54.0806509012589
x50=−47.7975151995624x_{50} = -47.7975151995624
x50=−37.9240494482814x_{50} = -37.9240494482814
x50=56.7734267059359x_{50} = 56.7734267059359
x50=96.2675868789753x_{50} = 96.2675868789753
x50=−91.7796077402351x_{50} = -91.7796077402351
x50=−72.0325148757039x_{50} = -72.0325148757039
x50=18.1774801895039x_{50} = 18.1774801895039
x50=−28.0506619473452x_{50} = -28.0506619473452
x50=70.2373263149385x_{50} = 70.2373263149385
x50=19.9725750897106x_{50} = 19.9725750897106
x50=28.0506619473452x_{50} = 28.0506619473452
x50=−55.8758345793656x_{50} = -55.8758345793656
x50=98.0627788001522x_{50} = 98.0627788001522
x50=26.2555158878374x_{50} = 26.2555158878374
x50=−19.9725750897106x_{50} = -19.9725750897106
x50=0.289822548301491x_{50} = 0.289822548301491
x50=52.2854680556737x_{50} = 52.2854680556737
x50=78.3156774526783x_{50} = 78.3156774526783
x50=62.1589829674753x_{50} = 62.1589829674753
x50=−99.857970860865x_{50} = -99.857970860865
x50=−65.7493566429938x_{50} = -65.7493566429938
x50=−3.82013085925533x_{50} = -3.82013085925533
x50=−98.0627788001522x_{50} = -98.0627788001522
x50=−46.0023360593325x_{50} = -46.0023360593325
x50=−63.9541695536103x_{50} = -63.9541695536103
x50=−21.7676866303789x_{50} = -21.7676866303789
x50=72.0325148757039x_{50} = 72.0325148757039
x50=44.2071582722925x_{50} = 44.2071582722925
x50=46.0023360593325x_{50} = 46.0023360593325
x50=−39.7192209289341x_{50} = -39.7192209289341
x50=8.30523760641417x_{50} = 8.30523760641417
x50=−89.9844163742712x_{50} = -89.9844163742712
x50=−1.13980938748761x_{50} = -1.13980938748761
x50=88.1892251889095x_{50} = 88.1892251889095
x50=−73.8277037886238x_{50} = -73.8277037886238
x50=63.9541695536103x_{50} = 63.9541695536103
x50=−29.8458139903334x_{50} = -29.8458139903334
x50=−13.6898586870028x_{50} = -13.6898586870028
x50=2.02963381788446x_{50} = 2.02963381788446
x50=−81.9060576338318x_{50} = -81.9060576338318
x50=−2.02963381788446x_{50} = -2.02963381788446
x50=−0.289822548301491x_{50} = -0.289822548301491
x50=−83.7012480918985x_{50} = -83.7012480918985
x50=80.1108674152685x_{50} = 80.1108674152685
Decrece en los intervalos
[100.306768896968,∞)\left[100.306768896968, \infty\right)
Crece en los intervalos
(−∞,−95.8187889214026]\left(-\infty, -95.8187889214026\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
−7xsin⁡(7x)+2cos⁡(7x)=0- 7 x \sin{\left(7 x \right)} + 2 \cos{\left(7 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=6.28967027132113x_{1} = 6.28967027132113
x2=59.6909442072692x_{2} = 59.6909442072692
x3=−15.7105609999856x_{3} = -15.7105609999856
x4=52.0614622543087x_{4} = 52.0614622543087
x5=−70.0132192570501x_{5} = -70.0132192570501
x6=92.0042284890364x_{6} = 92.0042284890364
x7=98.2873854364706x_{7} = 98.2873854364706
x8=−1.81747125310952x_{8} = -1.81747125310952
x9=−35.9050528020628x_{9} = -35.9050528020628
x10=17.954231182823x_{10} = 17.954231182823
x11=30.0708869804838x_{11} = 30.0708869804838
x12=−83.9258900817459x_{12} = -83.9258900817459
x13=50.2662944505908x_{13} = 50.2662944505908
x14=−4.04925383854091x_{14} = -4.04925383854091
x15=−92.0042284890364x_{15} = -92.0042284890364
x16=−43.9832251347853x_{16} = -43.9832251347853
x17=−27.8270016687724x_{17} = -27.8270016687724
x18=−37.700194477947x_{18} = -37.700194477947
x19=−19.7492204094199x_{19} = -19.7492204094199
x20=80.7843163408299x_{20} = 80.7843163408299
x21=−74.0523780137291x_{21} = -74.0523780137291
x22=−96.0434003864323x_{22} = -96.0434003864323
x23=61.9349141857625x_{23} = 61.9349141857625
x24=−93.7994158006192x_{24} = -93.7994158006192
x25=−21.993004348443x_{25} = -21.993004348443
x26=−23.7880601271722x_{26} = -23.7880601271722
x27=94.2482126790522x_{27} = 94.2482126790522
x28=−9.87770792521323x_{28} = -9.87770792521323
x29=−57.8957696071319x_{29} = -57.8957696071319
x30=96.0434003864323x_{30} = 96.0434003864323
x31=−75.8475607704332x_{31} = -75.8475607704332
x32=20.1979734512655x_{32} = 20.1979734512655
x33=32.31478748911x_{33} = 32.31478748911
x34=−52.0614622543087x_{34} = -52.0614622543087
x35=−26.0319070013448x_{35} = -26.0319070013448
x36=72.2571959052163x_{36} = 72.2571959052163
x37=70.0132192570501x_{37} = 70.0132192570501
x38=60.1397380550713x_{38} = 60.1397380550713
x39=−0.153839140901686x_{39} = -0.153839140901686
x40=−67.7692438077991x_{40} = -67.7692438077991
x41=−30.0708869804838x_{41} = -30.0708869804838
x42=0.520513881060772x_{42} = 0.520513881060772
x43=−31.8660063256979x_{43} = -31.8660063256979
x44=21.993004348443x_{44} = 21.993004348443
x45=−87.9650583050154x_{45} = -87.9650583050154
x46=−71.8084004850647x_{46} = -71.8084004850647
x47=−79.8867241166336x_{47} = -79.8867241166336
x48=−81.6819086897879x_{48} = -81.6819086897879
x49=−63.7300914246421x_{49} = -63.7300914246421
x50=46.2271748424979x_{50} = 46.2271748424979
x51=24.2368273119639x_{51} = 24.2368273119639
x52=68.2180387960574x_{52} = 68.2180387960574
x53=0.153839140901686x_{53} = 0.153839140901686
x54=−53.8566319244462x_{54} = -53.8566319244462
x55=−80.3355202128752x_{55} = -80.3355202128752
x56=−85.7210756989439x_{56} = -85.7210756989439
x57=56.1005963638318x_{57} = 56.1005963638318
x58=−89.760244827135x_{58} = -89.760244827135
x59=−65.9740643938236x_{59} = -65.9740643938236
x60=1.37565147761392x_{60} = 1.37565147761392
x61=100.08257378976x_{61} = 100.08257378976
x62=48.0223376394435x_{62} = 48.0223376394435
x63=65.9740643938236x_{63} = 65.9740643938236
x64=2.26194448784967x_{64} = 2.26194448784967
x65=−61.9349141857625x_{65} = -61.9349141857625
x66=38.1489806927767x_{66} = 38.1489806927767
x67=−11.2236096263714x_{67} = -11.2236096263714
x68=64.1788858966193x_{68} = 64.1788858966193
x69=74.0523780137291x_{69} = 74.0523780137291
x70=4.04925383854091x_{70} = 4.04925383854091
x71=39.9441284136678x_{71} = 39.9441284136678
x72=78.0915400597335x_{72} = 78.0915400597335
x73=28.2757773417391x_{73} = 28.2757773417391
x74=34.1099168227191x_{74} = 34.1099168227191
x75=−59.6909442072692x_{75} = -59.6909442072692
x76=76.2963565555275x_{76} = 76.2963565555275
x77=86.1698721695492x_{77} = 86.1698721695492
x78=82.1307049097742x_{78} = 82.1307049097742
x79=−0.520513881060772x_{79} = -0.520513881060772
x80=−48.0223376394435x_{80} = -48.0223376394435
x81=87.9650583050154x_{81} = 87.9650583050154
x82=−17.954231182823x_{82} = -17.954231182823
x83=−39.9441284136678x_{83} = -39.9441284136678
x84=−13.9157001672389x_{84} = -13.9157001672389
x85=42.1880688184805x_{85} = 42.1880688184805
x86=−5.84136825229155x_{86} = -5.84136825229155
x87=16.1592878293921x_{87} = 16.1592878293921
x88=54.3054246119194x_{88} = 54.3054246119194
x89=11.2236096263714x_{89} = 11.2236096263714
x90=−97.8385883908581x_{90} = -97.8385883908581
x91=26.0319070013448x_{91} = 26.0319070013448
x92=12.1209384633299x_{92} = 12.1209384633299
x93=90.2090415153744x_{93} = 90.2090415153744
x94=−45.7783845476972x_{94} = -45.7783845476972
x95=43.9832251347853x_{95} = 43.9832251347853
x96=−33.6611338247532x_{96} = -33.6611338247532
x97=−41.7392802700658x_{97} = -41.7392802700658
x98=−49.8175028149324x_{98} = -49.8175028149324
x99=8.08342839112335x_{99} = 8.08342839112335

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
[100.08257378976,∞)\left[100.08257378976, \infty\right)
Convexa en los intervalos
(−∞,−97.8385883908581]\left(-\infty, -97.8385883908581\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(7x)7)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{x \sin{\left(7 x \right)}}{7}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(7x)7)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{x \sin{\left(7 x \right)}}{7}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*sin(7*x))/7, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(7x)7)=⟨−17,17⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{7}\right) = \left\langle - \frac{1}{7}, \frac{1}{7}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−17,17⟩xy = \left\langle - \frac{1}{7}, \frac{1}{7}\right\rangle x
lim⁡x→∞(sin⁡(7x)7)=⟨−17,17⟩\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{7}\right) = \left\langle - \frac{1}{7}, \frac{1}{7}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−17,17⟩xy = \left\langle - \frac{1}{7}, \frac{1}{7}\right\rangle x
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:
xsin⁡(7x)7=xsin⁡(7x)7\frac{x \sin{\left(7 x \right)}}{7} = \frac{x \sin{\left(7 x \right)}}{7}
- Sí
xsin⁡(7x)7=−xsin⁡(7x)7\frac{x \sin{\left(7 x \right)}}{7} = - \frac{x \sin{\left(7 x \right)}}{7}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор