Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ -xsinxx-2

Gráfico de la función y = -xsinxx-2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = -x*sin(x)*x - 2
f(x)=x−xsin⁡(x)−2f{\left(x \right)} = x - x \sin{\left(x \right)} - 2 
f = x*((-x)*sin(x)) - 2
Gráfico de la función
02468-8-6-4-2-1010-100100
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:
x−xsin⁡(x)−2=0x - x \sin{\left(x \right)} - 2 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=59.6908217438756x_{1} = 59.6908217438756
x2=−37.700518977158x_{2} = -37.700518977158
x3=40.8419034938155x_{3} = 40.8419034938155
x4=−1.42208339294766x_{4} = -1.42208339294766
x5=28.2768351999549x_{5} = 28.2768351999549
x6=−21.987011448352x_{6} = -21.987011448352
x7=15.7160607006388x_{7} = 15.7160607006388
x8=47.1247904022886x_{8} = 47.1247904022886
x9=100.530767021156x_{9} = 100.530767021156
x10=−65.9729862124666x_{10} = -65.9729862124666
x11=−87.9648527714054x_{11} = -87.9648527714054
x12=−69.1154570564923x_{12} = -69.1154570564923
x13=3.32364732189927x_{13} = 3.32364732189927
x14=6.23166071387218x_{14} = 6.23166071387218
x15=−81.6817087579544x_{15} = -81.6817087579544
x16=−94.2480047648058x_{16} = -94.2480047648058
x17=−18.8551815474534x_{17} = -18.8551815474534
x18=34.559193758284x_{18} = 34.559193758284
x19=31.4138998493574x_{19} = 31.4138998493574
x20=−9.40215171277304x_{20} = -9.40215171277304
x21=75.3978718731992x_{21} = 75.3978718731992
x22=−43.9833309914671x_{22} = -43.9833309914671
x23=50.2646908606744x_{23} = 50.2646908606744
x24=69.1146196913135x_{24} = 69.1146196913135
x25=81.6811092243143x_{25} = 81.6811092243143
x26=12.5536795051862x_{26} = 12.5536795051862
x27=−97.3891613942219x_{27} = -97.3891613942219
x28=−15.6998491035096x_{28} = -15.6998491035096
x29=62.8313464576863x_{29} = 62.8313464576863
x30=9.44718895618355x_{30} = 9.44718895618355
x31=−50.2662740043369x_{31} = -50.2662740043369
x32=87.9643358265851x_{32} = 87.9643358265851
x33=−53.4063739078945x_{33} = -53.4063739078945
x34=84.8232796181751x_{34} = 84.8232796181751
x35=−47.1229891365535x_{35} = -47.1229891365535
x36=25.1295741382878x_{36} = 25.1295741382878
x37=−106.814325517644x_{37} = -106.814325517644
x38=−72.2562479616373x_{38} = -72.2562479616373
x39=37.6977044988782x_{39} = 37.6977044988782
x40=−25.1359067235501x_{40} = -25.1359067235501
x41=−34.555844296047x_{41} = -34.555844296047
x42=−62.8323596695666x_{42} = -62.8323596695666
x43=−59.689699071421x_{43} = -59.689699071421
x44=91.1064279068952x_{44} = 91.1064279068952
x45=−100.531162807032x_{45} = -100.531162807032
x46=18.8439235716796x_{46} = 18.8439235716796
x47=97.389583126519x_{47} = 97.389583126519
x48=43.9812632118327x_{48} = 43.9812632118327
x49=56.5480423113348x_{49} = 56.5480423113348
x50=−78.5394921092745x_{50} = -78.5394921092745
x51=−2.90178022717344x_{51} = -2.90178022717344
x52=−56.5492931902278x_{52} = -56.5492931902278
x53=94.2475544484302x_{53} = 94.2475544484302
x54=−75.3985754925447x_{54} = -75.3985754925447
x55=−40.8395053587023x_{55} = -40.8395053587023
x56=−28.2718316792198x_{56} = -28.2718316792198
x57=−91.1059459987638x_{57} = -91.1059459987638
x58=65.9739052255028x_{58} = 65.9739052255028
x59=−12.5790106584862x_{59} = -12.5790106584862
x60=72.2570140953699x_{60} = 72.2570140953699
x61=78.5401405648612x_{61} = 78.5401405648612
x62=21.9952825910084x_{62} = 21.9952825910084
x63=53.4077762773349x_{63} = 53.4077762773349
x64=−31.4179526995949x_{64} = -31.4179526995949
x65=−6.33307161321987x_{65} = -6.33307161321987
x66=−84.82272367203x_{66} = -84.82272367203
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(x))*x - 2.
−2+0⋅−0sin⁡(0)-2 + 0 \cdot - 0 \sin{\left(0 \right)}
Resultado:
f(0)=−2f{\left(0 \right)} = -2
Punto:
(0, -2)
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
−xsin⁡(x)+x(−xcos⁡(x)−sin⁡(x))=0- x \sin{\left(x \right)} + x \left(- x \cos{\left(x \right)} - \sin{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=89.5577188827244x_{1} = 89.5577188827244
x2=−92.6985552433969x_{2} = -92.6985552433969
x3=−29.9118938695518x_{3} = -29.9118938695518
x4=64.4336791037316x_{4} = 64.4336791037316
x5=−58.153842078645x_{5} = -58.153842078645
x6=23.6463238196036x_{6} = 23.6463238196036
x7=51.8748140534268x_{7} = 51.8748140534268
x8=92.6985552433969x_{8} = 92.6985552433969
x9=33.0471686947054x_{9} = 33.0471686947054
x10=−80.1355651940744x_{10} = -80.1355651940744
x11=70.7141100665485x_{11} = 70.7141100665485
x12=−17.3932439645948x_{12} = -17.3932439645948
x13=98.9803718651523x_{13} = 98.9803718651523
x14=−70.7141100665485x_{14} = -70.7141100665485
x15=11.17270586833x_{15} = 11.17270586833
x16=48.7357007949054x_{16} = 48.7357007949054
x17=−89.5577188827244x_{17} = -89.5577188827244
x18=−23.6463238196036x_{18} = -23.6463238196036
x19=14.2763529183365x_{19} = 14.2763529183365
x20=0x_{20} = 0
x21=26.7780870755585x_{21} = 26.7780870755585
x22=−48.7357007949054x_{22} = -48.7357007949054
x23=−55.0142096788381x_{23} = -55.0142096788381
x24=76.9949898891676x_{24} = 76.9949898891676
x25=86.4169374541167x_{25} = 86.4169374541167
x26=83.2762171649775x_{26} = 83.2762171649775
x27=−61.2936749662429x_{27} = -61.2936749662429
x28=−76.9949898891676x_{28} = -76.9949898891676
x29=−64.4336791037316x_{29} = -64.4336791037316
x30=−73.8545010149048x_{30} = -73.8545010149048
x31=−14.2763529183365x_{31} = -14.2763529183365
x32=−45.5969279840735x_{32} = -45.5969279840735
x33=20.5175229099417x_{33} = 20.5175229099417
x34=−120.967848975693x_{34} = -120.967848975693
x35=58.153842078645x_{35} = 58.153842078645
x36=55.0142096788381x_{36} = 55.0142096788381
x37=−51.8748140534268x_{37} = -51.8748140534268
x38=5.08698509410227x_{38} = 5.08698509410227
x39=3.95930141892882⋅10−7x_{39} = 3.95930141892882 \cdot 10^{-7}
x40=45.5969279840735x_{40} = 45.5969279840735
x41=73.8545010149048x_{41} = 73.8545010149048
x42=−42.458570771699x_{42} = -42.458570771699
x43=−8.09616360322292x_{43} = -8.09616360322292
x44=−36.1835330907526x_{44} = -36.1835330907526
x45=29.9118938695518x_{45} = 29.9118938695518
x46=−98.9803718651523x_{46} = -98.9803718651523
x47=2.2889297281034x_{47} = 2.2889297281034
x48=−11.17270586833x_{48} = -11.17270586833
x49=42.458570771699x_{49} = 42.458570771699
x50=−83.2762171649775x_{50} = -83.2762171649775
x51=−20.5175229099417x_{51} = -20.5175229099417
x52=−5.08698509410227x_{52} = -5.08698509410227
x53=67.573830670859x_{53} = 67.573830670859
x54=−2.2889297281034x_{54} = -2.2889297281034
x55=17.3932439645948x_{55} = 17.3932439645948
x56=−86.4169374541167x_{56} = -86.4169374541167
x57=36.1835330907526x_{57} = 36.1835330907526
x58=39.3207281322521x_{58} = 39.3207281322521
x59=−67.573830670859x_{59} = -67.573830670859
x60=80.1355651940744x_{60} = 80.1355651940744
x61=−33.0471686947054x_{61} = -33.0471686947054
x62=−95.839441141233x_{62} = -95.839441141233
x63=95.839441141233x_{63} = 95.839441141233
x64=−39.3207281322521x_{64} = -39.3207281322521
x65=8.09616360322292x_{65} = 8.09616360322292
x66=−26.7780870755585x_{66} = -26.7780870755585
x67=61.2936749662429x_{67} = 61.2936749662429
Signos de extremos en los puntos:
(89.55771888272442, -8020.58575924144)

(-92.69855524339692, -8593.02284218332)

(-29.911893869551772, -894.728075975236)

(64.43367910373156, -4151.70044687478)

(-58.153842078645, 3377.87112092779)

(23.64632381960362, 555.159297209023)

(51.874814053426775, -2690.99855997676)

(92.69855524339692, 8589.02284218332)

(33.04716869470536, -1092.12083594654)

(-80.13556519407445, -6421.70974281978)

(70.7141100665485, -5000.48656158818)

(-17.393243964594753, -302.544552657996)

(98.98037186515228, 9793.11462678078)

(-70.7141100665485, 4996.48656158818)

(11.172705868329984, 120.876173513916)

(48.73570079490539, 2371.17105456709)

(-89.55771888272442, 8016.58575924144)

(-23.64632381960362, -559.159297209023)

(14.276352918336478, -203.843217881861)

(0, -2)

(26.778087075558506, -717.074276149712)

(-48.73570079490539, -2375.17105456709)

(-55.01420967883812, -3026.56524685288)

(76.9949898891676, -5928.22947957101)

(86.4169374541167, 7463.88788203037)

(83.27621716497754, -6934.92921007843)

(-61.2936749662429, -3756.91618650696)

(-76.9949898891676, 5924.22947957101)

(-64.43367910373156, 4147.70044687478)

(-73.85450101490484, -5454.4884195005)

(-14.276352918336478, 199.843217881861)

(-45.59692798407349, 2075.08272285774)

(20.51752290994169, -420.982887272434)

(-120.96784897569329, 14629.2208957387)

(58.153842078645, -3381.87112092779)

(55.01420967883812, 3022.56524685288)

(-51.874814053426775, 2686.99855997676)

(5.08698509410227, 22.0829602230683)

(3.9593014189288195e-07, -2)

(45.59692798407349, -2079.08272285774)

(73.85450101490484, 5450.4884195005)

(-42.458570771699044, -1802.73355411815)

(-8.096163603222921, 61.6349819515545)

(-36.18353309075258, -1309.25263807613)

(29.911893869551772, 890.728075975236)

(-98.98037186515228, -9797.11462678078)

(2.2889297281034042, -5.94530162528433)

(-11.172705868329984, -124.876173513916)

(42.458570771699044, 1798.73355411815)

(-83.27621716497754, 6930.92921007843)

(-20.51752290994169, 416.982887272434)

(-5.08698509410227, -26.0829602230683)

(67.573830670859, 4562.22390457183)

(-2.2889297281034042, 1.94530162528433)

(17.393243964594753, 298.544552657996)

(-86.4169374541167, -7467.88788203037)

(36.18353309075258, 1305.25263807613)

(39.32072813225213, -1546.1235331857)

(-67.573830670859, -4566.22390457183)

(80.13556519407445, 6417.70974281978)

(-33.04716869470536, 1088.12083594654)

(-95.83944114123304, 9181.19913125177)

(95.83944114123304, -9185.19913125177)

(-39.32072813225213, 1542.1235331857)

(8.096163603222921, -65.6349819515545)

(-26.778087075558506, 713.074276149712)

(61.2936749662429, 3752.91618650696)


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=89.5577188827244x_{1} = 89.5577188827244
x2=−92.6985552433969x_{2} = -92.6985552433969
x3=−29.9118938695518x_{3} = -29.9118938695518
x4=64.4336791037316x_{4} = 64.4336791037316
x5=51.8748140534268x_{5} = 51.8748140534268
x6=33.0471686947054x_{6} = 33.0471686947054
x7=−80.1355651940744x_{7} = -80.1355651940744
x8=70.7141100665485x_{8} = 70.7141100665485
x9=−17.3932439645948x_{9} = -17.3932439645948
x10=−23.6463238196036x_{10} = -23.6463238196036
x11=14.2763529183365x_{11} = 14.2763529183365
x12=26.7780870755585x_{12} = 26.7780870755585
x13=−48.7357007949054x_{13} = -48.7357007949054
x14=−55.0142096788381x_{14} = -55.0142096788381
x15=76.9949898891676x_{15} = 76.9949898891676
x16=83.2762171649775x_{16} = 83.2762171649775
x17=−61.2936749662429x_{17} = -61.2936749662429
x18=−73.8545010149048x_{18} = -73.8545010149048
x19=20.5175229099417x_{19} = 20.5175229099417
x20=58.153842078645x_{20} = 58.153842078645
x21=45.5969279840735x_{21} = 45.5969279840735
x22=−42.458570771699x_{22} = -42.458570771699
x23=−36.1835330907526x_{23} = -36.1835330907526
x24=−98.9803718651523x_{24} = -98.9803718651523
x25=2.2889297281034x_{25} = 2.2889297281034
x26=−11.17270586833x_{26} = -11.17270586833
x27=−5.08698509410227x_{27} = -5.08698509410227
x28=−86.4169374541167x_{28} = -86.4169374541167
x29=39.3207281322521x_{29} = 39.3207281322521
x30=−67.573830670859x_{30} = -67.573830670859
x31=95.839441141233x_{31} = 95.839441141233
x32=8.09616360322292x_{32} = 8.09616360322292
Puntos máximos de la función:
x32=−58.153842078645x_{32} = -58.153842078645
x32=23.6463238196036x_{32} = 23.6463238196036
x32=92.6985552433969x_{32} = 92.6985552433969
x32=98.9803718651523x_{32} = 98.9803718651523
x32=−70.7141100665485x_{32} = -70.7141100665485
x32=11.17270586833x_{32} = 11.17270586833
x32=48.7357007949054x_{32} = 48.7357007949054
x32=−89.5577188827244x_{32} = -89.5577188827244
x32=86.4169374541167x_{32} = 86.4169374541167
x32=−76.9949898891676x_{32} = -76.9949898891676
x32=−64.4336791037316x_{32} = -64.4336791037316
x32=−14.2763529183365x_{32} = -14.2763529183365
x32=−45.5969279840735x_{32} = -45.5969279840735
x32=−120.967848975693x_{32} = -120.967848975693
x32=55.0142096788381x_{32} = 55.0142096788381
x32=−51.8748140534268x_{32} = -51.8748140534268
x32=5.08698509410227x_{32} = 5.08698509410227
x32=73.8545010149048x_{32} = 73.8545010149048
x32=−8.09616360322292x_{32} = -8.09616360322292
x32=29.9118938695518x_{32} = 29.9118938695518
x32=42.458570771699x_{32} = 42.458570771699
x32=−83.2762171649775x_{32} = -83.2762171649775
x32=−20.5175229099417x_{32} = -20.5175229099417
x32=67.573830670859x_{32} = 67.573830670859
x32=−2.2889297281034x_{32} = -2.2889297281034
x32=17.3932439645948x_{32} = 17.3932439645948
x32=36.1835330907526x_{32} = 36.1835330907526
x32=80.1355651940744x_{32} = 80.1355651940744
x32=−33.0471686947054x_{32} = -33.0471686947054
x32=−95.839441141233x_{32} = -95.839441141233
x32=−39.3207281322521x_{32} = -39.3207281322521
x32=−26.7780870755585x_{32} = -26.7780870755585
x32=61.2936749662429x_{32} = 61.2936749662429
Decrece en los intervalos
[95.839441141233,∞)\left[95.839441141233, \infty\right)
Crece en los intervalos
(−∞,−98.9803718651523]\left(-\infty, -98.9803718651523\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
x(xsin⁡(x)−2cos⁡(x))−2xcos⁡(x)−2sin⁡(x)=0x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) - 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9.81900340196872x_{1} = 9.81900340196872
x2=84.8701107016488x_{2} = 84.8701107016488
x3=−56.6192418251285x_{3} = -56.6192418251285
x4=−88.0100241275575x_{4} = -88.0100241275575
x5=−15.9554654297511x_{5} = -15.9554654297511
x6=69.1728243307457x_{6} = 69.1728243307457
x7=25.2900904960802x_{7} = 25.2900904960802
x8=−100.570724821846x_{8} = -100.570724821846
x9=94.290185945407x_{9} = 94.290185945407
x10=22.1703631077661x_{10} = 22.1703631077661
x11=62.895397234671x_{11} = 62.895397234671
x12=−31.5423183719258x_{12} = -31.5423183719258
x13=44.0729006762809x_{13} = 44.0729006762809
x14=81.7303260381702x_{14} = 81.7303260381702
x15=0x_{15} = 0
x16=−37.8046732869526x_{16} = -37.8046732869526
x17=−12.8711405784383x_{17} = -12.8711405784383
x18=56.6192418251285x_{18} = 56.6192418251285
x19=−97.4304127980508x_{19} = -97.4304127980508
x20=40.9382191715155x_{20} = 40.9382191715155
x21=91.1500530451789x_{21} = 91.1500530451789
x22=100.570724821846x_{22} = 100.570724821846
x23=−53.4817799880237x_{23} = -53.4817799880237
x24=53.4817799880237x_{24} = 53.4817799880237
x25=−81.7303260381702x_{25} = -81.7303260381702
x26=−78.5906855194896x_{26} = -78.5906855194896
x27=19.0575561537385x_{27} = 19.0575561537385
x28=−75.4512070764701x_{28} = -75.4512070764701
x29=−66.0339743721325x_{29} = -66.0339743721325
x30=−28.4145306971625x_{30} = -28.4145306971625
x31=−25.2900904960802x_{31} = -25.2900904960802
x32=34.6725661362236x_{32} = 34.6725661362236
x33=−3.99444471574142x_{33} = -3.99444471574142
x34=31.5423183719258x_{34} = 31.5423183719258
x35=−9.81900340196872x_{35} = -9.81900340196872
x36=72.3119117382824x_{36} = 72.3119117382824
x37=−50.3448303040845x_{37} = -50.3448303040845
x38=−40.9382191715155x_{38} = -40.9382191715155
x39=−6.83214574693118x_{39} = -6.83214574693118
x40=−69.1728243307457x_{40} = -69.1728243307457
x41=−72.3119117382824x_{41} = -72.3119117382824
x42=−47.2084939833195x_{42} = -47.2084939833195
x43=−44.0729006762809x_{43} = -44.0729006762809
x44=47.2084939833195x_{44} = 47.2084939833195
x45=6.83214574693118x_{45} = 6.83214574693118
x46=97.4304127980508x_{46} = 97.4304127980508
x47=59.7571356682663x_{47} = 59.7571356682663
x48=−94.290185945407x_{48} = -94.290185945407
x49=−22.1703631077661x_{49} = -22.1703631077661
x50=−59.7571356682663x_{50} = -59.7571356682663
x51=15.9554654297511x_{51} = 15.9554654297511
x52=3.99444471574142x_{52} = 3.99444471574142
x53=1.51985529843113x_{53} = 1.51985529843113
x54=−84.8701107016488x_{54} = -84.8701107016488
x55=−19.0575561537385x_{55} = -19.0575561537385
x56=50.3448303040845x_{56} = 50.3448303040845
x57=28.4145306971625x_{57} = 28.4145306971625
x58=12.8711405784383x_{58} = 12.8711405784383
x59=75.4512070764701x_{59} = 75.4512070764701
x60=−1.51985529843113x_{60} = -1.51985529843113
x61=37.8046732869526x_{61} = 37.8046732869526
x62=78.5906855194896x_{62} = 78.5906855194896
x63=−62.895397234671x_{63} = -62.895397234671
x64=−34.6725661362236x_{64} = -34.6725661362236
x65=−91.1500530451789x_{65} = -91.1500530451789
x66=88.0100241275575x_{66} = 88.0100241275575
x67=66.0339743721325x_{67} = 66.0339743721325

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
[100.570724821846,∞)\left[100.570724821846, \infty\right)
Convexa en los intervalos
(−∞,−100.570724821846]\left(-\infty, -100.570724821846\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(x−xsin⁡(x)−2)y = \lim_{x \to -\infty}\left(x - x \sin{\left(x \right)} - 2\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x−xsin⁡(x)−2)y = \lim_{x \to \infty}\left(x - x \sin{\left(x \right)} - 2\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((-x)*sin(x))*x - 2, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(x−xsin⁡(x)−2x)y = x \lim_{x \to -\infty}\left(\frac{x - x \sin{\left(x \right)} - 2}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x−xsin⁡(x)−2x)y = x \lim_{x \to \infty}\left(\frac{x - x \sin{\left(x \right)} - 2}{x}\right)
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:
x−xsin⁡(x)−2=x2sin⁡(x)−2x - x \sin{\left(x \right)} - 2 = x^{2} \sin{\left(x \right)} - 2
- No
x−xsin⁡(x)−2=−x2sin⁡(x)+2x - x \sin{\left(x \right)} - 2 = - x^{2} \sin{\left(x \right)} + 2
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор