Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ (x^2+x)*sin(x)

Gráfico de la función y = (x^2+x)*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       / 2    \       
f(x) = \x  + x/*sin(x)
f(x)=(x2+x)sin⁡(x)f{\left(x \right)} = \left(x^{2} + x\right) \sin{\left(x \right)} 
f = (x^2 + x)*sin(x)
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:
(x2+x)sin⁡(x)=0\left(x^{2} + x\right) \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−1x_{1} = -1
x2=0x_{2} = 0
x3=πx_{3} = \pi
Solución numérica
x1=69.1150383789755x_{1} = 69.1150383789755
x2=65.9734457253857x_{2} = 65.9734457253857
x3=−91.106186954104x_{3} = -91.106186954104
x4=−103.672557568463x_{4} = -103.672557568463
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=−21.9911485751286x_{6} = -21.9911485751286
x7=12.5663706143592x_{7} = 12.5663706143592
x8=21.9911485751286x_{8} = 21.9911485751286
x9=−69.1150383789755x_{9} = -69.1150383789755
x10=−100.530964914873x_{10} = -100.530964914873
x11=3.14159265358979x_{11} = 3.14159265358979
x12=−1x_{12} = -1
x13=−3.14159265358979x_{13} = -3.14159265358979
x14=−25.1327412287183x_{14} = -25.1327412287183
x15=−116.238928182822x_{15} = -116.238928182822
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=−15.707963267949x_{17} = -15.707963267949
x18=−72.2566310325652x_{18} = -72.2566310325652
x19=84.8230016469244x_{19} = 84.8230016469244
x20=−81.6814089933346x_{20} = -81.6814089933346
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=18.8495559215388x_{22} = 18.8495559215388
x23=−65.9734457253857x_{23} = -65.9734457253857
x24=94.2477796076938x_{24} = 94.2477796076938
x25=9.42477796076938x_{25} = 9.42477796076938
x26=−40.8407044966673x_{26} = -40.8407044966673
x27=34.5575191894877x_{27} = 34.5575191894877
x28=0x_{28} = 0
x29=97.3893722612836x_{29} = 97.3893722612836
x30=53.4070751110265x_{30} = 53.4070751110265
x31=147.65485471872x_{31} = 147.65485471872
x32=−62.8318530717959x_{32} = -62.8318530717959
x33=59.6902604182061x_{33} = 59.6902604182061
x34=−28.2743338823081x_{34} = -28.2743338823081
x35=−56.5486677646163x_{35} = -56.5486677646163
x36=91.106186954104x_{36} = 91.106186954104
x37=15.707963267949x_{37} = 15.707963267949
x38=−18.8495559215388x_{38} = -18.8495559215388
x39=6.28318530717959x_{39} = 6.28318530717959
x40=56.5486677646163x_{40} = 56.5486677646163
x41=87.9645943005142x_{41} = 87.9645943005142
x42=31.4159265358979x_{42} = 31.4159265358979
x43=25.1327412287183x_{43} = 25.1327412287183
x44=43.9822971502571x_{44} = 43.9822971502571
x45=−47.1238898038469x_{45} = -47.1238898038469
x46=72.2566310325652x_{46} = 72.2566310325652
x47=−34.5575191894877x_{47} = -34.5575191894877
x48=−97.3893722612836x_{48} = -97.3893722612836
x49=−50.2654824574367x_{49} = -50.2654824574367
x50=100.530964914873x_{50} = 100.530964914873
x51=81.6814089933346x_{51} = 81.6814089933346
x52=−75.398223686155x_{52} = -75.398223686155
x53=40.8407044966673x_{53} = 40.8407044966673
x54=−9.42477796076938x_{54} = -9.42477796076938
x55=78.5398163397448x_{55} = 78.5398163397448
x56=−87.9645943005142x_{56} = -87.9645943005142
x57=37.6991118430775x_{57} = 37.6991118430775
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−6.28318530717959x_{59} = -6.28318530717959
x60=50.2654824574367x_{60} = 50.2654824574367
x61=−37.6991118430775x_{61} = -37.6991118430775
x62=−43.9822971502571x_{62} = -43.9822971502571
x63=47.1238898038469x_{63} = 47.1238898038469
x64=28.2743338823081x_{64} = 28.2743338823081
x65=62.8318530717959x_{65} = 62.8318530717959
x66=−31.4159265358979x_{66} = -31.4159265358979
x67=−12.5663706143592x_{67} = -12.5663706143592
x68=75.398223686155x_{68} = 75.398223686155
x69=−84.8230016469244x_{69} = -84.8230016469244
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^2 + x)*sin(x).
02sin⁡(0)0^{2} \sin{\left(0 \right)}
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
(2x+1)sin⁡(x)+(x2+x)cos⁡(x)=0\left(2 x + 1\right) \sin{\left(x \right)} + \left(x^{2} + x\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−48.7361295392441x_{1} = -48.7361295392441
x2=−45.5974183148387x_{2} = -45.5974183148387
x3=20.5152887094394x_{3} = 20.5152887094394
x4=−42.4591369509051x_{4} = -42.4591369509051
x5=−92.6986728032932x_{5} = -92.6986728032932
x6=−23.6481708154477x_{6} = -23.6481708154477
x7=58.1535518920027x_{7} = 58.1535518920027
x8=−67.5740526651947x_{8} = -67.5740526651947
x9=14.2718933831386x_{9} = 14.2718933831386
x10=2.22109820882798x_{10} = 2.22109820882798
x11=−26.779523413937x_{11} = -26.779523413937
x12=64.4334422583637x_{12} = 64.4334422583637
x13=86.4168051847116x_{13} = 86.4168051847116
x14=−11.1810751482291x_{14} = -11.1810751482291
x15=29.910819411552x_{15} = 29.910819411552
x16=42.4580305922627x_{16} = 42.4580305922627
x17=−0.649755168374399x_{17} = -0.649755168374399
x18=45.5964586594414x_{18} = 45.5964586594414
x19=−55.0145455295839x_{19} = -55.0145455295839
x20=26.7767536014234x_{20} = 26.7767536014234
x21=51.8744502729041x_{21} = 51.8744502729041
x22=−17.3966801527861x_{22} = -17.3966801527861
x23=33.046284696792x_{23} = 33.046284696792
x24=−95.8395510876656x_{24} = -95.8395510876656
x25=39.3200997911221x_{25} = 39.3200997911221
x26=−83.2763629879876x_{26} = -83.2763629879876
x27=92.6984401916674x_{27} = 92.6984401916674
x28=55.0138858041882x_{28} = 55.0138858041882
x29=0x_{29} = 0
x30=−2.41352739608161x_{30} = -2.41352739608161
x31=−58.1541424076718x_{31} = -58.1541424076718
x32=98.9802708766234x_{32} = 98.9802708766234
x33=89.5575956718273x_{33} = 89.5575956718273
x34=−29.9130423127956x_{34} = -29.9130423127956
x35=11.1656732464925x_{35} = 11.1656732464925
x36=76.9948235344069x_{36} = 76.9948235344069
x37=5.06026836518236x_{37} = 5.06026836518236
x38=−89.5578448748316x_{38} = -89.5578448748316
x39=95.8393334645179x_{39} = 95.8393334645179
x40=−36.1843149453611x_{40} = -36.1843149453611
x41=−8.1119826164815x_{41} = -8.1119826164815
x42=−33.0481076541221x_{42} = -33.0481076541221
x43=−64.4339234092994x_{43} = -64.4339234092994
x44=70.7139131086823x_{44} = 70.7139131086823
x45=−14.2814705578331x_{45} = -14.2814705578331
x46=−80.1357227352106x_{46} = -80.1357227352106
x47=−76.9951606189843x_{47} = -76.9951606189843
x48=23.6446256872296x_{48} = 23.6446256872296
x49=−39.3213891805148x_{49} = -39.3213891805148
x50=−70.7143126702811x_{50} = -70.7143126702811
x51=−51.8751921133361x_{51} = -51.8751921133361
x52=67.5736151455506x_{52} = 67.5736151455506
x53=−158.663073635328x_{53} = -158.663073635328
x54=36.1827931655659x_{54} = 36.1827931655659
x55=83.276074800594x_{55} = 83.276074800594
x56=61.2934134764651x_{56} = 61.2934134764651
x57=80.1354115339569x_{57} = 80.1354115339569
x58=−20.519983804286x_{58} = -20.519983804286
x59=−86.4170728188749x_{59} = -86.4170728188749
x60=−5.12485035716986x_{60} = -5.12485035716986
x61=8.08365076227732x_{61} = 8.08365076227732
x62=48.7352892631046x_{62} = 48.7352892631046
x63=−98.9804749142348x_{63} = -98.9804749142348
x64=73.8543203258377x_{64} = 73.8543203258377
x65=17.3901768408315x_{65} = 17.3901768408315
x66=−61.2939451205206x_{66} = -61.2939451205206
x67=−73.854686660561x_{67} = -73.854686660561
Signos de extremos en los puntos:
(-48.73612953924414, 2324.47655388266)

(-45.59741831483875, -2031.52983921935)

(20.515288709439442, 439.404733492473)

(-42.45913695090512, 1758.3222926222)

(-92.6986728032932, 8498.34591348608)

(-23.648170815447703, 533.598022286833)

(58.153551892002696, 3437.99074695487)

(-67.57405266519466, 4496.67976267516)

(14.271893383138554, 215.983715663497)

(2.221098208827979, 5.69417506667367)

(-26.779523413937035, -688.371282181144)

(64.43344225836368, 4214.10322708716)

(86.41680518471165, -7552.28175128335)

(-11.181075148229146, 111.882406294904)

(29.91081941155198, -922.573866189311)

(42.45803059226273, -1843.14536818102)

(-0.6497551683743993, 0.137679962001981)

(45.59645865944136, 2122.63608586311)

(-55.0145455295839, 2969.58752323768)

(26.776753601423422, 741.778651182549)

(51.87445027290409, 2740.83504435784)

(-17.396680152786104, 283.266875092874)

(33.046284696792036, 1123.10809200774)

(-95.8395510876656, -9087.38060649368)

(39.32009979112207, 1583.39380980238)

(-83.27636298798761, -6849.67707188041)

(92.69844019166737, -8683.69988715932)

(55.01388580418817, -3079.54330011308)

(0, 0)

(-2.4135273960816113, -2.2701610308603)

(-58.154142407671834, -3321.75178998414)

(98.98027087662336, -9894.07484928877)

(89.55759567182727, 8108.12121609698)

(-29.913042312795604, 862.883394738015)

(11.16567324649254, -133.877521969079)

(76.99482353440692, 6003.19859003098)

(5.060268365182357, -28.8295870295239)

(-89.55784487483163, -7929.05042695635)

(95.83933346451788, 9279.01776479774)

(-36.18431494536108, 1271.12464275245)

(-8.111982616481502, -55.7827777203279)

(-33.04810765412208, -1057.13448970071)

(-64.43392340929937, -4085.29790712209)

(70.71391310868229, 5069.17250415237)

(-14.281470557833119, -187.707462567745)

(-80.1357227352106, 6339.59920242608)

(-76.99516061898431, -5849.2605375965)

(23.644625687229603, -580.722338398605)

(-39.32138918051478, -1504.85390043184)

(-70.71431267028109, -4927.80081872494)

(-51.875192113336084, -2637.16244624062)

(67.57361514555056, -4631.7682651326)

(-158.66307363532783, -25013.3080816123)

(36.18279316556589, -1343.38139313081)

(83.276074800594, 7016.18149232862)

(61.2934134764651, -3816.17738838667)

(80.13541153395687, -6499.82043876569)

(-20.519983804286046, -398.56337754808)

(-86.41707281887494, 7379.49414655867)

(-5.124850357169864, 19.3664389379329)

(8.083650762277319, 71.5009434734222)

(48.735289263104576, -2421.86597503638)

(-98.98047491423482, 9696.15450627079)

(73.85432032583775, -5526.31594543957)

(17.390176840831522, -317.825460652734)

(-61.29394512052059, 3693.65525030791)

(-73.854686660561, 5378.66107666168)


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=−45.5974183148387x_{1} = -45.5974183148387
x2=−26.779523413937x_{2} = -26.779523413937
x3=86.4168051847116x_{3} = 86.4168051847116
x4=29.910819411552x_{4} = 29.910819411552
x5=42.4580305922627x_{5} = 42.4580305922627
x6=−95.8395510876656x_{6} = -95.8395510876656
x7=−83.2763629879876x_{7} = -83.2763629879876
x8=92.6984401916674x_{8} = 92.6984401916674
x9=55.0138858041882x_{9} = 55.0138858041882
x10=0x_{10} = 0
x11=−2.41352739608161x_{11} = -2.41352739608161
x12=−58.1541424076718x_{12} = -58.1541424076718
x13=98.9802708766234x_{13} = 98.9802708766234
x14=11.1656732464925x_{14} = 11.1656732464925
x15=5.06026836518236x_{15} = 5.06026836518236
x16=−89.5578448748316x_{16} = -89.5578448748316
x17=−8.1119826164815x_{17} = -8.1119826164815
x18=−33.0481076541221x_{18} = -33.0481076541221
x19=−64.4339234092994x_{19} = -64.4339234092994
x20=−14.2814705578331x_{20} = -14.2814705578331
x21=−76.9951606189843x_{21} = -76.9951606189843
x22=23.6446256872296x_{22} = 23.6446256872296
x23=−39.3213891805148x_{23} = -39.3213891805148
x24=−70.7143126702811x_{24} = -70.7143126702811
x25=−51.8751921133361x_{25} = -51.8751921133361
x26=67.5736151455506x_{26} = 67.5736151455506
x27=−158.663073635328x_{27} = -158.663073635328
x28=36.1827931655659x_{28} = 36.1827931655659
x29=61.2934134764651x_{29} = 61.2934134764651
x30=80.1354115339569x_{30} = 80.1354115339569
x31=−20.519983804286x_{31} = -20.519983804286
x32=48.7352892631046x_{32} = 48.7352892631046
x33=73.8543203258377x_{33} = 73.8543203258377
x34=17.3901768408315x_{34} = 17.3901768408315
Puntos máximos de la función:
x34=−48.7361295392441x_{34} = -48.7361295392441
x34=20.5152887094394x_{34} = 20.5152887094394
x34=−42.4591369509051x_{34} = -42.4591369509051
x34=−92.6986728032932x_{34} = -92.6986728032932
x34=−23.6481708154477x_{34} = -23.6481708154477
x34=58.1535518920027x_{34} = 58.1535518920027
x34=−67.5740526651947x_{34} = -67.5740526651947
x34=14.2718933831386x_{34} = 14.2718933831386
x34=2.22109820882798x_{34} = 2.22109820882798
x34=64.4334422583637x_{34} = 64.4334422583637
x34=−11.1810751482291x_{34} = -11.1810751482291
x34=−0.649755168374399x_{34} = -0.649755168374399
x34=45.5964586594414x_{34} = 45.5964586594414
x34=−55.0145455295839x_{34} = -55.0145455295839
x34=26.7767536014234x_{34} = 26.7767536014234
x34=51.8744502729041x_{34} = 51.8744502729041
x34=−17.3966801527861x_{34} = -17.3966801527861
x34=33.046284696792x_{34} = 33.046284696792
x34=39.3200997911221x_{34} = 39.3200997911221
x34=89.5575956718273x_{34} = 89.5575956718273
x34=−29.9130423127956x_{34} = -29.9130423127956
x34=76.9948235344069x_{34} = 76.9948235344069
x34=95.8393334645179x_{34} = 95.8393334645179
x34=−36.1843149453611x_{34} = -36.1843149453611
x34=70.7139131086823x_{34} = 70.7139131086823
x34=−80.1357227352106x_{34} = -80.1357227352106
x34=83.276074800594x_{34} = 83.276074800594
x34=−86.4170728188749x_{34} = -86.4170728188749
x34=−5.12485035716986x_{34} = -5.12485035716986
x34=8.08365076227732x_{34} = 8.08365076227732
x34=−98.9804749142348x_{34} = -98.9804749142348
x34=−61.2939451205206x_{34} = -61.2939451205206
x34=−73.854686660561x_{34} = -73.854686660561
Decrece en los intervalos
[98.9802708766234,∞)\left[98.9802708766234, \infty\right)
Crece en los intervalos
(−∞,−158.663073635328]\left(-\infty, -158.663073635328\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(x+1)sin⁡(x)+2(2x+1)cos⁡(x)+2sin⁡(x)=0- x \left(x + 1\right) \sin{\left(x \right)} + 2 \left(2 x + 1\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x2+x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\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→∞((x2+x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\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^2 + x)*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x2+x)sin⁡(x)x)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞((x2+x)sin⁡(x)x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\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:
(x2+x)sin⁡(x)=−(x2−x)sin⁡(x)\left(x^{2} + x\right) \sin{\left(x \right)} = - \left(x^{2} - x\right) \sin{\left(x \right)}
- No
(x2+x)sin⁡(x)=(x2−x)sin⁡(x)\left(x^{2} + x\right) \sin{\left(x \right)} = \left(x^{2} - x\right) \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор