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sin(x)/(2*x+16)
Trazado el gráfico de la función/ sin(x)/(2*x+16)

Gráfico de la función y = sin(x)/(2*x+16)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        sin(x) 
f(x) = --------
       2*x + 16
f(x)=sin⁡(x)2x+16f{\left(x \right)} = \frac{\sin{\left(x \right)}}{2 x + 16} 
f = sin(x)/(2*x + 16)
Gráfico de la función
02468-8-6-4-2-1010-10050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−8x_{1} = -8
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:
sin⁡(x)2x+16=0\frac{\sin{\left(x \right)}}{2 x + 16} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=31.4159265358979x_{6} = 31.4159265358979
x7=69.1150383789755x_{7} = 69.1150383789755
x8=201.061929829747x_{8} = 201.061929829747
x9=−103.672557568463x_{9} = -103.672557568463
x10=−37.6991118430775x_{10} = -37.6991118430775
x11=−81.6814089933346x_{11} = -81.6814089933346
x12=−84.8230016469244x_{12} = -84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=47.1238898038469x_{14} = 47.1238898038469
x15=−15.707963267949x_{15} = -15.707963267949
x16=−12.5663706143592x_{16} = -12.5663706143592
x17=12.5663706143592x_{17} = 12.5663706143592
x18=−87.9645943005142x_{18} = -87.9645943005142
x19=53.4070751110265x_{19} = 53.4070751110265
x20=72.2566310325652x_{20} = 72.2566310325652
x21=−100.530964914873x_{21} = -100.530964914873
x22=−3.14159265358979x_{22} = -3.14159265358979
x23=34.5575191894877x_{23} = 34.5575191894877
x24=−94.2477796076938x_{24} = -94.2477796076938
x25=6.28318530717959x_{25} = 6.28318530717959
x26=−69.1150383789755x_{26} = -69.1150383789755
x27=97.3893722612836x_{27} = 97.3893722612836
x28=0x_{28} = 0
x29=65.9734457253857x_{29} = 65.9734457253857
x30=−50.2654824574367x_{30} = -50.2654824574367
x31=15.707963267949x_{31} = 15.707963267949
x32=3.14159265358979x_{32} = 3.14159265358979
x33=−25.1327412287183x_{33} = -25.1327412287183
x34=−18.8495559215388x_{34} = -18.8495559215388
x35=40.8407044966673x_{35} = 40.8407044966673
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=37.6991118430775x_{37} = 37.6991118430775
x38=−43.9822971502571x_{38} = -43.9822971502571
x39=18.8495559215388x_{39} = 18.8495559215388
x40=−78.5398163397448x_{40} = -78.5398163397448
x41=−6.28318530717959x_{41} = -6.28318530717959
x42=−40.8407044966673x_{42} = -40.8407044966673
x43=43.9822971502571x_{43} = 43.9822971502571
x44=56.5486677646163x_{44} = 56.5486677646163
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=−559.203492338983x_{46} = -559.203492338983
x47=−2463.0086404144x_{47} = -2463.0086404144
x48=25.1327412287183x_{48} = 25.1327412287183
x49=78.5398163397448x_{49} = 78.5398163397448
x50=−28.2743338823081x_{50} = -28.2743338823081
x51=−361.283155162826x_{51} = -361.283155162826
x52=75.398223686155x_{52} = 75.398223686155
x53=59.6902604182061x_{53} = 59.6902604182061
x54=−34.5575191894877x_{54} = -34.5575191894877
x55=81.6814089933346x_{55} = 81.6814089933346
x56=−47.1238898038469x_{56} = -47.1238898038469
x57=100.530964914873x_{57} = 100.530964914873
x58=−9.42477796076938x_{58} = -9.42477796076938
x59=−75.398223686155x_{59} = -75.398223686155
x60=−72.2566310325652x_{60} = -72.2566310325652
x61=−31.4159265358979x_{61} = -31.4159265358979
x62=28.2743338823081x_{62} = 28.2743338823081
x63=−91.106186954104x_{63} = -91.106186954104
x64=21.9911485751286x_{64} = 21.9911485751286
x65=62.8318530717959x_{65} = 62.8318530717959
x66=9.42477796076938x_{66} = 9.42477796076938
x67=50.2654824574367x_{67} = 50.2654824574367
x68=94.2477796076938x_{68} = 94.2477796076938
x69=91.106186954104x_{69} = 91.106186954104
x70=84.8230016469244x_{70} = 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 sin(x)/(2*x + 16).
sin⁡(0)0⋅2+16\frac{\sin{\left(0 \right)}}{0 \cdot 2 + 16}
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
cos⁡(x)2x+16−2sin⁡(x)(2x+16)2=0\frac{\cos{\left(x \right)}}{2 x + 16} - \frac{2 \sin{\left(x \right)}}{\left(2 x + 16\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=29.8186944197622x_{1} = 29.8186944197622
x2=83.2412458033685x_{2} = 83.2412458033685
x3=98.9508187684919x_{3} = 98.9508187684919
x4=23.5302399446763x_{4} = 23.5302399446763
x5=−89.5231247843125x_{5} = -89.5231247843125
x6=−39.2379067101878x_{6} = -39.2379067101878
x7=48.6770441334903x_{7} = 48.6770441334903
x8=−20.3394883287925x_{8} = -20.3394883287925
x9=45.5344160768147x_{9} = 45.5344160768147
x10=17.2391593368591x_{10} = 17.2391593368591
x11=89.5251372199914x_{11} = 89.5251372199914
x12=7.79073782639594x_{12} = 7.79073782639594
x13=296.877225767144x_{13} = 296.877225767144
x14=−86.3810404785128x_{14} = -86.3810404785128
x15=70.6731245692948x_{15} = 70.6731245692948
x16=−64.3849160180317x_{16} = -64.3849160180317
x17=−80.0967433038804x_{17} = -80.0967433038804
x18=51.8195634020601x_{18} = 51.8195634020601
x19=92.6670498706103x_{19} = 92.6670498706103
x20=−98.9491738790285x_{20} = -98.9491738790285
x21=−32.9466587840552x_{21} = -32.9466587840552
x22=39.2487467466437x_{22} = 39.2487467466437
x23=36.1056465662001x_{23} = 36.1056465662001
x24=−95.8071878375154x_{24} = -95.8071878375154
x25=−5.03798143596818x_{25} = -5.03798143596818
x26=67.5310032292546x_{26} = 67.5310032292546
x27=−92.665172598047x_{27} = -92.665172598047
x28=26.6747060811334x_{28} = 26.6747060811334
x29=4.63339841386275x_{29} = 4.63339841386275
x30=61.2466166099857x_{30} = 61.2466166099857
x31=−36.0927341468129x_{31} = -36.0927341468129
x32=−76.9545187167077x_{32} = -76.9545187167077
x33=−17.1701392036497x_{33} = -17.1701392036497
x34=−61.242276886312x_{34} = -61.242276886312
x35=76.9572499306778x_{35} = 76.9572499306778
x36=−45.5264519123192x_{36} = -45.5264519123192
x37=95.8089431510378x_{37} = 95.8089431510378
x38=−51.813458709347x_{38} = -51.813458709347
x39=−13.9712372494303x_{39} = -13.9712372494303
x40=14.0919324259525x_{40} = 14.0919324259525
x41=54.9619901747878x_{41} = 54.9619901747878
x42=−73.8122337852619x_{42} = -73.8122337852619
x43=−10.6325497313325x_{43} = -10.6325497313325
x44=73.8152053010897x_{44} = 73.8152053010897
x45=−114.658756435446x_{45} = -114.658756435446
x46=−70.6698794310721x_{46} = -70.6698794310721
x47=1.46554038881958x_{47} = 1.46554038881958
x48=80.0992623211471x_{48} = 80.0992623211471
x49=202.627978484871x_{49} = 202.627978484871
x50=32.9623150280616x_{50} = 32.9623150280616
x51=−29.7992892908393x_{51} = -29.7992892908393
x52=86.3832032646061x_{52} = 86.3832032646061
x53=10.9428328303325x_{53} = 10.9428328303325
x54=−58.0995064651176x_{54} = -58.0995064651176
x55=−83.2389151082586x_{55} = -83.2389151082586
x56=64.3888359923923x_{56} = 64.3888359923923
x57=−67.5274446580851x_{57} = -67.5274446580851
x58=−48.6701029977084x_{58} = -48.6701029977084
x59=−42.3824243930235x_{59} = -42.3824243930235
x60=−54.9565783858477x_{60} = -54.9565783858477
x61=−1160.81761806175x_{61} = -1160.81761806175
x62=58.1043376448773x_{62} = 58.1043376448773
x63=−26.6499694622544x_{63} = -26.6499694622544
x64=−23.4975077301914x_{64} = -23.4975077301914
x65=20.385137107631x_{65} = 20.385137107631
x66=−1.72892731735817x_{66} = -1.72892731735817
x67=42.3916588737312x_{67} = 42.3916588737312
Signos de extremos en los puntos:
(29.81869441976219, -0.0132163551678269)

(83.24124580336851, 0.00547964868461043)

(98.95081876849191, -0.00467484168082807)

(23.53023994467628, -0.015849822927082)

(-89.52312478431251, 0.00613276771551909)

(-39.2379067101878, 0.0159979989417826)

(48.67704413349028, -0.00882054100818635)

(-20.339488328792513, 0.040387910256165)

(45.53441607681472, 0.00933815716786801)

(17.23915933685907, -0.0197949543190484)

(89.52513721999138, 0.00512661382892545)

(7.790737826395936, 0.0316008278991694)

(296.87722576714367, 0.00163999560527707)

(-86.38104047851283, -0.00637857457720171)

(70.67312456929484, 0.00635489713149793)

(-64.38491601803167, 0.00886622556472135)

(-80.0967433038804, -0.00693445900681646)

(51.81956340206008, 0.00835730189580205)

(92.66704987061034, -0.00496662346503129)

(-98.94917387902854, -0.00549724376895022)

(-32.94665878405517, 0.0200266807430375)

(39.24874674664369, 0.0105799219479804)

(36.10564656620012, -0.0113335043037337)

(-95.80718783751544, 0.00569392541174881)

(-5.037981435968184, 0.159935085374477)

(67.53100322925458, -0.00661921810457567)

(-92.665172598047, -0.00590520413537255)

(26.674706081133444, 0.0144137400832445)

(4.633398413862747, -0.0394542242059762)

(61.24661660998567, -0.00721981659186086)

(-36.092734146812894, -0.0177869310376285)

(-76.95451871670768, 0.00725039400357598)

(-17.170139203649672, -0.0542034630579015)

(-61.24227688631198, -0.00938937740934138)

(76.95724993067785, 0.00588490525645087)

(-45.52645191231922, 0.0133192066337661)

(95.80894315103784, 0.00481631723986128)

(-51.81345870934699, 0.0114090471608567)

(-13.971237249430251, 0.0825846624278645)

(14.091932425952496, 0.0226095452967439)

(54.961990174787836, -0.00794029773055918)

(-73.81223378526194, -0.00759649477281035)

(-10.632549731332535, -0.177551645119497)

(73.81520530108966, -0.00611087699343733)

(-114.65875643544555, 0.00468764161745873)

(-70.66987943107209, 0.00797729887453037)

(1.4655403888195844, 0.0525308469032097)

(80.09926232114707, -0.00567505084138066)

(202.62797848487088, 0.00237382691753805)

(32.96231502806164, 0.0122027056198732)

(-29.799289290839333, -0.0229124325054616)

(86.38320326460605, -0.00529725545440014)

(10.942832830332486, -0.0263585045769199)

(-58.099506465117564, 0.00997815072173163)

(-83.23891510825857, 0.00664491031665852)

(64.38883599239234, 0.00690648345641789)

(-67.52744465808514, -0.00839830210112212)

(-48.67010299770837, -0.0122903284761894)

(-42.382424393023456, -0.0145361667774132)

(-54.95657838584767, -0.0106457214931494)

(-1160.8176180617456, -0.000433719787114089)

(58.10433764487731, 0.0075629348474873)

(-26.64996946225444, 0.0267712388155592)

(-23.497507730191376, -0.0321962944323512)

(20.385137107630992, 0.017603931367641)

(-1.7289273173581698, -0.0787363961574292)

(42.391658873731245, -0.00992032389208524)


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=29.8186944197622x_{1} = 29.8186944197622
x2=98.9508187684919x_{2} = 98.9508187684919
x3=23.5302399446763x_{3} = 23.5302399446763
x4=48.6770441334903x_{4} = 48.6770441334903
x5=17.2391593368591x_{5} = 17.2391593368591
x6=−86.3810404785128x_{6} = -86.3810404785128
x7=−80.0967433038804x_{7} = -80.0967433038804
x8=92.6670498706103x_{8} = 92.6670498706103
x9=−98.9491738790285x_{9} = -98.9491738790285
x10=36.1056465662001x_{10} = 36.1056465662001
x11=67.5310032292546x_{11} = 67.5310032292546
x12=−92.665172598047x_{12} = -92.665172598047
x13=4.63339841386275x_{13} = 4.63339841386275
x14=61.2466166099857x_{14} = 61.2466166099857
x15=−36.0927341468129x_{15} = -36.0927341468129
x16=−17.1701392036497x_{16} = -17.1701392036497
x17=−61.242276886312x_{17} = -61.242276886312
x18=54.9619901747878x_{18} = 54.9619901747878
x19=−73.8122337852619x_{19} = -73.8122337852619
x20=−10.6325497313325x_{20} = -10.6325497313325
x21=73.8152053010897x_{21} = 73.8152053010897
x22=80.0992623211471x_{22} = 80.0992623211471
x23=−29.7992892908393x_{23} = -29.7992892908393
x24=86.3832032646061x_{24} = 86.3832032646061
x25=10.9428328303325x_{25} = 10.9428328303325
x26=−67.5274446580851x_{26} = -67.5274446580851
x27=−48.6701029977084x_{27} = -48.6701029977084
x28=−42.3824243930235x_{28} = -42.3824243930235
x29=−54.9565783858477x_{29} = -54.9565783858477
x30=−1160.81761806175x_{30} = -1160.81761806175
x31=−23.4975077301914x_{31} = -23.4975077301914
x32=−1.72892731735817x_{32} = -1.72892731735817
x33=42.3916588737312x_{33} = 42.3916588737312
Puntos máximos de la función:
x33=83.2412458033685x_{33} = 83.2412458033685
x33=−89.5231247843125x_{33} = -89.5231247843125
x33=−39.2379067101878x_{33} = -39.2379067101878
x33=−20.3394883287925x_{33} = -20.3394883287925
x33=45.5344160768147x_{33} = 45.5344160768147
x33=89.5251372199914x_{33} = 89.5251372199914
x33=7.79073782639594x_{33} = 7.79073782639594
x33=296.877225767144x_{33} = 296.877225767144
x33=70.6731245692948x_{33} = 70.6731245692948
x33=−64.3849160180317x_{33} = -64.3849160180317
x33=51.8195634020601x_{33} = 51.8195634020601
x33=−32.9466587840552x_{33} = -32.9466587840552
x33=39.2487467466437x_{33} = 39.2487467466437
x33=−95.8071878375154x_{33} = -95.8071878375154
x33=−5.03798143596818x_{33} = -5.03798143596818
x33=26.6747060811334x_{33} = 26.6747060811334
x33=−76.9545187167077x_{33} = -76.9545187167077
x33=76.9572499306778x_{33} = 76.9572499306778
x33=−45.5264519123192x_{33} = -45.5264519123192
x33=95.8089431510378x_{33} = 95.8089431510378
x33=−51.813458709347x_{33} = -51.813458709347
x33=−13.9712372494303x_{33} = -13.9712372494303
x33=14.0919324259525x_{33} = 14.0919324259525
x33=−114.658756435446x_{33} = -114.658756435446
x33=−70.6698794310721x_{33} = -70.6698794310721
x33=1.46554038881958x_{33} = 1.46554038881958
x33=202.627978484871x_{33} = 202.627978484871
x33=32.9623150280616x_{33} = 32.9623150280616
x33=−58.0995064651176x_{33} = -58.0995064651176
x33=−83.2389151082586x_{33} = -83.2389151082586
x33=64.3888359923923x_{33} = 64.3888359923923
x33=58.1043376448773x_{33} = 58.1043376448773
x33=−26.6499694622544x_{33} = -26.6499694622544
x33=20.385137107631x_{33} = 20.385137107631
Decrece en los intervalos
[98.9508187684919,∞)\left[98.9508187684919, \infty\right)
Crece en los intervalos
(−∞,−1160.81761806175]\left(-\infty, -1160.81761806175\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
−sin⁡(x)2−cos⁡(x)x+8+sin⁡(x)(x+8)2x+8=0\frac{- \frac{\sin{\left(x \right)}}{2} - \frac{\cos{\left(x \right)}}{x + 8} + \frac{\sin{\left(x \right)}}{\left(x + 8\right)^{2}}}{x + 8} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=−8x_{1} = -8
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)2x+16)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{2 x + 16}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)2x+16)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{2 x + 16}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/(2*x + 16), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)x(2x+16))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(2 x + 16\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)x(2x+16))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(2 x + 16\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:
sin⁡(x)2x+16=−sin⁡(x)16−2x\frac{\sin{\left(x \right)}}{2 x + 16} = - \frac{\sin{\left(x \right)}}{16 - 2 x}
- No
sin⁡(x)2x+16=sin⁡(x)16−2x\frac{\sin{\left(x \right)}}{2 x + 16} = \frac{\sin{\left(x \right)}}{16 - 2 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(x)/(2*x+16)
    © Sr Examen – Калькулятор