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Trazado el gráfico de la función/ (absolute(x-7)*sin(x))/(x^3-x^2-42x)

Gráfico de la función y = (absolute(x-7)*sin(x))/(x^3-x^2-42x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       |x - 7|*sin(x)
f(x) = --------------
        3    2       
       x  - x  - 42*x
f(x)=sin⁡(x)∣x−7∣−42x+(x3−x2)f{\left(x \right)} = \frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} 
f = (sin(x)*|x - 7|)/(-42*x + x^3 - x^2)
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−6x_{1} = -6
x2=0x_{2} = 0
x3=7x_{3} = 7
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)∣x−7∣−42x+(x3−x2)=0\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=9.42477796076938x_{10} = 9.42477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=84.8230016469244x_{12} = 84.8230016469244
x13=2214.8228207808x_{13} = 2214.8228207808
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=6.28318530717959x_{16} = 6.28318530717959
x17=3.14159265358979x_{17} = 3.14159265358979
x18=−50.2654824574367x_{18} = -50.2654824574367
x19=28.2743338823081x_{19} = 28.2743338823081
x20=−75.398223686155x_{20} = -75.398223686155
x21=−28.2743338823081x_{21} = -28.2743338823081
x22=−56.5486677646163x_{22} = -56.5486677646163
x23=−65.9734457253857x_{23} = -65.9734457253857
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=1316.32732185513x_{25} = 1316.32732185513
x26=−91.106186954104x_{26} = -91.106186954104
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−69.1150383789755x_{28} = -69.1150383789755
x29=−100.530964914873x_{29} = -100.530964914873
x30=56.5486677646163x_{30} = 56.5486677646163
x31=−62.8318530717959x_{31} = -62.8318530717959
x32=−191.637151868977x_{32} = -191.637151868977
x33=40.8407044966673x_{33} = 40.8407044966673
x34=100.530964914873x_{34} = 100.530964914873
x35=−87.9645943005142x_{35} = -87.9645943005142
x36=18.8495559215388x_{36} = 18.8495559215388
x37=62.8318530717959x_{37} = 62.8318530717959
x38=−53.4070751110265x_{38} = -53.4070751110265
x39=94.2477796076938x_{39} = 94.2477796076938
x40=−3.14159265358979x_{40} = -3.14159265358979
x41=21.9911485751286x_{41} = 21.9911485751286
x42=12.5663706143592x_{42} = 12.5663706143592
x43=−84.8230016469244x_{43} = -84.8230016469244
x44=34.5575191894877x_{44} = 34.5575191894877
x45=47.1238898038469x_{45} = 47.1238898038469
x46=−15.707963267949x_{46} = -15.707963267949
x47=53.4070751110265x_{47} = 53.4070751110265
x48=65.9734457253857x_{48} = 65.9734457253857
x49=87.9645943005142x_{49} = 87.9645943005142
x50=91.106186954104x_{50} = 91.106186954104
x51=59.6902604182061x_{51} = 59.6902604182061
x52=69.1150383789755x_{52} = 69.1150383789755
x53=75.398223686155x_{53} = 75.398223686155
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=−12.5663706143592x_{55} = -12.5663706143592
x56=−18.8495559215388x_{56} = -18.8495559215388
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=78.5398163397448x_{59} = 78.5398163397448
x60=15.707963267949x_{60} = 15.707963267949
x61=72.2566310325652x_{61} = 72.2566310325652
x62=37.6991118430775x_{62} = 37.6991118430775
x63=25.1327412287183x_{63} = 25.1327412287183
x64=−47.1238898038469x_{64} = -47.1238898038469
x65=−1671.32729170977x_{65} = -1671.32729170977
x66=−9.42477796076938x_{66} = -9.42477796076938
x67=−34.5575191894877x_{67} = -34.5575191894877
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 - 7|*sin(x))/(x^3 - x^2 - 42*x).
sin⁡(0)∣−7∣(03−02)−0\frac{\sin{\left(0 \right)} \left|{-7}\right|}{\left(0^{3} - 0^{2}\right) - 0}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
sin⁡(x)sign⁡(x−7)+cos⁡(x)∣x−7∣−42x+(x3−x2)+(−3x2+2x+42)sin⁡(x)∣x−7∣(−42x+(x3−x2))2=0\frac{\sin{\left(x \right)} \operatorname{sign}{\left(x - 7 \right)} + \cos{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} + \frac{\left(- 3 x^{2} + 2 x + 42\right) \sin{\left(x \right)} \left|{x - 7}\right|}{\left(- 42 x + \left(x^{3} - x^{2}\right)\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=14.016453093047x_{1} = 14.016453093047
x2=26.6354574736327x_{2} = 26.6354574736327
x3=−42.3604359016656x_{3} = -42.3604359016656
x4=−61.2266303830853x_{4} = -61.2266303830853
x5=54.9432759558649x_{5} = 54.9432759558649
x6=−76.9419337897232x_{6} = -76.9419337897232
x7=83.2289873280938x_{7} = 83.2289873280938
x8=64.3729136691542x_{8} = 64.3729136691542
x9=−64.369993732511x_{9} = -64.369993732511
x10=−13.9420438831868x_{10} = -13.9420438831868
x11=70.6586440153259x_{11} = 70.6586440153259
x12=29.7836865268305x_{12} = 29.7836865268305
x13=36.0768763247778x_{13} = 36.0768763247778
x14=−32.919299828625x_{14} = -32.919299828625
x15=−51.7951587828659x_{15} = -51.7951587828659
x16=80.0865146143929x_{16} = 80.0865146143929
x17=−95.7970043123775x_{17} = -95.7970043123775
x18=−48.6507135445059x_{18} = -48.6507135445059
x19=42.3672514439482x_{19} = 42.3672514439482
x20=98.9405348240556x_{20} = 98.9405348240556
x21=−26.6176780767465x_{21} = -26.6176780767465
x22=−89.512248812553x_{22} = -89.512248812553
x23=−54.939255228509x_{23} = -54.939255228509
x24=−73.7991351656219x_{24} = -73.7991351656219
x25=−67.5131834431642x_{25} = -67.5131834431642
x26=10.8451478646932x_{26} = 10.8451478646932
x27=17.1777448915114x_{27} = 17.1777448915114
x28=95.7983167647195x_{28} = 95.7983167647195
x29=23.485599433969x_{29} = 23.485599433969
x30=−17.1316249007671x_{30} = -17.1316249007671
x31=−10.6983350907497x_{31} = -10.6983350907497
x32=−39.2143570561418x_{32} = -39.2143570561418
x33=76.9439724516011x_{33} = 76.9439724516011
x34=149.212507171659x_{34} = 149.212507171659
x35=−58.0830633547885x_{35} = -58.0830633547885
x36=73.8013522673499x_{36} = 73.8013522673499
x37=48.6558568457253x_{37} = 48.6558568457253
x38=−86.3697819811265x_{38} = -86.3697819811265
x39=89.5137528406582x_{39} = 89.5137528406582
x40=92.6560575082574x_{40} = 92.6560575082574
x41=146.07063730353x_{41} = 146.07063730353
x42=7.65282388683543x_{42} = 7.65282388683543
x43=−20.3017332658149x_{43} = -20.3017332658149
x44=−92.6546541656147x_{44} = -92.6546541656147
x45=−23.4623877903398x_{45} = -23.4623877903398
x46=67.5158358436257x_{46} = 67.5158358436257
x47=51.799688855665x_{47} = 51.799688855665
x48=58.0866563372379x_{48} = 58.0866563372379
x49=−98.9393046989388x_{49} = -98.9393046989388
x50=−7.00360982839007x_{50} = -7.00360982839007
x51=−80.0846336468534x_{51} = -80.0846336468534
x52=20.3334172344451x_{52} = 20.3334172344451
x53=−70.6562239130531x_{53} = -70.6562239130531
x54=86.3713979521823x_{54} = 86.3713979521823
x55=−45.5058407635821x_{55} = -45.5058407635821
x56=−83.2272464103056x_{56} = -83.2272464103056
x57=−36.0674064816266x_{57} = -36.0674064816266
x58=61.229860625401x_{58} = 61.229860625401
x59=4.39955639048156x_{59} = 4.39955639048156
x60=39.2223354468251x_{60} = 39.2223354468251
x61=−0.53859704600586x_{61} = -0.53859704600586
x62=45.5117316703557x_{62} = 45.5117316703557
x63=−29.7696122729537x_{63} = -29.7696122729537
x64=32.9307280335231x_{64} = 32.9307280335231
Signos de extremos en los puntos:
(14.016453093047046, 0.00353836638676052)

(26.635457473632727, 0.00114773837094551)

(-42.36043590166557, -0.000648401503723408)

(-61.2266303830853, -0.000295565464505356)

(54.943275955864934, -0.000298469329531356)

(-76.94193378972315, 0.000183136347207666)

(83.22898732809384, 0.000134617758893156)

(64.3729136691542, 0.000220647630392954)

(-64.369993732511, 0.00026600835720957)

(-13.942043883186841, 0.00885973686894448)

(70.65864401532592, 0.000184549555812555)

(29.783686526830525, -0.000936518043257309)

(36.07687632477782, -0.000657889201238731)

(-32.919299828625, 0.00112589471326007)

(-51.79515878286592, 0.000421234558721923)

(80.08651461439288, -0.000145003794971144)

(-95.79700431237748, 0.000116221150161414)

(-48.65071354450594, -0.000481464727267868)

(42.36725144394817, -0.000487520616923897)

(98.94053482405558, -9.62938965440315e-5)

(-26.61767807674652, 0.00181546269768391)

(-89.51224881255295, 0.000133736838158639)

(-54.939255228508955, -0.000371651579986343)

(-73.79913516562186, -0.000199779414092476)

(-67.51318344316424, -0.000240676511514366)

(10.845147864693171, -0.00541199541314581)

(17.17774489151138, -0.00249886693392249)

(95.79831676471952, 0.000102520896200968)

(23.485599433968986, -0.00143986404452334)

(-17.131624900767097, -0.00518710311846635)

(-10.698335090749685, -0.0190223979798966)

(-39.21435705614177, 0.000766581979435786)

(76.94397245160107, 0.000156640592186579)

(149.21250717165935, -4.31748206644313e-5)

(-58.08306335478854, 0.00033034381495904)

(73.80135226734993, -0.000169737488684419)

(48.655856845725324, -0.000375751471215424)

(-86.36978198112651, -0.000144019112679707)

(89.51375284065819, 0.000116934496654864)

(92.65605750825736, -0.000109372297110004)

(146.07063730352974, 4.50145170384153e-5)

(7.652823886835433, 0.00937797661890569)

(-20.3017332658149, 0.00341991769457062)

(-92.65465416561467, -0.000124518130309472)

(-23.46238779033976, -0.00242866893320253)

(67.51583584362565, -0.000201390142707435)

(51.799688855664975, 0.000333777119038759)

(58.08665633723788, 0.000268486368595447)

(-98.93930469893877, -0.000108726947503291)

(-7.003609828390072, 0.0938560163001481)

(-80.08463364685336, -0.000168490763387854)

(20.33341723444505, 0.00186054082102996)

(-70.65622391305313, 0.000218800755314414)

(86.37139795218235, -0.000125309376682272)

(-45.50584076358215, 0.000555631062996273)

(-83.22724641030557, 0.000155535180840878)

(-36.06740648162657, -0.00092041358758118)

(61.22986062540101, -0.000242808092240104)

(4.399556390481556, 0.0207955020018686)

(39.22233544682508, 0.000563147211410448)

(-0.5385970460058597, -0.174378019814268)

(45.511731670355665, 0.000426185714857683)

(-29.76961227295372, -0.00140917575807083)

(32.930728033523096, 0.000778798198554037)


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=−42.3604359016656x_{1} = -42.3604359016656
x2=−61.2266303830853x_{2} = -61.2266303830853
x3=54.9432759558649x_{3} = 54.9432759558649
x4=29.7836865268305x_{4} = 29.7836865268305
x5=36.0768763247778x_{5} = 36.0768763247778
x6=80.0865146143929x_{6} = 80.0865146143929
x7=−48.6507135445059x_{7} = -48.6507135445059
x8=42.3672514439482x_{8} = 42.3672514439482
x9=98.9405348240556x_{9} = 98.9405348240556
x10=−54.939255228509x_{10} = -54.939255228509
x11=−73.7991351656219x_{11} = -73.7991351656219
x12=−67.5131834431642x_{12} = -67.5131834431642
x13=10.8451478646932x_{13} = 10.8451478646932
x14=17.1777448915114x_{14} = 17.1777448915114
x15=23.485599433969x_{15} = 23.485599433969
x16=−17.1316249007671x_{16} = -17.1316249007671
x17=−10.6983350907497x_{17} = -10.6983350907497
x18=149.212507171659x_{18} = 149.212507171659
x19=73.8013522673499x_{19} = 73.8013522673499
x20=48.6558568457253x_{20} = 48.6558568457253
x21=−86.3697819811265x_{21} = -86.3697819811265
x22=92.6560575082574x_{22} = 92.6560575082574
x23=−92.6546541656147x_{23} = -92.6546541656147
x24=−23.4623877903398x_{24} = -23.4623877903398
x25=67.5158358436257x_{25} = 67.5158358436257
x26=−98.9393046989388x_{26} = -98.9393046989388
x27=−80.0846336468534x_{27} = -80.0846336468534
x28=86.3713979521823x_{28} = 86.3713979521823
x29=−36.0674064816266x_{29} = -36.0674064816266
x30=61.229860625401x_{30} = 61.229860625401
x31=−0.53859704600586x_{31} = -0.53859704600586
x32=−29.7696122729537x_{32} = -29.7696122729537
Puntos máximos de la función:
x32=14.016453093047x_{32} = 14.016453093047
x32=26.6354574736327x_{32} = 26.6354574736327
x32=−76.9419337897232x_{32} = -76.9419337897232
x32=83.2289873280938x_{32} = 83.2289873280938
x32=64.3729136691542x_{32} = 64.3729136691542
x32=−64.369993732511x_{32} = -64.369993732511
x32=−13.9420438831868x_{32} = -13.9420438831868
x32=70.6586440153259x_{32} = 70.6586440153259
x32=−32.919299828625x_{32} = -32.919299828625
x32=−51.7951587828659x_{32} = -51.7951587828659
x32=−95.7970043123775x_{32} = -95.7970043123775
x32=−26.6176780767465x_{32} = -26.6176780767465
x32=−89.512248812553x_{32} = -89.512248812553
x32=95.7983167647195x_{32} = 95.7983167647195
x32=−39.2143570561418x_{32} = -39.2143570561418
x32=76.9439724516011x_{32} = 76.9439724516011
x32=−58.0830633547885x_{32} = -58.0830633547885
x32=89.5137528406582x_{32} = 89.5137528406582
x32=146.07063730353x_{32} = 146.07063730353
x32=7.65282388683543x_{32} = 7.65282388683543
x32=−20.3017332658149x_{32} = -20.3017332658149
x32=51.799688855665x_{32} = 51.799688855665
x32=58.0866563372379x_{32} = 58.0866563372379
x32=−7.00360982839007x_{32} = -7.00360982839007
x32=20.3334172344451x_{32} = 20.3334172344451
x32=−70.6562239130531x_{32} = -70.6562239130531
x32=−45.5058407635821x_{32} = -45.5058407635821
x32=−83.2272464103056x_{32} = -83.2272464103056
x32=4.39955639048156x_{32} = 4.39955639048156
x32=39.2223354468251x_{32} = 39.2223354468251
x32=45.5117316703557x_{32} = 45.5117316703557
x32=32.9307280335231x_{32} = 32.9307280335231
Decrece en los intervalos
[149.212507171659,∞)\left[149.212507171659, \infty\right)
Crece en los intervalos
(−∞,−98.9393046989388]\left(-\infty, -98.9393046989388\right]
Asíntotas verticales
Hay:
x1=−6x_{1} = -6
x2=0x_{2} = 0
x3=7x_{3} = 7
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)∣x−7∣−42x+(x3−x2))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)∣x−7∣−42x+(x3−x2))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)}\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 (|x - 7|*sin(x))/(x^3 - x^2 - 42*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)∣x−7∣x(−42x+(x3−x2)))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{x \left(- 42 x + \left(x^{3} - x^{2}\right)\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)∣x−7∣x(−42x+(x3−x2)))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{x \left(- 42 x + \left(x^{3} - x^{2}\right)\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)∣x−7∣−42x+(x3−x2)=−sin⁡(x)∣x+7∣−x3−x2+42x\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} = - \frac{\sin{\left(x \right)} \left|{x + 7}\right|}{- x^{3} - x^{2} + 42 x}
- No
sin⁡(x)∣x−7∣−42x+(x3−x2)=sin⁡(x)∣x+7∣−x3−x2+42x\frac{\sin{\left(x \right)} \left|{x - 7}\right|}{- 42 x + \left(x^{3} - x^{2}\right)} = \frac{\sin{\left(x \right)} \left|{x + 7}\right|}{- x^{3} - x^{2} + 42 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор