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Trazado el gráfico de la función/ 10*cos(x)+5*x*sin(x)

Gráfico de la función y = 10*cos(x)+5*x*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = 10*cos(x) + 5*x*sin(x)
f(x)=5xsin⁡(x)+10cos⁡(x)f{\left(x \right)} = 5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)} 
f = (5*x)*sin(x) + 10*cos(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:
5xsin⁡(x)+10cos⁡(x)=05 x \sin{\left(x \right)} + 10 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=34.4996123350132x_{1} = 34.4996123350132
x2=−5.95939190757933x_{2} = -5.95939190757933
x3=9.21096438740149x_{3} = 9.21096438740149
x4=−31.3522215217643x_{4} = -31.3522215217643
x5=56.5132926241755x_{5} = 56.5132926241755
x6=−56.5132926241755x_{6} = -56.5132926241755
x7=−59.6567478435559x_{7} = -59.6567478435559
x8=94.2265573558031x_{8} = 94.2265573558031
x9=−50.2256832197934x_{9} = -50.2256832197934
x10=−47.0814357397523x_{10} = -47.0814357397523
x11=91.0842327848165x_{11} = 91.0842327848165
x12=−25.053079662454x_{12} = -25.053079662454
x13=−53.3696181339615x_{13} = -53.3696181339615
x14=−37.6460352959305x_{14} = -37.6460352959305
x15=65.9431258539286x_{15} = 65.9431258539286
x16=−18.7432530945386x_{16} = -18.7432530945386
x17=53.3696181339615x_{17} = 53.3696181339615
x18=−12.4065403639626x_{18} = -12.4065403639626
x19=84.7994209518635x_{19} = 84.7994209518635
x20=81.6569211705466x_{20} = 81.6569211705466
x21=47.0814357397523x_{21} = 47.0814357397523
x22=−28.2035393053095x_{22} = -28.2035393053095
x23=37.6460352959305x_{23} = 37.6460352959305
x24=−15.5802941824244x_{24} = -15.5802941824244
x25=−91.0842327848165x_{25} = -91.0842327848165
x26=75.3716947511882x_{26} = 75.3716947511882
x27=25.053079662454x_{27} = 25.053079662454
x28=−78.5143487963623x_{28} = -78.5143487963623
x29=40.7917141624847x_{29} = 40.7917141624847
x30=−97.3688346960149x_{30} = -97.3688346960149
x31=−9.21096438740149x_{31} = -9.21096438740149
x32=−72.2289483771681x_{32} = -72.2289483771681
x33=78.5143487963623x_{33} = 78.5143487963623
x34=−62.8000167068325x_{34} = -62.8000167068325
x35=28.2035393053095x_{35} = 28.2035393053095
x36=50.2256832197934x_{36} = 50.2256832197934
x37=72.2289483771681x_{37} = 72.2289483771681
x38=−34.4996123350132x_{38} = -34.4996123350132
x39=−81.6569211705466x_{39} = -81.6569211705466
x40=18.7432530945386x_{40} = 18.7432530945386
x41=12.4065403639626x_{41} = 12.4065403639626
x42=43.9368086315937x_{42} = 43.9368086315937
x43=−87.9418559209576x_{43} = -87.9418559209576
x44=100.511069234565x_{44} = 100.511069234565
x45=87.9418559209576x_{45} = 87.9418559209576
x46=21.9000773156394x_{46} = 21.9000773156394
x47=−21.9000773156394x_{47} = -21.9000773156394
x48=−94.2265573558031x_{48} = -94.2265573558031
x49=15.5802941824244x_{49} = 15.5802941824244
x50=59.6567478435559x_{50} = 59.6567478435559
x51=−100.511069234565x_{51} = -100.511069234565
x52=−75.3716947511882x_{52} = -75.3716947511882
x53=31.3522215217643x_{53} = 31.3522215217643
x54=−65.9431258539286x_{54} = -65.9431258539286
x55=−69.0860970774096x_{55} = -69.0860970774096
x56=69.0860970774096x_{56} = 69.0860970774096
x57=−43.9368086315937x_{57} = -43.9368086315937
x58=97.3688346960149x_{58} = 97.3688346960149
x59=−40.7917141624847x_{59} = -40.7917141624847
x60=5.95939190757933x_{60} = 5.95939190757933
x61=62.8000167068325x_{61} = 62.8000167068325
x62=−84.7994209518635x_{62} = -84.7994209518635
x63=2.45871417599962x_{63} = 2.45871417599962
x64=−2.45871417599962x_{64} = -2.45871417599962
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 10*cos(x) + (5*x)*sin(x).
0⋅5sin⁡(0)+10cos⁡(0)0 \cdot 5 \sin{\left(0 \right)} + 10 \cos{\left(0 \right)}
Resultado:
f(0)=10f{\left(0 \right)} = 10
Punto:
(0, 10)
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
5xcos⁡(x)−5sin⁡(x)=05 x \cos{\left(x \right)} - 5 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=70.6716857116195x_{1} = 70.6716857116195
x2=−48.6741442319544x_{2} = -48.6741442319544
x3=51.8169824872797x_{3} = 51.8169824872797
x4=48.6741442319544x_{4} = 48.6741442319544
x5=17.2207552719308x_{5} = 17.2207552719308
x6=−36.1006222443756x_{6} = -36.1006222443756
x7=0x_{7} = 0
x8=45.5311340139913x_{8} = 45.5311340139913
x9=98.9500628243319x_{9} = 98.9500628243319
x10=−89.5242209304172x_{10} = -89.5242209304172
x11=−4.49340945790906x_{11} = -4.49340945790906
x12=−64.3871195905574x_{12} = -64.3871195905574
x13=29.811598790893x_{13} = 29.811598790893
x14=61.2447302603744x_{14} = 61.2447302603744
x15=39.2444323611642x_{15} = 39.2444323611642
x16=80.0981286289451x_{16} = 80.0981286289451
x17=−7.72525183693771x_{17} = -7.72525183693771
x18=−20.3713029592876x_{18} = -20.3713029592876
x19=−86.3822220347287x_{19} = -86.3822220347287
x20=−32.9563890398225x_{20} = -32.9563890398225
x21=−10.9041216594289x_{21} = -10.9041216594289
x22=−70.6716857116195x_{22} = -70.6716857116195
x23=−61.2447302603744x_{23} = -61.2447302603744
x24=92.6661922776228x_{24} = 92.6661922776228
x25=−92.6661922776228x_{25} = -92.6661922776228
x26=67.5294347771441x_{26} = 67.5294347771441
x27=−14.0661939128315x_{27} = -14.0661939128315
x28=54.9596782878889x_{28} = 54.9596782878889
x29=−45.5311340139913x_{29} = -45.5311340139913
x30=−39.2444323611642x_{30} = -39.2444323611642
x31=89.5242209304172x_{31} = 89.5242209304172
x32=86.3822220347287x_{32} = 86.3822220347287
x33=4.49340945790906x_{33} = 4.49340945790906
x34=−42.3879135681319x_{34} = -42.3879135681319
x35=58.1022547544956x_{35} = 58.1022547544956
x36=−23.519452498689x_{36} = -23.519452498689
x37=36.1006222443756x_{37} = 36.1006222443756
x38=−80.0981286289451x_{38} = -80.0981286289451
x39=7.72525183693771x_{39} = 7.72525183693771
x40=−76.9560263103312x_{40} = -76.9560263103312
x41=73.8138806006806x_{41} = 73.8138806006806
x42=−98.9500628243319x_{42} = -98.9500628243319
x43=95.8081387868617x_{43} = 95.8081387868617
x44=26.6660542588127x_{44} = 26.6660542588127
x45=−17.2207552719308x_{45} = -17.2207552719308
x46=−26.6660542588127x_{46} = -26.6660542588127
x47=20.3713029592876x_{47} = 20.3713029592876
x48=102.091966464908x_{48} = 102.091966464908
x49=−54.9596782878889x_{49} = -54.9596782878889
x50=−73.8138806006806x_{50} = -73.8138806006806
x51=−58.1022547544956x_{51} = -58.1022547544956
x52=23.519452498689x_{52} = 23.519452498689
x53=83.2401924707234x_{53} = 83.2401924707234
x54=−95.8081387868617x_{54} = -95.8081387868617
x55=−29.811598790893x_{55} = -29.811598790893
x56=−51.8169824872797x_{56} = -51.8169824872797
x57=10.9041216594289x_{57} = 10.9041216594289
x58=14.0661939128315x_{58} = 14.0661939128315
x59=32.9563890398225x_{59} = 32.9563890398225
x60=64.3871195905574x_{60} = 64.3871195905574
x61=−67.5294347771441x_{61} = -67.5294347771441
x62=76.9560263103312x_{62} = 76.9560263103312
x63=−83.2401924707234x_{63} = -83.2401924707234
x64=42.3879135681319x_{64} = 42.3879135681319
Signos de extremos en los puntos:
(70.6716857116195, 353.464544243924)

(-48.674144231954386, -243.524779982621)

(51.81698248727967, 259.22963017313)

(48.674144231954386, -243.524779982621)

(17.22075527193077, -86.5386868495239)

(-36.10062224437561, -180.710797488168)

(0, 10)

(45.53113401399128, 227.820359424951)

(98.95006282433188, -494.826106704724)

(-89.52422093041719, 447.704876507814)

(-4.493409457909064, -24.1028623848146)

(-64.38711959055742, 322.052069172218)

(29.81159879089296, -149.309456030408)

(61.2447302603744, -306.346097216712)

(39.24443236116419, 196.41322003791)

(80.09812862894512, -400.584272210672)

(-7.725251836937707, 39.5904016050931)

(-20.37130295928756, 102.224310729287)

(-86.38222203472871, -431.997928746584)

(-32.956389039822476, 165.009431427628)

(-10.904121659428899, -55.2060253632466)

(-70.6716857116195, 353.464544243924)

(-61.2447302603744, -306.346097216712)

(92.66619227762284, -463.41189312734)

(-92.66619227762284, -463.41189312734)

(67.52943477714412, -337.758226419429)

(-14.066193912831473, 70.8630439495984)

(54.959678287888934, -274.93483630046)

(-45.53113401399128, 227.820359424951)

(-39.24443236116419, 196.41322003791)

(89.52422093041719, 447.704876507814)

(86.38222203472871, -431.997928746584)

(4.493409457909064, -24.1028623848146)

(-42.38791356813192, -212.116464054235)

(58.10225475449559, 290.640340613977)

(-23.519452498689006, -117.91590757357)

(36.10062224437561, -180.710797488168)

(-80.09812862894512, -400.584272210672)

(7.725251836937707, 39.5904016050931)

(-76.95602631033118, 384.877582950189)

(73.81388060068065, -369.17100213976)

(-98.95006282433188, -494.826106704724)

(95.8081387868617, 479.118971830918)

(26.666054258812675, 133.61136310898)

(-17.22075527193077, -86.5386868495239)

(-26.666054258812675, 133.61136310898)

(20.37130295928756, 102.224310729287)

(102.09196646490764, 510.533292562944)

(-54.959678287888934, -274.93483630046)

(-73.81388060068065, -369.17100213976)

(-58.10225475449559, 290.640340613977)

(23.519452498689006, -117.91590757357)

(83.2401924707234, 416.291057640705)

(-95.8081387868617, 479.118971830918)

(-29.81159879089296, -149.309456030408)

(-51.81698248727967, 259.22963017313)

(10.904121659428899, -55.2060253632466)

(14.066193912831473, 70.8630439495984)

(32.956389039822476, 165.009431427628)

(64.38711959055742, 322.052069172218)

(-67.52943477714412, -337.758226419429)

(76.95602631033118, 384.877582950189)

(-83.2401924707234, 416.291057640705)

(42.38791356813192, -212.116464054235)


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=−48.6741442319544x_{1} = -48.6741442319544
x2=48.6741442319544x_{2} = 48.6741442319544
x3=17.2207552719308x_{3} = 17.2207552719308
x4=−36.1006222443756x_{4} = -36.1006222443756
x5=98.9500628243319x_{5} = 98.9500628243319
x6=−4.49340945790906x_{6} = -4.49340945790906
x7=29.811598790893x_{7} = 29.811598790893
x8=61.2447302603744x_{8} = 61.2447302603744
x9=80.0981286289451x_{9} = 80.0981286289451
x10=−86.3822220347287x_{10} = -86.3822220347287
x11=−10.9041216594289x_{11} = -10.9041216594289
x12=−61.2447302603744x_{12} = -61.2447302603744
x13=92.6661922776228x_{13} = 92.6661922776228
x14=−92.6661922776228x_{14} = -92.6661922776228
x15=67.5294347771441x_{15} = 67.5294347771441
x16=54.9596782878889x_{16} = 54.9596782878889
x17=86.3822220347287x_{17} = 86.3822220347287
x18=4.49340945790906x_{18} = 4.49340945790906
x19=−42.3879135681319x_{19} = -42.3879135681319
x20=−23.519452498689x_{20} = -23.519452498689
x21=36.1006222443756x_{21} = 36.1006222443756
x22=−80.0981286289451x_{22} = -80.0981286289451
x23=73.8138806006806x_{23} = 73.8138806006806
x24=−98.9500628243319x_{24} = -98.9500628243319
x25=−17.2207552719308x_{25} = -17.2207552719308
x26=−54.9596782878889x_{26} = -54.9596782878889
x27=−73.8138806006806x_{27} = -73.8138806006806
x28=23.519452498689x_{28} = 23.519452498689
x29=−29.811598790893x_{29} = -29.811598790893
x30=10.9041216594289x_{30} = 10.9041216594289
x31=−67.5294347771441x_{31} = -67.5294347771441
x32=42.3879135681319x_{32} = 42.3879135681319
Puntos máximos de la función:
x32=70.6716857116195x_{32} = 70.6716857116195
x32=51.8169824872797x_{32} = 51.8169824872797
x32=45.5311340139913x_{32} = 45.5311340139913
x32=−89.5242209304172x_{32} = -89.5242209304172
x32=−64.3871195905574x_{32} = -64.3871195905574
x32=39.2444323611642x_{32} = 39.2444323611642
x32=−7.72525183693771x_{32} = -7.72525183693771
x32=−20.3713029592876x_{32} = -20.3713029592876
x32=−32.9563890398225x_{32} = -32.9563890398225
x32=−70.6716857116195x_{32} = -70.6716857116195
x32=−14.0661939128315x_{32} = -14.0661939128315
x32=−45.5311340139913x_{32} = -45.5311340139913
x32=−39.2444323611642x_{32} = -39.2444323611642
x32=89.5242209304172x_{32} = 89.5242209304172
x32=58.1022547544956x_{32} = 58.1022547544956
x32=7.72525183693771x_{32} = 7.72525183693771
x32=−76.9560263103312x_{32} = -76.9560263103312
x32=95.8081387868617x_{32} = 95.8081387868617
x32=26.6660542588127x_{32} = 26.6660542588127
x32=−26.6660542588127x_{32} = -26.6660542588127
x32=20.3713029592876x_{32} = 20.3713029592876
x32=102.091966464908x_{32} = 102.091966464908
x32=−58.1022547544956x_{32} = -58.1022547544956
x32=83.2401924707234x_{32} = 83.2401924707234
x32=−95.8081387868617x_{32} = -95.8081387868617
x32=−51.8169824872797x_{32} = -51.8169824872797
x32=14.0661939128315x_{32} = 14.0661939128315
x32=32.9563890398225x_{32} = 32.9563890398225
x32=64.3871195905574x_{32} = 64.3871195905574
x32=76.9560263103312x_{32} = 76.9560263103312
x32=−83.2401924707234x_{32} = -83.2401924707234
Decrece en los intervalos
[98.9500628243319,∞)\left[98.9500628243319, \infty\right)
Crece en los intervalos
(−∞,−98.9500628243319]\left(-\infty, -98.9500628243319\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
−5xsin⁡(x)=0- 5 x \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=πx_{2} = \pi

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
[π,∞)\left[\pi, \infty\right)
Convexa en los intervalos
(−∞,π]\left(-\infty, \pi\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→−∞(5xsin⁡(x)+10cos⁡(x))y = \lim_{x \to -\infty}\left(5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(5xsin⁡(x)+10cos⁡(x))y = \lim_{x \to \infty}\left(5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 10*cos(x) + (5*x)*sin(x), 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→−∞(5xsin⁡(x)+10cos⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(5xsin⁡(x)+10cos⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)}}{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:
5xsin⁡(x)+10cos⁡(x)=5xsin⁡(x)+10cos⁡(x)5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)} = 5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)}
- No
5xsin⁡(x)+10cos⁡(x)=−5xsin⁡(x)−10cos⁡(x)5 x \sin{\left(x \right)} + 10 \cos{\left(x \right)} = - 5 x \sin{\left(x \right)} - 10 \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор