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(3*e^x+x)*cos(x)
Trazado el gráfico de la función/ (3*e^x+x)*cos(x)

Gráfico de la función y = (3*e^x+x)*cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /   x    \       
f(x) = \3*E  + x/*cos(x)
f(x)=(3ex+x)cos⁡(x)f{\left(x \right)} = \left(3 e^{x} + x\right) \cos{\left(x \right)} 
f = (3*E^x + x)*cos(x)
Gráfico de la función
02468-8-6-4-2-1010-10000050000
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:
(3ex+x)cos⁡(x)=0\left(3 e^{x} + x\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=−W(3)x_{3} = - W\left(3\right)
Solución numérica
x1=4.71238898038469x_{1} = 4.71238898038469
x2=−1.04990889496404x_{2} = -1.04990889496404
x3=−10.9955742875643x_{3} = -10.9955742875643
x4=26.7035375555132x_{4} = 26.7035375555132
x5=−23.5619449019235x_{5} = -23.5619449019235
x6=−26.7035375555132x_{6} = -26.7035375555132
x7=−89.5353906273091x_{7} = -89.5353906273091
x8=−17.2787595947439x_{8} = -17.2787595947439
x9=−42.4115008234622x_{9} = -42.4115008234622
x10=−61.261056745001x_{10} = -61.261056745001
x11=−76.9690200129499x_{11} = -76.9690200129499
x12=−92.6769832808989x_{12} = -92.6769832808989
x13=−98.9601685880785x_{13} = -98.9601685880785
x14=−54.9778714378214x_{14} = -54.9778714378214
x15=−64.4026493985908x_{15} = -64.4026493985908
x16=−7.85398163397448x_{16} = -7.85398163397448
x17=−14.1371669411541x_{17} = -14.1371669411541
x18=14.1371669411541x_{18} = 14.1371669411541
x19=−1.5707963267949x_{19} = -1.5707963267949
x20=1.5707963267949x_{20} = 1.5707963267949
x21=29.845130209103x_{21} = 29.845130209103
x22=10.9955742875643x_{22} = 10.9955742875643
x23=17.2787595947439x_{23} = 17.2787595947439
x24=−51.8362787842316x_{24} = -51.8362787842316
x25=−29.845130209103x_{25} = -29.845130209103
x26=−48.6946861306418x_{26} = -48.6946861306418
x27=−73.8274273593601x_{27} = -73.8274273593601
x28=23.5619449019235x_{28} = 23.5619449019235
x29=20.4203522483337x_{29} = 20.4203522483337
x30=−86.3937979737193x_{30} = -86.3937979737193
x31=−67.5442420521806x_{31} = -67.5442420521806
x32=−4.71238898038469x_{32} = -4.71238898038469
x33=−45.553093477052x_{33} = -45.553093477052
x34=−70.6858347057703x_{34} = -70.6858347057703
x35=−83.2522053201295x_{35} = -83.2522053201295
x36=−114.668131856027x_{36} = -114.668131856027
x37=−95.8185759344887x_{37} = -95.8185759344887
x38=−39.2699081698724x_{38} = -39.2699081698724
x39=−32.9867228626928x_{39} = -32.9867228626928
x40=−20.4203522483337x_{40} = -20.4203522483337
x41=−36.1283155162826x_{41} = -36.1283155162826
x42=7.85398163397448x_{42} = 7.85398163397448
x43=−80.1106126665397x_{43} = -80.1106126665397
x44=−58.1194640914112x_{44} = -58.1194640914112
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 (3*E^x + x)*cos(x).
3e0cos⁡(0)3 e^{0} \cos{\left(0 \right)}
Resultado:
f(0)=3f{\left(0 \right)} = 3
Punto:
(0, 3)
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
−(3ex+x)sin⁡(x)+(3ex+1)cos⁡(x)=0- \left(3 e^{x} + x\right) \sin{\left(x \right)} + \left(3 e^{x} + 1\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.6452880386917x_{1} = -12.6452880386917
x2=29.0597320457045x_{2} = 29.0597320457045
x3=−40.8651703304881x_{3} = -40.8651703304881
x4=−22.0364967279767x_{4} = -22.0364967279767
x5=−65.9885986984904x_{5} = -65.9885986984904
x6=−97.3996388790738x_{6} = -97.3996388790738
x7=−53.4257904773947x_{7} = -53.4257904773947
x8=−87.9759605524932x_{8} = -87.9759605524932
x9=−31.4477146375462x_{9} = -31.4477146375462
x10=−72.270467060309x_{10} = -72.270467060309
x11=−78.5525459842429x_{11} = -78.5525459842429
x12=−9.52935914510957x_{12} = -9.52935914510957
x13=25.9181393920928x_{13} = 25.9181393920928
x14=−94.2583883450399x_{14} = -94.2583883450399
x15=−56.5663442798215x_{15} = -56.5663442798215
x16=19.6349540757142x_{16} = 19.6349540757142
x17=7.0677226670045x_{17} = 7.0677226670045
x18=−1.30555891104559x_{18} = -1.30555891104559
x19=22.7765467380603x_{19} = 22.7765467380603
x20=3.91747420743363x_{20} = 3.91747420743363
x21=−34.5864242152889x_{21} = -34.5864242152889
x22=−91.1171613944647x_{22} = -91.1171613944647
x23=−3.45637952124955x_{23} = -3.45637952124955
x24=−59.7070073053355x_{24} = -59.7070073053355
x25=−37.7256128277765x_{25} = -37.7256128277765
x26=16.4933612539199x_{26} = 16.4933612539199
x27=−50.2853663377737x_{27} = -50.2853663377737
x28=0.798698173732013x_{28} = 0.798698173732013
x29=−47.145097736761x_{29} = -47.145097736761
x30=−69.1295029738953x_{30} = -69.1295029738953
x31=−44.0050179208308x_{31} = -44.0050179208308
x32=−84.8347887180423x_{32} = -84.8347887180423
x33=−147.661626855354x_{33} = -147.661626855354
x34=−62.8477631944545x_{34} = -62.8477631944545
x35=−116.247530303932x_{35} = -116.247530303932
x36=−15.7712849032012x_{36} = -15.7712849032012
x37=−75.4114834888481x_{37} = -75.4114834888481
x38=13.3517655045151x_{38} = 13.3517655045151
x39=−18.9024099578865x_{39} = -18.9024099578865
x40=−81.6936492356017x_{40} = -81.6936492356017
x41=10.210119645356x_{41} = 10.210119645356
x42=−100.540910786842x_{42} = -100.540910786842
x43=−28.3096428544521x_{43} = -28.3096428544521
x44=−6.43812009324174x_{44} = -6.43812009324174
x45=−25.1724463266481x_{45} = -25.1724463266481
Signos de extremos en los puntos:
(-12.645288038691731, -12.6059216598502)

(29.059732045704468, -8852940185249.16)

(-40.86517033048807, 40.8529404645174)

(-22.036496727976683, 22.0138420783524)

(-65.98859869849039, 65.9810229367917)

(-97.39963887907376, 97.3945057956234)

(-53.42579047739466, 53.4164341598961)

(-87.97596055249322, -87.9702777324248)

(-31.447714637546234, -31.4318272785345)

(-72.27046706030896, 72.2635495982494)

(-78.55254598424293, 78.5461815917343)

(-9.529359145109572, 9.47707738933575)

(25.918139392092765, 382570233576.861)

(-94.25838834503986, -94.2530842251087)

(-56.56634427982152, -56.5575071728762)

(19.634954075714212, 714428015.910722)

(7.0677226670045, 2496.45512161844)

(-1.305558911045586, -0.129102072038201)

(22.776546738060297, -16532358819.2562)

(3.9174742074336257, -110.45215344464)

(-34.58642421528892, 34.5719767335884)

(-91.11716139446474, 91.1116744496469)

(-3.456379521249552, 3.19655996739232)

(-59.70700730533546, 59.6986348402658)

(-37.7256128277765, -37.71236621281)

(16.493361253919897, -30873244.9475011)

(-50.28536633777365, -50.2754260353972)

(0.798698173732013, 5.20902304798229)

(-47.14509773676103, 47.1344957575419)

(-69.12950297389526, -69.1222713069218)

(-44.005017920830845, -43.9936599791065)

(-84.83478871804229, 84.8288955236568)

(-147.66162685535437, 147.658240851742)

(-62.84776319445445, -62.8398089721545)

(-116.2475303039321, 116.243229375987)

(-15.771284903201174, 15.7396765386195)

(-75.41148348884815, -75.4048540732019)

(13.351765504515084, 1334162.82085398)

(-18.90240995788652, -18.8760136794635)

(-81.69364923560168, -81.6875294965246)

(10.210119645356034, -57661.2149709547)

(-100.54091078684232, -100.535938055826)

(-28.30964285445207, 28.2919975390928)

(-6.438120093241736, -6.35626093574794)

(-25.17244632664811, -25.1526068178365)


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=−12.6452880386917x_{1} = -12.6452880386917
x2=29.0597320457045x_{2} = 29.0597320457045
x3=−87.9759605524932x_{3} = -87.9759605524932
x4=−31.4477146375462x_{4} = -31.4477146375462
x5=−94.2583883450399x_{5} = -94.2583883450399
x6=−56.5663442798215x_{6} = -56.5663442798215
x7=−1.30555891104559x_{7} = -1.30555891104559
x8=22.7765467380603x_{8} = 22.7765467380603
x9=3.91747420743363x_{9} = 3.91747420743363
x10=−37.7256128277765x_{10} = -37.7256128277765
x11=16.4933612539199x_{11} = 16.4933612539199
x12=−50.2853663377737x_{12} = -50.2853663377737
x13=−69.1295029738953x_{13} = -69.1295029738953
x14=−44.0050179208308x_{14} = -44.0050179208308
x15=−62.8477631944545x_{15} = -62.8477631944545
x16=−75.4114834888481x_{16} = -75.4114834888481
x17=−18.9024099578865x_{17} = -18.9024099578865
x18=−81.6936492356017x_{18} = -81.6936492356017
x19=10.210119645356x_{19} = 10.210119645356
x20=−100.540910786842x_{20} = -100.540910786842
x21=−6.43812009324174x_{21} = -6.43812009324174
x22=−25.1724463266481x_{22} = -25.1724463266481
Puntos máximos de la función:
x22=−40.8651703304881x_{22} = -40.8651703304881
x22=−22.0364967279767x_{22} = -22.0364967279767
x22=−65.9885986984904x_{22} = -65.9885986984904
x22=−97.3996388790738x_{22} = -97.3996388790738
x22=−53.4257904773947x_{22} = -53.4257904773947
x22=−72.270467060309x_{22} = -72.270467060309
x22=−78.5525459842429x_{22} = -78.5525459842429
x22=−9.52935914510957x_{22} = -9.52935914510957
x22=25.9181393920928x_{22} = 25.9181393920928
x22=19.6349540757142x_{22} = 19.6349540757142
x22=7.0677226670045x_{22} = 7.0677226670045
x22=−34.5864242152889x_{22} = -34.5864242152889
x22=−91.1171613944647x_{22} = -91.1171613944647
x22=−3.45637952124955x_{22} = -3.45637952124955
x22=−59.7070073053355x_{22} = -59.7070073053355
x22=0.798698173732013x_{22} = 0.798698173732013
x22=−47.145097736761x_{22} = -47.145097736761
x22=−84.8347887180423x_{22} = -84.8347887180423
x22=−147.661626855354x_{22} = -147.661626855354
x22=−116.247530303932x_{22} = -116.247530303932
x22=−15.7712849032012x_{22} = -15.7712849032012
x22=13.3517655045151x_{22} = 13.3517655045151
x22=−28.3096428544521x_{22} = -28.3096428544521
Decrece en los intervalos
[29.0597320457045,∞)\left[29.0597320457045, \infty\right)
Crece en los intervalos
(−∞,−100.540910786842]\left(-\infty, -100.540910786842\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+3ex)cos⁡(x)−2(3ex+1)sin⁡(x)+3excos⁡(x)=0- \left(x + 3 e^{x}\right) \cos{\left(x \right)} - 2 \left(3 e^{x} + 1\right) \sin{\left(x \right)} + 3 e^{x} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=69.1150383789755x_{1} = 69.1150383789755
x2=−29.9118938695518x_{2} = -29.9118938695518
x3=−33.0471686947054x_{3} = -33.0471686947054
x4=−105.262351754877x_{4} = -105.262351754877
x5=−17.3932439740452x_{5} = -17.3932439740452
x6=−95.839441141233x_{6} = -95.839441141233
x7=−76.9949898891676x_{7} = -76.9949898891676
x8=−51.8748140534268x_{8} = -51.8748140534268
x9=12.5663633104458x_{9} = 12.5663633104458
x10=21.9911485740971x_{10} = 21.9911485740971
x11=−92.6985552433969x_{11} = -92.6985552433969
x12=−80.1355651940744x_{12} = -80.1355651940744
x13=−58.153842078645x_{13} = -58.153842078645
x14=−67.573830670859x_{14} = -67.573830670859
x15=−61.2936749662429x_{15} = -61.2936749662429
x16=−5.09284541648978x_{16} = -5.09284541648978
x17=−36.1835330907526x_{17} = -36.1835330907526
x18=−2.38888214877599x_{18} = -2.38888214877599
x19=−55.0142096788381x_{19} = -55.0142096788381
x20=−23.6463238196171x_{20} = -23.6463238196171
x21=−42.458570771699x_{21} = -42.458570771699
x22=−64.4336791037316x_{22} = -64.4336791037316
x23=0x_{23} = 0
x24=3.11895209519402x_{24} = 3.11895209519402
x25=−70.7141100665485x_{25} = -70.7141100665485
x26=−26.778087075559x_{26} = -26.778087075559
x27=15.7079628734126x_{27} = 15.7079628734126
x28=9.42465118730907x_{28} = 9.42465118730907
x29=−98.9803718651523x_{29} = -98.9803718651523
x30=−39.3207281322521x_{30} = -39.3207281322521
x31=18.8495559010794x_{31} = 18.8495559010794
x32=−86.4169374541167x_{32} = -86.4169374541167
x33=−14.2763531758453x_{33} = -14.2763531758453
x34=25.1327412286674x_{34} = 25.1327412286674
x35=−45.5969279840735x_{35} = -45.5969279840735
x36=−48.7357007949054x_{36} = -48.7357007949054
x37=−11.1727130687188x_{37} = -11.1727130687188
x38=28.2743338823057x_{38} = 28.2743338823057
x39=−8.09637042320182x_{39} = -8.09637042320182
x40=−20.5175229102959x_{40} = -20.5175229102959
x41=−73.8545010149048x_{41} = -73.8545010149048
x42=−89.5577188827244x_{42} = -89.5577188827244
x43=−83.2762171649775x_{43} = -83.2762171649775
x44=6.28122772768589x_{44} = 6.28122772768589

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
[28.2743338823057,∞)\left[28.2743338823057, \infty\right)
Convexa en los intervalos
(−∞,−95.839441141233]\left(-\infty, -95.839441141233\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((3ex+x)cos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(3 e^{x} + x\right) \cos{\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→∞((3ex+x)cos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(3 e^{x} + x\right) \cos{\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 (3*E^x + x)*cos(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((3ex+x)cos⁡(x)x)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{\left(3 e^{x} + x\right) \cos{\left(x \right)}}{x}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞((3ex+x)cos⁡(x)x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(3 e^{x} + x\right) \cos{\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:
(3ex+x)cos⁡(x)=(−x+3e−x)cos⁡(x)\left(3 e^{x} + x\right) \cos{\left(x \right)} = \left(- x + 3 e^{- x}\right) \cos{\left(x \right)}
- No
(3ex+x)cos⁡(x)=−(−x+3e−x)cos⁡(x)\left(3 e^{x} + x\right) \cos{\left(x \right)} = - \left(- x + 3 e^{- x}\right) \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (3*e^x+x)*cos(x)
    © Sr Examen – Калькулятор