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Trazado el gráfico de la función/ 2*sin(x)+3*cos(x)*x^2

Gráfico de la función y = 2*sin(x)+3*cos(x)*x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                            2
f(x) = 2*sin(x) + 3*cos(x)*x
f(x)=x2⋅3cos⁡(x)+2sin⁡(x)f{\left(x \right)} = x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)} 
f = x^2*(3*cos(x)) + 2*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-500500
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⋅3cos⁡(x)+2sin⁡(x)=0x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−14.1338297013084x_{1} = -14.1338297013084
x2=−76.9689074804168x_{2} = -76.9689074804168
x3=29.8458786196911x_{3} = 29.8458786196911
x4=39.2703404640447x_{4} = 39.2703404640447
x5=4.74202729139257x_{5} = 4.74202729139257
x6=−10.9900547255072x_{6} = -10.9900547255072
x7=23.5631456226234x_{7} = 23.5631456226234
x8=64.4028101293892x_{8} = 64.4028101293892
x9=−61.2608791043782x_{9} = -61.2608791043782
x10=−32.9861101646734x_{10} = -32.9861101646734
x11=−54.9776508729343x_{11} = -54.9776508729343
x12=−48.6944049725398x_{12} = -48.6944049725398
x13=1.77851814484338x_{13} = 1.77851814484338
x14=76.9691325448249x_{14} = 76.9691325448249
x15=58.1196614529777x_{15} = 58.1196614529777
x16=−86.3937086545851x_{16} = -86.3937086545851
x17=51.8365268900453x_{17} = 51.8365268900453
x18=−70.6857012781943x_{18} = -70.6857012781943
x19=−80.1105087870615x_{19} = -80.1105087870615
x20=17.2809919878968x_{20} = 17.2809919878968
x21=36.1288262576067x_{21} = 36.1288262576067
x22=42.4118714477262x_{22} = 42.4118714477262
x23=14.1405010328204x_{23} = 14.1405010328204
x24=89.5354737880726x_{24} = 89.5354737880726
x25=−95.8185033222253x_{25} = -95.8185033222253
x26=0x_{26} = 0
x27=−51.8360306736678x_{27} = -51.8360306736678
x28=86.3938872924842x_{28} = 86.3938872924842
x29=−83.2521091327153x_{29} = -83.2521091327153
x30=−36.1278047460758x_{30} = -36.1278047460758
x31=73.8275496721381x_{31} = 73.8275496721381
x32=−58.1192667271638x_{32} = -58.1192667271638
x33=80.1107165454791x_{33} = 80.1107165454791
x34=70.685968132339x_{34} = 70.685968132339
x35=−45.5527722004468x_{35} = -45.5527722004468
x36=−67.5440959240085x_{36} = -67.5440959240085
x37=11.0010827891937x_{37} = 11.0010827891937
x38=−39.2694758566641x_{38} = -39.2694758566641
x39=−20.418753242938x_{39} = -20.418753242938
x40=26.7044724026673x_{40} = 26.7044724026673
x41=−89.5353074662367x_{41} = -89.5353074662367
x42=−29.8443817234375x_{42} = -29.8443817234375
x43=7.86475920922712x_{43} = 7.86475920922712
x44=45.5534147445939x_{44} = 45.5534147445939
x45=−23.5607439364182x_{45} = -23.5607439364182
x46=32.9873355151944x_{46} = 32.9873355151944
x47=83.2523015070992x_{47} = 83.2523015070992
x48=67.5443881790881x_{48} = 67.5443881790881
x49=54.978091999169x_{49} = 54.978091999169
x50=−64.4024886661878x_{50} = -64.4024886661878
x51=61.2612343835633x_{51} = 61.2612343835633
x52=95.818648546532x_{52} = 95.818648546532
x53=−42.4111301862425x_{53} = -42.4111301862425
x54=−26.702602577431x_{54} = -26.702602577431
x55=−98.9601005129476x_{55} = -98.9601005129476
x56=−17.276526047309x_{56} = -17.276526047309
x57=98.960236663022x_{57} = 98.960236663022
x58=−92.6769056623194x_{58} = -92.6769056623194
x59=−73.8273050457717x_{59} = -73.8273050457717
x60=20.4219507530497x_{60} = 20.4219507530497
x61=−7.84314457847454x_{61} = -7.84314457847454
x62=48.6949672822505x_{62} = 48.6949672822505
x63=−4.68198610496305x_{63} = -4.68198610496305
x64=92.6770608992183x_{64} = 92.6770608992183
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 2*sin(x) + (3*cos(x))*x^2.
2sin⁡(0)+02⋅3cos⁡(0)2 \sin{\left(0 \right)} + 0^{2} \cdot 3 \cos{\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
−3x2sin⁡(x)+6xcos⁡(x)+2cos⁡(x)=0- 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
−3x2cos⁡(x)−12xsin⁡(x)−2sin⁡(x)+6cos⁡(x)=0- 3 x^{2} \cos{\left(x \right)} - 12 x \sin{\left(x \right)} - 2 \sin{\left(x \right)} + 6 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=14.4133792482364x_{1} = 14.4133792482364
x2=−23.7283781313504x_{2} = -23.7283781313504
x3=45.6409122588568x_{3} = 45.6409122588568
x4=−42.5050681149859x_{4} = -42.5050681149859
x5=77.0210368436624x_{5} = 77.0210368436624
x6=−73.8814133323596x_{6} = -73.8814133323596
x7=−51.9129896313131x_{7} = -51.9129896313131
x8=−36.237916093112x_{8} = -36.237916093112
x9=8.32153356439318x_{9} = 8.32153356439318
x10=83.3002971943809x_{10} = 83.3002971943809
x11=−55.0502336337086x_{11} = -55.0502336337086
x12=−86.4399631843049x_{12} = -86.4399631843049
x13=−58.1879432266192x_{13} = -58.1879432266192
x14=−80.1603832447484x_{14} = -80.1603832447484
x15=55.0506709974963x_{15} = 55.0506709974963
x16=−26.8509189401211x_{16} = -26.8509189401211
x17=20.6144117169035x_{17} = 20.6144117169035
x18=−92.7200297715138x_{18} = -92.7200297715138
x19=33.107775782714x_{19} = 33.107775782714
x20=36.23891765086x_{20} = 36.23891765086
x21=−33.1065790334073x_{21} = -33.1065790334073
x22=42.5057988132743x_{22} = 42.5057988132743
x23=70.74247305522x_{23} = 70.74247305522
x24=92.7201845398991x_{24} = 92.7201845398991
x25=−20.6114019737113x_{25} = -20.6114019737113
x26=17.5068895582383x_{26} = 17.5068895582383
x27=−45.6402776667111x_{27} = -45.6402776667111
x28=67.6035129101358x_{28} = 67.6035129101358
x29=2.70744937513831x_{29} = 2.70744937513831
x30=58.1883349371903x_{30} = 58.1883349371903
x31=23.7306723297619x_{31} = 23.7306723297619
x32=95.8603606101628x_{32} = 95.8603606101628
x33=26.8527229671753x_{33} = 26.8527229671753
x34=−39.3708623896056x_{34} = -39.3708623896056
x35=61.3264004081402x_{35} = 61.3264004081402
x36=29.9787970767767x_{36} = 29.9787970767767
x37=−11.3350930898675x_{37} = -11.3350930898675
x38=89.5801077985184x_{38} = 89.5801077985184
x39=−67.6032223112472x_{39} = -67.6032223112472
x40=−95.860215795961x_{40} = -95.860215795961
x41=48.7768562379425x_{41} = 48.7768562379425
x42=11.3441654506476x_{42} = 11.3441654506476
x43=−89.5799420143801x_{43} = -89.5799420143801
x44=−77.0208127627252x_{44} = -77.0208127627252
x45=39.3717126595863x_{45} = 39.3717126595863
x46=86.4401412020727x_{46} = 86.4401412020727
x47=5.39919307625504x_{47} = 5.39919307625504
x48=−83.3001055389754x_{48} = -83.3001055389754
x49=−14.4074767784673x_{49} = -14.4074767784673
x50=−99.0004907739355x_{50} = -99.0004907739355
x51=−2.66859348772284x_{51} = -2.66859348772284
x52=51.9134810918396x_{52} = 51.9134810918396
x53=−8.30635371816967x_{53} = -8.30635371816967
x54=80.1605901648458x_{54} = 80.1605901648458
x55=99.000626563491x_{55} = 99.000626563491
x56=−29.9773426450509x_{56} = -29.9773426450509
x57=−70.7422075825611x_{57} = -70.7422075825611
x58=64.4648089505737x_{58} = 64.4648089505737
x59=−5.37185517485809x_{59} = -5.37185517485809
x60=0.54190986940165x_{60} = 0.54190986940165
x61=−0.664310677408101x_{61} = -0.664310677408101
x62=−64.4644894900216x_{62} = -64.4644894900216
x63=73.8816567973019x_{63} = 73.8816567973019
x64=−17.5027816505754x_{64} = -17.5027816505754
x65=−48.7763000276406x_{65} = -48.7763000276406
x66=−61.3260475735095x_{66} = -61.3260475735095

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
[95.8603606101628,∞)\left[95.8603606101628, \infty\right)
Convexa en los intervalos
(−∞,−99.0004907739355]\left(-\infty, -99.0004907739355\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2⋅3cos⁡(x)+2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \cdot 3 \cos{\left(x \right)} + 2 \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⋅3cos⁡(x)+2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \cdot 3 \cos{\left(x \right)} + 2 \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 2*sin(x) + (3*cos(x))*x^2, 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→−∞(x2⋅3cos⁡(x)+2sin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x2⋅3cos⁡(x)+2sin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\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:
x2⋅3cos⁡(x)+2sin⁡(x)=x2⋅3cos⁡(x)−2sin⁡(x)x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)} = x^{2} \cdot 3 \cos{\left(x \right)} - 2 \sin{\left(x \right)}
- No
x2⋅3cos⁡(x)+2sin⁡(x)=−x2⋅3cos⁡(x)+2sin⁡(x)x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)} = - x^{2} \cdot 3 \cos{\left(x \right)} + 2 \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
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