Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ cos(3*x^2+5*x-4)

Gráfico de la función y = cos(3*x^2+5*x-4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /   2          \
f(x) = cos\3*x  + 5*x - 4/
f(x)=cos⁡((3x2+5x)−4)f{\left(x \right)} = \cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} 
f = cos(3*x^2 + 5*x - 4)
Gráfico de la función
02468-8-6-4-2-10102-2
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:
cos⁡((3x2+5x)−4)=0\cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−56+6π+736x_{1} = - \frac{5}{6} + \frac{\sqrt{6 \pi + 73}}{6}
x2=−56+18π+736x_{2} = - \frac{5}{6} + \frac{\sqrt{18 \pi + 73}}{6}
x3=−6π+736−56x_{3} = - \frac{\sqrt{6 \pi + 73}}{6} - \frac{5}{6}
x4=−18π+736−56x_{4} = - \frac{\sqrt{18 \pi + 73}}{6} - \frac{5}{6}
Solución numérica
x1=74.2500962603012x_{1} = 74.2500962603012
x2=−44.0128264125825x_{2} = -44.0128264125825
x3=2.13896575332596x_{3} = 2.13896575332596
x4=−0.157329301553285x_{4} = -0.157329301553285
x5=−10.6186263085014x_{5} = -10.6186263085014
x6=52.4527768901101x_{6} = 52.4527768901101
x7=69.0564825215673x_{7} = 69.0564825215673
x8=−7.80671621440522x_{8} = -7.80671621440522
x9=64.4548816431931x_{9} = 64.4548816431931
x10=−27.8583897557538x_{10} = -27.8583897557538
x11=−22.5518940473736x_{11} = -22.5518940473736
x12=−37.7502806009198x_{12} = -37.7502806009198
x13=−47.5091074344653x_{13} = -47.5091074344653
x14=−28.9410535370997x_{14} = -28.9410535370997
x15=59.2773024400408x_{15} = 59.2773024400408
x16=−29.237542455792x_{16} = -29.237542455792
x17=100.224706924333x_{17} = 100.224706924333
x18=45.9320969367869x_{18} = 45.9320969367869
x19=26.2304444838415x_{19} = 26.2304444838415
x20=15.6494196894751x_{20} = 15.6494196894751
x21=10.2089443931952x_{21} = 10.2089443931952
x22=−75.6512983724655x_{22} = -75.6512983724655
x23=−4.13923063405955x_{23} = -4.13923063405955
x24=79.9928709930815x_{24} = 79.9928709930815
x25=64.0768563899674x_{25} = 64.0768563899674
x26=−35.8280603475537x_{26} = -35.8280603475537
x27=−14.7308630012692x_{27} = -14.7308630012692
x28=12.249104600631x_{28} = 12.249104600631
x29=−23.7712964545801x_{29} = -23.7712964545801
x30=98.1358319546126x_{30} = 98.1358319546126
x31=84.1547746115663x_{31} = 84.1547746115663
x32=−59.7029662382575x_{32} = -59.7029662382575
x33=−11.8756110598619x_{33} = -11.8756110598619
x34=14.1519013130462x_{34} = 14.1519013130462
x35=−41.9633834990393x_{35} = -41.9633834990393
x36=5.91103291950589x_{36} = 5.91103291950589
x37=38.5937911804672x_{37} = 38.5937911804672
x38=54.2406268449245x_{38} = 54.2406268449245
x39=22.1274451748722x_{39} = 22.1274451748722
x40=29.496461117035x_{40} = 29.496461117035
x41=−44.4951762242237x_{41} = -44.4951762242237
x42=18.1861339729245x_{42} = 18.1861339729245
x43=75.9396098145325x_{43} = 75.9396098145325
x44=−69.651094906651x_{44} = -69.651094906651
x45=73.0834212358686x_{45} = 73.0834212358686
x46=43.9064867578079x_{46} = 43.9064867578079
x47=−13.9157712672976x_{47} = -13.9157712672976
x48=47.4090747762317x_{48} = 47.4090747762317
x49=17.5420467426348x_{49} = 17.5420467426348
x50=−56.6345456961348x_{50} = -56.6345456961348
x51=−53.267583145284x_{51} = -53.267583145284
x52=−57.7586019304445x_{52} = -57.7586019304445
x53=−54.0112457564606x_{53} = -54.0112457564606
x54=42.1395216359445x_{54} = 42.1395216359445
x55=−83.5927978451143x_{55} = -83.5927978451143
x56=25.2049806880311x_{56} = 25.2049806880311
x57=−32.1316829994068x_{57} = -32.1316829994068
x58=−34.0155817164464x_{58} = -34.0155817164464
x59=−2.05978308859422x_{59} = -2.05978308859422
x60=−99.6913351805163x_{60} = -99.6913351805163
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 cos(3*x^2 + 5*x - 4).
cos⁡(−4+(3⋅02+0⋅5))\cos{\left(-4 + \left(3 \cdot 0^{2} + 0 \cdot 5\right) \right)}
Resultado:
f(0)=cos⁡(4)f{\left(0 \right)} = \cos{\left(4 \right)}
Punto:
(0, cos(4))
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
−(6x+5)sin⁡(3x2+5x−4)=0- \left(6 x + 5\right) \sin{\left(3 x^{2} + 5 x - 4 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−56x_{1} = - \frac{5}{6}
x2=−56+736x_{2} = - \frac{5}{6} + \frac{\sqrt{73}}{6}
x3=−736−56x_{3} = - \frac{\sqrt{73}}{6} - \frac{5}{6}
Signos de extremos en los puntos:
          /73\ 
(-5/6, cos|--|)
          \12/ 

         ____    
   5   \/ 73     
(- - + ------, 1)
   6     6       

         ____    
   5   \/ 73     
(- - - ------, 1)
   6     6


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x3=−56x_{3} = - \frac{5}{6}
x3=−56+736x_{3} = - \frac{5}{6} + \frac{\sqrt{73}}{6}
x3=−736−56x_{3} = - \frac{\sqrt{73}}{6} - \frac{5}{6}
Decrece en los intervalos
(−∞,−736−56]\left(-\infty, - \frac{\sqrt{73}}{6} - \frac{5}{6}\right]
Crece en los intervalos
[−56+736,∞)\left[- \frac{5}{6} + \frac{\sqrt{73}}{6}, \infty\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
−((6x+5)2cos⁡(3x2+5x−4)+6sin⁡(3x2+5x−4))=0- (\left(6 x + 5\right)^{2} \cos{\left(3 x^{2} + 5 x - 4 \right)} + 6 \sin{\left(3 x^{2} + 5 x - 4 \right)}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.5759917392885x_{1} = -22.5759917392885
x2=56.9500794362886x_{2} = 56.9500794362886
x3=−77.1342388200672x_{3} = -77.1342388200672
x4=−7.80679812734477x_{4} = -7.80679812734477
x5=14.1519095678391x_{5} = 14.1519095678391
x6=−41.8358830609603x_{6} = -41.8358830609603
x7=0.407678991823459x_{7} = 0.407678991823459
x8=−71.948549477028x_{8} = -71.948549477028
x9=−16.7340191997537x_{9} = -16.7340191997537
x10=−73.9882134060027x_{10} = -73.9882134060027
x11=−42.6827218446709x_{11} = -42.6827218446709
x12=100.50927059732x_{12} = 100.50927059732
x13=−65.5010750339966x_{13} = -65.5010750339966
x14=−11.8756316908107x_{14} = -11.8756316908107
x15=−34.0471264503737x_{15} = -34.0471264503737
x16=94.2338631879939x_{16} = 94.2338631879939
x17=−5.68077511597242x_{17} = -5.68077511597242
x18=−85.790631501266x_{18} = -85.790631501266
x19=−1.57787049522169x_{19} = -1.57787049522169
x20=−13.8756993511859x_{20} = -13.8756993511859
x21=10.2089650241441x_{21} = 10.2089650241441
x22=−44.0128267576189x_{22} = -44.0128267576189
x23=18.1861380103283x_{23} = 18.1861380103283
x24=22.1046320895292x_{24} = 22.1046320895292
x25=−56.5030248482679x_{25} = -56.5030248482679
x26=26.2304458851449x_{26} = 26.2304458851449
x27=64.0929875095759x_{27} = 64.0929875095759
x28=1.06770212415681x_{28} = 1.06770212415681
x29=−35.8280609957248x_{29} = -35.8280609957248
x30=−45.9461014896306x_{30} = -45.9461014896306
x31=63.7290647675856x_{31} = 63.7290647675856
x32=−58.652980353083x_{32} = -58.652980353083
x33=−93.9247761410026x_{33} = -93.9247761410026
x34=−29.7671870555455x_{34} = -29.7671870555455
x35=45.9320972083825x_{35} = 45.9320972083825
x36=−2.99150252586691x_{36} = -2.99150252586691
x37=30.5151648923448x_{37} = 30.5151648923448
x38=84.0808123029767x_{38} = 84.0808123029767
x39=26.5764769470219x_{39} = 26.5764769470219
x40=87.7387494272427x_{40} = 87.7387494272427
x41=−45.643317876545x_{41} = -45.643317876545
x42=54.2406270112116x_{42} = 54.2406270112116
x43=−73.8377532674328x_{43} = -73.8377532674328
x44=36.0126304717147x_{44} = 36.0126304717147
x45=42.2247283628723x_{45} = 42.2247283628723
x46=6.98941217305221x_{46} = 6.98941217305221
x47=−98.0463349808894x_{47} = -98.0463349808894
x48=−69.9547643199512x_{48} = -69.9547643199512
x49=43.9298873351566x_{49} = 43.9298873351566
x50=6.06464019692482x_{50} = 6.06464019692482
x51=−75.6022942708514x_{51} = -75.6022942708514
x52=52.0780648304223x_{52} = 52.0780648304223
x53=74.2500963259255x_{53} = 74.2500963259255
x54=12.2491170065883x_{54} = 12.2491170065883
x55=−37.7502811530228x_{55} = -37.7502811530228
x56=98.1517022837559x_{56} = 98.1517022837559
x57=−43.03158981016x_{57} = -43.03158981016
x58=−23.7712987561958x_{58} = -23.7712987561958
x59=2.14002252842701x_{59} = 2.14002252842701
x60=10.3970330047682x_{60} = 10.3970330047682
x61=−0.088796171444973x_{61} = -0.088796171444973
x62=−50.6035976731857x_{62} = -50.6035976731857
x63=−10.1807936316384x_{63} = -10.1807936316384
x64=−57.7586020810297x_{64} = -57.7586020810297
x65=60.441948936822x_{65} = 60.441948936822
x66=−52.4321044181101x_{66} = -52.4321044181101
x67=72.2571017488591x_{67} = 72.2571017488591
x68=1.32483585920024x_{68} = 1.32483585920024
x69=76.0690825094222x_{69} = 76.0690825094222
x70=8.89833392735942x_{70} = 8.89833392735942
x71=−2.4373715767816x_{71} = -2.4373715767816
x72=2.31108675880332x_{72} = 2.31108675880332
x73=−47.0695443527351x_{73} = -47.0695443527351
x74=−86.6064047844505x_{74} = -86.6064047844505
x75=26.3655366375557x_{75} = 26.3655366375557
x76=−13.4263537019248x_{76} = -13.4263537019248
x77=−2.73436879082347x_{77} = -2.73436879082347
x78=−12.9603699202417x_{78} = -12.9603699202417
x79=67.5570260528611x_{79} = 67.5570260528611
x80=−4.58493254992695x_{80} = -4.58493254992695
x81=−66.5133432884798x_{81} = -66.5133432884798

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
[98.1517022837559,∞)\left[98.1517022837559, \infty\right)
Convexa en los intervalos
(−∞,−93.9247761410026]\left(-\infty, -93.9247761410026\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞cos⁡((3x2+5x)−4)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡((3x2+5x)−4)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(3*x^2 + 5*x - 4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡((3x2+5x)−4)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡((3x2+5x)−4)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)}}{x}\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:
cos⁡((3x2+5x)−4)=cos⁡(−3x2+5x+4)\cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} = \cos{\left(- 3 x^{2} + 5 x + 4 \right)}
- No
cos⁡((3x2+5x)−4)=−cos⁡(−3x2+5x+4)\cos{\left(\left(3 x^{2} + 5 x\right) - 4 \right)} = - \cos{\left(- 3 x^{2} + 5 x + 4 \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор