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Trazado el gráfico de la función/ (cos((pi*x)/4)^2*(1+x))/x

Gráfico de la función y = (cos((pi*x)/4)^2*(1+x))/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2/pi*x\        
       cos |----|*(1 + x)
           \ 4  /        
f(x) = ------------------
               x
f(x)=(x+1)cos⁡2(πx4)xf{\left(x \right)} = \frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x} 
f = ((x + 1)*cos((pi*x)/4)^2)/x
Gráfico de la función
02468-8-6-4-2-1010-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
(x+1)cos⁡2(πx4)x=0\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−1x_{1} = -1
x2=2x_{2} = 2
Solución numérica
x1=−29.9999993239155x_{1} = -29.9999993239155
x2=−85.9999994157479x_{2} = -85.9999994157479
x3=−53.9999993872206x_{3} = -53.9999993872206
x4=62x_{4} = 62
x5=58x_{5} = 58
x6=70.0000006054295x_{6} = 70.0000006054295
x7=38.0000006877626x_{7} = 38.0000006877626
x8=106.000000580183x_{8} = 106.000000580183
x9=−98x_{9} = -98
x10=42x_{10} = 42
x11=22.0000006954456x_{11} = 22.0000006954456
x12=22x_{12} = 22
x13=82x_{13} = 82
x14=−18x_{14} = -18
x15=−58x_{15} = -58
x16=34.0000007062372x_{16} = 34.0000007062372
x17=−70x_{17} = -70
x18=−45.9999993727629x_{18} = -45.9999993727629
x19=−6x_{19} = -6
x20=−13.9999992779697x_{20} = -13.9999992779697
x21=94.0000005856159x_{21} = 94.0000005856159
x22=18x_{22} = 18
x23=94x_{23} = 94
x24=10x_{24} = 10
x25=18.0000005397436x_{25} = 18.0000005397436
x26=−22x_{26} = -22
x27=−26x_{27} = -26
x28=−50x_{28} = -50
x29=−93.9999994192894x_{29} = -93.9999994192894
x30=46x_{30} = 46
x31=90.0000005879094x_{31} = 90.0000005879094
x32=−2.00000012051085x_{32} = -2.00000012051085
x33=−73.9999994085164x_{33} = -73.9999994085164
x34=26.0000007290226x_{34} = 26.0000007290226
x35=26x_{35} = 26
x36=−57.9999993928546x_{36} = -57.9999993928546
x37=70x_{37} = 70
x38=66x_{38} = 66
x39=−77.9999994112411x_{39} = -77.9999994112411
x40=−41.999999363466x_{40} = -41.999999363466
x41=−78x_{41} = -78
x42=54x_{42} = 54
x43=98x_{43} = 98
x44=−10x_{44} = -10
x45=−33.9999993392928x_{45} = -33.9999993392928
x46=−25.9999993064638x_{46} = -25.9999993064638
x47=30.0000007231778x_{47} = 30.0000007231778
x48=86x_{48} = 86
x49=−65.9999994018159x_{49} = -65.9999994018159
x50=−34x_{50} = -34
x51=−17.9999992741071x_{51} = -17.9999992741071
x52=50.0000006436889x_{52} = 50.0000006436889
x53=98.0000005835884x_{53} = 98.0000005835884
x54=14x_{54} = 14
x55=−9.99999934065966x_{55} = -9.99999934065966
x56=−94x_{56} = -94
x57=62.0000006170459x_{57} = 62.0000006170459
x58=6x_{58} = 6
x59=−66x_{59} = -66
x60=−74x_{60} = -74
x61=102.000000581788x_{61} = 102.000000581788
x62=−69.9999994053997x_{62} = -69.9999994053997
x63=58.0000006244519x_{63} = 58.0000006244519
x64=−49.999999380595x_{64} = -49.999999380595
x65=−38x_{65} = -38
x66=74x_{66} = 74
x67=−62x_{67} = -62
x68=−82x_{68} = -82
x69=86.0000005905157x_{69} = 86.0000005905157
x70=78x_{70} = 78
x71=42.0000006708038x_{71} = 42.0000006708038
x72=−5.99999957471976x_{72} = -5.99999957471976
x73=−21.9999992882976x_{73} = -21.9999992882976
x74=−61.9999993976724x_{74} = -61.9999993976724
x75=38x_{75} = 38
x76=66.0000006107726x_{76} = 66.0000006107726
x77=2x_{77} = 2
x78=−46x_{78} = -46
x79=54.0000006332367x_{79} = 54.0000006332367
x80=34x_{80} = 34
x81=90x_{81} = 90
x82=−2x_{82} = -2
x83=74.0000006008525x_{83} = 74.0000006008525
x84=−89.9999994176214x_{84} = -89.9999994176214
x85=−90x_{85} = -90
x86=46.0000006561216x_{86} = 46.0000006561216
x87=−86x_{87} = -86
x88=−37.9999993524093x_{88} = -37.9999993524093
x89=50x_{89} = 50
x90=14.0000000711243x_{90} = 14.0000000711243
x91=−14x_{91} = -14
x92=−30x_{92} = -30
x93=−42x_{93} = -42
x94=78.000000596909x_{94} = 78.000000596909
x95=−5.9999994409473x_{95} = -5.9999994409473
x96=−81.9999994136349x_{96} = -81.9999994136349
x97=30x_{97} = 30
x98=82.0000005934923x_{98} = 82.0000005934923
x99=−54x_{99} = -54
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((pi*x)/4)^2*(1 + x))/x.
cos⁡2(0π4)0\frac{\cos^{2}{\left(\frac{0 \pi}{4} \right)}}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−π(x+1)sin⁡(πx4)cos⁡(πx4)2+cos⁡2(πx4)x−(x+1)cos⁡2(πx4)x2=0\frac{- \frac{\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}}{2} + \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x} - \frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2x_{1} = -2
x2=2x_{2} = 2
Signos de extremos en los puntos:
(-2, 0)

(2, 0)


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=−2x_{1} = -2
x2=2x_{2} = 2
La función no tiene puntos máximos
Decrece en los intervalos
[2,∞)\left[2, \infty\right)
Crece en los intervalos
(−∞,−2]\left(-\infty, -2\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
π(π(x+1)(sin⁡2(πx4)−cos⁡2(πx4))−8sin⁡(πx4)cos⁡(πx4))8+(π(x+1)sin⁡(πx4)−2cos⁡(πx4))cos⁡(πx4)x+2(x+1)cos⁡2(πx4)x2x=0\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=44.9996028704428x_{1} = 44.9996028704428
x2=−31.0008536093723x_{2} = -31.0008536093723
x3=−85.0001143749339x_{3} = -85.0001143749339
x4=−15.0036931994676x_{4} = -15.0036931994676
x5=−23.0015572519595x_{5} = -23.0015572519595
x6=18.9979377413296x_{6} = 18.9979377413296
x7=−13.0054432541212x_{7} = -13.0054432541212
x8=96.9999141708959x_{8} = 96.9999141708959
x9=10.9942059886459x_{9} = 10.9942059886459
x10=52.9997133756277x_{10} = 52.9997133756277
x11=38.9994888660358x_{11} = 38.9994888660358
x12=−51.000313896626x_{12} = -51.000313896626
x13=−37.0006189745222x_{13} = -37.0006189745222
x14=−57.0002567704716x_{14} = -57.0002567704716
x15=74.9998590011155x_{15} = 74.9998590011155
x16=26.9989529792397x_{16} = 26.9989529792397
x17=−35.0006687223095x_{17} = -35.0006687223095
x18=−25.0013851152113x_{18} = -25.0013851152113
x19=−65.0001967550027x_{19} = -65.0001967550027
x20=80.9998770033365x_{20} = 80.9998770033365
x21=88.999898081113x_{21} = 88.999898081113
x22=40.9995219605996x_{22} = 40.9995219605996
x23=−3.0944985600091x_{23} = -3.0944985600091
x24=−27.0011272880282x_{24} = -27.0011272880282
x25=−89.0001042345009x_{25} = -89.0001042345009
x26=−91.0000982779299x_{26} = -91.0000982779299
x27=−67.0001815606124x_{27} = -67.0001815606124
x28=72.9998486406245x_{28} = 72.9998486406245
x29=−59.0002343114102x_{29} = -59.0002343114102
x30=−39.0005379923011x_{30} = -39.0005379923011
x31=−43.0004421616485x_{31} = -43.0004421616485
x32=−47.0003698293801x_{32} = -47.0003698293801
x33=−1.614150703963x_{33} = -1.614150703963
x34=102.999924798238x_{34} = 102.999924798238
x35=−69.000174348217x_{35} = -69.000174348217
x36=92.9999066437242x_{36} = 92.9999066437242
x37=50.9996981653844x_{37} = 50.9996981653844
x38=90.9999038579305x_{38} = 90.9999038579305
x39=−61.0002237758992x_{39} = -61.0002237758992
x40=2.94416537967508x_{40} = 2.94416537967508
x41=380.999994421377x_{41} = 380.999994421377
x42=70.9998428586052x_{42} = 70.9998428586052
x43=16.9972505899113x_{43} = 16.9972505899113
x44=−87.0001075427893x_{44} = -87.0001075427893
x45=−11.0069285099769x_{45} = -11.0069285099769
x46=104.999926730846x_{46} = 104.999926730846
x47=32.9992635820045x_{47} = 32.9992635820045
x48=58.9997734936612x_{48} = 58.9997734936612
x49=−93.0000953851793x_{49} = -93.0000953851793
x50=−9.01200286161397x_{50} = -9.01200286161397
x51=98.9999186505986x_{51} = 98.9999186505986
x52=12.9953233745264x_{52} = 12.9953233745264
x53=14.9967639467234x_{53} = 14.9967639467234
x54=−21.0019878533421x_{54} = -21.0019878533421
x55=−95.0000901607308x_{55} = -95.0000901607308
x56=−83.0001181823166x_{56} = -83.0001181823166
x57=48.9996648501043x_{57} = 48.9996648501043
x58=94.9999117171202x_{58} = 94.9999117171202
x59=−45.0004151584059x_{59} = -45.0004151584059
x60=86.9998949005943x_{60} = 86.9998949005943
x61=−49.0003490998428x_{61} = -49.0003490998428
x62=−19.0022898196266x_{62} = -19.0022898196266
x63=−5.04433678704151x_{63} = -5.04433678704151
x64=62.9998009955008x_{64} = 62.9998009955008
x65=76.9998639235117x_{65} = 76.9998639235117
x66=−63.000205420042x_{66} = -63.000205420042
x67=66.9998237765818x_{67} = 66.9998237765818
x68=−17.0030898885942x_{68} = -17.0030898885942
x69=−53.0002976385393x_{69} = -53.0002976385393
x70=−55.0002697559914x_{70} = -55.0002697559914
x71=68.999830630629x_{71} = 68.999830630629
x72=60.999783437987x_{72} = 60.999783437987
x73=56.9997520770949x_{73} = 56.9997520770949
x74=78.9998727782642x_{74} = 78.9998727782642
x75=84.9998882840567x_{75} = 84.9998882840567
x76=−79.0001304825544x_{76} = -79.0001304825544
x77=−71.0001616289714x_{77} = -71.0001616289714
x78=82.999884630574x_{78} = 82.999884630574
x79=64.9998092038565x_{79} = 64.9998092038565
x80=30.999199612962x_{80} = 30.999199612962
x81=54.99973987063x_{81} = 54.99973987063
x82=−41.0005019053572x_{82} = -41.0005019053572
x83=−75.0001448080093x_{83} = -75.0001448080093
x84=36.9994135457222x_{84} = 36.9994135457222
x85=−29.0010200422274x_{85} = -29.0010200422274
x86=42.9995779111432x_{86} = 42.9995779111432
x87=−73.00015556184x_{87} = -73.00015556184
x88=100.99992082309x_{88} = 100.99992082309
x89=−7.01740170165913x_{89} = -7.01740170165913
x90=−81.0001260704169x_{90} = -81.0001260704169
x91=46.999645563784x_{91} = 46.999645563784
x92=28.9990477660696x_{92} = 28.9990477660696
x93=34.9993683584672x_{93} = 34.9993683584672
x94=8.99032715593967x_{94} = 8.99032715593967
x95=−77.0001396559479x_{95} = -77.0001396559479
x96=−33.0007823359871x_{96} = -33.0007823359871
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(π(π(x+1)(sin⁡2(πx4)−cos⁡2(πx4))−8sin⁡(πx4)cos⁡(πx4))8+(π(x+1)sin⁡(πx4)−2cos⁡(πx4))cos⁡(πx4)x+2(x+1)cos⁡2(πx4)x2x)=−∞\lim_{x \to 0^-}\left(\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x}\right) = -\infty
lim⁡x→0+(π(π(x+1)(sin⁡2(πx4)−cos⁡2(πx4))−8sin⁡(πx4)cos⁡(πx4))8+(π(x+1)sin⁡(πx4)−2cos⁡(πx4))cos⁡(πx4)x+2(x+1)cos⁡2(πx4)x2x)=∞\lim_{x \to 0^+}\left(\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[380.999994421377,∞)\left[380.999994421377, \infty\right)
Convexa en los intervalos
(−∞,−95.0000901607308]\left(-\infty, -95.0000901607308\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x+1)cos⁡2(πx4)x)=⟨0,1⟩\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x}\right) = \left\langle 0, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞((x+1)cos⁡2(πx4)x)=⟨0,1⟩\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x}\right) = \left\langle 0, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (cos((pi*x)/4)^2*(1 + x))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x+1)cos⁡2(πx4)x2)=0\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((x+1)cos⁡2(πx4)x2)=0\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}\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:
(x+1)cos⁡2(πx4)x=−(1−x)cos⁡2(πx4)x\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x} = - \frac{\left(1 - x\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x}
- No
(x+1)cos⁡2(πx4)x=(1−x)cos⁡2(πx4)x\frac{\left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x} = \frac{\left(1 - x\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x}
- No
es decir, función
no es
par ni impar
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