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Trazado el gráfico de la función/ pi*(1-x)*tg(pi*x/2)

Gráfico de la función y = pi*(1-x)*tg(pi*x/2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                     /pi*x\
f(x) = pi*(1 - x)*tan|----|
                     \ 2  /
f(x)=π(1−x)tan⁡(πx2)f{\left(x \right)} = \pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)} 
f = (pi*(1 - x))*tan((pi*x)/2)
Gráfico de la función
02468-8-6-4-2-1010-50005000
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:
π(1−x)tan⁡(πx2)=0\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=28x_{1} = 28
x2=−22x_{2} = -22
x3=−54x_{3} = -54
x4=−32x_{4} = -32
x5=−38x_{5} = -38
x6=32x_{6} = 32
x7=82x_{7} = 82
x8=76x_{8} = 76
x9=−58x_{9} = -58
x10=−86x_{10} = -86
x11=−50x_{11} = -50
x12=86x_{12} = 86
x13=80x_{13} = 80
x14=−64x_{14} = -64
x15=−100x_{15} = -100
x16=36x_{16} = 36
x17=−12x_{17} = -12
x18=38x_{18} = 38
x19=−20x_{19} = -20
x20=−8x_{20} = -8
x21=−10x_{21} = -10
x22=−44x_{22} = -44
x23=66x_{23} = 66
x24=−62x_{24} = -62
x25=−46x_{25} = -46
x26=−48x_{26} = -48
x27=50x_{27} = 50
x28=−74x_{28} = -74
x29=4x_{29} = 4
x30=98x_{30} = 98
x31=−2x_{31} = -2
x32=−66x_{32} = -66
x33=2x_{33} = 2
x34=−28x_{34} = -28
x35=78x_{35} = 78
x36=−92x_{36} = -92
x37=20x_{37} = 20
x38=54x_{38} = 54
x39=40x_{39} = 40
x40=−40x_{40} = -40
x41=90x_{41} = 90
x42=74x_{42} = 74
x43=10x_{43} = 10
x44=−76x_{44} = -76
x45=60x_{45} = 60
x46=−18x_{46} = -18
x47=−98x_{47} = -98
x48=−36x_{48} = -36
x49=58x_{49} = 58
x50=−30x_{50} = -30
x51=34x_{51} = 34
x52=18x_{52} = 18
x53=−60x_{53} = -60
x54=70x_{54} = 70
x55=14x_{55} = 14
x56=30x_{56} = 30
x57=24x_{57} = 24
x58=64x_{58} = 64
x59=−84x_{59} = -84
x60=26x_{60} = 26
x61=84x_{61} = 84
x62=52x_{62} = 52
x63=56x_{63} = 56
x64=68x_{64} = 68
x65=44x_{65} = 44
x66=94x_{66} = 94
x67=96x_{67} = 96
x68=0x_{68} = 0
x69=−26x_{69} = -26
x70=48x_{70} = 48
x71=−14x_{71} = -14
x72=−78x_{72} = -78
x73=6x_{73} = 6
x74=−90x_{74} = -90
x75=16x_{75} = 16
x76=−82x_{76} = -82
x77=92x_{77} = 92
x78=−34x_{78} = -34
x79=42x_{79} = 42
x80=−4x_{80} = -4
x81=−56x_{81} = -56
x82=72x_{82} = 72
x83=−72x_{83} = -72
x84=−52x_{84} = -52
x85=−16x_{85} = -16
x86=−42x_{86} = -42
x87=−6x_{87} = -6
x88=8x_{88} = 8
x89=−24x_{89} = -24
x90=−68x_{90} = -68
x91=88x_{91} = 88
x92=−94x_{92} = -94
x93=46x_{93} = 46
x94=−88x_{94} = -88
x95=22x_{95} = 22
x96=−96x_{96} = -96
x97=−70x_{97} = -70
x98=−80x_{98} = -80
x99=100x_{99} = 100
x100=12x_{100} = 12
x101=62x_{101} = 62
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 (pi*(1 - x))*tan((pi*x)/2).
π(1−0)tan⁡(0π2)\pi \left(1 - 0\right) \tan{\left(\frac{0 \pi}{2} \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
π2(1−x)(tan⁡2(πx2)+1)2−πtan⁡(πx2)=0\frac{\pi^{2} \left(1 - x\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{2} - \pi \tan{\left(\frac{\pi x}{2} \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
−π2(π(x−1)tan⁡(πx2)2+1)(tan⁡2(πx2)+1)=0- \pi^{2} \left(\frac{\pi \left(x - 1\right) \tan{\left(\frac{\pi x}{2} \right)}}{2} + 1\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−29.9869226211336x_{1} = -29.9869226211336
x2=−47.9917279519966x_{2} = -47.9917279519966
x3=−55.9928891501073x_{3} = -55.9928891501073
x4=−49.9920524148283x_{4} = -49.9920524148283
x5=−11.9687742527727x_{5} = -11.9687742527727
x6=−89.9955461762757x_{6} = -89.9955461762757
x7=−77.994869591368x_{7} = -77.994869591368
x8=−35.9890441958502x_{8} = -35.9890441958502
x9=13.9687742527727x_{9} = 13.9687742527727
x10=−41.9905733959369x_{10} = -41.9905733959369
x11=19.9786532361719x_{11} = 19.9786532361719
x12=−61.993566470984x_{12} = -61.993566470984
x13=−53.9926305283549x_{13} = -53.9926305283549
x14=85.9952317659775x_{14} = 85.9952317659775
x15=−13.9729484959255x_{15} = -13.9729484959255
x16=55.9926305283549x_{16} = 55.9926305283549
x17=89.9954460835957x_{17} = 89.9954460835957
x18=95.995733717172x_{18} = 95.995733717172
x19=−93.995733717172x_{19} = -93.995733717172
x20=33.9877155958269x_{20} = 33.9877155958269
x21=7.94177944869404x_{21} = 7.94177944869404
x22=−99.9959871735669x_{22} = -99.9959871735669
x23=−97.9959061019405x_{23} = -97.9959061019405
x24=71.9942914583172x_{24} = 71.9942914583172
x25=−1.86059330624841x_{25} = -1.86059330624841
x26=91.9955461762757x_{26} = 91.9955461762757
x27=−57.9931302339508x_{27} = -57.9931302339508
x28=−51.9923523825456x_{28} = -51.9923523825456
x29=67.9939506099815x_{29} = 67.9939506099815
x30=−43.9909924705645x_{30} = -43.9909924705645
x31=−71.9944478728046x_{31} = -71.9944478728046
x32=93.9956419634817x_{32} = 93.9956419634817
x33=11.9630733012144x_{33} = 11.9630733012144
x34=75.9945959439502x_{34} = 75.9945959439502
x35=39.9896062370735x_{35} = 39.9896062370735
x36=29.986020171625x_{36} = 29.986020171625
x37=37.9890441958502x_{37} = 37.9890441958502
x38=−83.9952317659775x_{38} = -83.9952317659775
x39=−75.9947363220078x_{39} = -75.9947363220078
x40=−9.96307330121439x_{40} = -9.96307330121439
x41=−95.995821686872x_{41} = -95.995821686872
x42=23.9823699155424x_{42} = 23.9823699155424
x43=31.9869226211336x_{43} = 31.9869226211336
x44=−73.9945959439502x_{44} = -73.9945959439502
x45=−31.9877155958269x_{45} = -31.9877155958269
x46=73.9944478728046x_{46} = 73.9944478728046
x47=83.9951168598645x_{47} = 83.9951168598645
x48=61.9933555054535x_{48} = 61.9933555054535
x49=3.86059330624841x_{49} = 3.86059330624841
x50=9.95481716686503x_{50} = 9.95481716686503
x51=17.9761373921869x_{51} = 17.9761373921869
x52=57.9928891501073x_{52} = 57.9928891501073
x53=−33.9884178810314x_{53} = -33.9884178810314
x54=53.9923523825456x_{54} = 53.9923523825456
x55=−39.9901134171449x_{55} = -39.9901134171449
x56=97.995821686872x_{56} = 97.995821686872
x57=−63.9937644514482x_{57} = -63.9937644514482
x58=65.9937644514483x_{58} = 65.9937644514483
x59=−79.9949962786911x_{59} = -79.9949962786911
x60=−27.986020171625x_{60} = -27.986020171625
x61=59.9931302339508x_{61} = 59.9931302339508
x62=−5.94177944869404x_{62} = -5.94177944869404
x63=5.91804806591352x_{63} = 5.91804806591352
x64=−69.9942914583172x_{64} = -69.9942914583172
x65=−37.9896062370735x_{65} = -37.9896062370735
x66=43.9905733959369x_{66} = 43.9905733959369
x67=35.9884178810314x_{67} = 35.9884178810314
x68=−91.9956419634817x_{68} = -91.9956419634817
x69=−85.9953413884714x_{69} = -85.9953413884714
x70=−59.9933555054535x_{70} = -59.9933555054535
x71=−21.9823699155424x_{71} = -21.9823699155424
x72=77.9947363220078x_{72} = 77.9947363220078
x73=−45.9913758655731x_{73} = -45.9913758655731
x74=−23.9837815964593x_{74} = -23.9837815964593
x75=−7.95481716686503x_{75} = -7.95481716686503
x76=49.9917279519966x_{76} = 49.9917279519966
x77=−67.994125974822x_{77} = -67.994125974822
x78=−19.9806888885892x_{78} = -19.9806888885892
x79=63.993566470984x_{79} = 63.993566470984
x80=81.9949962786911x_{80} = 81.9949962786911
x81=41.9901134171449x_{81} = 41.9901134171449
x82=−15.9761373921869x_{82} = -15.9761373921869
x83=−17.9786532361719x_{83} = -17.9786532361719
x84=25.9837815964593x_{84} = 25.9837815964593
x85=45.9909924705645x_{85} = 45.9909924705645
x86=99.9959061019405x_{86} = 99.9959061019405
x87=87.9953413884714x_{87} = 87.9953413884714
x88=27.9849838868809x_{88} = 27.9849838868809
x89=−81.9951168598645x_{89} = -81.9951168598645
x90=−25.9849838868809x_{90} = -25.9849838868809
x91=47.9913758655731x_{91} = 47.9913758655731
x92=79.994869591368x_{92} = 79.994869591368
x93=51.9920524148283x_{93} = 51.9920524148283
x94=21.9806888885892x_{94} = 21.9806888885892
x95=−65.9939506099815x_{95} = -65.9939506099815
x96=69.994125974822x_{96} = 69.994125974822
x97=−3.91804806591352x_{97} = -3.91804806591352
x98=15.9729484959255x_{98} = 15.9729484959255
x99=−87.9954460835957x_{99} = -87.9954460835957

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
[−1.86059330624841,3.86059330624841]\left[-1.86059330624841, 3.86059330624841\right]
Convexa en los intervalos
(−∞,−99.9959871735669]\left(-\infty, -99.9959871735669\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→−∞(π(1−x)tan⁡(πx2))y = \lim_{x \to -\infty}\left(\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(π(1−x)tan⁡(πx2))y = \lim_{x \to \infty}\left(\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (pi*(1 - x))*tan((pi*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→−∞(π(1−x)tan⁡(πx2)x)y = x \lim_{x \to -\infty}\left(\frac{\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(π(1−x)tan⁡(πx2)x)y = x \lim_{x \to \infty}\left(\frac{\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \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:
π(1−x)tan⁡(πx2)=−π(x+1)tan⁡(πx2)\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)} = - \pi \left(x + 1\right) \tan{\left(\frac{\pi x}{2} \right)}
- No
π(1−x)tan⁡(πx2)=π(x+1)tan⁡(πx2)\pi \left(1 - x\right) \tan{\left(\frac{\pi x}{2} \right)} = \pi \left(x + 1\right) \tan{\left(\frac{\pi x}{2} \right)}
- No
es decir, función
no es
par ni impar
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