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Trazado el gráfico de la función/ y=√-1+sin2x

Gráfico de la función y = y=√-1+sin2x

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Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         ___               
f(x) = \/ x  - 1 + sin(2*x)
f(x)=(x−1)+sin⁡(2x)f{\left(x \right)} = \left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)} 
f = sqrt(x) - 1 + sin(2*x)
Gráfico de la función
02468-8-6-4-2-10105-5
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)+sin⁡(2x)=0\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=1.73147534492382x_{1} = 1.73147534492382
x2=0.257507547125266x_{2} = 0.257507547125266
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 sqrt(x) - 1 + sin(2*x).
(−1+0)+sin⁡(0⋅2)\left(-1 + \sqrt{0}\right) + \sin{\left(0 \cdot 2 \right)}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
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
2cos⁡(2x)+12x=02 \cos{\left(2 x \right)} + \frac{1}{2 \sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=8.5966951305364x_{1} = 8.5966951305364
x2=19.6631582802796x_{2} = 19.6631582802796
x3=11.7444651825614x_{3} = 11.7444651825614
x4=47.9273477595231x_{4} = 47.9273477595231
x5=76.1979436551916x_{5} = 76.1979436551916
x6=46.3201210597594x_{6} = 46.3201210597594
x7=98.1873866154547x_{7} = 98.1873866154547
x8=30.6079266796228x_{8} = 30.6079266796228
x9=41.6454773508261x_{9} = 41.6454773508261
x10=85.6219102787352x_{10} = 85.6219102787352
x11=77.7402391866666x_{11} = 77.7402391866666
x12=40.035545730025x_{12} = 40.035545730025
x13=33.7505979962805x_{13} = 33.7505979962805
x14=91.9046255070275x_{14} = 91.9046255070275
x15=90.307633577564x_{15} = 90.307633577564
x16=60.4917330571858x_{16} = 60.4917330571858
x17=99.7330487258588x_{17} = 99.7330487258588
x18=2.27289720229x_{18} = 2.27289720229
x19=96.5912540731082x_{19} = 96.5912540731082
x20=66.7741432614445x_{20} = 66.7741432614445
x21=27.4650749480003x_{21} = 27.4650749480003
x22=3.9897357129442x_{22} = 3.9897357129442
x23=62.030581139795x_{23} = 62.030581139795
x24=52.6044390279752x_{24} = 52.6044390279752
x25=24.3219861353816x_{25} = 24.3219861353816
x26=68.3145143664508x_{26} = 68.3145143664508
x27=18.0347063110013x_{27} = 18.0347063110013
x28=55.7465247112345x_{28} = 55.7465247112345
x29=10.2492607645079x_{29} = 10.2492607645079
x30=88.7632616524825x_{30} = 88.7632616524825
x31=63.6329238042827x_{31} = 63.6329238042827
x32=285.092129878266x_{32} = 285.092129878266
x33=84.0239651193739x_{33} = 84.0239651193739
x34=102.874834050437x_{34} = 102.874834050437
x35=38.5046598151723x_{35} = 38.5046598151723
x36=54.2094539898584x_{36} = 54.2094539898584
x37=69.9153881658923x_{37} = 69.9153881658923
x38=16.5241312439325x_{38} = 16.5241312439325
x39=82.4805725632931x_{39} = 82.4805725632931
x40=32.2233521960491x_{40} = 32.2233521960491
x41=74.598350939738x_{41} = 74.598350939738
x42=25.9426908359683x_{42} = 25.9426908359683
Signos de extremos en los puntos:
(8.596695130536403, 0.935653879039105)

(19.663158280279568, 4.43272545641072)

(11.744465182561358, 1.42968424630562)

(47.927347759523066, 6.92230577964855)

(76.19794365519158, 8.72873315155156)

(46.32012105975938, 4.80656358517176)

(98.18738661545474, 9.90863655033785)

(30.607926679622768, 3.53346460133345)

(41.64547735082608, 6.45258005425876)

(85.62191027873519, 9.25284576585701)

(77.74023918666657, 6.817444602508)

(40.035545730025, 4.32814569424704)

(33.75059799628046, 3.81045282498023)

(91.90462550702755, 9.58634993406283)

(90.30763357756403, 7.50337896136872)

(60.491733057185805, 7.77712642266359)

(99.73304872585881, 7.9869569020713)

(2.2728972022899994, -0.478542107500169)

(96.59125407310819, 7.82840854529554)

(66.77414326144448, 8.17107662412513)

(27.465074948000275, 3.24185166655725)

(3.9897357129441984, 1.98956876348278)

(62.030581139794954, 5.87645344966414)

(52.604439027975154, 5.25348600293076)

(24.321986135381643, 2.93301824153435)

(68.31451436645078, 6.26571703345779)

(18.034706311001298, 2.24846316935813)

(55.74652471123453, 5.46692026562848)

(10.249260764507943, 3.19839300463544)

(88.76326165248251, 9.42107354778489)

(63.63292380428267, 7.97653352775272)

(285.0921298782659, 14.8847810668489)

(84.02396511937387, 7.16683068801081)

(102.87483405043672, 8.14302702018185)

(38.5046598151723, 6.20440039055312)

(54.209453989858396, 7.36213034036371)

(69.91538816589225, 8.36109514339348)

(16.52413124393248, 4.06309550266006)

(82.48057256329305, 9.0815026052331)

(32.223352196049056, 5.67559138834095)

(74.59835093973801, 6.637452686544)

(25.942690835968342, 5.09219148301726)


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=8.5966951305364x_{1} = 8.5966951305364
x2=11.7444651825614x_{2} = 11.7444651825614
x3=46.3201210597594x_{3} = 46.3201210597594
x4=30.6079266796228x_{4} = 30.6079266796228
x5=77.7402391866666x_{5} = 77.7402391866666
x6=40.035545730025x_{6} = 40.035545730025
x7=33.7505979962805x_{7} = 33.7505979962805
x8=90.307633577564x_{8} = 90.307633577564
x9=99.7330487258588x_{9} = 99.7330487258588
x10=2.27289720229x_{10} = 2.27289720229
x11=96.5912540731082x_{11} = 96.5912540731082
x12=27.4650749480003x_{12} = 27.4650749480003
x13=62.030581139795x_{13} = 62.030581139795
x14=52.6044390279752x_{14} = 52.6044390279752
x15=24.3219861353816x_{15} = 24.3219861353816
x16=68.3145143664508x_{16} = 68.3145143664508
x17=18.0347063110013x_{17} = 18.0347063110013
x18=55.7465247112345x_{18} = 55.7465247112345
x19=285.092129878266x_{19} = 285.092129878266
x20=84.0239651193739x_{20} = 84.0239651193739
x21=102.874834050437x_{21} = 102.874834050437
x22=74.598350939738x_{22} = 74.598350939738
Puntos máximos de la función:
x22=19.6631582802796x_{22} = 19.6631582802796
x22=47.9273477595231x_{22} = 47.9273477595231
x22=76.1979436551916x_{22} = 76.1979436551916
x22=98.1873866154547x_{22} = 98.1873866154547
x22=41.6454773508261x_{22} = 41.6454773508261
x22=85.6219102787352x_{22} = 85.6219102787352
x22=91.9046255070275x_{22} = 91.9046255070275
x22=60.4917330571858x_{22} = 60.4917330571858
x22=66.7741432614445x_{22} = 66.7741432614445
x22=3.9897357129442x_{22} = 3.9897357129442
x22=10.2492607645079x_{22} = 10.2492607645079
x22=88.7632616524825x_{22} = 88.7632616524825
x22=63.6329238042827x_{22} = 63.6329238042827
x22=38.5046598151723x_{22} = 38.5046598151723
x22=54.2094539898584x_{22} = 54.2094539898584
x22=69.9153881658923x_{22} = 69.9153881658923
x22=16.5241312439325x_{22} = 16.5241312439325
x22=82.4805725632931x_{22} = 82.4805725632931
x22=32.2233521960491x_{22} = 32.2233521960491
x22=25.9426908359683x_{22} = 25.9426908359683
Decrece en los intervalos
[285.092129878266,∞)\left[285.092129878266, \infty\right)
Crece en los intervalos
(−∞,2.27289720229]\left(-\infty, 2.27289720229\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
−(4sin⁡(2x)+14x32)=0- (4 \sin{\left(2 x \right)} + \frac{1}{4 x^{\frac{3}{2}}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=21.9908455440609x_{1} = 21.9908455440609
x2=6.28120018514314x_{2} = 6.28120018514314
x3=12.5656690431817x_{3} = 12.5656690431817
x4=72.2565801541099x_{4} = 72.2565801541099
x5=15.7074612825354x_{5} = 15.7074612825354
x6=65.973387408161x_{6} = 65.973387408161
x7=89.5354275129773x_{7} = 89.5354275129773
x8=58.1195346202658x_{8} = 58.1195346202658
x9=92.6770183070118x_{9} = 92.6770183070118
x10=50.2653947681823x_{10} = 50.2653947681823
x11=23.5622181302895x_{11} = 23.5622181302895
x12=4.7154408731027x_{12} = 4.7154408731027
x13=31.4157490642828x_{13} = 31.4157490642828
x14=56.5485942764682x_{14} = 56.5485942764682
x15=34.5573653601779x_{15} = 34.5573653601779
x16=36.1284594212758x_{16} = 36.1284594212758
x17=73.8274766226017x_{17} = 73.8274766226017
x18=59.6901926547493x_{18} = 59.6901926547493
x19=204.203511774181x_{19} = 204.203511774181
x20=42.411613965195x_{20} = 42.411613965195
x21=28.2741260245446x_{21} = 28.2741260245446
x22=14.1377548086575x_{22} = 14.1377548086575
x23=20.4206908931308x_{23} = 20.4206908931308
x24=51.836362517539x_{24} = 51.836362517539
x25=94.2477454535371x_{25} = 94.2477454535371
x26=87.9645564223915x_{26} = 87.9645564223915
x27=48.6947780964597x_{27} = 48.6947780964597
x28=64.4027098621642x_{28} = 64.4027098621642
x29=70.6858872894474x_{29} = 70.6858872894474
x30=86.3938368895118x_{30} = 86.3938368895118
x31=80.1106562492855x_{31} = 80.1106562492855
x32=78.5397714429257x_{32} = 78.5397714429257
x33=81.6813666616522x_{33} = 81.6813666616522
x34=100.530933912107x_{34} = 100.530933912107
x35=7.85540101195812x_{35} = 7.85540101195812
x36=1.5864381063943x_{36} = 1.5864381063943
x37=37.6989768362656x_{37} = 37.6989768362656
x38=67.5442983468121x_{38} = 67.5442983468121
x39=26.7037640154149x_{39} = 26.7037640154149
x40=43.9821900144244x_{40} = 43.9821900144244
x41=29.8453218709336x_{41} = 29.8453218709336
x42=45.5531951187142x_{42} = 45.5531951187142
x43=95.8186092522051x_{43} = 95.8186092522051

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
[95.8186092522051,∞)\left[95.8186092522051, \infty\right)
Convexa en los intervalos
(−∞,1.5864381063943]\left(-\infty, 1.5864381063943\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x−1)+sin⁡(2x))=⟨−2,0⟩+∞i\lim_{x \to -\infty}\left(\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)}\right) = \left\langle -2, 0\right\rangle + \infty i
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,0⟩+∞iy = \left\langle -2, 0\right\rangle + \infty i
lim⁡x→∞((x−1)+sin⁡(2x))=∞\lim_{x \to \infty}\left(\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x) - 1 + sin(2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x−1)+sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((x−1)+sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\left(\sqrt{x} - 1\right) + \sin{\left(2 x \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:
(x−1)+sin⁡(2x)=−x−sin⁡(2x)−1\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)} = \sqrt{- x} - \sin{\left(2 x \right)} - 1
- No
(x−1)+sin⁡(2x)=−−x+sin⁡(2x)+1\left(\sqrt{x} - 1\right) + \sin{\left(2 x \right)} = - \sqrt{- x} + \sin{\left(2 x \right)} + 1
- No
es decir, función
no es
par ni impar
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