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Trazado el gráfico de la función/ abs(cos(x)+sin(x))

Gráfico de la función y = abs(cos(x)+sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = |cos(x) + sin(x)|
f(x)=∣sin⁡(x)+cos⁡(x)∣f{\left(x \right)} = \left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| 
f = Abs(sin(x) + cos(x))
Gráfico de la función
02468-8-6-4-2-101002
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:
∣sin⁡(x)+cos⁡(x)∣=0\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π4x_{1} = - \frac{\pi}{4}
Solución numérica
x1=93.4623814442964x_{1} = 93.4623814442964
x2=21.2057504117311x_{2} = 21.2057504117311
x3=43.1968989868597x_{3} = 43.1968989868597
x4=90.3207887907066x_{4} = 90.3207887907066
x5=−10.2101761241668x_{5} = -10.2101761241668
x6=−19.6349540849362x_{6} = -19.6349540849362
x7=71.4712328691678x_{7} = 71.4712328691678
x8=−44.7676953136546x_{8} = -44.7676953136546
x9=−41.6261026600648x_{9} = -41.6261026600648
x10=11.7809724509617x_{10} = 11.7809724509617
x11=8.63937979737193x_{11} = 8.63937979737193
x12=−66.7588438887831x_{12} = -66.7588438887831
x13=−60.4756585816035x_{13} = -60.4756585816035
x14=27.4889357189107x_{14} = 27.4889357189107
x15=−63618.0366333567x_{15} = -63618.0366333567
x16=74.6128255227576x_{16} = 74.6128255227576
x17=−29.0597320457056x_{17} = -29.0597320457056
x18=40.0553063332699x_{18} = 40.0553063332699
x19=−85.6083998103219x_{19} = -85.6083998103219
x20=33.7721210260903x_{20} = 33.7721210260903
x21=84.037603483527x_{21} = 84.037603483527
x22=−57.3340659280137x_{22} = -57.3340659280137
x23=−73.0420291959627x_{23} = -73.0420291959627
x24=−54.1924732744239x_{24} = -54.1924732744239
x25=−76.1836218495525x_{25} = -76.1836218495525
x26=−32.2013246992954x_{26} = -32.2013246992954
x27=30.6305283725005x_{27} = 30.6305283725005
x28=−82.4668071567321x_{28} = -82.4668071567321
x29=−38.484510006475x_{29} = -38.484510006475
x30=96.6039740978861x_{30} = 96.6039740978861
x31=−22.776546738526x_{31} = -22.776546738526
x32=−79.3252145031423x_{32} = -79.3252145031423
x33=2.35619449019234x_{33} = 2.35619449019234
x34=68.329640215578x_{34} = 68.329640215578
x35=−13.3517687777566x_{35} = -13.3517687777566
x36=−88.7499924639117x_{36} = -88.7499924639117
x37=80.8960108299372x_{37} = 80.8960108299372
x38=−51.0508806208341x_{38} = -51.0508806208341
x39=137.444678594553x_{39} = 137.444678594553
x40=36.9137136796801x_{40} = 36.9137136796801
x41=−7.06858347057703x_{41} = -7.06858347057703
x42=−16.4933614313464x_{42} = -16.4933614313464
x43=49.4800842940392x_{43} = 49.4800842940392
x44=62.0464549083984x_{44} = 62.0464549083984
x45=−98.174770424681x_{45} = -98.174770424681
x46=46.3384916404494x_{46} = 46.3384916404494
x47=52.621676947629x_{47} = 52.621676947629
x48=24.3473430653209x_{48} = 24.3473430653209
x49=−107.59954838545x_{49} = -107.59954838545
x50=−0.785398163397448x_{50} = -0.785398163397448
x51=5.49778714378214x_{51} = 5.49778714378214
x52=285.099533313274x_{52} = 285.099533313274
x53=−63.6172512351933x_{53} = -63.6172512351933
x54=99.7455667514759x_{54} = 99.7455667514759
x55=58.9048622548086x_{55} = 58.9048622548086
x56=14.9225651045515x_{56} = 14.9225651045515
x57=−91.8915851175014x_{57} = -91.8915851175014
x58=55.7632696012188x_{58} = 55.7632696012188
x59=−95.0331777710912x_{59} = -95.0331777710912
x60=−35.3429173528852x_{60} = -35.3429173528852
x61=77.7544181763474x_{61} = 77.7544181763474
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 Abs(cos(x) + sin(x)).
∣sin⁡(0)+cos⁡(0)∣\left|{\sin{\left(0 \right)} + \cos{\left(0 \right)}}\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
(−sin⁡(x)+cos⁡(x))sign⁡(sin⁡(x)+cos⁡(x))=0\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−52.621676947629x_{1} = -52.621676947629
x2=−33.7721210260903x_{2} = -33.7721210260903
x3=60.4756585816035x_{3} = 60.4756585816035
x4=−55.7632696012188x_{4} = -55.7632696012188
x5=−96.6039740978861x_{5} = -96.6039740978861
x6=13.3517687777566x_{6} = 13.3517687777566
x7=−106.028752058656x_{7} = -106.028752058656
x8=95.0331777710912x_{8} = 95.0331777710912
x9=32.2013246992954x_{9} = 32.2013246992954
x10=25.9181393921158x_{10} = 25.9181393921158
x11=66.7588438887831x_{11} = 66.7588438887831
x12=−6646.8246568326x_{12} = -6646.8246568326
x13=51.0508806208341x_{13} = 51.0508806208341
x14=−71.4712328691678x_{14} = -71.4712328691678
x15=41.6261026600648x_{15} = 41.6261026600648
x16=63.6172512351933x_{16} = 63.6172512351933
x17=−8.63937979737193x_{17} = -8.63937979737193
x18=88.7499924639117x_{18} = 88.7499924639117
x19=−99.7455667514759x_{19} = -99.7455667514759
x20=−58.9048622548086x_{20} = -58.9048622548086
x21=−18.0641577581413x_{21} = -18.0641577581413
x22=−354.214571692249x_{22} = -354.214571692249
x23=−1881.02860133689x_{23} = -1881.02860133689
x24=3.92699081698724x_{24} = 3.92699081698724
x25=−941.692397913541x_{25} = -941.692397913541
x26=73.0420291959627x_{26} = 73.0420291959627
x27=−14.9225651045515x_{27} = -14.9225651045515
x28=38.484510006475x_{28} = 38.484510006475
x29=−77.7544181763474x_{29} = -77.7544181763474
x30=22.776546738526x_{30} = 22.776546738526
x31=−36.9137136796801x_{31} = -36.9137136796801
x32=79.3252145031423x_{32} = 79.3252145031423
x33=82.4668071567321x_{33} = 82.4668071567321
x34=−62.0464549083984x_{34} = -62.0464549083984
x35=16.4933614313464x_{35} = 16.4933614313464
x36=10.2101761241668x_{36} = 10.2101761241668
x37=76.1836218495525x_{37} = 76.1836218495525
x38=85.6083998103219x_{38} = 85.6083998103219
x39=−2.35619449019234x_{39} = -2.35619449019234
x40=−21.2057504117311x_{40} = -21.2057504117311
x41=7.06858347057703x_{41} = 7.06858347057703
x42=57.3340659280137x_{42} = 57.3340659280137
x43=35.3429173528852x_{43} = 35.3429173528852
x44=−68.329640215578x_{44} = -68.329640215578
x45=98.174770424681x_{45} = 98.174770424681
x46=0.785398163397448x_{46} = 0.785398163397448
x47=−24.3473430653209x_{47} = -24.3473430653209
x48=−11.7809724509617x_{48} = -11.7809724509617
x49=−90.3207887907066x_{49} = -90.3207887907066
x50=−74.6128255227576x_{50} = -74.6128255227576
x51=−46.3384916404494x_{51} = -46.3384916404494
x52=19.6349540849362x_{52} = 19.6349540849362
x53=69.9004365423729x_{53} = 69.9004365423729
x54=−27.4889357189107x_{54} = -27.4889357189107
x55=44.7676953136546x_{55} = 44.7676953136546
x56=−30.6305283725005x_{56} = -30.6305283725005
x57=−259.966792084555x_{57} = -259.966792084555
x58=−84.037603483527x_{58} = -84.037603483527
x59=−49.4800842940392x_{59} = -49.4800842940392
x60=−40.0553063332699x_{60} = -40.0553063332699
x61=−65.1880475619882x_{61} = -65.1880475619882
x62=−5.49778714378214x_{62} = -5.49778714378214
x63=47.9092879672443x_{63} = 47.9092879672443
x64=−43.1968989868597x_{64} = -43.1968989868597
x65=7760.51925253019x_{65} = 7760.51925253019
x66=−80.8960108299372x_{66} = -80.8960108299372
x67=54.1924732744239x_{67} = 54.1924732744239
x68=101.316363078271x_{68} = 101.316363078271
x69=−93.4623814442964x_{69} = -93.4623814442964
x70=29.0597320457056x_{70} = 29.0597320457056
x71=91.8915851175014x_{71} = 91.8915851175014
x72=−87.1791961371168x_{72} = -87.1791961371168
Signos de extremos en los puntos:
(-52.621676947629034, 1.41421356237309)

(-33.772121026090275, 1.41421356237309)

(60.47565858160352, 1.41421356237309)

(-55.76326960121883, 1.41421356237309)

(-96.60397409788614, 1.41421356237309)

(13.351768777756622, 1.41421356237309)

(-106.02875205865553, 1.41421356237309)

(95.03317777109125, 1.41421356237309)

(32.201324699295384, 1.41421356237309)

(25.918139392115794, 1.41421356237309)

(66.7588438887831, 1.4142135623731)

(-6646.824656832605, 1.41421356237309)

(51.05088062083414, 1.41421356237309)

(-71.47123286916779, 1.41421356237309)

(41.62610266006476, 1.41421356237309)

(63.617251235193315, 1.41421356237309)

(-8.639379797371932, 1.41421356237309)

(88.74999246391165, 1.4142135623731)

(-99.74556675147593, 1.41421356237309)

(-58.90486225480862, 1.41421356237309)

(-18.06415775814131, 1.41421356237309)

(-354.2145716922492, 1.41421356237309)

(-1881.0286013368886, 1.41421356237309)

(3.9269908169872414, 1.4142135623731)

(-941.6923979135405, 1.41421356237309)

(73.0420291959627, 1.41421356237309)

(-14.922565104551518, 1.41421356237309)

(38.48451000647497, 1.41421356237309)

(-77.75441817634739, 1.41421356237309)

(22.776546738526, 1.4142135623731)

(-36.91371367968007, 1.41421356237309)

(79.32521450314228, 1.41421356237309)

(82.46680715673207, 1.41421356237309)

(-62.04645490839842, 1.41421356237309)

(16.493361431346415, 1.41421356237309)

(10.210176124166829, 1.41421356237309)

(76.18362184955248, 1.41421356237309)

(85.60839981032187, 1.41421356237309)

(-2.356194490192345, 1.41421356237309)

(-21.205750411731103, 1.4142135623731)

(7.0685834705770345, 1.41421356237309)

(57.33406592801373, 1.41421356237309)

(35.34291735288517, 1.41421356237309)

(-68.329640215578, 1.41421356237309)

(98.17477042468104, 1.41421356237309)

(0.7853981633974483, 1.41421356237309)

(-24.3473430653209, 1.4142135623731)

(-11.780972450961725, 1.41421356237309)

(-90.32078879070656, 1.41421356237309)

(-74.61282552275759, 1.41421356237309)

(-46.33849164044945, 1.41421356237309)

(19.634954084936208, 1.41421356237309)

(69.9004365423729, 1.4142135623731)

(-27.488935718910692, 1.41421356237309)

(44.767695313654556, 1.41421356237309)

(-30.630528372500486, 1.41421356237309)

(-259.9667920845554, 1.41421356237309)

(-84.03760348352696, 1.41421356237309)

(-49.480084294039244, 1.41421356237309)

(-40.05530633326986, 1.41421356237309)

(-65.18804756198821, 1.41421356237309)

(-5.497787143782138, 1.41421356237309)

(47.909287967244346, 1.41421356237309)

(-43.19689898685966, 1.41421356237309)

(7760.519252530186, 1.41421356237309)

(-80.89601082993718, 1.41421356237309)

(54.19247327442393, 1.41421356237309)

(101.31636307827083, 1.41421356237309)

(-93.46238144429635, 1.4142135623731)

(29.059732045705587, 1.41421356237309)

(91.89158511750145, 1.41421356237309)

(-87.17919613711676, 1.41421356237309)


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:
x72=−52.621676947629x_{72} = -52.621676947629
x72=−33.7721210260903x_{72} = -33.7721210260903
x72=60.4756585816035x_{72} = 60.4756585816035
x72=−55.7632696012188x_{72} = -55.7632696012188
x72=−96.6039740978861x_{72} = -96.6039740978861
x72=13.3517687777566x_{72} = 13.3517687777566
x72=−106.028752058656x_{72} = -106.028752058656
x72=95.0331777710912x_{72} = 95.0331777710912
x72=32.2013246992954x_{72} = 32.2013246992954
x72=25.9181393921158x_{72} = 25.9181393921158
x72=66.7588438887831x_{72} = 66.7588438887831
x72=−6646.8246568326x_{72} = -6646.8246568326
x72=51.0508806208341x_{72} = 51.0508806208341
x72=−71.4712328691678x_{72} = -71.4712328691678
x72=41.6261026600648x_{72} = 41.6261026600648
x72=63.6172512351933x_{72} = 63.6172512351933
x72=−8.63937979737193x_{72} = -8.63937979737193
x72=88.7499924639117x_{72} = 88.7499924639117
x72=−99.7455667514759x_{72} = -99.7455667514759
x72=−58.9048622548086x_{72} = -58.9048622548086
x72=−18.0641577581413x_{72} = -18.0641577581413
x72=−354.214571692249x_{72} = -354.214571692249
x72=−1881.02860133689x_{72} = -1881.02860133689
x72=3.92699081698724x_{72} = 3.92699081698724
x72=−941.692397913541x_{72} = -941.692397913541
x72=73.0420291959627x_{72} = 73.0420291959627
x72=−14.9225651045515x_{72} = -14.9225651045515
x72=38.484510006475x_{72} = 38.484510006475
x72=−77.7544181763474x_{72} = -77.7544181763474
x72=22.776546738526x_{72} = 22.776546738526
x72=−36.9137136796801x_{72} = -36.9137136796801
x72=79.3252145031423x_{72} = 79.3252145031423
x72=82.4668071567321x_{72} = 82.4668071567321
x72=−62.0464549083984x_{72} = -62.0464549083984
x72=16.4933614313464x_{72} = 16.4933614313464
x72=10.2101761241668x_{72} = 10.2101761241668
x72=76.1836218495525x_{72} = 76.1836218495525
x72=85.6083998103219x_{72} = 85.6083998103219
x72=−2.35619449019234x_{72} = -2.35619449019234
x72=−21.2057504117311x_{72} = -21.2057504117311
x72=7.06858347057703x_{72} = 7.06858347057703
x72=57.3340659280137x_{72} = 57.3340659280137
x72=35.3429173528852x_{72} = 35.3429173528852
x72=−68.329640215578x_{72} = -68.329640215578
x72=98.174770424681x_{72} = 98.174770424681
x72=0.785398163397448x_{72} = 0.785398163397448
x72=−24.3473430653209x_{72} = -24.3473430653209
x72=−11.7809724509617x_{72} = -11.7809724509617
x72=−90.3207887907066x_{72} = -90.3207887907066
x72=−74.6128255227576x_{72} = -74.6128255227576
x72=−46.3384916404494x_{72} = -46.3384916404494
x72=19.6349540849362x_{72} = 19.6349540849362
x72=69.9004365423729x_{72} = 69.9004365423729
x72=−27.4889357189107x_{72} = -27.4889357189107
x72=44.7676953136546x_{72} = 44.7676953136546
x72=−30.6305283725005x_{72} = -30.6305283725005
x72=−259.966792084555x_{72} = -259.966792084555
x72=−84.037603483527x_{72} = -84.037603483527
x72=−49.4800842940392x_{72} = -49.4800842940392
x72=−40.0553063332699x_{72} = -40.0553063332699
x72=−65.1880475619882x_{72} = -65.1880475619882
x72=−5.49778714378214x_{72} = -5.49778714378214
x72=47.9092879672443x_{72} = 47.9092879672443
x72=−43.1968989868597x_{72} = -43.1968989868597
x72=7760.51925253019x_{72} = 7760.51925253019
x72=−80.8960108299372x_{72} = -80.8960108299372
x72=54.1924732744239x_{72} = 54.1924732744239
x72=101.316363078271x_{72} = 101.316363078271
x72=−93.4623814442964x_{72} = -93.4623814442964
x72=29.0597320457056x_{72} = 29.0597320457056
x72=91.8915851175014x_{72} = 91.8915851175014
x72=−87.1791961371168x_{72} = -87.1791961371168
Decrece en los intervalos
(−∞,−6646.8246568326]\left(-\infty, -6646.8246568326\right]
Crece en los intervalos
[7760.51925253019,∞)\left[7760.51925253019, \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
2(sin⁡(x)−cos⁡(x))2δ(sin⁡(x)+cos⁡(x))−(sin⁡(x)+cos⁡(x))sign⁡(sin⁡(x)+cos⁡(x))=02 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞∣sin⁡(x)+cos⁡(x)∣=∣⟨−2,2⟩∣\lim_{x \to -\infty} \left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=∣⟨−2,2⟩∣y = \left|{\left\langle -2, 2\right\rangle}\right|
lim⁡x→∞∣sin⁡(x)+cos⁡(x)∣=∣⟨−2,2⟩∣\lim_{x \to \infty} \left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=∣⟨−2,2⟩∣y = \left|{\left\langle -2, 2\right\rangle}\right|
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(x) + sin(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(∣sin⁡(x)+cos⁡(x)∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right|}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(∣sin⁡(x)+cos⁡(x)∣x)=0\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\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:
∣sin⁡(x)+cos⁡(x)∣=∣sin⁡(x)−cos⁡(x)∣\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| = \left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right|
- No
∣sin⁡(x)+cos⁡(x)∣=−∣sin⁡(x)−cos⁡(x)∣\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| = - \left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right|
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор