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Trazado el gráfico de la función/ (2-x)*(sin(|x|)+cos(|x|))

Gráfico de la función y = (2-x)*(sin(|x|)+cos(|x|))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = (2 - x)*(sin(|x|) + cos(|x|))
f(x)=(2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))f{\left(x \right)} = \left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right) 
f = (2 - x)*(sin(|x|) + cos(|x|))
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
(2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))=0\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2x_{1} = 2
Solución numérica
x1=84.037603483527x_{1} = 84.037603483527
x2=−36.9137136796801x_{2} = -36.9137136796801
x3=87.1791961371168x_{3} = 87.1791961371168
x4=49.4800842940392x_{4} = 49.4800842940392
x5=−52.621676947629x_{5} = -52.621676947629
x6=−33.7721210260903x_{6} = -33.7721210260903
x7=52.621676947629x_{7} = 52.621676947629
x8=−8.63937979737193x_{8} = -8.63937979737193
x9=90.3207887907066x_{9} = 90.3207887907066
x10=11.7809724509617x_{10} = 11.7809724509617
x11=−14.9225651045515x_{11} = -14.9225651045515
x12=93.4623814442964x_{12} = 93.4623814442964
x13=−62.0464549083984x_{13} = -62.0464549083984
x14=−99.7455667514759x_{14} = -99.7455667514759
x15=74.6128255227576x_{15} = 74.6128255227576
x16=68.329640215578x_{16} = 68.329640215578
x17=−96.6039740978861x_{17} = -96.6039740978861
x18=−90.3207887907066x_{18} = -90.3207887907066
x19=−24.3473430653209x_{19} = -24.3473430653209
x20=−87.1791961371168x_{20} = -87.1791961371168
x21=−30.6305283725005x_{21} = -30.6305283725005
x22=36.9137136796801x_{22} = 36.9137136796801
x23=43.1968989868597x_{23} = 43.1968989868597
x24=8.63937979737193x_{24} = 8.63937979737193
x25=−80.8960108299372x_{25} = -80.8960108299372
x26=−71.4712328691678x_{26} = -71.4712328691678
x27=18.0641577581413x_{27} = 18.0641577581413
x28=−46.3384916404494x_{28} = -46.3384916404494
x29=−65.1880475619882x_{29} = -65.1880475619882
x30=−11.7809724509617x_{30} = -11.7809724509617
x31=58.9048622548086x_{31} = 58.9048622548086
x32=14.9225651045515x_{32} = 14.9225651045515
x33=−49.4800842940392x_{33} = -49.4800842940392
x34=46.3384916404494x_{34} = 46.3384916404494
x35=−93.4623814442964x_{35} = -93.4623814442964
x36=−74.6128255227576x_{36} = -74.6128255227576
x37=−2.35619449019234x_{37} = -2.35619449019234
x38=−77.7544181763474x_{38} = -77.7544181763474
x39=−43.1968989868597x_{39} = -43.1968989868597
x40=77.7544181763474x_{40} = 77.7544181763474
x41=−55.7632696012188x_{41} = -55.7632696012188
x42=96.6039740978861x_{42} = 96.6039740978861
x43=−40.0553063332699x_{43} = -40.0553063332699
x44=24.3473430653209x_{44} = 24.3473430653209
x45=−5.49778714378214x_{45} = -5.49778714378214
x46=−27.4889357189107x_{46} = -27.4889357189107
x47=−84.037603483527x_{47} = -84.037603483527
x48=80.8960108299372x_{48} = 80.8960108299372
x49=40.0553063332699x_{49} = 40.0553063332699
x50=−58.9048622548086x_{50} = -58.9048622548086
x51=30.6305283725005x_{51} = 30.6305283725005
x52=71.4712328691678x_{52} = 71.4712328691678
x53=−68.329640215578x_{53} = -68.329640215578
x54=65.1880475619882x_{54} = 65.1880475619882
x55=55.7632696012188x_{55} = 55.7632696012188
x56=62.0464549083984x_{56} = 62.0464549083984
x57=27.4889357189107x_{57} = 27.4889357189107
x58=21.2057504117311x_{58} = 21.2057504117311
x59=99.7455667514759x_{59} = 99.7455667514759
x60=5.49778714378214x_{60} = 5.49778714378214
x61=2.35619449019234x_{61} = 2.35619449019234
x62=−21.2057504117311x_{62} = -21.2057504117311
x63=−18.0641577581413x_{63} = -18.0641577581413
x64=2x_{64} = 2
x65=33.7721210260903x_{65} = 33.7721210260903
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 (2 - x)*(sin(|x|) + cos(|x|)).
(2−0)(sin⁡(∣0∣)+cos⁡(∣0∣))\left(2 - 0\right) \left(\sin{\left(\left|{0}\right| \right)} + \cos{\left(\left|{0}\right| \right)}\right)
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
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
(2−x)(−sin⁡(∣x∣)sign⁡(x)+cos⁡(∣x∣)sign⁡(x))−sin⁡(∣x∣)−cos⁡(∣x∣)=0\left(2 - x\right) \left(- \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} + \cos{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)}\right) - \sin{\left(\left|{x}\right| \right)} - \cos{\left(\left|{x}\right| \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=76.1970986495364x_{1} = 76.1970986495364
x2=−51.06972152751x_{2} = -51.06972152751
x3=10.3296572569379x_{3} = 10.3296572569379
x4=−41.6490086808161x_{4} = -41.6490086808161
x5=13.4389675712559x_{5} = 13.4389675712559
x6=51.0712563291639x_{6} = 51.0712563291639
x7=−104.467348006709x_{7} = -104.467348006709
x8=−19.6810446586041x_{8} = -19.6810446586041
x9=85.620358050592x_{9} = 85.620358050592
x10=−63.6324864134294x_{10} = -63.6324864134294
x11=−60.4916593514243x_{11} = -60.4916593514243
x12=16.5619260204649x_{12} = 16.5619260204649
x13=−73.0553519079212x_{13} = -73.0553519079212
x14=−82.4786439151159x_{14} = -82.4786439151159
x15=98.1851666635936x_{15} = 98.1851666635936
x16=57.3521301149554x_{16} = 57.3521301149554
x17=−76.1964094634414x_{17} = -76.1964094634414
x18=−91.9022340888408x_{18} = -91.9022340888408
x19=95.0439249692482x_{19} = 95.0439249692482
x20=−1.09766089776906x_{20} = -1.09766089776906
x21=−85.6198122587075x_{21} = -85.6198122587075
x22=−7.1771218627363x_{22} = -7.1771218627363
x23=73.0561016537038x_{23} = 73.0561016537038
x24=2.17903572333602x_{24} = 2.17903572333602
x25=−10.2913553205191x_{25} = -10.2913553205191
x26=−25.9538973310451x_{26} = -25.9538973310451
x27=29.0966202763459x_{27} = 29.0966202763459
x28=−16.547225697059x_{28} = -16.547225697059
x29=44.7910604279375x_{29} = 44.7910604279375
x30=−98.1847516521132x_{30} = -98.1847516521132
x31=47.9310562896495x_{31} = 47.9310562896495
x32=88.7615178008025x_{32} = 88.7615178008025
x33=−54.210261743828x_{33} = -54.210261743828
x34=25.9598516687689x_{34} = 25.9598516687689
x35=−22.8168202015565x_{35} = -22.8168202015565
x36=−35.3696706368231x_{36} = -35.3696706368231
x37=−38.5091907499526x_{37} = -38.5091907499526
x38=38.5118914986711x_{38} = 38.5118914986711
x39=−13.4165434160847x_{39} = -13.4165434160847
x40=32.2343875680821x_{40} = 32.2343875680821
x41=−88.7610099651303x_{41} = -88.7610099651303
x42=63.6334747609328x_{42} = 63.6334747609328
x43=79.3381440127403x_{43} = 79.3381440127403
x44=−95.0434820654718x_{44} = -95.0434820654718
x45=−57.3509132742806x_{45} = -57.3509132742806
x46=69.9151597315029x_{46} = 69.9151597315029
x47=101.326430551848x_{47} = 101.326430551848
x48=22.8245301698031x_{48} = 22.8245301698031
x49=7.25657505489695x_{49} = 7.25657505489695
x50=35.372872846264x_{50} = 35.372872846264
x51=−69.9143410785029x_{51} = -69.9143410785029
x52=−29.0918836969137x_{52} = -29.0918836969137
x53=−44.7890645756672x_{53} = -44.7890645756672
x54=0.263027254706413x_{54} = 0.263027254706413
x55=4.33207755419064x_{55} = 4.33207755419064
x56=−66.7733833731654x_{56} = -66.7733833731654
x57=−47.9293136044675x_{57} = -47.9293136044675
x58=66.7742808886442x_{58} = 66.7742808886442
x59=54.211623756215x_{59} = 54.211623756215
x60=−79.3375083346173x_{60} = -79.3375083346173
x61=−32.2305300800886x_{61} = -32.2305300800886
x62=19.6914185805939x_{62} = 19.6914185805939
x63=41.6513171583218x_{63} = 41.6513171583218
x64=−4.08974865556062x_{64} = -4.08974865556062
x65=91.9027077942952x_{65} = 91.9027077942952
x66=60.4927530512426x_{66} = 60.4927530512426
x67=82.4792320831881x_{67} = 82.4792320831881
Signos de extremos en los puntos:
(76.19709864953641, -104.921014383413)

(-51.06972152750996, 75.0385993731921)

(10.329657256937894, 11.695930741592)

(-41.64900868081614, -61.7128265988845)

(13.438967571255922, -16.1156795372477)

(51.07125632916393, -69.3828309142045)

(-104.46734800670926, -150.560926403858)

(-19.681044658604105, 30.6290652918276)

(85.62035805059199, 118.248589196674)

(-63.63248641342937, 92.8075805621684)

(-60.49165935142426, -88.3652391441028)

(16.561926020464906, 20.545285744725)

(-73.05535190792122, -106.134876713605)

(-82.47864391511587, 119.462474592143)

(98.1851666635936, 136.019016274793)

(57.352130114955415, -78.2669615424326)

(-76.19640946344138, 110.577381199668)

(-91.90223408884084, -132.790283381517)

(95.04392496924822, -131.576381536968)

(-1.0976608977690603, 4.16890458034267)

(-85.61981225870753, -123.90505744714)

(-7.177121862736295, 12.9020386046531)

(73.05610165370385, 100.478552738806)

(2.17903572333602, -0.0446214030337907)

(-10.291355320519108, -17.325356632034)

(-25.95389733104508, 39.5075095160384)

(29.09662027634593, 38.2943387597944)

(-16.547225697058977, -26.1916963792683)

(44.79106042793752, -60.4991801329014)

(-98.18475165211325, -141.675577028768)

(47.93105628964952, 64.9409332496534)

(88.76151780080245, -122.691165973158)

(-54.21026174382804, -79.4807378137109)

(25.959851668768867, -33.8548735308394)

(-22.81682020155651, -35.0678253101469)

(-35.36967063682312, -52.8297832490745)

(-38.509190749952644, 57.2711994676464)

(38.511891498671076, -51.6162565548779)

(-13.416543416084677, 21.7565622647035)

(32.23438756808213, -42.7345126185387)

(-88.76100996513028, 128.347661071567)

(63.6334747609328, -87.1514253961028)

(79.3381440127403, 109.363510244489)

(-95.04348206547185, 137.232922562819)

(-57.35091327428056, 83.92295502778)

(69.91515973150292, -96.0361300564761)

(101.32643055184802, -140.461666710328)

(22.824530169803058, 29.4164361347947)

(7.256575054896953, -7.30294567629717)

(35.37287284626396, 47.1751955814942)

(-69.91434107850294, 101.692405282587)

(-29.091883696913687, -43.9478387453793)

(-44.78906457566725, 66.1546222185845)

(0.2630272547064127, 2.12885495459675)

(4.33207755419064, 3.03113817900633)

(-66.7733833731654, -97.2499714341767)

(-47.929313604467495, -70.5965545616187)

(66.77428088864424, 91.5937520022301)

(54.211623756215026, 73.8248470632575)

(-79.33750833461727, -115.019914912285)

(-32.230530080088585, 48.3886359029811)

(19.691418580593886, -24.9795707027244)

(41.65131715832176, 56.0576058715679)

(-4.089748655560621, -8.49838757857203)

(91.90270779429521, 127.133764141988)

(60.49275305124256, 82.7091585219687)

(82.4792320831881, -113.806036316801)


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=76.1970986495364x_{1} = 76.1970986495364
x2=−41.6490086808161x_{2} = -41.6490086808161
x3=13.4389675712559x_{3} = 13.4389675712559
x4=51.0712563291639x_{4} = 51.0712563291639
x5=−104.467348006709x_{5} = -104.467348006709
x6=−60.4916593514243x_{6} = -60.4916593514243
x7=−73.0553519079212x_{7} = -73.0553519079212
x8=57.3521301149554x_{8} = 57.3521301149554
x9=−91.9022340888408x_{9} = -91.9022340888408
x10=95.0439249692482x_{10} = 95.0439249692482
x11=−85.6198122587075x_{11} = -85.6198122587075
x12=2.17903572333602x_{12} = 2.17903572333602
x13=−10.2913553205191x_{13} = -10.2913553205191
x14=−16.547225697059x_{14} = -16.547225697059
x15=44.7910604279375x_{15} = 44.7910604279375
x16=−98.1847516521132x_{16} = -98.1847516521132
x17=88.7615178008025x_{17} = 88.7615178008025
x18=−54.210261743828x_{18} = -54.210261743828
x19=25.9598516687689x_{19} = 25.9598516687689
x20=−22.8168202015565x_{20} = -22.8168202015565
x21=−35.3696706368231x_{21} = -35.3696706368231
x22=38.5118914986711x_{22} = 38.5118914986711
x23=32.2343875680821x_{23} = 32.2343875680821
x24=63.6334747609328x_{24} = 63.6334747609328
x25=69.9151597315029x_{25} = 69.9151597315029
x26=101.326430551848x_{26} = 101.326430551848
x27=7.25657505489695x_{27} = 7.25657505489695
x28=−29.0918836969137x_{28} = -29.0918836969137
x29=−66.7733833731654x_{29} = -66.7733833731654
x30=−47.9293136044675x_{30} = -47.9293136044675
x31=−79.3375083346173x_{31} = -79.3375083346173
x32=19.6914185805939x_{32} = 19.6914185805939
x33=−4.08974865556062x_{33} = -4.08974865556062
x34=82.4792320831881x_{34} = 82.4792320831881
Puntos máximos de la función:
x34=−51.06972152751x_{34} = -51.06972152751
x34=10.3296572569379x_{34} = 10.3296572569379
x34=−19.6810446586041x_{34} = -19.6810446586041
x34=85.620358050592x_{34} = 85.620358050592
x34=−63.6324864134294x_{34} = -63.6324864134294
x34=16.5619260204649x_{34} = 16.5619260204649
x34=−82.4786439151159x_{34} = -82.4786439151159
x34=98.1851666635936x_{34} = 98.1851666635936
x34=−76.1964094634414x_{34} = -76.1964094634414
x34=−1.09766089776906x_{34} = -1.09766089776906
x34=−7.1771218627363x_{34} = -7.1771218627363
x34=73.0561016537038x_{34} = 73.0561016537038
x34=−25.9538973310451x_{34} = -25.9538973310451
x34=29.0966202763459x_{34} = 29.0966202763459
x34=47.9310562896495x_{34} = 47.9310562896495
x34=−38.5091907499526x_{34} = -38.5091907499526
x34=−13.4165434160847x_{34} = -13.4165434160847
x34=−88.7610099651303x_{34} = -88.7610099651303
x34=79.3381440127403x_{34} = 79.3381440127403
x34=−95.0434820654718x_{34} = -95.0434820654718
x34=−57.3509132742806x_{34} = -57.3509132742806
x34=22.8245301698031x_{34} = 22.8245301698031
x34=35.372872846264x_{34} = 35.372872846264
x34=−69.9143410785029x_{34} = -69.9143410785029
x34=−44.7890645756672x_{34} = -44.7890645756672
x34=0.263027254706413x_{34} = 0.263027254706413
x34=4.33207755419064x_{34} = 4.33207755419064
x34=66.7742808886442x_{34} = 66.7742808886442
x34=54.211623756215x_{34} = 54.211623756215
x34=−32.2305300800886x_{34} = -32.2305300800886
x34=41.6513171583218x_{34} = 41.6513171583218
x34=91.9027077942952x_{34} = 91.9027077942952
x34=60.4927530512426x_{34} = 60.4927530512426
Decrece en los intervalos
[101.326430551848,∞)\left[101.326430551848, \infty\right)
Crece en los intervalos
(−∞,−104.467348006709]\left(-\infty, -104.467348006709\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
(x−2)(2sin⁡(∣x∣)δ(x)+sin⁡(∣x∣)sign⁡2(x)−2cos⁡(∣x∣)δ(x)+cos⁡(∣x∣)sign⁡2(x))+2sin⁡(∣x∣)sign⁡(x)−2cos⁡(∣x∣)sign⁡(x)=0\left(x - 2\right) \left(2 \sin{\left(\left|{x}\right| \right)} \delta\left(x\right) + \sin{\left(\left|{x}\right| \right)} \operatorname{sign}^{2}{\left(x \right)} - 2 \cos{\left(\left|{x}\right| \right)} \delta\left(x\right) + \cos{\left(\left|{x}\right| \right)} \operatorname{sign}^{2}{\left(x \right)}\right) + 2 \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} - 2 \cos{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \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→−∞((2−x)(sin⁡(∣x∣)+cos⁡(∣x∣)))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((2−x)(sin⁡(∣x∣)+cos⁡(∣x∣)))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (2 - x)*(sin(|x|) + cos(|x|)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))x)=⟨−2,2⟩\lim_{x \to -\infty}\left(\frac{\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)}{x}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−2,2⟩xy = \left\langle -2, 2\right\rangle x
lim⁡x→∞((2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))x)=⟨−2,2⟩\lim_{x \to \infty}\left(\frac{\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)}{x}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−2,2⟩xy = \left\langle -2, 2\right\rangle x
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:
(2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))=(x+2)(sin⁡(∣x∣)+cos⁡(∣x∣))\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right) = \left(x + 2\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)
- No
(2−x)(sin⁡(∣x∣)+cos⁡(∣x∣))=−(x+2)(sin⁡(∣x∣)+cos⁡(∣x∣))\left(2 - x\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right) = - \left(x + 2\right) \left(\sin{\left(\left|{x}\right| \right)} + \cos{\left(\left|{x}\right| \right)}\right)
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор