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Trazado el gráfico de la función/ cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))

Gráfico de la función y = cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)      /    ____\      /    ____\
f(x) = ------ + cos\x*\/ 19 / + sin\x*\/ 19 /
         18
f(x)=(cos⁡(x)18+cos⁡(19x))+sin⁡(19x)f{\left(x \right)} = \left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} 
f = cos(x)/18 + cos(sqrt(19)*x) + sin(sqrt(19)*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:
(cos⁡(x)18+cos⁡(19x))+sin⁡(19x)=0\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=50.261957629426x_{1} = 50.261957629426
x2=45.9500720824419x_{2} = 45.9500720824419
x3=−67.9322865974644x_{3} = -67.9322865974644
x4=−8.11050797950548x_{4} = -8.11050797950548
x5=−29.7311696744772x_{5} = -29.7311696744772
x6=−55.6706961951692x_{6} = -55.6706961951692
x7=92.0784360063655x_{7} = 92.0784360063655
x8=−34.0624614267879x_{8} = -34.0624614267879
x9=24.3184771377389x_{9} = 24.3184771377389
x10=42.3423105011817x_{10} = 42.3423105011817
x11=−78.0113231567737x_{11} = -78.0113231567737
x12=−80.1806688839985x_{12} = -80.1806688839985
x13=94.2445646011972x_{13} = 94.2445646011972
x14=−73.6935194584558x_{14} = -73.6935194584558
x15=89.9078852723621x_{15} = 89.9078852723621
x16=60.3682235156593x_{16} = 60.3682235156593
x17=−52.0707056975749x_{17} = -52.0707056975749
x18=−63.6109055061927x_{18} = -63.6109055061927
x19=−3.79101548200283x_{19} = -3.79101548200283
x20=74.0571401803626x_{20} = 74.0571401803626
x21=−81.6137673091875x_{21} = -81.6137673091875
x22=−23.9607984009412x_{22} = -23.9607984009412
x23=−11.7178337795199x_{23} = -11.7178337795199
x24=63.9686469875901x_{24} = 63.9686469875901
x25=17.8429060922582x_{25} = 17.8429060922582
x26=−60.0093970056837x_{26} = -60.0093970056837
x27=−70.08598069901x_{27} = -70.08598069901
x28=1.97843663832044x_{28} = 1.97843663832044
x29=71.9013470926892x_{29} = 71.9013470926892
x30=78.3705869951396x_{30} = 78.3705869951396
x31=53.8665468375114x_{31} = 53.8665468375114
x32=−26.1313710467411x_{32} = -26.1313710467411
x33=16.3896259659427x_{33} = 16.3896259659427
x34=58.1983010519632x_{34} = 58.1983010519632
x35=−85.9510076237066x_{35} = -85.9510076237066
x36=−88.1182467246423x_{36} = -88.1182467246423
x37=−16.0277024282486x_{37} = -16.0277024282486
x38=22.1535757048516x_{38} = 22.1535757048516
x39=−65.7755225942878x_{39} = -65.7755225942878
x40=4.1490153050931x_{40} = 4.1490153050931
x41=97.8472894323747x_{41} = 97.8472894323747
x42=56.0289944404762x_{42} = 56.0289944404762
x43=−75.8488005014799x_{43} = -75.8488005014799
x44=76.2110844013443x_{44} = 76.2110844013443
x45=−41.9787888379248x_{45} = -41.9787888379248
x46=43.0582331960684x_{46} = 43.0582331960684
x47=35.8587565877773x_{47} = 35.8587565877773
x48=34.423829083404x_{48} = 34.423829083404
x49=68.2953932558856x_{49} = 68.2953932558856
x50=84.1383367630414x_{50} = 84.1383367630414
x51=−83.7804112988407x_{51} = -83.7804112988407
x52=−13.871703113759x_{52} = -13.871703113759
x53=30.088339871775x_{53} = 30.088339871775
x54=−21.7932676046496x_{54} = -21.7932676046496
x55=−99.6467443289225x_{55} = -99.6467443289225
x56=86.3082811670737x_{56} = 86.3082811670737
x57=9.91798806002494x_{57} = 9.91798806002494
x58=12.0802103138978x_{58} = 12.0802103138978
x59=−1.62119029065689x_{59} = -1.62119029065689
x60=−39.8251268099257x_{60} = -39.8251268099257
x61=96.4027128688431x_{61} = 96.4027128688431
x62=40.1879016101227x_{62} = 40.1879016101227
x63=19.9965731669714x_{63} = 19.9965731669714
x64=−37.6671934984411x_{64} = -37.6671934984411
x65=6.31540430116095x_{65} = 6.31540430116095
x66=−44.1358520314859x_{66} = -44.1358520314859
x67=66.1359452473846x_{67} = 66.1359452473846
x68=27.9198639606168x_{68} = 27.9198639606168
x69=−5.9545609637231x_{69} = -5.9545609637231
x70=32.2587000138451x_{70} = 32.2587000138451
x71=81.9744966773527x_{71} = 81.9744966773527
x72=−45.586523321009x_{72} = -45.586523321009
x73=100.009210993045x_{73} = 100.009210993045
x74=−47.7410635035814x_{74} = -47.7410635035814
x75=38.027082758287x_{75} = 38.027082758287
x76=−19.633531325068x_{76} = -19.633531325068
x77=−91.7203588600024x_{77} = -91.7203588600024
x78=−49.9021808517224x_{78} = -49.9021808517224
x79=−57.8411217154807x_{79} = -57.8411217154807
x80=14.2353161195138x_{80} = 14.2353161195138
x81=11.3483955838268x_{81} = 11.3483955838268
x82=−26.8485238030702x_{82} = -26.8485238030702
x83=−71.5393227898632x_{83} = -71.5393227898632
x84=−62.1701414510933x_{84} = -62.1701414510933
x85=48.1037584023775x_{85} = 48.1037584023775
x86=−31.9003143336432x_{86} = -31.9003143336432
x87=−93.8836077933089x_{87} = -93.8836077933089
x88=−96.0393505752796x_{88} = -96.0393505752796
x89=67.5682942687045x_{89} = 67.5682942687045
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(x)/18 + cos(x*sqrt(19)) + sin(x*sqrt(19)).
sin⁡(019)+(cos⁡(0)18+cos⁡(019))\sin{\left(0 \sqrt{19} \right)} + \left(\frac{\cos{\left(0 \right)}}{18} + \cos{\left(0 \sqrt{19} \right)}\right)
Resultado:
f(0)=1918f{\left(0 \right)} = \frac{19}{18}
Punto:
(0, 19/18)
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
−(19sin⁡(19x)+cos⁡(x)18+19cos⁡(19x))=0- (19 \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + 19 \cos{\left(\sqrt{19} x \right)}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.8383386969039x_{1} = 17.8383386969039
x2=1.98182013808385x_{2} = 1.98182013808385
x3=86.307552259818x_{3} = 86.307552259818
x4=−35.4962713521321x_{4} = -35.4962713521321
x5=79.8207987925381x_{5} = 79.8207987925381
x6=42.3429010550322x_{6} = 42.3429010550322
x7=35.8564838675336x_{7} = 35.8564838675336
x8=16.3962589987325x_{8} = 16.3962589987325
x9=20.0000857309109x_{9} = 20.0000857309109
x10=−83.7847128409657x_{10} = -83.7847128409657
x11=34.4153644903525x_{11} = 34.4153644903525
x12=−45.5862378157438x_{12} = -45.5862378157438
x13=53.1543545959115x_{13} = 53.1543545959115
x14=−52.0726880597439x_{14} = -52.0726880597439
x15=12.0726583214759x_{15} = 12.0726583214759
x16=30.0903950640639x_{16} = 30.0903950640639
x17=78.3790045638792x_{17} = 78.3790045638792
x18=−29.7301992414662x_{18} = -29.7301992414662
x19=92.0736268149492x_{19} = 92.0736268149492
x20=−81.6222878012479x_{20} = -81.6222878012479
x21=−39.0997253739x_{21} = -39.0997253739
x22=−21.8016403804058x_{22} = -21.8016403804058
x23=−44.1442924071731x_{23} = -44.1442924071731
x24=1.26113432597172x_{24} = 1.26113432597172
x25=−99.6413295926589x_{25} = -99.6413295926589
x26=−91.7133818284164x_{26} = -91.7133818284164
x27=97.8396310051747x_{27} = 97.8396310051747
x28=60.3615745763682x_{28} = 60.3615745763682
x29=−47.7480295453976x_{29} = -47.7480295453976
x30=14.2344790722842x_{30} = 14.2344790722842
x31=68.2895634934508x_{31} = 68.2895634934508
x32=−63.6048312438269x_{32} = -63.6048312438269
x33=−95.3168743179986x_{33} = -95.3168743179986
x34=4.14445714699621x_{34} = 4.14445714699621
x35=48.1085174540459x_{35} = 48.1085174540459
x36=58.1989731458254x_{36} = 58.1989731458254
x37=989.38272092972x_{37} = 989.38272092972
x38=−8.10834067540243x_{38} = -8.10834067540243
x39=100.001805758604x_{39} = 100.001805758604
x40=−31.8927586202166x_{40} = -31.8927586202166
x41=−82.3438660914141x_{41} = -82.3438660914141
x42=6.30686856023374x_{42} = 6.30686856023374
x43=−70.0908028785674x_{43} = -70.0908028785674
x44=50.2704978681474x_{44} = 50.2704978681474
x45=−85.9473508085614x_{45} = -85.9473508085614
x46=−39.8206240406073x_{46} = -39.8206240406073
x47=−19.6395800198681x_{47} = -19.6395800198681
x48=−11.7121867996881x_{48} = -11.7121867996881
x49=−93.8756264972675x_{49} = -93.8756264972675
x50=−34.0549452295727x_{50} = -34.0549452295727
x51=53.8742026002769x_{51} = 53.8742026002769
x52=81.982673727597x_{52} = 81.982673727597
x53=−55.6761481279815x_{53} = -55.6761481279815
x54=89.9109913224088x_{54} = 89.9109913224088
x55=−23.9641129126381x_{55} = -23.9641129126381
x56=−1.62162016734916x_{56} = -1.62162016734916
x57=71.8933390918117x_{57} = 71.8933390918117
x58=9.91046775891682x_{58} = 9.91046775891682
x59=32.2530202145542x_{59} = 32.2530202145542
x60=−62.1634023800515x_{60} = -62.1634023800515
x61=−75.8564896405227x_{61} = -75.8564896405227
x62=96.3980020969127x_{62} = 96.3980020969127
x63=38.0189991331808x_{63} = 38.0189991331808
x64=−49.9101620992528x_{64} = -49.9101620992528
x65=66.1275181585432x_{65} = 66.1275181585432
x66=56.036405021783x_{66} = 56.036405021783
x67=−3.78421633182046x_{67} = -3.78421633182046
x68=84.144949641202x_{68} = 84.144949641202
x69=−26.1267495946787x_{69} = -26.1267495946787
x70=−5.94647666606874x_{70} = -5.94647666606874
x71=−88.1098077014585x_{71} = -88.1098077014585
x72=−13.8739441767131x_{72} = -13.8739441767131
x73=74.0551951767426x_{73} = 74.0551951767426
x74=45.9467694361715x_{74} = 45.9467694361715
x75=27.9278717385303x_{75} = 27.9278717385303
x76=−37.6586574155057x_{76} = -37.6586574155057
x77=−73.6946598879328x_{77} = -73.6946598879328
x78=−41.9823708166225x_{78} = -41.9823708166225
x79=62.5238475345719x_{79} = 62.5238475345719
x80=40.1811160542568x_{80} = 40.1811160542568
x81=−96.0374799932567x_{81} = -96.0374799932567
x82=−78.0186960137561x_{82} = -78.0186960137561
x83=−57.8387768251908x_{83} = -57.8387768251908
x84=63.9650580952362x_{84} = 63.9650580952362
x85=−60.0012893532383x_{85} = -60.0012893532383
x86=−16.0358108793466x_{86} = -16.0358108793466
x87=22.162004186949x_{87} = 22.162004186949
x88=76.2169563849449x_{88} = 76.2169563849449
x89=24.3243363662851x_{89} = 24.3243363662851
x90=−67.9290544628594x_{90} = -67.9290544628594
x91=−89.5507926247334x_{91} = -89.5507926247334
x92=94.2360243262519x_{92} = 94.2360243262519
x93=−65.7671483275469x_{93} = -65.7671483275469

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
[989.38272092972,∞)\left[989.38272092972, \infty\right)
Convexa en los intervalos
(−∞,−96.0374799932567]\left(-\infty, -96.0374799932567\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((cos⁡(x)18+cos⁡(19x))+sin⁡(19x))=⟨−3718,3718⟩\lim_{x \to -\infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−3718,3718⟩y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle
lim⁡x→∞((cos⁡(x)18+cos⁡(19x))+sin⁡(19x))=⟨−3718,3718⟩\lim_{x \to \infty}\left(\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}\right) = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−3718,3718⟩y = \left\langle - \frac{37}{18}, \frac{37}{18}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/18 + cos(x*sqrt(19)) + sin(x*sqrt(19)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((cos⁡(x)18+cos⁡(19x))+sin⁡(19x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((cos⁡(x)18+cos⁡(19x))+sin⁡(19x)x)=0\lim_{x \to \infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} 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:
(cos⁡(x)18+cos⁡(19x))+sin⁡(19x)=−sin⁡(19x)+cos⁡(x)18+cos⁡(19x)\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = - \sin{\left(\sqrt{19} x \right)} + \frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}
- No
(cos⁡(x)18+cos⁡(19x))+sin⁡(19x)=sin⁡(19x)−cos⁡(x)18−cos⁡(19x)\left(\frac{\cos{\left(x \right)}}{18} + \cos{\left(\sqrt{19} x \right)}\right) + \sin{\left(\sqrt{19} x \right)} = \sin{\left(\sqrt{19} x \right)} - \frac{\cos{\left(x \right)}}{18} - \cos{\left(\sqrt{19} x \right)}
- No
es decir, función
no es
par ni impar
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