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Trazado el gráfico de la función/ cos(x)/2-cos(x*sqrt(13))-sin(x*sqrt(13))

Gráfico de la función y = cos(x)/2-cos(x*sqrt(13))-sin(x*sqrt(13))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)      /    ____\      /    ____\
f(x) = ------ - cos\x*\/ 13 / - sin\x*\/ 13 /
         2
f(x)=(cos⁡(x)2−cos⁡(13x))−sin⁡(13x)f{\left(x \right)} = \left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)} 
f = cos(x)/2 - cos(sqrt(13)*x) - sin(sqrt(13)*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)2−cos⁡(13x))−sin⁡(13x)=0\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=72.2018834529011x_{1} = 72.2018834529011
x2=91.3675515154084x_{2} = 91.3675515154084
x3=30.2405193364248x_{3} = 30.2405193364248
x4=86.967855303755x_{4} = 86.967855303755
x5=64.2726427861001x_{5} = 64.2726427861001
x6=18.0135392440583x_{6} = 18.0135392440583
x7=−21.9010558381228x_{7} = -21.9010558381228
x8=−16.7206954343536x_{8} = -16.7206954343536
x9=−35.9218752245225x_{9} = -35.9218752245225
x10=39.812056486731x_{10} = 39.812056486731
x11=56.3205229168152x_{11} = 56.3205229168152
x12=−8.04143265828536x_{12} = -8.04143265828536
x13=−89.926409460813x_{13} = -89.926409460813
x14=78.1102999853726x_{14} = 78.1102999853726
x15=−48.0838529750941x_{15} = -48.0838529750941
x16=65.9026057420168x_{16} = 65.9026057420168
x17=−85.6726376574831x_{17} = -85.6726376574831
x18=22.3426745682996x_{18} = 22.3426745682996
x19=−25.5762885925397x_{19} = -25.5762885925397
x20=−71.7534976479548x_{20} = -71.7534976479548
x21=98.3019100717657x_{21} = 98.3019100717657
x22=−58.55482934989x_{22} = -58.55482934989
x23=5.79341730725067x_{23} = 5.79341730725067
x24=46.7408392432915x_{24} = 46.7408392432915
x25=−163.942922765403x_{25} = -163.942922765403
x26=24.2412847126875x_{26} = 24.2412847126875
x27=−89.0462900123944x_{27} = -89.0462900123944
x28=99.9035780264295x_{28} = 99.9035780264295
x29=2.4741757298227x_{29} = 2.4741757298227
x30=73.8429365993259x_{30} = 73.8429365993259
x31=−37.7844781522983x_{31} = -37.7844781522983
x32=−49.9790968551397x_{32} = -49.9790968551397
x33=−87.2734131220007x_{33} = -87.2734131220007
x34=46.0051937742398x_{34} = 46.0051937742398
x35=−28.200039327509x_{35} = -28.200039327509
x36=−55.9029786614563x_{36} = -55.9029786614563
x37=−99.4979229179526x_{37} = -99.4979229179526
x38=8.44073624961482x_{38} = 8.44073624961482
x39=−31.4854151009164x_{39} = -31.4854151009164
x40=12.0682554821657x_{40} = 12.0682554821657
x41=−51.644477782289x_{41} = -51.644477782289
x42=−69.849899879829x_{42} = -69.849899879829
x43=−15.9974838613164x_{43} = -15.9974838613164
x44=−0.118337829969558x_{44} = -0.118337829969558
x45=32.9007865790468x_{45} = 32.9007865790468
x46=−44.7280681700192x_{46} = -44.7280681700192
x47=44.1203132528385x_{47} = 44.1203132528385
x48=−17.6120946395667x_{48} = -17.6120946395667
x49=90.3288975649693x_{49} = 90.3288975649693
x50=86.0738560743879x_{50} = 86.0738560743879
x51=59.9986179467768x_{51} = 59.9986179467768
x52=50.4178020024072x_{52} = 50.4178020024072
x53=40.8344813627777x_{53} = 40.8344813627777
x54=−76.0986747497803x_{54} = -76.0986747497803
x55=70.3244080382417x_{55} = 70.3244080382417
x56=−91.6197173283789x_{56} = -91.6197173283789
x57=−3.78278759092504x_{57} = -3.78278759092504
x58=33.8383640965044x_{58} = 33.8383640965044
x59=52.0414376202635x_{59} = 52.0414376202635
x60=−9.70625026863057x_{60} = -9.70625026863057
x61=76.5026957573472x_{61} = 76.5026957573472
x62=−77.6993627493211x_{62} = -77.6993627493211
x63=−83.9263695905671x_{63} = -83.9263695905671
x64=−96.089417619132x_{64} = -96.089417619132
x65=−57.6834499852804x_{65} = -57.6834499852804
x66=−42.0738027071068x_{66} = -42.0738027071068
x67=−34.1094057231637x_{67} = -34.1094057231637
x68=−11.6012106245875x_{68} = -11.6012106245875
x69=−63.8739896336319x_{69} = -63.8739896336319
x70=−82.0279893218577x_{70} = -82.0279893218577
x71=92.0873125965294x_{71} = 92.0873125965294
x72=16.412889616247x_{72} = 16.412889616247
x73=36.4044268257053x_{73} = 36.4044268257053
x74=10.1647590089972x_{74} = 10.1647590089972
x75=31.9344822641224x_{75} = 31.9344822641224
x76=79.9256588466285x_{76} = 79.9256588466285
x77=−29.8429582769054x_{77} = -29.8429582769054
x78=93.9813775337235x_{78} = 93.9813775337235
x79=38.2073814099161x_{79} = 38.2073814099161
x80=−36.7553569469225x_{80} = -36.7553569469225
x81=42.471062741881x_{81} = 42.471062741881
x82=−2.00156893816052x_{82} = -2.00156893816052
x83=84.3909631880852x_{83} = 84.3909631880852
x84=−55.1254837713184x_{84} = -55.1254837713184
x85=−97.893099045985x_{85} = -97.893099045985
x86=4.188120702701x_{86} = 4.188120702701
x87=19.762089230874x_{87} = 19.762089230874
x88=25.9868106706493x_{88} = 25.9868106706493
x89=−68.1265939454709x_{89} = -68.1265939454709
x90=−43.6881455944144x_{90} = -43.6881455944144
x91=−65.4791296413507x_{91} = -65.4791296413507
x92=58.1651581014011x_{92} = 58.1651581014011
x93=−14.1613365745623x_{93} = -14.1613365745623
x94=−62.1593436090071x_{94} = -62.1593436090071
x95=−23.7631072417696x_{95} = -23.7631072417696
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)/2 - cos(x*sqrt(13)) - sin(x*sqrt(13)).
(−cos⁡(013)+cos⁡(0)2)−sin⁡(013)\left(- \cos{\left(0 \sqrt{13} \right)} + \frac{\cos{\left(0 \right)}}{2}\right) - \sin{\left(0 \sqrt{13} \right)}
Resultado:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
Punto:
(0, -1/2)
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
13sin⁡(13x)−cos⁡(x)2+13cos⁡(13x)=013 \sin{\left(\sqrt{13} x \right)} - \frac{\cos{\left(x \right)}}{2} + 13 \cos{\left(\sqrt{13} x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10.2328102266561x_{1} = 10.2328102266561
x2=−28.1075423835299x_{2} = -28.1075423835299
x3=18.9587313512208x_{3} = 18.9587313512208
x4=36.3795280568547x_{4} = 36.3795280568547
x5=−31.5779423218612x_{5} = -31.5779423218612
x6=39.8587476805068x_{6} = 39.8587476805068
x7=−55.9760360061067x_{7} = -55.9760360061067
x8=−21.134475535662x_{8} = -21.134475535662
x9=−0.210452883579795x_{9} = -0.210452883579795
x10=−14.1591324196394x_{10} = -14.1591324196394
x11=−8.05818988132525x_{11} = -8.05818988132525
x12=−69.9182747331007x_{12} = -69.9182747331007
x13=32.0148171859649x_{13} = 32.0148171859649
x14=−48.1364916540084x_{14} = -48.1364916540084
x15=−37.6921782549588x_{15} = -37.6921782549588
x16=−49.8901475835001x_{16} = -49.8901475835001
x17=86.0455260242817x_{17} = 86.0455260242817
x18=−52.5017495452869x_{18} = -52.5017495452869
x19=38.1271581871372x_{19} = 38.1271581871372
x20=31.1570192126231x_{20} = 31.1570192126231
x21=25.9270867044294x_{21} = 25.9270867044294
x22=−85.6126027827194x_{22} = -85.6126027827194
x23=−51.6273356777146x_{23} = -51.6273356777146
x24=−21.9933121844799x_{24} = -21.9933121844799
x25=−83.8690124683746x_{25} = -83.8690124683746
x26=−91.7002874872446x_{26} = -91.7002874872446
x27=253.33981352889x_{27} = 253.33981352889
x28=23.3097199920582x_{28} = 23.3097199920582
x29=22.4296866839236x_{29} = 22.4296866839236
x30=−29.8427631560723x_{30} = -29.8427631560723
x31=56.4105647227181x_{31} = 56.4105647227181
x32=52.0597598693124x_{32} = 52.0597598693124
x33=−57.7220991930938x_{33} = -57.7220991930938
x34=66.8785478775736x_{34} = 66.8785478775736
x35=−76.0288525167151x_{35} = -76.0288525167151
x36=98.2463848170604x_{36} = 98.2463848170604
x37=−82.1151620348295x_{37} = -82.1151620348295
x38=42.4764109776243x_{38} = 42.4764109776243
x39=30.2752604301401x_{39} = 30.2752604301401
x40=−17.6415737948684x_{40} = -17.6415737948684
x41=44.212196981543x_{41} = 44.212196981543
x42=−88.2285369038842x_{42} = -88.2285369038842
x43=−96.0649835319878x_{43} = -96.0649835319878
x44=−3.70947394616372x_{44} = -3.70947394616372
x45=11.9869765987075x_{45} = 11.9869765987075
x46=45.9652057161914x_{46} = 45.9652057161914
x47=−63.8283632352862x_{47} = -63.8283632352862
x48=50.3263194219786x_{48} = 50.3263194219786
x49=−35.9405847738249x_{49} = -35.9405847738249
x50=19.8182813386858x_{50} = 19.8182813386858
x51=59.9106821546735x_{51} = 59.9106821546735
x52=72.1092773770751x_{52} = 72.1092773770751
x53=92.1383294934171x_{53} = 92.1383294934171
x54=87.778180711809x_{54} = 87.778180711809
x55=33.7690072877756x_{55} = 33.7690072877756
x56=−75.1441373046402x_{56} = -75.1441373046402
x57=64.2609909662586x_{57} = 64.2609909662586
x58=−58.5929000511258x_{58} = -58.5929000511258
x59=2009.91404362727x_{59} = 2009.91404362727
x60=99.9776701936776x_{60} = 99.9776701936776
x61=65.9950259742992x_{61} = 65.9950259742992
x62=−11.5489681859229x_{62} = -11.5489681859229
x63=−9.79532987030075x_{63} = -9.79532987030075
x64=5.87449453145933x_{64} = 5.87449453145933
x65=−62.087170143839x_{65} = -62.087170143839
x66=93.8919121474639x_{66} = 93.8919121474639
x67=−99.5442680662756x_{67} = -99.5442680662756
x68=97.3625840299048x_{68} = 97.3625840299048
x69=24.1835513485605x_{69} = 24.1835513485605
x70=16.3433407193759x_{70} = 16.3433407193759
x71=2.40170399068693x_{71} = 2.40170399068693
x72=84.3068706820929x_{72} = 84.3068706820929
x73=−34.1923042474878x_{73} = -34.1923042474878
x74=−77.7600381097401x_{74} = -77.7600381097401
x75=−23.7448702025145x_{75} = -23.7448702025145
x76=53.7970964484986x_{76} = 53.7970964484986
x77=90.3938477797604x_{77} = 90.3938477797604
x78=70.3567353927848x_{78} = 70.3567353927848
x79=4.14284256662798x_{79} = 4.14284256662798
x80=−42.0439501806135x_{80} = -42.0439501806135
x81=−89.9607835487709x_{81} = -89.9607835487709
x82=−15.9090006986826x_{82} = -15.9090006986826
x83=−45.5263219425703x_{83} = -45.5263219425703
x84=−1.96335796700218x_{84} = -1.96335796700218
x85=78.1939654396765x_{85} = 78.1939654396765
x86=−71.6724473436302x_{86} = -71.6724473436302
x87=−78.6442257876116x_{87} = -78.6442257876116
x88=−43.776497164284x_{88} = -43.776497164284
x89=18.0745223983222x_{89} = 18.0745223983222
x90=−68.1764133605827x_{90} = -68.1764133605827
x91=−65.5599994724066x_{91} = -65.5599994724066
x92=76.4620826577859x_{92} = 76.4620826577859
x93=−97.8126636052305x_{93} = -97.8126636052305
x94=73.8443298637669x_{94} = 73.8443298637669
x95=58.1609919511325x_{95} = 58.1609919511325
x96=79.9424420343204x_{96} = 79.9424420343204

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
[97.3625840299048,∞)\left[97.3625840299048, \infty\right)
Convexa en los intervalos
(−∞,−99.5442680662756]\left(-\infty, -99.5442680662756\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((cos⁡(x)2−cos⁡(13x))−sin⁡(13x))=⟨−52,52⟩\lim_{x \to -\infty}\left(\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)}\right) = \left\langle - \frac{5}{2}, \frac{5}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−52,52⟩y = \left\langle - \frac{5}{2}, \frac{5}{2}\right\rangle
lim⁡x→∞((cos⁡(x)2−cos⁡(13x))−sin⁡(13x))=⟨−52,52⟩\lim_{x \to \infty}\left(\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)}\right) = \left\langle - \frac{5}{2}, \frac{5}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−52,52⟩y = \left\langle - \frac{5}{2}, \frac{5}{2}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/2 - cos(x*sqrt(13)) - sin(x*sqrt(13)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((cos⁡(x)2−cos⁡(13x))−sin⁡(13x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} 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)2−cos⁡(13x))−sin⁡(13x)x)=0\lim_{x \to \infty}\left(\frac{\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} 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)2−cos⁡(13x))−sin⁡(13x)=sin⁡(13x)+cos⁡(x)2−cos⁡(13x)\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)} = \sin{\left(\sqrt{13} x \right)} + \frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}
- No
(cos⁡(x)2−cos⁡(13x))−sin⁡(13x)=−sin⁡(13x)−cos⁡(x)2+cos⁡(13x)\left(\frac{\cos{\left(x \right)}}{2} - \cos{\left(\sqrt{13} x \right)}\right) - \sin{\left(\sqrt{13} x \right)} = - \sin{\left(\sqrt{13} x \right)} - \frac{\cos{\left(x \right)}}{2} + \cos{\left(\sqrt{13} x \right)}
- No
es decir, función
no es
par ni impar
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