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Trazado el gráfico de la función/ (tan(tan(x))-sin(sin(x)))/(tan(x)-sin(x))

Gráfico de la función y = (tan(tan(x))-sin(sin(x)))/(tan(x)-sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       tan(tan(x)) - sin(sin(x))
f(x) = -------------------------
            tan(x) - sin(x)
f(x)=−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x)f{\left(x \right)} = \frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}} 
f = (-sin(sin(x)) + tan(tan(x)))/(-sin(x) + tan(x))
Gráfico de la función
010000002000000300000040000005000000600000070000008000000900000010000000-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−3.14159265358979x_{1} = -3.14159265358979
x2=0x_{2} = 0
x3=3.14159265358979x_{3} = 3.14159265358979
x4=6.28318530717959x_{4} = 6.28318530717959
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⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x)=0\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=4.32728526758813x_{1} = 4.32728526758813
x2=52.2213824970282x_{2} = 52.2213824970282
x3=−29.6680711682306x_{3} = -29.6680711682306
x4=−102.486864954465x_{4} = -102.486864954465
x5=1688.64474362082x_{5} = 1688.64474362082
x6=−29.8796306126345x_{6} = -29.8796306126345
x7=28847.4191789135x_{7} = 28847.4191789135
x8=−1.74785536766736x_{8} = -1.74785536766736
x9=205.597259769259x_{9} = 205.597259769259
x10=42.666863498193x_{10} = 42.666863498193
x11=39.1714403624214x_{11} = 39.1714403624214
x12=96.2036796472853x_{12} = 96.2036796472853
x13=−20.1649895736029x_{13} = -20.1649895736029
x14=−23.4954171693482x_{14} = -23.4954171693482
x15=−17.5341222694746x_{15} = -17.5341222694746
x16=73.7723881454623x_{16} = 73.7723881454623
x17=−67.5953518933822x_{17} = -67.5953518933822
x18=−58.174503305309x_{18} = -58.174503305309
x19=−73.8883020269395x_{19} = -73.8883020269395
x20=−36.1268538873264x_{20} = -36.1268538873264
x21=−39.655011882669x_{21} = -39.655011882669
x22=92.2918795681023x_{22} = 92.2918795681023
x23=−96.2036796472853x_{23} = -96.2036796472853
x24=1.95590003959146x_{24} = 1.95590003959146
x25=70.4304720310396x_{25} = 70.4304720310396
x26=−89.9204943401057x_{26} = -89.9204943401057
x27=55.2332341125522x_{27} = 55.2332341125522
x28=375.390795688372x_{28} = 375.390795688372
x29=111.141435489641x_{29} = 111.141435489641
x30=20.414138088974x_{30} = 20.414138088974
x31=149.481013720246x_{31} = 149.481013720246
x32=30.1004928838338x_{32} = 30.1004928838338
x33=8.23908534677105x_{33} = 8.23908534677105
x34=−35.7432118034861x_{34} = -35.7432118034861
x35=98.783109547206x_{35} = 98.783109547206
x36=10.9115124455202x_{36} = 10.9115124455202
x37=64.2603406698377x_{37} = 64.2603406698377
x38=−58.1191822073437x_{38} = -58.1191822073437
x39=−300.04447518957x_{39} = -300.04447518957
x40=17.2962256345186x_{40} = 17.2962256345186
x41=−764.592707436318x_{41} = -764.592707436318
x42=26.4481748807825x_{42} = 26.4481748807825
x43=−61.5164194197317x_{43} = -61.5164194197317
x44=36.2706242450357x_{44} = 36.2706242450357
x45=−108.365327735712x_{45} = -108.365327735712
x46=−83.6373090329261x_{46} = -83.6373090329261
x47=79.7255089537432x_{47} = 79.7255089537432
x48=14.1734157621097x_{48} = 14.1734157621097
x49=−23.4778830598794x_{49} = -23.4778830598794
x50=10.6104705747677x_{50} = 10.6104705747677
x51=−95.8355261113998x_{51} = -95.8355261113998
x52=105.129268814369x_{52} = 105.129268814369
x53=−14.061922879902x_{53} = -14.061922879902
x54=−42.0263971106656x_{54} = -42.0263971106656
x55=27.0886412683098x_{55} = 27.0886412683098
x56=−73.8145812536186x_{56} = -73.8145812536186
x57=−86.0086942609228x_{57} = -86.0086942609228
x58=−33.3718265754894x_{58} = -33.3718265754894
x59=1151.77881125346x_{59} = 1151.77881125346
x60=67.5882851250914x_{60} = 67.5882851250914
x61=95.814453982878x_{61} = 95.814453982878
x62=−51.8486265932496x_{62} = -51.8486265932496
x63=36.1794253574842x_{63} = 36.1794253574842
x64=−7.75551382652349x_{64} = -7.75551382652349
x65=−51.81221032832x_{65} = -51.81221032832
x66=86.0086942609228x_{66} = 86.0086942609228
x67=410.15490033434x_{67} = 410.15490033434
x68=58.1192918665548x_{68} = 58.1192918665548
x69=−114.525823127274x_{69} = -114.525823127274
x70=−4.75108129669496x_{70} = -4.75108129669496
x71=14.5222706539506x_{71} = 14.5222706539506
x72=83.3187330527048x_{72} = 83.3187330527048
x73=−664.332761153352x_{73} = -664.332761153352
x74=−64.14728672386x_{74} = -64.14728672386
x75=48.3095824178452x_{75} = 48.3095824178452
x76=−79.9965275856509x_{76} = -79.9965275856509
x77=45.9381971898486x_{77} = 45.9381971898486
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 (tan(tan(x)) - sin(sin(x)))/(tan(x) - sin(x)).
tan⁡(tan⁡(0))−sin⁡(sin⁡(0))tan⁡(0)−sin⁡(0)\frac{\tan{\left(\tan{\left(0 \right)} \right)} - \sin{\left(\sin{\left(0 \right)} \right)}}{\tan{\left(0 \right)} - \sin{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=−3.14159265358979x_{1} = -3.14159265358979
x2=0x_{2} = 0
x3=3.14159265358979x_{3} = 3.14159265358979
x4=6.28318530717959x_{4} = 6.28318530717959
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x))y = \lim_{x \to -\infty}\left(\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x))y = \lim_{x \to \infty}\left(\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (tan(tan(x)) - sin(sin(x)))/(tan(x) - sin(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))x(−sin⁡(x)+tan⁡(x)))y = x \lim_{x \to -\infty}\left(\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{x \left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))x(−sin⁡(x)+tan⁡(x)))y = x \lim_{x \to \infty}\left(\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{x \left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right)}\right)
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⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x)=sin⁡(sin⁡(x))−tan⁡(tan⁡(x))sin⁡(x)−tan⁡(x)\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}} = \frac{\sin{\left(\sin{\left(x \right)} \right)} - \tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(x \right)} - \tan{\left(x \right)}}
- No
−sin⁡(sin⁡(x))+tan⁡(tan⁡(x))−sin⁡(x)+tan⁡(x)=−sin⁡(sin⁡(x))−tan⁡(tan⁡(x))sin⁡(x)−tan⁡(x)\frac{- \sin{\left(\sin{\left(x \right)} \right)} + \tan{\left(\tan{\left(x \right)} \right)}}{- \sin{\left(x \right)} + \tan{\left(x \right)}} = - \frac{\sin{\left(\sin{\left(x \right)} \right)} - \tan{\left(\tan{\left(x \right)} \right)}}{\sin{\left(x \right)} - \tan{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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