Sr Examen
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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3+3*x^2+2
-
-sqrt(3*x+1)
-
x*(-1-log(x))
-
-exp(-2*x)-exp(2*x)
-
Expresiones idénticas
- (sin(x^ dos)+cos(x^ dos))/(sin^ dos (x)+sin(x^ dos))
- ( seno de (x al cuadrado ) más coseno de (x al cuadrado )) dividir por ( seno de al cuadrado (x) más seno de (x al cuadrado ))
- ( seno de (x en el grado dos) más coseno de (x en el grado dos)) dividir por ( seno de en el grado dos (x) más seno de (x en el grado dos))
- (sin(x2)+cos(x2))/(sin2(x)+sin(x2))
- sinx2+cosx2/sin2x+sinx2
- (sin(x²)+cos(x²))/(sin²(x)+sin(x²))
- (sin(x en el grado 2)+cos(x en el grado 2))/(sin en el grado 2(x)+sin(x en el grado 2))
- sinx^2+cosx^2/sin^2x+sinx^2
- (sin(x^2)+cos(x^2)) dividir por (sin^2(x)+sin(x^2))
-
Expresiones semejantes
- (sin(x^2)+cos(x^2))/(sin^2(x)-sin(x^2))
- (sin(x^2)-cos(x^2))/(sin^2(x)+sin(x^2))
-
Expresiones con funciones
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(3*x)+x^4
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin(asin(x))
- sin(40*x+sin(40*x)^(2)*2)^(50)
- Coseno cos
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
- cos(3*x-1)
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(3*x)+x^4
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin(asin(x))
- sin(40*x+sin(40*x)^(2)*2)^(50)
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(3*x)+x^4
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin(asin(x))
- sin(40*x+sin(40*x)^(2)*2)^(50)
Gráfico de la función y = (sin(x^2)+cos(x^2))/(sin^2(x)+sin(x^2))
Solución
Ha introducido
[src]
/ 2\ / 2\
sin\x / + cos\x /
f(x) = -----------------
2 / 2\
sin (x) + sin\x /
$$f{\left(x \right)} = \frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}$$
f = (sin(x^2) + cos(x^2))/(sin(x)^2 + sin(x^2))
Gráfico de la función
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:
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -17.7908813679697$$
$$x_{2} = -92.9947266094659$$
$$x_{3} = -45.698727733645$$
$$x_{4} = -7.87695721128396$$
$$x_{5} = -21.7622400622882$$
$$x_{6} = -14.0404750280309$$
$$x_{7} = 42.2703302770299$$
$$x_{8} = -25.6700030987622$$
$$x_{9} = -78.6248621245298$$
$$x_{10} = -91.427526779625$$
$$x_{11} = 32.2347913003144$$
$$x_{12} = 4.25019502589485$$
$$x_{13} = 48.3704571899109$$
$$x_{14} = 40.2136148391978$$
$$x_{15} = 80.1876222601401$$
$$x_{16} = -119.318533798843$$
$$x_{17} = 20.2672948233568$$
$$x_{18} = -75.755556759522$$
$$x_{19} = 90.1471840751191$$
$$x_{20} = 78.3847551010925$$
$$x_{21} = -24.3510905990897$$
$$x_{22} = -99.4273940254077$$
$$x_{23} = -56.9051546671097$$
$$x_{24} = 55.3944999471412$$
$$x_{25} = 99.3957921512601$$
$$x_{26} = -53.5489373803448$$
$$x_{27} = -47.7823446577958$$
$$x_{28} = 128.166308316412$$
$$x_{29} = -97.73841548441$$
$$x_{30} = -19.7968099333031$$
$$x_{31} = 60.2308402459472$$
$$x_{32} = 12.2478997876743$$
$$x_{33} = 30.0141915336208$$
$$x_{34} = -3.86297360909333$$
$$x_{35} = -65.7661591232888$$
$$x_{36} = -5.81137858223763$$
$$x_{37} = -31.7931979563635$$
$$x_{38} = -29.698520796419$$
$$x_{39} = -83.1118333757628$$
$$x_{40} = 64.658135016134$$
$$x_{41} = -67.3007519149161$$
$$x_{42} = -55.8463616259042$$
$$x_{43} = -15.9274482192653$$
$$x_{44} = 76.989609293466$$
$$x_{45} = 68.2509817300108$$
$$x_{46} = 2.34473604991738$$
$$x_{47} = 78.0030741178835$$
$$x_{48} = 71.3123629393843$$
$$x_{49} = 52.2723656482058$$
$$x_{50} = -9.82873206969679$$
$$x_{51} = 81.1805148361846$$
$$x_{52} = -70.336498571087$$
$$x_{53} = 24.0916840995127$$
$$x_{54} = -11.8569081656283$$
$$x_{55} = -64.3903479146924$$
$$x_{56} = -39.821086021095$$
$$x_{57} = 98.2513553408707$$
$$x_{58} = 40.1354161947795$$
$$x_{59} = 74.2052998348468$$
$$x_{60} = 96.2484874544681$$
$$x_{61} = -83.7519611549641$$
$$x_{62} = 57.9174892055112$$
$$x_{63} = 22.1910938993597$$
$$x_{64} = -91.6334642195858$$
$$x_{65} = -1.53499006191973$$
$$x_{66} = 17.5240030212392$$
$$x_{67} = 92.1121944108522$$
$$x_{68} = 90.0774579240502$$
$$x_{69} = 46.2454235458089$$
$$x_{70} = 83.9393049997348$$
$$x_{71} = 48.1752174135152$$
$$x_{72} = 10.1433307845631$$
$$x_{73} = -48.8872847601341$$
$$x_{74} = 34.5855663948751$$
$$x_{75} = 50.2499263215659$$
$$x_{76} = -85.423265777689$$
$$x_{77} = -27.9548058941653$$
$$x_{78} = 29.0568929111882$$
$$x_{79} = -57.9987959688067$$
$$x_{80} = -61.8774276013682$$
$$x_{81} = 960.026766802951$$
$$x_{82} = 65.5748041903369$$
$$x_{83} = -33.7581506467653$$
$$x_{84} = 54.6522571848638$$
$$x_{85} = 114.714193662878$$
$$x_{86} = -95.7576292399995$$
$$x_{87} = 28.1786717784004$$
$$x_{88} = -71.7515502346136$$
$$x_{89} = 15.7289675907288$$
$$x_{90} = -67.6499452927859$$
$$x_{91} = -41.7468384980101$$
$$x_{92} = -35.8347038100614$$
$$x_{93} = 8.07391154038662$$
$$x_{94} = 26.0948310526355$$
$$x_{95} = -59.8121111003062$$
$$x_{96} = 70.1576103499243$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -17.7908813679697$$
$$x_{2} = -92.9947266094659$$
$$x_{3} = -45.698727733645$$
$$x_{4} = -7.87695721128396$$
$$x_{5} = -21.7622400622882$$
$$x_{6} = -14.0404750280309$$
$$x_{7} = 42.2703302770299$$
$$x_{8} = -25.6700030987622$$
$$x_{9} = -78.6248621245298$$
$$x_{10} = -91.427526779625$$
$$x_{11} = 32.2347913003144$$
$$x_{12} = 4.25019502589485$$
$$x_{13} = 48.3704571899109$$
$$x_{14} = 40.2136148391978$$
$$x_{15} = 80.1876222601401$$
$$x_{16} = -119.318533798843$$
$$x_{17} = 20.2672948233568$$
$$x_{18} = -75.755556759522$$
$$x_{19} = 90.1471840751191$$
$$x_{20} = 78.3847551010925$$
$$x_{21} = -24.3510905990897$$
$$x_{22} = -99.4273940254077$$
$$x_{23} = -56.9051546671097$$
$$x_{24} = 55.3944999471412$$
$$x_{25} = 99.3957921512601$$
$$x_{26} = -53.5489373803448$$
$$x_{27} = -47.7823446577958$$
$$x_{28} = 128.166308316412$$
$$x_{29} = -97.73841548441$$
$$x_{30} = -19.7968099333031$$
$$x_{31} = 60.2308402459472$$
$$x_{32} = 12.2478997876743$$
$$x_{33} = 30.0141915336208$$
$$x_{34} = -3.86297360909333$$
$$x_{35} = -65.7661591232888$$
$$x_{36} = -5.81137858223763$$
$$x_{37} = -31.7931979563635$$
$$x_{38} = -29.698520796419$$
$$x_{39} = -83.1118333757628$$
$$x_{40} = 64.658135016134$$
$$x_{41} = -67.3007519149161$$
$$x_{42} = -55.8463616259042$$
$$x_{43} = -15.9274482192653$$
$$x_{44} = 76.989609293466$$
$$x_{45} = 68.2509817300108$$
$$x_{46} = 2.34473604991738$$
$$x_{47} = 78.0030741178835$$
$$x_{48} = 71.3123629393843$$
$$x_{49} = 52.2723656482058$$
$$x_{50} = -9.82873206969679$$
$$x_{51} = 81.1805148361846$$
$$x_{52} = -70.336498571087$$
$$x_{53} = 24.0916840995127$$
$$x_{54} = -11.8569081656283$$
$$x_{55} = -64.3903479146924$$
$$x_{56} = -39.821086021095$$
$$x_{57} = 98.2513553408707$$
$$x_{58} = 40.1354161947795$$
$$x_{59} = 74.2052998348468$$
$$x_{60} = 96.2484874544681$$
$$x_{61} = -83.7519611549641$$
$$x_{62} = 57.9174892055112$$
$$x_{63} = 22.1910938993597$$
$$x_{64} = -91.6334642195858$$
$$x_{65} = -1.53499006191973$$
$$x_{66} = 17.5240030212392$$
$$x_{67} = 92.1121944108522$$
$$x_{68} = 90.0774579240502$$
$$x_{69} = 46.2454235458089$$
$$x_{70} = 83.9393049997348$$
$$x_{71} = 48.1752174135152$$
$$x_{72} = 10.1433307845631$$
$$x_{73} = -48.8872847601341$$
$$x_{74} = 34.5855663948751$$
$$x_{75} = 50.2499263215659$$
$$x_{76} = -85.423265777689$$
$$x_{77} = -27.9548058941653$$
$$x_{78} = 29.0568929111882$$
$$x_{79} = -57.9987959688067$$
$$x_{80} = -61.8774276013682$$
$$x_{81} = 960.026766802951$$
$$x_{82} = 65.5748041903369$$
$$x_{83} = -33.7581506467653$$
$$x_{84} = 54.6522571848638$$
$$x_{85} = 114.714193662878$$
$$x_{86} = -95.7576292399995$$
$$x_{87} = 28.1786717784004$$
$$x_{88} = -71.7515502346136$$
$$x_{89} = 15.7289675907288$$
$$x_{90} = -67.6499452927859$$
$$x_{91} = -41.7468384980101$$
$$x_{92} = -35.8347038100614$$
$$x_{93} = 8.07391154038662$$
$$x_{94} = 26.0948310526355$$
$$x_{95} = -59.8121111003062$$
$$x_{96} = 70.1576103499243$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{- 2 x \sin{\left(x^{2} \right)} + 2 x \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} + \frac{\left(- 2 x \cos{\left(x^{2} \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}\right)}{\left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 42.2420509425157$$
$$x_{2} = 36.1511086255456$$
$$x_{3} = -83.3618143109984$$
$$x_{4} = -63.9536851214823$$
$$x_{5} = -54.6872632012662$$
$$x_{6} = 35.8885004766764$$
$$x_{7} = -52.0385800201411$$
$$x_{8} = -98.9721469476567$$
$$x_{9} = -36.237778822583$$
$$x_{10} = 4.19024077075224$$
$$x_{11} = -7.9269010690929$$
$$x_{12} = 86.1415696407981$$
$$x_{13} = 11.2087112264728$$
$$x_{14} = -20.6688726311137$$
$$x_{15} = 70.2734736977571$$
$$x_{16} = 26.8801153827676$$
$$x_{17} = 7.9269010690929$$
$$x_{18} = 20.2076374397785$$
$$x_{19} = -29.9747014965618$$
$$x_{20} = -23.7791095690547$$
$$x_{21} = 58.2486504259788$$
$$x_{22} = 64.2497183020442$$
$$x_{23} = -80.6006600831109$$
$$x_{24} = 52.2181211688324$$
$$x_{25} = 64.4452041213861$$
$$x_{26} = 86.7948313052633$$
$$x_{27} = -11.2087112264728$$
$$x_{28} = -63.8496273267267$$
$$x_{29} = -79.7396570935865$$
$$x_{30} = 4.30706997401766$$
$$x_{31} = 26.4068862519245$$
$$x_{32} = -54.9174305424362$$
$$x_{33} = -5.00970729500514$$
$$x_{34} = -36.1511086255456$$
$$x_{35} = 23.9090740616131$$
$$x_{36} = -76.646296049036$$
$$x_{37} = -89.7496376072358$$
$$x_{38} = -39.7093805576759$$
$$x_{39} = 33.1591898578877$$
$$x_{40} = -8.30100224079401$$
$$x_{41} = 80.2511463209266$$
$$x_{42} = 20.6688726311137$$
Signos de extremos en los puntos:
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:
$$x_{1} = 4.19024077075224$$
$$x_{2} = -63.8496273267267$$
Puntos máximos de la función:
$$x_{2} = 42.2420509425157$$
$$x_{2} = 36.1511086255456$$
$$x_{2} = -83.3618143109984$$
$$x_{2} = -63.9536851214823$$
$$x_{2} = -54.6872632012662$$
$$x_{2} = 35.8885004766764$$
$$x_{2} = -52.0385800201411$$
$$x_{2} = -98.9721469476567$$
$$x_{2} = -36.237778822583$$
$$x_{2} = -7.9269010690929$$
$$x_{2} = 86.1415696407981$$
$$x_{2} = 11.2087112264728$$
$$x_{2} = -20.6688726311137$$
$$x_{2} = 70.2734736977571$$
$$x_{2} = 26.8801153827676$$
$$x_{2} = 7.9269010690929$$
$$x_{2} = 20.2076374397785$$
$$x_{2} = -29.9747014965618$$
$$x_{2} = -23.7791095690547$$
$$x_{2} = 58.2486504259788$$
$$x_{2} = 64.2497183020442$$
$$x_{2} = -80.6006600831109$$
$$x_{2} = 52.2181211688324$$
$$x_{2} = 64.4452041213861$$
$$x_{2} = 86.7948313052633$$
$$x_{2} = -11.2087112264728$$
$$x_{2} = -79.7396570935865$$
$$x_{2} = 4.30706997401766$$
$$x_{2} = 26.4068862519245$$
$$x_{2} = -54.9174305424362$$
$$x_{2} = -5.00970729500514$$
$$x_{2} = -36.1511086255456$$
$$x_{2} = 23.9090740616131$$
$$x_{2} = -76.646296049036$$
$$x_{2} = -89.7496376072358$$
$$x_{2} = -39.7093805576759$$
$$x_{2} = 33.1591898578877$$
$$x_{2} = -8.30100224079401$$
$$x_{2} = 80.2511463209266$$
$$x_{2} = 20.6688726311137$$
Decrece en los intervalos
$$\left[4.19024077075224, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -63.8496273267267\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{- 2 x \sin{\left(x^{2} \right)} + 2 x \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} + \frac{\left(- 2 x \cos{\left(x^{2} \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}\right)}{\left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 42.2420509425157$$
$$x_{2} = 36.1511086255456$$
$$x_{3} = -83.3618143109984$$
$$x_{4} = -63.9536851214823$$
$$x_{5} = -54.6872632012662$$
$$x_{6} = 35.8885004766764$$
$$x_{7} = -52.0385800201411$$
$$x_{8} = -98.9721469476567$$
$$x_{9} = -36.237778822583$$
$$x_{10} = 4.19024077075224$$
$$x_{11} = -7.9269010690929$$
$$x_{12} = 86.1415696407981$$
$$x_{13} = 11.2087112264728$$
$$x_{14} = -20.6688726311137$$
$$x_{15} = 70.2734736977571$$
$$x_{16} = 26.8801153827676$$
$$x_{17} = 7.9269010690929$$
$$x_{18} = 20.2076374397785$$
$$x_{19} = -29.9747014965618$$
$$x_{20} = -23.7791095690547$$
$$x_{21} = 58.2486504259788$$
$$x_{22} = 64.2497183020442$$
$$x_{23} = -80.6006600831109$$
$$x_{24} = 52.2181211688324$$
$$x_{25} = 64.4452041213861$$
$$x_{26} = 86.7948313052633$$
$$x_{27} = -11.2087112264728$$
$$x_{28} = -63.8496273267267$$
$$x_{29} = -79.7396570935865$$
$$x_{30} = 4.30706997401766$$
$$x_{31} = 26.4068862519245$$
$$x_{32} = -54.9174305424362$$
$$x_{33} = -5.00970729500514$$
$$x_{34} = -36.1511086255456$$
$$x_{35} = 23.9090740616131$$
$$x_{36} = -76.646296049036$$
$$x_{37} = -89.7496376072358$$
$$x_{38} = -39.7093805576759$$
$$x_{39} = 33.1591898578877$$
$$x_{40} = -8.30100224079401$$
$$x_{41} = 80.2511463209266$$
$$x_{42} = 20.6688726311137$$
Signos de extremos en los puntos:
(42.24205094251572, 1.02971793252094)
(36.15110862554562, 1.00051964197055)
(-83.36181431099843, 1.0121852710788)
(-63.9536851214823, 1.27919412710723)
(-54.68726320126624, 1.09425867266373)
(35.888500476676384, 1.06178671554201)
(-52.03858002014106, 1.04302482043874)
(-98.9721469476567, 1.00014349779555)
(-36.23777882258297, 1.01214861318927)
(4.190240770752238, 3.26460723851972)
(-7.926901069092898, 1.00530762464282)
(86.14156964079811, 1.06894475409017)
(11.208711226472843, 1.04784139811787)
(-20.668872631113732, 1.06668070533838)
(70.27347369775707, 1.21975974982824)
(26.88011538276757, 1.0323567461373)
(7.926901069092898, 1.00530762464282)
(20.207637439778537, 1.0477826430248)
(-29.974701496561764, 1.01711780550195)
(-23.77910956905474, 1.04994586078658)
(58.24865042597879, 1.01702133235174)
(64.24971830204419, 1.02405057246683)
(-80.60066008311091, 1.36936368961988)
(52.2181211688324, 1.17983656948106)
(64.44520412138611, 1.00181452348539)
(86.79483130526329, 1.20409212353676)
(-11.208711226472843, 1.04784139811787)
(-63.84962732672667, 2.56665191752467)
(-79.73965709358646, 1.16727405476512)
(4.307069974017655, 1.20209604565103)
(26.406886251924515, 1.098671029661)
(-54.917430542436236, 1.00366816002872)
(-5.0097072950051365, 1.09696144946127)
(-36.15110862554562, 1.00051964197055)
(23.909074061613143, 1.14211608918682)
(-76.646296049036, 1.11976890648106)
(-89.74963760723584, 1.04857569261922)
(-39.70938055767593, 1.26222035782209)
(33.159189857887704, 1.03081980222094)
(-8.301002240794014, 1.27254936120682)
(80.25114632092662, 1.02021958263179)
(20.668872631113732, 1.06668070533838)
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:
$$x_{1} = 4.19024077075224$$
$$x_{2} = -63.8496273267267$$
Puntos máximos de la función:
$$x_{2} = 42.2420509425157$$
$$x_{2} = 36.1511086255456$$
$$x_{2} = -83.3618143109984$$
$$x_{2} = -63.9536851214823$$
$$x_{2} = -54.6872632012662$$
$$x_{2} = 35.8885004766764$$
$$x_{2} = -52.0385800201411$$
$$x_{2} = -98.9721469476567$$
$$x_{2} = -36.237778822583$$
$$x_{2} = -7.9269010690929$$
$$x_{2} = 86.1415696407981$$
$$x_{2} = 11.2087112264728$$
$$x_{2} = -20.6688726311137$$
$$x_{2} = 70.2734736977571$$
$$x_{2} = 26.8801153827676$$
$$x_{2} = 7.9269010690929$$
$$x_{2} = 20.2076374397785$$
$$x_{2} = -29.9747014965618$$
$$x_{2} = -23.7791095690547$$
$$x_{2} = 58.2486504259788$$
$$x_{2} = 64.2497183020442$$
$$x_{2} = -80.6006600831109$$
$$x_{2} = 52.2181211688324$$
$$x_{2} = 64.4452041213861$$
$$x_{2} = 86.7948313052633$$
$$x_{2} = -11.2087112264728$$
$$x_{2} = -79.7396570935865$$
$$x_{2} = 4.30706997401766$$
$$x_{2} = 26.4068862519245$$
$$x_{2} = -54.9174305424362$$
$$x_{2} = -5.00970729500514$$
$$x_{2} = -36.1511086255456$$
$$x_{2} = 23.9090740616131$$
$$x_{2} = -76.646296049036$$
$$x_{2} = -89.7496376072358$$
$$x_{2} = -39.7093805576759$$
$$x_{2} = 33.1591898578877$$
$$x_{2} = -8.30100224079401$$
$$x_{2} = 80.2511463209266$$
$$x_{2} = 20.6688726311137$$
Decrece en los intervalos
$$\left[4.19024077075224, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -63.8496273267267\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(x^2) + cos(x^2))/(sin(x)^2 + sin(x^2)), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{x \left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{x \left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{x \left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{x \left(\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}\right)}\right)$$
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:
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = \frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}$$
- Sí
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = - \frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = \frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}$$
- Sí
$$\frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}} = - \frac{\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}}{\sin^{2}{\left(x \right)} + \sin{\left(x^{2} \right)}}$$
- No
es decir, función
es
par