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- Puntos de cruce de las funciones
-
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- Gráfico de la función y =:
-
x^4-x^2+2
-
(x^2-5)/(x-3)
-
(x^2-9)/(x^2-4)
-
(x+1)*(x-2)^2
-
Expresiones idénticas
- sin(x)-cos(x^ dos)+(uno / cuatro)
- seno de (x) menos coseno de (x al cuadrado ) más (1 dividir por 4)
- seno de (x) menos coseno de (x en el grado dos) más (uno dividir por cuatro)
- sin(x)-cos(x2)+(1/4)
- sinx-cosx2+1/4
- sin(x)-cos(x²)+(1/4)
- sin(x)-cos(x en el grado 2)+(1/4)
- sinx-cosx^2+1/4
- sin(x)-cos(x^2)+(1 dividir por 4)
-
Expresiones semejantes
- sin(x)-cos(x^2)-(1/4)
- sin(x)+cos(x^2)+(1/4)
- sinx-cos(x^2)+(1/4)
-
Expresiones con funciones
- Seno sin
- sin(x)+5
- sin(x)*2*x
- sin(x^(3)+x)
- sin(x)*3
- sin(x^2+x-2)
- Coseno cos
- cos(x)+cos(2x)
- cos(x^2+5)^(3)
- cos[x/3]
- cos(x*sqrt(6))+sin(x*sqrt(6))
- cos(x)/8
Gráfico de la función y = sin(x)-cos(x^2)+(1/4)
Solución
Ha introducido
[src]
/ 2\ 1
f(x) = sin(x) - cos\x / + -
4
$$f{\left(x \right)} = \left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}$$
f = sin(x) - cos(x^2) + 1/4
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:
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 40.1111824792713$$
$$x_{2} = -43.4238638224173$$
$$x_{3} = -9.65261285774575$$
$$x_{4} = -33.6919704330286$$
$$x_{5} = -78.6622401239618$$
$$x_{6} = 90.9385101983754$$
$$x_{7} = 54.3623963985464$$
$$x_{8} = -27.9964832675147$$
$$x_{9} = -69.7503172817115$$
$$x_{10} = 43.3212314723419$$
$$x_{11} = 0.691602448121161$$
$$x_{12} = -85.2944255047074$$
$$x_{13} = 73.876051319832$$
$$x_{14} = 6.24917558715415$$
$$x_{15} = 6.69651557104387$$
$$x_{16} = -24.7044915641557$$
$$x_{17} = 35.5686851346348$$
$$x_{18} = 25.4584447108766$$
$$x_{19} = -90.0071782775614$$
$$x_{20} = 21.8260959682456$$
$$x_{21} = -21.8848905883197$$
$$x_{22} = 68.2643913788867$$
$$x_{23} = 169.844666479171$$
$$x_{24} = -22.1635753055404$$
$$x_{25} = 2.42010171228507$$
$$x_{26} = 77.7441722010558$$
$$x_{27} = 98.2500719411958$$
$$x_{28} = -3.64841219611475$$
$$x_{29} = -25.9689454826438$$
$$x_{30} = -26.9248773198902$$
$$x_{31} = 34.0823189250325$$
$$x_{32} = 12.8221133045797$$
$$x_{33} = -47.7516263856737$$
$$x_{34} = -83.7139403250785$$
$$x_{35} = -1.55520891280466$$
$$x_{36} = -59.7503539523153$$
$$x_{37} = 40.9764116446742$$
$$x_{38} = 75.5244784928168$$
$$x_{39} = -11.7485351246587$$
$$x_{40} = -68.4142549325336$$
$$x_{41} = -90.8892327693225$$
$$x_{42} = 72.2496786240861$$
$$x_{43} = 68.2491628376529$$
$$x_{44} = -83.385133658414$$
$$x_{45} = -53.7697549442524$$
$$x_{46} = -10.05512490128$$
$$x_{47} = 30.2496446403622$$
$$x_{48} = -71.8104575408252$$
$$x_{49} = 28.0065685403739$$
$$x_{50} = -77.8794998506028$$
$$x_{51} = -97.5392361785001$$
$$x_{52} = 879.359550496925$$
$$x_{53} = -1.99782462685321$$
$$x_{54} = -21.8180927772482$$
$$x_{55} = 99.8040429118756$$
$$x_{56} = 4.0847382487479$$
$$x_{57} = -108.268255492474$$
$$x_{58} = 10.2347083141931$$
$$x_{59} = -18.5625074499192$$
$$x_{60} = 37.4528358578611$$
$$x_{61} = 367.362811896333$$
$$x_{62} = 65.5028621304597$$
$$x_{63} = 29.5130989528393$$
$$x_{64} = 87.9748646153398$$
$$x_{65} = 16.0000573107138$$
$$x_{66} = -24.4413856207385$$
$$x_{67} = 91.042700967986$$
$$x_{68} = -70.7821345465067$$
$$x_{69} = 94.3172556650567$$
$$x_{70} = 75.2503680674066$$
$$x_{71} = 82.3790523264691$$
$$x_{72} = -65.7124580075391$$
$$x_{73} = -45.4396168381865$$
$$x_{74} = 48.9030352869289$$
$$x_{75} = -55.9999908638865$$
$$x_{76} = -7.77289829567417$$
$$x_{77} = 25.6982715775585$$
$$x_{78} = 50.8848029425982$$
$$x_{79} = 37.7494005448309$$
$$x_{80} = -88.5267746421002$$
$$x_{81} = -93.9621491488001$$
$$x_{82} = -39.819348570996$$
$$x_{83} = -57.3069117873017$$
$$x_{84} = 35.5945980633381$$
$$x_{85} = 8.99137215079902$$
$$x_{86} = 106.101278877099$$
$$x_{87} = 23.829670676031$$
$$x_{88} = 61.0083417543066$$
$$x_{89} = -43.9055787200602$$
$$x_{90} = 13.2449784557428$$
$$x_{91} = -37.7847675885331$$
o sea hay que resolver la ecuación:
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 40.1111824792713$$
$$x_{2} = -43.4238638224173$$
$$x_{3} = -9.65261285774575$$
$$x_{4} = -33.6919704330286$$
$$x_{5} = -78.6622401239618$$
$$x_{6} = 90.9385101983754$$
$$x_{7} = 54.3623963985464$$
$$x_{8} = -27.9964832675147$$
$$x_{9} = -69.7503172817115$$
$$x_{10} = 43.3212314723419$$
$$x_{11} = 0.691602448121161$$
$$x_{12} = -85.2944255047074$$
$$x_{13} = 73.876051319832$$
$$x_{14} = 6.24917558715415$$
$$x_{15} = 6.69651557104387$$
$$x_{16} = -24.7044915641557$$
$$x_{17} = 35.5686851346348$$
$$x_{18} = 25.4584447108766$$
$$x_{19} = -90.0071782775614$$
$$x_{20} = 21.8260959682456$$
$$x_{21} = -21.8848905883197$$
$$x_{22} = 68.2643913788867$$
$$x_{23} = 169.844666479171$$
$$x_{24} = -22.1635753055404$$
$$x_{25} = 2.42010171228507$$
$$x_{26} = 77.7441722010558$$
$$x_{27} = 98.2500719411958$$
$$x_{28} = -3.64841219611475$$
$$x_{29} = -25.9689454826438$$
$$x_{30} = -26.9248773198902$$
$$x_{31} = 34.0823189250325$$
$$x_{32} = 12.8221133045797$$
$$x_{33} = -47.7516263856737$$
$$x_{34} = -83.7139403250785$$
$$x_{35} = -1.55520891280466$$
$$x_{36} = -59.7503539523153$$
$$x_{37} = 40.9764116446742$$
$$x_{38} = 75.5244784928168$$
$$x_{39} = -11.7485351246587$$
$$x_{40} = -68.4142549325336$$
$$x_{41} = -90.8892327693225$$
$$x_{42} = 72.2496786240861$$
$$x_{43} = 68.2491628376529$$
$$x_{44} = -83.385133658414$$
$$x_{45} = -53.7697549442524$$
$$x_{46} = -10.05512490128$$
$$x_{47} = 30.2496446403622$$
$$x_{48} = -71.8104575408252$$
$$x_{49} = 28.0065685403739$$
$$x_{50} = -77.8794998506028$$
$$x_{51} = -97.5392361785001$$
$$x_{52} = 879.359550496925$$
$$x_{53} = -1.99782462685321$$
$$x_{54} = -21.8180927772482$$
$$x_{55} = 99.8040429118756$$
$$x_{56} = 4.0847382487479$$
$$x_{57} = -108.268255492474$$
$$x_{58} = 10.2347083141931$$
$$x_{59} = -18.5625074499192$$
$$x_{60} = 37.4528358578611$$
$$x_{61} = 367.362811896333$$
$$x_{62} = 65.5028621304597$$
$$x_{63} = 29.5130989528393$$
$$x_{64} = 87.9748646153398$$
$$x_{65} = 16.0000573107138$$
$$x_{66} = -24.4413856207385$$
$$x_{67} = 91.042700967986$$
$$x_{68} = -70.7821345465067$$
$$x_{69} = 94.3172556650567$$
$$x_{70} = 75.2503680674066$$
$$x_{71} = 82.3790523264691$$
$$x_{72} = -65.7124580075391$$
$$x_{73} = -45.4396168381865$$
$$x_{74} = 48.9030352869289$$
$$x_{75} = -55.9999908638865$$
$$x_{76} = -7.77289829567417$$
$$x_{77} = 25.6982715775585$$
$$x_{78} = 50.8848029425982$$
$$x_{79} = 37.7494005448309$$
$$x_{80} = -88.5267746421002$$
$$x_{81} = -93.9621491488001$$
$$x_{82} = -39.819348570996$$
$$x_{83} = -57.3069117873017$$
$$x_{84} = 35.5945980633381$$
$$x_{85} = 8.99137215079902$$
$$x_{86} = 106.101278877099$$
$$x_{87} = 23.829670676031$$
$$x_{88} = 61.0083417543066$$
$$x_{89} = -43.9055787200602$$
$$x_{90} = 13.2449784557428$$
$$x_{91} = -37.7847675885331$$
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
$$2 x \sin{\left(x^{2} \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 22.349323885059$$
$$x_{2} = -47.4608457084011$$
$$x_{3} = -22.4903520904245$$
$$x_{4} = 6.3967526067572$$
$$x_{5} = 56.2735892190937$$
$$x_{6} = 96.0892263352259$$
$$x_{7} = -99.1940994186411$$
$$x_{8} = 42.2424741036786$$
$$x_{9} = 52.2497863224473$$
$$x_{10} = -43.8482155265515$$
$$x_{11} = -7.72541014397883$$
$$x_{12} = -19.816274863823$$
$$x_{13} = 90.6554540354125$$
$$x_{14} = -0.730978270667906$$
$$x_{15} = -9.87092112794122$$
$$x_{16} = 28.2485792398582$$
$$x_{17} = 14.7224702337865$$
$$x_{18} = -41.8314912305025$$
$$x_{19} = 98.0953333197619$$
$$x_{20} = -74.0622869224726$$
$$x_{21} = 87.7857185656868$$
$$x_{22} = -81.840586323002$$
$$x_{23} = -5.87191528001199$$
$$x_{24} = 87.8035463465879$$
$$x_{25} = 66.3427786515794$$
$$x_{26} = -72.1935532053449$$
$$x_{27} = -77.7257170387604$$
$$x_{28} = -83.9440058712679$$
$$x_{29} = 64.2499148858234$$
$$x_{30} = 16.2456215982105$$
$$x_{31} = -61.8584222114262$$
$$x_{32} = -39.7914116037745$$
$$x_{33} = 18.2479103002405$$
$$x_{34} = 69.3298616312379$$
$$x_{35} = 93.7559147144186$$
$$x_{36} = -53.8778557584569$$
$$x_{37} = 8.12140237825279$$
$$x_{38} = 29.2322260727598$$
$$x_{39} = 1.75748767956681$$
$$x_{40} = 79.7604098082138$$
$$x_{41} = -3.97400948361753$$
$$x_{42} = 27.2863953811205$$
$$x_{43} = 82.2044467599582$$
$$x_{44} = 60.885813954976$$
$$x_{45} = -15.3508947858058$$
$$x_{46} = 11.0691206807264$$
$$x_{47} = -15.7549206941732$$
$$x_{48} = 47.2616324020612$$
$$x_{49} = 46.1858745664205$$
$$x_{50} = -33.8628507747295$$
$$x_{51} = -46.928316630699$$
$$x_{52} = -1.78948236293348$$
$$x_{53} = -65.5808461895275$$
$$x_{54} = -51.1560410192306$$
$$x_{55} = 36.7972126661047$$
$$x_{56} = -91.9286709217352$$
$$x_{57} = 22.1384461152026$$
$$x_{58} = 6.13338537866179$$
$$x_{59} = -87.1571856052754$$
$$x_{60} = 3.95230187084535$$
$$x_{61} = 85.4278345722129$$
$$x_{62} = 57.5433221417068$$
$$x_{63} = -89.7498878995054$$
$$x_{64} = 48.1833155017359$$
$$x_{65} = 78.44981619248$$
$$x_{66} = -72.106467795218$$
$$x_{67} = 9.71067451011131$$
$$x_{68} = -11.621672446497$$
$$x_{69} = -40.1453184602676$$
$$x_{70} = -17.9879387647864$$
$$x_{71} = -89.8897901584974$$
$$x_{72} = -14.0683294162172$$
$$x_{73} = -22.0673844234238$$
$$x_{74} = 99.6364875148062$$
$$x_{75} = 44.7697244998911$$
$$x_{76} = 80.3686064857456$$
$$x_{77} = 94.1405234555763$$
$$x_{78} = 31.2578168207404$$
$$x_{79} = -49.2792605670951$$
$$x_{80} = -55.8533314248926$$
$$x_{81} = 57.7341585233858$$
$$x_{82} = -95.8600997718734$$
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:
$$x_{1} = 56.2735892190937$$
$$x_{2} = -99.1940994186411$$
$$x_{3} = 42.2424741036786$$
$$x_{4} = -43.8482155265515$$
$$x_{5} = 90.6554540354125$$
$$x_{6} = -0.730978270667906$$
$$x_{7} = 28.2485792398582$$
$$x_{8} = -74.0622869224726$$
$$x_{9} = -81.840586323002$$
$$x_{10} = 87.8035463465879$$
$$x_{11} = 64.2499148858234$$
$$x_{12} = 16.2456215982105$$
$$x_{13} = -61.8584222114262$$
$$x_{14} = -39.7914116037745$$
$$x_{15} = 18.2479103002405$$
$$x_{16} = 69.3298616312379$$
$$x_{17} = 93.7559147144186$$
$$x_{18} = -53.8778557584569$$
$$x_{19} = 29.2322260727598$$
$$x_{20} = 60.885813954976$$
$$x_{21} = -91.9286709217352$$
$$x_{22} = 22.1384461152026$$
$$x_{23} = 6.13338537866179$$
$$x_{24} = -87.1571856052754$$
$$x_{25} = 57.5433221417068$$
$$x_{26} = -89.7498878995054$$
$$x_{27} = 9.71067451011131$$
$$x_{28} = -89.8897901584974$$
$$x_{29} = 99.6364875148062$$
$$x_{30} = 44.7697244998911$$
$$x_{31} = 80.3686064857456$$
Puntos máximos de la función:
$$x_{31} = 22.349323885059$$
$$x_{31} = -47.4608457084011$$
$$x_{31} = -22.4903520904245$$
$$x_{31} = 6.3967526067572$$
$$x_{31} = 96.0892263352259$$
$$x_{31} = 52.2497863224473$$
$$x_{31} = -7.72541014397883$$
$$x_{31} = -19.816274863823$$
$$x_{31} = -9.87092112794122$$
$$x_{31} = 14.7224702337865$$
$$x_{31} = -41.8314912305025$$
$$x_{31} = 98.0953333197619$$
$$x_{31} = 87.7857185656868$$
$$x_{31} = -5.87191528001199$$
$$x_{31} = 66.3427786515794$$
$$x_{31} = -72.1935532053449$$
$$x_{31} = -77.7257170387604$$
$$x_{31} = -83.9440058712679$$
$$x_{31} = 8.12140237825279$$
$$x_{31} = 1.75748767956681$$
$$x_{31} = 79.7604098082138$$
$$x_{31} = -3.97400948361753$$
$$x_{31} = 27.2863953811205$$
$$x_{31} = 82.2044467599582$$
$$x_{31} = -15.3508947858058$$
$$x_{31} = 11.0691206807264$$
$$x_{31} = -15.7549206941732$$
$$x_{31} = 47.2616324020612$$
$$x_{31} = 46.1858745664205$$
$$x_{31} = -33.8628507747295$$
$$x_{31} = -46.928316630699$$
$$x_{31} = -1.78948236293348$$
$$x_{31} = -65.5808461895275$$
$$x_{31} = -51.1560410192306$$
$$x_{31} = 36.7972126661047$$
$$x_{31} = 3.95230187084535$$
$$x_{31} = 85.4278345722129$$
$$x_{31} = 48.1833155017359$$
$$x_{31} = 78.44981619248$$
$$x_{31} = -72.106467795218$$
$$x_{31} = -11.621672446497$$
$$x_{31} = -40.1453184602676$$
$$x_{31} = -17.9879387647864$$
$$x_{31} = -14.0683294162172$$
$$x_{31} = -22.0673844234238$$
$$x_{31} = 94.1405234555763$$
$$x_{31} = 31.2578168207404$$
$$x_{31} = -49.2792605670951$$
$$x_{31} = -55.8533314248926$$
$$x_{31} = 57.7341585233858$$
$$x_{31} = -95.8600997718734$$
Decrece en los intervalos
$$\left[99.6364875148062, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.1940994186411\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$2 x \sin{\left(x^{2} \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 22.349323885059$$
$$x_{2} = -47.4608457084011$$
$$x_{3} = -22.4903520904245$$
$$x_{4} = 6.3967526067572$$
$$x_{5} = 56.2735892190937$$
$$x_{6} = 96.0892263352259$$
$$x_{7} = -99.1940994186411$$
$$x_{8} = 42.2424741036786$$
$$x_{9} = 52.2497863224473$$
$$x_{10} = -43.8482155265515$$
$$x_{11} = -7.72541014397883$$
$$x_{12} = -19.816274863823$$
$$x_{13} = 90.6554540354125$$
$$x_{14} = -0.730978270667906$$
$$x_{15} = -9.87092112794122$$
$$x_{16} = 28.2485792398582$$
$$x_{17} = 14.7224702337865$$
$$x_{18} = -41.8314912305025$$
$$x_{19} = 98.0953333197619$$
$$x_{20} = -74.0622869224726$$
$$x_{21} = 87.7857185656868$$
$$x_{22} = -81.840586323002$$
$$x_{23} = -5.87191528001199$$
$$x_{24} = 87.8035463465879$$
$$x_{25} = 66.3427786515794$$
$$x_{26} = -72.1935532053449$$
$$x_{27} = -77.7257170387604$$
$$x_{28} = -83.9440058712679$$
$$x_{29} = 64.2499148858234$$
$$x_{30} = 16.2456215982105$$
$$x_{31} = -61.8584222114262$$
$$x_{32} = -39.7914116037745$$
$$x_{33} = 18.2479103002405$$
$$x_{34} = 69.3298616312379$$
$$x_{35} = 93.7559147144186$$
$$x_{36} = -53.8778557584569$$
$$x_{37} = 8.12140237825279$$
$$x_{38} = 29.2322260727598$$
$$x_{39} = 1.75748767956681$$
$$x_{40} = 79.7604098082138$$
$$x_{41} = -3.97400948361753$$
$$x_{42} = 27.2863953811205$$
$$x_{43} = 82.2044467599582$$
$$x_{44} = 60.885813954976$$
$$x_{45} = -15.3508947858058$$
$$x_{46} = 11.0691206807264$$
$$x_{47} = -15.7549206941732$$
$$x_{48} = 47.2616324020612$$
$$x_{49} = 46.1858745664205$$
$$x_{50} = -33.8628507747295$$
$$x_{51} = -46.928316630699$$
$$x_{52} = -1.78948236293348$$
$$x_{53} = -65.5808461895275$$
$$x_{54} = -51.1560410192306$$
$$x_{55} = 36.7972126661047$$
$$x_{56} = -91.9286709217352$$
$$x_{57} = 22.1384461152026$$
$$x_{58} = 6.13338537866179$$
$$x_{59} = -87.1571856052754$$
$$x_{60} = 3.95230187084535$$
$$x_{61} = 85.4278345722129$$
$$x_{62} = 57.5433221417068$$
$$x_{63} = -89.7498878995054$$
$$x_{64} = 48.1833155017359$$
$$x_{65} = 78.44981619248$$
$$x_{66} = -72.106467795218$$
$$x_{67} = 9.71067451011131$$
$$x_{68} = -11.621672446497$$
$$x_{69} = -40.1453184602676$$
$$x_{70} = -17.9879387647864$$
$$x_{71} = -89.8897901584974$$
$$x_{72} = -14.0683294162172$$
$$x_{73} = -22.0673844234238$$
$$x_{74} = 99.6364875148062$$
$$x_{75} = 44.7697244998911$$
$$x_{76} = 80.3686064857456$$
$$x_{77} = 94.1405234555763$$
$$x_{78} = 31.2578168207404$$
$$x_{79} = -49.2792605670951$$
$$x_{80} = -55.8533314248926$$
$$x_{81} = 57.7341585233858$$
$$x_{82} = -95.8600997718734$$
Signos de extremos en los puntos:
(22.34932388505898, 0.899214550998202)
(-47.46084570840113, 1.58056628769818)
(-22.490352090424498, 1.7285358978664)
(6.396752606757205, 1.36030314711518)
(56.27358921909367, -1.02158596885626)
(96.08922633522592, 2.21359624249138)
(-99.19409941864106, 0.222763416653808)
(42.242474103678596, -1.73574697954699)
(52.24978632244733, 2.16570965376942)
(-43.848215526551456, -0.616255911808996)
(-7.725410143978831, 0.258219501823969)
(-19.816274863823022, 0.426870799296581)
(90.65545403541253, -0.31436230477943)
(-0.7309782706679064, -1.27820871872329)
(-9.87092112794122, 1.68044483800528)
(28.24857923985821, -0.724091651171351)
(14.722470233786547, 2.0833685437479)
(-41.831491230502486, 2.08643593813823)
(98.09533331976195, 0.601226993957887)
(-74.06228692247262, 0.222548265536392)
(87.78571856568684, 1.07206093361989)
(-81.84058632300203, -0.908487796295731)
(-5.871915280011989, 1.64672316794882)
(87.8035463465879, -0.910336890591339)
(66.34277865157937, 0.888981880025799)
(-72.19355320534488, 1.18694010507801)
(-77.72571703876044, 0.522882705493363)
(-83.94400587126786, 0.479894133412259)
(64.2499148858234, 0.23835944207481)
(16.24562159821051, -1.26177664427882)
(-61.85842221142619, 0.0768306534387587)
(-39.79141160377455, -1.61705157908055)
(18.24791030024052, -1.3157447333335)
(69.32986163123788, -0.536800429133705)
(93.75591471441857, -1.22225947668199)
(-53.87785575845691, -0.296383649424088)
(8.121402378252785, 2.21432331894488)
(29.23222607275975, -1.56793247208485)
(1.757487679566813, 2.23122856178744)
(79.76040980821381, 0.310694553304226)
(-3.9740094836175293, 1.98596791565122)
(27.286395381120506, 2.08484221611745)
(82.20444675995824, 1.74950019091033)
(60.88581395497595, -1.68041413795555)
(-15.3508947858058, 0.900005123632314)
(11.069120680726392, 0.252697808719373)
(-15.754920694173173, 1.29643756401958)
(47.26163240206123, 1.1126376460065)
(46.18587456642046, 2.05636542658033)
(-33.8628507747295, 0.609804928108061)
(-46.92831663069898, 1.05561656898399)
(-1.7894823629334766, 0.271977728650254)
(-65.58084618952749, 0.867383727832319)
(-51.15604101923061, 0.472557911280981)
(36.79721266610471, 0.465458455391031)
(-91.92867092173525, -0.0171549645821363)
(22.138446115202626, -0.89651589398298)
(6.13338537866179, -0.895986177863084)
(-87.15718560527543, -0.0274940903763204)
(3.9523018708453495, 0.521418115405953)
(85.42783457221287, 0.68136376085504)
(57.54332214170678, 0.0885819277160163)
(-89.74988789950542, -1.72708282333138)
(48.183315501735876, 0.377912527419599)
(78.44981619247996, 1.33985854897891)
(-72.10646779521804, 1.10037696142293)
(9.710674510111312, -1.03079681074621)
(-11.621672446496989, 2.06000225359449)
(-40.145318460267575, 0.609272386516736)
(-17.98793876478641, 2.00873282272673)
(-89.88979015849739, -1.6878531739228)
(-14.068329416217198, 0.252365378903732)
(-22.067384423423757, 1.32590679062373)
(99.63648751480619, -1.52987714544096)
(44.76972449989114, -0.0414287677900198)
(80.36860648574562, -1.71690252331832)
(94.1405234555763, 1.14293543038608)
(31.257816820740402, 1.09242344557681)
(-49.27926056709511, 2.08393133091633)
(-55.85333142489263, 1.89062010610832)
(57.73415852338577, 2.17667833321499)
(-95.86009977187342, 0.25086196722793)
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:
$$x_{1} = 56.2735892190937$$
$$x_{2} = -99.1940994186411$$
$$x_{3} = 42.2424741036786$$
$$x_{4} = -43.8482155265515$$
$$x_{5} = 90.6554540354125$$
$$x_{6} = -0.730978270667906$$
$$x_{7} = 28.2485792398582$$
$$x_{8} = -74.0622869224726$$
$$x_{9} = -81.840586323002$$
$$x_{10} = 87.8035463465879$$
$$x_{11} = 64.2499148858234$$
$$x_{12} = 16.2456215982105$$
$$x_{13} = -61.8584222114262$$
$$x_{14} = -39.7914116037745$$
$$x_{15} = 18.2479103002405$$
$$x_{16} = 69.3298616312379$$
$$x_{17} = 93.7559147144186$$
$$x_{18} = -53.8778557584569$$
$$x_{19} = 29.2322260727598$$
$$x_{20} = 60.885813954976$$
$$x_{21} = -91.9286709217352$$
$$x_{22} = 22.1384461152026$$
$$x_{23} = 6.13338537866179$$
$$x_{24} = -87.1571856052754$$
$$x_{25} = 57.5433221417068$$
$$x_{26} = -89.7498878995054$$
$$x_{27} = 9.71067451011131$$
$$x_{28} = -89.8897901584974$$
$$x_{29} = 99.6364875148062$$
$$x_{30} = 44.7697244998911$$
$$x_{31} = 80.3686064857456$$
Puntos máximos de la función:
$$x_{31} = 22.349323885059$$
$$x_{31} = -47.4608457084011$$
$$x_{31} = -22.4903520904245$$
$$x_{31} = 6.3967526067572$$
$$x_{31} = 96.0892263352259$$
$$x_{31} = 52.2497863224473$$
$$x_{31} = -7.72541014397883$$
$$x_{31} = -19.816274863823$$
$$x_{31} = -9.87092112794122$$
$$x_{31} = 14.7224702337865$$
$$x_{31} = -41.8314912305025$$
$$x_{31} = 98.0953333197619$$
$$x_{31} = 87.7857185656868$$
$$x_{31} = -5.87191528001199$$
$$x_{31} = 66.3427786515794$$
$$x_{31} = -72.1935532053449$$
$$x_{31} = -77.7257170387604$$
$$x_{31} = -83.9440058712679$$
$$x_{31} = 8.12140237825279$$
$$x_{31} = 1.75748767956681$$
$$x_{31} = 79.7604098082138$$
$$x_{31} = -3.97400948361753$$
$$x_{31} = 27.2863953811205$$
$$x_{31} = 82.2044467599582$$
$$x_{31} = -15.3508947858058$$
$$x_{31} = 11.0691206807264$$
$$x_{31} = -15.7549206941732$$
$$x_{31} = 47.2616324020612$$
$$x_{31} = 46.1858745664205$$
$$x_{31} = -33.8628507747295$$
$$x_{31} = -46.928316630699$$
$$x_{31} = -1.78948236293348$$
$$x_{31} = -65.5808461895275$$
$$x_{31} = -51.1560410192306$$
$$x_{31} = 36.7972126661047$$
$$x_{31} = 3.95230187084535$$
$$x_{31} = 85.4278345722129$$
$$x_{31} = 48.1833155017359$$
$$x_{31} = 78.44981619248$$
$$x_{31} = -72.106467795218$$
$$x_{31} = -11.621672446497$$
$$x_{31} = -40.1453184602676$$
$$x_{31} = -17.9879387647864$$
$$x_{31} = -14.0683294162172$$
$$x_{31} = -22.0673844234238$$
$$x_{31} = 94.1405234555763$$
$$x_{31} = 31.2578168207404$$
$$x_{31} = -49.2792605670951$$
$$x_{31} = -55.8533314248926$$
$$x_{31} = 57.7341585233858$$
$$x_{31} = -95.8600997718734$$
Decrece en los intervalos
$$\left[99.6364875148062, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.1940994186411\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}\right) = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}\right) = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}\right) = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
$$\lim_{x \to \infty}\left(\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}\right) = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{7}{4}, \frac{9}{4}\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x) - cos(x^2) + 1/4, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4}}{x}\right) = 0$$
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:
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = - \sin{\left(x \right)} - \cos{\left(x^{2} \right)} + \frac{1}{4}$$
- No
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = \sin{\left(x \right)} + \cos{\left(x^{2} \right)} - \frac{1}{4}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = - \sin{\left(x \right)} - \cos{\left(x^{2} \right)} + \frac{1}{4}$$
- No
$$\left(\sin{\left(x \right)} - \cos{\left(x^{2} \right)}\right) + \frac{1}{4} = \sin{\left(x \right)} + \cos{\left(x^{2} \right)} - \frac{1}{4}$$
- No
es decir, función
no es
par ni impar