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- Gráfico de la función implícita
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- Integrales paso a paso
- Gráfico de la función paramétrica
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- Superficie especificada paramétricamente
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- Superficie de una figura limitada con líneas
- 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^3)
-
Expresiones idénticas
- - uno +(- uno -x)*cos(x*sqrt(dos)/ dos)+(- uno -x)*sin(x*sqrt(dos)/ dos)
- menos 1 más ( menos 1 menos x) multiplicar por coseno de (x multiplicar por raíz cuadrada de (2) dividir por 2) más ( menos 1 menos x) multiplicar por seno de (x multiplicar por raíz cuadrada de (2) dividir por 2)
- menos uno más ( menos uno menos x) multiplicar por coseno de (x multiplicar por raíz cuadrada de (dos) dividir por dos) más ( menos uno menos x) multiplicar por seno de (x multiplicar por raíz cuadrada de (dos) dividir por dos)
- -1+(-1-x)*cos(x*√(2)/2)+(-1-x)*sin(x*√(2)/2)
- -1+(-1-x)cos(xsqrt(2)/2)+(-1-x)sin(xsqrt(2)/2)
- -1+-1-xcosxsqrt2/2+-1-xsinxsqrt2/2
- -1+(-1-x)*cos(x*sqrt(2) dividir por 2)+(-1-x)*sin(x*sqrt(2) dividir por 2)
-
Expresiones semejantes
- -1+(-1+x)*cos(x*sqrt(2)/2)+(-1-x)*sin(x*sqrt(2)/2)
- -1+(-1-x)*cos(x*sqrt(2)/2)-(-1-x)*sin(x*sqrt(2)/2)
- 1+(-1-x)*cos(x*sqrt(2)/2)+(-1-x)*sin(x*sqrt(2)/2)
- -1-(-1-x)*cos(x*sqrt(2)/2)+(-1-x)*sin(x*sqrt(2)/2)
- -1+(-1-x)*cos(x*sqrt(2)/2)+(-1+x)*sin(x*sqrt(2)/2)
- -1+(-1-x)*cos(x*sqrt(2)/2)+(1-x)*sin(x*sqrt(2)/2)
- -1+(1-x)*cos(x*sqrt(2)/2)+(-1-x)*sin(x*sqrt(2)/2)
-
Expresiones con funciones
- Coseno cos
- cos(x)+cos(2x)
- cos(x^2+5)^(3)
- cos[x/3]
- cos(x)/8
- cos(5*x)+sin(5*x)
- Raíz cuadrada sqrt
- sqrt(1-x*x)
- sqrt(x)/(sqrt(x)+1)
- sqrt(x+4)^3
- sqrt(x)^2+5*x
- sqrt(x^2-1)+sqrt(1-x^2)
- Seno sin
- sin(x)+5
- sin(x)-cos(x^2)+(1/4)
- sin(x)+5*cos(x)
- sin(x^(3)+x)
- sin(x)*3
- Raíz cuadrada sqrt
- sqrt(1-x*x)
- sqrt(x)/(sqrt(x)+1)
- sqrt(x+4)^3
- sqrt(x)^2+5*x
- sqrt(x^2-1)+sqrt(1-x^2)
Gráfico de la función y = -1+(-1-x)*cos(x*sqrt(2)/2)+(-1-x)*sin(x*sqrt(2)/2)
Solución
Ha introducido
[src]
/ ___\ / ___\
|x*\/ 2 | |x*\/ 2 |
f(x) = -1 + (-1 - x)*cos|-------| + (-1 - x)*sin|-------|
\ 2 / \ 2 /
$$f{\left(x \right)} = \left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)$$
f = (-x - 1)*sin((sqrt(2)*x)/2) + (-x - 1)*cos((sqrt(2)*x)/2) - 1
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:
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 78.8486482950879$$
$$x_{2} = 87.7356684839146$$
$$x_{3} = 96.6224602710347$$
$$x_{4} = -98.8439249264527$$
$$x_{5} = -36.6257127826445$$
$$x_{6} = 21.1488506987593$$
$$x_{7} = 34.4040955727992$$
$$x_{8} = 56.6640997085722$$
$$x_{9} = -76.652949131055$$
$$x_{10} = -72.1827991763375$$
$$x_{11} = -81.0701244063939$$
$$x_{12} = -18.8261403727126$$
$$x_{13} = 30.0216981262351$$
$$x_{14} = -54.4065911647164$$
$$x_{15} = -32.2429112952846$$
$$x_{16} = 65.54755046973$$
$$x_{17} = 61.0735301248675$$
$$x_{18} = -67.7689421101633$$
$$x_{19} = -9.88380300947023$$
$$x_{20} = 83.315915386568$$
$$x_{21} = 38.9002894957406$$
$$x_{22} = -50.0028407427305$$
$$x_{23} = 12.2931900885414$$
$$x_{24} = -89.9571380104765$$
$$x_{25} = -85.5373257918033$$
$$x_{26} = 101.085382643154$$
$$x_{27} = -23.3698459078779$$
$$x_{28} = 92.2005506200682$$
$$x_{29} = -14.513403960098$$
$$x_{30} = 16.6039904294345$$
$$x_{31} = 52.1850717008618$$
$$x_{32} = 7.65943544932799$$
$$x_{33} = 69.9613138572449$$
$$x_{34} = -27.7306036954298$$
$$x_{35} = 43.295532047774$$
$$x_{36} = -41.1215927032987$$
$$x_{37} = 74.4315464611969$$
$$x_{38} = -58.8854748512754$$
$$x_{39} = -45.5170858861585$$
$$x_{40} = 25.508849167392$$
$$x_{41} = -5.76427742810406$$
$$x_{42} = -94.4219666586963$$
$$x_{43} = -63.2950288787638$$
$$x_{44} = 47.7814918811987$$
$$x_{45} = 3.55270468296405$$
$$x_{46} = -209.931005106154$$
o sea hay que resolver la ecuación:
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 78.8486482950879$$
$$x_{2} = 87.7356684839146$$
$$x_{3} = 96.6224602710347$$
$$x_{4} = -98.8439249264527$$
$$x_{5} = -36.6257127826445$$
$$x_{6} = 21.1488506987593$$
$$x_{7} = 34.4040955727992$$
$$x_{8} = 56.6640997085722$$
$$x_{9} = -76.652949131055$$
$$x_{10} = -72.1827991763375$$
$$x_{11} = -81.0701244063939$$
$$x_{12} = -18.8261403727126$$
$$x_{13} = 30.0216981262351$$
$$x_{14} = -54.4065911647164$$
$$x_{15} = -32.2429112952846$$
$$x_{16} = 65.54755046973$$
$$x_{17} = 61.0735301248675$$
$$x_{18} = -67.7689421101633$$
$$x_{19} = -9.88380300947023$$
$$x_{20} = 83.315915386568$$
$$x_{21} = 38.9002894957406$$
$$x_{22} = -50.0028407427305$$
$$x_{23} = 12.2931900885414$$
$$x_{24} = -89.9571380104765$$
$$x_{25} = -85.5373257918033$$
$$x_{26} = 101.085382643154$$
$$x_{27} = -23.3698459078779$$
$$x_{28} = 92.2005506200682$$
$$x_{29} = -14.513403960098$$
$$x_{30} = 16.6039904294345$$
$$x_{31} = 52.1850717008618$$
$$x_{32} = 7.65943544932799$$
$$x_{33} = 69.9613138572449$$
$$x_{34} = -27.7306036954298$$
$$x_{35} = 43.295532047774$$
$$x_{36} = -41.1215927032987$$
$$x_{37} = 74.4315464611969$$
$$x_{38} = -58.8854748512754$$
$$x_{39} = -45.5170858861585$$
$$x_{40} = 25.508849167392$$
$$x_{41} = -5.76427742810406$$
$$x_{42} = -94.4219666586963$$
$$x_{43} = -63.2950288787638$$
$$x_{44} = 47.7814918811987$$
$$x_{45} = 3.55270468296405$$
$$x_{46} = -209.931005106154$$
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
$$- \frac{\sqrt{2} \left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\sqrt{2} \left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -56.6826675610891$$
$$x_{2} = -25.6276971876651$$
$$x_{3} = -30.0582328781461$$
$$x_{4} = 63.3421606824646$$
$$x_{5} = 50.0216220875817$$
$$x_{6} = 1.77687936809853$$
$$x_{7} = -38.9279328760707$$
$$x_{8} = 54.4613693161347$$
$$x_{9} = 10.1745187671212$$
$$x_{10} = -3.96210099634574$$
$$x_{11} = 89.9903580789576$$
$$x_{12} = -8.05483051308111$$
$$x_{13} = 32.2709775887511$$
$$x_{14} = 98.8741692336653$$
$$x_{15} = 45.5824715372825$$
$$x_{16} = -78.8868476073282$$
$$x_{17} = 58.9015808298421$$
$$x_{18} = -70.0043858839905$$
$$x_{19} = -16.7871590134849$$
$$x_{20} = 5.84186222701659$$
$$x_{21} = 23.4069876500919$$
$$x_{22} = 27.8373174223413$$
$$x_{23} = 67.7830376469223$$
$$x_{24} = -12.3925873529074$$
$$x_{25} = -87.7699854329339$$
$$x_{26} = -92.211746210428$$
$$x_{27} = -52.242894420254$$
$$x_{28} = 41.144105598611$$
$$x_{29} = -1.05537450400253$$
$$x_{30} = -83.3283456710998$$
$$x_{31} = 36.7068002210725$$
$$x_{32} = 72.224157740075$$
$$x_{33} = 81.1069696811089$$
$$x_{34} = 14.5674907317375$$
$$x_{35} = -65.5634956420564$$
$$x_{36} = -43.3652995706866$$
$$x_{37} = 94.4322181849273$$
$$x_{38} = 85.5486029062939$$
$$x_{39} = -61.1228994620739$$
$$x_{40} = 18.9821750786416$$
$$x_{41} = -47.8037102427204$$
$$x_{42} = -34.4920230298981$$
$$x_{43} = 76.6654793043163$$
$$x_{44} = -101.095566418709$$
$$x_{45} = -74.4455169192843$$
$$x_{46} = -96.6536111584108$$
$$x_{47} = -21.2025302264653$$
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:
$$x_{1} = -56.6826675610891$$
$$x_{2} = -30.0582328781461$$
$$x_{3} = 63.3421606824646$$
$$x_{4} = 1.77687936809853$$
$$x_{5} = -38.9279328760707$$
$$x_{6} = 54.4613693161347$$
$$x_{7} = 10.1745187671212$$
$$x_{8} = -3.96210099634574$$
$$x_{9} = 89.9903580789576$$
$$x_{10} = 98.8741692336653$$
$$x_{11} = 45.5824715372825$$
$$x_{12} = 27.8373174223413$$
$$x_{13} = -12.3925873529074$$
$$x_{14} = -92.211746210428$$
$$x_{15} = -83.3283456710998$$
$$x_{16} = 36.7068002210725$$
$$x_{17} = 72.224157740075$$
$$x_{18} = 81.1069696811089$$
$$x_{19} = -65.5634956420564$$
$$x_{20} = 18.9821750786416$$
$$x_{21} = -47.8037102427204$$
$$x_{22} = -101.095566418709$$
$$x_{23} = -74.4455169192843$$
$$x_{24} = -21.2025302264653$$
Puntos máximos de la función:
$$x_{24} = -25.6276971876651$$
$$x_{24} = 50.0216220875817$$
$$x_{24} = -8.05483051308111$$
$$x_{24} = 32.2709775887511$$
$$x_{24} = -78.8868476073282$$
$$x_{24} = 58.9015808298421$$
$$x_{24} = -70.0043858839905$$
$$x_{24} = -16.7871590134849$$
$$x_{24} = 5.84186222701659$$
$$x_{24} = 23.4069876500919$$
$$x_{24} = 67.7830376469223$$
$$x_{24} = -87.7699854329339$$
$$x_{24} = -52.242894420254$$
$$x_{24} = 41.144105598611$$
$$x_{24} = -1.05537450400253$$
$$x_{24} = 14.5674907317375$$
$$x_{24} = -43.3652995706866$$
$$x_{24} = 94.4322181849273$$
$$x_{24} = 85.5486029062939$$
$$x_{24} = -61.1228994620739$$
$$x_{24} = -34.4920230298981$$
$$x_{24} = 76.6654793043163$$
$$x_{24} = -96.6536111584108$$
Decrece en los intervalos
$$\left[98.8741692336653, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -101.095566418709\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{\sqrt{2} \left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\sqrt{2} \left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -56.6826675610891$$
$$x_{2} = -25.6276971876651$$
$$x_{3} = -30.0582328781461$$
$$x_{4} = 63.3421606824646$$
$$x_{5} = 50.0216220875817$$
$$x_{6} = 1.77687936809853$$
$$x_{7} = -38.9279328760707$$
$$x_{8} = 54.4613693161347$$
$$x_{9} = 10.1745187671212$$
$$x_{10} = -3.96210099634574$$
$$x_{11} = 89.9903580789576$$
$$x_{12} = -8.05483051308111$$
$$x_{13} = 32.2709775887511$$
$$x_{14} = 98.8741692336653$$
$$x_{15} = 45.5824715372825$$
$$x_{16} = -78.8868476073282$$
$$x_{17} = 58.9015808298421$$
$$x_{18} = -70.0043858839905$$
$$x_{19} = -16.7871590134849$$
$$x_{20} = 5.84186222701659$$
$$x_{21} = 23.4069876500919$$
$$x_{22} = 27.8373174223413$$
$$x_{23} = 67.7830376469223$$
$$x_{24} = -12.3925873529074$$
$$x_{25} = -87.7699854329339$$
$$x_{26} = -92.211746210428$$
$$x_{27} = -52.242894420254$$
$$x_{28} = 41.144105598611$$
$$x_{29} = -1.05537450400253$$
$$x_{30} = -83.3283456710998$$
$$x_{31} = 36.7068002210725$$
$$x_{32} = 72.224157740075$$
$$x_{33} = 81.1069696811089$$
$$x_{34} = 14.5674907317375$$
$$x_{35} = -65.5634956420564$$
$$x_{36} = -43.3652995706866$$
$$x_{37} = 94.4322181849273$$
$$x_{38} = 85.5486029062939$$
$$x_{39} = -61.1228994620739$$
$$x_{40} = 18.9821750786416$$
$$x_{41} = -47.8037102427204$$
$$x_{42} = -34.4920230298981$$
$$x_{43} = 76.6654793043163$$
$$x_{44} = -101.095566418709$$
$$x_{45} = -74.4455169192843$$
$$x_{46} = -96.6536111584108$$
$$x_{47} = -21.2025302264653$$
Signos de extremos en los puntos:
/ ___\ / ___\ (-56.682667561089104, -1 + 55.6826675610891*cos\28.3413337805446*\/ 2 / - 55.6826675610891*sin\28.3413337805446*\/ 2 /)
/ ___\ / ___\ (-25.627697187665067, -1 + 24.6276971876651*cos\12.8138485938325*\/ 2 / - 24.6276971876651*sin\12.8138485938325*\/ 2 /)
/ ___\ / ___\ (-30.05823287814606, -1 + 29.0582328781461*cos\15.029116439073*\/ 2 / - 29.0582328781461*sin\15.029116439073*\/ 2 /)
/ ___\ / ___\ (63.3421606824646, -1 - 64.3421606824646*cos\31.6710803412323*\/ 2 / - 64.3421606824646*sin\31.6710803412323*\/ 2 /)
/ ___\ / ___\ (50.0216220875817, -1 - 51.0216220875817*cos\25.0108110437908*\/ 2 / - 51.0216220875817*sin\25.0108110437908*\/ 2 /)
/ ___\ / ___\ (1.7768793680985289, -1 - 2.77687936809853*cos\0.888439684049264*\/ 2 / - 2.77687936809853*sin\0.888439684049264*\/ 2 /)
/ ___\ / ___\ (-38.927932876070706, -1 + 37.9279328760707*cos\19.4639664380354*\/ 2 / - 37.9279328760707*sin\19.4639664380354*\/ 2 /)
/ ___\ / ___\ (54.46136931613474, -1 - 55.4613693161347*cos\27.2306846580674*\/ 2 / - 55.4613693161347*sin\27.2306846580674*\/ 2 /)
/ ___\ / ___\ (10.174518767121183, -1 - 11.1745187671212*cos\5.08725938356059*\/ 2 / - 11.1745187671212*sin\5.08725938356059*\/ 2 /)
/ ___\ / ___\ (-3.962100996345738, -1 + 2.96210099634574*cos\1.98105049817287*\/ 2 / - 2.96210099634574*sin\1.98105049817287*\/ 2 /)
/ ___\ / ___\ (89.99035807895756, -1 - 90.9903580789576*cos\44.9951790394788*\/ 2 / - 90.9903580789576*sin\44.9951790394788*\/ 2 /)
/ ___\ / ___\ (-8.054830513081113, -1 + 7.05483051308111*cos\4.02741525654056*\/ 2 / - 7.05483051308111*sin\4.02741525654056*\/ 2 /)
/ ___\ / ___\ (32.270977588751094, -1 - 33.2709775887511*cos\16.1354887943755*\/ 2 / - 33.2709775887511*sin\16.1354887943755*\/ 2 /)
/ ___\ / ___\ (98.87416923366531, -1 - 99.8741692336653*cos\49.4370846168327*\/ 2 / - 99.8741692336653*sin\49.4370846168327*\/ 2 /)
/ ___\ / ___\ (45.582471537282466, -1 - 46.5824715372825*cos\22.7912357686412*\/ 2 / - 46.5824715372825*sin\22.7912357686412*\/ 2 /)
/ ___\ / ___\ (-78.8868476073282, -1 + 77.8868476073282*cos\39.4434238036641*\/ 2 / - 77.8868476073282*sin\39.4434238036641*\/ 2 /)
/ ___\ / ___\ (58.901580829842096, -1 - 59.9015808298421*cos\29.450790414921*\/ 2 / - 59.9015808298421*sin\29.450790414921*\/ 2 /)
/ ___\ / ___\ (-70.00438588399047, -1 + 69.0043858839905*cos\35.0021929419952*\/ 2 / - 69.0043858839905*sin\35.0021929419952*\/ 2 /)
/ ___\ / ___\ (-16.787159013484853, -1 + 15.7871590134849*cos\8.39357950674243*\/ 2 / - 15.7871590134849*sin\8.39357950674243*\/ 2 /)
/ ___\ / ___\ (5.841862227016587, -1 - 6.84186222701659*cos\2.92093111350829*\/ 2 / - 6.84186222701659*sin\2.92093111350829*\/ 2 /)
/ ___\ / ___\ (23.40698765009192, -1 - 24.4069876500919*cos\11.703493825046*\/ 2 / - 24.4069876500919*sin\11.703493825046*\/ 2 /)
/ ___\ / ___\ (27.837317422341297, -1 - 28.8373174223413*cos\13.9186587111706*\/ 2 / - 28.8373174223413*sin\13.9186587111706*\/ 2 /)
/ ___\ / ___\ (67.78303764692231, -1 - 68.7830376469223*cos\33.8915188234612*\/ 2 / - 68.7830376469223*sin\33.8915188234612*\/ 2 /)
/ ___\ / ___\ (-12.392587352907352, -1 + 11.3925873529074*cos\6.19629367645368*\/ 2 / - 11.3925873529074*sin\6.19629367645368*\/ 2 /)
/ ___\ / ___\ (-87.76998543293394, -1 + 86.7699854329339*cos\43.884992716467*\/ 2 / - 86.7699854329339*sin\43.884992716467*\/ 2 /)
/ ___\ / ___\ (-92.21174621042796, -1 + 91.211746210428*cos\46.105873105214*\/ 2 / - 91.211746210428*sin\46.105873105214*\/ 2 /)
/ ___\ / ___\ (-52.242894420254046, -1 + 51.242894420254*cos\26.121447210127*\/ 2 / - 51.242894420254*sin\26.121447210127*\/ 2 /)
/ ___\ / ___\ (41.144105598610956, -1 - 42.144105598611*cos\20.5720527993055*\/ 2 / - 42.144105598611*sin\20.5720527993055*\/ 2 /)
/ ___\ / ___\ (-1.0553745040025266, -1 + 0.0553745040025266*cos\0.527687252001263*\/ 2 / - 0.0553745040025266*sin\0.527687252001263*\/ 2 /)
/ ___\ / ___\ (-83.32834567109983, -1 + 82.3283456710998*cos\41.6641728355499*\/ 2 / - 82.3283456710998*sin\41.6641728355499*\/ 2 /)
/ ___\ / ___\ (36.706800221072506, -1 - 37.7068002210725*cos\18.3534001105363*\/ 2 / - 37.7068002210725*sin\18.3534001105363*\/ 2 /)
/ ___\ / ___\ (72.22415774007503, -1 - 73.224157740075*cos\36.1120788700375*\/ 2 / - 73.224157740075*sin\36.1120788700375*\/ 2 /)
/ ___\ / ___\ (81.1069696811089, -1 - 82.1069696811089*cos\40.5534848405544*\/ 2 / - 82.1069696811089*sin\40.5534848405544*\/ 2 /)
/ ___\ / ___\ (14.567490731737491, -1 - 15.5674907317375*cos\7.28374536586875*\/ 2 / - 15.5674907317375*sin\7.28374536586875*\/ 2 /)
/ ___\ / ___\ (-65.56349564205641, -1 + 64.5634956420564*cos\32.7817478210282*\/ 2 / - 64.5634956420564*sin\32.7817478210282*\/ 2 /)
/ ___\ / ___\ (-43.36529957068655, -1 + 42.3652995706866*cos\21.6826497853433*\/ 2 / - 42.3652995706866*sin\21.6826497853433*\/ 2 /)
/ ___\ / ___\ (94.43221818492732, -1 - 95.4322181849273*cos\47.2161090924637*\/ 2 / - 95.4322181849273*sin\47.2161090924637*\/ 2 /)
/ ___\ / ___\ (85.54860290629391, -1 - 86.5486029062939*cos\42.774301453147*\/ 2 / - 86.5486029062939*sin\42.774301453147*\/ 2 /)
/ ___\ / ___\ (-61.12289946207391, -1 + 60.1228994620739*cos\30.561449731037*\/ 2 / - 60.1228994620739*sin\30.561449731037*\/ 2 /)
/ ___\ / ___\ (18.98217507864162, -1 - 19.9821750786416*cos\9.49108753932081*\/ 2 / - 19.9821750786416*sin\9.49108753932081*\/ 2 /)
/ ___\ / ___\ (-47.80371024272038, -1 + 46.8037102427204*cos\23.9018551213602*\/ 2 / - 46.8037102427204*sin\23.9018551213602*\/ 2 /)
/ ___\ / ___\ (-34.492023029898064, -1 + 33.4920230298981*cos\17.246011514949*\/ 2 / - 33.4920230298981*sin\17.246011514949*\/ 2 /)
/ ___\ / ___\ (76.66547930431632, -1 - 77.6654793043163*cos\38.3327396521582*\/ 2 / - 77.6654793043163*sin\38.3327396521582*\/ 2 /)
/ ___\ / ___\ (-101.09556641870877, -1 + 100.095566418709*cos\50.5477832093544*\/ 2 / - 100.095566418709*sin\50.5477832093544*\/ 2 /)
/ ___\ / ___\ (-74.44551691928426, -1 + 73.4455169192843*cos\37.2227584596421*\/ 2 / - 73.4455169192843*sin\37.2227584596421*\/ 2 /)
/ ___\ / ___\ (-96.65361115841083, -1 + 95.6536111584108*cos\48.3268055792054*\/ 2 / - 95.6536111584108*sin\48.3268055792054*\/ 2 /)
/ ___\ / ___\ (-21.202530226465328, -1 + 20.2025302264653*cos\10.6012651132327*\/ 2 / - 20.2025302264653*sin\10.6012651132327*\/ 2 /)
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:
$$x_{1} = -56.6826675610891$$
$$x_{2} = -30.0582328781461$$
$$x_{3} = 63.3421606824646$$
$$x_{4} = 1.77687936809853$$
$$x_{5} = -38.9279328760707$$
$$x_{6} = 54.4613693161347$$
$$x_{7} = 10.1745187671212$$
$$x_{8} = -3.96210099634574$$
$$x_{9} = 89.9903580789576$$
$$x_{10} = 98.8741692336653$$
$$x_{11} = 45.5824715372825$$
$$x_{12} = 27.8373174223413$$
$$x_{13} = -12.3925873529074$$
$$x_{14} = -92.211746210428$$
$$x_{15} = -83.3283456710998$$
$$x_{16} = 36.7068002210725$$
$$x_{17} = 72.224157740075$$
$$x_{18} = 81.1069696811089$$
$$x_{19} = -65.5634956420564$$
$$x_{20} = 18.9821750786416$$
$$x_{21} = -47.8037102427204$$
$$x_{22} = -101.095566418709$$
$$x_{23} = -74.4455169192843$$
$$x_{24} = -21.2025302264653$$
Puntos máximos de la función:
$$x_{24} = -25.6276971876651$$
$$x_{24} = 50.0216220875817$$
$$x_{24} = -8.05483051308111$$
$$x_{24} = 32.2709775887511$$
$$x_{24} = -78.8868476073282$$
$$x_{24} = 58.9015808298421$$
$$x_{24} = -70.0043858839905$$
$$x_{24} = -16.7871590134849$$
$$x_{24} = 5.84186222701659$$
$$x_{24} = 23.4069876500919$$
$$x_{24} = 67.7830376469223$$
$$x_{24} = -87.7699854329339$$
$$x_{24} = -52.242894420254$$
$$x_{24} = 41.144105598611$$
$$x_{24} = -1.05537450400253$$
$$x_{24} = 14.5674907317375$$
$$x_{24} = -43.3652995706866$$
$$x_{24} = 94.4322181849273$$
$$x_{24} = 85.5486029062939$$
$$x_{24} = -61.1228994620739$$
$$x_{24} = -34.4920230298981$$
$$x_{24} = 76.6654793043163$$
$$x_{24} = -96.6536111584108$$
Decrece en los intervalos
$$\left[98.8741692336653, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -101.095566418709\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\left(x + 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\left(x + 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \sqrt{2} \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \sqrt{2} \cos{\left(\frac{\sqrt{2} x}{2} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 4.05374579292121$$
$$x_{2} = 78.9112068225516$$
$$x_{3} = 38.9751214656651$$
$$x_{4} = 105.555999915198$$
$$x_{5} = 52.2788807662597$$
$$x_{6} = 8.19698757617941$$
$$x_{7} = 136.64770595186$$
$$x_{8} = -54.5000127889216$$
$$x_{9} = 83.3514579684065$$
$$x_{10} = 61.1539523559803$$
$$x_{11} = 87.791971918382$$
$$x_{12} = 34.5446406916175$$
$$x_{13} = 12.5097939402944$$
$$x_{14} = 30.1176519761613$$
$$x_{15} = -76.692551465808$$
$$x_{16} = 25.6958559733325$$
$$x_{17} = 92.2327112087521$$
$$x_{18} = -72.2529563082306$$
$$x_{19} = -85.5727754819736$$
$$x_{20} = -98.8949941177303$$
$$x_{21} = 70.0316894356098$$
$$x_{22} = 43.408060880844$$
$$x_{23} = 47.8427956197$$
$$x_{24} = 65.5925540321573$$
$$x_{25} = -67.8137969569567$$
$$x_{26} = 16.8826537404212$$
$$x_{27} = -143.311078494836$$
$$x_{28} = -94.4540511570912$$
$$x_{29} = -19.1014566650618$$
$$x_{30} = -36.765391918438$$
$$x_{31} = 21.2822540587397$$
$$x_{32} = -58.9371844511484$$
$$x_{33} = -6.2522080294602$$
$$x_{34} = -32.3381961683913$$
$$x_{35} = 0.443398334876145$$
$$x_{36} = -10.4094401282718$$
$$x_{37} = 74.4712647261291$$
$$x_{38} = 101.114748448919$$
$$x_{39} = -50.0638693094348$$
$$x_{40} = -14.7267492968965$$
$$x_{41} = 56.7160069247046$$
$$x_{42} = -23.5019702167022$$
$$x_{43} = -63.3751660551827$$
$$x_{44} = -41.1960157741251$$
$$x_{45} = -2.60298606603129$$
$$x_{46} = -45.6290581098304$$
$$x_{47} = -90.0133014871093$$
$$x_{48} = 96.6736451688094$$
$$x_{49} = -81.132510224783$$
$$x_{50} = -27.9160849748559$$
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
$$\left[105.555999915198, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.4540511570912\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\left(x + 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\left(x + 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \sqrt{2} \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \sqrt{2} \cos{\left(\frac{\sqrt{2} x}{2} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 4.05374579292121$$
$$x_{2} = 78.9112068225516$$
$$x_{3} = 38.9751214656651$$
$$x_{4} = 105.555999915198$$
$$x_{5} = 52.2788807662597$$
$$x_{6} = 8.19698757617941$$
$$x_{7} = 136.64770595186$$
$$x_{8} = -54.5000127889216$$
$$x_{9} = 83.3514579684065$$
$$x_{10} = 61.1539523559803$$
$$x_{11} = 87.791971918382$$
$$x_{12} = 34.5446406916175$$
$$x_{13} = 12.5097939402944$$
$$x_{14} = 30.1176519761613$$
$$x_{15} = -76.692551465808$$
$$x_{16} = 25.6958559733325$$
$$x_{17} = 92.2327112087521$$
$$x_{18} = -72.2529563082306$$
$$x_{19} = -85.5727754819736$$
$$x_{20} = -98.8949941177303$$
$$x_{21} = 70.0316894356098$$
$$x_{22} = 43.408060880844$$
$$x_{23} = 47.8427956197$$
$$x_{24} = 65.5925540321573$$
$$x_{25} = -67.8137969569567$$
$$x_{26} = 16.8826537404212$$
$$x_{27} = -143.311078494836$$
$$x_{28} = -94.4540511570912$$
$$x_{29} = -19.1014566650618$$
$$x_{30} = -36.765391918438$$
$$x_{31} = 21.2822540587397$$
$$x_{32} = -58.9371844511484$$
$$x_{33} = -6.2522080294602$$
$$x_{34} = -32.3381961683913$$
$$x_{35} = 0.443398334876145$$
$$x_{36} = -10.4094401282718$$
$$x_{37} = 74.4712647261291$$
$$x_{38} = 101.114748448919$$
$$x_{39} = -50.0638693094348$$
$$x_{40} = -14.7267492968965$$
$$x_{41} = 56.7160069247046$$
$$x_{42} = -23.5019702167022$$
$$x_{43} = -63.3751660551827$$
$$x_{44} = -41.1960157741251$$
$$x_{45} = -2.60298606603129$$
$$x_{46} = -45.6290581098304$$
$$x_{47} = -90.0133014871093$$
$$x_{48} = 96.6736451688094$$
$$x_{49} = -81.132510224783$$
$$x_{50} = -27.9160849748559$$
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
$$\left[105.555999915198, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.4540511570912\right]$$
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:
$$y = \lim_{x \to -\infty}\left(\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -1 + (-1 - x)*cos((x*sqrt(2))/2) + (-1 - x)*sin((x*sqrt(2))/2), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right)}{x}\right)$$
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:
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = - \left(x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1$$
- No
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = \left(x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \left(x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} + 1$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = - \left(x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1$$
- No
$$\left(- x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} + \left(\left(- x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1\right) = \left(x - 1\right) \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \left(x - 1\right) \cos{\left(\frac{\sqrt{2} x}{2} \right)} + 1$$
- No
es decir, función
no es
par ni impar