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- Integrales paso a paso
- Gráfico de la función paramétrica
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- Curva en el espacio especificada paramétricamente
- 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))
-
x+1/(x-1)^2
-
Expresiones idénticas
- sinc(x)^(tres)
- seno de c(x) en el grado (3)
- seno de c(x) en el grado (tres)
- sinc(x)(3)
- sincx3
- sincx^3
-
Expresiones con funciones
- sinc
- sinc(e)+sin(2*x)
Gráfico de la función y = sinc(x)^(3)
Solución
Ha introducido
[src]
3 f(x) = sinc (x)
$$f{\left(x \right)} = \operatorname{sinc}^{3}{\left(x \right)}$$
f = sinc(x)^3
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:
$$\operatorname{sinc}^{3}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 47.1240184351636$$
$$x_{2} = 50.2654784071452$$
$$x_{3} = -65.9734547010727$$
$$x_{4} = -9.42484580431789$$
$$x_{5} = -34.5573997308378$$
$$x_{6} = -56.5485509101693$$
$$x_{7} = -15.7079741004534$$
$$x_{8} = -12.566271765656$$
$$x_{9} = -47.1237906903152$$
$$x_{10} = 53.4071921831066$$
$$x_{11} = -91.1060869923014$$
$$x_{12} = 87.9646063006236$$
$$x_{13} = 84.8230161580009$$
$$x_{14} = -6.28310528806017$$
$$x_{15} = 47.123752025495$$
$$x_{16} = 40.8405885020816$$
$$x_{17} = 72.2566292955063$$
$$x_{18} = -18.8496732970081$$
$$x_{19} = -62.8317143308171$$
$$x_{20} = -100.530851290256$$
$$x_{21} = -81.6813390363158$$
$$x_{22} = 100.530900883867$$
$$x_{23} = 91.1063177615201$$
$$x_{24} = -84.8228641262343$$
$$x_{25} = -40.8405633424584$$
$$x_{26} = -97.3894502058245$$
$$x_{27} = -28.2742613366093$$
$$x_{28} = -91.1063043543888$$
$$x_{29} = -40.8408291405924$$
$$x_{30} = 28.2743275276783$$
$$x_{31} = 75.3981469179945$$
$$x_{32} = 109.956008382299$$
$$x_{33} = -56.5486319390578$$
$$x_{34} = -78.5397012369791$$
$$x_{35} = 18.8494350189175$$
$$x_{36} = -53.4071495562697$$
$$x_{37} = 15.7080350584218$$
$$x_{38} = 18.8495458115422$$
$$x_{39} = 3.14133975637727$$
$$x_{40} = 31.4158434908652$$
$$x_{41} = 69.1149024808305$$
$$x_{42} = -12.566242795222$$
$$x_{43} = 34.5574493139676$$
$$x_{44} = -21.9911516388309$$
$$x_{45} = -47.1240040395702$$
$$x_{46} = 97.3892920448606$$
$$x_{47} = -9.42492732546751$$
$$x_{48} = 12.5662969282502$$
$$x_{49} = -25.1328522224088$$
$$x_{50} = -62.8319470609358$$
$$x_{51} = 78.5397504916067$$
$$x_{52} = 9.42463043710858$$
$$x_{53} = -72.2565629434837$$
$$x_{54} = 84.8228894866492$$
$$x_{55} = -43.9823032287297$$
$$x_{56} = 37.6991868392924$$
$$x_{57} = -75.3982999386403$$
$$x_{58} = 9.42488254660626$$
$$x_{59} = 97.3894922754282$$
$$x_{60} = -84.8231289595522$$
$$x_{61} = 50.2654847506095$$
$$x_{62} = -31.4159988545404$$
$$x_{63} = -50.2654123352875$$
$$x_{64} = -69.1149400901718$$
$$x_{65} = 75.3983423999932$$
$$x_{66} = -37.6991249286554$$
$$x_{67} = -62.8319796795721$$
$$x_{68} = 53.4069994036149$$
$$x_{69} = 25.1325982306676$$
$$x_{70} = -34.5577267319505$$
$$x_{71} = 9.42452189854482$$
$$x_{72} = -18.8494066063646$$
$$x_{73} = 59.6903374809205$$
$$x_{74} = -81.6814264795845$$
$$x_{75} = 62.831873453669$$
$$x_{76} = 25.1328652975779$$
$$x_{77} = -59.6902757177337$$
$$x_{78} = 43.9823032484069$$
$$x_{79} = -25.1326315806211$$
$$x_{80} = -78.5397738502329$$
$$x_{81} = 3.14165099600769$$
$$x_{82} = -3.14167458117952$$
$$x_{83} = -94.2477134529675$$
$$x_{84} = 91.1060520727976$$
$$x_{85} = 21.991151639729$$
$$x_{86} = 56.5486000260377$$
$$x_{87} = 40.8407241631044$$
$$x_{88} = 65.9734548059398$$
$$x_{89} = -87.9646059631775$$
$$x_{90} = 69.115168546552$$
$$x_{91} = 62.8317392771584$$
$$x_{92} = 31.4160409768417$$
$$x_{93} = -69.1151544097582$$
$$x_{94} = 72.2565536602152$$
$$x_{95} = 6.28317660646225$$
$$x_{96} = -34.5574781292524$$
$$x_{97} = 21.991232203185$$
$$x_{98} = 81.6814878812647$$
$$x_{99} = 94.2477801894617$$
$$x_{100} = -100.531086252313$$
o sea hay que resolver la ecuación:
$$\operatorname{sinc}^{3}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 47.1240184351636$$
$$x_{2} = 50.2654784071452$$
$$x_{3} = -65.9734547010727$$
$$x_{4} = -9.42484580431789$$
$$x_{5} = -34.5573997308378$$
$$x_{6} = -56.5485509101693$$
$$x_{7} = -15.7079741004534$$
$$x_{8} = -12.566271765656$$
$$x_{9} = -47.1237906903152$$
$$x_{10} = 53.4071921831066$$
$$x_{11} = -91.1060869923014$$
$$x_{12} = 87.9646063006236$$
$$x_{13} = 84.8230161580009$$
$$x_{14} = -6.28310528806017$$
$$x_{15} = 47.123752025495$$
$$x_{16} = 40.8405885020816$$
$$x_{17} = 72.2566292955063$$
$$x_{18} = -18.8496732970081$$
$$x_{19} = -62.8317143308171$$
$$x_{20} = -100.530851290256$$
$$x_{21} = -81.6813390363158$$
$$x_{22} = 100.530900883867$$
$$x_{23} = 91.1063177615201$$
$$x_{24} = -84.8228641262343$$
$$x_{25} = -40.8405633424584$$
$$x_{26} = -97.3894502058245$$
$$x_{27} = -28.2742613366093$$
$$x_{28} = -91.1063043543888$$
$$x_{29} = -40.8408291405924$$
$$x_{30} = 28.2743275276783$$
$$x_{31} = 75.3981469179945$$
$$x_{32} = 109.956008382299$$
$$x_{33} = -56.5486319390578$$
$$x_{34} = -78.5397012369791$$
$$x_{35} = 18.8494350189175$$
$$x_{36} = -53.4071495562697$$
$$x_{37} = 15.7080350584218$$
$$x_{38} = 18.8495458115422$$
$$x_{39} = 3.14133975637727$$
$$x_{40} = 31.4158434908652$$
$$x_{41} = 69.1149024808305$$
$$x_{42} = -12.566242795222$$
$$x_{43} = 34.5574493139676$$
$$x_{44} = -21.9911516388309$$
$$x_{45} = -47.1240040395702$$
$$x_{46} = 97.3892920448606$$
$$x_{47} = -9.42492732546751$$
$$x_{48} = 12.5662969282502$$
$$x_{49} = -25.1328522224088$$
$$x_{50} = -62.8319470609358$$
$$x_{51} = 78.5397504916067$$
$$x_{52} = 9.42463043710858$$
$$x_{53} = -72.2565629434837$$
$$x_{54} = 84.8228894866492$$
$$x_{55} = -43.9823032287297$$
$$x_{56} = 37.6991868392924$$
$$x_{57} = -75.3982999386403$$
$$x_{58} = 9.42488254660626$$
$$x_{59} = 97.3894922754282$$
$$x_{60} = -84.8231289595522$$
$$x_{61} = 50.2654847506095$$
$$x_{62} = -31.4159988545404$$
$$x_{63} = -50.2654123352875$$
$$x_{64} = -69.1149400901718$$
$$x_{65} = 75.3983423999932$$
$$x_{66} = -37.6991249286554$$
$$x_{67} = -62.8319796795721$$
$$x_{68} = 53.4069994036149$$
$$x_{69} = 25.1325982306676$$
$$x_{70} = -34.5577267319505$$
$$x_{71} = 9.42452189854482$$
$$x_{72} = -18.8494066063646$$
$$x_{73} = 59.6903374809205$$
$$x_{74} = -81.6814264795845$$
$$x_{75} = 62.831873453669$$
$$x_{76} = 25.1328652975779$$
$$x_{77} = -59.6902757177337$$
$$x_{78} = 43.9823032484069$$
$$x_{79} = -25.1326315806211$$
$$x_{80} = -78.5397738502329$$
$$x_{81} = 3.14165099600769$$
$$x_{82} = -3.14167458117952$$
$$x_{83} = -94.2477134529675$$
$$x_{84} = 91.1060520727976$$
$$x_{85} = 21.991151639729$$
$$x_{86} = 56.5486000260377$$
$$x_{87} = 40.8407241631044$$
$$x_{88} = 65.9734548059398$$
$$x_{89} = -87.9646059631775$$
$$x_{90} = 69.115168546552$$
$$x_{91} = 62.8317392771584$$
$$x_{92} = 31.4160409768417$$
$$x_{93} = -69.1151544097582$$
$$x_{94} = 72.2565536602152$$
$$x_{95} = 6.28317660646225$$
$$x_{96} = -34.5574781292524$$
$$x_{97} = 21.991232203185$$
$$x_{98} = 81.6814878812647$$
$$x_{99} = 94.2477801894617$$
$$x_{100} = -100.531086252313$$
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
$$3 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \operatorname{sinc}^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -50.2654823091191$$
$$x_{2} = -42.3879135681319$$
$$x_{3} = -12.5663704308121$$
$$x_{4} = 7.72525183693771$$
$$x_{5} = -65.9734457649905$$
$$x_{6} = 40.8407044193221$$
$$x_{7} = 56.5486676222718$$
$$x_{8} = -59.6902604570707$$
$$x_{9} = 31.4159264831211$$
$$x_{10} = 80.0981286289451$$
$$x_{11} = 3.14159186591764$$
$$x_{12} = 100.530964779253$$
$$x_{13} = 28.2743338651534$$
$$x_{14} = 20.3713029592876$$
$$x_{15} = 50.2654824463423$$
$$x_{16} = 87.9645943353247$$
$$x_{17} = 62.8318529973144$$
$$x_{18} = -37.6991118766425$$
$$x_{19} = 72.256631027719$$
$$x_{20} = 18.8495558026501$$
$$x_{21} = 86.3822220347287$$
$$x_{22} = 51.8169824872797$$
$$x_{23} = -51.8169824872797$$
$$x_{24} = 72.2566310953004$$
$$x_{25} = -86.3822220347287$$
$$x_{26} = 31.4159261175591$$
$$x_{27} = 43.9822971692734$$
$$x_{28} = 75.3982236347907$$
$$x_{29} = -369.134427763239$$
$$x_{30} = -43.9822971745409$$
$$x_{31} = 95.8081387868617$$
$$x_{32} = 6.28318528359219$$
$$x_{33} = -72.256630888283$$
$$x_{34} = -7.72525183693771$$
$$x_{35} = 65.9734457526203$$
$$x_{36} = -100.530964792421$$
$$x_{37} = -6.28318510886302$$
$$x_{38} = 28.2743336706419$$
$$x_{39} = -3.141591675475$$
$$x_{40} = 94.2477794951749$$
$$x_{41} = 0$$
$$x_{42} = 29.811598790893$$
$$x_{43} = -87.9645943587528$$
$$x_{44} = -78.5398162211095$$
$$x_{45} = -36.1006222443756$$
$$x_{46} = 78.5398162010092$$
$$x_{47} = -23.519452498689$$
$$x_{48} = 97.3893721871461$$
$$x_{49} = -80.0981286289451$$
$$x_{50} = 15.7079633984768$$
$$x_{51} = -56.5486676474249$$
$$x_{52} = 50.26548251683$$
$$x_{53} = -67.5294347771441$$
$$x_{54} = -20.3713029592876$$
$$x_{55} = 9.42477753273002$$
$$x_{56} = -91.10618668612$$
$$x_{57} = -97.3893724067995$$
$$x_{58} = -29.811598790893$$
$$x_{59} = -89.5242209304172$$
$$x_{60} = -9.42477807462313$$
$$x_{61} = -84.8230016206649$$
$$x_{62} = 81.6814091426717$$
$$x_{63} = -28.2743337272277$$
$$x_{64} = -59.690263002773$$
$$x_{65} = 37.6991119854857$$
$$x_{66} = -94.247779466738$$
$$x_{67} = 53.4070750718002$$
$$x_{68} = 94.2477796093525$$
$$x_{69} = -75.3982238297323$$
$$x_{70} = 58.1022547544956$$
$$x_{71} = -58.1022547544956$$
$$x_{72} = -15.7079632959088$$
$$x_{73} = -81.6814090373249$$
$$x_{74} = -47.1238895886687$$
$$x_{75} = -31.41592667158$$
$$x_{76} = -69.1150381490577$$
$$x_{77} = 34.5575190418347$$
$$x_{78} = 14.0661939128315$$
$$x_{79} = -45.5311340139913$$
$$x_{80} = -25.1327409474033$$
$$x_{81} = 64.3871195905574$$
$$x_{82} = 36.1006222443756$$
$$x_{83} = -21.9911485864199$$
$$x_{84} = 39.2444323611642$$
$$x_{85} = 4.49340945790906$$
$$x_{86} = 54.9596782878889$$
$$x_{87} = -34.5575190652112$$
$$x_{88} = -53.4070752517995$$
$$x_{89} = -95.8081387868617$$
$$x_{90} = 12.5663704496531$$
$$x_{91} = 84.8230014700644$$
$$x_{92} = -14.0661939128315$$
$$x_{93} = -73.8138806006806$$
$$x_{94} = 59.6902605648944$$
$$x_{95} = 73.8138806006806$$
$$x_{96} = 56.5486673804109$$
$$x_{97} = 42.3879135681319$$
$$x_{98} = 21.9911485851347$$
$$x_{99} = 84.8230015667727$$
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:
$$x_{1} = -42.3879135681319$$
$$x_{2} = 80.0981286289451$$
$$x_{3} = 86.3822220347287$$
$$x_{4} = -86.3822220347287$$
$$x_{5} = -369.134427763239$$
$$x_{6} = 29.811598790893$$
$$x_{7} = -36.1006222443756$$
$$x_{8} = -23.519452498689$$
$$x_{9} = -80.0981286289451$$
$$x_{10} = -67.5294347771441$$
$$x_{11} = -29.811598790893$$
$$x_{12} = 36.1006222443756$$
$$x_{13} = 4.49340945790906$$
$$x_{14} = 54.9596782878889$$
$$x_{15} = -73.8138806006806$$
$$x_{16} = 73.8138806006806$$
$$x_{17} = 42.3879135681319$$
Puntos máximos de la función:
$$x_{17} = 7.72525183693771$$
$$x_{17} = 20.3713029592876$$
$$x_{17} = 51.8169824872797$$
$$x_{17} = -51.8169824872797$$
$$x_{17} = 95.8081387868617$$
$$x_{17} = -7.72525183693771$$
$$x_{17} = 0$$
$$x_{17} = -20.3713029592876$$
$$x_{17} = -89.5242209304172$$
$$x_{17} = 58.1022547544956$$
$$x_{17} = -58.1022547544956$$
$$x_{17} = 14.0661939128315$$
$$x_{17} = -45.5311340139913$$
$$x_{17} = 64.3871195905574$$
$$x_{17} = 39.2444323611642$$
$$x_{17} = -95.8081387868617$$
$$x_{17} = -14.0661939128315$$
Decrece en los intervalos
$$\left[86.3822220347287, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -369.134427763239\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$3 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \operatorname{sinc}^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -50.2654823091191$$
$$x_{2} = -42.3879135681319$$
$$x_{3} = -12.5663704308121$$
$$x_{4} = 7.72525183693771$$
$$x_{5} = -65.9734457649905$$
$$x_{6} = 40.8407044193221$$
$$x_{7} = 56.5486676222718$$
$$x_{8} = -59.6902604570707$$
$$x_{9} = 31.4159264831211$$
$$x_{10} = 80.0981286289451$$
$$x_{11} = 3.14159186591764$$
$$x_{12} = 100.530964779253$$
$$x_{13} = 28.2743338651534$$
$$x_{14} = 20.3713029592876$$
$$x_{15} = 50.2654824463423$$
$$x_{16} = 87.9645943353247$$
$$x_{17} = 62.8318529973144$$
$$x_{18} = -37.6991118766425$$
$$x_{19} = 72.256631027719$$
$$x_{20} = 18.8495558026501$$
$$x_{21} = 86.3822220347287$$
$$x_{22} = 51.8169824872797$$
$$x_{23} = -51.8169824872797$$
$$x_{24} = 72.2566310953004$$
$$x_{25} = -86.3822220347287$$
$$x_{26} = 31.4159261175591$$
$$x_{27} = 43.9822971692734$$
$$x_{28} = 75.3982236347907$$
$$x_{29} = -369.134427763239$$
$$x_{30} = -43.9822971745409$$
$$x_{31} = 95.8081387868617$$
$$x_{32} = 6.28318528359219$$
$$x_{33} = -72.256630888283$$
$$x_{34} = -7.72525183693771$$
$$x_{35} = 65.9734457526203$$
$$x_{36} = -100.530964792421$$
$$x_{37} = -6.28318510886302$$
$$x_{38} = 28.2743336706419$$
$$x_{39} = -3.141591675475$$
$$x_{40} = 94.2477794951749$$
$$x_{41} = 0$$
$$x_{42} = 29.811598790893$$
$$x_{43} = -87.9645943587528$$
$$x_{44} = -78.5398162211095$$
$$x_{45} = -36.1006222443756$$
$$x_{46} = 78.5398162010092$$
$$x_{47} = -23.519452498689$$
$$x_{48} = 97.3893721871461$$
$$x_{49} = -80.0981286289451$$
$$x_{50} = 15.7079633984768$$
$$x_{51} = -56.5486676474249$$
$$x_{52} = 50.26548251683$$
$$x_{53} = -67.5294347771441$$
$$x_{54} = -20.3713029592876$$
$$x_{55} = 9.42477753273002$$
$$x_{56} = -91.10618668612$$
$$x_{57} = -97.3893724067995$$
$$x_{58} = -29.811598790893$$
$$x_{59} = -89.5242209304172$$
$$x_{60} = -9.42477807462313$$
$$x_{61} = -84.8230016206649$$
$$x_{62} = 81.6814091426717$$
$$x_{63} = -28.2743337272277$$
$$x_{64} = -59.690263002773$$
$$x_{65} = 37.6991119854857$$
$$x_{66} = -94.247779466738$$
$$x_{67} = 53.4070750718002$$
$$x_{68} = 94.2477796093525$$
$$x_{69} = -75.3982238297323$$
$$x_{70} = 58.1022547544956$$
$$x_{71} = -58.1022547544956$$
$$x_{72} = -15.7079632959088$$
$$x_{73} = -81.6814090373249$$
$$x_{74} = -47.1238895886687$$
$$x_{75} = -31.41592667158$$
$$x_{76} = -69.1150381490577$$
$$x_{77} = 34.5575190418347$$
$$x_{78} = 14.0661939128315$$
$$x_{79} = -45.5311340139913$$
$$x_{80} = -25.1327409474033$$
$$x_{81} = 64.3871195905574$$
$$x_{82} = 36.1006222443756$$
$$x_{83} = -21.9911485864199$$
$$x_{84} = 39.2444323611642$$
$$x_{85} = 4.49340945790906$$
$$x_{86} = 54.9596782878889$$
$$x_{87} = -34.5575190652112$$
$$x_{88} = -53.4070752517995$$
$$x_{89} = -95.8081387868617$$
$$x_{90} = 12.5663704496531$$
$$x_{91} = 84.8230014700644$$
$$x_{92} = -14.0661939128315$$
$$x_{93} = -73.8138806006806$$
$$x_{94} = 59.6902605648944$$
$$x_{95} = 73.8138806006806$$
$$x_{96} = 56.5486673804109$$
$$x_{97} = 42.3879135681319$$
$$x_{98} = 21.9911485851347$$
$$x_{99} = 84.8230015667727$$
Signos de extremos en los puntos:
(-50.26548230911909, -2.56902595265939e-26)
(-42.38791356813192, -1.31193225845809e-5)
(-12.566370430812075, -3.11611153713364e-24)
(7.725251836937707, 0.00211561597894324)
(-65.97344576499046, -2.16339581004841e-28)
(40.84070441932212, 6.79235307631041e-27)
(56.548667622271765, -1.59497935910326e-26)
(-59.690260457070664, -2.76027239789523e-28)
(31.415926483121066, -4.74112270146226e-27)
(80.09812862894512, -1.94550056235955e-6)
(3.141591865917642, 1.57611238333761e-20)
(100.53096477925327, -2.45512035513654e-27)
(28.27433386515337, 2.23345653932216e-28)
(20.37130295928756, 0.000117862510803959)
(50.26548244634234, -1.07521741742519e-29)
(87.96459433532469, 6.19735424017428e-29)
(62.831852997314414, -1.66573067239596e-27)
(-37.69911187664254, 7.05776873676609e-28)
(72.25663102771895, 3.01714011012449e-31)
(18.849555802650073, -2.509101462397e-25)
(86.38222203472871, -1.55109925520621e-6)
(51.81698248727967, 7.183582214747e-6)
(-51.81698248727967, 7.183582214747e-6)
(72.25663109530039, -6.54484920728489e-28)
(-86.38222203472871, -1.55109925520621e-6)
(31.41592611755909, -2.36121196307999e-24)
(43.982297169273394, 8.08246038408359e-29)
(75.39822363479065, -3.16157081086531e-28)
(-369.13442776323865, -1.98811527885619e-8)
(-43.98229717454091, 1.68312549334634e-28)
(95.8081387868617, 1.13689884218094e-6)
(6.283185283592194, -5.29054257973859e-26)
(-72.25663088828301, 7.96169304103459e-27)
(-7.725251836937707, 0.00211561597894324)
(65.97344575262035, -7.03493190786403e-29)
(-100.53096479242076, -1.80719377412012e-27)
(-6.283185108863016, -3.14439753268755e-23)
(28.27433367064189, 4.1954417904638e-25)
(-3.141591675474998, 3.01800720306874e-20)
(94.24777949517491, -1.70162124648341e-27)
(0, 1)
(29.81159879089296, -3.76800618419885e-5)
(-87.96459435875285, 2.90208345531379e-28)
(-78.53981622110952, 3.44645248279774e-27)
(-36.10062224437561, -2.12303068518345e-5)
(78.5398162010092, 5.51181177258165e-27)
(-23.519452498689006, -7.66552274610893e-5)
(97.3893721871461, 4.41142651314252e-28)
(-80.09812862894512, -1.94550056235955e-6)
(15.707963398476767, -5.73785310313736e-25)
(-56.548667647424885, -8.90061506773078e-27)
(50.26548251683002, 1.64969402724626e-27)
(-67.52943477714412, -3.24621902462579e-6)
(-20.37130295928756, 0.000117862510803959)
(9.424777532730024, 9.36780368970191e-23)
(-91.10618668611997, 2.54497287404149e-26)
(-97.38937240679952, -3.33577650593265e-27)
(-29.81159879089296, -3.76800618419885e-5)
(-89.52422093041719, 1.39346823158858e-6)
(-9.424778074623129, -1.76290378887572e-24)
(-84.82300162066493, 2.96700348293795e-29)
(81.6814091426717, 6.11129487263598e-27)
(-28.27433372722774, 1.65003660305682e-25)
(-59.69026300277295, -8.11807102921252e-23)
(37.69911198548568, 5.39027926836945e-26)
(-94.24777946673798, -3.34530423390033e-27)
(53.407075071800236, 3.96217787511325e-28)
(94.24777960935249, 5.45112006952741e-33)
(-75.3982238297323, 6.90515891906261e-27)
(58.10225475449559, 5.09598451854886e-6)
(-58.10225475449559, 5.09598451854886e-6)
(-15.707963295908767, -5.63952587340883e-27)
(-81.6814090373249, 1.56206935568771e-28)
(-47.123889588668746, 9.520749420828e-26)
(-31.415926671579964, 8.05596162123372e-26)
(-69.1150381490577, -3.6812886758792e-26)
(34.55751904183472, 7.80008217737221e-26)
(14.066193912831473, 0.000356603841835024)
(-45.53113401399128, 1.05866958550917e-5)
(-25.13274094740334, -1.4023587374037e-24)
(64.38711959055742, 3.74494885788318e-6)
(36.10062224437561, -2.12303068518345e-5)
(-21.99114858641993, -1.35362028598572e-28)
(39.24443236116419, 1.6528863850759e-5)
(4.493409457909064, -0.0102513525022404)
(54.959678287888934, -6.02076701016006e-6)
(-34.55751906521119, 4.65092954885001e-26)
(-53.40707525179951, -1.83131038953297e-26)
(-95.8081387868617, 1.13689884218094e-6)
(12.566370449653139, -2.25164000398326e-24)
(84.82300147006437, 9.064592994945e-27)
(-14.066193912831473, 0.000356603841835024)
(-73.81388060068065, -2.4858008560758e-6)
(59.69026056489435, -1.48414327775062e-26)
(73.81388060068065, -2.4858008560758e-6)
(56.54866738041086, -3.13634135529366e-25)
(42.38791356813192, -1.31193225845809e-5)
(21.991148585134653, -9.41999723282504e-29)
(84.82300156677272, 8.4371798582798e-28)
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:
$$x_{1} = -42.3879135681319$$
$$x_{2} = 80.0981286289451$$
$$x_{3} = 86.3822220347287$$
$$x_{4} = -86.3822220347287$$
$$x_{5} = -369.134427763239$$
$$x_{6} = 29.811598790893$$
$$x_{7} = -36.1006222443756$$
$$x_{8} = -23.519452498689$$
$$x_{9} = -80.0981286289451$$
$$x_{10} = -67.5294347771441$$
$$x_{11} = -29.811598790893$$
$$x_{12} = 36.1006222443756$$
$$x_{13} = 4.49340945790906$$
$$x_{14} = 54.9596782878889$$
$$x_{15} = -73.8138806006806$$
$$x_{16} = 73.8138806006806$$
$$x_{17} = 42.3879135681319$$
Puntos máximos de la función:
$$x_{17} = 7.72525183693771$$
$$x_{17} = 20.3713029592876$$
$$x_{17} = 51.8169824872797$$
$$x_{17} = -51.8169824872797$$
$$x_{17} = 95.8081387868617$$
$$x_{17} = -7.72525183693771$$
$$x_{17} = 0$$
$$x_{17} = -20.3713029592876$$
$$x_{17} = -89.5242209304172$$
$$x_{17} = 58.1022547544956$$
$$x_{17} = -58.1022547544956$$
$$x_{17} = 14.0661939128315$$
$$x_{17} = -45.5311340139913$$
$$x_{17} = 64.3871195905574$$
$$x_{17} = 39.2444323611642$$
$$x_{17} = -95.8081387868617$$
$$x_{17} = -14.0661939128315$$
Decrece en los intervalos
$$\left[86.3822220347287, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -369.134427763239\right]$$
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
$$3 \left(2 \left(\begin{cases} \frac{\left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)^{2}}{x^{4}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) + \left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \operatorname{sinc}{\left(x \right)}\right) \operatorname{sinc}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -43.9822971502571$$
$$x_{2} = 92.0507095319312$$
$$x_{3} = 28.2743338823081$$
$$x_{4} = 32.3408855747494$$
$$x_{5} = -81.6814089933346$$
$$x_{6} = -15.707963267949$$
$$x_{7} = -92.0507095319312$$
$$x_{8} = 72.2566310325652$$
$$x_{9} = 44.9156418006958$$
$$x_{10} = -33.5718932434468$$
$$x_{11} = 192.58729192346$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = -48.0586535729426$$
$$x_{14} = 68.1449202782273$$
$$x_{15} = 120.327567503223$$
$$x_{16} = -120.327567503223$$
$$x_{17} = 100.530964914873$$
$$x_{18} = 54.3441899793913$$
$$x_{19} = -57.4867673470335$$
$$x_{20} = -21.9911485751286$$
$$x_{21} = -78.5398163397448$$
$$x_{22} = 48.0586535729426$$
$$x_{23} = 96.4236213671802$$
$$x_{24} = 39.8599293062401$$
$$x_{25} = 61.860217016315$$
$$x_{26} = 19.7557617553282$$
$$x_{27} = -70.0562007906555$$
$$x_{28} = -72.2566310325652$$
$$x_{29} = 90.139703929183$$
$$x_{30} = 50.2654824574367$$
$$x_{31} = -28.2743338823081$$
$$x_{32} = 30.4271084775819$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = 3.14159265358979$$
$$x_{35} = -63.7716336070576$$
$$x_{36} = 52.4324720546584$$
$$x_{37} = -87.9645943005142$$
$$x_{38} = -98.3345804511699$$
$$x_{39} = -20.9868472968342$$
$$x_{40} = -30.4271084775819$$
$$x_{41} = 0$$
$$x_{42} = 74.4293651545191$$
$$x_{43} = -8.34119393827257$$
$$x_{44} = -26.0505384126728$$
$$x_{45} = -65.9734457253857$$
$$x_{46} = -41.7724194444636$$
$$x_{47} = 6.28318530717959$$
$$x_{48} = 3.87684257324433$$
$$x_{49} = 70.0562007906555$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 65.9734457253857$$
$$x_{52} = 98.3345804511699$$
$$x_{53} = -99.5655452247495$$
$$x_{54} = -19.7557617553282$$
$$x_{55} = 21.9911485751286$$
$$x_{56} = -35.4851227077876$$
$$x_{57} = 78.5398163397448$$
$$x_{58} = -39.8599293062401$$
$$x_{59} = 20.9868472968342$$
$$x_{60} = 26.0505384126728$$
$$x_{61} = 87.9645943005142$$
$$x_{62} = 12.5663706143592$$
$$x_{63} = -37.6991118430775$$
$$x_{64} = -7.10936834077326$$
$$x_{65} = -17.836325752306$$
$$x_{66} = -24.1349805588138$$
$$x_{67} = -79.4826448590369$$
$$x_{68} = -77.5715104754082$$
$$x_{69} = 63.7716336070576$$
$$x_{70} = -83.8556759865409$$
$$x_{71} = -3.87684257324433$$
$$x_{72} = 17.836325752306$$
$$x_{73} = 43.9822971502571$$
$$x_{74} = -46.1466265071908$$
$$x_{75} = -50.2654824574367$$
$$x_{76} = 76.3405422031993$$
$$x_{77} = 46.1466265071908$$
$$x_{78} = 38.6289358533096$$
$$x_{79} = 24.1349805588138$$
$$x_{80} = -59.6902604182061$$
$$x_{81} = -68.1449202782273$$
$$x_{82} = 8.34119393827257$$
$$x_{83} = 41.7724194444636$$
$$x_{84} = 62.8318530717959$$
$$x_{85} = -11.5198306381068$$
$$x_{86} = -55.5751667555014$$
$$x_{87} = 85.7667388325444$$
$$x_{88} = -61.860217016315$$
$$x_{89} = 211.437308054913$$
$$x_{90} = -85.7667388325444$$
$$x_{91} = 10.2884332837981$$
$$x_{92} = -94.2477796076938$$
$$x_{93} = 83.8556759865409$$
$$x_{94} = -90.139703929183$$
$$x_{95} = -13.4505870133108$$
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
$$\left[211.437308054913, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -120.327567503223\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$3 \left(2 \left(\begin{cases} \frac{\left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)^{2}}{x^{4}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) + \left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \operatorname{sinc}{\left(x \right)}\right) \operatorname{sinc}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -43.9822971502571$$
$$x_{2} = 92.0507095319312$$
$$x_{3} = 28.2743338823081$$
$$x_{4} = 32.3408855747494$$
$$x_{5} = -81.6814089933346$$
$$x_{6} = -15.707963267949$$
$$x_{7} = -92.0507095319312$$
$$x_{8} = 72.2566310325652$$
$$x_{9} = 44.9156418006958$$
$$x_{10} = -33.5718932434468$$
$$x_{11} = 192.58729192346$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = -48.0586535729426$$
$$x_{14} = 68.1449202782273$$
$$x_{15} = 120.327567503223$$
$$x_{16} = -120.327567503223$$
$$x_{17} = 100.530964914873$$
$$x_{18} = 54.3441899793913$$
$$x_{19} = -57.4867673470335$$
$$x_{20} = -21.9911485751286$$
$$x_{21} = -78.5398163397448$$
$$x_{22} = 48.0586535729426$$
$$x_{23} = 96.4236213671802$$
$$x_{24} = 39.8599293062401$$
$$x_{25} = 61.860217016315$$
$$x_{26} = 19.7557617553282$$
$$x_{27} = -70.0562007906555$$
$$x_{28} = -72.2566310325652$$
$$x_{29} = 90.139703929183$$
$$x_{30} = 50.2654824574367$$
$$x_{31} = -28.2743338823081$$
$$x_{32} = 30.4271084775819$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = 3.14159265358979$$
$$x_{35} = -63.7716336070576$$
$$x_{36} = 52.4324720546584$$
$$x_{37} = -87.9645943005142$$
$$x_{38} = -98.3345804511699$$
$$x_{39} = -20.9868472968342$$
$$x_{40} = -30.4271084775819$$
$$x_{41} = 0$$
$$x_{42} = 74.4293651545191$$
$$x_{43} = -8.34119393827257$$
$$x_{44} = -26.0505384126728$$
$$x_{45} = -65.9734457253857$$
$$x_{46} = -41.7724194444636$$
$$x_{47} = 6.28318530717959$$
$$x_{48} = 3.87684257324433$$
$$x_{49} = 70.0562007906555$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 65.9734457253857$$
$$x_{52} = 98.3345804511699$$
$$x_{53} = -99.5655452247495$$
$$x_{54} = -19.7557617553282$$
$$x_{55} = 21.9911485751286$$
$$x_{56} = -35.4851227077876$$
$$x_{57} = 78.5398163397448$$
$$x_{58} = -39.8599293062401$$
$$x_{59} = 20.9868472968342$$
$$x_{60} = 26.0505384126728$$
$$x_{61} = 87.9645943005142$$
$$x_{62} = 12.5663706143592$$
$$x_{63} = -37.6991118430775$$
$$x_{64} = -7.10936834077326$$
$$x_{65} = -17.836325752306$$
$$x_{66} = -24.1349805588138$$
$$x_{67} = -79.4826448590369$$
$$x_{68} = -77.5715104754082$$
$$x_{69} = 63.7716336070576$$
$$x_{70} = -83.8556759865409$$
$$x_{71} = -3.87684257324433$$
$$x_{72} = 17.836325752306$$
$$x_{73} = 43.9822971502571$$
$$x_{74} = -46.1466265071908$$
$$x_{75} = -50.2654824574367$$
$$x_{76} = 76.3405422031993$$
$$x_{77} = 46.1466265071908$$
$$x_{78} = 38.6289358533096$$
$$x_{79} = 24.1349805588138$$
$$x_{80} = -59.6902604182061$$
$$x_{81} = -68.1449202782273$$
$$x_{82} = 8.34119393827257$$
$$x_{83} = 41.7724194444636$$
$$x_{84} = 62.8318530717959$$
$$x_{85} = -11.5198306381068$$
$$x_{86} = -55.5751667555014$$
$$x_{87} = 85.7667388325444$$
$$x_{88} = -61.860217016315$$
$$x_{89} = 211.437308054913$$
$$x_{90} = -85.7667388325444$$
$$x_{91} = 10.2884332837981$$
$$x_{92} = -94.2477796076938$$
$$x_{93} = 83.8556759865409$$
$$x_{94} = -90.139703929183$$
$$x_{95} = -13.4505870133108$$
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
$$\left[211.437308054913, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -120.327567503223\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty} \operatorname{sinc}^{3}{\left(x \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \operatorname{sinc}^{3}{\left(x \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty} \operatorname{sinc}^{3}{\left(x \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \operatorname{sinc}^{3}{\left(x \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sinc(x)^3, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sinc}^{3}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{sinc}^{3}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sinc}^{3}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{sinc}^{3}{\left(x \right)}}{x}\right) = 0$$
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:
$$\operatorname{sinc}^{3}{\left(x \right)} = \operatorname{sinc}^{3}{\left(x \right)}$$
- Sí
$$\operatorname{sinc}^{3}{\left(x \right)} = - \operatorname{sinc}^{3}{\left(x \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\operatorname{sinc}^{3}{\left(x \right)} = \operatorname{sinc}^{3}{\left(x \right)}$$
- Sí
$$\operatorname{sinc}^{3}{\left(x \right)} = - \operatorname{sinc}^{3}{\left(x \right)}$$
- No
es decir, función
es
par