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Trazado el gráfico de la función/ arcsin^n*(1/n)

Gráfico de la función y = arcsin^n*(1/n)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           n/1\
f(n) = asin |-|
            \n/
f(n)=asin⁡n(1n)f{\left(n \right)} = \operatorname{asin}^{n}{\left(\frac{1}{n} \right)} 
f = asin(1/n)^n
Gráfico de la función
02468-8-6-4-2-1010020000000000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
n1=0n_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje N con f = 0
o sea hay que resolver la ecuación:
asin⁡n(1n)=0\operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje N:

Solución numérica
n1=76.1413029694416n_{1} = 76.1413029694416
n2=68.227642355023n_{2} = 68.227642355023
n3=90.0172047696751n_{3} = 90.0172047696751
n4=62.3024106697315n_{4} = 62.3024106697315
n5=70.2047622120685n_{5} = 70.2047622120685
n6=56.3882280719695n_{6} = 56.3882280719695
n7=12.7190298267371n_{7} = 12.7190298267371
n8=92.0016681864692n_{8} = 92.0016681864692
n9=80.1027737756433n_{9} = 80.1027737756433
n10=95.9719549750227n_{10} = 95.9719549750227
n11=36.8088117204549n_{11} = 36.8088117204549
n12=31.0091761785888n_{12} = 31.0091761785888
n13=60.3296536460115n_{13} = 60.3296536460115
n14=25.2802461454154n_{14} = 25.2802461454154
n15=52.4531102259721n_{15} = 52.4531102259721
n16=97.9577328069831n_{16} = 97.9577328069831
n17=101.930455249072n_{17} = 101.930455249072
n18=84.0668468715313n_{18} = 84.0668468715313
n19=42.6524139163152n_{19} = 42.6524139163152
n20=64.2763878895567n_{20} = 64.2763878895567
n21=40.7007093796399n_{21} = 40.7007093796399
n22=99.9439058964489n_{22} = 99.9439058964489
n23=17.8610268222444n_{23} = 17.8610268222444
n24=14.3586503712144n_{24} = 14.3586503712144
n25=50.4883051754613n_{25} = 50.4883051754613
n26=78.121692644277n_{26} = 78.121692644277
n27=72.1827851540364n_{27} = 72.1827851540364
n28=66.2514927490357n_{28} = 66.2514927490357
n29=74.1616504376086n_{29} = 74.1616504376086
n30=19.6791295916708n_{30} = 19.6791295916708
n31=23.3945887504459n_{31} = 23.3945887504459
n32=58.3582205455502n_{32} = 58.3582205455502
n33=93.9865927927436n_{33} = 93.9865927927436
n34=29.0897697856963n_{34} = 29.0897697856963
n35=38.7526719952482n_{35} = 38.7526719952482
n36=34.8697445725517n_{36} = 34.8697445725517
n37=21.5259528694806n_{37} = 21.5259528694806
n38=88.0332280151518n_{38} = 88.0332280151518
n39=86.0497654492348n_{39} = 86.0497654492348
n40=32.936222793468n_{40} = 32.936222793468
n41=103.917363146584n_{41} = 103.917363146584
n42=82.0845046042045n_{42} = 82.0845046042045
n43=27.1794862575478n_{43} = 27.1794862575478
n44=16.082035099177n_{44} = 16.082035099177
n45=46.5651860777106n_{45} = 46.5651860777106
n46=48.5255885956478n_{46} = 48.5255885956478
n47=54.4198080819267n_{47} = 54.4198080819267
n48=44.6073593155809n_{48} = 44.6073593155809
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando n es igual a 0:
sustituimos n = 0 en asin(1/n)^n.
asin⁡0(10)\operatorname{asin}^{0}{\left(\frac{1}{0} \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddnf(n)=0\frac{d}{d n} f{\left(n \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddnf(n)=\frac{d}{d n} f{\left(n \right)} =
primera derivada
(log⁡(asin⁡(1n))−1n1−1n2asin⁡(1n))asin⁡n(1n)=0\left(\log{\left(\operatorname{asin}{\left(\frac{1}{n} \right)} \right)} - \frac{1}{n \sqrt{1 - \frac{1}{n^{2}}} \operatorname{asin}{\left(\frac{1}{n} \right)}}\right) \operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
n1=50.4920213999051n_{1} = 50.4920213999051
n2=74.1636156233944n_{2} = 74.1636156233944
n3=36.8152609275401n_{3} = 36.8152609275401
n4=21.5440365547278n_{4} = 21.5440365547278
n5=60.3324095913534n_{5} = 60.3324095913534
n6=68.2298930259002n_{6} = 68.2298930259002
n7=78.1234995325494n_{7} = 78.1234995325494
n8=29.0997186861361n_{8} = 29.0997186861361
n9=101.931639304185n_{9} = 101.931639304185
n10=16.1168996933363n_{10} = 16.1168996933363
n11=32.9441144547345n_{11} = 32.9441144547345
n12=31.0179994407077n_{12} = 31.0179994407077
n13=46.5694536646329n_{13} = 46.5694536646329
n14=97.9589932129564n_{14} = 97.9589932129564
n15=25.2932952508302n_{15} = 25.2932952508302
n16=44.6119555143769n_{16} = 44.6119555143769
n17=64.2788698823777n_{17} = 64.2788698823777
n18=93.9879382366637n_{18} = 93.9879382366637
n19=80.1045093574331n_{19} = 80.1045093574331
n20=19.7010265653842n_{20} = 19.7010265653842
n21=92.0030598477981n_{21} = 92.0030598477981
n22=34.8768554472142n_{22} = 34.8768554472142
n23=52.4565935736786n_{23} = 52.4565935736786
n24=23.4098251816329n_{24} = 23.4098251816329
n25=14.4053437377151n_{25} = 14.4053437377151
n26=72.1848384292403n_{26} = 72.1848384292403
n27=95.9732567302792n_{27} = 95.9732567302792
n28=99.9451271209593n_{28} = 99.9451271209593
n29=86.0513131469282n_{29} = 86.0513131469282
n30=40.7061035281569n_{30} = 40.7061035281569
n31=84.0684532399976n_{31} = 84.0684532399976
n32=82.0861734807751n_{32} = 82.0861734807751
n33=12.7855746109492n_{33} = 12.7855746109492
n34=42.6573827658331n_{34} = 42.6573827658331
n35=103.918511902677n_{35} = 103.918511902677
n36=17.8882141546746n_{36} = 17.8882141546746
n37=48.5295647161957n_{37} = 48.5295647161957
n38=76.1431862208168n_{38} = 76.1431862208168
n39=88.0347205581939n_{39} = 88.0347205581939
n40=54.4230818259256n_{40} = 54.4230818259256
n41=66.2538543554076n_{41} = 66.2538543554076
n42=27.1908150755455n_{42} = 27.1908150755455
n43=70.2069103960084n_{43} = 70.2069103960084
n44=58.3611329794969n_{44} = 58.3611329794969
n45=56.3913123771491n_{45} = 56.3913123771491
n46=62.3050236595806n_{46} = 62.3050236595806
n47=90.0186453866301n_{47} = 90.0186453866301
n48=38.7585549633988n_{48} = 38.7585549633988
Signos de extremos en los puntos:
(50.492021399905134, 1.00524939834085e-86)

(74.16361562339439, 1.99920539656424e-139)

(36.8152609275401, 2.22967468072109e-58)

(21.544036554727807, 1.8972086984024e-29)

(60.33240959135339, 3.76953240049036e-108)

(68.22989302590021, 7.39884634013172e-126)

(78.12349953254937, 1.34957540637356e-148)

(29.099718686136086, 2.53359081743753e-43)

(101.93163930418518, 1.95200371628247e-205)

(16.116899693336347, 3.5225937113623e-20)

(32.944114454734525, 1.00080865974225e-50)

(31.0179994407077, 5.4382471322251e-47)

(46.56945366463294, 2.0844700827568e-78)

(97.95899321295637, 9.12100290601291e-196)

(25.293295250830226, 3.28271515429515e-36)

(44.611955514376945, 2.60841406419252e-74)

(64.27886988237775, 6.03314141986229e-117)

(93.98793823666374, 3.59574876909215e-186)

(80.10450935743306, 3.23144307209686e-153)

(19.70102656538417, 3.16897027592555e-26)

(92.00305984779807, 2.11333754427403e-181)

(34.8768554472142, 1.59670548950702e-54)

(52.45659357367864, 6.1168235531938e-91)

(23.409825181632886, 8.8251533855494e-33)

(14.40534373771506, 2.07063016266843e-17)

(72.18483842924027, 7.06934833242825e-135)

(95.97325673027918, 5.85249569099651e-191)

(99.94512712095931, 1.36236806607938e-200)

(86.0513131469282, 3.25399980035706e-167)

(40.706103528156916, 3.0116949243113e-66)

(84.06845323999755, 1.58466287314508e-162)

(82.08617348077509, 7.3414460557382e-158)

(12.785574610949201, 7.17165940348463e-15)

(42.65738276583309, 2.95399956697055e-70)

(103.91851190267707, 2.68515548563672e-210)

(17.88821415467463, 3.96154450055025e-23)

(48.52956471619568, 1.51473004531965e-82)

(76.14318622081683, 5.34032315118709e-144)

(88.03472055819391, 6.36431472146587e-172)

(54.423081825925635, 3.42501744757311e-95)

(66.25385435540755, 2.18189776775241e-121)

(27.19081507554549, 9.98973215239634e-40)

(70.20691039600837, 2.35724963434491e-130)

(58.3611329794969, 8.47733667005571e-104)

(56.39131237714911, 1.77063647182785e-99)

(62.30502365958062, 1.56084512766927e-112)

(90.01864538663011, 1.18697681183847e-176)

(38.758554963398815, 2.74812160852837e-62)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
Asíntotas verticales
Hay:
n1=0n_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con n->+oo y n->-oo
lim⁡n→−∞asin⁡n(1n)=∞\lim_{n \to -\infty} \operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡n→∞asin⁡n(1n)=0\lim_{n \to \infty} \operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función asin(1/n)^n, dividida por n con n->+oo y n ->-oo
lim⁡n→−∞(asin⁡n(1n)n)=−∞\lim_{n \to -\infty}\left(\frac{\operatorname{asin}^{n}{\left(\frac{1}{n} \right)}}{n}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡n→∞(asin⁡n(1n)n)=0\lim_{n \to \infty}\left(\frac{\operatorname{asin}^{n}{\left(\frac{1}{n} \right)}}{n}\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(-n) и f = -f(-n).
Pues, comprobamos:
asin⁡n(1n)=(−asin⁡(1n))−n\operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = \left(- \operatorname{asin}{\left(\frac{1}{n} \right)}\right)^{- n}
- No
asin⁡n(1n)=−(−asin⁡(1n))−n\operatorname{asin}^{n}{\left(\frac{1}{n} \right)} = - \left(- \operatorname{asin}{\left(\frac{1}{n} \right)}\right)^{- n}
- No
es decir, función
no es
par ni impar
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