159:@0.937671:0.974518:0.968375:0.974518:0.968375:0.956333:0.937671:0.956333:0.000000:0.000000:0.000000
Pensamiento espacial, numérico y variacional:@0.522009:0.043661:0.926913:0.043661:0.926913:0.024916:0.522009:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
De manera que la distancia total recorrida por la partícula en los primeros 8 :@0.134754:0.136836:0.709962:0.136836:0.709962:0.120402:0.134754:0.120402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
segundos es 7 metros + 32 metros + 81 metros = 120 metros.:@0.134754:0.153478:0.580876:0.153478:0.580876:0.137044:0.134754:0.137044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 2.:@0.073089:0.184418:0.152609:0.184418:0.152609:0.167748:0.073089:0.167748:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.152498:0.184418:0.156529:0.184418:0.156529:0.167984:0.152498:0.167984:0.000000
La frecuencia de vibración de una cuerda de violín está dada por :@0.156456:0.184418:0.627648:0.184418:0.627648:0.167984:0.156456:0.167984:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.079495:0.216939:0.084189:0.216939:0.084189:0.200533:0.079495:0.200533:0.000000
 =      :@0.084115:0.216939:0.136808:0.216935:0.136808:0.200501:0.084115:0.200505:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.107448:0.210098:0.116431:0.210098:0.116431:0.193664:0.107448:0.193664:0.000000
2:@0.103675:0.226490:0.112658:0.226490:0.112658:0.210057:0.103675:0.210057:0.000000
L:@0.112584:0.226490:0.120205:0.226490:0.120205:0.210084:0.112584:0.210084:0.000000
T:@0.140872:0.210098:0.148989:0.210098:0.148989:0.193692:0.140872:0.193692:0.000000
ρ:@0.140135:0.226490:0.149707:0.226490:0.149707:0.210084:0.140135:0.210084:0.000000
 donde  es la longitud de la cuerda es la tensión y  es la densidad lineal.:@0.162408:0.226947:0.700095:0.226947:0.700095:0.210513:0.162408:0.210513:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 L:@0.214574:0.226947:0.225434:0.226947:0.225434:0.210541:0.214574:0.210541:0.000000:0.000000
,T :@0.419815:0.226947:0.433370:0.226947:0.433370:0.210541:0.419815:0.210541:0.000000:0.000000:0.000000
ρ :@0.539039:0.226947:0.551850:0.226947:0.551850:0.210541:0.539039:0.210541:0.000000:0.000000
1.  Encontremos la rapidez de cambio de la frecuencia respecto a::@0.073089:0.254310:0.574554:0.254310:0.574554:0.237640:0.073089:0.237640:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a.  La longitud (cuando  y   son constantes).:@0.104633:0.279172:0.439743:0.279172:0.439743:0.262738:0.104633:0.262738:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
T  ρ:@0.285479:0.279172:0.318317:0.279172:0.318317:0.262766:0.285479:0.262766:0.000000:0.000000:0.000000:0.000000
b.  La tensión (cuando  y   son constantes).:@0.104633:0.299387:0.431331:0.299387:0.431331:0.282953:0.104633:0.282953:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
L  ρ:@0.277564:0.299387:0.309905:0.299387:0.309905:0.282981:0.277564:0.282981:0.000000:0.000000:0.000000:0.000000
c.  La densidad lineal (cuando  y  son constantes).:@0.104633:0.319603:0.484528:0.319603:0.484528:0.303169:0.104633:0.303169:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
L  T :@0.332914:0.319603:0.367041:0.319603:0.367041:0.303197:0.332914:0.303197:0.000000:0.000000:0.000000:0.000000:0.000000
2.  El tono de una nota (qué tan altas y bajas son las notas) está determinado por la :@0.073089:0.344465:0.710436:0.344465:0.710436:0.327795:0.073089:0.327795:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
frecuencia  . Cuanto mayor sea la frecuencia, mayor será el tono). Utilicemos los :@0.103214:0.361106:0.709987:0.361106:0.709987:0.344437:0.103214:0.344437:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.186452:0.361106:0.191661:0.361106:0.191661:0.344423:0.186452:0.344423:0.000000
signos de las derivadas en el numeral 1 para determinar lo que sucede con el :@0.103214:0.377748:0.710563:0.377748:0.710563:0.361079:0.103214:0.361079:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
tono de una nota::@0.103214:0.394390:0.238080:0.394390:0.238080:0.377720:0.103214:0.377720:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a.  Cuando se reduce la longitud efectiva al poner un dedo sobre la cuerda para :@0.104633:0.414605:0.710120:0.414605:0.710120:0.398171:0.104633:0.398171:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
hacer que vibre solo una porción menor de la cuerda.:@0.134760:0.431247:0.522470:0.431247:0.522470:0.414813:0.134760:0.414813:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
b.  Cuando se incrementa la tensión al girar la llave de ajuste.:@0.104633:0.451463:0.551721:0.451463:0.551721:0.435029:0.104633:0.435029:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c.  Cuando aumenta la densidad lineal para cambiar la otra cuerda.:@0.104633:0.471679:0.596770:0.471679:0.596770:0.455245:0.104633:0.455245:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Nota: :@0.073096:0.496544:0.119280:0.496544:0.119280:0.479875:0.073096:0.479875:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
se habla de derivación parcial con respecto a una variable cuando las otras :@0.121157:0.496544:0.710191:0.496544:0.710191:0.480111:0.121157:0.480111:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
variables que intervienen son consideradas constantes.:@0.073096:0.513186:0.472335:0.513186:0.472335:0.496752:0.073096:0.496752:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejercicio tomado del libro :@0.073096:0.540548:0.270387:0.540548:0.270387:0.523879:0.073096:0.523879:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Cálculo de varias variables:@0.270219:0.540548:0.462211:0.540548:0.462211:0.523865:0.270219:0.523865:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
. Stewart (2012).:@0.462299:0.540548:0.583881:0.540548:0.583881:0.523879:0.462299:0.523879:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Respuestas::@0.073114:0.567910:0.157394:0.567910:0.157394:0.551476:0.073114:0.551476:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1.  Derivadas de   respecto a la longitud   (con   y   constante).:@0.073089:0.595271:0.554730:0.595271:0.554730:0.578601:0.073089:0.578601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.205835:0.595271:0.211044:0.595271:0.211044:0.578587:0.205835:0.578587:0.000000
L:@0.383078:0.595271:0.391214:0.595271:0.391214:0.578587:0.383078:0.578587:0.000000
T:@0.432170:0.595271:0.440895:0.595271:0.440895:0.578587:0.432170:0.578587:0.000000
ρ:@0.457057:0.595271:0.466629:0.595271:0.466629:0.578865:0.457057:0.578865:0.000000
f(L,T)=:@0.340429:0.625295:0.384786:0.625295:0.384786:0.608889:0.340429:0.608889:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
      :@0.384753:0.625295:0.422633:0.625289:0.422633:0.608855:0.384753:0.608862:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.393273:0.618452:0.402256:0.618452:0.402256:0.602018:0.393273:0.602018:0.000000
2:@0.389500:0.634844:0.398483:0.634844:0.398483:0.618411:0.389500:0.618411:0.000000
L:@0.398409:0.634844:0.406030:0.634844:0.406030:0.618438:0.398409:0.618438:0.000000
T:@0.426696:0.618452:0.434814:0.618452:0.434814:0.602046:0.426696:0.602046:0.000000
ρ:@0.425978:0.634844:0.435550:0.634844:0.435550:0.618438:0.425978:0.618438:0.000000
Tomamos la derivadas parcial respecto a  .:@0.103210:0.664452:0.410059:0.664452:0.410059:0.648018:0.103210:0.648018:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
L:@0.399309:0.664452:0.406929:0.664452:0.406929:0.648046:0.399309:0.648046:0.000000
   = – :@0.321520:0.696709:0.375459:0.696708:0.375459:0.680274:0.321520:0.680275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.335572:0.689870:0.340266:0.689870:0.340266:0.673464:0.335572:0.673464:0.000000
L:@0.334108:0.706259:0.341728:0.706259:0.341728:0.689853:0.334108:0.689853:0.000000
1:@0.384020:0.689240:0.393002:0.689240:0.393002:0.672807:0.384020:0.672807:0.000000
2:@0.377669:0.705633:0.386652:0.705633:0.386652:0.689199:0.377669:0.689199:0.000000
L:@0.386578:0.705633:0.394199:0.705633:0.394199:0.689227:0.386578:0.689227:0.000000
2:@0.394120:0.699566:0.399357:0.699566:0.399357:0.689985:0.394120:0.689985:0.000000
    :@0.401592:0.696708:0.417496:0.696708:0.417496:0.680274:0.401592:0.680274:0.000000:0.000000:0.000000:0.000000
T:@0.421562:0.689870:0.429680:0.689870:0.429680:0.673464:0.421562:0.673464:0.000000
ρ:@0.420844:0.706262:0.430416:0.706262:0.430416:0.689856:0.420844:0.689856:0.000000
Conclusión::@0.103210:0.737920:0.191727:0.737920:0.191727:0.721251:0.103210:0.721251:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 la derivada es negativa, lo que indica que si se:@0.191601:0.737920:0.531963:0.737920:0.531963:0.721487:0.191601:0.721487:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 reduce la longitud :@0.531639:0.737920:0.678893:0.737920:0.678893:0.721251:0.531639:0.721251:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
L:@0.679136:0.737920:0.686757:0.737920:0.686757:0.721514:0.679136:0.721514:0.000000
, la :@0.686683:0.737920:0.710373:0.737920:0.710373:0.721487:0.686683:0.721487:0.000000:0.000000:0.000000:0.000000:0.000000
frecuencias  aumenta.:@0.103191:0.754562:0.265893:0.754562:0.265893:0.738128:0.103191:0.738128:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f :@0.189282:0.754562:0.197216:0.754562:0.197216:0.738156:0.189282:0.738156:0.000000:0.000000
2.  Derivada de   respecto a la tensión   (con  y   constantes)::@0.073089:0.781925:0.546170:0.781925:0.546170:0.765256:0.073089:0.765256:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.198619:0.781925:0.203828:0.781925:0.203828:0.765242:0.198619:0.765242:0.000000
T:@0.367506:0.781925:0.376231:0.781925:0.376231:0.765242:0.367506:0.765242:0.000000
L  ρ:@0.417187:0.781925:0.450854:0.781925:0.450854:0.765242:0.417187:0.765242:0.000000:0.000000:0.000000:0.000000
f(L,T)= :@0.338753:0.812144:0.386351:0.812144:0.386351:0.795738:0.338753:0.795738:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
      :@0.386318:0.812144:0.424194:0.812135:0.424194:0.795702:0.386318:0.795710:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.394835:0.805298:0.403817:0.805298:0.403817:0.788865:0.394835:0.788865:0.000000
2:@0.391061:0.821691:0.400044:0.821691:0.400044:0.805257:0.391061:0.805257:0.000000
L:@0.399970:0.821691:0.407591:0.821691:0.407591:0.805284:0.399970:0.805284:0.000000
T:@0.428257:0.805298:0.436375:0.805298:0.436375:0.788892:0.428257:0.788892:0.000000
ρ:@0.427521:0.821691:0.437093:0.821691:0.437093:0.805284:0.427521:0.805284:0.000000
Tomamos la derivada parcial respecto a T::@0.103210:0.854422:0.403287:0.854422:0.403287:0.837988:0.103210:0.837988:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.307392:0.878153:0.311994:0.878153:0.311994:0.861164:0.307392:0.861164:0.000000
f:@0.309601:0.878569:0.314295:0.878569:0.314295:0.862163:0.309601:0.862163:0.000000
 :@0.305928:0.894795:0.310530:0.894795:0.310530:0.877806:0.305928:0.877806:0.000000
L:@0.308137:0.895211:0.315757:0.895211:0.315757:0.878805:0.308137:0.878805:0.000000
 = :@0.316401:0.885406:0.335287:0.885406:0.335287:0.868972:0.316401:0.868972:0.000000:0.000000:0.000000
1:@0.343872:0.877940:0.352855:0.877940:0.352855:0.861506:0.343872:0.861506:0.000000
2:@0.337521:0.894332:0.346504:0.894332:0.346504:0.877898:0.337521:0.877898:0.000000
L:@0.346430:0.894332:0.354051:0.894332:0.354051:0.877926:0.346430:0.877926:0.000000
2:@0.353972:0.888266:0.359209:0.888266:0.359209:0.878685:0.353972:0.878685:0.000000
                  :@0.361444:0.885407:0.460212:0.885406:0.460212:0.868972:0.361444:0.868974:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
.:@0.365402:0.884991:0.370003:0.884991:0.370003:0.868003:0.365402:0.868003:0.000000
1:@0.378407:0.878569:0.387390:0.878569:0.387390:0.862135:0.378407:0.862135:0.000000
2:@0.378407:0.894961:0.387390:0.894961:0.387390:0.878527:0.378407:0.878527:0.000000
T:@0.407880:0.878569:0.415998:0.878569:0.415998:0.862163:0.407880:0.862163:0.000000
ρ:@0.407144:0.894961:0.416716:0.894961:0.416716:0.878555:0.407144:0.878555:0.000000
.:@0.451671:0.884990:0.456273:0.884990:0.456273:0.868002:0.451671:0.868002:0.000000
1:@0.464700:0.878569:0.473683:0.878569:0.473683:0.862135:0.464700:0.862135:0.000000
ρ:@0.464406:0.894961:0.473978:0.894961:0.473978:0.878555:0.464406:0.878555:0.000000
–1/2:@0.430034:0.872897:0.451295:0.872897:0.451295:0.862439:0.430034:0.862439:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000