118:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000
Derivadas:@0.073089:0.043661:0.163191:0.043661:0.163191:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Asumimos, sin demostración, que las funciones polinomiales de grado ≤ 4 son funciones :@0.293660:0.249588:0.930479:0.249588:0.930479:0.233155:0.293660:0.233155:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
continuas. Con esta premisa, las nociones que introducimos en esta sección :@0.293660:0.266230:0.930903:0.266230:0.930903:0.249796:0.293660:0.249796:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
corresponden a los primeros pasos que se dan en el cálculo diferencial.:@0.293660:0.282872:0.807580:0.282872:0.807580:0.266438:0.293660:0.266438:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Definición  :@0.293660:0.310234:0.381628:0.310234:0.381628:0.293287:0.293660:0.293287:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
. Sea  ,      .  La distancia de   a   se denota  ( ,  ), y se lee “distancia de   a :@0.374689:0.310234:0.930938:0.310234:0.930938:0.293800:0.374689:0.293800:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.411632:0.310234:0.433795:0.310234:0.433795:0.293828:0.411632:0.293828:0.000000:0.000000:0.000000
∈ ℝ:@0.438286:0.310817:0.471750:0.310817:0.471750:0.293162:0.438286:0.293162:0.000000:0.000000:0.000000
x y:@0.595281:0.310234:0.626795:0.310234:0.626795:0.293828:0.595281:0.293828:0.000000:0.000000:0.000000
d x y:@0.707087:0.310234:0.743239:0.310234:0.743239:0.293828:0.707087:0.293828:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.906971:0.310234:0.914426:0.310234:0.914426:0.293828:0.906971:0.293828:0.000000
y:@0.293642:0.326876:0.301115:0.326876:0.301115:0.310470:0.293642:0.310470:0.000000
”. Se define como  ( ,  ) = |  –  |.:@0.301041:0.326876:0.530911:0.326876:0.530911:0.310442:0.301041:0.310442:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x y:@0.428825:0.326876:0.464737:0.326876:0.464737:0.310470:0.428825:0.310470:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.492072:0.326876:0.523953:0.326876:0.523953:0.310470:0.492072:0.310470:0.000000:0.000000:0.000000
De la definición de valor absoluto y de la distancia entre dos números reales, tenemos::@0.293605:0.354238:0.917497:0.354238:0.917497:0.337804:0.293605:0.337804:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x y:@0.317313:0.385178:0.353226:0.385178:0.353226:0.368772:0.317313:0.368772:0.000000:0.000000:0.000000:0.000000:0.000000
( ,  ) =   –  , si   ≥  ,:@0.326499:0.385178:0.466891:0.385178:0.466891:0.368744:0.326499:0.368744:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.376732:0.385178:0.408613:0.385178:0.408613:0.368772:0.376732:0.368772:0.000000:0.000000:0.000000
x:@0.430113:0.385178:0.437568:0.385178:0.437568:0.368772:0.430113:0.368772:0.000000
y:@0.456288:0.385178:0.463762:0.385178:0.463762:0.368772:0.456288:0.368772:0.000000
d x y:@0.317258:0.405398:0.353171:0.405398:0.353171:0.388992:0.317258:0.388992:0.000000:0.000000:0.000000:0.000000:0.000000
( ,  ) =   –  , si   <  .:@0.326443:0.405398:0.466836:0.405398:0.466836:0.388964:0.326443:0.388964:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y x:@0.376677:0.405398:0.408558:0.405398:0.408558:0.388992:0.376677:0.388992:0.000000:0.000000:0.000000
x:@0.430058:0.405398:0.437513:0.405398:0.437513:0.388992:0.430058:0.388992:0.000000
y:@0.456233:0.405398:0.463706:0.405398:0.463706:0.388992:0.456233:0.388992:0.000000
A la distancia entre dos números reales arriba definida la denominaremos métrica usual :@0.293494:0.436338:0.930706:0.436338:0.930706:0.419904:0.293494:0.419904:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
en  , y, como se dijo, esta nos permite medir la proximidad o lejanía entre dos números :@0.293494:0.452979:0.931173:0.452979:0.931173:0.436546:0.293494:0.436546:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ℝ:@0.316209:0.453562:0.329499:0.453562:0.329499:0.435908:0.316209:0.435908:0.000000
reales. Sea 0 <  < 1,  ,      , se dice que   es próximo a   respecto de   si y solo si se :@0.293494:0.469621:0.930841:0.469614:0.930841:0.453180:0.293494:0.453187:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
δ:@0.402596:0.470012:0.411381:0.470012:0.411381:0.453160:0.402596:0.453160:0.000000
 :@0.411434:0.469456:0.415282:0.469456:0.415282:0.453769:0.411434:0.453769:0.000000
x y:@0.448147:0.469614:0.470788:0.469614:0.470788:0.453208:0.448147:0.453208:0.000000:0.000000:0.000000
∈ ℝ:@0.475335:0.470196:0.508523:0.470196:0.508523:0.452542:0.475335:0.452542:0.000000:0.000000:0.000000
y:@0.605106:0.469614:0.612579:0.469614:0.612579:0.453208:0.605106:0.453208:0.000000
x:@0.716267:0.469614:0.723722:0.469614:0.723722:0.453208:0.716267:0.453208:0.000000
δ:@0.819847:0.470012:0.828632:0.470012:0.828632:0.453160:0.819847:0.453160:0.000000
verifica::@0.293655:0.486255:0.347281:0.486255:0.347281:0.469822:0.293655:0.469822:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x y:@0.546405:0.510039:0.581857:0.510039:0.581857:0.493633:0.546405:0.493633:0.000000:0.000000:0.000000:0.000000:0.000000
( ,  ) = |  –  | <  .:@0.555498:0.510039:0.674165:0.510044:0.674165:0.493611:0.555498:0.493606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.608640:0.510039:0.640153:0.510039:0.640153:0.493633:0.608640:0.493633:0.000000:0.000000:0.000000
δ:@0.662265:0.510443:0.671051:0.510443:0.671051:0.493591:0.662265:0.493591:0.000000
En unidades anteriores se estudió, de forma intuitiva, la derivada de funciones cuadráticas, :@0.293668:0.540984:0.931110:0.540984:0.931110:0.524551:0.293668:0.524551:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
llamadas también funciones polinomiales de grado ≤ 2. En esta sección continuaremos :@0.293668:0.557626:0.931342:0.557626:0.931342:0.541192:0.293668:0.541192:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
con la metodología establecida para funciones polinomiales de la forma::@0.293668:0.574268:0.813002:0.574268:0.813002:0.557834:0.293668:0.557834:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p x:@0.458542:0.605208:0.479987:0.605208:0.479987:0.588802:0.458542:0.588802:0.000000:0.000000:0.000000
( ) =   + :@0.467727:0.605208:0.536768:0.605202:0.536768:0.588768:0.467727:0.588774:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.503493:0.605208:0.512752:0.605208:0.512752:0.588802:0.503493:0.588802:0.000000
0:@0.512693:0.608374:0.517930:0.608374:0.517930:0.598793:0.512693:0.598793:0.000000
a x:@0.536676:0.605202:0.558510:0.605202:0.558510:0.588796:0.536676:0.588796:0.000000:0.000000:0.000000
1:@0.545866:0.608374:0.551103:0.608374:0.551103:0.598793:0.545866:0.598793:0.000000
 + :@0.558437:0.605202:0.577323:0.605202:0.577323:0.588768:0.558437:0.588768:0.000000:0.000000:0.000000
a x:@0.577231:0.605202:0.599073:0.605202:0.599073:0.588796:0.577231:0.588796:0.000000:0.000000:0.000000
2:@0.586429:0.608374:0.591666:0.608374:0.591666:0.598793:0.586429:0.598793:0.000000
2:@0.599006:0.599138:0.604243:0.599138:0.604243:0.589557:0.599006:0.589557:0.000000
 + :@0.604197:0.605202:0.623083:0.605202:0.623083:0.588768:0.604197:0.588768:0.000000:0.000000:0.000000
a x:@0.622991:0.605202:0.644825:0.605202:0.644825:0.588796:0.622991:0.588796:0.000000:0.000000:0.000000
3:@0.632181:0.608374:0.637418:0.608374:0.637418:0.598793:0.632181:0.598793:0.000000
3:@0.644760:0.599138:0.649997:0.599138:0.649997:0.589557:0.644760:0.589557:0.000000
 + :@0.649949:0.605202:0.668835:0.605202:0.668835:0.588768:0.649949:0.588768:0.000000:0.000000:0.000000
a x:@0.668743:0.605202:0.690577:0.605202:0.690577:0.588796:0.668743:0.588796:0.000000:0.000000:0.000000
4:@0.677933:0.608374:0.683170:0.608374:0.683170:0.598793:0.677933:0.598793:0.000000
4:@0.690512:0.599138:0.695749:0.599138:0.695749:0.589557:0.690512:0.589557:0.000000
,  :@0.695701:0.605202:0.706819:0.605202:0.706819:0.588768:0.695701:0.588768:0.000000:0.000000:0.000000
∀:@0.706727:0.605784:0.719005:0.605784:0.719005:0.588130:0.706727:0.588130:0.000000
x:@0.718913:0.605202:0.726368:0.605202:0.726368:0.588796:0.718913:0.588796:0.000000
   :@0.726299:0.605202:0.748830:0.605202:0.748830:0.588768:0.726299:0.588768:0.000000:0.000000:0.000000
∈:@0.730250:0.605600:0.744886:0.605600:0.744886:0.588748:0.730250:0.588748:0.000000
ℝ:@0.748756:0.605784:0.762046:0.605784:0.762046:0.588130:0.748756:0.588130:0.000000
donde  ,  ,  ,       fijos. El conjunto de todas las funciones polinomiales de grado ≤ :@0.293674:0.636142:0.931205:0.636139:0.931205:0.619705:0.293674:0.619708:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a a a a:@0.345692:0.636142:0.418911:0.636139:0.418911:0.619733:0.345692:0.619736:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.354879:0.639312:0.360116:0.639312:0.360116:0.629731:0.354879:0.629731:0.000000
1:@0.376202:0.639312:0.381438:0.639312:0.381438:0.629731:0.376202:0.629731:0.000000
2:@0.397524:0.639312:0.402761:0.639312:0.402761:0.629731:0.397524:0.629731:0.000000
4:@0.418845:0.630075:0.424081:0.630075:0.424081:0.620494:0.418845:0.620494:0.000000
∈:@0.427852:0.636538:0.442489:0.636538:0.442489:0.619686:0.427852:0.619686:0.000000
ℝ:@0.446211:0.636722:0.459501:0.636722:0.459501:0.619067:0.446211:0.619067:0.000000
4 con coeficientes reales se denota con  [ ]. Se tiene la siguiente equivalencia::@0.293652:0.652781:0.869459:0.652781:0.869459:0.636347:0.293652:0.636347:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
P:@0.581836:0.652781:0.590598:0.652781:0.590598:0.636375:0.581836:0.636375:0.000000
4:@0.590531:0.655953:0.595768:0.655953:0.595768:0.646372:0.590531:0.646372:0.000000
ℝ:@0.600526:0.653363:0.613816:0.653363:0.613816:0.635709:0.600526:0.635709:0.000000
p x:@0.475320:0.687285:0.496764:0.687285:0.496764:0.670879:0.475320:0.670879:0.000000:0.000000:0.000000
( )    [  ]   :@0.484505:0.687285:0.595106:0.687291:0.595106:0.670858:0.484505:0.670851:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.505441:0.687690:0.520077:0.687690:0.520077:0.670838:0.505441:0.670838:0.000000
P:@0.523947:0.687291:0.532708:0.687291:0.532708:0.670885:0.523947:0.670885:0.000000
4 :@0.532635:0.690464:0.540179:0.690464:0.540179:0.680883:0.532635:0.680883:0.000000:0.000000
ℝ :@0.548872:0.687874:0.566671:0.687874:0.566671:0.670220:0.548872:0.670220:0.000000:0.000000
∃ :@0.595025:0.687690:0.611050:0.687690:0.611050:0.670838:0.595025:0.670838:0.000000:0.000000
a a a a a:@0.610961:0.687291:0.706038:0.687291:0.706038:0.670885:0.610961:0.670885:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.620155:0.690464:0.625391:0.690464:0.625391:0.680883:0.620155:0.680883:0.000000
,  ,  ,  , :@0.625344:0.687291:0.696871:0.687291:0.696871:0.670858:0.625344:0.670858:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.641609:0.690464:0.646846:0.690464:0.646846:0.680883:0.641609:0.680883:0.000000
2:@0.663065:0.690464:0.668302:0.690464:0.668302:0.680883:0.663065:0.680883:0.000000
3:@0.684520:0.690464:0.689757:0.690464:0.689757:0.680883:0.684520:0.680883:0.000000
4 :@0.705976:0.690464:0.713520:0.690464:0.713520:0.680883:0.705976:0.680883:0.000000:0.000000
∈:@0.713469:0.687690:0.728106:0.687690:0.728106:0.670838:0.713469:0.670838:0.000000
 :@0.728018:0.687291:0.732049:0.687291:0.732049:0.670858:0.728018:0.670858:0.000000
ℝ:@0.731975:0.687874:0.745265:0.687874:0.745265:0.670220:0.731975:0.670220:0.000000
tal que  ( ) =   + :@0.293662:0.721795:0.425936:0.721802:0.425936:0.705368:0.293662:0.705362:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p x:@0.347705:0.721795:0.369150:0.721795:0.369150:0.705389:0.347705:0.705389:0.000000:0.000000:0.000000
a:@0.392656:0.721795:0.401915:0.721795:0.401915:0.705389:0.392656:0.705389:0.000000
0:@0.401861:0.724974:0.407098:0.724974:0.407098:0.715393:0.401861:0.715393:0.000000
a x:@0.425844:0.721802:0.447679:0.721802:0.447679:0.705396:0.425844:0.705396:0.000000:0.000000:0.000000
1:@0.435035:0.724974:0.440271:0.724974:0.440271:0.715393:0.435035:0.715393:0.000000
 + :@0.447605:0.721802:0.466491:0.721802:0.466491:0.705368:0.447605:0.705368:0.000000:0.000000:0.000000
a x:@0.466399:0.721802:0.488240:0.721802:0.488240:0.705396:0.466399:0.705396:0.000000:0.000000:0.000000
2:@0.475596:0.724974:0.480833:0.724974:0.480833:0.715393:0.475596:0.715393:0.000000
2:@0.488175:0.715738:0.493411:0.715738:0.493411:0.706157:0.488175:0.706157:0.000000
 + :@0.493364:0.721802:0.512250:0.721802:0.512250:0.705368:0.493364:0.705368:0.000000:0.000000:0.000000
a x:@0.512158:0.721802:0.533992:0.721802:0.533992:0.705396:0.512158:0.705396:0.000000:0.000000:0.000000
3:@0.521348:0.724974:0.526585:0.724974:0.526585:0.715393:0.521348:0.715393:0.000000
3:@0.533925:0.715738:0.539162:0.715738:0.539162:0.706157:0.533925:0.706157:0.000000
 + :@0.539116:0.721802:0.558002:0.721802:0.558002:0.705368:0.539116:0.705368:0.000000:0.000000:0.000000
a x:@0.557910:0.721802:0.579744:0.721802:0.579744:0.705396:0.557910:0.705396:0.000000:0.000000:0.000000
4:@0.567098:0.724974:0.572335:0.724974:0.572335:0.715393:0.567098:0.715393:0.000000
4:@0.579677:0.715738:0.584914:0.715738:0.584914:0.706157:0.579677:0.706157:0.000000
,   :@0.584868:0.721802:0.599943:0.721802:0.599943:0.705368:0.584868:0.705368:0.000000:0.000000:0.000000:0.000000
∀:@0.599833:0.722384:0.612111:0.722384:0.612111:0.704730:0.599833:0.704730:0.000000
x:@0.612019:0.721802:0.619474:0.721802:0.619474:0.705396:0.612019:0.705396:0.000000
    .:@0.619400:0.721802:0.658273:0.721802:0.658273:0.705368:0.619400:0.705368:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.623366:0.722200:0.638002:0.722200:0.638002:0.705348:0.623366:0.705348:0.000000
ℝ:@0.641872:0.722384:0.655162:0.722384:0.655162:0.704730:0.641872:0.704730:0.000000
Recordemos que  [ ], con las operaciones habituales de adición y producto de :@0.293662:0.752742:0.930608:0.752739:0.930608:0.736305:0.293662:0.736308:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
P:@0.431422:0.752742:0.440184:0.752742:0.440184:0.736336:0.431422:0.736336:0.000000
4:@0.440446:0.755911:0.445683:0.755911:0.445683:0.746331:0.440446:0.746331:0.000000
ℝ:@0.451061:0.753322:0.464351:0.753322:0.464351:0.735667:0.451061:0.735667:0.000000
polinomios por números reales, tiene una estructura denominada espacio vectorial real.:@0.293660:0.769381:0.927370:0.769381:0.927370:0.752947:0.293660:0.752947:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Definición  :@0.293660:0.796743:0.381408:0.796743:0.381408:0.779796:0.293660:0.779796:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
. Sean :@0.374486:0.796743:0.420357:0.796743:0.420357:0.780309:0.374486:0.780309:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p(x):@0.419842:0.796743:0.446063:0.796743:0.446063:0.780337:0.419842:0.780337:0.000000:0.000000:0.000000:0.000000
    [ ], :@0.445989:0.796743:0.513362:0.796744:0.513362:0.780310:0.445989:0.780309:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.449508:0.797143:0.464144:0.797143:0.464144:0.780291:0.449508:0.780291:0.000000
P:@0.467569:0.796744:0.476330:0.796744:0.476330:0.780338:0.467569:0.780338:0.000000
4 :@0.476242:0.799916:0.483770:0.799916:0.483770:0.790336:0.476242:0.790336:0.000000:0.000000
ℝ:@0.488263:0.797327:0.501553:0.797327:0.501553:0.779672:0.488263:0.779672:0.000000
a:@0.512855:0.796744:0.522114:0.796744:0.522114:0.780338:0.512855:0.780338:0.000000
 :@0.522014:0.799916:0.524364:0.799916:0.524364:0.790336:0.522014:0.790336:0.000000
∈:@0.524070:0.797143:0.538707:0.797143:0.538707:0.780291:0.524070:0.780291:0.000000
   fijo, :@0.538597:0.796744:0.587654:0.796744:0.587654:0.780310:0.538597:0.780310:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ℝ:@0.542131:0.797327:0.555421:0.797327:0.555421:0.779672:0.542131:0.779672:0.000000
h:@0.587137:0.796744:0.596635:0.796744:0.596635:0.780338:0.587137:0.780338:0.000000
 :@0.596543:0.799916:0.598893:0.799916:0.598893:0.790336:0.596543:0.790336:0.000000
∈:@0.598600:0.797143:0.613236:0.797143:0.613236:0.780291:0.598600:0.780291:0.000000
  ,  tal que   ≠ 0. El cociente incremental de :@0.613126:0.796744:0.931086:0.796744:0.931086:0.780310:0.613126:0.780310:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ℝ:@0.616661:0.797327:0.629951:0.797327:0.629951:0.779672:0.616661:0.779672:0.000000
h:@0.693069:0.796744:0.702567:0.796744:0.702567:0.780338:0.693069:0.780338:0.000000
la función polinomial :@0.293668:0.813386:0.451198:0.813386:0.451198:0.796952:0.293668:0.796952:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p(x):@0.450977:0.813386:0.477226:0.813386:0.477226:0.796980:0.450977:0.796980:0.000000:0.000000:0.000000:0.000000
 en el punto   se denota    ( ) y se define como sigue::@0.477189:0.813386:0.870883:0.813386:0.870883:0.796952:0.477189:0.796952:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.569059:0.813386:0.578318:0.813386:0.578318:0.796980:0.569059:0.796980:0.000000
Q h:@0.665790:0.813386:0.691891:0.813386:0.691891:0.796980:0.665790:0.796980:0.000000:0.000000:0.000000
Q h:@0.528582:0.852494:0.554683:0.852494:0.554683:0.836088:0.528582:0.836088:0.000000:0.000000:0.000000
( ) = :@0.540381:0.852494:0.578300:0.852494:0.578300:0.836061:0.540381:0.836061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p a + h:@0.583589:0.844429:0.633639:0.844429:0.633639:0.828023:0.583589:0.828023:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.592775:0.844429:0.597653:0.844429:0.597653:0.827996:0.592775:0.827996:0.000000
) –  ( ):@0.633602:0.844429:0.683467:0.844429:0.683467:0.827996:0.633602:0.827996:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p a:@0.655415:0.844429:0.678663:0.844429:0.678663:0.828023:0.655415:0.828023:0.000000:0.000000:0.000000
h:@0.628779:0.864400:0.638277:0.864400:0.638277:0.847993:0.628779:0.847993:0.000000
.:@0.688787:0.852501:0.691990:0.852501:0.691990:0.836067:0.688787:0.836067:0.000000
En general, los valores de | | ≠ 0 son suficientemente pequeños; esto es, si 0 <   < 1, 0 :@0.293657:0.890583:0.930971:0.890578:0.930971:0.874144:0.293657:0.874149:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
h:@0.487375:0.890583:0.496873:0.890583:0.496873:0.874177:0.487375:0.874177:0.000000
δ:@0.872873:0.890976:0.881658:0.890976:0.881658:0.874124:0.872873:0.874124:0.000000
< | | <  . Este cociente incremental tiene muchas aplicaciones, y es un paso previo al :@0.293657:0.907220:0.930843:0.907219:0.930843:0.890785:0.293657:0.890786:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
h:@0.313261:0.907220:0.322759:0.907220:0.322759:0.890813:0.313261:0.890813:0.000000
δ:@0.347003:0.907617:0.355788:0.907617:0.355788:0.890766:0.347003:0.890766:0.000000
cálculo de la derivada de una función.:@0.293663:0.923861:0.571587:0.923861:0.571587:0.907427:0.293663:0.907427:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Tema 1. Derivada de una función:@0.073089:0.098522:0.557698:0.098522:0.557698:0.067281:0.073089:0.067281:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
¿Cómo explicas qué es una función continua? ¿Podrías dar un ejemplo de esto en la :@0.293660:0.167374:0.907605:0.167374:0.907605:0.150940:0.293660:0.150940:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
vida real?:@0.293660:0.184016:0.360986:0.184016:0.360986:0.167582:0.293660:0.167582:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Reto matemático:@0.302768:0.139600:0.437491:0.139600:0.437491:0.122653:0.302768:0.122653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
La derivada nos revela :@0.079767:0.299518:0.221991:0.299518:0.221991:0.285325:0.079767:0.285325:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la velocidad a la que :@0.079767:0.313890:0.210596:0.313890:0.210596:0.299697:0.079767:0.299697:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
una función aumenta o :@0.079767:0.328263:0.230615:0.328263:0.230615:0.314070:0.079767:0.314070:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
disminuye en un punto :@0.079767:0.342635:0.230411:0.342635:0.230411:0.328442:0.079767:0.328442:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
particular. Esta tasa de :@0.079767:0.357008:0.221509:0.357008:0.221509:0.342815:0.079767:0.342815:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cambio se visualiza como :@0.079767:0.371380:0.243182:0.371380:0.243182:0.357187:0.079767:0.357187:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la pendiente de la recta :@0.079767:0.385753:0.230580:0.385753:0.230580:0.371560:0.079767:0.371560:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
tangente a la gráfica de :@0.079767:0.400125:0.230141:0.400125:0.230141:0.385932:0.079767:0.385932:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la función en ese punto :@0.079767:0.414498:0.231467:0.414498:0.231467:0.400305:0.079767:0.400305:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
exacto. ¿Para qué sirve?:@0.079767:0.428870:0.226192:0.428870:0.226192:0.414677:0.079767:0.414677:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
• :@0.079767:0.448602:0.087111:0.448602:0.087111:0.434409:0.079767:0.434409:0.000000:0.000000
Encontrar picos y valles: :@0.109888:0.448602:0.261903:0.448602:0.261903:0.434409:0.109888:0.434409:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ayuda a identificar los :@0.109888:0.462975:0.248727:0.462975:0.248727:0.448782:0.109888:0.448782:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
puntos más altos y más :@0.109888:0.477347:0.260000:0.477347:0.260000:0.463154:0.109888:0.463154:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
bajos de una gráfica.:@0.109888:0.491720:0.238362:0.491720:0.238362:0.477527:0.109888:0.477527:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
• :@0.079767:0.511453:0.087111:0.511453:0.087111:0.497260:0.079767:0.497260:0.000000:0.000000
Calcular velocidades :@0.109888:0.511453:0.241201:0.511453:0.241201:0.497260:0.109888:0.497260:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y aceleraciones: en :@0.109888:0.525825:0.231215:0.525825:0.231215:0.511632:0.109888:0.511632:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
física, es esencial :@0.109888:0.540198:0.217511:0.540198:0.217511:0.526005:0.109888:0.526005:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
para entender el :@0.109888:0.554570:0.216678:0.554570:0.216678:0.540377:0.109888:0.540377:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
movimiento.:@0.109888:0.568943:0.189466:0.568943:0.189466:0.554750:0.109888:0.554750:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
• :@0.079767:0.588676:0.087111:0.588676:0.087111:0.574483:0.079767:0.574483:0.000000:0.000000
Modelar el mundo: :@0.109888:0.588676:0.232830:0.588676:0.232830:0.574483:0.109888:0.574483:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
se usa para describir :@0.109888:0.603048:0.239952:0.603048:0.239952:0.588856:0.109888:0.588856:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cómo cambian las :@0.109888:0.617421:0.228082:0.617421:0.228082:0.603228:0.109888:0.603228:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cosas con el tiempo, :@0.109888:0.631793:0.241049:0.631793:0.241049:0.617601:0.109888:0.617601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
desde el crecimiento :@0.109888:0.646166:0.244413:0.646166:0.244413:0.631973:0.109888:0.631973:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
de una población hasta :@0.109888:0.660538:0.260799:0.660538:0.260799:0.646346:0.109888:0.646346:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
el movimiento de un :@0.109888:0.674911:0.244812:0.674911:0.244812:0.660718:0.109888:0.660718:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
planeta.:@0.109888:0.689283:0.159932:0.689283:0.159932:0.675091:0.109888:0.675091:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
¿Cómo se calcula? Hay :@0.079762:0.709022:0.224986:0.709022:0.224986:0.694829:0.079762:0.694829:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
reglas específicas para :@0.079762:0.723394:0.222354:0.723394:0.222354:0.709201:0.079762:0.709201:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
calcular derivadas. :@0.079762:0.737767:0.197325:0.737767:0.197325:0.723574:0.079762:0.723574:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Piensa en ellas como :@0.079762:0.752139:0.213674:0.752139:0.213674:0.737946:0.079762:0.737946:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
recetas para encontrar la :@0.079762:0.766512:0.237536:0.766512:0.237536:0.752319:0.079762:0.752319:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
pendiente en cualquier :@0.079762:0.780884:0.229730:0.780884:0.229730:0.766691:0.079762:0.766691:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
punto de una función.:@0.079762:0.795257:0.219763:0.795257:0.219763:0.781064:0.079762:0.781064:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Competencia :@0.078581:0.259440:0.187141:0.259440:0.187141:0.242493:0.078581:0.242493:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática :@0.078581:0.276082:0.176057:0.276082:0.176057:0.259135:0.078581:0.259135:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
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