183:@0.916858:0.966074:0.945034:0.966074:0.945034:0.949450:0.916858:0.949450:0.000000:0.000000:0.000000
Competencia :@0.684606:0.571796:0.782229:0.571796:0.782229:0.556324:0.684606:0.556324:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
comunicacional:@0.684606:0.584364:0.796565:0.584364:0.796565:0.568892:0.684606:0.568892:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Matemática :@0.678034:0.609260:0.766371:0.609260:0.766371:0.593787:0.678034:0.593787:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y otras disciplinas:@0.638316:0.625102:0.763027:0.625102:0.763027:0.609629:0.638316:0.609629:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
En la actualidad, el cálculo dife-:@0.638316:0.640521:0.834246:0.640521:0.834246:0.625458:0.638316:0.625458:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
rencial e integral no solo consti-:@0.638316:0.656363:0.837996:0.656363:0.837996:0.641300:0.638316:0.641300:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
tuye un instrumento de cálculo :@0.638316:0.672205:0.841108:0.672205:0.841108:0.657142:0.638316:0.657142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
en las ciencias y la técnica, sino :@0.638316:0.688047:0.836378:0.688047:0.836378:0.672984:0.638316:0.672984:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
que es también un conjunto de :@0.638316:0.703889:0.842181:0.703889:0.842181:0.688826:0.638316:0.688826:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ideas y problemas que han sido :@0.638316:0.719731:0.840594:0.719731:0.840594:0.704668:0.638316:0.704668:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
objeto de estudio como parte :@0.638316:0.735573:0.831625:0.735573:0.831625:0.720510:0.638316:0.720510:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
del pensamiento. En ese sentido, :@0.638316:0.751415:0.846652:0.751415:0.846652:0.736352:0.638316:0.736352:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
forman parte del lenguaje que :@0.638316:0.767257:0.832117:0.767257:0.832117:0.752194:0.638316:0.752194:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
facilita la expresión de leyes y :@0.638316:0.783099:0.825732:0.783099:0.825732:0.768036:0.638316:0.768036:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
principios de manera matemáti-:@0.638316:0.798941:0.841516:0.798941:0.841516:0.783878:0.638316:0.783878:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ca. Son la base para el desarrollo :@0.638316:0.814783:0.844833:0.814783:0.844833:0.799720:0.638316:0.799720:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
de otras áreas de la matemática, :@0.638316:0.830625:0.843642:0.830625:0.843642:0.815562:0.638316:0.815562:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la física, la química, la biología, :@0.638316:0.846467:0.831449:0.846467:0.831449:0.831404:0.638316:0.831404:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
las ciencias económicas y socia-:@0.638316:0.862309:0.838028:0.862309:0.838028:0.847246:0.638316:0.847246:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
les, así como las distintas ramas :@0.638316:0.878151:0.838960:0.878151:0.838960:0.863088:0.638316:0.863088:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
de la ingeniería e industria.:@0.638316:0.893993:0.804770:0.893993:0.804770:0.878930:0.638316:0.878930:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Para   < 0,   +   <  . El punto   = (  +  ,   (  +  )) se localiza a la :@0.108233:0.096194:0.599909:0.096194:0.599909:0.079697:0.108233:0.079697:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
h:@0.143084:0.096310:0.152485:0.096310:0.152485:0.079841:0.143084:0.079841:0.000000
x:@0.188068:0.096310:0.195543:0.096310:0.195543:0.079841:0.188068:0.079841:0.000000
0:@0.195543:0.099472:0.200496:0.099472:0.200496:0.089854:0.195543:0.089854:0.000000
h:@0.220321:0.096310:0.229722:0.096310:0.229722:0.079841:0.220321:0.079841:0.000000
x:@0.249546:0.096310:0.257021:0.096310:0.257021:0.079841:0.249546:0.079841:0.000000
0:@0.257021:0.099472:0.261974:0.099472:0.261974:0.089854:0.257021:0.089854:0.000000
P:@0.334603:0.096310:0.343388:0.096310:0.343388:0.079841:0.334603:0.079841:0.000000
1:@0.343388:0.099472:0.348341:0.099472:0.348341:0.089854:0.343388:0.089854:0.000000
x:@0.374119:0.096310:0.381594:0.096310:0.381594:0.079841:0.374119:0.079841:0.000000
0:@0.381594:0.099472:0.386547:0.099472:0.386547:0.089854:0.381594:0.089854:0.000000
h f x:@0.406370:0.096310:0.445266:0.096310:0.445266:0.079841:0.406370:0.079841:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.445266:0.099472:0.450219:0.099472:0.450219:0.089854:0.445266:0.089854:0.000000
h:@0.470044:0.096310:0.479445:0.096310:0.479445:0.079841:0.470044:0.079841:0.000000
izquierda de  . Un razonamiento similar se da en el caso   > 0. Cuan-:@0.108242:0.113545:0.595759:0.113545:0.595759:0.097047:0.108242:0.097047:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
P:@0.198038:0.113661:0.206823:0.113661:0.206823:0.097192:0.198038:0.097192:0.000000
1:@0.206814:0.116823:0.211767:0.116823:0.211767:0.107205:0.206814:0.107205:0.000000
h:@0.509702:0.113661:0.519103:0.113661:0.519103:0.097192:0.509702:0.097192:0.000000
do la función es derivable en el punto  , se tiene: :@0.108235:0.130896:0.460245:0.130896:0.460245:0.114398:0.108235:0.114398:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.379755:0.131011:0.387230:0.131011:0.387230:0.114543:0.379755:0.114543:0.000000
0:@0.387227:0.134174:0.392181:0.134174:0.392181:0.124556:0.387227:0.124556:0.000000
m:@0.189357:0.159047:0.204152:0.159047:0.204152:0.142579:0.189357:0.142579:0.000000
 = lím  ( ) = lím                             =       ( ).:@0.204152:0.158932:0.514636:0.158939:0.514636:0.142441:0.204152:0.142434:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
m h:@0.249907:0.159047:0.280057:0.159047:0.280057:0.142579:0.249907:0.142579:0.000000:0.000000:0.000000
x:@0.493497:0.159047:0.500972:0.159047:0.500972:0.142579:0.493497:0.142579:0.000000
0:@0.500975:0.162216:0.505928:0.162216:0.505928:0.152598:0.500975:0.152598:0.000000
  :@0.108236:0.176289:0.116520:0.176289:0.116520:0.159792:0.108236:0.159792:0.000000:0.000000
La recta tangente a la gráfica de la función   en el punto   = ( ,  ( )) :@0.108236:0.193640:0.599910:0.193640:0.599910:0.177142:0.108236:0.177142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.410120:0.193756:0.414416:0.193756:0.414416:0.177287:0.410120:0.177287:0.000000
P:@0.503730:0.193756:0.512515:0.193756:0.512515:0.177287:0.503730:0.177287:0.000000
0:@0.512455:0.196918:0.517408:0.196918:0.517408:0.187300:0.512455:0.187300:0.000000
x f x:@0.542067:0.193756:0.578913:0.193756:0.578913:0.177287:0.542067:0.177287:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.549533:0.196918:0.554486:0.196918:0.554486:0.187300:0.549533:0.187300:0.000000
0:@0.578908:0.196918:0.583862:0.196918:0.583862:0.187300:0.578908:0.187300:0.000000
es el conjunto::@0.108241:0.210991:0.210711:0.210991:0.210711:0.194493:0.108241:0.194493:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
L =   ( ,  )     |   –  ( ) =       ( )(  –  )   :@0.195455:0.235470:0.505700:0.235470:0.505700:0.218973:0.195455:0.218972:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.236818:0.235586:0.258876:0.235586:0.258876:0.219117:0.236818:0.219117:0.000000:0.000000:0.000000
:@0.268962:0.234952:0.283615:0.234952:0.283615:0.221579:0.268962:0.221579:0.000000
:@0.287757:0.234384:0.301062:0.234384:0.301062:0.221521:0.287757:0.221521:0.000000
2:@0.301062:0.229117:0.306015:0.229117:0.306015:0.219499:0.301062:0.219499:0.000000
y f x:@0.319308:0.235586:0.363503:0.235586:0.363503:0.219117:0.319308:0.219117:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.363501:0.238747:0.368454:0.238747:0.368454:0.229129:0.363501:0.229129:0.000000
x x x:@0.424305:0.235586:0.482370:0.235586:0.482370:0.219117:0.424305:0.219117:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.431777:0.238747:0.436730:0.238747:0.436730:0.229129:0.431777:0.229129:0.000000
0:@0.482369:0.238747:0.487323:0.238747:0.487323:0.229129:0.482369:0.229129:0.000000
.:@0.505700:0.235586:0.508552:0.235586:0.508552:0.219117:0.505700:0.219117:0.000000
 :@0.108239:0.252821:0.112381:0.252821:0.112381:0.236324:0.108239:0.236324:0.000000
La ecuación cartesiana de la recta tangente a la gráfica de la función   :@0.108239:0.270172:0.599884:0.270172:0.599884:0.253674:0.108239:0.253674:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.591446:0.270288:0.595742:0.270288:0.595742:0.253819:0.591446:0.253819:0.000000
en el punto   = ( ,  ( )) está definida como: :@0.108239:0.287523:0.434413:0.287523:0.434413:0.271025:0.108239:0.271025:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
P:@0.194181:0.287638:0.202966:0.287638:0.202966:0.271170:0.194181:0.271170:0.000000
0:@0.202959:0.290799:0.207913:0.290799:0.207913:0.281181:0.202959:0.281181:0.000000
x f x:@0.232958:0.287638:0.270006:0.287638:0.270006:0.271170:0.232958:0.271170:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.240431:0.290799:0.245384:0.290799:0.245384:0.281181:0.240431:0.281181:0.000000
0:@0.270004:0.290799:0.274957:0.290799:0.274957:0.281181:0.270004:0.281181:0.000000
( ,  )     tales que   –  ( ) =       ( )(  –  ):@0.190056:0.315559:0.511085:0.315566:0.511085:0.299068:0.190056:0.299061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.196009:0.315674:0.218068:0.315674:0.218068:0.299206:0.196009:0.299206:0.000000:0.000000:0.000000
:@0.228170:0.315047:0.242823:0.315047:0.242823:0.301674:0.228170:0.301674:0.000000
:@0.246965:0.314479:0.260269:0.314479:0.260269:0.301616:0.246965:0.301616:0.000000
2:@0.260269:0.309213:0.265223:0.309213:0.265223:0.299594:0.260269:0.299594:0.000000
y f x:@0.337119:0.315681:0.381314:0.315681:0.381314:0.299212:0.337119:0.299212:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.381312:0.318842:0.386265:0.318842:0.386265:0.309224:0.381312:0.309224:0.000000
x x x .:@0.442114:0.315681:0.513937:0.315681:0.513937:0.299212:0.442114:0.299212:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.449588:0.318842:0.454541:0.318842:0.454541:0.309224:0.449588:0.309224:0.000000
0:@0.500179:0.318842:0.505132:0.318842:0.505132:0.309224:0.500179:0.309224:0.000000
Problema 2 :@0.108230:0.350730:0.202707:0.350730:0.202707:0.333784:0.108230:0.333784:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
. Sea   una función real definida en el intervalo cerrado :@0.195925:0.350267:0.583866:0.350267:0.583866:0.333769:0.195925:0.333769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.230969:0.350383:0.235265:0.350383:0.235265:0.333914:0.230969:0.333914:0.000000
[ ,  ] de  . Suponemos que  ( ) ≥ 0,       [ ,  ].:@0.108230:0.367618:0.460131:0.367618:0.460131:0.351120:0.108230:0.351120:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a b:@0.114086:0.367734:0.139227:0.367734:0.139227:0.351265:0.114086:0.351265:0.000000:0.000000:0.000000
:@0.171229:0.366531:0.184534:0.366531:0.184534:0.353669:0.171229:0.353669:0.000000
f x:@0.312087:0.367734:0.329811:0.367734:0.329811:0.351265:0.312087:0.351265:0.000000:0.000000:0.000000
∀:@0.378530:0.375204:0.390102:0.375204:0.390102:0.356205:0.378530:0.356205:0.000000
x:@0.390111:0.367734:0.397585:0.367734:0.397585:0.351265:0.390111:0.351265:0.000000
:@0.401727:0.367099:0.416379:0.367099:0.416379:0.353726:0.401727:0.353726:0.000000
a b:@0.426379:0.367734:0.451520:0.367734:0.451520:0.351265:0.426379:0.351265:0.000000:0.000000:0.000000
 :@0.108232:0.384969:0.112374:0.384969:0.112374:0.368471:0.108232:0.368471:0.000000
Denotamos con   la región comprendida por el eje   y la gráfica de la :@0.108232:0.402319:0.599818:0.402319:0.599818:0.385822:0.108232:0.385822:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
S:@0.227482:0.402435:0.235362:0.402435:0.235362:0.385966:0.227482:0.385966:0.000000
x:@0.472610:0.402435:0.480084:0.402435:0.480084:0.385966:0.472610:0.385966:0.000000
función  . Esto es::@0.108232:0.419670:0.230393:0.419670:0.230393:0.403172:0.108232:0.403172:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.167357:0.419786:0.171653:0.419786:0.171653:0.403317:0.167357:0.403317:0.000000
S:@0.210569:0.444265:0.218449:0.444265:0.218449:0.427796:0.210569:0.427796:0.000000
 = {( ,  )     | 0 ≤   ≤  ( ),    ≤   ≤  }.:@0.218449:0.444149:0.493439:0.444148:0.493439:0.427651:0.218449:0.427651:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.249350:0.444265:0.271409:0.444265:0.271409:0.427796:0.249350:0.427796:0.000000:0.000000:0.000000
:@0.281498:0.443630:0.296151:0.443630:0.296151:0.430257:0.281498:0.430257:0.000000
:@0.300293:0.443062:0.313597:0.443062:0.313597:0.430199:0.300293:0.430199:0.000000
2:@0.313597:0.437795:0.318551:0.437795:0.318551:0.428177:0.313597:0.428177:0.000000
y f x a x b:@0.359430:0.444264:0.484828:0.444264:0.484828:0.427795:0.359430:0.427795:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.108231:0.461499:0.112373:0.461499:0.112373:0.445002:0.108231:0.445002:0.000000
En la Figura 6.2. se muestran regiones de dos tipos de funciones. La :@0.108231:0.478850:0.599876:0.478850:0.599876:0.462352:0.108231:0.462352:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
primera presenta discontinuidades en la función  , y la segunda, en el :@0.108231:0.496201:0.599990:0.496201:0.599990:0.479703:0.108231:0.479703:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.456080:0.496316:0.460376:0.496316:0.460376:0.479848:0.456080:0.479848:0.000000
hecho de que la función   es continua.:@0.108231:0.513552:0.379193:0.513552:0.379193:0.497054:0.108231:0.497054:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.285373:0.513667:0.289669:0.513667:0.289669:0.497198:0.285373:0.497198:0.000000
Integral indefinida o primitiva de una función:@0.108233:0.549651:0.474180:0.549651:0.474180:0.531968:0.108233:0.531968:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
En temas anteriores se calcularon las derivadas de algunas funciones :@0.108233:0.566358:0.599979:0.566358:0.599979:0.549860:0.108233:0.549860:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
de los tipos que presentamos a continuación.:@0.108233:0.583709:0.430323:0.583709:0.430323:0.567211:0.108233:0.567211:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1.:@0.108233:0.618627:0.120505:0.618627:0.120505:0.601913:0.108233:0.601913:0.000000:0.000000
 :@0.120506:0.618410:0.124648:0.618410:0.124648:0.601913:0.120506:0.601913:0.000000
Función cuadrática:@0.131968:0.618627:0.273683:0.618627:0.273683:0.601913:0.131968:0.601913:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Considera ( ) = :@0.131968:0.635761:0.249158:0.635761:0.249158:0.619263:0.131968:0.619263:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.206967:0.635877:0.224692:0.635877:0.224692:0.619408:0.206967:0.619408:0.000000:0.000000:0.000000
ax:@0.248580:0.635877:0.265052:0.635877:0.265052:0.619408:0.248580:0.619408:0.000000:0.000000
2:@0.265074:0.629408:0.270027:0.629408:0.270027:0.619790:0.265074:0.619790:0.000000
 +   +  ,        , con  ,  ,   constantes reales :@0.270028:0.635761:0.599897:0.635761:0.599897:0.619263:0.270028:0.619263:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
bx c:@0.287964:0.635877:0.329827:0.635877:0.329827:0.619408:0.287964:0.619408:0.000000:0.000000:0.000000:0.000000
∀:@0.343309:0.643348:0.354882:0.643348:0.354882:0.624349:0.343309:0.624349:0.000000
x:@0.354890:0.635877:0.362365:0.635877:0.362365:0.619408:0.354890:0.619408:0.000000
:@0.365933:0.635243:0.380586:0.635243:0.380586:0.621869:0.365933:0.621869:0.000000
:@0.384157:0.634674:0.397462:0.634674:0.397462:0.621812:0.384157:0.621812:0.000000
a b c:@0.434293:0.635877:0.472382:0.635877:0.472382:0.619408:0.434293:0.619408:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.131985:0.653228:0.140962:0.653228:0.140962:0.636759:0.131985:0.636759:0.000000
 ≠ 0. La derivada de esta función está definida como::@0.140962:0.653112:0.513566:0.653112:0.513566:0.636614:0.140962:0.636614:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
( ) = 2  +  ,        .:@0.281367:0.677590:0.446386:0.677590:0.446386:0.661093:0.281367:0.661093:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.287320:0.677706:0.294795:0.677706:0.294795:0.661237:0.287320:0.661237:0.000000
ax b:@0.328336:0.677706:0.373166:0.677706:0.373166:0.661237:0.328336:0.661237:0.000000:0.000000:0.000000:0.000000
∀:@0.388335:0.685177:0.399907:0.685177:0.399907:0.666178:0.388335:0.666178:0.000000
x:@0.399916:0.677706:0.407391:0.677706:0.407391:0.661237:0.399916:0.661237:0.000000
:@0.411532:0.677072:0.426184:0.677072:0.426184:0.663698:0.411532:0.663698:0.000000
:@0.430326:0.676503:0.443631:0.676503:0.443631:0.663641:0.430326:0.663641:0.000000
 :@0.108242:0.694941:0.112384:0.694941:0.112384:0.678443:0.108242:0.678443:0.000000
 :@0.108242:0.712292:0.112384:0.712292:0.112384:0.695794:0.108242:0.695794:0.000000
A esta derivada se la denota también como  ( ), esto es,:@0.131978:0.712292:0.530227:0.712292:0.530227:0.695794:0.131978:0.695794:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f' x:@0.445148:0.712407:0.466153:0.712407:0.466153:0.695939:0.445148:0.695939:0.000000:0.000000:0.000000:0.000000
 :@0.108242:0.736771:0.112384:0.736771:0.112384:0.720274:0.108242:0.720274:0.000000
      ( ) =  ( ). Recuerda que esta derivada define una nueva:@0.131978:0.736771:0.549356:0.736771:0.549356:0.720274:0.131978:0.720274:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.160857:0.736887:0.168331:0.736887:0.168331:0.720418:0.160857:0.720418:0.000000
f' x:@0.193376:0.736887:0.214375:0.736887:0.214375:0.720418:0.193376:0.720418:0.000000:0.000000:0.000000:0.000000
 :@0.108242:0.761250:0.112384:0.761250:0.112384:0.744752:0.108242:0.744752:0.000000
función que es un polinomio de grado ≤ 1. :@0.131978:0.761250:0.440756:0.761250:0.440756:0.744752:0.131978:0.744752:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2. :@0.108243:0.796168:0.124598:0.796168:0.124598:0.779454:0.108243:0.779454:0.000000:0.000000:0.000000
Funciones polinomiales de grado ≤ 4:@0.131978:0.796168:0.403516:0.796168:0.403516:0.779454:0.131978:0.779454:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.108242:0.820431:0.112384:0.820431:0.112384:0.803933:0.108242:0.803933:0.000000
p x:@0.208017:0.820547:0.230711:0.820547:0.230711:0.804078:0.208017:0.804078:0.000000:0.000000:0.000000
( ) =   + :@0.217284:0.820431:0.288759:0.820430:0.288759:0.803932:0.217284:0.803933:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.255756:0.820547:0.264734:0.820547:0.264734:0.804078:0.255756:0.804078:0.000000
0:@0.264716:0.823708:0.269669:0.823708:0.269669:0.814090:0.264716:0.814090:0.000000
a x a x:@0.288759:0.820545:0.350662:0.820545:0.350662:0.804077:0.288759:0.804077:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.297738:0.823708:0.302691:0.823708:0.302691:0.814090:0.297738:0.814090:0.000000
 + :@0.310165:0.820430:0.329257:0.820430:0.329257:0.803932:0.310165:0.803932:0.000000:0.000000:0.000000
2:@0.338233:0.823708:0.343187:0.823708:0.343187:0.814090:0.338233:0.814090:0.000000
2:@0.350662:0.814078:0.355615:0.814078:0.355615:0.804460:0.350662:0.804460:0.000000
 + :@0.355614:0.820430:0.374706:0.820430:0.374706:0.803932:0.355614:0.803932:0.000000:0.000000:0.000000
a x:@0.374706:0.820545:0.396111:0.820545:0.396111:0.804077:0.374706:0.804077:0.000000:0.000000:0.000000
3:@0.383683:0.823708:0.388636:0.823708:0.388636:0.814090:0.383683:0.814090:0.000000
3:@0.396111:0.814078:0.401064:0.814078:0.401064:0.804460:0.396111:0.804460:0.000000
 + :@0.401063:0.820430:0.420155:0.820430:0.420155:0.803932:0.401063:0.803932:0.000000:0.000000:0.000000
a x:@0.420155:0.820545:0.441561:0.820545:0.441561:0.804077:0.420155:0.804077:0.000000:0.000000:0.000000
4:@0.429134:0.823708:0.434087:0.823708:0.434087:0.814090:0.429134:0.814090:0.000000
4:@0.441561:0.814078:0.446514:0.814078:0.446514:0.804460:0.441561:0.804460:0.000000
,        ,:@0.446514:0.820430:0.519745:0.820430:0.519745:0.803932:0.446514:0.803932:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∀:@0.461694:0.828016:0.473266:0.828016:0.473266:0.809017:0.461694:0.809017:0.000000
x:@0.473274:0.820545:0.480749:0.820545:0.480749:0.804077:0.473274:0.804077:0.000000
:@0.484891:0.819911:0.499543:0.819911:0.499543:0.806538:0.484891:0.806538:0.000000
:@0.503685:0.819343:0.516990:0.819343:0.516990:0.806481:0.503685:0.806481:0.000000
siendo ,  ,  ,  ,      fijos. La derivada de   es::@0.131975:0.848466:0.495038:0.848473:0.495038:0.831975:0.131975:0.831969:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a a a a a:@0.183124:0.848582:0.275412:0.848588:0.275412:0.832120:0.183124:0.832113:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.192102:0.851751:0.197055:0.851751:0.197055:0.842132:0.192102:0.842132:0.000000
1:@0.212929:0.851751:0.217882:0.851751:0.217882:0.842132:0.212929:0.842132:0.000000
2:@0.233757:0.851751:0.238710:0.851751:0.238710:0.842132:0.233757:0.842132:0.000000
3:@0.254584:0.851751:0.259537:0.851751:0.259537:0.842132:0.254584:0.842132:0.000000
4:@0.275410:0.851751:0.280363:0.851751:0.280363:0.842132:0.275410:0.842132:0.000000
:@0.284507:0.847954:0.299159:0.847954:0.299159:0.834581:0.284507:0.834581:0.000000
 :@0.303300:0.847386:0.321211:0.847386:0.321211:0.834523:0.303300:0.834523:0.000000:0.000000
p:@0.464213:0.848588:0.473480:0.848588:0.473480:0.832120:0.464213:0.832120:0.000000
( ) =  ( ) =   + 2:@0.197587:0.876509:0.327629:0.876515:0.327629:0.860018:0.197587:0.860011:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.203540:0.876625:0.211015:0.876625:0.211015:0.860156:0.203540:0.860156:0.000000
p' x:@0.236060:0.876625:0.261066:0.876625:0.261066:0.860156:0.236060:0.860156:0.000000:0.000000:0.000000:0.000000
a:@0.286111:0.876625:0.295088:0.876625:0.295088:0.860156:0.286111:0.860156:0.000000
1:@0.295088:0.879792:0.300041:0.879792:0.300041:0.870174:0.295088:0.870174:0.000000
a x:@0.327629:0.876631:0.349034:0.876631:0.349034:0.860162:0.327629:0.860162:0.000000:0.000000:0.000000
2:@0.336605:0.879792:0.341558:0.879792:0.341558:0.870174:0.336605:0.870174:0.000000
 + 3:@0.349034:0.876515:0.376622:0.876515:0.376622:0.860018:0.349034:0.860018:0.000000:0.000000:0.000000:0.000000
a x:@0.376622:0.876631:0.398026:0.876631:0.398026:0.860162:0.376622:0.860162:0.000000:0.000000:0.000000
3:@0.385597:0.879792:0.390551:0.879792:0.390551:0.870174:0.385597:0.870174:0.000000
2:@0.398026:0.870162:0.402979:0.870162:0.402979:0.860544:0.398026:0.860544:0.000000
 + 4:@0.402980:0.876515:0.430568:0.876515:0.430568:0.860018:0.402980:0.860018:0.000000:0.000000:0.000000:0.000000
a x:@0.430568:0.876631:0.451972:0.876631:0.451972:0.860162:0.430568:0.860162:0.000000:0.000000:0.000000
4:@0.439544:0.879792:0.444497:0.879792:0.444497:0.870174:0.439544:0.870174:0.000000
3:@0.451972:0.870162:0.456925:0.870162:0.456925:0.860544:0.451972:0.860544:0.000000
,        ,:@0.456924:0.876515:0.530155:0.876515:0.530155:0.860018:0.456924:0.860018:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∀:@0.472105:0.884102:0.483678:0.884102:0.483678:0.865103:0.472105:0.865103:0.000000
x:@0.483684:0.876631:0.491159:0.876631:0.491159:0.860162:0.483684:0.860162:0.000000
:@0.495302:0.875997:0.509955:0.875997:0.509955:0.862624:0.495302:0.862624:0.000000
:@0.514097:0.875429:0.527401:0.875429:0.527401:0.862566:0.514097:0.862566:0.000000
 :@0.108227:0.904551:0.112369:0.904551:0.112369:0.888053:0.108227:0.888053:0.000000
que define una nueva función, o sea, un polinomio de grado ≤ 3.:@0.131961:0.904551:0.591468:0.904551:0.591468:0.888053:0.131961:0.888053:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
p:@0.629204:0.513492:0.640685:0.513492:0.640685:0.503467:0.629204:0.503467:0.000000
 :@0.640685:0.514200:0.643566:0.514200:0.643566:0.502723:0.640685:0.502723:0.000000
Figura 6.2.:@0.643566:0.514200:0.692134:0.514200:0.692134:0.502723:0.643566:0.502723:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
h:@0.224508:0.166433:0.230639:0.166433:0.230639:0.155693:0.224508:0.155693:0.000000
→:@0.230639:0.165751:0.241387:0.165751:0.241387:0.155740:0.230639:0.155740:0.000000
0:@0.241384:0.166358:0.246925:0.166358:0.246925:0.155599:0.241384:0.155599:0.000000
h:@0.306656:0.166433:0.312787:0.166433:0.312787:0.155693:0.306656:0.155693:0.000000
→:@0.312793:0.165751:0.323541:0.165751:0.323541:0.155740:0.312793:0.155740:0.000000
0:@0.323538:0.166358:0.329079:0.166358:0.329079:0.155599:0.323538:0.155599:0.000000
df:@0.263034:0.670282:0.276500:0.670282:0.276500:0.653813:0.263034:0.653813:0.000000:0.000000
dx:@0.261512:0.686620:0.278003:0.686620:0.278003:0.670151:0.261512:0.670151:0.000000:0.000000
df:@0.135610:0.729159:0.149076:0.729159:0.149076:0.712690:0.135610:0.712690:0.000000:0.000000
dx:@0.134088:0.745497:0.150579:0.745497:0.150579:0.729028:0.134088:0.729028:0.000000:0.000000
dp:@0.176507:0.869243:0.194867:0.869243:0.194867:0.852774:0.176507:0.852774:0.000000:0.000000
dx:@0.177451:0.885582:0.193943:0.885582:0.193943:0.869113:0.177451:0.869113:0.000000:0.000000
f x:@0.335174:0.152157:0.352898:0.152157:0.352898:0.135688:0.335174:0.135688:0.000000:0.000000:0.000000
(  +  ) –  ( ):@0.339471:0.152041:0.439710:0.152041:0.439710:0.135543:0.339471:0.135543:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.352898:0.152803:0.357852:0.152803:0.357852:0.143185:0.352898:0.143185:0.000000
h:@0.376942:0.152157:0.386344:0.152157:0.386344:0.135688:0.376942:0.135688:0.000000
f x:@0.411080:0.152157:0.428804:0.152157:0.428804:0.135688:0.411080:0.135688:0.000000:0.000000:0.000000
0:@0.428804:0.152803:0.433757:0.152803:0.433757:0.143185:0.428804:0.143185:0.000000
h:@0.382743:0.168495:0.392144:0.168495:0.392144:0.152027:0.382743:0.152027:0.000000
df:@0.468810:0.151552:0.482276:0.151552:0.482276:0.135083:0.468810:0.135083:0.000000:0.000000
dx:@0.467288:0.167891:0.483779:0.167891:0.483779:0.151422:0.467288:0.151422:0.000000:0.000000
df:@0.398741:0.227998:0.412208:0.227998:0.412208:0.211530:0.398741:0.211530:0.000000:0.000000
dx:@0.397219:0.244337:0.413710:0.244337:0.413710:0.227868:0.397219:0.227868:0.000000:0.000000
df:@0.417524:0.308190:0.430991:0.308190:0.430991:0.291721:0.417524:0.291721:0.000000:0.000000
dx:@0.416002:0.324529:0.432493:0.324529:0.432493:0.308060:0.416002:0.308060:0.000000:0.000000
S:@0.738120:0.458197:0.744286:0.458197:0.744286:0.445308:0.738120:0.445308:0.000000
0:@0.683939:0.491904:0.689849:0.491904:0.689849:0.480427:0.683939:0.480427:0.000000
x:@0.848017:0.494014:0.854622:0.494014:0.854622:0.480386:0.848017:0.480386:0.000000
y f(x):@0.729013:0.422248:0.767686:0.422248:0.767686:0.409360:0.729013:0.409360:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 = :@0.735029:0.422158:0.749970:0.422158:0.749970:0.409246:0.735029:0.409246:0.000000:0.000000:0.000000
a:@0.636694:0.492104:0.642149:0.492104:0.642149:0.480627:0.636694:0.480627:0.000000
b:@0.827497:0.492104:0.834091:0.492104:0.834091:0.480627:0.827497:0.480627:0.000000
x:@0.718030:0.492104:0.723592:0.492104:0.723592:0.480627:0.718030:0.480627:0.000000
1:@0.723580:0.494384:0.727026:0.494384:0.727026:0.487693:0.723580:0.487693:0.000000
x:@0.777004:0.492104:0.782566:0.492104:0.782566:0.480627:0.777004:0.480627:0.000000
2:@0.782566:0.494384:0.786011:0.494384:0.786011:0.487693:0.782566:0.487693:0.000000
y:@0.681787:0.395419:0.688248:0.395419:0.688248:0.381791:0.681787:0.381791:0.000000
x:@0.847731:0.359370:0.854335:0.359370:0.854335:0.345741:0.847731:0.345741:0.000000
y f(x):@0.728847:0.304618:0.767520:0.304618:0.767520:0.291729:0.728847:0.291729:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 = :@0.734863:0.304527:0.749805:0.304527:0.749805:0.291616:0.734863:0.291616:0.000000:0.000000:0.000000
a:@0.634649:0.356960:0.640103:0.356960:0.640103:0.345484:0.634649:0.345484:0.000000
0:@0.657097:0.355723:0.663007:0.355723:0.663007:0.344246:0.657097:0.344246:0.000000
x:@0.685107:0.356960:0.690669:0.356960:0.690669:0.345484:0.685107:0.345484:0.000000
1:@0.690669:0.359240:0.694114:0.359240:0.694114:0.352549:0.690669:0.352549:0.000000
x:@0.716936:0.356960:0.722498:0.356960:0.722498:0.345484:0.716936:0.345484:0.000000
2:@0.722498:0.359240:0.725944:0.359240:0.725944:0.352549:0.722498:0.352549:0.000000
x:@0.774844:0.356960:0.780406:0.356960:0.780406:0.345484:0.774844:0.345484:0.000000
3:@0.780406:0.359240:0.783852:0.359240:0.783852:0.352549:0.780406:0.352549:0.000000
b:@0.822959:0.356960:0.829553:0.356960:0.829553:0.345484:0.822959:0.345484:0.000000
y:@0.654696:0.259516:0.661158:0.259516:0.661158:0.245888:0.654696:0.245888:0.000000
 ©:@0.079553:0.731952:0.079553:0.720342:0.061505:0.720342:0.061505:0.731952:0.000000:0.000000
maya:@0.080774:0.720340:0.080774:0.684191:0.055786:0.684191:0.055786:0.720340:0.000000:0.000000:0.000000:0.000000
®EDUCACIÓN – Libro resuelto solo para fines didácticos – Prohibida su reproducción :@0.079553:0.684191:0.079553:0.268044:0.061505:0.268044:0.061505:0.684191:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000