201:@0.937671:0.974518:0.968375:0.974518:0.968375:0.956333:0.937671:0.956333:0.000000:0.000000:0.000000 Pensamiento numérico y variacional:@0.608443:0.043660:0.931638:0.043660:0.931638:0.024915:0.608443:0.024915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Para 0, . El punto ( , ( )) se localiza a la izquierda de . Un :@0.073089:0.236975:0.710385:0.236975:0.710385:0.221262:0.073089:0.221262:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 h:@0.107216:0.236975:0.116714:0.236975:0.116714:0.221539:0.107216:0.221539:0.000000 ,:@0.120248:0.236420:0.135581:0.236420:0.135581:0.222704:0.120248:0.222704:0.000000 x:@0.154725:0.236975:0.162180:0.236975:0.162180:0.221539:0.154725:0.221539:0.000000 0:@0.162120:0.240147:0.167357:0.240147:0.167357:0.230987:0.162120:0.230987:0.000000 1 ,:@0.170917:0.236420:0.218132:0.236420:0.218132:0.222704:0.170917:0.222704:0.000000:0.000000:0.000000 h:@0.189766:0.236975:0.199264:0.236975:0.199264:0.221539:0.189766:0.221539:0.000000 x:@0.221648:0.236975:0.229103:0.236975:0.229103:0.221539:0.221648:0.221539:0.000000 0:@0.229047:0.240147:0.234284:0.240147:0.234284:0.230987:0.229047:0.230987:0.000000 P:@0.305272:0.236975:0.314034:0.236975:0.314034:0.221539:0.305272:0.221539:0.000000 1:@0.313968:0.240147:0.319205:0.240147:0.319205:0.230987:0.313968:0.230987:0.000000 5:@0.322765:0.236420:0.338099:0.236420:0.338099:0.222704:0.322765:0.222704:0.000000 x:@0.346419:0.236975:0.353874:0.236975:0.353874:0.221539:0.346419:0.221539:0.000000 0:@0.353803:0.240147:0.359040:0.240147:0.359040:0.230987:0.353803:0.230987:0.000000 1:@0.362600:0.236420:0.377934:0.236420:0.377934:0.222704:0.362600:0.222704:0.000000 h f x:@0.381449:0.236975:0.418098:0.236975:0.418098:0.221539:0.381449:0.221539:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.418036:0.240147:0.423273:0.240147:0.423273:0.230987:0.418036:0.230987:0.000000 1:@0.426835:0.236420:0.442168:0.236420:0.442168:0.222704:0.426835:0.222704:0.000000 h:@0.445684:0.236975:0.455182:0.236975:0.455182:0.221539:0.445684:0.221539:0.000000 P:@0.664399:0.236975:0.673161:0.236975:0.673161:0.221539:0.664399:0.221539:0.000000 1:@0.673111:0.240147:0.678347:0.240147:0.678347:0.230987:0.673111:0.230987:0.000000 razonamiento similar se da en el caso 0. Cuando la función es derivable en el punto :@0.073089:0.255253:0.710566:0.255253:0.710566:0.239540:0.073089:0.239540:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 h:@0.346308:0.255253:0.355806:0.255253:0.355806:0.239818:0.346308:0.239818:0.000000 .:@0.359617:0.254698:0.374950:0.254698:0.374950:0.240983:0.359617:0.240983:0.000000 x:@0.073089:0.273531:0.080544:0.273531:0.080544:0.258096:0.073089:0.258096:0.000000 0:@0.080478:0.276709:0.085714:0.276709:0.085714:0.267548:0.080478:0.267548:0.000000 , se tiene: :@0.085667:0.273536:0.156480:0.273536:0.156480:0.257824:0.085667:0.257824:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 m:@0.236257:0.312742:0.250872:0.312742:0.250872:0.297306:0.236257:0.297306:0.000000 lím ( ) lím :@0.250799:0.312742:0.383828:0.312742:0.383828:0.297029:0.250799:0.297029:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.254756:0.312187:0.270090:0.312187:0.270090:0.298471:0.254756:0.298471:0.000000 m h:@0.300480:0.312742:0.329324:0.312742:0.329324:0.297306:0.300480:0.297306:0.000000:0.000000:0.000000 5:@0.337994:0.312187:0.353327:0.312187:0.353327:0.298471:0.337994:0.298471:0.000000 f x:@0.388273:0.305409:0.403384:0.305409:0.403384:0.291377:0.388273:0.291377:0.000000:0.000000:0.000000 (:@0.392356:0.305409:0.396791:0.305409:0.396791:0.291125:0.392356:0.291125:0.000000 0:@0.403200:0.308293:0.407961:0.308293:0.407961:0.299965:0.403200:0.299965:0.000000 1 2:@0.407854:0.304905:0.448617:0.304905:0.448617:0.292436:0.407854:0.292436:0.000000:0.000000:0.000000 h:@0.421793:0.305409:0.430428:0.305409:0.430428:0.291377:0.421793:0.291377:0.000000 ):@0.430428:0.305409:0.434862:0.305409:0.434862:0.291125:0.430428:0.291125:0.000000 f x:@0.448617:0.305409:0.463728:0.305409:0.463728:0.291377:0.448617:0.291377:0.000000:0.000000:0.000000 ( ):@0.452700:0.305409:0.472631:0.305409:0.472631:0.291125:0.452700:0.291125:0.000000:0.000000:0.000000 0:@0.463542:0.308293:0.468303:0.308293:0.468303:0.299965:0.463542:0.299965:0.000000 h:@0.426127:0.321799:0.434762:0.321799:0.434762:0.307767:0.426127:0.307767:0.000000 ( ).:@0.477085:0.312740:0.543172:0.312740:0.543172:0.297028:0.477085:0.297028:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.481043:0.312186:0.496376:0.312186:0.496376:0.298470:0.481043:0.298470:0.000000 df:@0.502798:0.305173:0.515299:0.305173:0.515299:0.291141:0.502798:0.291141:0.000000:0.000000 dx:@0.501443:0.321563:0.516637:0.321563:0.516637:0.307531:0.501443:0.307531:0.000000:0.000000 x:@0.522593:0.312740:0.530048:0.312740:0.530048:0.297305:0.522593:0.297305:0.000000 0:@0.529976:0.315913:0.535213:0.315913:0.535213:0.306752:0.529976:0.306752:0.000000 lím 0:@0.274804:0.320967:0.310909:0.320967:0.310909:0.309539:0.274804:0.309539:0.000000:0.000000:0.000000:0.000000:0.000000 →:@0.291230:0.320664:0.304443:0.320664:0.304443:0.308309:0.291230:0.308309:0.000000 lím 0:@0.355448:0.320967:0.391553:0.320967:0.391553:0.309539:0.355448:0.309539:0.000000:0.000000:0.000000:0.000000:0.000000 →:@0.371874:0.320664:0.385087:0.320664:0.385087:0.308309:0.371874:0.308309:0.000000 La recta tangente a la gráfica de la función en el punto ( , ( )) es el conjunto::@0.073089:0.352269:0.682263:0.352269:0.682263:0.336556:0.073089:0.336556:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.383067:0.352269:0.387761:0.352269:0.387761:0.336833:0.383067:0.336833:0.000000 P:@0.479558:0.352269:0.488320:0.352269:0.488320:0.336833:0.479558:0.336833:0.000000 0:@0.488253:0.355441:0.493490:0.355441:0.493490:0.346280:0.488253:0.346280:0.000000 5:@0.497400:0.351714:0.512733:0.351714:0.512733:0.337998:0.497400:0.337998:0.000000 x f x:@0.521385:0.352269:0.557908:0.352269:0.557908:0.336833:0.521385:0.336833:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.528771:0.355441:0.534008:0.355441:0.534008:0.346280:0.528771:0.346280:0.000000 0:@0.557846:0.355441:0.563083:0.355441:0.563083:0.346280:0.557846:0.346280:0.000000 L ( , ) ( ) ( )( ):@0.238370:0.391640:0.533815:0.391634:0.533815:0.375921:0.238370:0.375928:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.250482:0.391086:0.265815:0.391086:0.265815:0.377370:0.250482:0.377370:0.000000 5:@0.269680:0.392223:0.273767:0.392223:0.273767:0.374569:0.269680:0.374569:0.000000 x y:@0.278479:0.391640:0.300402:0.391640:0.300402:0.376205:0.278479:0.376205:0.000000:0.000000:0.000000 [:@0.309084:0.391344:0.323720:0.391344:0.323720:0.377934:0.309084:0.377934:0.000000 R:@0.327590:0.390774:0.340880:0.390774:0.340880:0.377877:0.327590:0.377877:0.000000 2:@0.340781:0.385570:0.346018:0.385570:0.346018:0.376410:0.340781:0.376410:0.000000 Z:@0.349928:0.392217:0.353002:0.392217:0.353002:0.374562:0.349928:0.374562:0.000000 y:@0.356867:0.391634:0.364341:0.391634:0.364341:0.376199:0.356867:0.376199:0.000000 2:@0.368225:0.391079:0.383558:0.391079:0.383558:0.377364:0.368225:0.377364:0.000000 f x:@0.387423:0.391634:0.404303:0.391634:0.404303:0.376199:0.387423:0.376199:0.000000:0.000000:0.000000 0:@0.404216:0.394806:0.409453:0.394806:0.409453:0.385646:0.404216:0.385646:0.000000 5:@0.418148:0.391079:0.433482:0.391079:0.433482:0.377364:0.418148:0.377364:0.000000 df:@0.439916:0.384067:0.452416:0.384067:0.452416:0.370035:0.439916:0.370035:0.000000:0.000000 dx:@0.438560:0.400457:0.453755:0.400457:0.453755:0.386425:0.438560:0.386425:0.000000:0.000000 x x:@0.463651:0.391634:0.493278:0.391634:0.493278:0.376199:0.463651:0.376199:0.000000:0.000000:0.000000 0:@0.471044:0.394806:0.476281:0.394806:0.476281:0.385646:0.471044:0.385646:0.000000 2:@0.497162:0.391079:0.512496:0.391079:0.512496:0.377364:0.497162:0.377364:0.000000 x .:@0.516361:0.391634:0.541067:0.391634:0.541067:0.376199:0.516361:0.376199:0.000000:0.000000:0.000000 0:@0.523747:0.394806:0.528984:0.394806:0.528984:0.385646:0.523747:0.385646:0.000000 6:@0.533741:0.392217:0.537827:0.392217:0.537827:0.374562:0.533741:0.374562:0.000000 La ecuación cartesiana de la recta tangente a la gráfica de la función en el punto :@0.073099:0.431006:0.710395:0.431006:0.710395:0.415293:0.073099:0.415293:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.606450:0.431006:0.611143:0.431006:0.611143:0.415571:0.606450:0.415571:0.000000 P:@0.073118:0.449284:0.081880:0.449284:0.081880:0.433849:0.073118:0.433849:0.000000 0:@0.081785:0.452454:0.087021:0.452454:0.087021:0.443293:0.081785:0.443293:0.000000 ( , ( )) está definida como :@0.086974:0.449282:0.313379:0.449282:0.313379:0.433569:0.086974:0.433569:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.090932:0.448727:0.106265:0.448727:0.106265:0.435011:0.090932:0.435011:0.000000 x f x:@0.114917:0.449282:0.151440:0.449282:0.151440:0.433846:0.114917:0.433846:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.122302:0.452454:0.127539:0.452454:0.127539:0.443293:0.122302:0.443293:0.000000 0:@0.151379:0.452454:0.156616:0.452454:0.156616:0.443293:0.151379:0.443293:0.000000 ( , ) tales que ( ) ( )( ):@0.226644:0.488653:0.549533:0.488647:0.549533:0.472934:0.226644:0.472941:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.231449:0.488653:0.253372:0.488653:0.253372:0.473218:0.231449:0.473218:0.000000:0.000000:0.000000 [:@0.262060:0.488357:0.276696:0.488357:0.276696:0.474947:0.262060:0.474947:0.000000 R:@0.280566:0.487787:0.293856:0.487787:0.293856:0.474890:0.280566:0.474890:0.000000 2:@0.293757:0.482583:0.298994:0.482583:0.298994:0.473423:0.293757:0.473423:0.000000 y:@0.372575:0.488647:0.380049:0.488647:0.380049:0.473212:0.372575:0.473212:0.000000 2:@0.383933:0.488092:0.399266:0.488092:0.399266:0.474377:0.383933:0.474377:0.000000 f x:@0.403131:0.488647:0.420011:0.488647:0.420011:0.473212:0.403131:0.473212:0.000000:0.000000:0.000000 0:@0.419932:0.491820:0.425169:0.491820:0.425169:0.482659:0.419932:0.482659:0.000000 5:@0.433867:0.488092:0.449200:0.488092:0.449200:0.474377:0.433867:0.474377:0.000000 df:@0.455634:0.481080:0.468134:0.481080:0.468134:0.467048:0.455634:0.467048:0.000000:0.000000 dx:@0.454278:0.497470:0.469473:0.497470:0.469473:0.483438:0.454278:0.483438:0.000000:0.000000 x x:@0.479368:0.488647:0.508995:0.488647:0.508995:0.473212:0.479368:0.473212:0.000000:0.000000:0.000000 0:@0.486761:0.491820:0.491997:0.491820:0.491997:0.482659:0.486761:0.482659:0.000000 2:@0.512879:0.488092:0.528212:0.488092:0.528212:0.474377:0.512879:0.474377:0.000000 x .:@0.532078:0.488647:0.552791:0.488647:0.552791:0.473212:0.532078:0.473212:0.000000:0.000000:0.000000 0:@0.539464:0.491820:0.544701:0.491820:0.544701:0.482659:0.539464:0.482659:0.000000 Problema 2:@0.073098:0.528019:0.158001:0.528019:0.158001:0.511349:0.073098:0.511349:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Sea una función real definida en el intervalo cerrado , de . Suponemos :@0.157845:0.528019:0.710384:0.528019:0.710384:0.512306:0.157845:0.512306:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.160864:0.528019:0.164674:0.528019:0.164674:0.511072:0.160864:0.511072:0.000000 f:@0.192690:0.528019:0.197383:0.528019:0.197383:0.512584:0.192690:0.512584:0.000000 F:@0.542115:0.528601:0.546201:0.528601:0.546201:0.510947:0.542115:0.510947:0.000000 a b:@0.545980:0.528019:0.570683:0.528019:0.570683:0.512584:0.545980:0.512584:0.000000:0.000000:0.000000 G:@0.570535:0.528601:0.574622:0.528601:0.574622:0.510947:0.570535:0.510947:0.000000 R:@0.599619:0.527159:0.612909:0.527159:0.612909:0.514262:0.599619:0.514262:0.000000 que ( ) 0, :@0.073079:0.546297:0.174190:0.546297:0.174190:0.530585:0.073079:0.530585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.105458:0.546297:0.122337:0.546297:0.122337:0.530862:0.105458:0.530862:0.000000:0.000000:0.000000 $:@0.131007:0.545742:0.146340:0.545742:0.146340:0.532027:0.131007:0.532027:0.000000 ;:@0.174080:0.545742:0.186358:0.545742:0.186358:0.532027:0.174080:0.532027:0.000000 x:@0.186266:0.546297:0.193720:0.546297:0.193720:0.530862:0.186266:0.530862:0.000000 , .:@0.193647:0.546297:0.252764:0.546295:0.252764:0.530582:0.193647:0.530585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [:@0.197628:0.546005:0.212264:0.546005:0.212264:0.532595:0.197628:0.532595:0.000000 F:@0.216134:0.546877:0.220220:0.546877:0.220220:0.529223:0.216134:0.529223:0.000000 a b:@0.220128:0.546295:0.245641:0.546295:0.245641:0.530859:0.220128:0.530859:0.000000:0.000000:0.000000 G:@0.245567:0.546877:0.249653:0.546877:0.249653:0.529223:0.245567:0.529223:0.000000 Denotamos con la región comprendida por el eje y la gráfica de la función . Esto es::@0.073073:0.585666:0.702006:0.585666:0.702006:0.569954:0.073073:0.569954:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 S:@0.194266:0.585666:0.202328:0.585666:0.202328:0.570231:0.194266:0.570231:0.000000 x:@0.447016:0.585666:0.454471:0.585666:0.454471:0.570231:0.447016:0.570231:0.000000 f:@0.636979:0.585666:0.641673:0.585666:0.641673:0.570231:0.636979:0.570231:0.000000 S:@0.242475:0.625038:0.250537:0.625038:0.250537:0.609603:0.242475:0.609603:0.000000 {( , ) 0 ( ), }.:@0.250463:0.625038:0.536939:0.625026:0.536939:0.609313:0.250463:0.609326:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.254421:0.624484:0.269754:0.624484:0.269754:0.610768:0.254421:0.610768:0.000000 x y:@0.283210:0.625038:0.305133:0.625038:0.305133:0.609603:0.283210:0.609603:0.000000:0.000000:0.000000 [:@0.313863:0.624736:0.328499:0.624736:0.328499:0.611326:0.313863:0.611326:0.000000 R:@0.332369:0.624166:0.345659:0.624166:0.345659:0.611268:0.332369:0.611268:0.000000 2:@0.345559:0.618962:0.350795:0.618962:0.350795:0.609801:0.345559:0.609801:0.000000 Z:@0.354707:0.625608:0.357781:0.625608:0.357781:0.607954:0.354707:0.607954:0.000000 # #:@0.374495:0.624471:0.420384:0.624471:0.420384:0.610755:0.374495:0.610755:0.000000:0.000000:0.000000 y:@0.393694:0.625026:0.401167:0.625026:0.401167:0.609590:0.393694:0.609590:0.000000 f x a:@0.424250:0.625026:0.466126:0.625026:0.466126:0.609590:0.424250:0.609590:0.000000:0.000000:0.000000:0.000000:0.000000 # #:@0.470010:0.624471:0.515881:0.624471:0.515881:0.610755:0.470010:0.610755:0.000000:0.000000:0.000000 x:@0.489209:0.625026:0.496664:0.625026:0.496664:0.609590:0.489209:0.609590:0.000000 b:@0.519746:0.625026:0.529005:0.625026:0.529005:0.609590:0.519746:0.609590:0.000000 En la Figura 1.2. se muestran regiones de dos tipos de funciones. La primera :@0.073113:0.664397:0.662806:0.664397:0.662806:0.648685:0.073113:0.648685:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 presenta discontinuidades en la función , y la segunda, en el hecho de que la :@0.073113:0.682676:0.662955:0.682676:0.662955:0.666963:0.073113:0.666963:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.378195:0.682676:0.382888:0.682676:0.382888:0.667240:0.378195:0.667240:0.000000 función es continua.:@0.073113:0.700954:0.228875:0.700954:0.228875:0.685241:0.073113:0.685241:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.132679:0.700954:0.137372:0.700954:0.137372:0.685519:0.132679:0.685519:0.000000 1.1 Integral indefinida o primitiva de una función:@0.073113:0.740326:0.457552:0.740326:0.457552:0.723379:0.073113:0.723379:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En temas anteriores se calcularon las derivadas de algunas funciones de los tipos :@0.073113:0.779698:0.663336:0.779698:0.663336:0.763985:0.073113:0.763985:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que presentamos a continuación.:@0.073113:0.797976:0.316101:0.797976:0.316101:0.782263:0.073113:0.782263:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1. Función cuadrática:@0.073089:0.837333:0.247367:0.837333:0.247367:0.820663:0.073089:0.820663:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Considera ( ) :@0.103214:0.876704:0.224150:0.876704:0.224150:0.860992:0.103214:0.860992:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.179623:0.876704:0.196502:0.876704:0.196502:0.861269:0.179623:0.861269:0.000000:0.000000:0.000000 5:@0.204878:0.876150:0.220211:0.876150:0.220211:0.862434:0.204878:0.862434:0.000000 ax:@0.223764:0.876704:0.240404:0.876704:0.240404:0.861269:0.223764:0.861269:0.000000:0.000000 2:@0.240351:0.870634:0.245588:0.870634:0.245588:0.861474:0.240351:0.861474:0.000000 , :@0.245540:0.876699:0.321765:0.876699:0.321765:0.860987:0.245540:0.860987:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.249185:0.876145:0.264518:0.876145:0.264518:0.862429:0.249185:0.862429:0.000000 bx:@0.268071:0.876699:0.284545:0.876699:0.284545:0.861264:0.268071:0.861264:0.000000:0.000000 1 ;:@0.288135:0.876145:0.333656:0.876145:0.333656:0.862429:0.288135:0.862429:0.000000:0.000000:0.000000 c:@0.307020:0.876699:0.314678:0.876699:0.314678:0.861264:0.307020:0.861264:0.000000 x:@0.333564:0.876699:0.341019:0.876699:0.341019:0.861264:0.333564:0.861264:0.000000 , con , , constantes reales 0. La :@0.340945:0.876699:0.653420:0.876699:0.653420:0.860987:0.340945:0.860987:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [:@0.344631:0.876409:0.359268:0.876409:0.359268:0.863000:0.344631:0.863000:0.000000 R:@0.362825:0.875840:0.376115:0.875840:0.376115:0.862942:0.362825:0.862942:0.000000 a b c:@0.414070:0.876699:0.453646:0.876699:0.453646:0.861264:0.414070:0.861264:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.585460:0.876699:0.594719:0.876699:0.594719:0.861264:0.585460:0.861264:0.000000 ±:@0.598290:0.877240:0.613623:0.877240:0.613623:0.862068:0.598290:0.862068:0.000000 derivada de esta función está definida como:@0.103172:0.894978:0.425006:0.894978:0.425006:0.879265:0.103172:0.879265:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 df:@0.282460:0.926778:0.294960:0.926778:0.294960:0.912746:0.282460:0.912746:0.000000:0.000000 dx:@0.281105:0.943168:0.296299:0.943168:0.296299:0.929136:0.281105:0.929136:0.000000:0.000000 ( ) 2 , :@0.297450:0.934346:0.414410:0.934346:0.414410:0.918633:0.297450:0.918633:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.306194:0.934346:0.313649:0.934346:0.313649:0.918910:0.306194:0.918910:0.000000 5:@0.322319:0.933791:0.337652:0.933791:0.337652:0.920075:0.322319:0.920075:0.000000 ax:@0.350408:0.934346:0.367048:0.934346:0.367048:0.918910:0.350408:0.918910:0.000000:0.000000 1:@0.370951:0.933791:0.386284:0.933791:0.386284:0.920075:0.370951:0.920075:0.000000 b:@0.390149:0.934346:0.399408:0.934346:0.399408:0.918910:0.390149:0.918910:0.000000 ;:@0.414300:0.933791:0.426577:0.933791:0.426577:0.920075:0.414300:0.920075:0.000000 x:@0.426485:0.934346:0.433940:0.934346:0.433940:0.918910:0.426485:0.918910:0.000000 .:@0.433867:0.934346:0.472749:0.934346:0.472749:0.918633:0.433867:0.918633:0.000000:0.000000:0.000000:0.000000:0.000000 [:@0.437842:0.934056:0.452479:0.934056:0.452479:0.920646:0.437842:0.920646:0.000000 R:@0.456348:0.933486:0.469638:0.933486:0.469638:0.920588:0.456348:0.920588:0.000000 DBA 1 (Ev. 2) Propone y justifica diferentes estrategias para resolver problemas con números naturales en contextos escolares y extraescolares.:@0.083241:0.971338:0.828344:0.971338:0.828344:0.959911:0.083241:0.959911:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En la actualidad, el :@0.749032:0.205621:0.871278:0.205621:0.871278:0.191337:0.749032:0.191337:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cálculo diferencial :@0.749032:0.220750:0.868331:0.220750:0.868331:0.206466:0.749032:0.206466:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 e integral no solo :@0.749032:0.235879:0.863980:0.235879:0.863980:0.221595:0.749032:0.221595:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 constituyen un :@0.749032:0.251008:0.849797:0.251008:0.849797:0.236724:0.749032:0.236724:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 instrumento de cálculo :@0.749032:0.266137:0.902191:0.266137:0.902191:0.251853:0.749032:0.251853:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en las ciencias y la :@0.749032:0.281266:0.868730:0.281266:0.868730:0.266982:0.749032:0.266982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 técnica, sino también :@0.749032:0.296395:0.890326:0.296395:0.890326:0.282111:0.749032:0.282111:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 un conjunto de ideas y :@0.749032:0.311524:0.900166:0.311524:0.900166:0.297240:0.749032:0.297240:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 problemas que han sido :@0.749032:0.326653:0.909455:0.326653:0.909455:0.312368:0.749032:0.312368:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 objeto de estudio como :@0.749032:0.341782:0.908683:0.341782:0.908683:0.327497:0.749032:0.327497:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 parte del pensamiento. :@0.749032:0.356911:0.902274:0.356911:0.902274:0.342626:0.749032:0.342626:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En ese sentido, forman :@0.749032:0.372040:0.898966:0.372040:0.898966:0.357755:0.749032:0.357755:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 parte del lenguaje que :@0.749032:0.387168:0.898906:0.387168:0.898906:0.372884:0.749032:0.372884:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 facilita la expresión de :@0.749032:0.402297:0.894426:0.402297:0.894426:0.388013:0.749032:0.388013:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 leyes y principios de :@0.749032:0.417426:0.882500:0.417426:0.882500:0.403142:0.749032:0.403142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 manera matemática. Son :@0.749032:0.432555:0.914165:0.432555:0.914165:0.418271:0.749032:0.418271:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la base para el desarrollo :@0.749032:0.447684:0.910618:0.447684:0.910618:0.433400:0.749032:0.433400:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de otras áreas de la :@0.749032:0.462813:0.875634:0.462813:0.875634:0.448529:0.749032:0.448529:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática, la física, la :@0.749032:0.477942:0.898375:0.477942:0.898375:0.463658:0.749032:0.463658:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 química, la biología, las :@0.749032:0.493071:0.900283:0.493071:0.900283:0.478787:0.749032:0.478787:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ciencias económicas y :@0.749032:0.508200:0.896407:0.508200:0.896407:0.493916:0.749032:0.493916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sociales, así como las :@0.749032:0.523329:0.886797:0.523329:0.886797:0.509045:0.749032:0.509045:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 distintas ramas de la :@0.749032:0.538458:0.882937:0.538458:0.882937:0.524174:0.749032:0.524174:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ingeniería e industria.:@0.749032:0.553587:0.886656:0.553587:0.886656:0.539303:0.749032:0.539303:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Interdisciplinaria :@0.740294:0.141152:0.863703:0.141152:0.863703:0.125745:0.740294:0.125745:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática :@0.740294:0.156281:0.825787:0.156281:0.825787:0.141996:0.740294:0.141996:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y otras disciplinas:@0.740294:0.171410:0.855523:0.171410:0.855523:0.157125:0.740294:0.157125:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Figura 1.2.:@0.873047:0.940308:0.926702:0.940308:0.926702:0.928881:0.873047:0.928881:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 S:@0.798703:0.872876:0.805300:0.872876:0.805300:0.860247:0.798703:0.860247:0.000000 0:@0.744487:0.906810:0.751020:0.906810:0.751020:0.895383:0.744487:0.895383:0.000000 x:@0.908588:0.908860:0.915408:0.908860:0.915408:0.895290:0.908588:0.895290:0.000000 y:@0.787115:0.836828:0.793230:0.836828:0.793230:0.824199:0.787115:0.824199:0.000000 :@0.793230:0.836828:0.812372:0.836828:0.812372:0.823973:0.793230:0.823973:0.000000:0.000000:0.000000 5:@0.796528:0.836374:0.809073:0.836374:0.809073:0.825153:0.796528:0.825153:0.000000 f x:@0.812372:0.836828:0.826589:0.836828:0.826589:0.824199:0.812372:0.824199:0.000000:0.000000:0.000000 ( ):@0.816212:0.836828:0.831016:0.836828:0.831016:0.823190:0.816212:0.823190:0.000000:0.000000:0.000000 a:@0.697294:0.907011:0.703372:0.907011:0.703372:0.895583:0.697294:0.895583:0.000000 b:@0.887793:0.907011:0.895182:0.907011:0.895182:0.895583:0.887793:0.895583:0.000000 x:@0.778567:0.907011:0.784310:0.907011:0.784310:0.895583:0.778567:0.895583:0.000000 1:@0.784310:0.909318:0.788119:0.909318:0.788119:0.902656:0.784310:0.902656:0.000000 x:@0.837487:0.907011:0.843230:0.907011:0.843230:0.895583:0.837487:0.895583:0.000000 2:@0.843230:0.909318:0.847039:0.909318:0.847039:0.902656:0.843230:0.902656:0.000000 y:@0.742380:0.809995:0.749375:0.809995:0.749375:0.796425:0.742380:0.796425:0.000000 x:@0.908302:0.773847:0.915122:0.773847:0.915122:0.760277:0.908302:0.760277:0.000000 y:@0.789660:0.718876:0.795775:0.718876:0.795775:0.706247:0.789660:0.706247:0.000000 :@0.795775:0.718876:0.814917:0.718876:0.814917:0.706020:0.795775:0.706020:0.000000:0.000000:0.000000 5:@0.799073:0.718422:0.811618:0.718422:0.811618:0.707200:0.799073:0.707200:0.000000 f x:@0.814917:0.718876:0.829134:0.718876:0.829134:0.706247:0.814917:0.706247:0.000000:0.000000:0.000000 ( ):@0.818757:0.718876:0.833562:0.718876:0.833562:0.705237:0.818757:0.705237:0.000000:0.000000:0.000000 a:@0.695251:0.771497:0.701329:0.771497:0.701329:0.760069:0.695251:0.760069:0.000000 0:@0.717675:0.770256:0.724207:0.770256:0.724207:0.758829:0.717675:0.758829:0.000000 x:@0.745694:0.771497:0.751437:0.771497:0.751437:0.760069:0.745694:0.760069:0.000000 1:@0.751435:0.773804:0.755244:0.773804:0.755244:0.767142:0.751435:0.767142:0.000000 x:@0.777486:0.771497:0.783229:0.771497:0.783229:0.760069:0.777486:0.760069:0.000000 2:@0.783229:0.773804:0.787038:0.773804:0.787038:0.767142:0.783229:0.767142:0.000000 x:@0.835330:0.771497:0.841073:0.771497:0.841073:0.760069:0.835330:0.760069:0.000000 3:@0.841073:0.773804:0.844882:0.773804:0.844882:0.767142:0.841073:0.767142:0.000000 b:@0.883266:0.771497:0.890656:0.771497:0.890656:0.760069:0.883266:0.760069:0.000000 y:@0.715322:0.673719:0.722317:0.673719:0.722317:0.660149:0.715322:0.660149:0.000000 que ( ) se aproxima a conforme :@0.144240:0.136836:0.423957:0.136836:0.423957:0.121124:0.144240:0.121124:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.178019:0.136149:0.186804:0.136149:0.186804:0.123057:0.178019:0.123057:0.000000 h:@0.191836:0.136836:0.201334:0.136836:0.201334:0.121401:0.191836:0.121401:0.000000 u:@0.319392:0.136149:0.328177:0.136149:0.328177:0.123057:0.319392:0.123057:0.000000 h:@0.410372:0.136836:0.419870:0.136836:0.419870:0.121401:0.410372:0.121401:0.000000 →:@0.424803:0.136420:0.442971:0.136420:0.442971:0.119432:0.424803:0.119432:0.000000 0 . En la Figura 1.1 se muestra una :@0.443045:0.136836:0.710360:0.136836:0.710360:0.121124:0.443045:0.121124:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.456998:0.130449:0.465937:0.130449:0.465937:0.122453:0.456998:0.122453:0.000000 porción de la gráfica de la función , una porción de la recta que pasa por :@0.144245:0.155115:0.710372:0.155117:0.710372:0.139404:0.144245:0.139402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.394381:0.155115:0.399075:0.155115:0.399075:0.139679:0.394381:0.139679:0.000000 L:@0.577386:0.155115:0.585007:0.155115:0.585007:0.139679:0.577386:0.139679:0.000000 h:@0.584955:0.158289:0.590492:0.158289:0.590492:0.149291:0.584955:0.149291:0.000000 P:@0.693514:0.155117:0.702276:0.155117:0.702276:0.139682:0.693514:0.139682:0.000000 0:@0.701103:0.158289:0.706340:0.158289:0.706340:0.149129:0.701103:0.149129:0.000000 y , el triángulo rectángulo :@0.144238:0.173395:0.351596:0.173398:0.351596:0.157685:0.144238:0.157683:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 P:@0.157031:0.173395:0.165793:0.173395:0.165793:0.157960:0.157031:0.157960:0.000000 1:@0.164729:0.176570:0.169966:0.176570:0.169966:0.167410:0.164729:0.167410:0.000000 P RP:@0.352255:0.173398:0.384058:0.173398:0.384058:0.157963:0.352255:0.157963:0.000000:0.000000:0.000000:0.000000 0:@0.361054:0.176570:0.366291:0.176570:0.366291:0.167410:0.361054:0.167410:0.000000 1:@0.384083:0.176570:0.389320:0.176570:0.389320:0.167410:0.384083:0.167410:0.000000 , y la recta tangente a la gráfica de en el :@0.389343:0.173398:0.710365:0.173398:0.710365:0.157685:0.389343:0.157685:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 L:@0.540687:0.173398:0.548308:0.173398:0.548308:0.157963:0.540687:0.157963:0.000000 f:@0.660555:0.173398:0.665249:0.173398:0.665249:0.157963:0.660555:0.157963:0.000000 punto ( , ( )).:@0.144250:0.191676:0.289249:0.191679:0.289249:0.175966:0.144250:0.175963:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 P:@0.192974:0.191676:0.201736:0.191676:0.201736:0.176241:0.192974:0.176241:0.000000 0:@0.201654:0.194851:0.206891:0.194851:0.206891:0.185690:0.201654:0.185690:0.000000 5:@0.210802:0.191124:0.226136:0.191124:0.226136:0.177408:0.210802:0.177408:0.000000 x f x:@0.234787:0.191679:0.271310:0.191679:0.271310:0.176243:0.234787:0.176243:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.242172:0.194851:0.247409:0.194851:0.247409:0.185690:0.242172:0.185690:0.000000 0:@0.271248:0.194851:0.276485:0.194851:0.276485:0.185690:0.271248:0.185690:0.000000 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