86:@0.058194:0.966765:0.075166:0.966765:0.075166:0.950778:0.058194:0.950778:0.000000:0.000000 Problemas relativos a la parábola:@0.404236:0.097328:0.668945:0.097328:0.668945:0.079645:0.404236:0.079645:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Problema 1:@0.404236:0.114498:0.491932:0.114498:0.491932:0.097552:0.404236:0.097552:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Consideremos la parábola:@0.491932:0.114035:0.682808:0.114035:0.682808:0.097538:0.491932:0.097538:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.404236:0.138972:0.417651:0.138972:0.417651:0.119973:0.404236:0.119973:0.000000:0.000000 (0, ) = {( , ) | = 4 }, donde > 0. Se tiene la siguiente :@0.417643:0.131386:0.868106:0.131386:0.868106:0.114888:0.417643:0.114888:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.424868:0.129362:0.431283:0.129362:0.431283:0.113544:0.424868:0.113544:0.000000 c:@0.438084:0.131502:0.445269:0.131502:0.445269:0.115033:0.438084:0.115033:0.000000 x y:@0.482124:0.131502:0.504183:0.131502:0.504183:0.115033:0.482124:0.115033:0.000000:0.000000:0.000000 :@0.514274:0.130867:0.528927:0.130867:0.528927:0.117494:0.514274:0.117494:0.000000 :@0.533069:0.130299:0.546374:0.130299:0.546374:0.117437:0.533069:0.117437:0.000000 2:@0.546374:0.125034:0.551327:0.125034:0.551327:0.115416:0.546374:0.115416:0.000000 y:@0.564619:0.131502:0.572305:0.131502:0.572305:0.115033:0.564619:0.115033:0.000000 2:@0.572306:0.125034:0.577259:0.125034:0.577259:0.115416:0.572306:0.115416:0.000000 cx:@0.604846:0.131502:0.619507:0.131502:0.619507:0.115033:0.604846:0.115033:0.000000:0.000000 c:@0.683294:0.131502:0.690480:0.131502:0.690480:0.115033:0.683294:0.115033:0.000000 equivalencia: ( , ) (0, ) = 4 .:@0.404237:0.148737:0.684665:0.148737:0.684665:0.132239:0.404237:0.132239:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.504956:0.148852:0.527014:0.148852:0.527014:0.132384:0.504956:0.132384:0.000000:0.000000:0.000000 :@0.537106:0.148218:0.551759:0.148218:0.551759:0.134845:0.537106:0.134845:0.000000 :@0.555901:0.156323:0.569316:0.156323:0.569316:0.137324:0.555901:0.137324:0.000000:0.000000 :@0.576532:0.146713:0.582947:0.146713:0.582947:0.130894:0.576532:0.130894:0.000000 c:@0.589748:0.148852:0.596934:0.148852:0.596934:0.132384:0.589748:0.132384:0.000000 y:@0.627022:0.148852:0.634709:0.148852:0.634709:0.132384:0.627022:0.132384:0.000000 2:@0.634709:0.142385:0.639662:0.142385:0.639662:0.132767:0.634709:0.132767:0.000000 cx:@0.667249:0.148852:0.681910:0.148852:0.681910:0.132384:0.667249:0.132384:0.000000:0.000000 Sea ( , ) (0, ), es decir, = 4:@0.404240:0.179665:0.659940:0.179666:0.659940:0.163169:0.404240:0.163167:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.438455:0.179780:0.466023:0.179782:0.466023:0.163313:0.438455:0.163311:0.000000:0.000000:0.000000 0:@0.445925:0.182944:0.450878:0.182944:0.450878:0.173326:0.445925:0.173326:0.000000 0:@0.466022:0.182944:0.470976:0.182944:0.470976:0.173326:0.466022:0.173326:0.000000 :@0.481071:0.179148:0.495724:0.179148:0.495724:0.165775:0.481071:0.165775:0.000000 :@0.499866:0.187253:0.513281:0.187253:0.513281:0.168254:0.499866:0.168254:0.000000:0.000000 :@0.520497:0.177642:0.526912:0.177642:0.526912:0.161824:0.520497:0.161824:0.000000 c:@0.533713:0.179782:0.540899:0.179782:0.540899:0.163313:0.533713:0.163313:0.000000 y:@0.613471:0.179782:0.621158:0.179782:0.621158:0.163313:0.613471:0.163313:0.000000 0 :@0.621155:0.182944:0.628523:0.182944:0.628523:0.173326:0.621155:0.173326:0.000000:0.000000 2:@0.627399:0.173318:0.632353:0.173318:0.632353:0.163700:0.627399:0.163700:0.000000 cx:@0.659940:0.179782:0.674601:0.179782:0.674601:0.163313:0.659940:0.163313:0.000000:0.000000 0:@0.674600:0.182944:0.679553:0.182944:0.679553:0.173326:0.674600:0.173326:0.000000 . Determinemos una recta L :@0.679552:0.179666:0.880775:0.179666:0.880775:0.163169:0.679552:0.163169:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que pasa por ( , ) y es tangente a (0, ) en dicho punto.:@0.404232:0.197017:0.824082:0.197017:0.824082:0.180520:0.404232:0.180520:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.507106:0.197133:0.534683:0.197133:0.534683:0.180664:0.507106:0.180664:0.000000:0.000000:0.000000 0:@0.514584:0.200295:0.519537:0.200295:0.519537:0.190677:0.514584:0.190677:0.000000 0:@0.534682:0.200295:0.539635:0.200295:0.539635:0.190677:0.534682:0.190677:0.000000 :@0.659734:0.204604:0.673149:0.204604:0.673149:0.185605:0.659734:0.185605:0.000000:0.000000 :@0.680365:0.194993:0.686780:0.194993:0.686780:0.179175:0.680365:0.179175:0.000000 c:@0.693581:0.197133:0.700767:0.197133:0.700767:0.180664:0.693581:0.180664:0.000000 Si = = 0, esto es, ( , ) es el vértice de la parábola, el eje es la :@0.404236:0.227945:0.895975:0.227947:0.895975:0.211449:0.404236:0.211447:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.420727:0.228061:0.428202:0.228061:0.428202:0.211592:0.420727:0.211592:0.000000 0:@0.428195:0.231225:0.433149:0.231225:0.433149:0.221607:0.428195:0.221607:0.000000 y:@0.453881:0.228062:0.461568:0.228062:0.461568:0.211594:0.453881:0.211594:0.000000 0:@0.461556:0.231225:0.466509:0.231225:0.466509:0.221607:0.461556:0.221607:0.000000 x y:@0.565013:0.228062:0.593095:0.228062:0.593095:0.211594:0.565013:0.211594:0.000000:0.000000:0.000000 0:@0.572457:0.231225:0.577410:0.231225:0.577410:0.221607:0.572457:0.221607:0.000000 0:@0.593089:0.231225:0.598042:0.231225:0.598042:0.221607:0.593089:0.221607:0.000000 Y:@0.846311:0.228062:0.856444:0.228062:0.856444:0.211594:0.846311:0.211594:0.000000 recta tangente a (0, ) en (0, 0) cuya ecuación vectorial está definida :@0.404233:0.245298:0.895931:0.245298:0.895931:0.228800:0.404233:0.228800:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.522326:0.252884:0.535741:0.252884:0.535741:0.233885:0.522326:0.233885:0.000000:0.000000 :@0.542957:0.243273:0.549372:0.243273:0.549372:0.227455:0.542957:0.227455:0.000000 c:@0.555691:0.245413:0.562877:0.245413:0.562877:0.228944:0.555691:0.228944:0.000000 como ( ) = (0, 1), . En la Figura 2.26. se muestran la parábola :@0.404247:0.262648:0.895966:0.262648:0.895966:0.246151:0.404247:0.246151:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y t:@0.449809:0.262764:0.470520:0.262764:0.470520:0.246295:0.449809:0.246295:0.000000:0.000000:0.000000 :@0.451717:0.263140:0.458132:0.263140:0.458132:0.247322:0.451717:0.247322:0.000000 t:@0.494813:0.262764:0.500477:0.262764:0.500477:0.246295:0.494813:0.246295:0.000000 ∀:@0.546171:0.270235:0.557743:0.270235:0.557743:0.251236:0.546171:0.251236:0.000000 t:@0.557752:0.262764:0.563416:0.262764:0.563416:0.246295:0.557752:0.246295:0.000000 :@0.567183:0.262130:0.581836:0.262130:0.581836:0.248756:0.567183:0.248756:0.000000 :@0.585604:0.261561:0.598909:0.261561:0.598909:0.248699:0.585604:0.248699:0.000000 :@0.404236:0.287585:0.417651:0.287585:0.417651:0.268586:0.404236:0.268586:0.000000:0.000000 (0, ) y la recta tangente en (0, 0).:@0.417643:0.279999:0.650924:0.279999:0.650924:0.263501:0.417643:0.263501:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.424868:0.277975:0.431283:0.277975:0.431283:0.262157:0.424868:0.262157:0.000000 c:@0.438084:0.280115:0.445269:0.280115:0.445269:0.263646:0.438084:0.263646:0.000000 Supongamos que ≠ 0, ≠ 0 y la recta tangente a la parábola:@0.404235:0.310927:0.862268:0.310929:0.862268:0.294431:0.404235:0.294429:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.533136:0.311042:0.540611:0.311042:0.540611:0.294574:0.533136:0.294574:0.000000 0:@0.540612:0.314207:0.545566:0.314207:0.545566:0.304588:0.540612:0.304588:0.000000 y:@0.580612:0.311044:0.588299:0.311044:0.588299:0.294576:0.580612:0.294576:0.000000 0:@0.588298:0.314207:0.593251:0.314207:0.593251:0.304588:0.588298:0.304588:0.000000 L:@0.636944:0.311044:0.644785:0.311044:0.644785:0.294576:0.636944:0.294576:0.000000 :@0.404236:0.335866:0.417651:0.335866:0.417651:0.316867:0.404236:0.316867:0.000000:0.000000 (0, ) en el punto ( , ) (0, ), la ecuación cartesiana es::@0.417643:0.328279:0.839313:0.328279:0.839313:0.311782:0.417643:0.311782:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.424868:0.326255:0.431283:0.326255:0.431283:0.310437:0.424868:0.310437:0.000000 c:@0.438084:0.328395:0.445269:0.328395:0.445269:0.311926:0.438084:0.311926:0.000000 x y:@0.547258:0.328395:0.574830:0.328395:0.574830:0.311926:0.547258:0.311926:0.000000:0.000000:0.000000 0:@0.554733:0.331557:0.559686:0.331557:0.559686:0.321939:0.554733:0.321939:0.000000 0:@0.574831:0.331557:0.579784:0.331557:0.579784:0.321939:0.574831:0.321939:0.000000 :@0.589878:0.327761:0.604530:0.327761:0.604530:0.314387:0.589878:0.314387:0.000000 :@0.608672:0.335866:0.622088:0.335866:0.622088:0.316867:0.608672:0.316867:0.000000:0.000000 :@0.629304:0.326255:0.635719:0.326255:0.635719:0.310437:0.629304:0.310437:0.000000 c:@0.642520:0.328395:0.649706:0.328395:0.649706:0.311926:0.642520:0.311926:0.000000 ( , ) tal que – = ( – ),:@0.518548:0.358455:0.777468:0.358455:0.777468:0.341957:0.518548:0.341958:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.524501:0.358571:0.546560:0.358571:0.546560:0.342102:0.524501:0.342102:0.000000:0.000000:0.000000 :@0.556644:0.357936:0.571297:0.357936:0.571297:0.344563:0.556644:0.344563:0.000000 :@0.575439:0.357368:0.588744:0.357368:0.588744:0.344506:0.575439:0.344506:0.000000 2:@0.588744:0.352103:0.593697:0.352103:0.593697:0.342485:0.588744:0.342485:0.000000 y y:@0.651125:0.358570:0.685283:0.358570:0.685283:0.342102:0.651125:0.342102:0.000000:0.000000:0.000000 0:@0.685281:0.361733:0.690234:0.361733:0.690234:0.352114:0.685281:0.352114:0.000000 m x x:@0.709327:0.358570:0.763809:0.358570:0.763809:0.342102:0.709327:0.342102:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.763808:0.361733:0.768761:0.361733:0.768761:0.352114:0.763808:0.352114:0.000000 donde ≠ 0 es la pendiente.:@0.404242:0.390886:0.611534:0.390886:0.611534:0.374388:0.404242:0.374388:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 m:@0.455276:0.391002:0.470071:0.391002:0.470071:0.374533:0.455276:0.374533:0.000000 De la ecuación cartesiana de se obtiene = + ( – ). Reempla-:@0.404242:0.423318:0.891776:0.423331:0.891776:0.406834:0.404242:0.406820:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 L:@0.608163:0.423433:0.616004:0.423433:0.616004:0.406964:0.608163:0.406964:0.000000 y y:@0.696321:0.423433:0.729938:0.423433:0.729938:0.406964:0.696321:0.406964:0.000000:0.000000:0.000000 0:@0.729873:0.426609:0.734826:0.426609:0.734826:0.416991:0.729873:0.416991:0.000000 m x x:@0.753071:0.423447:0.806705:0.423447:0.806705:0.406978:0.753071:0.406978:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.806681:0.426609:0.811634:0.426609:0.811634:0.416991:0.806681:0.416991:0.000000 zando en la ecuación cartesiana de la parábola, resulta::@0.404232:0.440682:0.791077:0.440682:0.791077:0.424184:0.404232:0.424184:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [ + ( – )] = 4 (:@0.464706:0.473114:0.650825:0.473120:0.650825:0.456623:0.464706:0.456616:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.470562:0.473229:0.478249:0.473229:0.478249:0.456760:0.470562:0.456760:0.000000 0:@0.478247:0.476398:0.483200:0.476398:0.483200:0.466780:0.478247:0.466780:0.000000 m x x:@0.502292:0.473236:0.556774:0.473236:0.556774:0.456767:0.502292:0.456767:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.556772:0.476398:0.561725:0.476398:0.561725:0.466780:0.556772:0.466780:0.000000 2:@0.573534:0.466769:0.578487:0.466769:0.578487:0.457151:0.573534:0.457151:0.000000 cx:@0.606076:0.473236:0.620737:0.473236:0.620737:0.456767:0.606076:0.456767:0.000000:0.000000 mx y:@0.650825:0.473236:0.699893:0.473236:0.699893:0.456767:0.650825:0.456767:0.000000:0.000000:0.000000:0.000000 + – :@0.673115:0.473120:0.723628:0.473120:0.723628:0.456623:0.673115:0.456623:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.699891:0.476398:0.704844:0.476398:0.704844:0.466780:0.699891:0.466780:0.000000 mx:@0.723628:0.473236:0.745918:0.473236:0.745918:0.456767:0.723628:0.456767:0.000000:0.000000 0:@0.745916:0.476398:0.750870:0.476398:0.750870:0.466780:0.745916:0.466780:0.000000 ) – 4 = 0:@0.750870:0.473120:0.831304:0.473120:0.831304:0.456623:0.750870:0.456623:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.756824:0.466769:0.761777:0.466769:0.761777:0.457151:0.756824:0.457151:0.000000 cx:@0.789055:0.473236:0.803716:0.473236:0.803716:0.456767:0.789055:0.456767:0.000000:0.000000 :@0.494108:0.490471:0.498250:0.490471:0.498250:0.473974:0.494108:0.473974:0.000000 m x:@0.498250:0.490587:0.525473:0.490587:0.525473:0.474118:0.498250:0.474118:0.000000:0.000000:0.000000 2 2:@0.513045:0.484119:0.530426:0.484119:0.530426:0.474501:0.513045:0.474501:0.000000:0.000000:0.000000 + 2 ( – :@0.530425:0.490471:0.610184:0.490471:0.610184:0.473974:0.530425:0.473974:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 m y:@0.558013:0.490587:0.586449:0.490587:0.586449:0.474118:0.558013:0.474118:0.000000:0.000000:0.000000 0:@0.586449:0.493749:0.591402:0.493749:0.591402:0.484131:0.586449:0.484131:0.000000 mx x:@0.610184:0.490587:0.650854:0.490587:0.650854:0.474118:0.610184:0.474118:0.000000:0.000000:0.000000:0.000000 0:@0.632474:0.493749:0.637427:0.493749:0.637427:0.484131:0.632474:0.484131:0.000000 ) + ( – :@0.637426:0.490471:0.707321:0.490471:0.707321:0.473974:0.637426:0.473974:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.675899:0.490587:0.683586:0.490587:0.683586:0.474118:0.675899:0.474118:0.000000 0:@0.683586:0.493749:0.688539:0.493749:0.688539:0.484131:0.683586:0.484131:0.000000 mx:@0.707321:0.490587:0.729611:0.490587:0.729611:0.474118:0.707321:0.474118:0.000000:0.000000 0:@0.729611:0.493749:0.734564:0.493749:0.734564:0.484131:0.729611:0.484131:0.000000 ) – 4 = 0.:@0.734563:0.490471:0.817752:0.490471:0.817752:0.473974:0.734563:0.473974:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.740517:0.484119:0.745470:0.484119:0.745470:0.474501:0.740517:0.474501:0.000000 cx:@0.772749:0.490587:0.787410:0.490587:0.787410:0.474118:0.772749:0.474118:0.000000:0.000000 El discriminante de esta ecuación debe ser cero::@0.404243:0.525173:0.741709:0.525173:0.741709:0.508675:0.404243:0.508675:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [2 ( – :@0.499066:0.542523:0.565583:0.542523:0.565583:0.526026:0.499066:0.526026:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 m y:@0.513419:0.542639:0.541855:0.542639:0.541855:0.526170:0.513419:0.526170:0.000000:0.000000:0.000000 0:@0.541847:0.545801:0.546800:0.545801:0.546800:0.536183:0.541847:0.536183:0.000000 mx:@0.565583:0.542639:0.587873:0.542639:0.587873:0.526170:0.565583:0.526170:0.000000:0.000000 0:@0.587873:0.545801:0.592826:0.545801:0.592826:0.536183:0.587873:0.536183:0.000000 ) – 4 ] – 4:@0.592826:0.542523:0.671333:0.542523:0.671333:0.526026:0.592826:0.526026:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c:@0.626059:0.542639:0.633245:0.542639:0.633245:0.526170:0.626059:0.526170:0.000000 2:@0.639100:0.536172:0.644053:0.536172:0.644053:0.526554:0.639100:0.526554:0.000000 m y:@0.671333:0.542639:0.704721:0.542639:0.704721:0.526170:0.671333:0.526170:0.000000:0.000000:0.000000 2:@0.686127:0.536172:0.691080:0.536172:0.691080:0.526554:0.686127:0.526554:0.000000 ( – :@0.691081:0.542523:0.728456:0.542523:0.728456:0.526026:0.691081:0.526026:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.704721:0.545801:0.709674:0.545801:0.709674:0.536183:0.704721:0.536183:0.000000 mx:@0.728456:0.542639:0.750746:0.542639:0.750746:0.526170:0.728456:0.526170:0.000000:0.000000 0:@0.750746:0.545801:0.755699:0.545801:0.755699:0.536183:0.750746:0.536183:0.000000 ) = 0.:@0.755698:0.542523:0.796947:0.542523:0.796947:0.526026:0.755698:0.526026:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.761652:0.536172:0.766605:0.536172:0.766605:0.526554:0.761652:0.526554:0.000000 :@0.404244:0.559874:0.408386:0.559874:0.408386:0.543377:0.404244:0.543377:0.000000 La pendiente de la recta tangente es = . La ecuación cartesia-:@0.404244:0.577225:0.891782:0.577225:0.891782:0.560727:0.404244:0.560727:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 L:@0.644535:0.577341:0.652376:0.577341:0.652376:0.560872:0.644535:0.560872:0.000000 m:@0.674339:0.577341:0.689134:0.577341:0.689134:0.560872:0.674339:0.560872:0.000000 na de la recta tangente es::@0.404244:0.594576:0.600516:0.594576:0.600516:0.578078:0.404244:0.578078:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 L:@0.571118:0.594691:0.578958:0.594691:0.578958:0.578223:0.571118:0.578223:0.000000 ( , ) tal que – = ( – ).:@0.511455:0.619055:0.784568:0.619055:0.784568:0.602558:0.511455:0.602557:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.517408:0.619170:0.539467:0.619170:0.539467:0.602702:0.517408:0.602702:0.000000:0.000000:0.000000 :@0.549545:0.618537:0.564197:0.618537:0.564197:0.605163:0.549545:0.605163:0.000000 :@0.568339:0.617968:0.581644:0.617968:0.581644:0.605106:0.568339:0.605106:0.000000 2:@0.581644:0.612702:0.586597:0.612702:0.586597:0.603084:0.581644:0.603084:0.000000 y y:@0.644027:0.619171:0.678185:0.619171:0.678185:0.602702:0.644027:0.602702:0.000000:0.000000:0.000000 0:@0.678183:0.622332:0.683136:0.622332:0.683136:0.612714:0.678183:0.612714:0.000000 x x:@0.737174:0.619171:0.770908:0.619171:0.770908:0.602702:0.737174:0.602702:0.000000:0.000000:0.000000 0:@0.770906:0.622332:0.775859:0.622332:0.775859:0.612714:0.770906:0.612714:0.000000 En la Figura 2.27. se muestran la parábola (0, ), y las rectas , , , :@0.404233:0.651487:0.895911:0.651494:0.895911:0.634996:0.404233:0.634989:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.693872:0.659080:0.707287:0.659080:0.707287:0.640081:0.693872:0.640081:0.000000:0.000000 :@0.714503:0.649470:0.720918:0.649470:0.720918:0.633651:0.714503:0.633651:0.000000 c:@0.727295:0.651609:0.734481:0.651609:0.734481:0.635141:0.727295:0.635141:0.000000 L L L L:@0.826165:0.651609:0.886827:0.651609:0.886827:0.635141:0.826165:0.635141:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.848302:0.654770:0.853255:0.654770:0.853255:0.645152:0.848302:0.645152:0.000000 2:@0.867559:0.654770:0.872512:0.654770:0.872512:0.645152:0.867559:0.645152:0.000000 3:@0.886817:0.654770:0.891771:0.654770:0.891771:0.645152:0.886817:0.645152:0.000000 del haz de rectas que pasan por el punto ( , ) (0, ).:@0.404243:0.668845:0.814941:0.668845:0.814941:0.652347:0.404243:0.652347:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.703798:0.668960:0.731359:0.668960:0.731359:0.652491:0.703798:0.652491:0.000000:0.000000:0.000000 0:@0.711261:0.672121:0.716214:0.672121:0.716214:0.662503:0.711261:0.662503:0.000000 0:@0.731359:0.672121:0.736312:0.672121:0.736312:0.662503:0.731359:0.662503:0.000000 :@0.746406:0.668326:0.761058:0.668326:0.761058:0.654953:0.746406:0.654953:0.000000 :@0.765200:0.676431:0.778615:0.676431:0.778615:0.657432:0.765200:0.657432:0.000000:0.000000 :@0.785831:0.666820:0.792247:0.666820:0.792247:0.651002:0.785831:0.651002:0.000000 c:@0.799047:0.668960:0.806233:0.668960:0.806233:0.652491:0.799047:0.652491:0.000000 Ejercicio resuelto:@0.404245:0.703922:0.522223:0.703922:0.522223:0.687135:0.404245:0.687135:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si (0, 2) = {( , ) | = 8 } y = 2, entonces, = 8 = 16. Elegi-:@0.404245:0.720897:0.891779:0.720897:0.891779:0.704399:0.404245:0.704399:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.419369:0.728483:0.432784:0.728483:0.432784:0.709484:0.419369:0.709484:0.000000:0.000000 :@0.440000:0.718873:0.446415:0.718873:0.446415:0.703054:0.440000:0.703054:0.000000 x y:@0.496120:0.721012:0.517350:0.721012:0.517350:0.704544:0.496120:0.704544:0.000000:0.000000:0.000000 :@0.526628:0.720378:0.541280:0.720378:0.541280:0.707005:0.526628:0.707005:0.000000 :@0.544605:0.719810:0.557909:0.719810:0.557909:0.706948:0.544605:0.706948:0.000000 2:@0.557909:0.714544:0.562862:0.714544:0.562862:0.704926:0.557909:0.704926:0.000000 y:@0.574528:0.721012:0.582215:0.721012:0.582215:0.704544:0.574528:0.704544:0.000000 2:@0.582205:0.714544:0.587158:0.714544:0.587158:0.704926:0.582205:0.704926:0.000000 x:@0.613119:0.721012:0.620594:0.721012:0.620594:0.704544:0.613119:0.704544:0.000000 x:@0.640909:0.721012:0.648384:0.721012:0.648384:0.704544:0.640909:0.704544:0.000000 0:@0.648386:0.724173:0.653339:0.724173:0.653339:0.714555:0.648386:0.714555:0.000000 y:@0.756408:0.721012:0.764095:0.721012:0.764095:0.704544:0.756408:0.704544:0.000000 0 :@0.764096:0.724173:0.771464:0.724173:0.771464:0.714555:0.764096:0.714555:0.000000:0.000000 2:@0.769869:0.714547:0.774822:0.714547:0.774822:0.704929:0.769869:0.704929:0.000000 x:@0.800777:0.721012:0.808252:0.721012:0.808252:0.704544:0.800777:0.704544:0.000000 0:@0.808244:0.724173:0.813197:0.724173:0.813197:0.714555:0.808244:0.714555:0.000000 mos = –4. La pendiente de la recta tangente a la parábola (0, 2):@0.404233:0.738248:0.891772:0.738248:0.891772:0.721750:0.404233:0.721750:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.438987:0.738363:0.446674:0.738363:0.446674:0.721894:0.438987:0.721894:0.000000 0:@0.446677:0.741524:0.451630:0.741524:0.451630:0.731906:0.446677:0.731906:0.000000 L:@0.737947:0.738363:0.745788:0.738363:0.745788:0.721894:0.737947:0.721894:0.000000 :@0.843591:0.745834:0.857007:0.745834:0.857007:0.726835:0.843591:0.726835:0.000000:0.000000 :@0.864223:0.736223:0.870638:0.736223:0.870638:0.720405:0.864223:0.720405:0.000000 en el punto (2, –4) es m = = –1.:@0.404245:0.762727:0.662959:0.762727:0.662959:0.746229:0.404245:0.746229:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La ecuación cartesiana de la recta tangente a la parábola (0, 2) en el :@0.404245:0.795158:0.895894:0.795164:0.895894:0.778667:0.404245:0.778660:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.806334:0.802751:0.819749:0.802751:0.819749:0.783752:0.806334:0.783752:0.000000:0.000000 :@0.826965:0.793140:0.833381:0.793140:0.833381:0.777322:0.826965:0.777322:0.000000 punto (2, –4) es: ( , ) tal que + 4 = – + 2, o bien, = – – 2.:@0.404228:0.812515:0.890242:0.812515:0.890242:0.796017:0.404228:0.796017:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.530589:0.812631:0.552647:0.812631:0.552647:0.796162:0.530589:0.796162:0.000000:0.000000:0.000000 :@0.562747:0.811996:0.577400:0.811996:0.577400:0.798623:0.562747:0.798623:0.000000 :@0.581542:0.811428:0.594847:0.811428:0.594847:0.798566:0.581542:0.798566:0.000000 2:@0.594847:0.806163:0.599800:0.806163:0.599800:0.796545:0.594847:0.796545:0.000000 y:@0.657228:0.812631:0.664915:0.812631:0.664915:0.796162:0.657228:0.796162:0.000000 x:@0.722094:0.812631:0.729569:0.812631:0.729569:0.796162:0.722094:0.796162:0.000000 y:@0.815454:0.812631:0.823141:0.812631:0.823141:0.796162:0.815454:0.796162:0.000000 x:@0.852732:0.812631:0.860207:0.812631:0.860207:0.796162:0.852732:0.796162:0.000000 En la Figura 2.28. se muestran la parábola (0, 2), el punto (2, –4) y la :@0.404237:0.847217:0.895881:0.847217:0.895881:0.830719:0.404237:0.830719:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.697835:0.854803:0.711251:0.854803:0.711251:0.835804:0.697835:0.835804:0.000000:0.000000 :@0.718467:0.845192:0.724882:0.845192:0.724882:0.829374:0.718467:0.829374:0.000000 recta que pasa por (2, –4) y es tangente a (0, 2). :@0.404231:0.864567:0.768784:0.864567:0.768784:0.848070:0.404231:0.848070:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 L:@0.444245:0.864683:0.452086:0.864683:0.452086:0.848214:0.444245:0.848214:0.000000 :@0.713593:0.872154:0.727008:0.872154:0.727008:0.853155:0.713593:0.853155:0.000000:0.000000 :@0.734222:0.862543:0.740638:0.862543:0.740638:0.846725:0.734222:0.846725:0.000000 Problema 2:@0.404228:0.899732:0.491923:0.899732:0.491923:0.882786:0.404228:0.882786:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Consideremos la parábola:@0.491923:0.899269:0.682800:0.899269:0.682800:0.882771:0.491923:0.882771:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.404236:0.924206:0.417651:0.924206:0.417651:0.905207:0.404236:0.905207:0.000000:0.000000 (0, ) = {( , ) | = 4 }, donde > 0, y ( , ) , tal que :@0.417643:0.916620:0.895919:0.916620:0.895919:0.900122:0.417643:0.900122:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.424868:0.914595:0.431283:0.914595:0.431283:0.898777:0.424868:0.898777:0.000000 c:@0.438719:0.916735:0.445905:0.916735:0.445905:0.900267:0.438719:0.900267:0.000000 x y:@0.484021:0.916735:0.506706:0.916735:0.506706:0.900267:0.484021:0.900267:0.000000:0.000000:0.000000 :@0.517412:0.916101:0.532065:0.916101:0.532065:0.902728:0.517412:0.902728:0.000000 :@0.536835:0.915533:0.550140:0.915533:0.550140:0.902671:0.536835:0.902671:0.000000 2:@0.550138:0.910268:0.555091:0.910268:0.555091:0.900650:0.550138:0.900650:0.000000 y:@0.569637:0.916735:0.577324:0.916735:0.577324:0.900267:0.569637:0.900267:0.000000 2:@0.577325:0.910268:0.582278:0.910268:0.582278:0.900650:0.577325:0.900650:0.000000 cx:@0.611129:0.916735:0.625790:0.916735:0.625790:0.900267:0.611129:0.900267:0.000000:0.000000 c:@0.690819:0.916735:0.698005:0.916735:0.698005:0.900267:0.690819:0.900267:0.000000 x y:@0.752950:0.916735:0.781128:0.916735:0.781128:0.900267:0.752950:0.900267:0.000000:0.000000:0.000000 0:@0.760394:0.919897:0.765347:0.919897:0.765347:0.910279:0.760394:0.910279:0.000000 0:@0.781118:0.919897:0.786071:0.919897:0.786071:0.910279:0.781118:0.910279:0.000000 :@0.796793:0.916101:0.811446:0.916101:0.811446:0.902728:0.796793:0.902728:0.000000 :@0.816216:0.915533:0.829521:0.915533:0.829521:0.902671:0.816216:0.902671:0.000000 2:@0.829521:0.910268:0.834474:0.910268:0.834474:0.900650:0.829521:0.900650:0.000000 ( , ) (0, ) y si > 0, > 4:@0.404241:0.933970:0.641412:0.933970:0.641412:0.917473:0.404241:0.917473:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.410194:0.934086:0.437935:0.934086:0.437935:0.917617:0.410194:0.917617:0.000000:0.000000:0.000000 0:@0.417663:0.937248:0.422616:0.937248:0.422616:0.927630:0.417663:0.927630:0.000000 0:@0.437939:0.937248:0.442892:0.937248:0.442892:0.927630:0.437939:0.927630:0.000000 :@0.453162:0.933452:0.467814:0.933452:0.467814:0.920079:0.453162:0.920079:0.000000 :@0.472134:0.941557:0.485549:0.941557:0.485549:0.922558:0.472134:0.922558:0.000000:0.000000 :@0.492765:0.931946:0.499180:0.931946:0.499180:0.916128:0.492765:0.916128:0.000000 c:@0.506154:0.934086:0.513340:0.934086:0.513340:0.917617:0.506154:0.917617:0.000000 x:@0.550118:0.934086:0.557593:0.934086:0.557593:0.917617:0.550118:0.917617:0.000000 0:@0.557603:0.937248:0.562556:0.937248:0.562556:0.927630:0.557603:0.927630:0.000000 y:@0.598123:0.934086:0.605809:0.934086:0.605809:0.917617:0.598123:0.917617:0.000000 0:@0.605818:0.937248:0.610771:0.937248:0.610771:0.927630:0.605818:0.927630:0.000000 2:@0.608525:0.927621:0.613478:0.927621:0.613478:0.918003:0.608525:0.918003:0.000000 cx:@0.641412:0.934086:0.656072:0.934086:0.656072:0.917617:0.641412:0.917617:0.000000:0.000000 0:@0.656078:0.937248:0.661031:0.937248:0.661031:0.927630:0.656078:0.927630:0.000000 . En la Figura 2.29. se muestran la :@0.661030:0.933970:0.895891:0.933970:0.895891:0.917473:0.661030:0.917473:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 parábola (0, ), el punto ( , ) y , , que son tres rectas de un :@0.404244:0.951321:0.895914:0.951321:0.895914:0.934824:0.404244:0.934824:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.471373:0.958908:0.484788:0.958908:0.484788:0.939909:0.471373:0.939909:0.000000:0.000000 :@0.492004:0.949297:0.498420:0.949297:0.498420:0.933479:0.492004:0.933479:0.000000 c:@0.506010:0.951437:0.513196:0.951437:0.513196:0.934968:0.506010:0.934968:0.000000 x y:@0.598618:0.951437:0.626976:0.951437:0.626976:0.934968:0.598618:0.934968:0.000000:0.000000:0.000000 0:@0.606089:0.954598:0.611042:0.954598:0.611042:0.944980:0.606089:0.944980:0.000000 0:@0.626976:0.954598:0.631929:0.954598:0.631929:0.944980:0.626976:0.944980:0.000000 L L L:@0.655567:0.951437:0.699413:0.951437:0.699413:0.934968:0.655567:0.934968:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.678934:0.954598:0.683887:0.954598:0.683887:0.944980:0.678934:0.944980:0.000000 2:@0.699413:0.954598:0.704367:0.954598:0.704367:0.944980:0.699413:0.944980:0.000000 p:@0.143988:0.346447:0.155468:0.346447:0.155468:0.336422:0.143988:0.336422:0.000000 :@0.155468:0.347155:0.158350:0.347155:0.158350:0.335678:0.155468:0.335678:0.000000 Figura 2.26.:@0.158350:0.347155:0.212828:0.347155:0.212828:0.335678:0.158350:0.335678:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 p:@0.143988:0.624755:0.155468:0.624755:0.155468:0.614731:0.143988:0.614731:0.000000 :@0.155468:0.625463:0.158350:0.625463:0.158350:0.613987:0.155468:0.613987:0.000000 Figura 2.27.:@0.158350:0.625463:0.212828:0.625463:0.212828:0.613987:0.158350:0.613987:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 p:@0.143988:0.923488:0.155468:0.923488:0.155468:0.913464:0.143988:0.913464:0.000000 :@0.155468:0.924196:0.158350:0.924196:0.158350:0.912720:0.155468:0.912720:0.000000 Figura 2.28.:@0.158350:0.924196:0.212828:0.924196:0.212828:0.912720:0.158350:0.912720:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.601727:0.755489:0.609414:0.755489:0.609414:0.739020:0.601727:0.739020:0.000000 0:@0.609413:0.757394:0.614366:0.757394:0.614366:0.747776:0.609413:0.747776:0.000000 2:@0.597586:0.771718:0.606082:0.771718:0.606082:0.755220:0.597586:0.755220:0.000000 x:@0.606082:0.771834:0.613557:0.771834:0.613557:0.755365:0.606082:0.755365:0.000000 0:@0.613556:0.773739:0.618509:0.773739:0.618509:0.764121:0.613556:0.764121:0.000000 y:@0.714970:0.569902:0.722657:0.569902:0.722657:0.553434:0.714970:0.553434:0.000000 0:@0.722656:0.571807:0.727609:0.571807:0.727609:0.562189:0.722656:0.562189:0.000000 2:@0.710828:0.586132:0.719324:0.586132:0.719324:0.569634:0.710828:0.569634:0.000000 x:@0.719324:0.586247:0.726799:0.586247:0.726799:0.569779:0.719324:0.569779:0.000000 0:@0.726799:0.588152:0.731752:0.588152:0.731752:0.578534:0.726799:0.578534:0.000000 y:@0.708322:0.611844:0.716009:0.611844:0.716009:0.595375:0.708322:0.595375:0.000000 0:@0.716008:0.613747:0.720962:0.613747:0.720962:0.604129:0.716008:0.604129:0.000000 2:@0.704179:0.628073:0.712675:0.628073:0.712675:0.611575:0.704179:0.611575:0.000000 x:@0.712675:0.628188:0.720150:0.628188:0.720150:0.611720:0.712675:0.611720:0.000000 0:@0.720150:0.630092:0.725103:0.630092:0.725103:0.620474:0.720150:0.620474:0.000000 0:@0.155664:0.284706:0.161981:0.284706:0.161981:0.271817:0.155664:0.271817:0.000000  :@0.157081:0.283744:0.177571:0.285000:0.177571:0.272621:0.157081:0.271365:0.000000:0.000000:0.000000 = (0, 0):@0.162102:0.284615:0.205585:0.284615:0.205585:0.271704:0.162102:0.271704:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 j:@0.155664:0.247398:0.158680:0.247398:0.158680:0.234509:0.155664:0.234509:0.000000  :@0.155664:0.245802:0.176154:0.247692:0.176154:0.235313:0.155664:0.233423:0.000000:0.000000:0.000000 = (0, 1):@0.160685:0.247307:0.204167:0.247307:0.204167:0.234396:0.160685:0.234396:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.194369:0.213212:0.200830:0.213212:0.200830:0.199584:0.194369:0.199584:0.000000 x:@0.358116:0.285930:0.364721:0.285930:0.364721:0.272301:0.358116:0.272301:0.000000 y:@0.273412:0.306793:0.280096:0.306793:0.280096:0.292472:0.273412:0.292472:0.000000 2:@0.280096:0.301168:0.284403:0.301168:0.284403:0.292805:0.280096:0.292805:0.000000 = 4:@0.284401:0.306692:0.308391:0.306692:0.308391:0.292346:0.284401:0.292346:0.000000:0.000000:0.000000:0.000000 cx:@0.308391:0.306793:0.321139:0.306793:0.321139:0.292472:0.308391:0.292472:0.000000:0.000000 ( , ):@0.265071:0.526422:0.295589:0.526422:0.295589:0.514945:0.265071:0.514945:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.269212:0.526502:0.288003:0.526502:0.288003:0.515046:0.269212:0.515046:0.000000:0.000000:0.000000 0:@0.274412:0.528702:0.277857:0.528702:0.277857:0.522011:0.274412:0.522011:0.000000 0:@0.288003:0.528702:0.291449:0.528702:0.291449:0.522011:0.288003:0.522011:0.000000 y:@0.193174:0.491871:0.199636:0.491871:0.199636:0.478242:0.193174:0.478242:0.000000 x:@0.356922:0.564588:0.363526:0.564588:0.363526:0.550959:0.356922:0.550959:0.000000 0:@0.197428:0.561135:0.204077:0.561135:0.204077:0.548223:0.197428:0.548223:0.000000 y:@0.303211:0.592019:0.309895:0.592019:0.309895:0.577698:0.303211:0.577698:0.000000 2:@0.309895:0.586395:0.314202:0.586395:0.314202:0.578031:0.309895:0.578031:0.000000 = 4:@0.314202:0.591918:0.338192:0.591918:0.338192:0.577573:0.314202:0.577573:0.000000:0.000000:0.000000:0.000000 cx:@0.338192:0.592019:0.350940:0.592019:0.350940:0.577698:0.338192:0.577698:0.000000:0.000000 L:@0.160482:0.517096:0.167300:0.517096:0.167300:0.502776:0.160482:0.502776:0.000000 1:@0.167302:0.519846:0.171609:0.519846:0.171609:0.511482:0.167302:0.511482:0.000000 L:@0.152587:0.566131:0.159405:0.566131:0.159405:0.551811:0.152587:0.551811:0.000000 L:@0.175554:0.592535:0.182372:0.592535:0.182372:0.578214:0.175554:0.578214:0.000000 3:@0.182379:0.595284:0.186686:0.595284:0.186686:0.586920:0.182379:0.586920:0.000000 L:@0.229169:0.500752:0.235987:0.500752:0.235987:0.486431:0.229169:0.486431:0.000000 2:@0.235987:0.503501:0.240294:0.503501:0.240294:0.495137:0.235987:0.495137:0.000000 1:@0.209825:0.812665:0.215735:0.812665:0.215735:0.801189:0.209825:0.801189:0.000000 –1:@0.202990:0.841895:0.216204:0.841895:0.216204:0.830418:0.202990:0.830418:0.000000:0.000000 –2:@0.202990:0.855796:0.216204:0.855796:0.216204:0.844319:0.202990:0.844319:0.000000:0.000000 –3:@0.202990:0.870049:0.216204:0.870049:0.216204:0.858572:0.202990:0.858572:0.000000:0.000000 –4:@0.202990:0.884301:0.216204:0.884301:0.216204:0.872825:0.202990:0.872825:0.000000:0.000000 –5:@0.202990:0.898554:0.216204:0.898554:0.216204:0.887077:0.202990:0.887077:0.000000:0.000000 2:@0.209825:0.798402:0.215735:0.798402:0.215735:0.786926:0.209825:0.786926:0.000000 3:@0.209825:0.784150:0.215735:0.784150:0.215735:0.772673:0.209825:0.772673:0.000000 4:@0.209825:0.769897:0.215735:0.769897:0.215735:0.758420:0.209825:0.758420:0.000000 5:@0.209825:0.755644:0.215735:0.755644:0.215735:0.744168:0.209825:0.744168:0.000000 1 2 3 4:@0.238733:0.836826:0.301628:0.836826:0.301628:0.825349:0.238733:0.825349:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –1:@0.194721:0.836826:0.207935:0.836826:0.207935:0.825349:0.194721:0.825349:0.000000:0.000000 –2:@0.175730:0.836826:0.188944:0.836826:0.188944:0.825349:0.175730:0.825349:0.000000:0.000000 –3:@0.156740:0.836826:0.169954:0.836826:0.169954:0.825349:0.156740:0.825349:0.000000:0.000000 0:@0.212929:0.833770:0.219947:0.833770:0.219947:0.820141:0.212929:0.820141:0.000000 y:@0.208536:0.740735:0.214998:0.740735:0.214998:0.727107:0.208536:0.727107:0.000000 x:@0.362225:0.837544:0.368830:0.837544:0.368830:0.823916:0.362225:0.823916:0.000000 y:@0.289397:0.762587:0.295413:0.762587:0.295413:0.749698:0.289397:0.749698:0.000000 2:@0.295413:0.757525:0.299289:0.757525:0.299289:0.749998:0.295413:0.749998:0.000000 = 8:@0.299287:0.762496:0.320878:0.762496:0.320878:0.749585:0.299287:0.749585:0.000000:0.000000:0.000000:0.000000 x:@0.320878:0.762587:0.326728:0.762587:0.326728:0.749698:0.320878:0.749698:0.000000 y:@0.146601:0.794758:0.152617:0.794758:0.152617:0.781869:0.146601:0.781869:0.000000 = – +2:@0.152617:0.794667:0.196733:0.794667:0.196733:0.781756:0.152617:0.781756:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.175775:0.794758:0.181625:0.794758:0.181625:0.781869:0.175775:0.781869:0.000000 u:@0.268674:0.881695:0.275881:0.881695:0.275881:0.868806:0.268674:0.868806:0.000000  :@0.270981:0.881989:0.291471:0.881989:0.291471:0.869610:0.270981:0.869610:0.000000:0.000000:0.000000 = (2, –4):@0.276002:0.881604:0.327701:0.881604:0.327701:0.868693:0.276002:0.868693:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000