173:@0.937671:0.974518:0.968375:0.974518:0.968375:0.956333:0.937671:0.956333:0.000000:0.000000:0.000000 Pensamiento espacial, numérico y variacional:@0.522009:0.043661:0.926913:0.043661:0.926913:0.024916:0.522009:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 3.:@0.073089:0.140040:0.152608:0.140040:0.152608:0.123370:0.073089:0.123370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.152498:0.140040:0.156529:0.140040:0.156529:0.123606:0.152498:0.123606:0.000000 Encontremos los puntos críticos de la función ( ) = :@0.156455:0.140040:0.531356:0.140040:0.531356:0.123606:0.156455:0.123606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.490860:0.140040:0.507739:0.140040:0.507739:0.123634:0.490860:0.123634:0.000000:0.000000:0.000000 x:@0.531245:0.140040:0.538700:0.140040:0.538700:0.123634:0.531245:0.123634:0.000000 4 :@0.538627:0.133976:0.546171:0.133976:0.546171:0.124395:0.538627:0.124395:0.000000:0.000000 – :@0.546121:0.140622:0.559834:0.140622:0.559834:0.122968:0.546121:0.122968:0.000000:0.000000 2 .:@0.559834:0.140040:0.584426:0.140040:0.584426:0.123606:0.559834:0.123606:0.000000:0.000000:0.000000 x:@0.568651:0.140040:0.576106:0.140040:0.576106:0.123634:0.568651:0.123634:0.000000 2:@0.576032:0.133976:0.581269:0.133976:0.581269:0.124395:0.576032:0.124395:0.000000 Empezamos por hallar la derivada de la función ´( ) = 4:@0.073091:0.167402:0.478124:0.167402:0.478124:0.150968:0.073091:0.150968:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.420031:0.167402:0.445599:0.167402:0.445599:0.150996:0.420031:0.150996:0.000000:0.000000:0.000000:0.000000 x:@0.478014:0.167402:0.485469:0.167402:0.485469:0.150996:0.478014:0.150996:0.000000 3 :@0.485414:0.161339:0.492958:0.161339:0.492958:0.151758:0.485414:0.151758:0.000000:0.000000 – :@0.492905:0.167985:0.506619:0.167985:0.506619:0.150331:0.492905:0.150331:0.000000:0.000000 4 . :@0.506619:0.167403:0.530051:0.167403:0.530051:0.150969:0.506619:0.150969:0.000000:0.000000:0.000000:0.000000 x:@0.515436:0.167403:0.522891:0.167403:0.522891:0.150997:0.515436:0.150997:0.000000 Para determinar los puntos críticos, igualamos a 0 la derivada :@0.073091:0.194765:0.517497:0.194765:0.517497:0.178331:0.073091:0.178331:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de y solucionamos la ecuación obtenida.:@0.073091:0.211407:0.381111:0.211407:0.381111:0.194973:0.073091:0.194973:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.099763:0.211407:0.104457:0.211407:0.104457:0.195001:0.099763:0.195001:0.000000 4:@0.073073:0.238769:0.082055:0.238769:0.082055:0.222335:0.073073:0.222335:0.000000 x:@0.081982:0.238769:0.089437:0.238769:0.089437:0.222363:0.081982:0.222363:0.000000 3 :@0.089380:0.232707:0.096924:0.232707:0.096924:0.223126:0.089380:0.223126:0.000000:0.000000 – :@0.096873:0.239354:0.110586:0.239354:0.110586:0.221699:0.096873:0.221699:0.000000:0.000000 4 = 0:@0.110586:0.238771:0.153880:0.238771:0.153880:0.222337:0.110586:0.222337:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.119403:0.238771:0.130098:0.238771:0.130098:0.222365:0.119403:0.222365:0.000000:0.000000 4 ( 1) = 0 :@0.073091:0.258991:0.169966:0.258986:0.169966:0.242552:0.073091:0.242557:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x x:@0.082000:0.258991:0.101640:0.258991:0.101640:0.242585:0.082000:0.242585:0.000000:0.000000:0.000000 2:@0.101566:0.252922:0.106802:0.252922:0.106802:0.243341:0.101566:0.243341:0.000000 – :@0.110713:0.259568:0.124427:0.259568:0.124427:0.241914:0.110713:0.241914:0.000000:0.000000 4 ( + 1)(x 1) = 0. :@0.073089:0.279206:0.212303:0.279206:0.212303:0.262772:0.073089:0.262772:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x x :@0.081998:0.279206:0.104878:0.279206:0.104878:0.262800:0.081998:0.262800:0.000000:0.000000:0.000000:0.000000 – :@0.149921:0.279788:0.163634:0.279788:0.163634:0.262134:0.149921:0.262134:0.000000:0.000000 De aquí concluimos que las soluciones de la ecuación son :@0.073089:0.306568:0.501591:0.306568:0.501591:0.290134:0.073089:0.290134:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.073089:0.323792:0.082292:0.323792:0.082292:0.306138:0.073089:0.306138:0.000000 1, 0 y 1. Por lo tanto, los puntos críticos que buscamos son 1, 0 y 1.:@0.082200:0.323210:0.567968:0.323210:0.567968:0.306776:0.082200:0.306776:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.505898:0.323792:0.515102:0.323792:0.515102:0.306138:0.505898:0.306138:0.000000 La gráfica de la función mostrada en la Figura 2.8 corrobora el :@0.073089:0.350572:0.573983:0.350572:0.573983:0.334138:0.073089:0.334138:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 resultado.:@0.073089:0.367213:0.143833:0.367213:0.143833:0.350780:0.073089:0.350780:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 4.:@0.073089:0.394575:0.151890:0.394575:0.151890:0.377906:0.073089:0.377906:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Calculemos el máximo y el mínimo absoluto de la función :@0.151780:0.394575:0.574129:0.394575:0.574129:0.378142:0.151780:0.378142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 como la figura 2.9:@0.073089:0.411217:0.202473:0.411217:0.202473:0.394783:0.073089:0.394783:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = 2:@0.073089:0.438579:0.161996:0.438579:0.161996:0.422145:0.073089:0.422145:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.112591:0.438579:0.129470:0.438579:0.129470:0.422173:0.112591:0.422173:0.000000:0.000000:0.000000 x:@0.161885:0.438579:0.169340:0.438579:0.169340:0.422173:0.161885:0.422173:0.000000 3 :@0.169287:0.432516:0.176831:0.432516:0.176831:0.422936:0.169287:0.422936:0.000000:0.000000 – :@0.176781:0.439163:0.190494:0.439163:0.190494:0.421509:0.176781:0.421509:0.000000:0.000000 3:@0.190494:0.438580:0.199477:0.438580:0.199477:0.422147:0.190494:0.422147:0.000000 x:@0.199311:0.438580:0.206766:0.438580:0.206766:0.422174:0.199311:0.422174:0.000000 2:@0.206693:0.432516:0.211929:0.432516:0.211929:0.422936:0.206693:0.422936:0.000000 – :@0.211883:0.439163:0.225597:0.439163:0.225597:0.421509:0.211883:0.421509:0.000000:0.000000 1 si 1 ≤ ≤ 2:@0.225597:0.438580:0.328088:0.438580:0.328088:0.422147:0.225597:0.422147:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.256815:0.439163:0.266019:0.439163:0.266019:0.421509:0.256815:0.421509:0.000000 x :@0.293612:0.438580:0.304306:0.438580:0.304306:0.422174:0.293612:0.422174:0.000000:0.000000 Solución::@0.073093:0.465942:0.139517:0.465942:0.139517:0.449509:0.073093:0.449509:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ´( ) = 6:@0.073093:0.493304:0.170688:0.493304:0.170688:0.476870:0.073093:0.476870:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.112595:0.493304:0.138162:0.493304:0.138162:0.476898:0.112595:0.476898:0.000000:0.000000:0.000000:0.000000 x:@0.170578:0.493304:0.178032:0.493304:0.178032:0.476898:0.170578:0.476898:0.000000 2 :@0.177976:0.487243:0.185520:0.487243:0.185520:0.477662:0.177976:0.477662:0.000000:0.000000 – :@0.185469:0.493889:0.199182:0.493889:0.199182:0.476235:0.185469:0.476235:0.000000:0.000000 6 = 6 (:@0.199182:0.493307:0.254643:0.493307:0.254643:0.476873:0.199182:0.476873:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.207999:0.493307:0.218694:0.493307:0.218694:0.476901:0.207999:0.476901:0.000000:0.000000 x x :@0.242384:0.493307:0.265264:0.493307:0.265264:0.476901:0.242384:0.476901:0.000000:0.000000:0.000000:0.000000 – :@0.265209:0.493889:0.278923:0.493889:0.278923:0.476235:0.265209:0.476235:0.000000:0.000000 1) = 0.:@0.278923:0.493307:0.323560:0.493307:0.323560:0.476873:0.278923:0.476873:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Entonces, en = 0 y = 1 se presentan los valores extremos, puesto que hacen que ´( ) :@0.073111:0.520669:0.710352:0.520669:0.710352:0.504235:0.073111:0.504235:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.169491:0.520669:0.180186:0.520669:0.180186:0.504263:0.169491:0.504263:0.000000:0.000000 x :@0.219578:0.520669:0.230272:0.520669:0.230272:0.504263:0.219578:0.504263:0.000000:0.000000 f x:@0.676004:0.520669:0.701516:0.520669:0.701516:0.504263:0.676004:0.504263:0.000000:0.000000:0.000000:0.000000 = 0. Por lo tanto, los puntos críticos son 0 y 1.:@0.073074:0.537311:0.397244:0.537311:0.397244:0.520877:0.073074:0.520877:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Los valores de en estos puntos críticos son::@0.073074:0.564673:0.392105:0.564673:0.392105:0.548239:0.073074:0.548239:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.178879:0.564673:0.183573:0.564673:0.183573:0.548266:0.178879:0.548266:0.000000 (0) = 1 y (1) = 2.:@0.073056:0.592035:0.227842:0.592035:0.227842:0.575601:0.073056:0.575601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.077013:0.592035:0.081707:0.592035:0.081707:0.575628:0.077013:0.575628:0.000000 –:@0.118927:0.592617:0.128130:0.592617:0.128130:0.574963:0.118927:0.574963:0.000000 f:@0.164706:0.592035:0.169399:0.592035:0.169399:0.575628:0.164706:0.575628:0.000000 –:@0.206619:0.592617:0.215823:0.592617:0.215823:0.574963:0.206619:0.574963:0.000000 Y los valores de en los extremos de los intervalos son::@0.073056:0.619396:0.466647:0.619396:0.466647:0.602963:0.073056:0.602963:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.187972:0.619396:0.192666:0.619396:0.192666:0.602990:0.187972:0.602990:0.000000 f:@0.073056:0.646758:0.077750:0.646758:0.077750:0.630352:0.073056:0.630352:0.000000 ( 1) = 6 y (2) = 3.:@0.077676:0.646758:0.212067:0.646758:0.212067:0.630325:0.077676:0.630325:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.082480:0.647341:0.091684:0.647341:0.091684:0.629687:0.082480:0.629687:0.000000 –:@0.124081:0.647341:0.133284:0.647341:0.133284:0.629687:0.124081:0.629687:0.000000 f:@0.158005:0.646758:0.162699:0.646758:0.162699:0.630352:0.158005:0.630352:0.000000 Al comparar los cuatro valores, podemos concluir que el mínimo :@0.073037:0.674120:0.550895:0.674120:0.550895:0.657687:0.073037:0.657687:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 absoluto es ( 1)= 6 y el máximo absoluto es (2) = 3.:@0.073037:0.690762:0.465750:0.690768:0.465750:0.674335:0.073037:0.674328:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f :@0.160141:0.690762:0.168074:0.690762:0.168074:0.674356:0.160141:0.674356:0.000000:0.000000 –:@0.172872:0.691351:0.182075:0.691351:0.182075:0.673697:0.172872:0.673697:0.000000 –:@0.210515:0.691351:0.219718:0.691351:0.219718:0.673697:0.210515:0.673697:0.000000 f:@0.411687:0.690768:0.416381:0.690768:0.416381:0.674362:0.411687:0.674362:0.000000 Los teoremas que se presentan a continuación nos ayudarán a :@0.073068:0.718130:0.550585:0.718130:0.550585:0.701697:0.073068:0.701697:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 resolver problemas, mismos que, a su vez, contribuyen a demostrar :@0.073068:0.734772:0.550876:0.734772:0.550876:0.718338:0.073068:0.718338:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 otros teoremas. :@0.073068:0.751414:0.346401:0.751414:0.346401:0.734980:0.073068:0.734980:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2.3 Teorema de Rolle:@0.073068:0.778776:0.229345:0.778776:0.229345:0.762106:0.073068:0.762106:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Figura 2.8:@0.758801:0.164796:0.815995:0.164796:0.815995:0.151350:0.758801:0.151350:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.878186:0.331302:0.885086:0.331302:0.885086:0.317121:0.878186:0.317121:0.000000 y:@0.722397:0.201410:0.729297:0.201410:0.729297:0.187229:0.722397:0.187229:0.000000 3:@0.732400:0.201637:0.738967:0.201637:0.738967:0.190044:0.732400:0.190044:0.000000 2:@0.732400:0.240215:0.738967:0.240215:0.738967:0.228622:0.732400:0.228622:0.000000 1:@0.732400:0.279093:0.738967:0.279093:0.738967:0.267500:0.732400:0.267500:0.000000 0:@0.732400:0.323401:0.738967:0.323401:0.738967:0.311808:0.732400:0.311808:0.000000 1:@0.807212:0.323401:0.813779:0.323401:0.813779:0.311808:0.807212:0.311808:0.000000 -1:@0.668636:0.323401:0.679134:0.323401:0.679134:0.311808:0.668636:0.311808:0.000000:0.000000 2:@0.875623:0.323401:0.882190:0.323401:0.882190:0.311808:0.875623:0.311808:0.000000 -1:@0.730442:0.357003:0.740939:0.357003:0.740939:0.345410:0.730442:0.345410:0.000000:0.000000 (0,0):@0.706003:0.309180:0.736743:0.309180:0.736743:0.293723:0.706003:0.293723:0.000000:0.000000:0.000000:0.000000:0.000000 (-1,-1):@0.626800:0.369996:0.668020:0.369996:0.668020:0.354539:0.626800:0.354539:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (1,-1):@0.767746:0.369996:0.803726:0.369996:0.803726:0.354539:0.767746:0.354539:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f(x) = x:@0.878775:0.198984:0.920910:0.198984:0.920910:0.184803:0.878775:0.184803:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4 :@0.920911:0.195233:0.930192:0.195233:0.930192:0.183640:0.920911:0.183640:0.000000:0.000000 - 2:@0.930190:0.198984:0.946336:0.198984:0.946336:0.184815:0.930190:0.184815:0.000000:0.000000:0.000000 x:@0.946336:0.198984:0.953236:0.198984:0.953236:0.184803:0.946336:0.184803:0.000000 2:@0.953237:0.195233:0.959804:0.195233:0.959804:0.183640:0.953237:0.183640:0.000000 Figura 2.9:@0.729549:0.554337:0.786743:0.554337:0.786743:0.540891:0.729549:0.540891:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.868398:0.655084:0.874600:0.655084:0.874600:0.642338:0.868398:0.642338:0.000000 y:@0.695773:0.584085:0.701975:0.584085:0.701975:0.571339:0.695773:0.571339:0.000000 (-1,-6):@0.617219:0.752778:0.654269:0.752778:0.654269:0.738885:0.617219:0.738885:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (1,-2):@0.728053:0.687066:0.760393:0.687066:0.760393:0.673172:0.728053:0.673172:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (2,3):@0.830981:0.604445:0.858612:0.604445:0.858612:0.590552:0.830981:0.590552:0.000000:0.000000:0.000000:0.000000:0.000000 (0,-1):@0.670219:0.659508:0.702559:0.659508:0.702559:0.645615:0.670219:0.645615:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Mínimo absoluto:@0.579582:0.734925:0.662737:0.734925:0.662737:0.724331:0.579582:0.724331:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Mínimo relativo:@0.772000:0.681577:0.849414:0.681577:0.849414:0.670983:0.772000:0.670983:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Máximo relativo:@0.718231:0.653316:0.797647:0.653316:0.797647:0.642723:0.718231:0.642723:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Máximo absoluto:@0.738264:0.585889:0.823421:0.585889:0.823421:0.575296:0.738264:0.575296:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.708600:0.647502:0.714502:0.647502:0.714502:0.637082:0.708600:0.637082:0.000000 -5:@0.705551:0.722956:0.714986:0.722956:0.714986:0.712536:0.705551:0.712536:0.000000:0.000000 1:@0.766246:0.647502:0.772148:0.647502:0.772148:0.637082:0.766246:0.637082:0.000000 -1:@0.657662:0.647502:0.667097:0.647502:0.667097:0.637082:0.657662:0.637082:0.000000:0.000000 -2:@0.605884:0.647502:0.615319:0.647502:0.615319:0.637082:0.605884:0.637082:0.000000:0.000000 2:@0.818772:0.647502:0.824674:0.647502:0.824674:0.637082:0.818772:0.637082:0.000000 3:@0.869273:0.647502:0.875175:0.647502:0.875175:0.637082:0.869273:0.637082:0.000000 f(x) = x:@0.835521:0.586172:0.880608:0.586172:0.880608:0.573426:0.835521:0.573426:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.867191:0.586172:0.874406:0.586172:0.874406:0.573437:0.867191:0.573437:0.000000 3 :@0.880606:0.582801:0.888948:0.582801:0.888948:0.572381:0.880606:0.572381:0.000000:0.000000 - 3:@0.888947:0.586172:0.903460:0.586172:0.903460:0.573437:0.888947:0.573437:0.000000:0.000000:0.000000 x:@0.903460:0.586172:0.909662:0.586172:0.909662:0.573426:0.903460:0.573426:0.000000 2 :@0.909662:0.582801:0.918004:0.582801:0.918004:0.572381:0.909662:0.572381:0.000000:0.000000 -1:@0.918004:0.586172:0.929536:0.586172:0.929536:0.573437:0.918004:0.573437:0.000000:0.000000 Sea una función continua en el intervalo cerrado [a,b] y :@0.091667:0.814958:0.520762:0.814958:0.520762:0.798524:0.091667:0.798524:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.122755:0.814958:0.127449:0.814958:0.127449:0.798552:0.122755:0.798552:0.000000 derivable en el intervalo abierto ( , ).:@0.091665:0.831600:0.364111:0.831600:0.364111:0.815166:0.091665:0.815166:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.330665:0.831600:0.356178:0.831600:0.356178:0.815194:0.330665:0.815194:0.000000:0.000000:0.000000 Si se cumple que ( ) = ( ), entonces existe :@0.091647:0.858962:0.402096:0.858962:0.402096:0.842528:0.091647:0.842528:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f a:@0.216356:0.858962:0.234892:0.858962:0.234892:0.842556:0.216356:0.842556:0.000000:0.000000:0.000000 f b:@0.257220:0.858962:0.275756:0.858962:0.275756:0.842556:0.257220:0.842556:0.000000:0.000000:0.000000 c a b:@0.401625:0.858962:0.459626:0.858962:0.459626:0.842556:0.401625:0.842556:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ∈:@0.412007:0.859544:0.427340:0.859544:0.427340:0.841890:0.412007:0.841890:0.000000 ( , ) tal que :@0.430009:0.858962:0.520959:0.858962:0.520959:0.842528:0.430009:0.842528:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f c:@0.091647:0.875604:0.117417:0.875604:0.117417:0.859198:0.091647:0.859198:0.000000:0.000000:0.000000:0.000000 ´( ) = 0. :@0.099525:0.875604:0.173213:0.875604:0.173213:0.859170:0.099525:0.859170:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 MUESTRA EDITORIAL:@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