174:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000 Aplicaciones de la derivada:@0.073089:0.043661:0.316384:0.043661:0.316384:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 5. :@0.293660:0.140040:0.376438:0.140040:0.376438:0.123370:0.293660:0.123370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Consideremos la función ( ) = (:@0.375941:0.140040:0.602939:0.140040:0.602939:0.123606:0.375941:0.123606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.558541:0.140040:0.575402:0.140040:0.575402:0.123634:0.558541:0.123634:0.000000:0.000000:0.000000 x :@0.602828:0.140040:0.613523:0.140040:0.613523:0.123634:0.602828:0.123634:0.000000:0.000000 – :@0.613100:0.140622:0.626795:0.140622:0.626795:0.122968:0.613100:0.122968:0.000000:0.000000 1) + 1. Hallemos un número en ( 1,3) tal :@0.626224:0.140040:0.930968:0.140040:0.930968:0.123606:0.626224:0.123606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2 :@0.639889:0.133976:0.647422:0.133976:0.647422:0.124395:0.639889:0.124395:0.000000:0.000000 c:@0.832839:0.140040:0.840496:0.140040:0.840496:0.123634:0.832839:0.123634:0.000000 –:@0.870868:0.140622:0.880072:0.140622:0.880072:0.122968:0.870868:0.122968:0.000000 que ´( ) = 0.:@0.293653:0.156682:0.387457:0.156682:0.387457:0.140248:0.293653:0.140248:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f c:@0.326032:0.156682:0.351802:0.156682:0.351802:0.140276:0.326032:0.140276:0.000000:0.000000:0.000000:0.000000 Entonces: ( 1) = 5 = (3), y es derivable y existe ( 1,3) de manera que ´( ) = 0 de :@0.293635:0.184044:0.930903:0.184044:0.930903:0.167610:0.293635:0.167610:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.374535:0.184044:0.379229:0.184044:0.379229:0.167637:0.374535:0.167637:0.000000 –:@0.384199:0.184626:0.393402:0.184626:0.393402:0.166972:0.384199:0.166972:0.000000 f:@0.457552:0.184044:0.462245:0.184044:0.462245:0.167637:0.457552:0.167637:0.000000 f:@0.502116:0.184044:0.506809:0.184044:0.506809:0.167637:0.502116:0.167637:0.000000 c:@0.665572:0.184044:0.673229:0.184044:0.673229:0.167637:0.665572:0.167637:0.000000 –:@0.682948:0.184626:0.692152:0.184626:0.692152:0.166972:0.682948:0.166972:0.000000 f c:@0.841729:0.184044:0.868549:0.184044:0.868549:0.167637:0.841729:0.167637:0.000000:0.000000:0.000000:0.000000 acuerdo con el teorema de Rolle.:@0.293598:0.200685:0.532193:0.200685:0.532193:0.184252:0.293598:0.184252:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si igualamos a 0 la derivada, ´( ) = 2:@0.293598:0.228047:0.560172:0.228047:0.560172:0.211614:0.293598:0.211614:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.501398:0.228047:0.527168:0.228047:0.527168:0.211641:0.501398:0.211641:0.000000:0.000000:0.000000:0.000000 x :@0.560062:0.228047:0.570756:0.228047:0.570756:0.211641:0.560062:0.211641:0.000000:0.000000 – :@0.570903:0.228630:0.584617:0.228630:0.584617:0.210976:0.570903:0.210976:0.000000:0.000000 2 = 2(:@0.584801:0.228047:0.626788:0.228047:0.626788:0.211614:0.584801:0.211614:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.626677:0.228047:0.637372:0.228047:0.637372:0.211641:0.626677:0.211641:0.000000:0.000000 – :@0.637519:0.228630:0.651233:0.228630:0.651233:0.210976:0.637519:0.210976:0.000000:0.000000 1) = 0 tendremos que x = 1. Por tanto, :@0.651417:0.228047:0.930894:0.228047:0.930894:0.211614:0.651417:0.211614:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f :@0.293617:0.244689:0.301550:0.244689:0.301550:0.228283:0.293617:0.228283:0.000000:0.000000 ´(1) = 0, como lo podemos ver en la gráfica de la función mostrada en la Figura 2.10.:@0.301495:0.244689:0.907662:0.244689:0.907662:0.228255:0.301495:0.228255:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Figura 2.10:@0.578069:0.271425:0.642552:0.271425:0.642552:0.257979:0.578069:0.257979:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.538254:0.296185:0.545154:0.296185:0.545154:0.282004:0.538254:0.282004:0.000000 x:@0.726994:0.415538:0.733894:0.415538:0.733894:0.401357:0.726994:0.401357:0.000000 5:@0.552520:0.325302:0.559087:0.325302:0.559087:0.313709:0.552520:0.313709:0.000000 5:@0.735529:0.406028:0.742096:0.406028:0.742096:0.394435:0.735529:0.394435:0.000000 0:@0.552520:0.405671:0.559087:0.405671:0.559087:0.394078:0.552520:0.394078:0.000000 f(x) = x:@0.652662:0.299933:0.699241:0.299933:0.699241:0.285752:0.652662:0.285752:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ( -1) + 1:@0.687897:0.299933:0.746463:0.299933:0.746463:0.285764:0.687897:0.285764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2 :@0.716513:0.296182:0.725794:0.296182:0.725794:0.284589:0.716513:0.284589:0.000000:0.000000 f:@0.467710:0.331548:0.472529:0.331548:0.472529:0.317144:0.467710:0.317144:0.000000 (-1) = 5:@0.472467:0.331548:0.518952:0.331548:0.518952:0.317144:0.472467:0.317144:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.682485:0.329945:0.687304:0.329945:0.687304:0.315540:0.682485:0.315540:0.000000 (3) = 5:@0.687242:0.329945:0.728860:0.329945:0.728860:0.315540:0.687242:0.315540:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f’:@0.580785:0.393176:0.589093:0.393176:0.589093:0.378771:0.580785:0.378771:0.000000:0.000000 (1) = 0:@0.589031:0.393176:0.630649:0.393176:0.630649:0.378771:0.589031:0.378771:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 6. :@0.293660:0.464664:0.377321:0.464664:0.377321:0.447994:0.293660:0.447994:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Consideremos la función ( ) = 3 + 2 + 5. Hallemos un número en ( 1,1) :@0.377524:0.464664:0.930971:0.464664:0.930971:0.448230:0.377524:0.448230:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.562682:0.464664:0.579562:0.464664:0.579562:0.448258:0.562682:0.448258:0.000000:0.000000:0.000000 x:@0.612639:0.464664:0.620094:0.464664:0.620094:0.448258:0.612639:0.448258:0.000000 2 :@0.620026:0.458600:0.627570:0.458600:0.627570:0.449019:0.620026:0.449019:0.000000:0.000000 x :@0.651787:0.464664:0.662481:0.464664:0.662481:0.448258:0.651787:0.448258:0.000000:0.000000 c:@0.852426:0.464664:0.860083:0.464664:0.860083:0.448258:0.852426:0.448258:0.000000 –:@0.892001:0.465246:0.901205:0.465246:0.901205:0.447592:0.892001:0.447592:0.000000 tal que :@0.293674:0.489876:0.347846:0.489876:0.347846:0.473443:0.293674:0.473443:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.502116:0.489879:0.505318:0.489879:0.505318:0.473445:0.502116:0.473445:0.000000 La pendiente de la secante que pasa por ( 1, ( 1)) y (1, (1)) es::@0.293660:0.527963:0.749442:0.527963:0.749442:0.511530:0.293660:0.511530:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.599883:0.528546:0.609087:0.528546:0.609087:0.510892:0.599883:0.510892:0.000000 f:@0.624972:0.527963:0.629666:0.527963:0.629666:0.511557:0.624972:0.511557:0.000000 –:@0.634397:0.528546:0.643600:0.528546:0.643600:0.510892:0.634397:0.510892:0.000000 f:@0.698693:0.527963:0.703387:0.527963:0.703387:0.511557:0.698693:0.511557:0.000000 Observamos que satisface las condiciones del teorema del valor medio, por tanto, existe:@0.293660:0.599114:0.926979:0.599114:0.926979:0.582680:0.293660:0.582680:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.419941:0.599114:0.424634:0.599114:0.424634:0.582708:0.419941:0.582708:0.000000 c :@0.293667:0.626476:0.322511:0.626476:0.322511:0.610069:0.293667:0.610069:0.000000:0.000000:0.000000:0.000000 ∈ –:@0.304141:0.627058:0.336077:0.627058:0.336077:0.609404:0.304141:0.609404:0.000000:0.000000:0.000000 ( 1,1) tal que ´( ) = 2. Al resolver la ecuación ´( ) = 6 + 2 = 2, obtenemos que = 0.:@0.322253:0.626476:0.919090:0.626476:0.919090:0.610042:0.322253:0.610042:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f c:@0.417124:0.626476:0.442158:0.626476:0.442158:0.610069:0.417124:0.610069:0.000000:0.000000:0.000000:0.000000 f x:@0.641122:0.626476:0.665954:0.626476:0.665954:0.610069:0.641122:0.610069:0.000000:0.000000:0.000000:0.000000 x :@0.697264:0.626476:0.707775:0.626476:0.707775:0.610069:0.697264:0.610069:0.000000:0.000000 x :@0.882404:0.626476:0.892915:0.626476:0.892915:0.610069:0.882404:0.610069:0.000000:0.000000 Finalmente, :@0.293648:0.659898:0.382167:0.659898:0.382167:0.643464:0.293648:0.643464:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 como se ve en la gráfica de la Figura 2.11.:@0.585561:0.659904:0.888338:0.659904:0.888338:0.643470:0.585561:0.643470:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Figura 2.11:@0.578069:0.690210:0.642552:0.690210:0.642552:0.676764:0.578069:0.676764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.539246:0.724498:0.546146:0.724498:0.546146:0.710317:0.539246:0.710317:0.000000 x:@0.664682:0.903864:0.671582:0.903864:0.671582:0.889683:0.664682:0.889683:0.000000 0:@0.561335:0.892391:0.567902:0.892391:0.567902:0.880798:0.561335:0.880798:0.000000 2:@0.644544:0.892391:0.651111:0.892391:0.651111:0.880798:0.644544:0.880798:0.000000 -2:@0.491094:0.892391:0.501591:0.892391:0.501591:0.880798:0.491094:0.880798:0.000000:0.000000 10:@0.555203:0.756295:0.568337:0.756295:0.568337:0.744702:0.555203:0.744702:0.000000:0.000000 5:@0.560861:0.819882:0.567429:0.819882:0.567429:0.808289:0.560861:0.808289:0.000000 f’:@0.522256:0.863011:0.530565:0.863011:0.530565:0.848606:0.522256:0.848606:0.000000:0.000000 (0) = 2:@0.530502:0.863011:0.572121:0.863011:0.572121:0.848606:0.530502:0.848606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (-1, 6):@0.475102:0.807874:0.514377:0.807874:0.514377:0.792417:0.475102:0.792417:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (0,5):@0.583203:0.830022:0.613721:0.830022:0.613721:0.814564:0.583203:0.814564:0.000000:0.000000:0.000000:0.000000:0.000000 (1, 10):@0.615206:0.758579:0.657997:0.758579:0.657997:0.743122:0.615206:0.743122:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f(x) = x:@0.655724:0.720209:0.705886:0.720209:0.705886:0.706028:0.655724:0.706028:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.690959:0.720209:0.698986:0.720209:0.698986:0.706040:0.690959:0.706040:0.000000 2 :@0.705884:0.716458:0.715165:0.716458:0.715165:0.704865:0.705884:0.704865:0.000000:0.000000 + 2 + 5:@0.715164:0.720209:0.766719:0.720209:0.766719:0.706040:0.715164:0.706040:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.735833:0.720209:0.742733:0.720209:0.742733:0.706028:0.735833:0.706028:0.000000 Competencia :@0.078581:0.455864:0.187141:0.455864:0.187141:0.438917:0.078581:0.438917:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.078581:0.472506:0.176057:0.472506:0.176057:0.455559:0.078581:0.455559:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Teorema del valor :@0.084947:0.498279:0.223983:0.498279:0.223983:0.481610:0.084947:0.481610:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 medio:@0.084947:0.514921:0.134021:0.514921:0.134021:0.498251:0.084947:0.498251:0.000000:0.000000:0.000000:0.000000:0.000000 Si es una función :@0.084947:0.542283:0.220811:0.542283:0.220811:0.525849:0.084947:0.525849:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.101127:0.542283:0.105821:0.542283:0.105821:0.525877:0.101127:0.525877:0.000000 continua en el :@0.084947:0.558925:0.193108:0.558925:0.193108:0.542491:0.084947:0.542491:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 intervalo cerrado [ , ] :@0.084947:0.575567:0.251570:0.575567:0.251570:0.559133:0.084947:0.559133:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.217222:0.575567:0.242734:0.575567:0.242734:0.559161:0.217222:0.559161:0.000000:0.000000:0.000000 y derivable en el :@0.084929:0.592208:0.207606:0.592208:0.207606:0.575775:0.084929:0.575775:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 intervalo abierto ( , ), :@0.084929:0.608850:0.250962:0.608850:0.250962:0.592417:0.084929:0.592417:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.213485:0.608850:0.238998:0.608850:0.238998:0.592444:0.213485:0.592444:0.000000:0.000000:0.000000 entonces existe :@0.084911:0.625492:0.205957:0.625492:0.205957:0.609058:0.084911:0.609058:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c :@0.084911:0.642134:0.095808:0.642134:0.095808:0.625728:0.084911:0.625728:0.000000:0.000000 ∈:@0.095753:0.642716:0.111086:0.642716:0.111086:0.625062:0.095753:0.625062:0.000000 ( , ) tal que :@0.110994:0.642134:0.208221:0.642134:0.208221:0.625700:0.110994:0.625700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.115798:0.642134:0.141310:0.642134:0.141310:0.625728:0.115798:0.625728:0.000000:0.000000:0.000000 f b:@0.084911:0.658776:0.103594:0.658776:0.103594:0.642370:0.084911:0.642370:0.000000:0.000000:0.000000 ( ) ( ) = ´( )(:@0.089531:0.658776:0.202901:0.658776:0.202901:0.642342:0.089531:0.642342:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.112264:0.659358:0.121467:0.659358:0.121467:0.641704:0.112264:0.641704:0.000000 f a:@0.125333:0.658776:0.144016:0.658776:0.144016:0.642370:0.125333:0.642370:0.000000:0.000000:0.000000 f c b a:@0.167522:0.658776:0.236863:0.658776:0.236863:0.642370:0.167522:0.642370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –:@0.215252:0.659358:0.224456:0.659358:0.224456:0.641704:0.215252:0.641704:0.000000 ). :@0.236807:0.658776:0.254515:0.658776:0.254515:0.642342:0.236807:0.642342:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 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