8/22/2023 0 Comments Ejemplo de intervalo abiertoAplicaciones de la derivada /Relaciones entre funcionesĮjemplos 1 Demostrar que cos2( x ) + sen2( x ) = 1. This point has always to be taken into account when using the Mean Value Theorem. The point here is that the function f is not defined for x=0, that is, the domain of definition of the function is the union of two intervals, not a single interval. Clearly f(x)=1 if x is positive, and f(x) = -1 if x is negative. For example, the derivative of the function f(x) = absolute value of x divided by x is everywhere 0, but the function is not a constant function. What you just said is true only if the function is defined on an interval. The other is that if the derivative of a function is positive, then the function is increasing. One is the fact that, if the derivative of a function vanishes, then the function is a constant function. (M) The Mean Value Theorem has two important consequences. Por ejemplo, la función f(x) = |x|/x definida para x ≠ 0, es derivable en su dominio, su derivada es nula, pero la función no es constante. Por lo tanto estas conclusiones no se pueden aplicar en casos generales donde la función no esté definida en un intervalo determinado. NOTA: Se supone que f(x) es derivable en un intervalo abierto. Teorema B Si f es derivable en un intervalo abierto y f’(x) > 0 para todo x excepto por un numero finito de puntos, entonces f es estrictamente creciente. Las aplicaciones de la derivación para probar igualdades y desigualdades están basadas en una consecuencia del Teorema del Valor Medio: Teorema A Si f es derivable en un intervalo abierto y f’(x) = 0 para todo x, entonces f es una función constante. Aplicaciones de la derivada /Relaciones entre funciones We use the Mean Value Theorem to show that certain equalities between functions are true. This has to do with proving complicated formulae. (F) In this module we consider one particular application of the Mean Value Theorem. The study of functions depends on the Mean Value Theorem, and many of the fundamental results of calculus need the Mean Value Theorem. (M) The Mean Value Theorem is a corner stone of calculus.
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