Some Basic Properties of Differentiable Functions

Abstract

Rolle’s theorem and the mean value theorems of Lagrange and Cauchy are proved for differentiable functions. Next, Fermat’s theorem concerning relative extreme values is given. Then Taylor’s formula is investigated, and in particular, its remainder term is given in the different forms due to Lagrange, Cauchy, and Peano. Furthermore, it is indicated why sometimes the Taylor series of a function does not converge to that fuction. The Maclaurin series for elementary functions is derived and power series in both real and complex variables are studied. Finally, Euler’s formulas are proved.