Teaching calculus: Construction of exponential and trigonometric functions as Taylor series solutions of differential equations with initial conditions

Abstract

Abstract. In this work we show how to introduce the exponential and trigonometric functions based on the concepts and methods of differential calculus. We present a construction of these functions as natural generalizations of the class of polynomial functions, by solving certain elementary differential equations that describe some of the most important dynamical laws that govern phenomena in nature. With this approach, the meaning and purpose of calculus is rescued as the branch of mathematics for the description of continuous change and movement. Starting from the problem of finding the solutions of differential equations that describe the growth of populations and the dynamics of the harmonic oscillator, we obtain from the initial conditions of the problem and the application of Taylor's Theorem, the coefficients of the power series that define the exponential function and the trigonometric functions, showing in parallel the importance of the concept of limit as the foundation of differential calculus.