A new perspective on teaching the natural exponential to engineering students

Abstract

The natural exponential and logarithm are typically introduced to undergraduate engineering students in a calculus course using the notion of limits. We here present an approach to introduce the natural exponential/logarithm through a novel interpretation of derivatives. This approach does not rely on limits, allowing an early and intuitive introduction of these functions. The question behind our contribution is whether one can introduce derivatives using only polynomials and power series? Motivated by an earlier exposure of engineering students to differential equations, we demonstrate that the natural exponential/logarithm can arise from two common differential equations. Our limit-free approach to derivatives provides an intuitive interpretation of $\mathrm{e}$, the Euler number, and an intuitive introduction of time constants in first-order dynamical systems.

Keywords:

• Exponential function
• logarithmic function
• power series
• ordinary differential equations

2010 Mathematics Subject Classifications:

• 97E02
• 97M02

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Notice the subtle difference between the usage of the two terms: slope of a line versus derivative of a function

2 Since $\mathrm{d}x$ can be chosen arbitrarily (large or small) just like $\delta x$, there is no point in distinguishing between the two and we can take $\mathrm{d}x=\delta x$.