Without derivatives or limits: from visual and geometrical points of view to algebraic methods for identifying tangent lines

Abstract

Usually, the tangent line is considered to be a calculus notion. However, it is also a graphical and an algebraic notion. The graphical frame, where our primary conceptions are conceived, could give rise to algebraic methods to obtain the tangent line to a curve. In this pre-calculus perspective, two methods are described and discussed according to their potential for secondary students and teacher training.

Keywords:

• tangent
• graphical
• calculus
• algebraic
• Descartes
• transition line

Notes

Some students drew a line perpendicular to an imaginary radius while others said that there was no tangent, since the curve was not a circle. (The tangent to a circle appears in most syllabi around grade 8.)

This is directly related to the visual perception of two non-vertical half-tangents at point A.

In Crombie and Grant,[7] the example is a generic quadratic function.

In this paper, we do not consider the case of vertical tangents, although they can be equally, but separately, determined using the method [6].

Various examples are given in Vivier.[5]