All functions considered here are real valued functions of a real variable. If f is a power series function but not a polynomial function, then its graph intersects the graph of any polynomial function only finitely many times over any closed bounded interval. If f is given by f(x) = Ae^kx for some nonzero real numbers A and k, then the preceding assertion can be strengthened by replacing “over any closed bounded interval” with “over (-∞, ∞)". If f and g are C² functions whose graphs intersect at a point P₀, a notion of “the graphs of f and g intersect tangentially at P₀" is defined, it implies that the graphs of f and g have the same tangent line at P₀, the converse fails even if f and g are power series functions, but that converse holds if f is of the above form f(x) = Ae^kx with g being a linear function. Examples are given to indicate the sharpness of our results. Various portions of this note could be used as enrichment material in courses ranging from honors calculus to real analysis and advanced calculus.

analytic continuation, L’Hôpital’s rule, exponential function, C² function, Taylor’s theorem, tangential intersection.

Received: April 11, 2021; Accepted: October 22, 2021; Published: November 11, 2021

How to cite this article: David E. Dobbs, On the real intersection points of the graphs of an exponential function and a polynomial function, Far East Journal of Mathematical Education 21(2) (2021), 175-206. DOI: 10.17654/ME021020175

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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