IJPAM: Volume 75, No. 3 (2012)

ILLUMINATION BY TANGENT LINES

Alan Horwitz
Penn State Brandywine
25, Yearsley Mill Rd.
Media, PA 19063, USA


Abstract. Let $f$ be a differentiable function on the real line, and let $P\in G_{f}^{C}=$ all points not on the graph of $f$. We say that the illumination index of $P$, denoted by $I_{f}(P)$, is $k$ if there are $k$ distinct tangents to the graph of $f$ which pass through $P$. In section 2 we prove results about the illumination index of $f$ with $f\,^{\prime \prime }(x)\geq 0$ on $\Re $. In particular, suppose that $y=L_{1}(x)$ and $% y=L_{2}(x)$ are distinct oblique asymptotes of $f$ and let $P=(s,t)\in G_{f}^{C}$. If $\max \left( L_{1}(s),L_{2}(s)\right) <t<f(s)$, then $% I_{f}(P)=2$. If $L_{1}(s)\neq L_{2}(s)$ $\ $and $\min \left( L_{1}(s),L_{2}(s)\right) <t\leq \max \left( L_{1}(s),L_{2}(s)\right) $, then $I_{f}(P)=1$.

Finally, if $t\leq \min \left( L_{1}(s),L_{2}(s)\right) $, then $I_{f}(P)=0$. We also show that any point below the graph of a convex rational function or exponential polynomial must have illumination index equal to $2$. In section 3 we also prove results about the illumination index of polynomials.

Received: July 28, 2011

AMS Subject Classification: 26A06

Key Words and Phrases: tangent line, oblique asymptote, illumination index

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 75
Issue: 3