We introduce a new proportional alpha-derivative with parameter alpha in [0,1], explore its calculus properties, and give several examples of our results. We begin with an introduction to our proportional alpha-derivative and some of its basic calculus properties. We next investigate the system of alpha-lines which make up our curved yet Euclidean geometry, as well as address traditional calculus concepts such as Rolle's Theorem and the Mean Value Theorem in terms of our alpha-derivative. We also introduce a new alpha-integral to be paired with our alpha-derivative, which leads to proofs of the Fundamental Theorem of Calculus Parts I and II, as applied to our formulas. Finally, we provide instructions on how to locate alpha-maximum and alpha-minimum values as they are related to our type of Euclidean geometry, including an increasing and decreasing test, concavity test, and first and second alpha-derivative tests.
Douglas R. Anderson
LeGare, Laura A. and Bryan, Grace K.
"The Calculus of Proportional 𝛼-Derivatives,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 18:
1, Article 2.
Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/2