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.

Author Bio

Laura LeGare is a junior at Concordia College in Moorhead, MN, majoring in Mathematics with a minor in Music. Some of her favorite hobbies include singing, reading, cycling, and traveling. She is grateful for the opportunities she's had during her time at Concordia to study abroad in Italy, as well as travel nationally and internationally as a member of The Concordia Choir, most notably during their 2017 tour to Germany and Austria. Laura hopes to participate in a mathematics research program next summer, and after her time at Concordia plans to attend graduate school in mathematics.

Grace Bryan is a junior at Concordia College majoring in Mathematics. She enjoys growing indoor desert plants, and is looking forward to her second year of working for Concordia's Res Life as a DA.