Generalized Approximaters and Approximaters for Exponential Calculus

Anthology of Computers and the Humanities: Under Review Generalized Approximaters and Approximaters for Exponential Calculus Anonymous Submission Abstract In this paper we will be introducing the notion of tangents lines being linear approximators to a curve. We generalize this notion to Multiplicative Calculus to show that we get in analogy a tangent exponential curve or more precisely we get a multiplcative approximator. We generalize this to give a recursive pattern for generating higher approximators and find an analogy in Exponential Calculus Keywords: computers, humanities proceedings 1 Introduction The first thing we must do is we must define all the required differential operators . Let us say our ordinary calculus is a special case of this generalization called Additive Calculus. DAdd =d dx (1) Similarly we define for Multiplicative Calculus: DMult =e d dx (2) The analogous notion for Exponential Calculus is not clear. But the respective "Tangent curves" are still well defined via simple analytic reasoning let TAbe the tangent line or linear approximator then TMis the multiplicative approximator and TEis the exponential approximator 1.1 Approximator Curves So the Tangent line to a curve f(x) is given by TA=f′(a)·(x−a) + f(a)(3) Similarly the Multiplicative Approximator to a curve f(x) is given by TM=f′(a)(x−a)·f(a)(4) Note: We prefer "Approximator" over "Tangent" So the Tangent line to a curve f(x) is given by TE= ((x−a)f′(a))f(a)(5) where abis bexponentiated atimes Paper under review. ©2025 by the authors. Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). 1 Figure 1: All the approximators graphed 2 Calculating TEwithout direct tetration 2.1 Tetrating without Tetration TE=(x−a)f′(a)f(a) Now, xz=ezx−1+zx−2Rx 0ln t dt (x−a)z=ezx−a−1+zx−a−2Rx 0ln t dt z=f′(a) TE=e[f(a)f′(a)(x−a−1)+f(a)f′(a)x−a−2Rx−a 0ln(t)dt] Take a special case, Let f(x) = x2⇒f′(x) = 2x f(a) = a2, f′(a)=2a TA= 2a(x−a) + a2 Tm= (2a)(x−a)·a2 TE=e[a2(2a)(x−a−1)+a2(2a)x−a−2Rx−a 0ln(t)dt] When graphed side by side the tangents look like this Acknowledgements This unnumbered section should be blank when submitting your paper. After review, you may include lists of people and organizations who supported the work. 2 A First Appendix Section Appendix sections should be ordered by letters rather than numbers, and their contents do not count towards the paper’s length limit. Appendix sections may also contain additional tables and figures. 3