Geometric Similarity of Functions
This is wonderful! What an interesting set of ideas. It looks like you're saying:
Functions of the same form (such as quadratics) can be geometrically similar to each other— they are related to each other by a change of scale.
By extension, you identify three important geometric properties of these (graphs of) functions: they have proportional (a) width, (b) height, and (c) area, and the proportionality factor for area is the square of the factor for length.
Based on these four factors, you can define similarity between functions of different kinds (such as a quadratic and an exponential). They are similar, by definition, whenever they have proportional (a) width, (b) height, and (c) area. In fact, because the functions have different forms, it is even interesting when they are just equal (proportional with a factor of 1).
You suggest an interesting theorem, such as: arguing from similar (infinitesimal) triangles, if two (graphs of) functions are similar, then their average arc lengths are proportional by the same factor.
You deploy a suite of calculation methods to find intervals over which $x^2$ and $e^x$ are similar to each other in this sense.
You suggest a theorem that the derivatives of similar functions are similar over the same regions.
These are all interesting explorations of the idea, and your writing does a good job of communicating them. I have read some mathematical papers that prioritize definitions and symbols before telling the reader the most important ideas and explaining where they come from. I liked that you do a good job of explaining that your ideas of geometric similarity of function graphs come from extracting ideas from geometric similarity of functions. Also, I liked
I think you could make your paper even better by clearly separating out the definition of geometric similarity of different functions. That is, you could write, in its own paragraph, "Definition: Two functions are geometrically similar if ... "
This modification has three advantages. First, it highlights your innovative contribution (the definition) by separating it a little from the discussion of concepts that have come before, i.e. similarity of triangles, and similarity of functions of the same form. Second, it helps readers (like me) remember the difference between which properties are your definition of similarity (proportional length, width, area), and which properties are theorems that you derive from your definition (proportional arc length). Third, often readers like me want to look back up to check your definition; separating out the definition makes it easy to refer back to.
All in all, I enjoyed reading your paper, and the different directions and approaches you took in investigating the ideas.