Consider a plotted curve $C$ on an unlabelled axes that are linearly scaled. $C$ corresponds to either the exponential curve $2^x -1$ or the square curve $x^2$. I wish to know if there is a tell-tale difference between the two, based on their shape purely.
To clarify, I am not talking about two curves plotted together and telling which is which, to accomplish that I can just see which one is asymptotically greater. Nor am I talking about a mathematical method to tell them apart. For example, I could measure the ratio of increase and expect it to stay constant for the exponential curve and decrease for the polynomial one. I don't want to do this.
In the same way that you can tell the difference between a linear curve and the curve $ x^2$ very easily visually, is there some telltale aspect of exponential curve that immediately distinguishes them to the eye? Note I used $2^x -1$ because I want it to pass through the origin to better resemble the polynomial curves.
I want strictly a visual,not magnitude related telltale difference between a polynomial's shape and an exponential one in the positive XY quadrant. Below are two images to clarify corresponding to a polynomial and an exponential. Again, comparative analysis is not desired; take each curve as if it were presented in isolation.
For further clarification, there must not exist a shifting, or modification using sub-quadratic polynomial terms that renders the visual difference null.
if you look closely around zero you can see that one of the curves is initially horizontal, whereas the other has a nonzero slope.
Slope is related to differentiation:
the derivative of $x^2$ is $2x$, which at $x=0$ is zero
the derivative of $2^x - 1 = e^{x \log(2)} - 1$ is $\log(2) e^{x \log(2)}$, which as $x=0$ is simply $\log(2)$.
Hence in this case the curve that is horizontal at $x=0$ is the quadratic polynomial, whereas the exponential starts with a slope.
Note that if the polynomial were offset, for example considering $(x+1)^2 - 1$, this visual clue would fail.