It is known that, given a function $f(x)$, plugging in the dual number $x+\varepsilon$, where $\varepsilon^2=0$, yields $f(x) + f'(x)\varepsilon$. For example: $$f(x) = x^3\\f(x+\varepsilon)=(x+\varepsilon)(x+\varepsilon)(x+\varepsilon)\\f(x+\varepsilon) = (x^2+2x\varepsilon)(x+\varepsilon)\\f(x+\varepsilon) = x^3 + 3x^2\varepsilon$$ We observe that $x^3$ is the value of $f(x)$ and $3x^2$ is the derivative of $f(x)$. I would like to know if the process is reversible in such a way that, given a few data points demonstrating the derivative of some function $g(x)$, such as $$g(1+\varepsilon) = a_1 + \varepsilon\\g(2+\varepsilon) = a_2 + 4\varepsilon\\g(3+\varepsilon) = a_3 + 9\varepsilon\\g(4+\varepsilon) = a_4+16\varepsilon,$$could we algorithmically recover $g(x)$? We as humans can see that the pattern among the dual parts is the series of squares, so $g'(x)=x^2$, telling us that $g(x)$ must be $\int{g'(x)}dx$, which would be $\frac{1}{3}x^3+C$. I don't have much hope for it due to the significant hurdles in the way, such as the data-fitting aspect of trying to decide what function is being represented by the dual parts and the fact that we are losing information when we multiply $\varepsilon$ by itself. It would be extremely convenient to have a quick and simple way of computing antiderivatives though.
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