Linear Fit Issue

Question

Consider a quantity $Q$ that changes as a function of time. The function $Q(t)$ is not explicitly known. We know that $Q(t_{0})=Q_{0}$. Assume for $t-t_{0}\le\epsilon$, we have a method to estimate one additional point on the $Q(t)$ curve. This estimation gets better for small $\epsilon$. Given the inital point and another point, for a fixed epsilon, it seems that one can approximate $Q(t)$ by a linear function. However the nature of the linear function seems to change based on $\epsilon$ that one picks. What is the best way to estimate $Q(t)$ in this case? Thanks -abiyo

Answer 1

Here is what you are missing: because you have chosen a linear model, $T = at + c$, so $\frac{t_1 - t_0}{T_1 - T_0} = \frac{1}{a}$. By the nature of the linear model that you have chosen, $T(t) \in O(t)$, so the limit of their ratio must be real-valued.