Taylor series express functions locally as a power series. They can be applied to find unknown value f(x) in terms of known value f(x0) when x is close to x0. This is quite commonplace.
Consider the formula for continuous compounding. A loan of principal P continuously compounded grows to f(t) = P exp(rt) at time t. This can be approximated as f(t) ~= P (1 + rt) for small t, but the error grows exponentially in t.
More generally, in a neighborhood of x0, any analytic function f(x) can be written as f(x; x0) = sum_{i=1}^n (x-x0)^n f^n(x0)/n! where f^n is the nth derivative of f. This works on multivariable functions (Black-Scholes) as well as on multidimensional functions (Linear Regression).