In my AP Calculus BC class, we recently explored Taylor series—a topic that combines mathematical precision with philosophical depth. As we worked through how derivatives at a single point can approximate a function's behavior nearby, I started thinking about how this process reflects larger ideas about knowledge and prediction. While Taylor series are tools for solving problems in calculus, they are also windows into how local information can illuminate broader patterns and behaviors. What Is a Taylor Series? The Taylor series uses information about a function's derivatives at a single point to approximate that function near the point. The formula for a Taylor series expansion around a point a is: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) + f ′ ′ ( a ) 2 ! ( x − a ) 2 + f ( 3 ) ( a ) 3 ! ( x − a ) 3 + ⋯ This formula is built step by step, incorporating more terms to reflect higher-order behaviors of the function. The zeroth term, f ( a ) f(a) , gives the function's value at...