Mathematics
Taylor Series: Approximating Functions with Polynomials
Quick fact
The Taylor series can approximate any smooth function to any desired accuracy by including enough terms—no matter how complex the function, it can be reduced to simple algebra.
Why this is interesting
You've probably punched sin(x) into a calculator without thinking—but how does it actually compute that? It doesn't use a circle; it uses a polynomial that mimics the sine wave, based on a Taylor series.
Read the full explanation
Understanding Taylor Series: Approximating Functions with Polynomials
Imagine you need to describe the height of a roller coaster track at any point. You only have one small snapshot of the track (its height, slope, curvature, and higher-order changes) at a single point. A Taylor series uses this snapshot—specifically the derivatives at that point—to reconstruct the entire path. At first, you might just use the slope (a straight line). Adding the curvature (second derivative) makes a parabola. Adding more derivatives creates a more flexible polynomial that hugs the true shape more closely. The idea is that the more local information about the function you use, the better you can predict its behavior away from that point.
A deeper explanation
Formally, the Taylor series of a function f centered at a point a is: f(x) = f(a) + f'(a)(x-a) + f''(a)/2! (x-a)^2 + f'''(a)/3! (x-a)^3 + ... The nth term is the nth derivative evaluated at a, divided by n!, multiplied by (x-a)^n. This works because polynomials are easy to evaluate, and by matching all derivatives at the center, the polynomial has the same local 'shape' as the function—up to infinitesimally small differences. The infinite sum often converges to the exact function within an interval (the radius of convergence). In practice, we truncate the series after a few terms; the error is given by the next term (remainder), which we can bound. This is how calculators compute sin, cos, e^x, and logarithms—by using precomputed Taylor polynomials. It's also used in physics to linearize equations (e.g., small-angle approximation), in optimization, and in numerical methods for solving differential equations.