Local Polynomial Approximation

Calculus 2 · Axiom Academy

Local Polynomial Approximation A GPS chip can't re-solve a satellite's orbit every millisecond — so it does what Taylor did: trade the hard curve for a polynomial that matches it at one instant. A satellite broadcasts its position and a few derivatives at one moment ; your phone must fill in every instant between — match more derivatives at that point , and a simple polynomial tracks the real orbit. Three moves with one idea. Match more derivatives, hug the curve longer The blue curve is the satellite's true position. The red curve is the Taylor polynomial built at t = 0 — order 1 is just the tangent line. Crank the order and watch each extra term grab the curve a little farther out. Pins only the position at t = 0 — a flat guess. Adds the velocity — the tangent line. Adds the acceleration — now it curves the right way. Matches the third derivative — one more derivative pinned down. How far from the point can you trust it? Take the tangent-line fit — position and velocity at t = 0, the order-1 polynomial. Now drag how far past the broadcast moment you ask about. Right at t = 0 the error is zero; the farther you reach, the more that straight-line guess drifts off the curving orbit. How many terms does the chip actually need? Every term costs the processor multiplications. So an engineer truncates: keep just enough terms to hit the accuracy spec over the window between broadcasts. Drag the order and watch the worst-case error fall — and the payoff from each new term shrink.

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