Taylor Series Calculator

This calculator builds a Taylor polynomial for sin(x), cos(x), eˣ, or ln(x) — chosen via Function — around a chosen Center Point, using the requested Number of Terms, then evaluates that polynomial at x and compares it against the true function value as the Error. Of the numeric inputs, Evaluate at x is the only one with a measurable effect on the Taylor Approximation under a small nudge at the default settings — though switching Function itself moves the approximation more than any single numeric input does, since it swaps the underlying series entirely. That also means Evaluate at x's direction of pull is not fixed: for the default sin(x) series it raises the approximation, but for the cos(x) series the same nudge can lower it instead, since sine and cosine move in opposite directions around most points — a fact about which function is selected, not about Evaluate at x on its own.

Center Point defaults to 0 and gets excluded from that kind of check entirely (a percentage nudge of zero stays zero), while Number of Terms happens to show zero measured effect for an unrelated reason: for the default sin(x) series centered at 0, every even-indexed term is exactly zero (sin's even derivatives at 0 vanish), so nudging the term count from 5 down to 4 drops a term that was already contributing nothing — a coincidence of this particular function and center, not evidence that the term count is generally unimportant. What the calculator does NOT account for: with Number of Terms capped at 15 and no warning when x is evaluated far from the Center Point, the polynomial can diverge wildly from the true function — the calculator's own default sin(x) series reports an "approximation" of about -666.67 at x = 16, even though sin(x) is mathematically bounded between -1 and 1 everywhere.