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Taylor Series

Copy a function with a polynomial, one derivative at a time. sin x and eˣ give way everywhere; ln(1 + x) only inside a radius of 1.

sin x · step 1 of 25

— f(x) = sin x— Taylor polynomial, terms up to x¹| error at the probe

Speed

A Taylor polynomial copies a function using only what happens at one point: its value, its slope, its curvature, and so on. Each new term matches one more derivative at x = 0, and the copy hugs the curve over a wider range.

\sin x \approx x

P(1) = 1, \quad \sin(1) = 0.841471, \quad |\text{error}| = 0.158529

error at x = 10.158529

error at x = 32.85888

error at x = 66.279415

Taylor Series

Copy a function with a polynomial, one derivative at a time. sin x and eˣ give way everywhere; ln(1 + x) only inside a radius of 1.

sin x · step 1 of 25

— f(x) = sin x— Taylor polynomial, terms up to x¹| error at the probe

Speed

\sin x \approx x

P(1) = 1, \quad \sin(1) = 0.841471, \quad |\text{error}| = 0.158529

error at x = 10.158529

error at x = 32.85888

error at x = 66.279415

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Taylor Series

See also

Taylor Series

See also