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Taylor approximation and error

How well a polynomial stands in for a function, and how far you can trust it.

Near the centre the Taylor polynomial is nearly the function; the error is controlled by the next derivative and the distance from the centre. This is why calculators can compute sin: a handful of terms inside a small interval, plus symmetry.

ตัวอย่างที่ทำงานแล้ว: taylor series of sin(x)

Calculus · Taylor-Series

Taylor series of sin(x)

\sin{\left(x \right)}

ก้าวต่อก้าว

  1. f(x) = \sin{\left(x \right)}

    Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

  2. f^{(0)}(0) = \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 0 at the centre.

  3. f^{(1)}(0) = \cos{\left(x \right)}\big|_{x=0} = 1

    Derivative 1 at the centre.

  4. f^{(2)}(0) = - \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 2 at the centre.

  5. f^{(3)}(0) = - \cos{\left(x \right)}\big|_{x=0} = -1

    Derivative 3 at the centre.

  6. f^{(4)}(0) = \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 4 at the centre.

  7. f^{(5)}(0) = \cos{\left(x \right)}\big|_{x=0} = 1

    Derivative 5 at the centre.

  8. \frac{x^{5}}{120} - \frac{x^{3}}{6} + x

    Assemble the terms up to degree 5.

เปิดเผยคำตอบ
\frac{x^{5}}{120} - \frac{x^{3}}{6} + x + O((x-0)^{6})

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Sequences and their limitsConvergence of seriesImproper integralsFunctions of a complex variable