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Taylor series

Approximating a function by polynomials built from its derivatives.

A Taylor series rebuilds a function from its derivatives at one point: f(x₀) + f′(x₀)(x − x₀) + f″(x₀)(x − x₀)²/2! + …. The dashed curve is the polynomial; watch how it hugs the function near the centre and drifts away from it.

Иштөө мисалы: taylor series of e^x

Calculus · Taylor-Series

Taylor series of e^(x)

e^{x}

Аткаруу

  1. f(x) = e^{x}

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

  2. f^{(0)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 0 at the centre.

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

    Derivative 1 at the centre.

  4. f^{(2)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 2 at the centre.

  5. f^{(3)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 3 at the centre.

  6. f^{(4)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 4 at the centre.

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

    Derivative 5 at the centre.

  8. \frac{x^{5}}{120} + \frac{x^{4}}{24} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1

    Assemble the terms up to degree 5.

Жауап
\frac{x^{5}}{120} + \frac{x^{4}}{24} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1 + O((x-0)^{6})

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Кээ бирлери Calculus

LimitsDerivativesIntegralsDefinite integralsSeries and sumsMaxima and minima