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Kalkulator serii Taylor i Maclaurin (Power)

Znajdź serię Taylora/Maclaurina krok po kroku

Kalkulator znajdzie rozwinięcie szeregu Taylora (lub potęgowego) danej funkcji wokół podanego punktu, z pokazanymi krokami. Można określić rząd wielomianu Taylora. Aby uzyskać wielomian Maclaurina, wystarczy ustawić punkt na 0.

Enter a function:

Enter a point:

For Maclaurin series, set the point to .

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Evaluate the series and find the error at the point

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Solution

Your input: calculate the Taylor (Maclaurin) series of x33x2 up to n=5

A Maclaurin series is given by f(x)=k=0f(k)(a)k!xk

In our case, f(x)P(x)=nk=0f(k)(a)k!xk=5k=0f(k)(a)k!xk

So, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.

f(0)(x)=f(x)=x33x2

Evaluate the function at the point: f(0)=0

  1. Find the 1st derivative: f(1)(x)=(f(0)(x))=(x33x2)=3x(x2) (steps can be seen here).

    Evaluate the 1st derivative at the given point: (f(0))=0

  2. Find the 2nd derivative: f(2)(x)=(f(1)(x))=(3x(x2))=6x6 (steps can be seen here).

    Evaluate the 2nd derivative at the given point: (f(0))=6

  3. Find the 3rd derivative: f(3)(x)=(f(2)(x))=(6x6)=6 (steps can be seen here).

    Evaluate the 3rd derivative at the given point: (f(0))=6

  4. Find the 4th derivative: f(4)(x)=(f(3)(x))=(6)=0 (steps can be seen here).

    Evaluate the 4th derivative at the given point: (f(0))=0

  5. Find the 5th derivative: f(5)(x)=(f(4)(x))=(0)=0 (steps can be seen here).

    Evaluate the 5th derivative at the given point: (f(0))(5)=0

Now, use the calculated values to get a polynomial:

f(x)00!x0+01!x1+62!x2+63!x3+04!x4+05!x5

Finally, after simplifying we get the final answer:

f(x)P(x)=3x2+x3

Answer: the Taylor (Maclaurin) series of x33x2 up to n=5 is x33x2P(x)=3x2+x3