Calcolatrice delle serie di Taylor e Maclaurin (Power)
Trovare la serie di Taylor/Maclaurin passo dopo passo
La calcolatrice troverà l'espansione in serie di Taylor (o potenza) della funzione data nell'intorno del punto dato, con i passi indicati. È possibile specificare l'ordine del polinomio di Taylor. Se si desidera il polinomio di Maclaurin, è sufficiente impostare il punto su 0.
Solution
Your input: calculate the Taylor (Maclaurin) series of sin(x) 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)=n∑k=0f(k)(a)k!xk=5∑k=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)=sin(x)
Evaluate the function at the point: f(0)=0
Find the 1st derivative: f(1)(x)=(f(0)(x))′=(sin(x))′=cos(x) (steps can be seen here).
Evaluate the 1st derivative at the given point: (f(0))′=1
Find the 2nd derivative: f(2)(x)=(f(1)(x))′=(cos(x))′=−sin(x) (steps can be seen here).
Evaluate the 2nd derivative at the given point: (f(0))′′=0
Find the 3rd derivative: f(3)(x)=(f(2)(x))′=(−sin(x))′=−cos(x) (steps can be seen here).
Evaluate the 3rd derivative at the given point: (f(0))′′′=−1
Find the 4th derivative: f(4)(x)=(f(3)(x))′=(−cos(x))′=sin(x) (steps can be seen here).
Evaluate the 4th derivative at the given point: (f(0))′′′′=0
Find the 5th derivative: f(5)(x)=(f(4)(x))′=(sin(x))′=cos(x) (steps can be seen here).
Evaluate the 5th derivative at the given point: (f(0))(5)=1
Now, use the calculated values to get a polynomial:
f(x)≈00!x0+11!x1+02!x2+−13!x3+04!x4+15!x5
Finally, after simplifying we get the final answer:
f(x)≈P(x)=x−16x3+1120x5
Answer: the Taylor (Maclaurin) series of sin(x) up to n=5 is sin(x)≈P(x)=x−16x3+1120x5