Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the series to approximate $x \sin \left(x^{2}\right)$ for 0

Question

Find the 12th-degree Taylor polynomial for $x \sin \left(x^{2}\right)$ centered at x = 0.Suppose you use the first two non-zero terms of the series to approximate $x \sin \left(x^{2}\right)$ for 0 < x < 1.Is the magnitude of the error always less than 0.011?

Quizplus · Accepted Answer

The first two non-zero terms of the Taylor series for $x \sin(x^2)$ are $x^3$ and $-\frac{x^7}{6}$. The error term for a Taylor polynomial is given by the next term in the series, which is $\frac{x^{11}}{120}$. For $0 < x < 1$, $\frac{x^{11}}{120} < 0.011$, ensuring the error is always less than 0.011.

Find the 12th-Degree Taylor Polynomial For xsin⁡(x2)x \sin \left(x^{2}\right)xsin(x2) Centered at X = 0

Find the 12th-Degree Taylor Polynomial For $x \sin \left(x^{2}\right)$ Centered at X = 0