{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.


sin 0.3, n = 4

Verified step by step guidance

1. Identify the function and the point of expansion: here, the function is \(f(x) = \sin x\), and the Taylor polynomial is centered at \(0\) (Maclaurin series).
2. Recall the Taylor series for \(\sin x\) centered at \(0\): \(\sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}\). The 4th-order Taylor polynomial means including terms up to degree 4, which for \(\sin x\) corresponds to terms up to \(x^3\) because the series has only odd powers.
3. Determine the remainder term (Lagrange form) for the Taylor polynomial of order \(n=4\) centered at 0: \(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\), where \(c\) is some value between \(0\) and \(x\).
4. Since \(n=4\), the remainder involves the 5th derivative of \(\sin x\). Find \(f^{(5)}(x)\) and note its maximum absolute value on the interval between \(0\) and \(0.3\) to bound the error.
5. Use the bound on \(|f^{(5)}(c)|\) and plug into the remainder formula: \(|R_4(0.3)| \leq \frac{M}{5!} |0.3|^5\), where \(M\) is the maximum of \(|f^{(5)}(c)|\) for \(c\) in \([0, 0.3]\). This gives a bound on the error in approximating \(\sin 0.3\) by the 4th-order Taylor polynomial.

Key Concepts

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using a finite sum of its derivatives at that point. The nth-order Taylor polynomial centered at 0 (Maclaurin polynomial) uses derivatives up to order n to estimate the function's value close to zero.

Remainder (Error) Term in Taylor Series

The remainder term quantifies the difference between the actual function value and its Taylor polynomial approximation. It provides an upper bound on the error, ensuring the approximation's accuracy within a specified range.

Bounding the Error for sin(x)

For sin(x), the remainder after the nth-order polynomial can be bounded using the next derivative's maximum value on the interval. Since derivatives of sin(x) are bounded by 1, this helps estimate the maximum possible error when approximating sin(0.3) with a 4th-order polynomial.