{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


sin x ≈ x − x³/6 on [π/4, π/4]

Verified step by step guidance

1. Identify the Taylor polynomial used for the approximation. Here, \( \sin x \) is approximated by \( x - \frac{x^3}{6} \), which is the Taylor polynomial of degree 3 centered at 0 (Maclaurin polynomial).
2. Recall the remainder term (Lagrange form) for the Taylor polynomial of degree 3: \( R_3(x) = \frac{f^{(4)}(c)}{4!} x^4 \), where \( c \) is some value between 0 and \( x \).
3. Determine the fourth derivative of \( \sin x \). Since \( f(x) = \sin x \), the derivatives cycle every four steps: \( f^{(4)}(x) = \sin x \).
4. Find the maximum value of \( |f^{(4)}(c)| = |\sin c| \) on the interval \( [\frac{\pi}{4}, \frac{\pi}{4}] \). Since the interval is a single point, \( c = \frac{\pi}{4} \), so evaluate \( |\sin(\frac{\pi}{4})| \).
5. Use the remainder formula to write the error bound: \( |R_3(x)| \leq \frac{|\sin c|}{4!} |x|^4 \). Substitute the values for \( c \) and \( x \) from the interval to get the bound on the error.

Key Concepts

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using derivatives at that point. For sin x, the polynomial x − x³/6 is the third-degree Taylor polynomial centered at 0, providing an approximation of sin x near 0 by matching its value and derivatives up to the third order.

Remainder Term (Lagrange Form)

The remainder term quantifies the error between the actual function and its Taylor polynomial approximation. The Lagrange form expresses this error as a function of the next derivative evaluated at some point in the interval, allowing us to bound the maximum error on the given interval.

Error Bound on a Closed Interval

To find an error bound on [π/4, π/4], we evaluate the maximum absolute value of the relevant derivative in the remainder term over the interval. This maximum value, combined with the formula for the remainder, gives a guaranteed upper bound on the approximation error.