66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)

Verified step by step guidance

1. Step 1: Recall Theorem 8.1, which provides a formula for estimating the error in a Taylor polynomial approximation. The error bound is given by: $\left|R_{\mathrm{n}}\right(x\left)\right|\le \frac{M}{(n+1)!}\cdot |x-a|^n+1$, where M is the maximum value of the absolute derivative of order n+1 on the interval [a, x].
2. Step 2: Identify the function f(x) = √(sin x) and the interval [1, 2]. The problem states that |f''''(x)| ≤ 1 on this interval, which means M = 1 for the fourth derivative.
3. Step 3: Determine the degree of the Taylor polynomial used in part (a). If the polynomial is of degree n, the error bound involves the (n+1)-th derivative. For this problem, we are using the fourth derivative, so n = 3.
4. Step 4: Substitute the values into the error formula. Using n = 3, M = 1, and the interval [1, 2], the error bound becomes: $\left|R_3\right(x\left)\right|\le \frac{1}{4!}\cdot |x-1|^4$.
5. Step 5: Simplify the factorial term and the power term. Factorial calculations involve multiplying integers up to the given number, so 4! = 4 × 3 × 2 × 1 = 24. The error bound simplifies to: $\left|R_3\right(x\left)\right|\le \frac{1}{24}\cdot |x-1|^4$. Evaluate this expression for the specific x-value used in part (a) to find the upper bound on the error.

Key Concepts

Theorem 8.1 (Taylor's Theorem)

Theorem 8.1, commonly known as Taylor's Theorem, provides a way to approximate a function using its derivatives at a specific point. It states that a function can be expressed as a Taylor series, and the remainder term quantifies the error in this approximation. Understanding this theorem is crucial for estimating the accuracy of polynomial approximations of functions.

Absolute Error

Absolute error measures the difference between the true value of a function and its approximation. It is defined as the absolute value of the difference between the actual function value and the estimated value. In the context of Theorem 8.1, finding an upper bound on the absolute error helps in assessing how close the approximation is to the actual function over a specified interval.

Higher-Order Derivatives

Higher-order derivatives are derivatives of a function taken multiple times. In the context of Taylor's Theorem, the behavior of these derivatives, particularly the fourth derivative in this case, is essential for determining the error bound. The hint provided in the question indicates that the fourth derivative of the function is bounded by 1 on the interval [1,2], which is key to calculating the upper bound on the absolute error.