66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

Verified step by step guidance

1. Step 1: Recall Theorem 8.1, which provides a method to estimate the error in numerical integration. The error bound is given by \( E \leq \frac{K(b-a)^3}{24n^2} \), where \( K \) is the maximum value of the second derivative of \( f(x) \) on the interval \([a, b]\), \( n \) is the number of subintervals, and \( [a, b] \) is the interval of integration.
2. Step 2: Compute the second derivative of \( f(x) = \sin(e^x) \). Start by finding the first derivative: \( f'(x) = \cos(e^x) \cdot e^x \). Then, differentiate again to find \( f''(x) = -\sin(e^x) \cdot (e^x)^2 + \cos(e^x) \cdot e^x \).
3. Step 3: Determine the maximum value of \( f''(x) \) on the interval \([a, b]\). This involves analyzing \( f''(x) \) and finding its critical points by setting \( f'''(x) = 0 \) and solving for \( x \). Evaluate \( f''(x) \) at these critical points and endpoints of the interval to find \( K \).
4. Step 4: Plug the values of \( K \), \( b-a \) (the length of the interval), and \( n \) (the number of subintervals) into the error formula \( E \leq \frac{K(b-a)^3}{24n^2} \). This will give the upper bound on the absolute error.
5. Step 5: Simplify the expression for \( E \) as much as possible without calculating the final numerical value. This provides the theoretical upper bound for the error in the estimate.

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 polynomials. It states that a function can be expressed as a Taylor series around a point, and the remainder term gives an estimate of the error involved in this approximation. Understanding this theorem is crucial for estimating the accuracy of function approximations, particularly when dealing with functions like f(x) = sin(eˣ).

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, calculating the upper bound on absolute error helps determine how close the approximation is to the actual function, which is essential for assessing the reliability of the estimate.

Upper Bound

An upper bound is a value that serves as a limit on the size of a quantity, ensuring that the actual value does not exceed this limit. In the context of estimating errors using Theorem 8.1, finding an upper bound on the absolute error provides a way to quantify the worst-case scenario for the approximation's accuracy. This concept is vital for understanding the reliability of numerical estimates in calculus.