66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Verified step by step guidance

1. Understand Theorem 8.1: The theorem provides a way to estimate the error in approximating a definite integral using numerical methods. Specifically, it involves the second derivative of the function and the width of the interval.
2. Identify the function and its derivatives: The given function is f(x) = cos(x²). Compute the second derivative of f(x), which will be needed to apply Theorem 8.1. Start by finding the first derivative f'(x) = -2x*sin(x²), and then compute the second derivative f''(x).
3. Determine the maximum value of |f''(x)|: To apply Theorem 8.1, find the maximum absolute value of the second derivative |f''(x)| over the interval of integration. This may involve analyzing critical points or using technology to evaluate the maximum value numerically.
4. Apply Theorem 8.1 formula: The error bound is given by the formula E ≤ (K(b-a)³)/(12n²), where K is the maximum value of |f''(x)|, [a, b] is the interval of integration, and n is the number of subintervals used in the numerical method (e.g., trapezoidal rule or Simpson's rule). Substitute the values into this formula.
5. Interpret the result: The computed upper bound provides an estimate of the maximum possible error in the numerical approximation of the integral. This helps assess the accuracy of the method used in part (a).

Key Concepts

Theorem 8.1 (Taylor's Theorem)

Theorem 8.1, often referred to as Taylor's Theorem, provides a framework for approximating 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 estimated value. It is calculated as the absolute value of the difference, providing a straightforward way to assess the accuracy of an approximation. In the context of Theorem 8.1, finding an upper bound on the absolute error helps determine how close the approximation is to the actual function.

Upper Bound

An upper bound is a value that serves as a limit, indicating that the actual value will not exceed this threshold. In calculus, particularly when estimating errors, finding an upper bound on the absolute error allows mathematicians to understand the worst-case scenario for the accuracy of their approximations. This concept is essential for ensuring that estimates remain within acceptable limits.