Question

The error formula for Simpson's rule depends on a. \(f(x)\) b. \(f^{\prime}(x)\) c. \(f^{(4)}(x)\) d. the number of steps

Short answer

c. \(f^{(4)}(x)\)

Step-by-step solution

Understand Simpson's Rule

Simpson's rule is a method for numerical integration, which approximates the integral of a function by dividing the area under the curve into an even number of intervals. It uses a quadratic polynomial to approximate sections of the curve.

Define the Error Formula for Simpson's Rule

The error involved in Simpson's Rule can be expressed with the formula: \[E = \frac{-(b-a)^5}{180n^4}f^{(4)}(c)\]where \(a\) and \(b\) are the limits of integration, \(n\) is the number of subintervals, and \(c\) is some number in the interval \([a, b]\). The derivative \(f^{(4)}(c)\) indicates the fourth derivative of the function.

Identify the Right Answer

From the error formula presented in Step 2, note that the error in Simpson’s rule directly depends on the fourth derivative of the function. This means the correct answer is related to the term that involves the fourth derivative of \(f(x)\).

Key concepts

Numerical Integration and Simpson's Rule

Numerical integration is a technique used to estimate the value of definite integrals when an exact analytical solution is difficult or impossible to obtain. One of the popular methods for numerical integration is Simpson's Rule. It provides an approximation to the integral of a function over a given interval. This is achieved by breaking the interval into an even number of subintervals and using parabolic arcs instead of straight lines to approximate the curve. Simpson's Rule pairs sub-intervals and approximates the area under the curve using a quadratic polynomial for every two sub-intervals. This results in more accurate calculations compared to linear approximations. By using higher-order polynomials, Simpson's rule can yield results with smaller errors for functions that are smooth and well-behaved over the interval. The basic idea is to compute integrals using:

• Subdivision of the integral into smaller parts for easier approximation.
• Using quadratic polynomials to estimate the area under graphical curves.
• A specific weight combination of the endpoints and midpoints of each subinterval.

Employing Simpson’s Rule can be highly effective for integrals over functions that can be well approximated by polynomials.

Error Formula in Simpson's Rule

The accuracy of an approximation using Simpson's Rule can be expressed through its error formula. This provides a way to quantify how close the approximate result is to the actual integral value. It is crucial to evaluate the potential error involved to understand the reliability of the integration result.Simpson's Rule poses the following error formula:\[E = \frac{-(b-a)^5}{180n^4}f^{(4)}(c)\]Here,

• \((b-a)\) is the length of the integration interval.
• \(n\) represents the number of subintervals, which must be even.
• \(f^{(4)}(c)\) is the fourth derivative of the function evaluated at some point \(c\) within the integration interval \([a, b]\).

This formula indicates that the error decreases as the number of intervals increases and is inversely proportional to the fourth power of the number of subintervals. Thus, increasing the subintervals decreases the potential error. Evaluating the fourth derivative allows us to estimate how significant the error might be, especially for functions that may not have equally distributed changes across the interval.

Fourth Derivative and Its Role in Simpson's Rule

The fourth derivative is a crucial term in understanding the error in Simpson's Rule. The error formula of Simpson's Rule includes a factor of the fourth derivative of the function involved. But why is this the case? The fourth derivative measures the rate at which the third derivative changes. It essentially gives information about the variation of the function's curvature. When leveraging Simpson's Rule, using quadratic polynomials naturally involves considering higher derivatives to understand the smoothness and curvature of the function. The presence of the fourth derivative in the error estimation formula provides insight into how the function behaves in higher dimensions. If the fourth derivative is small or does not vary significantly, the error in Simpson's Rule will also be smaller, making it highly accurate for smooth functions. Therefore, knowing the nature of the fourth derivative is pivotal for predicting and minimizing error when applying Simpson’s Rule. In practical scenarios, understanding and calculating the fourth derivative helps tailor the application of Simpson's Rule. It ensures optimal selection of subintervals, enhancing the preciseness of numerical integration results for complex functions.