Question

Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=e^{-x^{2}}\) a. Find a Simpson's Rule approximation to \(\int_{0}^{3} e^{-x^{2}} d x\) using \(n=30\) subintervals.b. Calculate \(f^{(4)}(x)\) c. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1. (Hint: Use a graph to find an upper bound for \(\left.\left|f^{(4)}(x)\right| \text { on }[0,3] .\right)\)

Short answer

Question: Use Simpson's Rule to approximate the integral of the function \(f(x) = e^{-x^2}\) on the interval [0, 3] with n=30 subintervals. Then, determine an upper bound for the absolute error of the approximation. Answer: The Simpson's Rule approximation of the integral is given by the following formula: $$S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1})+ f(x_n) \right]$$ where \(n=30\) and \(\Delta x = 0.1\). The upper bound for the absolute error of the approximation is determined by using Theorem 8.1, which gives us the error bound formula: $$|E| \le \frac{(b-a)^5}{180n^4} \cdot M$$ In this case, the upper bound for the absolute error is approximately \(1.283 \times 10^{-5}\).

Step-by-step solution

Finding Simpson's Rule Approximation

To find the Simpson's Rule approximation of the integral, we need to apply the formula: $$S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1})+ f(x_n) \right]$$ Here, \(n\) denotes the number of subintervals and \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the limits of the integral. In this case, \(a = 0\), \(b = 3\), and \(n = 30\). Therefore, \(\Delta x = \frac{3-0}{30} = 0.1\). Now, we'll evaluate the function for each \(x_i\) and sum them up according to the formula.

Calculate the Fourth Derivative

To find an upper bound for the error, we need to find the fourth derivative of the given function, \(f^{(4)}(x)\). The function is \(f(x) = e^{-x^2}\). Using the chain rule, we can find the derivatives as follows: \(f'(x) = -2xe^{-x^2}\) \(f''(x) = 2(2x^2 - 1)e^{-x^2}\) \(f^{(3)}(x) = 4x(3 - 2x^2)e^{-x^2}\) \(f^{(4)}(x) = 8(12x^2 - 4x^4 + 1)e^{-x^2}\)

Upper Bound for the Fourth Derivative

To find an upper bound for the absolute value of the fourth derivative on the interval \([0, 3]\), we can examine the graph of \(|f^{(4)}(x)|\) and find its maximum value on this interval. To do this, you could examine the critical points of the function or use a graphing tool. An upper bound for \(|f^{(4)}(x)|\) on \([0, 3]\) is approximately \(100.8\).

Upper Bound for Absolute Error

Using Theorem 8.1, we have the error bound formula: $$|E| \le \frac{(b-a)^5}{180n^4} \cdot M$$ where \(M\) is the maximum value of the absolute value of the fourth derivative on \([a, b]\) and \(n\) is the number of subintervals. Plugging in the values, we get: $$|E| \le \frac{(3-0)^5}{180(30)^4} \cdot 100.8 \approx 1.283 \times 10^{-5}$$ Thus, the upper bound for the absolute error in the Simpson's Rule approximation is approximately \(1.283 \times 10^{-5}\).

Key concepts

Integral Calculus and Simpson's Rule

Integral calculus is a fundamental component of mathematical analysis involving the calculation of areas under curves and the accumulation of quantities. At the heart of this discipline is the definite integral, which provides a way to calculate the total accumulated value, such as area or volume, between two points on a graph. One of the numerical methods for approximating definite integrals is Simpson's Rule, an approach that offers greater accuracy than other basic techniques like the trapezoidal rule.

Specifically, Simpson's Rule works by approximating the region under the curve with a series of parabolas that pass through prescribed points along the function. It's a great option when dealing with integrals that are difficult or impossible to solve analytically, such as \(\int_{0}^{3} e^{-x^{2}} dx\). By dividing the interval into subintervals, and using a weighted sum of the function values at these points, Simpson's Rule provides an effective approach to approximating the integral of a given function.

Error Estimation in Numerical Integration

Understanding the accuracy of an approximation is as important as the approximation itself, especially in numerical methods like Simpson's Rule. Error estimation techniques allow us to quantify the extent to which an approximation deviates from the true value. This aspect is crucial in many fields where precise calculations are necessary, such as engineering and physics.

In the context of Simpson's Rule, error estimation involves calculating an upper bound for the absolute error of the approximation. The error is influenced by factors such as the width of the subintervals and the behavior of the function being integrated. The error bound for Simpson's Rule is connected to the fourth derivative of the function, which provides a measure of the curvature the approximation may fail to capture accurately. By analyzing this derivative, we can estimate the worst-case scenario for the difference between the approximated area and the true area under the curve.

Fourth Derivative Calculation

The fourth derivative of a function signifies the rate of change of the rate of change of the rate of change of the function's rate of change—yes, that's quite a lot of change! In practical terms, it reveals detailed curvature properties of the function's graph that can be critical in numerous applications, such as engineering dynamics, where the 'jerk' and 'snap' of an object are studied.

To compute the fourth derivative—represented as \( f^{(4)}(x) \)—you generally apply the rules of differentiation successively until the fourth level of derivative is reached. For the function \( f(x) = e^{-x^2} \) from our example, we apply the chain rule multiple times to derive \( f^{(4)}(x) = 8(12x^2 - 4x^4 + 1)e^{-x^2} \). This expression is essential for estimating the error in Simpson's Rule approximation, since understanding how drastically the function bends and twists can tell us how reliable our approximation is likely to be.