How Large Should $n$ Be To Guarantee That The Simpson's Rule Approximation To $\int_{-4}^2\left(-x^4-4 X^3+48 X^2-4 X-3\right) Dx$ Is Accurate To Within 0.1?$n =$

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Introduction

Simpson's Rule is a method for approximating the value of a definite integral. It is a powerful tool for numerical integration, but it requires careful consideration of the number of subintervals, $n$, to achieve the desired level of accuracy. In this article, we will explore how to determine the minimum value of $n$ required to guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1.

Background

The given integral is $\int_{-4}^2\left(-x^4-4 x^3+48 x^2-4 x-3\right) dx$. To approximate this integral using Simpson's Rule, we need to divide the interval $[-4, 2]$ into $n$ subintervals of equal width. The width of each subinterval is given by $\Delta x = \frac{2 - (-4)}{n} = \frac{6}{n}$.

Simpson's Rule

Simpson's Rule states that the approximation to the integral is given by:

$$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]$$

where $x_i = a + i \Delta x$ for $i = 0, 1, 2, \ldots, n$.

Error Bound

The error bound for Simpson's Rule is given by:

$$\left| \int_{a}^{b} f(x) dx - \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] \right| \leq \frac{M(b-a)^5}{180n^4}$$

where $M$ is the maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[a, b]$.

Determining the Minimum Value of n

To determine the minimum value of $n$ required to guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1, we need to find the maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[-4, 2]$.

Finding the Fourth Derivative

To find the fourth derivative of $f(x)$, we need to differentiate $f(x)$ four times.

$$f(x) = -x^4 - 4x^3 + 48x^2 - 4x - 3$$

$$f'(x) = -4x^3 - 12x^2 + 96x - 4$$

$$f''(x) = -12x^2 - 24x + 96$$

$$f'''(x) = -24x - 24$$

$$f''''(x) = -24$$

Finding the Maximum Value of the Absolute Value of the Fourth Derivative

The maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[-4, 2]$ is simply the absolute value of the fourth derivative, which is 24.

Determining the Minimum Value of n

Now that we have found the maximum value of the absolute value of the fourth derivative of $f(x)$, we can use the error bound formula to determine the minimum value of $n$ required to guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1.

$$\frac{M(b-a)^5}{180n^4} \leq 0.1$$

$$\frac{24(2-(-4))^5}{180n^4} \leq 0.1$$

$$\frac{24(6)^5}{180n^4} \leq 0.1$$

$$\frac{24(7776)}{180n^4} \leq 0.1$$

$$\frac{186624}{180n^4} \leq 0.1$$

$$1048.8 \leq \frac{1}{n^4}$$

$$n^4 \geq \frac{1}{1048.8}$$

$$n^4 \geq 0.0009537$$

$$n \geq \sqrt[4]{0.0009537}$$

$$n \geq 0.098$$

Conclusion

To guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1, we need to divide the interval $[-4, 2]$ into at least 3 subintervals. This means that the minimum value of $n$ required is 3.

Recommendations

Based on the analysis above, we recommend using at least 3 subintervals to approximate the given integral using Simpson's Rule. This will guarantee that the approximation is accurate to within 0.1.

Future Work

In future work, we plan to explore other methods for approximating the given integral, such as the trapezoidal rule and Gaussian quadrature. We also plan to investigate the use of adaptive quadrature methods to improve the accuracy of the approximation.

References

• [1] Burden, R. L., & Faires, J. D. (2016). Numerical analysis (10th ed.). Brooks Cole.
• [2] Atkinson, K. E. (1989). An introduction to numerical analysis (2nd ed.). John Wiley & Sons.
• [3] Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis (4th ed.). Springer.
  

Q: What is Simpson's Rule?

A: Simpson's Rule is a method for approximating the value of a definite integral. It is a powerful tool for numerical integration, but it requires careful consideration of the number of subintervals, $n$, to achieve the desired level of accuracy.

Q: How does Simpson's Rule work?

A: Simpson's Rule states that the approximation to the integral is given by:

$$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]$$

where $x_i = a + i \Delta x$ for $i = 0, 1, 2, \ldots, n$.

Q: What is the error bound for Simpson's Rule?

A: The error bound for Simpson's Rule is given by:

$$\left| \int_{a}^{b} f(x) dx - \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] \right| \leq \frac{M(b-a)^5}{180n^4}$$

where $M$ is the maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[a, b]$.

Q: How do I determine the minimum value of n required to guarantee that the Simpson's Rule approximation is accurate to within a certain tolerance?

A: To determine the minimum value of $n$ required to guarantee that the Simpson's Rule approximation is accurate to within a certain tolerance, you need to find the maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[a, b]$. Then, you can use the error bound formula to determine the minimum value of $n$ required.

Q: What is the maximum value of the absolute value of the fourth derivative of f(x) on the interval [-4, 2]?

A: The maximum value of the absolute value of the fourth derivative of $f(x)$ on the interval $[-4, 2]$ is simply the absolute value of the fourth derivative, which is 24.

Q: How do I find the fourth derivative of f(x)?

A: To find the fourth derivative of $f(x)$, you need to differentiate $f(x)$ four times.

$$f(x) = -x^4 - 4x^3 + 48x^2 - 4x - 3$$

$$f'(x) = -4x^3 - 12x^2 + 96x - 4$$

$$f''(x) = -12x^2 - 24x + 96$$

$$f'''(x) = -24x - 24$$

$$f''''(x) = -24$$

Q: What is the minimum value of n required to guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1?

A: To guarantee that the Simpson's Rule approximation to the given integral is accurate to within 0.1, we need to divide the interval $[-4, 2]$ into at least 3 subintervals. This means that the minimum value of $n$ required is 3.

Q: What are some other methods for approximating the value of a definite integral?

A: Some other methods for approximating the value of a definite integral include the trapezoidal rule and Gaussian quadrature. Adaptive quadrature methods can also be used to improve the accuracy of the approximation.

Q: What is adaptive quadrature?

A: Adaptive quadrature is a method for approximating the value of a definite integral by dividing the interval into smaller subintervals and using a different method for each subinterval. This can improve the accuracy of the approximation by allowing the method to adapt to the shape of the function.

Q: What are some common applications of numerical integration?

A: Numerical integration has many common applications, including:

• Physics and engineering: Numerical integration is used to solve problems in physics and engineering, such as calculating the trajectory of a projectile or the stress on a beam.
• Economics: Numerical integration is used in economics to calculate the value of a function over a given interval.
• Computer graphics: Numerical integration is used in computer graphics to calculate the value of a function over a given interval, such as the value of a lighting function.
• Signal processing: Numerical integration is used in signal processing to calculate the value of a function over a given interval, such as the value of a filter function.

Q: What are some common challenges associated with numerical integration?

A: Some common challenges associated with numerical integration include:

• Choosing the correct method: Choosing the correct method for numerical integration can be challenging, as different methods have different strengths and weaknesses.
• Choosing the correct parameters: Choosing the correct parameters for the method can be challenging, as different parameters can affect the accuracy and efficiency of the method.
• Dealing with singularities: Dealing with singularities, such as points where the function is not defined, can be challenging.
• Dealing with oscillatory functions: Dealing with oscillatory functions, such as functions that have many local maxima and minima, can be challenging.