Finding the Maximum Value of the Third Derivative

Oct 13, 2025 by windailys.com 21 views

Introduction

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. It is a simple and widely used method, but it can be inaccurate if the number of subintervals, $n$ , is too small. In this article, we will discuss how to determine the minimum value of $n$ required to guarantee that the Trapezoidal Rule approximation to the given integral is accurate to within 0.001.

The Trapezoidal Rule

The Trapezoidal Rule is a method for approximating the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small trapezoids and summing the areas of these trapezoids. The Trapezoidal Rule states that the integral of a function $f(x)$ from $a$ to $b$ can be approximated as:

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

where $h = \frac{b-a}{n}$ is the width of each subinterval, and $x_i = a + ih$ is the $i$ th point in the subinterval.

Error Analysis

The error in the Trapezoidal Rule approximation is given by:

E = \left| \int_{a}^{b} f(x) dx - \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \right|

Using the Taylor series expansion of $f(x)$ around the midpoint of each subinterval, we can show that the error is bounded by:

|E| \leq \frac{M(b-a)^3}{12n^2}

where $M$ is the maximum value of the absolute value of the third 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 Trapezoidal Rule approximation is accurate to within 0.001, we need to find the smallest value of $n$ such that:

\frac{M(b-a)^3}{12n^2} \leq 0.001

In this case, we have $a = -4$ , $b = -2$ , and $f(x) = -x^4-12x^3-48x^2-x+3$ . We need to find the maximum value of the absolute value of the third derivative of $f(x)$ on the interval $[-4,-2]$ .

Finding the Maximum Value of the Third Derivative

To find the maximum value of the absolute value of the third derivative of $f(x)$ on the interval $[-4,-2]$ , we need to find the critical points of the third derivative and evaluate the function at these points.

The third derivative of $f(x)$ is given by:

f'''(x) = -24x^2 - 144x - 192

To find the critical points, we set the derivative equal to zero and solve for $x$ :

f'''(x) = -24x^2 - 144x - 192 = 0

Solving this equation, we find that the critical points are $x = -2$ and $x = -4$ .

Evaluating the Third Derivative at the Critical Points

We now evaluate the third derivative at the critical points:

f'''(-2) = -24(-2)^2 - 144(-2) - 192 = 192 + 288 - 192 = 288

f'''(-4) = -24(-4)^2 - 144(-4) - 192 = 384 + 576 - 192 = 768

Determining the Minimum Value of $n$

We now have the maximum value of the absolute value of the third derivative, which is $M = 768$ . We can now use the error bound formula to determine the minimum value of $n$ :

\frac{M(b-a)^3}{12n^2} \leq 0.001

Substituting the values, we get:

\frac{768(-4-(-2))^3}{12n^2} \leq 0.001

Simplifying, we get:

\frac{768(2)^3}{12n^2} \leq 0.001

\frac{768 \cdot 8}{12n^2} \leq 0.001

\frac{6144}{12n^2} \leq 0.001

n^2 \geq \frac{6144}{12 \cdot 0.001}

n^2 \geq 512000

n \geq \sqrt{512000}

n \geq 716.128

Conclusion

To guarantee that the Trapezoidal Rule approximation to the given integral is accurate to within 0.001, we need to choose $n \geq 716.128$ . This means that we need to divide the interval $[-4,-2]$ into at least 716 subintervals.

Q: What is the Trapezoidal Rule?

A: The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. It is a simple and widely used method, but it can be inaccurate if the number of subintervals, $n$ , is too small.

Q: How does the Trapezoidal Rule work?

A: The Trapezoidal Rule works by dividing the area under a curve into small trapezoids and summing the areas of these trapezoids. The Trapezoidal Rule states that the integral of a function $f(x)$ from $a$ to $b$ can be approximated as:

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

where $h = \frac{b-a}{n}$ is the width of each subinterval, and $x_i = a + ih$ is the $i$ th point in the subinterval.

Q: What is the error in the Trapezoidal Rule approximation?

A: The error in the Trapezoidal Rule approximation is given by:

E = \left| \int_{a}^{b} f(x) dx - \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \right|

Using the Taylor series expansion of $f(x)$ around the midpoint of each subinterval, we can show that the error is bounded by:

|E| \leq \frac{M(b-a)^3}{12n^2}

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

Q: How do I determine the minimum value of $n$ required to guarantee that the Trapezoidal Rule approximation is accurate to within a certain error?

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

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

A: The maximum value of the absolute value of the third derivative of $f(x)$ on the interval $[-4,-2]$ is 768.

Q: How do I find the maximum value of the absolute value of the third derivative of $f(x)$ on the interval $[-4,-2]$ ?

A: To find the maximum value of the absolute value of the third derivative of $f(x)$ on the interval $[-4,-2]$ , you need to find the critical points of the third derivative and evaluate the function at these points.

Q: What is the minimum value of $n$ required to guarantee that the Trapezoidal Rule approximation to the given integral is accurate to within 0.001?

A: The minimum value of $n$ required to guarantee that the Trapezoidal Rule approximation to the given integral is accurate to within 0.001 is 716.128.

Q: How do I choose the number of subintervals, $n$ , for the Trapezoidal Rule?

A: To choose the number of subintervals, $n$ , for the Trapezoidal Rule, you need to determine the minimum value of $n$ required to guarantee that the approximation is accurate to within a certain error. You can then choose a value of $n$ that is greater than or equal to this minimum value.

Q: What are the advantages and disadvantages of the Trapezoidal Rule?

A: The advantages of the Trapezoidal Rule are that it is simple to implement and can be used to approximate the value of a definite integral. The disadvantages of the Trapezoidal Rule are that it can be inaccurate if the number of subintervals, $n$ , is too small, and it can be computationally expensive for large values of $n$ .

Q: When should I use the Trapezoidal Rule?

A: You should use the Trapezoidal Rule when you need to approximate the value of a definite integral and you do not have access to a more accurate method, such as Simpson's Rule. You should also use the Trapezoidal Rule when you need to approximate the value of a definite integral and you are working with a large dataset.

Q: How do I implement the Trapezoidal Rule in a programming language?

A: To implement the Trapezoidal Rule in a programming language, you need to write a function that takes the function $f(x)$ , the interval $[a,b]$ , and the number of subintervals $n$ as input. You can then use the Trapezoidal Rule formula to approximate the value of the definite integral.

Q: What are some common applications of the Trapezoidal Rule?

A: Some common applications of the Trapezoidal Rule include approximating the value of definite integrals in physics, engineering, and economics. The Trapezoidal Rule is also used in computer graphics to approximate the value of definite integrals that are used to calculate the area under curves.