The Formula and Its Derivations

Oct 13, 2025 by windailys.com 45 views

Introduction

In the realm of numerical analysis and approximation theory, the concept of approximating definite integrals using various methods is crucial. One such method is the approximation of the definite integral using the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ . This formula is a fundamental concept in understanding the behavior of polynomials and their derivatives. In this article, we will delve into the world of polynomials and explore the maximum degree for which this formula is exact.

The Formula and Its Derivations

The given formula is a result of the Taylor series expansion of the function $f(x)$ around the point $a$ . The Taylor series expansion of a function $f(x)$ around a point $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots

By integrating the Taylor series expansion of $f(x)$ from $a$ to $a+h$ , we obtain:

\int_a^{a+h} f(x) \, dx = \int_a^{a+h} \left( f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots \right) \, dx

Evaluating the integral, we get:

\int_a^{a+h} f(x) \, dx = h f(a) + \frac{h^2}{2} f^{\prime}(a) + \frac{h^3}{3\cdot2} f^{\prime\prime}(a) + \ldots

Now, let's consider the given formula:

\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)

Comparing the two expressions, we can see that the given formula is an approximation of the integral, and it is exact only up to a certain degree of the polynomial.

Exactness for Polynomials

To determine the maximum degree for which the formula is exact, we need to consider the Taylor series expansion of the function $f(x)$ around the point $a$ . Let's assume that the function $f(x)$ is a polynomial of degree $n$ . Then, the Taylor series expansion of $f(x)$ around $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n

Integrating the Taylor series expansion of $f(x)$ from $a$ to $a+h$ , we obtain:

\int_a^{a+h} f(x) \, dx = h f(a) + \frac{h^2}{2} f^{\prime}(a) + \frac{h^3}{3\cdot2} f^{\prime\prime}(a) + \ldots + \frac{h^{n+1}}{(n+1)!} f^{(n)}(a)

Now, let's consider the given formula:

\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)

Comparing the two expressions, we can see that the given formula is an approximation of the integral, and it is exact only up to the degree $n=1$ .

Conclusion

In conclusion, the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ is exact for polynomials of degree $n=1$ . This means that the formula is an accurate approximation of the definite integral for linear polynomials, but it is not exact for polynomials of degree greater than 1.

Future Work

Future research in this area could focus on developing new approximation formulas that are exact for higher-degree polynomials. This could involve using more advanced techniques, such as the use of orthogonal polynomials or the application of numerical methods.

References

[1] Taylor, B. (1715). Methodus Incrementorum Directa et Inversa. London: J. Streater.
[2] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. London: J. Streater.
[3] Bernoulli, J. (1690). Ars Conjectandi. Basel: J.J. de Tournes.

Glossary

Taylor series expansion: A mathematical representation of a function as an infinite sum of terms, each term being a power of the variable.
Polynomial: A mathematical expression consisting of variables and coefficients, with each term being a power of the variable.
Derivative: A measure of how a function changes as its input changes.
Integral: A measure of the area under a curve or the accumulation of a quantity over a given interval.

Introduction

In our previous article, we explored the concept of approximating definite integrals using the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ . We determined that this formula is exact for polynomials of degree $n=1$ . In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the Taylor series expansion of a function?

A: The Taylor series expansion of a function $f(x)$ around a point $a$ is a mathematical representation of the function as an infinite sum of terms, each term being a power of the variable. It is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots

Q: What is the significance of the Taylor series expansion in approximating definite integrals?

A: The Taylor series expansion is a powerful tool in approximating definite integrals. By integrating the Taylor series expansion of a function from $a$ to $a+h$ , we can obtain an approximation of the definite integral.

Q: Why is the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ not exact for polynomials of degree greater than 1?

A: The formula is not exact for polynomials of degree greater than 1 because it only takes into account the first two terms of the Taylor series expansion of the function. For higher-degree polynomials, the formula is not accurate enough to provide an exact approximation of the definite integral.

Q: Can we develop new approximation formulas that are exact for higher-degree polynomials?

A: Yes, it is possible to develop new approximation formulas that are exact for higher-degree polynomials. This could involve using more advanced techniques, such as the use of orthogonal polynomials or the application of numerical methods.

Q: What are some potential applications of the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ ?

A: The formula has several potential applications in fields such as physics, engineering, and economics. For example, it can be used to approximate the area under a curve, which is essential in many scientific and engineering applications.

Q: How can we improve the accuracy of the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ ?

A: To improve the accuracy of the formula, we can use more advanced techniques, such as the use of orthogonal polynomials or the application of numerical methods. We can also use higher-order approximations of the Taylor series expansion of the function.

Conclusion

In conclusion, the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ is a powerful tool in approximating definite integrals. However, it is not exact for polynomials of degree greater than 1. By understanding the limitations of the formula and exploring new approximation techniques, we can improve its accuracy and make it more useful in various scientific and engineering applications.

Future Work

References

[1] Taylor, B. (1715). Methodus Incrementorum Directa et Inversa. London: J. Streater.
[2] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. London: J. Streater.
[3] Bernoulli, J. (1690). Ars Conjectandi. Basel: J.J. de Tournes.

Glossary

Taylor series expansion: A mathematical representation of a function as an infinite sum of terms, each term being a power of the variable.
Polynomial: A mathematical expression consisting of variables and coefficients, with each term being a power of the variable.
Derivative: A measure of how a function changes as its input changes.
Integral: A measure of the area under a curve or the accumulation of a quantity over a given interval.