Suppose That ∫ A A + H F ( X ) D X ≈ H F ( A + H ) − H 2 2 F ′ ( A \int_a^{a+h} F(x) \, Dx \approx H F(a+h) - \frac{h^2}{2} F^{\prime}(a ∫ A A + H F ( X ) D X ≈ H F ( A + H ) − 2 H 2 F ′ ( A ]. For Polynomials Of What Maximum Degree Is This Formula Exact?

Mar 9, 2025 by ADMIN 260 views

Suppose that $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a$ ] For Polynomials of What Maximum Degree is This Formula Exact?

In the field of numerical analysis, approximation formulas for definite integrals are crucial for solving various mathematical problems. One such approximation formula is given by $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ . This formula is a simple yet effective method for approximating the value of a definite integral. However, the question remains: for polynomials of what maximum degree is this formula exact?

To understand the formula, let's break it down into its components. The formula is an approximation of the definite integral $\int_a^{a+h} f(x) \, dx$ . The approximation is given by the expression $h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ . This expression consists of two terms: the first term is $h f(a+h)$ , and the second term is $-\frac{h^2}{2} f^{\prime}(a)$ .

The First Term: $h f(a+h)$

The first term, $h f(a+h)$ , is a simple multiplication of the function value at the point $a+h$ and the step size $h$ . This term represents the area of a rectangle with height $f(a+h)$ and width $h$ . The area of this rectangle is given by the product of the height and the width, which is $h f(a+h)$ .

The Second Term: $-\frac{h^2}{2} f^{\prime}(a)$

The second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is a more complex expression. This term represents the area of a trapezoid with height $f(a)$ and width $h$ . The area of this trapezoid is given by the formula $\frac{h}{2} (f(a) + f(a+h))$ . However, in this formula, we are using the approximation $f(a+h) \approx f(a) + h f^{\prime}(a)$ , which is a first-order Taylor series expansion of the function $f(x)$ around the point $a$ . Substituting this approximation into the formula for the area of the trapezoid, we get $\frac{h}{2} (f(a) + f(a) + h f^{\prime}(a)) = h f(a) + \frac{h^2}{2} f^{\prime}(a)$ .

The Approximation Formula

Now that we have broken down the two terms of the formula, we can see that the first term represents the area of a rectangle, while the second term represents the area of a trapezoid. The approximation formula is obtained by subtracting the area of the trapezoid from the area of the rectangle. This gives us the formula $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ .

To determine the maximum degree of polynomials for which this formula is exact, we need to consider the properties of polynomials. A polynomial of degree $n$ is a function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ , where $a_n \neq 0$ . The degree of a polynomial is the highest power of $x$ in the function.

Polynomials of Degree 0

For polynomials of degree 0, the function is a constant, i.e., $f(x) = a_0$ . In this case, the formula is exact because the first term, $h f(a+h)$ , is equal to $h a_0$ , and the second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is equal to 0. Therefore, the formula reduces to $h a_0$ , which is the exact value of the definite integral.

Polynomials of Degree 1

For polynomials of degree 1, the function is of the form $f(x) = a_1 x + a_0$ . In this case, the formula is not exact because the first term, $h f(a+h)$ , is equal to $h (a_1 (a+h) + a_0)$ , and the second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is equal to $-\frac{h^2}{2} a_1$ . Therefore, the formula reduces to $h (a_1 (a+h) + a_0) - \frac{h^2}{2} a_1$ , which is not equal to the exact value of the definite integral.

Polynomials of Degree 2

For polynomials of degree 2, the function is of the form $f(x) = a_2 x^2 + a_1 x + a_0$ . In this case, the formula is not exact because the first term, $h f(a+h)$ , is equal to $h (a_2 (a+h)^2 + a_1 (a+h) + a_0)$ , and the second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is equal to $-\frac{h^2}{2} (2 a_2 a + a_1)$ . Therefore, the formula reduces to $h (a_2 (a+h)^2 + a_1 (a+h) + a_0) - \frac{h^2}{2} (2 a_2 a + a_1)$ , which is not equal to the exact value of the definite integral.

Polynomials of Degree 3

For polynomials of degree 3, the function is of the form $f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0$ . In this case, the formula is not exact because the first term, $h f(a+h)$ , is equal to $h (a_3 (a+h)^3 + a_2 (a+h)^2 + a_1 (a+h) + a_0)$ , and the second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is equal to $-\frac{h^2}{2} (3 a_3 a^2 + 2 a_2 a + a_1)$ . Therefore, the formula reduces to $h (a_3 (a+h)^3 + a_2 (a+h)^2 + a_1 (a+h) + a_0) - \frac{h^2}{2} (3 a_3 a^2 + 2 a_2 a + a_1)$ , which is not equal to the exact value of the definite integral.

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 0. However, for polynomials of degree 1, 2, and 3, the formula is not exact. Therefore, the maximum degree of polynomials for which this formula is exact is 0.

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Q&A: Suppose that $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a$ ] For Polynomials of What Maximum Degree is This Formula Exact?

Q: What is the formula for approximating the definite integral? A: The formula for approximating the definite integral is given by $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ .

Q: What is the maximum degree of polynomials for which this formula is exact? A: The maximum degree of polynomials for which this formula is exact is 0.

Q: Why is the formula not exact for polynomials of degree 1, 2, and 3? A: The formula is not exact for polynomials of degree 1, 2, and 3 because the first term, $h f(a+h)$ , is not equal to the exact value of the definite integral, and the second term, $-\frac{h^2}{2} f^{\prime}(a)$ , is not equal to the exact value of the definite integral.

Q: What is the significance of the formula being exact for polynomials of degree 0? A: The formula being exact for polynomials of degree 0 means that the formula can be used to approximate the definite integral of a constant function.

Q: Can the formula be used to approximate the definite integral of a polynomial of degree 1, 2, or 3? A: No, the formula cannot be used to approximate the definite integral of a polynomial of degree 1, 2, or 3.

Q: What are some common applications of the formula? A: The formula has many common applications in numerical analysis, such as approximating the definite integral of a function, and solving problems in physics and engineering.

Q: How can the formula be modified to make it exact for polynomials of degree 1, 2, and 3? A: The formula can be modified by adding additional terms to the expression, such as higher-order Taylor series expansions of the function.

Q: What are some limitations of the formula? A: The formula has several limitations, including the fact that it is not exact for polynomials of degree 1, 2, and 3, and that it requires the function to be differentiable at the point $a$ .

Q: Can the formula be used to approximate the definite integral of a function that is not a polynomial? A: Yes, the formula can be used to approximate the definite integral of a function that is not a polynomial, but the accuracy of the approximation will depend on the degree of the polynomial used to approximate the function.

Q: How can the formula be used in practice? A: The formula can be used in practice by substituting the function and the point $a$ into the formula, and then evaluating the expression to obtain an approximation of the definite integral.

Q: What are some common mistakes to avoid when using the formula? A: Some common mistakes to avoid when using the formula include using the formula for polynomials of degree 1, 2, or 3, and not checking the accuracy of the approximation.

Q: Can the formula be used to solve problems in physics and engineering? A: Yes, the formula can be used to solve problems in physics and engineering, such as approximating the definite integral of a function that represents a physical quantity.

Q: How can the formula be modified to make it more accurate? A: The formula can be modified by adding additional terms to the expression, such as higher-order Taylor series expansions of the function, or by using a more accurate method for approximating the definite integral.

Q: What are some common applications of the formula in physics and engineering? A: The formula has many common applications in physics and engineering, such as approximating the definite integral of a function that represents a physical quantity, and solving problems in mechanics, electromagnetism, and thermodynamics.

Q: Can the formula be used to solve problems in computer science? A: Yes, the formula can be used to solve problems in computer science, such as approximating the definite integral of a function that represents a physical quantity, and solving problems in computer graphics and game development.

Q: How can the formula be modified to make it more efficient? A: The formula can be modified by using a more efficient method for approximating the definite integral, such as the trapezoidal rule or the Simpson's rule.

Q: What are some common mistakes to avoid when using the formula in practice? A: Some common mistakes to avoid when using the formula in practice include using the formula for polynomials of degree 1, 2, or 3, and not checking the accuracy of the approximation.