Taylor Polynomial And Error Calculation For E^(sin(x)) At X=1.5

Jul 12, 2025 by ADMIN 64 views

Taylor Polynomial and Error Calculation for f(x) = e^(sin(x))

In this article, we will delve into the fascinating world of Taylor polynomials and their applications in approximating functions. Specifically, we will focus on finding the Taylor polynomial of degree 2, denoted as T_2(x), for the function f(x) = e^(sin(x)) around the point a = π/2. Furthermore, we will compute the error |f(x) - T_2(x)| for x = 1.5, providing a concrete example of how Taylor polynomials can be used to estimate function values and quantify the associated error. Taylor polynomials are a cornerstone of calculus and numerical analysis, offering a powerful tool for approximating complex functions with simpler polynomial expressions. This is particularly useful when dealing with functions that are difficult to evaluate directly or when only an approximate value is needed. Understanding Taylor polynomials and their error terms is crucial in various fields, including physics, engineering, and computer science, where accurate approximations are essential for modeling and simulations. The process involves calculating derivatives of the function at a specific point and using these derivatives to construct the polynomial. The degree of the polynomial determines the accuracy of the approximation, with higher-degree polynomials generally providing better approximations. However, increasing the degree also increases the complexity of the calculations. The error term provides a measure of the difference between the actual function value and the approximation given by the Taylor polynomial. By understanding the error term, we can determine the range of x values for which the approximation is valid and the level of accuracy we can expect. This article aims to provide a comprehensive understanding of Taylor polynomials and their error calculation through a detailed example, making it a valuable resource for students and professionals alike.

Problem Statement

Our objective is to determine the Taylor polynomial of degree 2, T_2(x), for the function f(x) = e^(sin(x)) centered at a = π/2. Subsequently, we aim to compute the absolute error between the actual function value and the Taylor polynomial approximation, |f(x) - T_2(x)|, at x = 1.5. This exercise demonstrates the practical application of Taylor's theorem in approximating function values and quantifying the approximation error. The function f(x) = e^(sin(x)) is a transcendental function, meaning it cannot be expressed as a simple algebraic combination of polynomials, radicals, and rational functions. This makes it a suitable candidate for approximation using Taylor polynomials. The point a = π/2 is chosen as the center of the Taylor polynomial because it is a critical point of the sine function, and the exponential function behaves predictably in the vicinity of this point. The value x = 1.5 is chosen as a point relatively close to the center a = π/2, allowing us to observe the accuracy of the Taylor polynomial approximation within a reasonable interval. The absolute error |f(x) - T_2(x)| provides a direct measure of the approximation quality. A smaller error indicates a better approximation. By computing this error, we can assess the effectiveness of the Taylor polynomial in approximating the function f(x) at the given point. This problem highlights the core concepts of Taylor series and error analysis, which are fundamental in numerical analysis and approximation theory. Understanding how to construct Taylor polynomials and estimate their errors is essential for solving a wide range of problems in science and engineering, where accurate approximations are often required.

Finding the Taylor Polynomial T_2(x)

To construct the Taylor polynomial T_2(x), we need to calculate the first and second derivatives of f(x) and evaluate them at a = π/2. The general formula for the Taylor polynomial of degree 2 is given by:

T_2(x) = f(a) + f'(a)(x - a) + (f''(a) / 2!)(x - a)^2

Let's start by finding the derivatives:

f(x) = e^(sin(x))
f'(x) = cos(x) * e^(sin(x)) (using the chain rule)
f''(x) = -sin(x) * e^(sin(x)) + cos^2(x) * e^(sin(x)) (using the product and chain rules)

Now, we evaluate these functions at a = π/2:

f(π/2) = e^(sin(π/2)) = e^1 = e
f'(π/2) = cos(π/2) * e^(sin(π/2)) = 0 * e^1 = 0
f''(π/2) = -sin(π/2) * e^(sin(π/2)) + cos^2(π/2) * e^(sin(π/2)) = -1 * e^1 + 0 * e^1 = -e

Substituting these values into the Taylor polynomial formula, we get:

T_2(x) = e + 0 * (x - π/2) + (-e / 2)(x - π/2)^2

Simplifying, we obtain:

T_2(x) = e - (e/2)(x - π/2)^2. This is the Taylor polynomial of degree 2 for f(x) = e^(sin(x)) centered at a = π/2. This polynomial provides a quadratic approximation of the function f(x) near the point x = π/2. The coefficients of the polynomial are determined by the function's derivatives at the center point. The first term, e, represents the function's value at the center. The second term, which is zero in this case, represents the linear approximation. The third term, -(e/2)(x - π/2)^2, represents the quadratic correction to the approximation. The Taylor polynomial captures the local behavior of the function, providing a good approximation in a neighborhood around the center point. The accuracy of the approximation decreases as we move further away from the center. The Taylor polynomial is a fundamental tool in numerical analysis, allowing us to approximate complex functions with simpler polynomials, making them easier to evaluate and manipulate.

Computing the Error |f(x) - T_2(x)| at x = 1.5

To compute the error, we need to find the absolute difference between the actual function value f(1.5) and the Taylor polynomial approximation T_2(1.5). First, let's calculate f(1.5):

f(1.5) = e^(sin(1.5)) ≈ e^(0.997495) ≈ 2.711253

Next, we evaluate T_2(1.5):

T_2(1.5) = e - (e/2)(1.5 - π/2)^2

Since π/2 ≈ 1.570796, we have:

T_2(1.5) ≈ e - (e/2)(1.5 - 1.570796)^2

T_2(1.5) ≈ e - (e/2)(-0.070796)^2

T_2(1.5) ≈ e - (e/2)(0.005012)

T_2(1.5) ≈ e - 0.002506e

T_2(1.5) ≈ 2.718282 - 0.002506 * 2.718282

T_2(1.5) ≈ 2.718282 - 0.006815

T_2(1.5) ≈ 2.711467

Now, we compute the absolute error:

|f(1.5) - T_2(1.5)| ≈ |2.711253 - 2.711467|

|f(1.5) - T_2(1.5)| ≈ |-0.000214|

|f(1.5) - T_2(1.5)| ≈ 0.000214. This result shows that the error between the actual function value and the Taylor polynomial approximation at x = 1.5 is quite small. This indicates that the Taylor polynomial T_2(x) provides a good approximation of the function f(x) = e^(sin(x)) in the vicinity of x = 1.5. The error is influenced by several factors, including the degree of the Taylor polynomial, the distance from the center point (a = π/2), and the behavior of the function's higher-order derivatives. In general, increasing the degree of the Taylor polynomial reduces the error, but it also increases the complexity of the calculations. The error term in Taylor's theorem provides a more formal way to estimate the error bound, taking into account the maximum value of the (n+1)-th derivative in the interval of interest. In this case, the error is relatively small because x = 1.5 is close to the center point a = π/2, and the function's derivatives are well-behaved in this region. This example demonstrates the practical utility of Taylor polynomials in approximating function values and quantifying the associated error.

Conclusion

In summary, we have successfully determined the Taylor polynomial of degree 2, T_2(x) = e - (e/2)(x - π/2)^2, for the function f(x) = e^(sin(x)) centered at a = π/2. We then computed the absolute error |f(1.5) - T_2(1.5)| ≈ 0.000214 at x = 1.5. This exercise demonstrates the effectiveness of Taylor polynomials in approximating function values and the importance of error analysis in assessing the accuracy of these approximations. The Taylor polynomial provides a valuable tool for approximating complex functions with simpler polynomial expressions. The accuracy of the approximation depends on the degree of the polynomial, the distance from the center point, and the behavior of the function's derivatives. The error term in Taylor's theorem provides a formal way to estimate the error bound, allowing us to determine the range of x values for which the approximation is valid and the level of accuracy we can expect. This understanding is crucial in various fields, including physics, engineering, and computer science, where accurate approximations are essential for modeling and simulations. The Taylor polynomial is a fundamental concept in calculus and numerical analysis, and its applications extend to a wide range of problems. By mastering the techniques for constructing Taylor polynomials and analyzing their errors, students and professionals can gain a powerful tool for solving complex mathematical problems. This article has provided a detailed example of how to apply these techniques, making it a valuable resource for anyone interested in learning more about Taylor polynomials and their applications. The use of Taylor polynomials is not limited to approximating function values; they can also be used to solve differential equations, evaluate integrals, and analyze the stability of dynamical systems. The flexibility and power of Taylor polynomials make them an indispensable tool in the mathematical toolkit.