Problem 23 - Calculus

Question

To approximate \(\pi\) using a Taylor polynomial, we could use the series for the arctangent or the series for the arcsine. In this problem, we compare the two methods. (a) Using the fact that \(d(\arctan x) / d x=1 /\left(1+x^{2}\right)\) and arctan \(1=\pi / 4,\) approximate the value of \(\pi\) using the third-degree Taylor polynomial of 4 arctan \(x\) about \(x=0\). (b) Using the fact that \(d(\arcsin x) / d x=1 / \sqrt{1-x^{2}}\) and arcsin \(1=\pi / 2,\) approximate the value of \(\pi\) using the third-degree Taylor polynomial of \(2 \arcsin x\) about \(x=0\). (c) Estimate the maximum error of the approximation you found in part (a). (d) Explain the problem in estimating the error in the arcsine approximation.

Step-by-Step Solution

Answer

The arctan method approximates π as 3.43, while the arcsin method gives 4.67. Maximum error for arctan is near zero; arcsin error difficult to estimate.

Step 1: Calculating the Third-degree Taylor Polynomial for Arctan

The Taylor series for arctan(x) is the Maclaurin series expansion. To find the third-degree polynomial, we consider the first four derivatives:1. Arctan(x) at x=0: 02. Derivative: \(\frac{1}{1+x^2}\) at x=0 is 13. Second derivative: \(-\frac{2x}{(1+x^2)^2}\) at x=0 is 04. Third derivative: \(-\frac{2(1-x^2)}{(1+x^2)^3}\) at x=0 is -2Thus, the third-degree Taylor polynomial for \(\text{arctan}(x)\) is \(x - \frac{x^3}{3}\). For \(4\text{ arctan}(x)\), it becomes: \(4x - \frac{4x^3}{3}\). Evaluated at \(x=1\), we get: \[4(1) - \frac{4(1)^3}{3} = 4 - \frac{4}{3} = \frac{8}{3}.\] Therefore, approximating \(\pi\) results in \(\pi \approx \frac{8}{3} \times 4 = \frac{32}{3}.\)

Step 2: Calculating the Third-degree Taylor Polynomial for Arcsin

The Taylor series for arcsin(x) can similarly be expanded as a Maclaurin series:1. Arcsin(x) at x=0: 02. Derivative: \(\frac{1}{\sqrt{1-x^2}}\) at x=0 is 13. Second derivative: \(\frac{x}{(1-x^2)^{3/2}}\) at x=0 is 04. Third derivative: \(\frac{3x^2 + 1}{(1-x^2)^{5/2}}\) at x=0 is 1Thus, the third-degree Taylor polynomial for \(\text{arcsin}(x)\) is \(x + \frac{x^3}{6}\). For \(2\text{ arcsin}(x)\), it becomes: \(2x + \frac{x^3}{3}\). Evaluated at \(x=1\), we get: \[2(1) + \frac{(1)^3}{3} = 2 + \frac{1}{3} = \frac{7}{3}.\] Therefore, approximating \(\pi\) results in \(\pi \approx \frac{7}{3} \times 2 = \frac{14}{3}.\)

Step 3: Estimate Maximum Error for Arctan Taylor Polynomial

To estimate the maximum error for the approximation using arctan, we consider the remainder term in Taylor's Theorem: \(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\). The fourth derivative of arctan at any point is zero or near zero, hence the maximum error is very small. For the Taylor polynomial \(f(x) = 4x - \frac{4x^3}{3}\) up to the third degree, the error term is: \(-\frac{f^{(4)}(c)}{4!}(1)^4 \approx 0\). Thus, the maximum error is near zero.

Step 4: Understanding Error Estimation for Arcsin Taylor Polynomial

Estimating the error for arcsin is challenging because the series does not converge as easily on the interval [0, 1] as the series for arctan. Specifically, as x approaches 1, the value \(\frac{1}{\sqrt{1-x^2}}\) becomes undefined, which complicates finding an accurate remainder bound. This non-convergence near x=1 makes reliable error estimates difficult.

Key Concepts

Taylor Series

Taylor Series are powerful tools in mathematics. They allow us to approximate functions using polynomials. These series exploit the derivatives of a function at a specific point to predict the function's value in an interval close to that point. The Taylor Series for a function \( f(x) \) at a given point \( a \) is expressed through an infinite sum of terms. Each term involves the derivatives of \( f \) evaluated at \( a \). The formula is:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \] Taylor series are especially useful for approximations when computing an infinite number of terms is not practical.
Typically, a Maclaurin series is used, which is just a special case of the Taylor series at \( a = 0 \). This simplifies calculations, and it's often used for functions like sine, cosine, and tangent.

ArcTan Function

The ArcTan Function, or arctangent, is the inverse of the tangent function. It's usually denoted by \( \arctan(x) \). This function is significant because it maps any real number \( x \) to an angle whose tangent is \( x \).
When using Taylor series to approximate \( \arctan(x) \), the expansion begins with a Maclaurin series. For \( \arctan(x) \), up to the third degree, this series is:\[ \arctan(x) \approx x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots \] For smaller values of \( x \), the series converges quickly, making it efficient for calculations.
To approximate \( \pi \) using this series, we can use the known property that \( \arctan(1) = \frac{\pi}{4} \). By evaluating the polynomial at \( x = 1 \), and multiplying by 4, this allows us to approximate the value of \( \pi \).

ArcSin Function

The ArcSin Function, also known as the arcsine function, is the inverse function of sine. It's denoted \( \arcsin(x) \) and maps a real number within the interval \([-1, 1]\) to an angle whose sine is \( x \).
Like arctangent, arcsine can also be expanded using a Maclaurin series to ease calculations. The Taylor series for \( \arcsin(x) \) up to the third-degree polynomial is:\[ \arcsin(x) \approx x + \frac{x^3}{6} + \ldots \] When approximating \( \pi \) using this function, the relation \( \arcsin(1) = \frac{\pi}{2} \) is used. Evaluating the polynomial at \( x = 1 \), and then multiplying by 2, can therefore help approximate \( \pi \).
However, due to the behavior of \( \arcsin(x) \) near \( x = 1 \), where it becomes less defined, the series does not converge as well as \( \arctan(x) \). This poses challenges, especially in determining accurate error estimates.

Error Estimation

Estimating the error in a Taylor polynomial approximation is crucial for understanding its precision. Error estimation allows us to gauge how close the polynomial approximation is to the actual function value.
The error or remainder for a Taylor series is given by Taylor's theorem as:\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \]where \( c \) is some point in the interval between \( a \) and \( x \). This formula is essential in approximating the maximum error.
With the arctangent function, the error was found to be negligible due to the convergence of its series. This is due to the higher derivatives diminishing rapidly.
In contrast, the error estimation for the arcsine function is more complex due to the convergence issues near \( x = 1 \). The derivatives grow excessively large, causing potential inaccuracies. This situation highlights why arctangent might be the more reliable choice for approximating \( \pi \) using Taylor polynomials.