Question

Best center point Suppose you wish to approximate \(\cos (\pi / 12)\) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at \(\pi / 6 ?\) Use a calculator for numerical experiments and check for consistency with Theorem \(11.2 .\) Does the answer depend on the order of the polynomial?

Short answer

Based on the given step-by-step solution, the short answer is: When approximating the value of \(\cos(\pi/12)\) using Taylor polynomials, it is more accurate to use a polynomial centered at \(\pi/6\) than at \(0\). Additionally, while higher-order polynomials provide better approximations, the polynomial centered at \(\pi/6\) consistently outperforms the one centered at \(0\) for a given order.

Step-by-step solution

Taylor polynomial formula

Recall the Taylor polynomial formula for a function \(f(x)\) centered at \(a\) and of order \(n\): \(P_n(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n\) Since we are dealing with the cosine function, let's find its derivatives: \(f(x) = \cos(x) \Rightarrow f'(x) = -\sin(x) \Rightarrow f''(x) = -\cos(x) \Rightarrow f'''(x) = \sin(x) \Rightarrow f^{(4)}(x) = \cos(x)\) This pattern will continue for higher-order derivatives.

Taylor polynomials centered at 0 and \(\pi / 6\)

For \(a = 0\), we can construct the following Taylor polynomial: \(P_n^0(x) = \cos(0) + \frac{-\sin(0)}{1!}(x-0) + \frac{-\cos(0)}{2!}(x-0)^2 + \frac{\sin(0)}{3!}(x-0)^3 + \frac{\cos(0)}{4!}(x-0)^4 + \cdots\) For \(a = \pi / 6\), construct the Taylor polynomial: \(P_n^{\pi/6}(x) = \cos(\pi / 6) + \frac{-\sin(\pi / 6)}{1!}(x-\pi/6) + \frac{-\cos(\pi / 6)}{2!}(x-\pi/6)^2 + \frac{\sin(\pi / 6)}{3!}(x-\pi/6)^3 + \frac{\cos(\pi / 6)}{4!}(x-\pi/6)^4 + \cdots\)

Comparison of approximations

Next, let's compare the approximations of \(\cos(\pi/12)\) using \(P_n^0(x)\) and \(P_n^{\pi/6}(x)\). We'll evaluate the polynomials at \(x=\pi/12\), trying different orders (such as n = 2, 4, 6...). For each order n, calculate and compare \(P_n^0(\pi/12)\) and \(P_n^{\pi/6}(\pi/12)\) using a calculator. You'll find that the approximation is consistently more accurate when using the Taylor polynomial centered at \(\pi / 6\), as this is closer to the point of interest, \(\pi/12\).

Investigate the effect of polynomial orders

To examine the effect of the polynomial order, compare higher-order polynomial approximations for both cases, \(P_n^0(\pi/12)\) and \(P_n^{\pi/6}(\pi/12)\). You'll notice that the difference in accuracy between the two center points decreases as the order increases, but the polynomial centered at \(\pi/6\) continues to yield a more accurate approximation for any given order n. In conclusion, the approximation is more accurate when using Taylor polynomials centered at \(\pi / 6\) when approximating \(\cos(\pi/12)\). The answer does depend on the order of the polynomial, with higher orders generally providing better approximations, but the polynomial centered at \(\pi/6\) remains more accurate for any given order.

Key concepts

Cosine Function Approximation

When attempting to approximate the value of the cosine function at a specific point, Taylor polynomials are an invaluable tool. Essentially, a Taylor polynomial takes the form of a sum whose terms are derived from the function's derivatives at a single point. This allows us to create a polynomial that closely mimics the behavior of the cosine function around that point.

As a concrete example, for \(\cos (x)\), we attempt to approximate \(\cos (\pi / 12)\) by constructing a Taylor polynomial. Through careful evaluation using various polynomial orders, we might select \(\pi / 6\) as a center point because it's closer to \(\pi / 12\) than 0 is, enhancing the precision of our approximation.

Each incremental term in the Taylor polynomial represents an added layer of precision, making the polynomial a more accurate representation of the function within a certain radius from the center point. When using Taylor polynomials for function approximation, such as the cosine function, students should be mindful of choosing the center point that yields the highest accuracy, often one that's nearest to the target value we're trying to estimate.

Derivative of Cosine

Understanding the derivatives of cosine is crucial for constructing Taylor polynomials. The derivative pattern for the cosine function follows a cyclical nature, where \(f(x) = \cos(x)\) transitions to \(f'(x) = -\sin(x)\), then to \(f''(x) = -\cos(x)\), and it continues to alternate.

This pattern repeats every four derivatives, meaning the fourth derivative—and every fourth one after that—is the same as the original function, \(\cos(x)\). This repetitive nature simplifies calculating higher-order derivatives when constructing Taylor polynomials for the cosine function. Armed with this knowledge, students can predict the behavior of subsequent terms in the series, enabling them to compile the terms necessary for their Taylor polynomial with greater ease.

Polynomial Order Effect

The order of the Taylor polynomial significantly impacts the approximation's accuracy. In essence, the higher the order of the polynomial, the more terms it includes, and the more precisely it approximates the function near the center point. A first-order polynomial simply takes the function's value and slope into account, whereas a second-order polynomial also considers the curvature, and higher orders account for more subtle changes in the function's behavior at that point.

As we increase the polynomial order, the approximation becomes progressively better, due to the inclusion of these higher-order derivatives. However, beyond a certain point, the returns on accuracy begin to diminish. It's worth noting that the proximity of the center to the point of interest also plays a role; a higher-order polynomial centered near the point of interest will capture the function's behavior more effectively than one centered further away, even if it's of the same order.

Centering a Taylor Series

The choice of the center point, often denoted as \(a\) in the Taylor polynomial formula, is another critical factor determining the approximation's precision. Centering the series at a point close to where we need the approximation helps in achieving a more accurate result for a given polynomial order.

The centering of a Taylor series affects not just the accuracy but also the interval of convergence. For example, in the exercise provided, choosing \(\pi/6\) instead of 0 allowed for a more accurate approximation of \(\cos(\pi/12)\) because \(\pi/6\) is closer to \(\pi/12\), reducing the error introduced in each term of the polynomial.

Moreover, the convergence of the Taylor series to the actual function may be faster when centered at \(\pi/6\), meaning fewer terms may be needed to achieve a comparable level of accuracy as with a series centered at 0. This concept teaches students the importance of the center's proximity to the target evaluation point in optimizing Taylor polynomial approximations.