Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For \(x\) near zero, \(\cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\cdots+(-1)^{n} \frac{x^{2 n}}{(2 n) !} .\) What degree Taylor polyno- mial must be used to approximate \(\cos (0.2)\) with error less than \(\frac{1}{10^{8}}\) ?

Step-by-Step Solution

Answer

The smallest degree \(n\) of the Taylor polynomial is found as per the inequality \(\frac{{|0.2|}^{n+1}}{(n+1)!} < \frac{1}{10^{8}}\). By checking for sufficient values of \(n\), we can find the smallest degree that will provide the desired level of error for approximating \(\cos (0.2)\).

Step 1: Understand the Taylor series error estimation

When we truncate a Taylor series, we introduce some error. To find a bound on the error, we can employ the Lagrange form of the remainder, which states that the error \(R_{n}(x)\) is given by \(R_{n}(x) = \frac{f^{(n+1)}(c) x^{n+1}}{(n+1)!}\), where \(c\) is somewhere between 0 and \(x\). Let's apply this to our problem.

Step 2: Apply Taylor series error estimation for cosine function

The Taylor series expansion for the cosine function around 0 is \(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^{2n}}{(2n)!} + ...\). The \((n+1)\)th derivative of the cosine function is either \(\sin(x)\), \(-\cos(x)\), \(-\sin(x)\), or \(\cos(x)\) depending on whether \(n\) is 0, 1, 2, or 3 (mod 4) respectively. However, given that the maximum possible value for \(\cos(x)\) and \(\sin(x)\) is 1, the Taylor remainder can be bounded as \(|R_{n}(x)| = |\frac{f^{(n+1)}(c) x^{n+1}} {(n+1)!}| ≤ \frac{{|x|}^{n+1}}{(n+1)!}\).

Step 3: Find the smallest degree for the required error

We want the absolute error to be less than \(\frac{1}{10^{8}}\). That is, we must have \(\frac{{|0.2|}^{n+1}}{(n+1)!} < \frac{1}{10^{8}}\). Now, we will set this inequality and start computing values of \(n\) until we find a degree \(n\) such that it satisfies this inequality. This will result in the smallest integer degree of the Taylor polynomial that provides an approximation with the desired level of error.

Key Concepts

Taylor Polynomial

The Taylor polynomial is a finite approximation of a Taylor series. It's used to estimate the value of functions that are difficult to compute directly. In this context, we're talking about approximating trigonometric functions like cosine. The Taylor polynomial of degree \( n \) will consist of a finite number of terms from the full, infinitely expanding Taylor series.

• The general form for the Taylor polynomial of a function \( f(x) \) at \( x=a \) is given by:\[P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n\]
• For the cosine function at \( x=0 \), this specific Taylor polynomial looks like:\[P_n(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (-1)^n \frac{x^{2n}}{(2n)!}\]
• This polynomial offers an approximation by adding correction terms to refine the estimate.

It is crucial to choose the right degree of the polynomial, so the approximation error remains within acceptable limits.

Lagrange Remainder

The Lagrange remainder provides a mathematical tool to estimate the error introduced when using a Taylor polynomial instead of the full Taylor series. It's particularly useful when deciding how many terms are necessary for a given level of precision.

• The Lagrange form of the remainder for a Taylor series is:\[R_n(x) = \frac{f^{(n+1)}(c) x^{n+1}}{(n+1)!}\]where \( c \) is some value between the point of expansion and the point of interest.
• For functions like cosine, where the derivatives are bounded, calculating \( R_n(x) \) helps determine the accuracy of the polynomial.
• In many situations, you can bound \( R_n(x) \) by finding the maximum value of the derivatives, which for cosine is 1, simplifying the calculations.

This estimation aids in ensuring that the polynomial chosen isn’t just detailed but meets your accuracy needs.

Error Estimation

Error estimation involves evaluating the accuracy of a Taylor polynomial approximation. This is particularly important when high precision is required for tasks like scientific computations.
Estimating the error is about ensuring your approximation is reliable. By employing the Lagrange remainder, you calculate:

• An error bound that provides a maximum possible error for the polynomial approximation.
• By using the inequality:\[\frac{{|x|}^{n+1}}{(n+1)!} < desired \ error \ margin\]you can find an \( n \) that suits your precision needs.
• For example, with \( x = 0.2 \) and a desired error less than \( \frac{1}{10^{8}} \), you increment \( n \) until the inequality is satisfied.

Choosing a polynomial with the smallest degree \( n \) that meets error criteria ensures computational efficiency and accuracy.

Cosine Function

The cosine function, denoted \(\cos(x)\), is an essential trigonometric function used extensively in mathematics and its applications.

• It is periodic with a period of \(2\pi\) and oscillates between -1 and 1.
• The Taylor series for \(\cos(x)\) is unique because all its terms containing odd powers (x^1, x^3,and so on) are zero. This happens due to\( \cos(0) = 1\) and its symmetry about the y-axis.\[\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\]
• This characteristic makes cosine especially suitable for approximations using even-degree polynomials.

The simplicity of the cosine's Taylor series makes it a valuable tool for approximations around small values of \( x \), leveraging its inherent symmetry and periodicity for efficient calculations.