Question

Find the \(n^{\text {th }}\) term of the indicated Taylor polynomial. Find a formula for the \(n^{\text {th }}\) term of the Maclaurin polynomial for \(f(x)=\cos x\).

Short answer

For even \(n\), the term is \(\frac{(-1)^{n/2}}{n!}x^n\); \(0\) for odd \(n\).

Step-by-step solution

Understand the Taylor Series

The Taylor series of a function at a point is given by the formula \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \). For the Maclaurin series, which is a Taylor series about \( a = 0 \), we have \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \).

Calculate Derivatives of \(\cos x\)

To find the Maclaurin polynomial terms, calculate derivatives of \(\cos x\) evaluated at \(x = 0\): \( f(x) = \cos x \), \( f'(x) = -\sin x \), \( f''(x) = -\cos x \), \( f'''(x) = \sin x \), and repeats with periodicity of 4.

Evaluate Derivatives at \(x = 0\)

Evaluate the derivatives at \(x=0\): \( f(0) = \cos(0) = 1 \), \( f'(0) = -\sin(0) = 0 \), \( f''(0) = -\cos(0) = -1 \), \( f'''(0) = \sin(0) = 0 \), and continues in the same pattern.

Generalize the Pattern

Observe the pattern in the evaluated derivatives: Even derivatives are non-zero and alternate starting from 1 ( \(1, -1, 1, -1, \dots\) ) while odd derivatives are zero.

Construct the Formula for \(n^{\text{th}}\) Term

The pattern observed leads to the general formula for even \(n\): \( a_n = \frac{(-1)^{n/2}}{n!}x^n \) for even \( n = 0, 2, 4, \dots \), and zero for odd \(n\).

Key concepts

Taylor series

The Taylor series is a powerful mathematical tool that allows us to express a wide range of functions as an infinite sum of terms calculated from the values of the function's derivatives at a single point. This means any function that can be differentiated enough times can potentially be represented as a Taylor series.
In general, the Taylor series of a function \( f(x) \) about a point \( a \) is given by the formula:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \)

The Maclaurin series is a special case of the Taylor series where the function is expanded around the point \( a = 0 \). This simplifies the formula to:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)

Using a Maclaurin series can make it easier to approximate functions near \( x = 0 \) using a polynomial.

derivatives of cosine

To find the Maclaurin series for the cosine function, we need to determine its derivatives and their values at zero. The function \( f(x) = \cos x \) has a repeating cycle of derivatives:

• \( f(x) = \cos x \)
• \( f'(x) = -\sin x \)
• \( f''(x) = -\cos x \)
• \( f'''(x) = \sin x \)
• Cycle repeats...

Evaluating these at \( x = 0 \):

• \( f(0) = 1 \)
• \( f'(0) = 0 \)
• \( f''(0) = -1 \)
• \( f'''(0) = 0 \)

The pattern continues indefinitely, with even derivatives (0th, 2nd, 4th, etc.) giving non-zero values and alternating between \( 1 \) and \(-1\), while odd derivatives (1st, 3rd, 5th, etc.) evaluate to zero.

nth term formula

The observation of patterns in the derivatives guides us to develop a general formula for the \( n^{ ext{th}} \) term of the Maclaurin series for \( \cos x \).
For cosine, the only terms that need to be considered are those from even derivatives, since the odd derivatives are zero at \( x = 0 \).
The nth term formula can be expressed as:

• For even \( n \): \( a_n = \frac{(-1)^{n/2}}{n!}x^n \)
• For odd \( n \): \( a_n = 0 \)

This expression outlines that only even powers of \( x \) have contributing terms in the polynomial derived from the Maclaurin series of \( \cos x \). The \((-1)^{n/2}\) factor accounts for the alternating sign observed in the evaluated derivatives.

polynomial approximation

Polynomial approximation using the Maclaurin series allows us to approximate functions like \( \cos x \) with polynomials for values of \( x \) near zero. This is particularly useful for calculating trigonometric functions without a calculator or in computer algorithms.
The more terms you include from the series, the better the approximation will be. Usually, only a few terms are needed to achieve a fairly accurate approximation for small values of \( x \).
For example:

• The 0th-term approximation is \( 1 \)
• The 2nd-term approximation is \( 1 - \frac{x^2}{2} \)
• The 4th-term approximation is \( 1 - \frac{x^2}{2} + \frac{x^4}{24} \)

As more terms are added, the polynomial gets closer to the value of \( \cos x \) for different \( x \) values, particularly for those close to zero.