Problem 1 - Essential Calculus Early Transcendentals

Question

(a) Find the Taylor polynomials up to degree 6 for \(f(x)=\cos x\) centered at \(a=0 .\) Graph \(f\) and these polynomials on a common screen. (b) Evaluate \(f\) and these polynomials at \(x=\pi / 4, \pi / 2,\) and \(\pi .\) (c) Comment on how the Taylor polynomials converge to \(f(x) .\)

Step-by-Step Solution

Answer

The Taylor polynomial \(P_6(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}\). It approximates \(\cos x\) well near \(x = 0\), especially around smaller values of \(x\).

Step 1: Recall Taylor Series Definition

The Taylor series expansion of a function \(f(x)\) about \(a = 0\) is given by \(f(x) = \sum_{n=0}^{ ext{degree}} \frac{f^{(n)}(0)}{n!} x^n\). For \(f(x) = \cos x\), calculate the derivatives and evaluate them at \(x = 0\).

Step 2: Calculate Derivatives of Cosine

Calculate the first few derivatives of \(\cos x\) and evaluate them at \(x = 0\): \(f(x) = \cos x\), \(f'(x) = -\sin x\), \(f''(x) = -\cos x\), \(f'''(x) = \sin x\), \(f''''(x) = \cos x\), and so forth. Evaluate these: \(f(0) = 1\), \(f''(0) = -1\), \(f''''(0) = 1\), etc.

Step 3: Construct Taylor Polynomial

Using the derivatives, construct the Taylor polynomial up to degree 6: \(P_6(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}\).

Step 4: Graph the Function and Polynomial

Graph \(f(x) = \cos x\) and \(P_6(x)\) on the same coordinate plane to visualize how closely the polynomial approximates the function.

Step 5: Evaluate Function and Polynomial at Given Points

Calculate \(\cos x\), \(P_6(x)\) at \(x = \pi/4, \pi/2,\) and \(\pi\). For example, \(P_6(\pi/4) = 1 - \frac{(\pi/4)^2}{2} + \frac{(\pi/4)^4}{24} - \frac{(\pi/4)^6}{720}\). Repeat similarly for other values.

Step 6: Analyze Convergence

Compare values from Step 5 to see how closely \(P_6(x)\) approximates \(\cos x\). Notice how the approximation improves closer to the center point \(x = 0\) and degrades as we move farther away, such as at \(x = \pi\).

Key Concepts

Taylor Series Expansion

The Taylor series expansion is a fundamental mathematical tool used to approximate complex functions by polynomials. It breaks down a function into an infinite sum of terms coming from the function's derivatives evaluated at a single point. The general formula for the Taylor series of a function \( f(x) \) centered at \( a \) is given by:

• \( f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \)
• This continues infinitely, taking higher derivatives at \( a \) and dividing by factorial of the order of derivative.

In practical applications, a finite number of terms are taken, resulting in a Taylor polynomial. The degree of this polynomial influences the accuracy of the approximation.

Cosine Function

The cosine function, \( \cos x \), is a periodic function often encountered in trigonometry and calculus. It describes the x-coordinate of a point on the unit circle as an angle \( x \) is varied.

• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• The fundamental period of cosine is \( 2\pi \), and it oscillates between -1 and 1.

For the purpose of Taylor series, the cosine function is ideal due to its well-known derivatives: \( \cos x \) and its derivatives form a repeating cycle. This cycle makes cosine a clear choice for polynomial approximation using Taylor series.

Polynomial Approximation

Polynomial approximation involves representing a complicated function with a simpler polynomial. This enables easier computation and analysis of the function's behavior around a certain point. Taylor polynomials serve this purpose exceptionally well.

• The approximation polynomials are constructed using derivatives of the function at a specific point, known as the center.
• For instance, the 6th-degree Taylor polynomial for \( \cos x \) centered at 0, \( P_6(x) \), is given by: \( 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} \).

By calculating the value of \( P_6(x) \) at various points, one can observe how this polynomial mimics the behavior of the cosine function near its center point.

Convergence Analysis

Convergence analysis is crucial when discussing the efficiency of Taylor polynomials in approximating functions. It evaluates how closely the polynomial models the target function as the degree increases. A higher degree generally means better approximation.

• The success of convergence depends on proximity to the expansion point. Close to this point, the polynomial will resemble the function accurately.
• As distance increases, especially away from the center, discrepancies can become noticeable.

Effective convergence can be seen in the graph of \( \cos x \) overlapping the polynomial \( P_6(x) \), especially around \( x = 0 \). Moving further such as towards \( x = \pi \), the deviation increases, reflecting a common pattern in polynomial approximations.