Problem 20 - Calculus

Question

For \(|x| \leq 0.1,\) graph the error $$E_{0}=\cos x-P_{0}(x)=\cos x-1$$ Explain the shape of the graph, using the Taylor expansion of \(\cos x .\) Find a bound for \(\left|E_{0}\right|\) for \(|x| \leq 0.1\).

Step-by-Step Solution

Answer

The maximum error \(|E_0|\) is approximately 0.005 for \(|x| \leq 0.1\).

Step 1: Understanding the Function and Polynomial

The function is given as \( \cos x \) and the polynomial \( P_0(x) \) based on the Taylor expansion of \( \cos x \) at \( x = 0 \). The polynomial \( P_0(x) = 1 \) is the first-term of the Taylor series.

Step 2: Writing the Error Function

The error function is \( E_0 = \cos x - P_0(x) = \cos x - 1 \). We need to analyze this function within the interval \( |x| \leq 0.1 \).

Step 3: Calculate the Taylor Expansion of \( \cos x \)

The Taylor series of \( \cos x \) around \( x = 0 \) is \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \). This expansion helps in understanding the behavior of \( E_0 \).

Step 4: Determine the Error Using the Taylor Series

By substituting \( x = 0 \), which gives \( E_0 = \cos(0) - 1 = 0 \). As \( x \) increases within the interval, \( E_0 = \cos x - 1 \) can be approximated as \( -\frac{x^2}{2} \) since higher-order terms have lesser influence near zero.

Step 5: Graphing the Error Function

The graph of \( E_0 = \cos x - 1 \) will be a downward-opening parabola within the interval because \( \cos x \) is approximated by a parabola near zero, and it's negative because \( \cos x < 1 \).

Step 6: Determining the Bound of the Error

For \( |x| \leq 0.1 \), the error can be approximated as \( E_0 \approx -\frac{x^2}{2} \). The maximum value \( E_0 \) can achieve is at \( x = \pm 0.1 \), therefore, \( |E_0| \leq \frac{(0.1)^2}{2} = 0.005 \).

Key Concepts

Error Function

In the realm of mathematics, an error function is pivotal when evaluating the difference between a true function and its approximation. In our exercise, the error function is denoted as \( E_0 = \cos x - 1 \). This calculation specifically examines how well the simplest Taylor polynomial approximates the cosine function. The simplest polynomial in this scenario is a constant value of 1, which directly comes from the Taylor expansion of \( \cos x \) at \( x = 0 \).Understanding the error function involves mapping its value across an interval. Here, we are considering \( |x| \leq 0.1 \). Due to the approximation \( \cos x \approx 1 - \frac{x^2}{2} \), the error function simplifies to \( E_0 \approx -\frac{x^2}{2} \). This tells us that near zero, the approximation error forms a downward-opening parabola. The error function indicates the deviation of \( \cos x \) from its polynomial approximation, thus providing insight into accuracy.

Cosine Function

The cosine function, \( \cos x \), is one of the fundamental trigonometric functions. It is periodic with a period of \( 2\pi \) and is essential in various fields including geometry, physics, and engineering. In our context, we focus on its approximation near \( x = 0 \) using a Taylor series expansion.For small values of \( x \), the Taylor series expansion of the cosine function is very useful. It is given by:\[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \]This expansion shows that the cosine function is symmetric and can be approximated to a polynomial for small intervals around zero. This makes the cosine function an excellent candidate for simplification in calculations by providing a polynomial approximation, especially useful where high precision is not mandatory, or computational efficiency is more critical.The first term, \(1\), is why for very small \( x \), \( \cos x \) is close to 1. The accuracy of the approximation increases as more terms are included, but just the first term gives a quick and reasonably accurate estimation in a small vicinity around zero.

Polynomial Approximation

Polynomial approximation is a method in mathematics that aims to approximate complex functions using polynomials. This is incredibly advantageous because polynomials are simpler to evaluate, analyze, and differentiate. In our situation with \( \cos x \), the polynomial approximation begins with its Taylor series expansion where higher-order terms diminish as \( x \to 0 \).Key benefits of polynomial approximation include:

• Simplicity: Polynomials are made of basic operations: addition, subtraction, and multiplication.
• Computational Efficiency: Easier to compute compared to more complex functions.
• Analytical Convenience: Differentiating or integrating polynomials is straightforward.

In the exercise, the approximation \( P_0(x) = 1 \) is a result of truncating the Taylor series of \( \cos x \) after the zero-order term. Although this is the simplest possible polynomial approximation, it effectively illustrates basic trends of \( \cos x \) over small intervals. The key idea here is recognizing that such approximations are highly useful for estimating function values and determining error bounds, especially in engineering and physics applications where small errors are acceptable.